Digital Halftoning and the Physical
ReconstructionFunction
RLE Technical Report No. 520
June 1986
Robert A. Ulichney
Research Laboratory of Electronics
Massachusetts Institute of Technology
Cambridge, MA 02139 USA
This work has been supported by the Digital Equipment Corporation.
__ill--LIII--
-·^
YI----·IY
--rl
I
P
It
I.
L
·
Digital Halftoning and
the Physical Reconstruction Function
by
Robert A. Ulichney
Submitted to the Department of Electrical Engineering and Computer Science
on April 10, 1986 in partial fulfillment of the requirements for the degree of
Doctor of Philosophy.
Abstract
Digital halftoning, also referred to as spatial dithering, is the method of rendering the illusion of continuous-tone pictures on displays that are capable of only
producing binary picture elements. This report addresses the problem of developing such algorithms that best match the specific parameters of any target
display device, modeled as the Physical Reconstruction Function, particularly
for nonstandard grid geometries. Techniques are organized by computational
complexity and according to the nature of the dots produced, dispersed or clustered. The point process of dispersed-dot ordered dither is generalized for both
rectangular and hexagonal grids, by means of the Method of Recursive Tessellation, a sub-tiling algorithm. Hexagonal ordered dither proves to be the solution
for asymmetric rectangular grids. The concept of blue noise-high frequency
white noise-is introduced and found to have desirable properties for halftoning.
Very efficient algorithms for dithering with blue noise are developed, based on
perturbed error diffusion. The nature of dither patterns produced is extensively
examined in the frequency domain. New metrics for analyzing the frequency
content of periodic and aperiodic patterns for both rectangular and hexagonal
grids are developed. Generalized sampling grids are also examined in detail;
presented is a new "aspect ratio immunity" argument in favor of hexagonal
grids. While some techniques benefit from the use of hexagonal grids, others
are found to be ideally suited for rectangular grids. Several carefully selected
digitally produced examples are included.
Name and Title of Thesis Supervisor:
William F. Schreiber
Professor of Electrical Engineering
-
1)·b·-I
1^·-
--
1
---
_
-
A:
lb
r
1
·
-
Digital Halftoning and
the Physical Reconstruction Function
by
Robert A. Ulichney
S.B., University of Dayton
(1976)
S.M., Massachusetts Institute of Technology
(1979)
E.E., Massachusetts Institute of Technology
(1984)
Submitted to the Department of
Electrical Engineering and Computer Science
in Partial Fulfillment of the Requirements for the Degree of
DOCTOR OF PHILOSOPHY
at the
OF TECHNOLOGY
INSTITUTE
MASSACHUSETTS
June, 1986
(Robert A. Ulichney, 1986
The author hereby grants to M.I.T. permission to reproduce and
to distribute copies of this thesis document in whole or in part.
..........
.
Signature of Author .....
Department of Electrical Engineering and 'omputer Science
April 10, 1986
Certified by
William F. Schreiber
Thesis Supervisor
Accepted by
,..o.........o............................o...............
Arther C. Smith
Chairman, Departmental Committee on Graduate Students
_.. I _I1_L1_IIII_·______I___·
.- I1 111 -1 I ·--II^LII_--l----·LI
·CI·I-IPI-·
-.-
r+
t0
dl
I
Contents
1
Introduction
15
1.1
Choice of Halftone Techniques.
...................
18
1.2
Image Rendering Systems . . . ...................
23
1.3
Image Presentation Strategy .
...................
27
Tone Scale Adjustment
...................
28
1.3.1
33
2 Physical Reconstruction Functioi
2.1
Grid Geometries
.
2.1.1
Periodic Sampling Grids
2.1.2
Semi-Regular Grids
2.1.2.1
3
........
.
.
.
.
37
..............
..............
37
..............
40
Effect of Aspect Ratio . ..............
41
..............
47
2.2
Dot Function and Tone Map
2.3
Reflectance and Luminance
2.3.1
.
Direct Measurement
.
.
.
.
.
.
..
..............
56
.
.
..
..............
57
61
Tools for Fourier Analysis
3.1
62
Periodic (Ordered) Patterns ....................
3.1.1
Continuous-space Fourier Transform Computation . . .
64
3.1.2
Composite Fourier Transform ...............
71
5
~_
w·
_
__1_
C·_
-- -
I_
.-
IIC
Z--
-
3.2
Estimating the Power Spectrum
3.2.2
Radially Averaged Power Spectra and Anisotropy
76
Quality of Measurement ..........
82
Rectangular Grids
4.1.1
4.2
......
The Mezzotint
Hexagonal Grids
85
....
.......
...................
86
...................
90
...................
95
Clustered-Dot Ordered Dither
5.1
5.2
5.3
99
The Classical Screen ................
............
101
...
.
Rectangular Grids .................
108
..
.
5.2.1
Classical Screen at 45 °...........
108
..
.
5.2.2
Classical Screen at 0° ...........
123
..
.
5.2.3
The Spiral and Line Screens ........
128
..
.
5.2.4
Asymmetric Correction
136
..
.
142
..
.
..........
Hexagonal Grids ..................
5.3.1
Hexagonal Version of the Classical Screen
142
..
.
5.3.2
Spiral Screen ................
148
...
.
153
6 Dispersed-Dot Ordered Dither
6.1
100
Orientation Sensitivity.
5.1.1
6.1.1
. . . . . . . . . 154
Tessellating Regular Grids . . . . . . . . . . . . . . . . . . 154
6.1.2
Generation of Threshold Arrays ......
Method of Recursive Tessellation .........
. . . . . . . . . 157
6.2
Examples for Regular Grids ............
. . . . . . . . . 169
6.3
Exposure Plots ...................
. . . . . . . . . 197
6
I
75
Dithering with White Noise
4.1
5
..........
3.2.1
3.2.2.1
4
75
Aperiodic Patterns ......................
6.3.1
7
.. . .. .
Analysis . . . . . . . . . . . . .
. . . .
Ordered Dither on Asymmetric Grids
7.1
Hexagonal Grid Solution .......
7.1.1
7.2
Generalized Procedure.
Rectangular Grid Solution ......
7.3. Examples for a, = 1
. 211
215
...........
...........
....
220
....
225
...........
....
226
...........
....
230
...........
....
240
7.4
Examples for a,
7.5
Crossover Points ...........
...........
....
251
7.5.1
Examples for a, = 1.
...........
...
. 251
7.5.2
Examples for a, =
...........
....
256
...........
....
266
7.6
6
......
Summary ...............
8 Dithering with Blue Noise
8.1
Principal Wavelength .
8.2
8.3
8.4
267
. . . . . . . . . . 268
...
The Error Diffusion Algorithm ......
. . . . . . . . . . 274
...
8.2.1
The Floyd and Steinberg Filter . .
. . . . . . . . . . 277
...
8.2.2
Filters with 12 Weights.
. . . . . . . . . . 289
...
Error Diffusion with Perturbation .....
. . . . . . . . . . 302
...
8.3.1
One Weight .............
. . . . . . . . . . 307
...
8.3.2
Two Weights ............
. . . . . . . . . . 318
...
8.3.3
Four Weights ............
. . . . . . . . . . 332
...
8.3.4
Asymmetric Robustness ......
. . . . . . . . . . 346
...
Hexagonal Case ...............
. . . . . . . . . . 349
...
8.4.1
. . . . . . . . . . 359
...
...........
Error Diffusion with Perturbation
9 Concluding Remarks
9.1
373
. 373
Other Neighborhood Processes
7
_·_
__·__1_11··_111___C·
9.1.1
Sharpening
..........................
374
9.2
Blue Noise is Pleasant ........................
379
9.3
Beyond Binary Displays .......................
382
9.4
Suggestions for Future Research
385
9.5
Summary ...............................
.
..................
387
Glossary of Principal Symbols
391
References
397
8
List of Figures
19
.....
1.1
Quantizing with a Fixed Threshold (2 pages)
1.2
Image Rendering System ................
24
1.3
Tone Scale Adjustment used for the Scanned Picture.
29
1.4
"Scanned Picture" without Tone Scale Adjustment. .
31
2.1
The Physical Reconstruction Function.
...............
34
2.2
General Periodic Sampling Grid ...................
2.3
Pixel Shape vs. Aspect Ratio on Rectangular Grids
2.4
Pixel Shape vs. Aspect Ratio on Hexagonal Grids
........
43
2.5
Covering Efficiency of Pixels on Semi-regular Grids ........
46
2.6
Photographic Enlargements of Binary Display Output (2 pages) .
49
2.7
Cascaded Components of a "Gaussian" Dot Function ......
51
2.8
Diversity and Nonlinearity in Laser Printer Output (3 pages) . .
53
2.9
Example Reflectance Measurements (2 pages) ...........
58
3.1
Odd and Even Spatial Periods
3.2
Rhomboidal Tiling of a Rectangular Array ............
69
3.3
Rhomboidal Tiling of a Hexagonal Array with an even period..
70
3.4
Packing odd periods into an even period .............
72
3.5
Segmentation Strategy for Spectral Estimation.
39
42
.......
63
..................
77
.........
9
-.
_·L111
I--.
.-l-LL-l----
-.-------l.ll--PI
-·--
3.6
Segmenting the Spectral Estimate into Concentric Annuli (2 pages) 78
3.7
The dependence of
3.8
Number of Frequency Samples within each Annulus .......
83
4.1
Rectangular Random Dither of a Gray Scale Ramp .......
87
4.2
Rectangular Random Dither of a Scanned Picture ........
88
4.3
Rectangular Random Dither of a Synthesized Image........
89
4.4
Rectangular Radial Spectra for Random Dither (3 pages)
4.5
Detail from a 1695 Mezzotint.
4.6
Hexagonal Random Dither of a Gray Scale Ramp ........
96
4.7
Hexagonal Random Dither of a Scanned Picture.
97
4.8
Hexagonal Radial Spectrum for Random Dither .........
5.1
Microphotographs of printed classical halftones (3 pages)
5.2
Orientation Perception of a 65 lpi screen.
5.3
Spatial Frequency Sensitivity vs. Orientation ......
.....
.107
5.4
Disparity in sensitivity to oblique and vertical gratings
.....
.107
5.5
Threshold arrays for 45° Classical Screens (2 pages)
.....
.109
5.6
45° Classical Screen on a Gray Scale Ramp (3 pages)
.....
.112
5.7
450 Classical Screen on a Scanned Picture (3 pages)
.....
5.8
450 Classical Screen on a Synthesized Image (2 pages)
.....
5.9
Composite Fourier Transform of the 45° Classical Screen (3 pages) 120
2
on gray level ................
81
....
91
....................
94
.........
98
....
.............
103
.106
115
.118
5.10 Threshold array for a 0° Classical Screen ........
.....
124
5.11 0° Classical Screen on a Gray Scale Ramp.........
.....
125
5.12 0° Classical Screen on a Scanned Picture ........
.....
126
5.13 Composite Fourier Transform of the 0 ° Classical Screen. .....
127
.....
129
5.14 Threshold arrays for Spiral and Line Screens.
10
......
5.15 Spiral Screen on a Gray Scale Ramp .................
130
5.16 Spiral Screen on a Scanned Picture.
131
...............
5.17 Composite Fourier Transform of the Spiral Screen. ........
132
5.18 Line Screen on a Gray Scale Ramp .................
133
5.19 Line Screen on a Scanned Picture.
.
.
.......
....
5.20 Composite Fourier Transform of the Line Screen
...
134
......
135
5.21 Uncorrected Classical Screen on a Scanned Picture .........
. 138
......
5.22 Corrected Threshold Array for a grid with a =
5.23 Corrected Classical Screen on a Gray Scale Ramp, a =
5.24 Corrected Classical Screen on a Scanned Picture, a =
5.25 Composite Fourier Transforms with
=
.
137
.
139
.
.
140
141
...........
5.26 Tri-state Ordering Scheme for the Hexagonal Classical Screen.. . 144
5.27 Threshold array for a 27 element Hexagonal Classical Screen. ..
144
5.28 Hexagonal Classical Screen on a Gray Scale Ramp .......
145
5.29 Hexagonal Classical Screen on a Scanned Picture..........
146
5.30 Composite Fourier Transform of the Hexagonal Classical Screen.
147
5.31 Threshold array for a Hexagonal Spiral Screen
149
..........
.........
5.32 Hexagonal Spiral Screen on a Gray Scale Ramp.
5.33 Hexagonal Spiral Screen on a Scanned Picture.
150
..........
151
5.34 Composite Fourier Transform of the Hexagonal Spiral Screen. ..
152
6.1
The first 8 Orders of Rectangular Tiles. ..............
155
6.2
The first 5 Orders of Hexagonal Tiles.
156
6.3
Rectangular Recursive Tessellation ...............
6.4
Hexagonal Recursive Tessellation (2 pages)
6.5
Rectangular Threshold Arrays (3 pages) .............
162
6.6
Hexagonal Threshold Arrays (2 pages)
165
...............
158
............
160
.............
11
.- SS
1II
_II·__
Ir.--·L-
lll
6.7
Packing two odd ordered periods for rectangular storage ....
167
6.8
Packing two even ordered periods for rectangular storage ....
168
6.9
Rectangular Ordered Dither of a Gray Scale Ramp (8 pages)
6.10 Hexagonal Ordered Dither of a Gray Scale Ramp (5 pages)
.
170
.
178
6.11 Rectangular Ordered Dither of a Scanned Picture (5 pages) . . . 184
6.12 Rectangular Ordered Dither of a Synthesized Image (2 pages) . . 189
6.13 Hexagonal Ordered Dither of a Scanned Picture (4 pages) ...
. 191
6.14 Importance of Hexagonal Kind ....................
195
6.15 Detail from a 1844 silk weaving by the Jacquard loom ......
196
6.16 Rectangular Composite Fourier Transforms (8 pages) ........
198
6.17 Hexagonal Composite Fourier Transforms (5 pages) ........
206
6.18 Superposition of Square and Regular Hexagonal Basebands . . . 213
7.1
Failure of ordered dither on an asymmetric grid (2 pages) ...
. 216
7.2
Sacrificing Resolution for Symmetry with ar=..........
. 219
7.3
Complement Aspect Ratios Pairs ..................
221
7.4
Ternary Subsampling of an Asymmetric Hexagonal Grid ....
223
7.5
Compensating Threshold Arrays by Ternary Replication ....
224
7.6
Hexagonal Subsampling of an Asymmetric Rectangular Grid. ..
227
7.7
Compensating Rectangular Arrays by Hexagonal Subsampling
7.8
Compensated Threshold Array for
7.9
Comparison of Gray Scale Ramp for a,
230
< a, < 1. ..........
= 2
(3 pages)
......
(3 pages) .......
7.10 Comparison of Scanned Picture for a,
7.11 Comparison of Exposure Plots for a, =
(2 pages) ........
7.12 Compensated Threshold Array for
<.
<
7.13 Comparison of Gray Scale Ramp for ca,
7.14 Comparison of Scanned Picture for a,
12
= 6
=
G
..........
(4 pages)
(4 pages)
. 229
......
.......
232
235
238
241
242
247
7.15 Synthesized Image with r7h
=
2, /c = 1, and ar =
7.16 Comparison of Exposure Plots for a, =
7.17 Effect of 77h = 3 and
.........
249
.............
250
= 0 on an a, = 1 Grid (3 pages) .....
7.18 Threshold Array designed for ar =
.................
7.19 Comparison of Gray Scale Ramp for ar =
7.20 Comparison of Scanned Picture for a,
=
(3 pages)
. 252
257
.....
. 258
(3 pages) .......
261
7.21 Comparison of Exposure Plots for a, =.
..
265
8.1
Principal Frequency, fg, as a function of g..............
270
8.2
Patterns with the highest possible Spatial Frequency .......
271
8.3
Spectral Characteristics of a Well Formed Dither Pattern.
272
8.4
Point Processes.
274
8.5
The Error Diffusion Algorithm.
8.6
Error Filters reported in the Literature.
8.7
Error Diffusion with the Floyd and Steinberg Filter (3 pages)
278
8.8
Radial Spectra for the Floyd and Steinberg Filter (7 pages) .
281
8.9
Error Diffusion with the Jarvis, et al. Filter (3 pages)
290
............................
275
...................
..............
276
......
8.10 Error Diffusion with the Stucki Filter (3 pages) ..........
293
8.11 Radial Spectra for the Jarvis, et al. Filter (7 pages)
297
.......
8.12 Two Processing Path Options ...................
304
8.13 Failure of One Deterministic Weight Location ..........
308
8.14 Effect of One Randomly Positioned Weight (2 page)
.......
309
8.15 Radial Spectra for One Randomly Positioned Weight (6 pages)
312
8.16 Deterministic Part of a Two Weight Error Filter..........
318
8.17 Effect of Two Deterministic Weights.
319
................
8.18 Radial Spectra for Two Deterministic Weights ..........
320
8.19 Effect of Two Weights on a Serpentine Raster (2 pages)
.....
321
13
In-ant
S
-
-_.-L_~~
IIX---·---L1
- I------L--l_
__-------
14
8.20 Effect of Two 100% Random Weights (2 pages) ..........
323
8.21 Effect of Two 50% Random Weights (2 pages) ..........
. 325
327
8.22 Radial Spectra for Two 50% Random Weights (5 pages) .....
8.23 Floyd and Steinberg Filter with a 30% Random Threshold (2 pages) 333
8.24 Floyd and Steinberg Filter with 50% Random Weights (3 pages)
335
8.25 Spectra for the F&S Filter with Random Weights (7 pages) . . . 338
8.26 Asymmetric Robustness of Error Diffusion (2 pages) .......
347
. 350
8.27 Error Diffusion with the Stevenson and Arce Filter (2 pages)
8.28 Radial Spectra for the Stevenson and Arce Filter (7 pages)
. . . 352
8.29 Two Hexagonal Error Filters to be Perturbed ............
359
8.30 Effect of Two 50% Random Weights on a Hexagonal Grid (2 pages)360
8.31 Effect of Four 50% Random Weights on a Hexagonal Grid (2 pages)362
8.32 Radial Spectra for the Stochastic Hexagonal Filter (7 pages) . . . 365
9.1
Digital "Laplacian" Filters,
[nl . ...................
376
9.2
Sharpening with a Rectangular Laplacian Filter.
9.3
Sharpening with a Hexagonal Laplacian Filter. ..........
378
9.4
Comparison
of Halftone Patterns for g = o.
.
Comparison~~~~~~
380
.........
Haftn
Pat
.o en
. . ... . . ......
377
Chapter 1
Introduction
Along with text and graphics, images will become a generic data type in general
purpose computer systems. This poses new problems for the system designer.
To the user, displaying an image on any of a wide variety of devices must be as
transparent as displaying ASCII documents. Office video displays with differing
gray level capacity, laser printers, and home dot-matrix printers, all of various
resolutions and aspect ratios must render a given image in a similar way.
A solution requires that associated with each device is a dedicated display
preprocessor that transforms the digital image data to a form tailored to the
characteristics peculiar to that device.
Digital halftoning, a key component of such a preprocessor, refers to any
algorithmic process which creates the illusion of continuous-tone images from
the judicious arrangement of binary picture elements. It is often called spatial
dithering. This report addresses the problem of developing such algorithms that
best match the specific parameters of any target display device, modeled as the
Physical Reconstruction Function.
Outside of using photographic film and some thermal sensitive materials,
15
_._·I
··-.
I-Y1--··--·IIUIII-L·II)-LIII--Y----
-
_IILXIIWIYIII···ICI_·l
Il*LII
X
-_--_
-.
·
__
16
CHAPTER 1. INTRODUCTION
there does not exist a practical method of producing true continuous-tone hard
copy. Computer hard copy devices are almost exclusively binary in nature.
While the video displays associated with workstations and terminals are certainly capable of true continuous-tone, they are often implemented with frame
buffers that provide high spatial resolution rather that full gray scale capability.
Such devices are designed for dot-matrix text and graphics; digital halftoning
provides the mechanism to display images on them.
The literature is replete with approaches to this problem, but almost all
of them implicitly assume target displays with nonoverlapping symmetrically
spaced dots on a rectangular raster. Often the grids are not symmetric, especially in the case of low resolution displays. It is frequently the case that
resolution can be easily increased in one direction and not the other due to
different physical constraints. Also, a conventional rectangular display can be
made hexagonal by simply introducing a half pixel offset on every other line.
Generalized techniques for halftoning on such grids are introduced in this report.
Understanding the nature of the dither patterns created by various algorithms is enhanced by examining their representation in the frequency domain.
New techniques for summarizing the frequency content of periodic and aperiodic
patterns on both rectangular and hexagonal grids are also presented.
In describing the parameters of the Physical Reconstruction Function in the
next chapter, a new aspect ratio immunity argument in favor of hexagonal grids
is developed. While some halftoning schemes benefit from the use of hexagonal
grids, it will be shown in Chapter 8 through revelations in the frequency domain,
that a rectangular grid is preferred for others.
A list of the major symbols used throughout this report is organized in a
Glossary starting on page 391. An explanation of the notational conventions
17
used is also included there.
_____II____LYIli--.-I_(
.-..l____tIICLIIYPI^--··II-III-l--LL·
.I.I*IUFI-LI··_l·-·IIIC-LIIY--.IILIII
·I IIC---
---
18
CHAPTER 1. INTRODUCTION
1.1
Choice of Halftone Techniques
This is a study of controlled noise.
Contouringis a well known noise form resulting from coarse amplitude quantizing; artificial contours or boundaries develop in slowly varying regions of pictures that are truncated to a limited number gray levels. An extreme example
illustrated in Figure 1.1 is when the number of gray levels is limited to two. (A
good rendition of the original images can be seen on pages 336 and 337.)
Here, the two test pictures that will be used throughout this text demonstrate dependence on image content. While the perception of gray level is completely gone, much of the detail of the scanned image survives. But, because
many of the details in the synthesized image fluctuate entirely above or below
the threshold, much of its content is obliterated.
Roberts [62] was the first to point out that dither does not increase noise
energy, but simply redistributes that induced by fixed quantization in a way
which makes it less visible. In the frequency domain, all of the error in coarse
quantizing a fixed gray level is in the zero frequency (dc) term. A dithered
rendition should have an error-free zero frequency term with all of the error
scattered in higher frequency components.
Table 1.1 categorically lists the chapters which address particular halftone
techniques. A gray level can be rendered by covering a small area with either a
clustered or dispersed "dot". If a display device can successfully accommodate
an isolated black or white pixel, then by far the preferred choice is dispersed-dot
·d
I
1.1.
CHOICE OF HALFTONE TECHNIQUES
19
Figure 1.1: Quantizing with a Fixed Threshold.
(a) Scanned Picture. (See page 336.)
-
.··II·
-- ----·-C-rl·i .·itXrrlrrry·C*LI··-L----U---'lsl--L-
1111111_ ----11111
20
CHAPTER 1. INTRODUCTION
0
-4
X
M
vxn Co.
cd
~C
w
bO
w
XD
E
0-4
ce
w
w
0.
b E
"v a"
0 =
_=
.I
-
-
21
1.1. CHOICE OF HALFTONE TECHNIQUES
Chapter
4
white noise
5
ordered dither
6, 7
ordered dither
8
blue noise
Type of
Pattern
Computational
Complexity
(Type of operation)
Type of
"Dot"
aperiodic
point
dispersed
periodic
point
clustered
periodic
point
dispersed
aperiodic
neighborhood
dispersed
Table 1.1: Categorization of Halftone Techniques.
halftoning which maximizes the use of resolution. A clustered-dot halftone mimics the photoengraving process used in printing, where tiny pixels collectively
comprise dots of various sizes.
There is a choice of computationally complexity that can be accepted. A
point operation in image processing refers to any algorithm which produces
output for a given location based only on the single input pixel at that location, independent of its neighbors. For applications where the minimization
of computation time and/or hardware is a premium, then a point operation is
preferred. Of course, neighborhood operations generally produce higher quality
results.
All of the methods listed in Table 1.1 are viable options with the exception
of dithering with white noise, which is presented for heuristic reasons only. The
concept of dithering with blue noise, introduced in Chapter 8, achieves the
uncorrelated features of white noise without the low frequency artifacts.
The preferred choice of a dispersed-dot point operation is ordered dither.
The solution for hexagonal grids included in Chapter 6 proves to be the solution
""-I
11-- 114--11'"1-^-~11-1"
- --·
I-I
22
CHAPTER 1. INTRODUCTION
for asymmetric rectangular grids in Chapter 7.
All of these techniques can be used to augment a display with limited gray
scale capability. To best examine the nature of patterns produced, the focus of
this work will be on the worst case, that of binary displays. For devices that
can display more that two levels of gray, a simple extrapolation is explained at
the end of this report in section 9.3.
Several overviews of existing halftone algorithms have been written. Among
them are surveys be Allebach [71, Jarvis, et al. [39], Stoffel and Moreland [81],
and Stucki [82].
1.2. IMAGE RENDERING SYSTEMS
1.2
23
Image Rendering Systems
The display of high quality images on displays with only two levels of gray
by halftoning can only be successful when performed as a component of the
total image rendering system. The elements of such a system are identified in
Figure 1.2.
The Physical Reconstruction Function is a system model of a given binary
display device. It takes as its input a binary discrete-space image, I[n], and
produces the continuous-space visual image, I(x). What happens in this step
varies widely from device to device. It is here that the image data is given the
physical dimensions of a grid geometry, with a "resolution" and aspect ratio. It
is here that actual luminance values and dot structure are realized.
In a distributed system network, a given digital image may be displayed on
any of several different hard copy and video devices. The role of the device
dependent display preprocessor is to perform all the image processing manipulations necessary to transform a given continuous-tone digital image into a suitable
intermediate binary digital image, I[n], that will yield the visible image when
presented to the device.
The input to a halftoning process is a preprocessed continuous-tone digital
image, J[n]. Halftoning is the last step in the display preprocessor seen in Figure 1.2, after the other device dependent operations of Retrospective Resampler,
Tone Scale Adjust, and optionally, Sharpen are executed.
The Retrospective Resampler is, in most cases, a digital scaler. Except for
_ILIY1111·-l.·*liX-1IPIL·III-·Y--·
I;
_
·_ilI___II
_--·III^IYI--ICCI
-C
I
·I
CHAPTER 1. INTRODUCTION
24
Continuous-tone Digital Image
I_
___ _
_____
l
Retrospectiv re Resampler
Tone Scaile' Adjust
Sha rpen
l
J [n]
1ftone
Hall ftone
-I
Display Preprocessor
.
I[n]
I
Physical Reconstruction
Function
I
I(x)
Figure 1.2: Image Rendering System.
25
1.2. IMAGE RENDERING SYSTEMS
the special simple case of scaling by integral factors on a rectangular grid, some
form of interpolation must be carried out. This is especially true if the given
digital image and display device have different grid geometries, for example,
one rectangular and the other hexagonal. The term retrospective resampling,
adopted from an earlier work on scaling [851, describes the conceptual process
of reconstructing the original continuous image from the given samples, then
resampling this reconstruction.
A common method of performing such a reconstruction is through convolution with an appropriate interpolation function. Schreiber and Troxel have
recently reported on the merits of such functions [73].
While resampling onto a new grid establishes the frame or "backbone" of the
digital image, probably the most significant (and most underrated) contribution
to image quality is tone scale. The simple point operation of mapping the
gray level values of an image onto another distribution can have tremendous
impact on the perceived quality of an image. The Tone Scale Adjust segment
of the display preprocessor must compensate for the tone scale modification
peculiar to the intended Physical Reconstruction Function in combination with
the halftoning method to be used.
Once a halftoning algorithm has been selected for use on a particular device, a gray scale ramp should be generated on that device for the purpose of
calibration. Physical measurements of the reflectance (for hardcopy devices)
or luminance (for luminous devices) should be made of the output gray scale
to determine the compensating tone scale table to be employed in the display
preprocessor.
The next element seen in Figure 1.2 is the Sharpen operation. Sharpening
is optional but if it is to occur it should take place after resampling and before
I----__-·_
-----
- -I11--I----^-l, -----LLL
il-*·~- C
~
I*WL-L-i~l-·
·_II·I---L-^I·IIYII
IIL-C----Yillll^llUY·--·III
---···-·I-·-----·I---·
26
CHAPTER 1. INTRODUCTION
halftoning. At this point, a distinction should be made between image enhancement and the integrity of an image rendering system. In most cases digital
images "look better" if some sharpening is performed. Such an operation can
be classified as image enhancement, a process that creates a changed image
which, by some criteria, is better than the original. The technique of creating
the illusion of a gray level by the judicious distribution of binary pixels, the
essence of halftoning, tends to unsharpen an image; fine image detail can be
lost in halftone patterns. Presharpening an image to compensate for this effect,
to maintain as closely as possible the unadulterated integrity of the original
image, is not enhancement in the usual sense.
In a well designed practical implementation, all of the operations in the
display preprocessor should be carried out in one processing loop; there is no
need to store intermediate images. However, these separate processes should
not be compounded in the halftoning algorithm. They need to be controlled in
a manner decoupled from one another.
The virtues of a halftoning method should be evaluated in terms of its ability
to render the illusion of gray scale with minimum visibility of algorithmic artifacts. Care must be used when evaluating methods which intrinsically sharpen.
The "enhancement" perceived in the sharpened output can misleadingly outweigh other shortcomings in the algorithms. The precise degree of sharpening
should be controlled independently of halftoning.
.·
I
27
1.3. IMAGE PRESENTATION STRATEGY
1.3
Image Presentation Strategy
To assure a consistent and fair evaluation of all halftoning techniques to be
presented, the same three specially selected source images are used throughout
this text.
A historic picture of M.I.T. [55] of very high quality was digitized on both
a rectangular and hexagonal grid with a constant number of samples per unit
area. The image is a good test picture with textured and uniform regions, as
well as areas of high detail such as the stone cut letters.
Scanned image data can sometimes benefit from noise inherent in the sampling process when halftoned. For that reason, a noiseless computer synthesized
image borrowed from a work by Garcia [26] is occasionally shown for comparison.
While a halftoning scheme may perform well rendering the gray levels seen
in a particular picture, it may fail on others. For this reason, a wrapped tone
scale ramp revealing all gray levels is always shown for both rectangular and
hexagonal grids. The ramp proceeds linearly from white to black marked at the
beginning with a one pixel wide black line for reference.
All digitally generated images were printed by an ECRM Autokon' 8400, a
device whose design and operation is described by Schreiber [69,71]. The device
is capable of very high resolution output, but by means of pixel replication,
the images in this text are displayed at very low resolution (about 67 dots per
'Autokon is a trademark of ECRM.
~..
-I~
-. ,,.,
S
I-l_-III
_,I-ULI
28
CHAPTER 1. INTRODUCTION
inch for the case of a square grid). The reasons for this are to allow the reader
to easily examine each of the dot patterns, and to assure that the images will
survive reproduction.
Higher resolution can be simulated by increasing the
viewing distance. It should be noted that in the case of symmetric grids, the
images in this text at the resolution shown would cover just over one square
inch on a 300 dpi by 300 dpi display.
Several images are also shown on asymmetric rectangular grids, particularly
in Chapter 7. To avoid overly small pixels for the same reasons stated above, the
smaller dimension of any rectangular pixel is fixed to that of the square grids, at
about 67 dots per inch. The other dimension will then have a lower resolution.
Therefor, the overall resolution in pixels per unit area in this report is lower
for images on asymmetric grids, by an amount proportional to the degree of
asymmetry. For example, a picture shown on a grid with an aspect ratio of
will have
1.3.1
the number of pixels per unit area as that on a symmetric grid.
Tone Scale Adjustment
The process of printing and reproducing this report can itself be modeled with
a Physical Reconstruction Function. One important characteristic that must
be compensated for is the darkening due to broadening of black pixels which
darkens images.
Figure 1.3 shows the actual tone scale adjustment curve used to prepare
the rectangularly and hexagonally scanned images for the figures in this report.
The transformation maps midrange gray levels to lighter values. (The straight
superimposed line is a reference for a no-change transformation.)
Of particular importance are the horizontal portions at the top and bottom
of the curve. They define the light and dark input ranges that are mapped to
29
1.3. IMAGE PRESENTATION STRATEGY
1
Q"
so
C5
aU
Go
-+,
;F?
0
0
1
Input Gray Level
Figure 1.3: Tone Scale Adjustment used for the Scanned Picture.
30
CHAPTER 1. INTRODUCTION
complete white and complete black. Such a tone scale clipping creates an effect
referred to as "snap" or "punch" in the graphic arts to pictures that would
otherwise be described as "flat". The adjustment is especially important for
low resolution images as the ones in this report.
As an illustration of the dramatic difference a tone scale adjustment can
make, Figure 1.4 is the scanned image used in all of the examples without
the compensation of Figure 1.3. This should be compared to the identically
halftoned picture on page 188.
1.3. IMAGE PRESENTATION STRATEGY
Figure 1.4: "Scanned Picture" without Tone Scale Adjustment.
Compare with Figure 6.11(e) (page 188).
31
32
CHAPTER 1. INTRODUCTION
Chapter 2
Physical Reconstruction
Function
Understanding the nature of binary displays is central to designing high quality halftoned images. The Physical Reconstruction Function, a general system
model of a binary display shown in figure 2.1, is the topic of this chapter. The
only feature of this model which makes the display binary is the fact that it will
accept only binary input, that is, the input digital image, I[n], is a discrete set
of ones and zeros.
The first section will address the details of the first block, D/C or the
Discrete-to-Continuous Space Converter, which is a mathematically convenient
mechanism to map a set of numbers to physical two-dimensional space. It is in
that section that a new argument in favor of hexagonal grids is made in terms
of aspect ratio.
In section 2.2 attention will be paid to the linear shift invariant function,
d(x), which governs the nature of an individual output dot along with the
generally nonlinear Tone Map, which assigns physical output luminances to
33
34
CHAPTER 2. PHYSICAL RECONSTRUCTION FUNCTION
I[n]
c(x)
I(x)
Figure 2.1: The Physical Reconstruction Function.
35
input values. In this model d(x) is convolved with a two-dimensional array of
impulses.
Noise
For completeness, two linear but space-varying components in the model of
figure 2.1 are included to describe the stochastic characteristics of real physical
devices.
Irregularities in the locations of dot centers is described by the Position
Noise impulse. Dot-matrix impact printers have print wires that may wander
in their solenoids. Perturbations in the dot positions of ink-jet printers are due
to both aerodynamic and electrostatic interactions of drops in flight [44].
(x)
would usually be a zero mean random process, or could have a nonzero mean
at spatially dependent locations, such as seen in column misalignments.
Ink spread, dot size fluctuations, and other local degradations are described
by the Dot Noise, w(x). The nature of this function can depend on anything
from the size of the toner particles in laser printers [76] to the type of paper
used [37].
Finally, to complete the Physical Reconstruction Function model, a "Background", b(x), is added to describe the luminance from the paper or video
phosphor used, along with any global degradations.
If the space varying, that is, the noise components are ignored, the relationship between the input and output of the Physical Reconstruction Function can
be succinctly expressed as:
I(x) = TONE MAP
(I[nl6(x
- Vn)) * d(x) + b(x)
where the sum is taken over all input locations, n, "*" denotes convolution, and
36
CHAPTER 2. PHYSICAL RECONSTRUCTION FUNCTION
the matrix, V, is described in the next section.
I
_
37
2.1. GRID GEOMETRIES
2.1
Grid Geometries
The content of this section is relevant not only to binary images and halftoning,
but to digital image processing in general.
Perhaps the most important component of the Physical Reconstruction
Function is the Discrete-to-Continuous Space Converter (D/C). It maps the
input digital image, I[n], into a weighted set of delta functions in continuous
space, c(x), expressed in vector notation as
c(x) = E I[n]6(x - Vn).
(2.1)
n
It is in this step that the set of numbers, I[n], takes on real dimensions; the
Discrete-to-Continuous Space Converter establishes resolution and aspect ratio.
The nature of the impulses in equation (2.1) is the topic of this section.
2.1.1
Periodic Sampling Grids
A Periodic Sampling Grid is a two dimensional impulse train,
E 6(x - Vn),
(2.2)
n
where the Sampling Matrix, V = [vl 1 v 2], is composed of two linearly independent Sampling Vectors,
VI
i
I
V21
V12
V22
(2.3)
38
CHAPTER 2. PHYSICAL RECONSTRUCTION FUNCTION
with reference coordinate system, x, and index vector, n.
Figure 2.2 shows a periodic sampling grid in its most general form. The
sampling vectors, vl and v 2 , can be thought of as grid generating vectors. Note
that if they were not linearly independent, the sampling grid would be only one
dimensional.
Since antiquity, the format (frame shape) used for the overwhelming majority of painted and printed pictures produced has been rectangular, rather than
rhomboidal as suggested by the sampling vectors or any other shape. So, the
reference coordinate system (,x
2)
should be orthogonal, as shown, regardless
of the grid pattern generated. The reference system can, without loss of generality, be aligned anywhere. In this paper, the orthogonal coordinate system
(x1',x 2 ') is adopted.
Image data is almost always organized in lines for digitizing, storing, and
displaying.
It is convenient to refer to the coordinates, xl' and x 2', as the
sample and line directions, respectively. Most often(but not always) the sample
axis refers to the horizontal direction, and the line axis refers to the vertical
dimension. The sample period, S, is the distance between grid points in the
sample direction, and the line period, L, is the distance between lines. In terms
of the sampling vectors, S = Ivl and L = Iv2 1cos 6, where 0 is the angle between
vl and v 2.
Figure 2.2 also illustrates a natural way of defining pixel shape. Pixel Shape
is defined as the smallest circumscribing polygon about a given grid point constructed from the perpendicular bisectors of lines between that point and all
other grid points. Actual dot functions are most often circular in shape. The
size of the circles is chosen to be just big enough to achieve complete coverage
of the plane. For dots of this size and shape, an equivalent definition of pixel
2.1. GRID GEOMETRIES
39
0
,-I
S
l
Figure 2.2: General Periodic Sampling Grid
with associated Pixel Shape.
-__
_L
11_1__
_
1___11·___-
X--
~--
CHAPTER 2. PHYSICAL RECONSTRUCTION FUNCTION
40
shape results: The polygon surrounding a grid point whose vertices are the
intersections of circles of this size centered at each grid point. Note that for
periodic sampling grids, pixels will always be either "hexagonal parallelograms"
or rectangles. It should be noted that the area of such pixels is always S x L
regardless of its shape.
2.1.2
Semi-Regular Grids
A general periodic sampling grid, that is, one with v 2 uncontrained, has two
shortcomings. Firstly, there is a lack of symmetry in the pixel shape and in the
neighborhood surrounding it. Secondly, the offset for every line on an orthogonal
coordinate system is different. This fact adds considerable complexity to simple
image operations like cropping and scaling, or to the design of displays.
Semi-regular Grids are defined as Periodic sampling grids whose corresponding pixel shapes are symmetric about at least two axes. There are two classes
of semi-regular grids:
1. Rectangular Grids, where vl v2 = 0,
(that is, vl I v 2 ).
2. Semi-regular Hexagonal Grids, where vl
(or IV21 COS
.2
=
v 1 12 /2,
= Iv1/2).
Note that semi-regular hexagonal grids require only one offset of exactly S/2
for every other line on an orthogonal coordinate system. Such grids have been
referred to as "offset sampled" or "quincuncial". The familiar case of rectangular
grids requires no offset.
An appropriate name for the shape of a pixel on semi-regular hexagonal grids
is semi-regular hexagon. Of course, the shape of pixels on rectangular grids is
rectangular. A useful metric which completely describes the shape of pixels on
2.1.
41
GRID GEOMETRIES
semi-regular grids is aspect ratio. Aspect Ratio is defined as the ratio of the
sample period, S, to the line period, L. That is, a _ S/L.
2.1.2.1
Effect of Aspect Ratio
Aspect ratio is a parameter that often varies among display devices, particularly
binary devices, but is seldom addressed theoretically. The effect of aspect ratio
on pixel shape is shown for rectangular grid in figure 2.3 and for semi-regular
hexagonal grids in figure 2.4.
The shape of the pixel on a rectangular grid is a regular polygon, a square,
for only one value of a (a
1). The shape of pixels on semi-regular hexagonal
grids is much more interesting. There are two cases where the pixel is a regular
hexagon, for a = 2 and a = 23,
and one special case where it is square,
a = 2.
To avoid confusion between the two kinds of hexagons, a Hexagonal Grid of
the First Kind is defined to be a semi-regular hexagonal grid with a < 2 (see
the top row of figure 2.4). A Hexagonal Grid of the Second Kind is defined to
be a semi-regular hexagonal grid with a > 2. (see the bottom row of figure 2.4).
Mersereau [53,54] has shown that for a circularly band-limited waveform,
sampling with a regular hexagonal grid involves 13.4% fewer samples to avoid
aliasing (spectral overlap) than sampling with a square grid. This packing efficiency argument has long been recognized as one of the most important features
of regular hexagonal grids. More recently, it has been shown [52] that hexagonal
sampling produces samples with greater intersample dependency which allows
"lost" samples to be more accurately recovered or restored. Another important
argument can be made for semi-regular hexagons.
In practical display devices, the physical constraints governing the size of
I__I1_Y__LY__Y____ILI_·*·LI
^-1U····ls---------
--C(-l
,,_
I
42
CHAPTER 2. PHYSICAL RECONSTRUCTION FUNCTION
_ _
a -. 5
aI-I
a -2
Figure 2.3: Pixel Shape on Rectangular Grids as a function of Aspect Ratio, a.
2.1. GRID GEOMETRIES
43
o
-
-
.
-
-
C =
a=.7
2
C73
5
_
-
a = 2.5
_
-
-
-
-
a = 1.6
~ ~~
*
a = 23
a = 5.7
Figure 2.4: Pixel Shape on Hexagonal Grids as a function of Aspect Ratio, a.
_.ll_.lll·LI
1II__1II
1^1
44
CHAPTER 2. PHYSICAL RECONSTRUCTION FUNCTION
the sample and line periods, S and L, are often very different. Accepting these
values as fixed, a given rectangular pixel device can be converted into a hexagonal device by means of a very simple modification, introducing an alternate
line offset of S/2. Assuming that the actual dots produced by the device are
circles with a radius just large enough to achieve complete coverage of the image
plane, a reasonable measure of performance is to compare the covering efficiency
of the two arrangements. Note that in each case, rectangular and semi-regular
hexagonal, the aspect ratio and number of pixels per unit area remain constant.
Covering Efficiency as a function of aspect ratio is defined as
Pixel Area
Circumscribing Circle Area
A high covering efficiency is a desirable property for several reasons. Higher
E(a) means
1. less dot overlap and thus a more linear tone scale rendition,
2. more similarly sized isolated black and white pixels, and
3. less spectral overlap (aliasing) for circularly band limited images.
As stated earlier, it can easily be shown that in all cases, pixel area is S x L.
The radius, r, of the circumscribing circle is equal to the distance from the pixel
center to any of its vertices. Simple geometry reveals that
2+
r =+
8L
4L+2
4S
L2
for rectangular grids.
S
for hexagonal grids, a < 2.
2
for hexagonal grids,
a > 2.
2.1. GRID GEOMETRIES
45
So, the resulting covering efficiency is
for rectangular grids.
E(a) =
64a- ' 1
r(4a-1 + ay)2
16
- ~r(4a+
1
·4a
+ a):
for hexagonal grids, a < 2.
for hexagonal grids,
(2.4)
> 2.
for hexagonal grids, a> 2.
This function, plotted in figure 2.5, reveals several interesting features. On
a plot as this one where the abscissa, x, is equal to the logarithm of a, a is
proportional to eZ and the rectangular function of equation (2.4) takes the form
of a hyperbolic secant, as evidenced by its shape in figure 2.5. The hexagonal
curve is bimodal, peaking at E =
~ z .827 at the two aspect ratios where the
pixel shapes are regular hexagons. At the cusp (a = 2), where the hexagonal
grid has square pixels, the covering efficiency, E =
.637, is precisely that of
the peak of the rectangular grid at a = 1.
Probably the most important observation is that while semi-regular hexagonal grids outperform rectangular grids at all aspect ratios, they achieve covering
efficiencies better than or equal to the best rectangular case for aspect ratios between .591 and 6.77!
Digital halftoning employs binary displays where, particularly in the case
of lower resolution devices, each pixel adds an appreciable contribution to the
quality of the output. Along with the increased symmetry and decreased aliasing
arguments, the relative "immunity" of hexagonal grids to aspect ratio make it
an important alternative to rectangular grids.
.-.--llllllllll·^LIII·YIIII·IIPIIYLilX
__YI1
II^II_-_·L-
~
I
1I-
I~I-~--^---)-
46
CHAPTER 2. PHYSICAL RECONSTRUCTION F UNCTION
2
22v'i
I. A
2r
J
a)
a)
2
Q
VI
eE
b
n n
1('
ilI
Aspect Ratio, a
Figure 2.5: Covering Efficiency of Pixels on Semi-regular Grids
as a function of Aspect Ratio.
2.2. DOT FUNCTION AND TONE MAP
Class
d(x)
I No Overlap
47
Tone Map Examples
-
Liquid Crystal
Electroluminescent
Plasma Panel
II
Pillbox
Hard Step
Wire Impact (carbon ribbon)
III
Pillbox
Soft Step
Ink-jet
Wire Impact
IV
Gaussian
Hard Step
Electrophotographic (Laser)
Offset Printing
V
Gaussian
Soft Step
Cathode Ray Tube (Video)
Table 2.1: Some Major Display Classes
2.2
Dot Function and Tone Map
The dot function, d(x), is the linear space invariant component of the display
with superposition described by convolution, while the generally nonlinear Tone
Map assigns physical output luminances to input values. Next to the Discreteto-Continuous Space Converter, it is these two components that most distinguish
display devices.
Table 2.1 identifies some basic classes of binary displays. The five classes
shown are not expected to be exhaustive, but exemplify target models for
halftone display.
48
CHAPTER 2. PHYSICAL RECONSTRUCTION FUNCTION
Most existing halftoning algorithms are implicitly designed for Class I displays (with square grids). This, the simplest class, includes any display with
nonoverlapping dots. The resulting luminance depends only on the number of
dots turned on, independent of any Tone Map. The other classes in table 2.1
all have overlapping dot functions. The Tone Maps for these classes are all described as "steps" since they eventually clip the output at some minimum and
maximum. "Hard" steps have no transition region between the two extremes,
and thus do not increase density at areas of overlap. Photographic close-ups of
a Class II device (a), and Class V devices (b and c) are shown in figure 2.6.
A very important type of binary display is Class IV, which includes the popular laser printers. Plain paper laser printers operate by charging or discharging
a photosensitive drum with a scanning laser illumination, leaving a latent image
to which oppositely charged toner particles are attracted. Existing products are
described as "positive" or "negative" printers depending on whether the laser
erases white or writes black.
An example of how the dot function, d(x), can itself be composed of several
linear components cascaded together in convolution is shown qualitatively in
figure 2.7 for class IV and V devices. Sonnenberg [78,79] of Xerox has carefully
studied the tradeoffs that must be considered in setting the parameters for d(x)
in laser printers. To maintain the integrity of primitives occurring most often
in text, thin horizontal and vertical lines are usually favored at the expense
of isolated pixels, and thus dispersed-dot halftoning on such devices yield very
nonlinear results.
Figure 2.8 illustrates both the nonlinearities and diversity in laser printer
output. Since this is a cursory rather than thorough comparison of two products,
the names "Product X" and "Product Y" is used to protect the innocent. The
49
2.2. DOT FUNCTION AND TONE MAP
I
Y
l
l
I
1111111
--
(a) Detail of output from a Printronix 600 Dot-matrix Printer.
(60 by 72 dpi, reticle is ruled in millimeters.)
,
Figure 2.6: Photographic Enlargements of Binary Display Output.
--l---------1L---
------ --II--·
- I
50
_________ _ _____I__ · _
______I______
CHAPTER 2. PHYSICAL RECONSTRUC.TION FUNCTION
Figure 2.6 (continued)
(c) Detail of output from an ECRM Autok4:On.
(740 dpi, reticle is ruled in millimeters.)
2.2. DOT FUNCTION AND TONE MAP
51
Beam Shape
J;1
One Period Pulse
31
,1
Video Bandwidth
2
X1
Figure 2.7: Cascaded Components of a "Gaussian" Dot Function.
Llllllyllllllllll___14111 .-.*LII*ll----·l--XI-_·^__·---·l
I_____
II--II
52
CHAPTER 2. PHYSICAL RECONSTRUCTION FUNCTION
difference between figure 2.8 (b) and (c) is due to a tighter dot size and smaller
toner particles in (c); both printers are "positive" in that the laser erases white
area.
53
2.2. DOT FUNCTION AND TONE MAP
Figure 2.8: Diversity and Nonlinearity in Laser Printer Output.
(a) Exact data to be printed, simulated by the Autokon.
____l__laIl__lll__I1I__I--L---·^Y·I
_ -- I
54
CHAPTER 2. PHYSICAL RECONSTRUCTION FUNCTION
i9-.
4"-~j
.o ,
:W.
~~·
CLU
s.-
Figure 2.8: Diversity and Nonlinearity in Laser Printer Output (continued).
(b) Detail of output from "Product X".
(300 dpi, reticle is ruled in millimeters.)
_
____________________·
2.2. DOT FUNCTION AND TONE MAP
55
Figure 2.8: Diversity and Nonlinearity in Laser Printer Output (continued).
(c) Detail of output from "Product Y".
(300 dpi, reticle is ruled in millimeters.)
56
CHAPTER 2. PHYSICAL RECONSTRUCTION FUNCTION
2.3
Reflectance and Luminance
It is customary to assign zero to the amplitude of a digital image sample if
no output is to be produced at that location. This leads to two conventions;
the hardcopy convention assigns zero to the unmarked white paper, while the
convention for luminous displays, such as CRTs, assigns zero to the dark screen.
In this report, the hardcopy convention is adopted, and the gray level, g,
is proportional to reflectance. Reflectance is defined as the ratio of reflected to
incident radiant power. g = 0 corresponds to the reflectance of the unmarked
white paper, Rw, and g = 1 to the reflectance, RB, generated by all black output
pixels. The desired macroscopic output reflectance is then
R = Rw + (RB - Rw)
(2.5)
For the case of perfectly diffuse reflection copy, the luminance,
, observed
at any angle and given illumination is directly proportional to the reflectance.
£
=
£w + g(£B + w)
(2.6)
=
LB + (1 - g)(£w + £B)
(2.7)
For the luminous display convention, equation (2.7) would best describe the
relationship between the desired output luminance,
, and the "gray level"
amplitude, (1 - g).
In either case, if dots do not overlap, then a halftoning scheme should create
the desired output by setting the ratio of black dots to total dots in a given
region of uniform gray level to g. However, dots usually do overlap.
2.3. REFLECTANCE AND LUMINANCE
57
The fact that the linear relationship of equation (2.5) does not hold for most
devices has been only rarely addressed in the literature. Those halftoning techniques which have taken dot overlap into account only consider Class II devices,
that is, displays with perfect circular dots where density does not increase at
areas of dot overlap.
Allebach [6] describes a solution for an imaginary device that has overlapping
pixels that are perfectly square and have sizes that are exact integer multiples of
the grid period. Roetling [67] examined one particular plotter with a fixed dot
size. He proposed computing the overlap into a cell due to all combinations of
its eight nearest neighbors, then using that information to create a compensated
classical halftone screen. Similar geometric solutions were proposed to compensate for dot overlap in the error diffusion algorithm, a dispersed-dot technique
for rectangular [831 and hexagonal [80] grids.
2.3.1
Direct Measurement
It appears the the most reliable means of compensating for nonlinearity in tone
scale (reflectance or luminance) is to use the information from direct measurement of output from a candidate device with a given halftone technique. The
suitability of a clustered of dispersed-dot method can be established, and a
compensating tone scale transformation can then be made prior to halftoning.
The reflectance of three hard copy devices was measured and plotted in
figure 2.9, as a function of covering 4 x 4 periods with increasing number of
black output pixels with the dispersed-dot patterns of figure 6.9(d) (page 173).
In each of the three plots, Rw and RB are slightly different, due to the differences
in whiteness of the papers used, and blackness of the maximum densities which
can be achieved.
58
CHAPTER 2. PHYSICAL RECONSTRUCTION FUNCTION
Black Units per 4 x 4 block
(a) Xerox 2700 laser printer.
(Exhibits failure to accommodate dispersed-dot patterns.)
1.8
8.8%s
Black Units per 4 x 4 block
(b) Printronix 600 dot-matrix printer.
Figure 2.9: Example Reflectance Measurements.
59
2.3. REFLECTANCE AND LUMINANCE
a
nfn
#UN
I
Q
Q
0
0)
+D
U
w
V
1
V
Mn
- 0It
IX*
BS
.l.,S
S
4.
9
12.SS
16.M
Black Units per 4 x 4 block
Figure 2.9: Example Reflectance Measurements (continued).
(c) ECRM Autokon.
"X" for individual dots, "0" for groups of 3 x 3 dots.
1_-1--_11_--
·I-- ··--_IIUI-·I-C---·i·---------------·
--11--^-
60
CHAPTER 2. PHYSICAL RECONSTRUCTION FUNCTION
Figure 2.9(a) illustrates the failure of a particular laser printer to accommodate dispersed-dot halftoning. The inability of this device to reliably print
isolated black pixels results in a tone scale which is not only nonlinear but
nonmonotonic. Clustered-dot halftoning is necessary for this device.
The devices measured in (b) and (c) of figure 2.9 can support dispersed-dot
halftoning, but are nonlinear. Accounting for dot overlap and even the increase
in density at regions of overlap is only part of the reason for this nonlinearity.
To minimize the effect of dot overlap, the 720 dpi Autokon generated the
same patterns with "super pixels" composed of 3 x 3 Autokon dots. The output
was effectively from a 240 dpi device with nearly square dots. The measured
reflectance curve for this case is shown on the same plot as the measurements
for patterns generated with individual dots in figure 2.9(c). The disparity from
linearity in reflectance was reduced but not nearly to the degree that would be
predicted by the reduction in dot overlap.
If the surface onto which a halftone was printed was completely opaque, then
the reflectance could indeed be exactly calculated based on the percentage of
area covered. Surprisingly, multiple internal reflections within paper contributes
appreciably to the nonlinearity of reflectance, a phenomenon recognized over 30
years ago [16j. This effect depends on the translucency and thickness of the
paper, as well as the size and distribution of dots on the top surface.
Considering the complexity and number of parameters that contribute to the
relationship between gray level, g, and output reflectance, R, the best method
of calibrating tone scale is direct measurement.
Chapter 3
Tools for Fourier Analysis
The Fourier Transform has been employed in the past to evaluate halftoned
images, but only very special cases of even period ordered dither on square grids
were considered [3,41,63] In this chapter, comprehensive methods for analyzing
the nature of all types of patterns produced by halftoning will be developed in
the frequency domain for both rectangular and hexagonal grids.
The most characteristic feature of a halftone technique is the texture generated in areas of uniform gray. The rendition of high frequency detail depends
primarily on how sharp the image was (or to what extent high pass filtering was
performed) prior to halftoning. As stated earlier, some degree of presharpening
will usually produce higher quality halftoned pictures.
The best measure of the virtues of a halftone algorithm, then, is its ability
to render areas of uniform gray. The approach used to examine this ability in
the frequency domain depends on whether or not the resulting binary texture
patterns are periodic. This chapter is divided into two part to address each case
separately. Introduced are "exposure plots" of Composite Fourier Transforms
which will present insight into the nature of the periodic output of ordered
61
_
LI
·-LI-·--
I·l-----_L-^_l^-^ll_------__1111·-^1-
^.1__ILlltllll-l-llIl_._iXI_·lll-----_
-
ll__l___ ___ __.___._ _ ___,_II_-X
------
62
CHAPTER 3. TOOLS FOR FOURIER ANALYSIS
dither, and Radially Averaged Power Spectra along with a measure of anisotropy
to provide a mechanism for studying aperiodic patterns.
3.1
Periodic (Ordered) Patterns
The ordered dither algorithms of Chapters 5, 6, and 7 halftone by thresholding
or "screening" with periodic threshold arrays. The binary output from such
halftone processes will also be periodic with the same spatial period as the
threshold array. The spatial period will be specified by two vectors, pi and P2,
in terms of the spatial sampling vectors, vl and v 2, described in section 2.1.1.
The spatial periods can be thought of as tiles which cover all of two-space.
Two types of periods are of interest. Figure 3.1 shows examples of odd and even
periods' (tiles) on semi-regular rectangular and hexagonal grids of the first kind.
Since rectangular tiles share each vertex with 4 other tiles, and each edge with
2 other tiles, only 1 vertex and 2 edges are unique to each tile. By a similar
argument, only 2 vertices and 3 edges are unique to each hexagonal tile. The
outlined edge and vertices in figure 3.1 show those points which are not part of
the unique period.
Even periods are replicated by period vectors that are collinear with the
sampling vectors,
pi=Nv,
(3.1)
P 2 =NV 2 ,
for some integer, N. Odd periods are defined by
p,=N(v, + v 2 )
(3.2)
P 2=N(v2 - v 2 ).
'Describing periods as "odd" or "even" is consistent with period order, r1, introduced in
Chapter 6.
I
3.1. PERIODIC (ORDERED) PATTERNS
63
.
*
I
·
·
a
a
I
I
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
Odd
*
a
*
·
·
*
-
·
·
·
·
·
·
·
·
Even
Figure 3.1: Odd and Even Spatial Periods
for rectangular grids (top) and hexagonal grids of the first kind (bottom).
Boundary locations which are not part of the unique period are outlined.
·--------
---------
--------·---·- ·---
-
-
64
CHAPTER 3. TOOLS FOR FOURIER ANALYSIS
It is important to note that two odd periods on a rectangular grid and three
odd periods on a hexagonal grid can always be packed into one even period.
The derivation of a general expression for the Fourier Transform of periodic
patterns with an even period is addressed in the following section.
3.1.1
Continuous-space Fourier Transform Computation
For nonrectangular grids, it is not immediately clear how a discrete Fourier
Transform can be computed, or what its dimensions in continuous frequency
space are. So, a method of computation along with an explicit expression in
continuous space is sought.
Probably the first formalization of the Fourier representation of nonrectangularly sampled spaces was by Petersen and Middleton in 1962 [60]. Mersereau
later specifically addressed hexagonally sampled signals with his derivation of
the Hexagonal Discrete Fourier Transform (HDFT) [54). But this expression is
complicated, primarily because hexagonal periods can not in general be rearranged to repeat in a rectangular (rhomboidal) fashion.
For the case where the number of elements in a hexagonal period, or any
shaped period, on a general periodic sampling grid is a perfect square, the
following is a proof inspired by the multidimensional sampling theorem [19] that
the canonical rectangular DFT can be used to compute its Fourier Transform.
Proof
The two-dimensional continuous space Fourier Transform, C(fl, f2), of an image, c(xl,:x2 ), can be expressed as
C(fl, f2)
=
Y{C(XI, X 2 )}
J /
C(X1, X2 )e-j2r(flzl+f2z2)dxldX2
65
3.1. PERIODIC (ORDERED) PATTERNS
C(X1,x
2)
=
-l(C(fl,
f2)}
f
=
-CO
C(fl,
f 2 )ej2r(fzl+f2z2)dfidf
2
OO
or, more conveniently in vector form:
fc(x)}
C(f)=
=
c(x)e-2fTxdx
(3.3)
C(f)eJ2rfTXdf
(3.4)
_00
c(X) =
f-{C(f)}
= f
-00
The units of the frequency components in these expressions are cycles/unitlength as opposed to the more common radians/unit-length.
One important identity that will be used is the transform of a twodimensional impulse train:
{6(x-An
)
where
I detAl
BTA
= I,
detA
=
and
aja
22
m
(
(3.)
- a2a2l,
represents
E
n
E.
nl=-oo n2=-00oo
Note that the spatial pixel area as defined in section 2.1.1 is equal to I det Al,
the frequency "pixel area" is det B , and I det Al =
det BI
-
. The matrix B
can be expressed explicitly as
B = (A-')T
a22
)T
=
detA
-a 1
2
-a2l
(3.6)
all
Recalling that V has been defined as the spatial sampling matrix (equations
(2.2) and (2.3), page 37),
.
where U = [ul
Iu2]
6(x - Vn)
n
=
(V-1)T
Idet VI
Jdet~ m
(f U
)
is the frequency bandpass replication matrix.
(3.7)
66
CHAPTER 3. TOOLS FOR FOURIER ANALYSIS
Suppose c(x) is a periodic array of weighted impulses on the sampling grid of
equation (2.2) with an N x N rhomboidal shaped period. It could be expressed
as
0*
n)) *
c(x) = (I[nl 6 (x -
(3.8)
6(x-Pl)
I
where "*" denotes two-dimensional convolution,
N
N
N
represents
n
n
l -=-0n2=0
I[n) is a discrete-space image, and P = [pi P2 = NV is the spatial period
replication matrix.
Note that
5r
E00
I 6(x -
P1l)
1
00
d e PI E
Z c(f
-
(3.9)
- Qk)
k
where Q = [ql ' q 2] = (p-)T is the frequency sampling matrix. Q also equals
-U,
since P = NV and U = (V-1)T.
The Fourier Transform of this image is
{c(x)}
C(f) =
1
1: I[njb(x - Vn)
e j2,rfTxdx)
det P
nC
(na I[n] e - i2 ~rfTVn)
n
X
1
1
>det6(f - Qk)
k
°°
ldet l
(3.10)
6(f - Qk).
This expression is zero everywhere except at f = Qk, so the frequency term in
the exponent becomes
fT = kTQT
where
QT =
V
since Q = (p-1)T and P = NV.
Thus, equation (3.10) becomes simply
C(f) =~t
E 1: I(n]e
-
kTn)
6(f
-
Qk)
(3.11)
67
3.1. PERIODIC (ORDERED) PATTERNS
or,
det
Idet PI
C(f) =
k
I[k16(f
-
I[ni,n2
leN(k
Qk)
where, in scalar notation,
I[k
= I[k 1 ,k2l
=
N
N
E
E
(3.12)
+k2n2)
n I =O n2=O
is recognized as the familiar two-dimensional discrete Fourier Transform!
Equation (3.11) satisfies the need for an explicit expression in continuous
frequency space and a simple mechanism for computation.
Check of Proof
For completeness and as a check for this proof, the reverse Fourier Transform
can be computed in a similar way. Since I[k] is also periodic with period N x N,
1
N
dt
d=
C(f)
0o
I[kl6(f - Qk))
*
>Z(f - Um)
(3.13)
and its inverse Fourier transform is
c(x)
- 1 {C(f)}
=
=det PI
}ddetPI
- Qk) e 2'rfdf) detVIZ6(x -Vn)
E
(6t I[kle 2
k
)7 (x
-
Vn)
(3.14)
1 V 1 again simplifies the exponent.
where QT
The expression is further simplified by observing that Idet PI = N
2
det Vl,
so
0oN
0
c(x) = E I[n]6(x - Vn) =
n
o
E I[n16(x - Vn)
n
* E
1
6(x - P1)
· _I__I__·
(3.15)
II
_
_· _I _
_^
68
CHAPTER 3. TOOLS FOR FOURIER ANALYSIS
because I[n] is periodic with period N x N, where in scalar notation
I[n]-- Intl,n 2
1
N2
N
N
E
ki =O nk =
I[ki, k 2 le3j 2(k ' nIk2n2)
is the two-dimensional inverse discrete Fourier Transform.
(3.16)
F[
This proof applies to periodic grids in general. For the case of semi-regular
grids considered in this text, the matrices involved can be expressed in terms of
the sample and line periods.
For the rectangular case:
V=
U
0
L,
NS,
0
0
NL,
(Sr) -
01
0O
(Lr)
(NSr)-'L
(3.17)
1-
0
(3.18)
(NL,) - l
o
For the semi-regular hexagonal case:
V =
Sh
L0
Sh/2
Lh
NSh
NSh/2
0
NLh
U
(3.19)
-
(-2Lh)
1
o
(NSh)(-2NLh)-1
(3.19)
(Lh)
(NLh)
The associated vectors are displayed in figures 3.2 and 3.3.
(3.20)
Recalling that
equation (3.11) is only valid for periods which have a perfect square number
of elements, only even tiles meet that condition. This is most certainly true
for rectangular tiles as shown in figure 3.2, but figure 3.3 illustrates why the
condition can be met for hexagonal grids. The rhomboidal tiles shown cover the
plane with precisely the same data as the hexagonal tiles.
In frequency space, the rhombus described by the vectors, ul and u 2 , define
how the Fourier Transform is tiled but does not necessarily describe the shape
I
_
3.1. PERIODIC (ORDERED) PATTERNS
'· P2.
...
69
...
*
·
*
*
*
*
*
*
*
*
.
*
*
·
*V2
P1
vi
*
*
*
·
*
*
X
*
*
Spatial Domain
·
112
.
.
.
.
.
.
.
·
2
·
·
·
·
q
·
Frequency Domain
Figure 3.2: Rhomboidal Tiling of a Rectangular Array
coincides with the rectangularly shaped period when the period is even.
-·--------- -----lsblY·UI--ll-
---··-c·-·-····----·--I
·------------
·1·1---1--------·--
-----------------
CHAPTER 3. TOOLS FOR FOURIER ANALYSIS
70
m
Rhomboidal and Hexagonal Tiling
of equivalent data in the Spatial Domain.
Resulting Rhomboidal tiling in the Frequency Domain.
Baseband has Hexagonal Shape.
Figure 3.3: Rhomboidal Tiling of a Hexagonal Array with an even period.
3.1. PERIODIC (ORDERED) PATTERNS
of the baseband.
71
In the hexagonal case (figure 3.3), the baseband has the
hexagonal shape as shown. It is interesting to note that when the period is
even, a spatial hexagonal grid of the first kind has a transform on a frequency
hexagonal grid of the second kind (and vice versa).
For the case where the period is odd, equation (3.11) can still be used by
invoking the Similarity Theorem [13, p. 370]. Figure 3.4 shows how 2 odd rectangular tiles and 3 odd hexagonal tiles can be packed into a single even period.
The Similarity Theorem states that when exactly K periods are transformed
as one period, each nonzero frequency component in the resulting DFT will be
accompanied by K zero coefficients and have a magnitude K times as large as
the DFT of a single period. So, packing odd tiles in this way only requires the
additional step of dividing the resulting DFT by K.
This step is automatically handled by the normalization term in equation (3.11), det P - , where
Idet PI
= Z x (pixel area) = (spatial period area).
where Z is the number of elements in the period used.
3.1.2
Composite Fourier Transform
The DFT is a strictly valid description of the Fourier Transform only for periodic
arrays of weighted impulses of which equation (3.8) is one. It is impossible for a
signal to be both of finite spatial extent and truly band limited. When the DFT
is applied to real signals, some degree of spectral overlap must be accepted. In
well designed systems, this error is made inappreciable, but never zero.
In this section, the Fourier Transform of the images resulting from halftoning a two-dimensional plane of one gray level are examined. I[n] in this case
__^·I1IIU)
·Y-----_lll·l--l-(UWL·Y·LII
i-.·.--l.
72
CHlAPTER 3. TOOLS FOR FOURIER ANALYSIS
/
/
.11
Figure 3.4: Packing an integral number of odd periods
into an even period.
73
3.1. PERIODIC (ORDERED) PATTERNS
happens to consist of only 1's and O's. Such images are precisely as described
by equation (3.8), and thus the results are theoretically exact. (This could be
the only practical use of the DFT that can make that claim.)
The output binary image is capable of rendering one of Z + 1 gray levels
for ordered dither with a Z element threshold array. The binary image, I[n; g],
resulting from halftoning a continuous-tone image consisting of one constant
gray level, g, with a given threshold array has a DFT equation (3.12) denoted
by I[k; g.
The phase of I[n] relative to an origin is not important. Of interest is the
magnitude of Ilk; g], which will provide insights into the relative distribution of
energy in frequency space. To summarize this information over all gray levels
for a given threshold matrix, an average is used. The Composite DFT is defined
as having frequency components,
I[k]
Z+
1
(3.21)
I[k;g 1
all g
Plugging I [k] into equation (3.11) yields a specification of the location and
magnitude of Fourier Transform impulses. This Composite Fourier Transform
is defined as
(3.22)
(3.22)
1
Cr(f) =
det P l
k
I[k6(f - Qk)
Note that in all cases, the zero frequency term of Cr(f) will be an impulse with
area
ldetP
2SL'
(3.23)
2SL'
Idet PI
since I[O] will always equal Z/2, the average number of black output pixels in
a period.
A reasonable means of displaying Cz(f), is with dots of an area proportional
to the magnitude of the impulses. Such a display is similar to a photograph
_
___
i
____n·
L_
1___1_
_YL__LaJ__^I__LIL--r--·-^l··lll·
(----I-11(·.-.
11--_lLlllllD-·
--·--ll--·P-·
II----.
74
CHAPTER 3. TOOLS FOR FOURIER ANALYSIS
of an ideal optical Fourier Transform consisting of points of light of different
intensity; the resulting size of the exposed points would be proportional to their
magnitude. Thus, the name "exposure plots" will be used to describe them. An
exposure plot of the Composite Fourier Transform will be presented with each
ordered dither threshold array explored in Chapters 5, 6, and 7.
_
I
___
3.2. APERIODIC PATTERNS
3.2
75
Aperiodic Patterns
Halftone processes which do not produce output by thresholding with a deterministic, periodic threshold array will in. general be aperiodic. The analysis of
section 3.1 will be inappropriate. Such aperiodic dither patterns can be modeled
as stochastic processes.
The unconditional probability mass function of any individual binary output
pixel, I[n], is
p, (I[n]) =
g
for
for
for
(1 - g)
I[n] = 1
I[nJ=
I[n] =O.
(3.24)
Since this is true for all n, I[n] is a stationary random process with
E{I[n]}
and,
var{I[n]}
= g
a2
(3.25)
=
(1-
9).
(3.26)
The mean of I[n] is exactly what is expected, since the gray level g is being
represented. The variance of I[n] varies with g, and has a maximum at g =
1
midway between the extremes of zero variance at solid black and white.
3.2.1
Estimating the Power Spectrum
The Fourier Transform of the autocorrelation function of a stationary random
process is the Power Spectrum, P(f). In most cases, the autocorrelation function
of a given aperiodic halftone process will not be known, so a method of spectral
I-·-c·Vlrrlllllsrr1I0--·---·D-·--r
-Ir)--_-r-^--·--*-U-I·--_I··V---Y--
.I-.·_-_
·-^ill·^-*r_-n-rrrr-------·lll-
---- _ ·-*··-·------I----
76
CHAPTER 3. TOOLS FOR FOURIER ANALYSIS
estimation must be employed to produce an estimate, P(f), of P(f). Bartlett's
Method [10] of averaging periodograms, named after the one who first suggested
the technique for the one-dimensional case, will be used in this study to produce
P(f).
A periodogram is the magnitude squared of the Fourier transform of sample output, I[n;g], divided by the sample size. All spectral estimates in this
text will be produced by averaging 10 periodograms of 256 x 256 output pixels
from a given halftone rendering of a fixed gray level. Figure 3.5 illustrates how
the sample output will be segmented for the purpose of computing the 10 periodograms for both rectangular and hexagonal grids. Since some of the processes
of Chapter 8 have transient behavior near edges or boundaries, the segments
are cropped sufficiently far from output edges to avoid such artifacts; only the
"steady state" output will be measured.
It can be shown 571 that a spectral estimate formed by averaging K periodograms has an expectation equal to P(f) smoothed by convolution with the
Fourier Transform of a triangle function with a span equal to the size of the
sample segments, and variance
var{P(f)}
3.2.2
1
p2(f)
(3.27)
Radially Averaged Power Spectra and Anisotropy
A desirable attribute of a well produced aperiodic halftone of a fixed gray level
is radial symmetry; directional artifacts are perceptually disturbing. P(f) is a
function of two dimensions. Although anisotropies in I[n; g] can be qualitatively
observed by studying 3-D plots of P(f), a more quantitative metric is proposed.
Figure 3.6 shows how spectral estimates, P(f), can be partitioned into annuli
of width A for regular (a) rectangular and (b) hexagonal grids.
Each annulus
77
3.2. APERIODIC PATTERNS
256
Rectangular Grid Case
Hexagonal Grid Case
Figure 3.5: Segmentation Strategy for Spectral Estimation.
_^___I*I_
--·IYIIIII--·IYU
--l·l-L-
----·11---
78
CHAPTER 3. TOOLS FOR FOURIER ANALYSIS
I
I
I
I
I
I
0
1
1
2
Radial Frequency, fr (units of S - 1 )
A;
Figure 3.6: Segmenting the Spectral Estimate into Concentric Annuli.
(a) Regular Rectangular Grid, S = L (a = 1).
181 annuli actually used.
3.2. APERIODIC PATTERNS
9
79
I
I
I
I
I
I
I
L
I
I
I
I
1
2
VV-
3
I
0
Radial Frequency, fr (units of S-')
Figure 3.6: Segmenting the Spectral Estimate into Concentric Annuli.
(b) Regular Hexagonal Grid, S =
L ( = ).
148 annuli actually used.
80
CHAPTER 3. TOOLS FOR FOURIER ANALYSIS
has a central radius fr, the radial frequency, and Nr(fr) frequency samples.
Two useful one-dimensional statistics can be derived from averages within
these annuli. The sample mean of the frequency samples of iP(f) in the annulus,
IIfI - frl < A/2 about fr, is defined as the Radially Averaged Power Spectrum,
1
r(fr)
N,(fr)
Nr(f)
(f).
E
(f)
(3.28)
=1
The sample variance of the same frequency samples is defined as
1
(f) =
(f-
Nr(fr)
1
(P(f) - P(f)) 2
(3.29)
Note that the sum is divided by Nr(fr) - 1 and not Nr(fr), so as to yield an
unbiased estimate of the variance (see [24] or [581).
For each gray level output of a halftone process to be analyzed, two plots will
be presented. First, the Radially Averaged Power Spectrum divided by oU will
be shown. Since spectral energy increases with a2 equation (3.26), normalizing
Pr(fr) by this amount will render all plots on the same relative scale. Because of
the importance of or,
its relationship to gray level, g, is now shown in figure 3.7.
Secondly, the Anisotropy of P(f) will be plotted. Anisotropy is defined as
S~~~2
~(3.30)
(fr)
P. (fr)'
a measure of the relative variance of frequency samples within a given annulus.
The zero frequency term is assumed to be close to g in all cases; the spike
at this frequency will not be shown since it does not contribute to the structure
of the dither pattern.
__
___
81
3.2. APERIODIC PATTERNS
.3
b
v
CZ
e.
.2
.r"
X,4
.1
0
0
1
2
Gray Level, g
Figure 3.7: The dependence of r2 on gray level.
1
CHAPTER 3. TOOLS FOR FOURIER ANALYSIS
82
3.2.2.1
Quality of Measurement
To what extent will Pr(f,) and
S 2 (f,)/P,2(f,)
be meaningful metrics?
From
equation (3.27) and the fact that K = 10 segments are used in the estimate,
P(f),
var{P(f)}
-
P 2 (f)
10
(3.31)
If P(f) is perfectly radially symmetric, the measure of anisotropy, S2 (fr)/P,2(fr),
is merely an estimate of the above ratio. 'Therefore, an anisotropy of i! or -10
dB should be considered "background noise", and a reference line at this level
will appear in each plot.
Also, if anisotropy is low, that is, close to -10 dB, indicating good radial
symmetry, then P(f) is effectively a function of one independent variable, f,
instead of two variables, f. The variance of Pr(fr) is that of equation (3.31)
divided by Nr(fr), assuming that each of the Nr(f,) samples are independent.
This reduction in variance as Nr(fr) increases is indeed observed in the experimental data of Chapters 4 and 8.
Nr(fr) depends on the width of the annuli, A. As indicated earlier, in this
report all estimates, P(f), will consist of 2562 frequency samples. The size of A
was chosen so that exactly one sample along each frequency axis fell into each
annulus; that is, A = ql, where q = ql = q 2 , since the grids are assumed to be
regular. A plot of Nr(fr) for (a) rectangular and (b) hexagonal grids is plotted
in figure 3.8.
The irregularities in the shape of these plots are a consequence of rectangular
and hexagonal grids not being perfectly radially symmetric.
The number of
grid points that fall into a particular annulus essentially increases linearly, as
one would expect, up to the largest annulus that will completely fit within the
shape of the baseband; this occurs at f/S
- 1
= !2 for rectangular grids and
3.2. APERIODIC PATTERNS
83
1000
-
a
500
C- 4
0
To
4a)
0d
rw
0
0
1
12
Radial Frequency, f, (units of S-')
(a) Regular Rectangular Grid.
1000
E
co
C*.
4
04
500
,0
$4
W
;Z;
0
0
1
Jli
2
3
Radial Frequency, f, (units of S - ')
(b) Regular Hexagonal Grid.
Figure 3.8: Number of Frequency Samples within each Annulus.
_ _·_____X_III111_____^_^--·-·-ly·ll
-li^--·----
II
--_-__
84
fr/S-
CHAPTER 3. TOOLS FOR FOURIER ANALYSIS
1
=
for hexagonal grids.
W
Chapter 4
Dithering with White Noise
In this chapter, the process of creating a dispersed-dot halftone by the point
process of thresholding an input image with uniformly distributed, uncorrelated
(white) noise is investigated. The quality of output from this method does not
deserve consideration for practical use; as will be seen in Chapter 6, another
point process taking no more computational effort performs much better than
this one.
So why should a chapter be devoted to this so called technique of "random
dither"? There are two reasons.
The first is historical. The idea was the first used to exploit the fact that
electronic displays can have independently addressable dots. Goodall [29 in
1951 and Roberts [62] in 1962 demonstrated how contouring due to insufficient
gray levels can be corrected by adding noise of this type. This is perhaps the
first technique that comes to mind to correct the shortcomings of using a fixed
threshold, and in the early days of digital halftoning it was always referenced for
comparison; in fact the name "ordered dither" was meant to contrast random
dither.
85
_·I_
_
_·II
11_ II^I _X·
·I
86
CHAPTER 4. DITHERING WITH WHITE NOISE
Secondly, investigating the output of single gray levels, I[n; g], dithered in
this way, provides a means to check the validity of the newly introduced metrics
of Radially Averaged Power Spectrum and Anisotropy. Since I[n;g] is white
noise, it has a known autocorrelation function, namely an impulse at the origin
with area u2. So, the power spectrum should be radially symmetric with fixed
amplitude, aJ. Such radial symmetry has been observed optically in Fraunhofer
diffraction patterns of randomly distributed apertures [32].
The random numbers used in this chapter (and in Chapter 8) are, strictly
speaking, pseudo-random.
They are produced by means of a multiplicative
congruential random number generator [431 available with many programming
libraries. This is a very efficient scheme requiring only one multiplication and
division per random number, and in the case used in this study, has a repeat
period of 232.
The effects of dithering with white noise on regular rectangular and hexagonal grids will now be considered separately.
4.1
Rectangular Grids
A random dithered gray scale ramp is shown in Figure 4.1. Examples of the
effect of random dither on a scanned and synthesized image are given in Figures 4.2 and 4.3. They suffer from a grainy appearance. This is the case at any
displayed resolution because of the presence of long wavelengths (low frequencies) at all gray levels.
The Radially Averaged Power Spectrum and Anisotropy for gray levels,
(a) g = , (b) g = , (c) g =
7
are displayed in Figure 4.4. With each plot a
small portion of the sample image, I[n;g], is shown at the top. The well be-
I
_
4.1. RECTANGULAR
GRIDS
~
·.
·
.
~,
"
·
87
,.
·
r~~~~
~
·
.I~
·
·
=
.
'"
"''"'.'
4'
.
,
.'t°l'~.'.;
...
·.·-._
N"
·
'5
~·
·+
.
· ·
.··
:i '· :"'
'':'
~
.·
'
.. . ''
r,$· '·,l,.~);
· 'CY
X"
.:Y-...;/..
·.: C' '~.'ea%
·
·
·~,lt
....-
_-
,I'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-
-··..~x.)-~,,','.;..~?_
- k
,.:~
~~~~~
~
~.. ~% ~...........
~ ~ ~~~~~~~~X·
~~z(.'.:='
,C··
~~.,,~~-,~,
:),'~..
"l
~ .
/
[kee
d':..'.'~'~
_~,C·..
li. .;..,
·-
.,% .~~j
~. -.~
,.·.
~..
..
~...··:-.
,..-,.
, -..,,,:.-.-.,
-. , ,":. .~.,,
'..~,.-~'.,U?..',,.,-...~',':,.
:..,,,,:,~·
....~..':::,.~~.~~',~~.57.-'._~.-','
-,,,
i .,.~ -', .
,. ~~'.,J.:-.., :.'.':...
':.r.''.,,
·
,~,'
'~.'
? '':T -. '.~.~:,-.d',.'P..~
.7
: I
r-
p
.
., , r- .·· -~~~~~~
x. ~ ~ ~ ~ ~ ~~~~
;r=~~~L.ZI
r,~~~~
5-lb;I·k
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~·
~ ~ ~ .,~~~~~~ :r
...
.
~
i- .
Figure 4.1: Rectangular Random Dither of a Gray Scale Ramp.
-
-
-
-
--
-
-
-·-
-
·-
·-
·-
-
88
CHAPTER 4. DITHERING WITH WHITE NOISE
To
Figure 4.2: Rectangular Random Dither of a Scanned Picture.
I
-
-
4.1. RECTANGULAR GRIDS
89
a)
do
co
E
U2
a)
0
a)
w
4.
S
c0n
QO
.
4
a)
E
o
ad
..
CD
._
O
ed
uu
CT~i
CHAPTER 4. DITHERING WITH WHITE NOISE
90
haved nature of these plots validate four things:
1. The amplitude of Pr(fr) is correct, that is, flat, as expected for white
noise.
2. The values of a2 are as predicted by equation (3.26), Figure 3.7 (page 81).
3. The apparent variance of Pr(fh) decreases with Nr(fr) (Figure 3.8) with
a minimum at , (IS for the hexagonal case).
4. The anisotropy measure is correct. White noise is radially symmetric so
the anisotropy should be at the "background noise" minimum of -10 dB.
The characteristic features seen here will serve as a reference for the many
plots to be studied in Chapter 8.
4.1.1
The Mezzotint
It should be mentioned that a randomly dithered binary image is often called a
mezzotint, after a print making technique invented in the seventeenth century
[36]. The etymology of this Italian-rooted word is that of "halftone", a word
used to differentiate the modern mechanical printing technique from the older
art. The dark regions of an image were roughened or ground on a copper plate
by a skilled craftsman in a somewhat random fashion by hand. The resulting
scratches acted as tiny wells which held ink, much the same as in modern day
gravure.
A photographic enlargement detailing an actual 1695 mezzotint from a collection by Holman [361 is shown in Figure 4.5. The patterns are not as structured
as that of a periodic screen, but do not have frequency components which go
to zero as in random dither. The ancient mezzotint engravers would probably
4.1. RECTANGULAR GRIDS
91
I
+10
0o
A- -
Aon,
AM44---
4-1 M
A- A
0
0o
r_
-10
W. V
V",."v
I
- -
-V- --I
I
2.88
1.
1.88
ti
_
_'
, ,
.
._
,
'
.
.
I
_
.
0
r
'
.
Ct
R.n
.
.
.
.
1
.
-- a.. Iam
1
2
Radial Frequency, f, (units of S - 1)
Figure 4.4: Rectangular Radial Spectra for Random Dither.
(a)
=
2= 7
9
64
.1094
7;-
92
-
CHAPTER 4. DITHERING WITH WHITE NOISE
+10
10
N-
4-o
w
o
I
.
I
.-
r_
-10
2.88
i|I
I
tI
I
·
eq M
b
%-
CI5 1.88
a an
'u." a'
m8
I
!
2
Radial Frequency, f, (units of S - ')
Figure 4.4: Rectangular Radial Spectra for Random Dither.
(b)
1 ,2-2 = ~4 = .25
1
%/-2
93
4.1. RECTANGULAR GRIDS
I
m
M
+10
I--,
ok
o
0
e"
ma
A;
-10
----- MAA
-AA
-v - -
-K-
N
-,A-
- -
- .
- - - -,- 0,in VIA -V
M - .. ..
v
1.v vyvq
A
I
2.88
I,
I
l
II
1
2
,/2
C4 t
-b
i.
CC
1.8
ImOR
w8 .
88
Radial Frequency, f, (units of SFigure 4.4: Rectangular Radial Spectra for Random Dither.
(c),
3 2O=g- - .1094
1
l)
94
CHAPTER 4. DITHERING WITH WHITE NOISE
Figure 4.5: Detail from a 1695 Mezzotint.
--
--
.__.....
.___
..I
95
4.2. HEXAGONAL GRIDS
be outraged at the association. A true mezzotint beautifully renders delicate
shades of gray without the graininess seen in white noise.
4.2
Hexagonal Grids
Hexagonal Radially Averaged Power Spectra exhibit the same well behaved
features as in the rectangular case. A random dithered gray scale ramp and
scanned image are shown in Figures 4.6 and 4.7 for regular hexagonal grids of
the first kind. The Radially Average Power Spectrum and Anisotropy have been
observed for several gray levels. One representative sample at g =
Figure 4.8.
8
is shown in
96
CHAPTER 4. DITHERING WITH WHITE NOISE
~
,,....
~-:. ~ ~
-:~ ~ ~
* /:.. .
-- ..-
;..
".;"
~ ~ ~~.,Y...
~
)' .~
~~~~~~~~~~~~~~~~~~~
..
· · · ·
'.Qb~~~~~~~~
~
'a~
~
· · · T..
.&
·-B-URMW-1-m
I
MI
m-~
I
ri-IW
Pi
Am
m
an
Mi
iii,
Ia~
i I IIm I,
U0
ax-VIA
W MRno t
e
-
-
.J
-J-
RSEM
mffi
e'M
wamH
rm
M
I MM ME
Sm
--
AM
--- - IFR
-
w-
~-0II-Q~--
-
-
- -
-
-
PIC
Figure 4.6: Hexagonal Random Dither of a Gray Scale Ramp.
97
4.2. HEXAGONAL GRIDS
Figure 4.7: Hexagonal Random Dither of a Scanned Picture.
____11_1_
_11
I
_I_
I
_
_
_
_
CHAPTER 4. DITHERING WITH WHITE NOISE
98
+10
0
>o
I.n
kD
4%
0
-10
.. ^
.-----
01 e
j..f
2.8
eq bM
t
145
cl
I.8e
I
!
i
,
laaI
I I
1
Radial Frequency, f,. (units of S - ')
Figure 4.8: Hexagonal Radial Spectrum for Random Dither.
9r =
= g'
LO'i ~--
.1094
2
Chapter 5
Clustered-Dot Ordered Dither
Halftoning by ordered dither is the topic of this and the next two chapters. An
ordered dither algorithm generates a binary halftone image by comparing pixels
from an original continuous-tone image to a threshold value from a deterministic,
periodic array. The thresholds are "ordered" rather than "random". Ordered
dither is a point operation, that is, the output depends only on the state of
the current pixel. Once a suitable threshold array is determined, the extreme
simplicity of implementing halftoning by ordered dither makes it an important
practical design candidate; it is for this reason that three chapters are dedicated
to it.
The approach should not be confused with a method called Pulse-SurfaceArea Modulation (PSAM) which maps an input pixel into a cell with a black to
white distribution proportional to the original gray value. This is probably the
earliest approach to computer halftoning [31,42,59] where variations ranged from
overprinting of characters, use of special "gray-scale fonts", and arrangements
of dots in fixed size cells. In the latter case, the mapping from input to output
pixels is not one to one; an implied scale change occurs. There are no advantages
99
100
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
of halftoning with PSAM, especially compared to the quality achieved with
ordered dither.
Ordered dithering techniques can be divided into two classes by the nature of
the "dots" produced, clustered and dispersed. In general, dispersed-dot ordered
dither (to be addressed in Chapters 6 and 7) is preferred, but clustered-dot
dither must be used for those binary display devices whose physical reconstruction functions cannot properly display isolated pixels. An example of such an
electronic display was seen in section 2.2.
The most well known example of a system that must use clustered-dot
halftones is the printing process.
It is the printing process that gave us the
word "halftoning". Clustered-dot ordered dither is by far the most widely used
halftoning technique, electronic or otherwise, and is in fact the type assumed
when the word halftoning is used without further qualification. It is interesting
to note that to date, it is the only type of halftoning that is supported by a
popular page description language [1].
5.1
The Classical Screen
The art of printing halftones is over a century old [36], and has experienced
almost no change since it first came into practice. Besides being old, the printing
industry today is one of the largest in the country, three times as large as the
semiconductor industry. The vast majority of pictures produced every day are
made on printing presses [72]. We should take serious note of those practices
which have survived the decades.
Around 1850, the feasibility of the process was demonstrated by photographing an image through a loosely woven fabric or "screen" placed some distance
5.1.
THE CLASSICAL SCREEN
101
from the focal plane. It came into practical use in the 1890's when the halftone
screen became commercially available, consisting of two ruled glass plates cemented together. The quality of the resulting image depended a great deal on
the skill of the printer, since the optimum distance at which the screen should be
placed from the focal plane depended on a complex combination of parameters.
The only significant advance occurred in the 1940's with the introduction
of the contact screen, a film bearing a properly exposed light distribution of
a conventional screen. Not only did the contact screen eliminate cumbersome
geometric considerations, but eliminated diffraction effects. Halftone dots produced with such a screen do not only vary in size but also in shape, thus allowing
more high frequency detail to be produced at a given screen period.
I call this screen the "classical" halftone screen, also referred to as the graphic
arts or printer's screen.
Today this ancient screen is the best available for
preparing pictures that are to be reproduced by means of the printing process.
Naturally, for low visibility of the screen, as small a period as possible is
desired.
But this is limited by factors such as the viscosity of the ink, pa-
per coarseness, and the minimum printing plate area that will hold a dot of
ink. Details of the use of this screen in the printing process are reviewed by
Schreiber [74, ch. 6 and Yule [93]. Roetling [651 has quantified the number
of perceptually detectable gray levels possible for this screen as a function of
spatial frequency.
5.1.1
Orientation Sensitivity
The halftoning process shares many of the perceptual concerns common with
most image processing problems. However, general texts such as by Graham [30]
or Cornsweet [17] omit one issue that is specifically important in halftoning.
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
102
Figure 5.1 shows the detail of actual printed halftones produced with a
(a) 125 pi screen in 1902, (b) 150 Ipi screen in 1916, and (c) 88 lpi screen in
1984. These pictures reveal that since the dawn of halftoning it was recognized
that the resulting images "looked better" if the screen was oriented at a 45
degree angle.
We can demonstrate the lack of symmetry in the frequency response of the visual system by looking at Figure 5.2. This phenomenon has been quantified only
relatively recently [14,34,841. Figure 5.3 [84] shows the sharp asymmetry with
regard to our ability to detect a fine grating as a function of orientation. Also,
the disparity between sensitivity to oblique and horizontal gratings increases
with spatial frequency, as seen in Figure 5.4 [141 indicating the importance of
contrast in this phenomenon.
_
---
---
5..THE CLA SICAL SCREEN
IV9
V.r
*
*
aA*
103
*
4
a
A
I'
I
IV
I
r_
Figure 5.1: Microphotographs of printed classical halftones.
(a) Detail from a book published in 1902 [451.
(Reticle is ruled in millimeters.)
104
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
l
!~~~~~~~~~~~~
_~~~~~~~~~~
!~~~~~~~~~~~
l ~ ~~~~~~~
0
d
Figure 5.1: Microphotographs of printed classical halftones (continued).
(b) Detail from a book published in 1916 [50].
(Reticle is ruled in millimeters.)
5.1. THE CLASSICAL SCREEN
105
I
0
,;
4
Figure 5.1: Microphotographs of printed classical halftones (continued).
(c) Detail from a contemporary newspaper (the Boston Globe).
(Reticle is ruled in millimeters.)
__I
L___l_
_lI---
-- ·--
l_
·
C
106
C- -!--
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
--- -- --- --
^~
~~
~
a Scee at 45dgeeage
A
(b) Horizontal screen.
Figure 5.2: Orientation Perception of a 65 pi screen.
(17 cycles/degree at 40 cm)
107
5.1. THE CLASSICAL SCREEN
3
0
oP9
a
-%
0
0
I
,/
w
"-2
-J
aJ
Li
IL
w
w
o
21
Q:
i
0
0_/
z
W
M
Z
O
z
In
a
U.
VERT
NOR
I
HOR
ORIEN TAT I ON
Figure 5.3: Spatial Frequency Sensitivity vs. Orientation [84].
.
A
1C0
A
10
1
1
2
5
10
20
5(
Figure 5.4: Disparity in sensitivity to oblique and vertical gratings [14].
(, = vertical, A = obliq
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
108
5.2
Rectangular Grids
Several major classes of clustered-dot halftones will now be demonstrated digitally, first for rectangular, then for hexagonal grids. In all cases, the threshold
array, halftoned gray scale ramp, halftoned scanned picture, and exposure plot
of the Composite Fourier Transform will be shown.
5.2.1
Classical Screen at 45 °
Three sizes of this, the most popular halftone screen will be examined in this
section. Generating the ordered threshold array for this screen is very simple.
At middle gray (g =
) the output plane will be covered by a checkerboard
pattern of alternating black and white squares of size M x M pixels. For gray
levels lighter than g =
,
the black squares must diminish is size, and for darker
gray levels, the white squares contract as black pixels spiral in.
The threshold array will have Z = 2M 2 elements, and thus 2M 2 + 1 gray
levels can be represented. The tradeoff in the selection of any ordered dither
threshold array size is always between the visibility of the low frequency due
to the period or size of the array and the number of gray levels which can be
rendered. Threshold arrays for M = 3, 4, and 8 are shown in Figure 5.5.
Halftoning with threshold arrays such as these simulate the contact screen
in that they allow high contrast, high frequency detail to punch through the
screen. Simulating the pre-1940's screen would require averaging or low-pass
filtering the image data in the threshold period prior to screening.
5.2. RECTANGULAR GRIDS
109
I
n
2
4 3
I
1
.X u
(a) M = 3 (19 levels of gray).
I F%
29
32 31
26 25 18
]
17 14
12
23
5
4
22
6
1
18
9
7
9
.l,.
(b) M = 4 (33 levels of gray).
Figure 5.5: Threshold arrays for 450 Classical Screens.
1 1_·1___11_
11___1_·________111I- --
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
110
Or
127
121 114
106
97
93
92
82
74
67
(
5
(c) M = 8 (129 levels of gray).
Figure 5.5: Threshold arrays for 45 ° Classical Screens (continued).
5.2. RECTANGULAR GRIDS
111
Note that the period is odd, and as is the convention in this text, the period
is shown with nonunique edges (see Figure 3.1) to better illustrate the nature of
the periodicity. Also, borders between the dark and light halves of the threshold
array have been drawn. In this and all other illustrations of threshold arrays,
the values shown are actually the order in which the thresholds are arranged
rather that their absolute value.
The effect of halftoning a gray scale ramp is shown in Figure 5.6, a scanned
image in Figure 5.7, and a synthesized image in Figure 5.8. These illustrations
underline the extremely low resolution used to display results in this report.
The extremely coarse screen with M = 8, would look quite reasonable if the
longer dimension of the images shown were shrunk to about 1.5 cm at normal
viewing distance, or if the figure was viewed at a distance of 5 meters.
Exposure plots of Composite Fourier Transforms (see section 3.1) are shown
in Figure 5.9. At the bottom of each exposure plot, a scale defining the actual
dimensions (in cycles/unit-length) of the plot in terms of the original sample
period, S, is provided. The shape of the baseband reflects the shape of the
pixel, which remains constant (square) in this case. Because the spatial period
was odd, the frequency samples are arranged in an odd fashion. Also, for all
period sizes, the distribution of energy in the frequency domain is concentrated
at the low frequency center in a symmetric fashion.
_
·II----·IYIYI·-*II.I_-I
--11-C.I-_-_·X--·L--_PIIIIIII-IIII..-1·111
-.1-.11111_-1-1
..1IL-
·._
112
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
. ..................
. . . . . .o ·.
.
. . . . . . . . ............
. . . . . . . . . . .
o .
. . . . . . . .
·..
·
.
. . . . . . . . . . .
. . . . . . . . o o
. . . . .
o o o..
. . . . . . o .
....................
..........
oo... . . . . . . . . .
o . . . .
.
. . . . .
· · .
.. o............o......
......
...............
.o..o....o....
1ooo.oo...
.......
.............
. . . . . .· · . .
. . . . . . . . .
· .
o .
· .
· ·..
. ·.
·..
oo... . . . . . . . . . . . . .
. .
·.. . . . . .
.
. . . . . . . . . . . . .o..
. .
. .
·.. . . . . . .
.
aaaaiaaaa.mm
........................................
mm
. . . .. . . . .. .
.
..
..
.. .
..
..
. ..
..
..
mmmmmmmmmmmm
mmmmmmllAll
. . . .
.
. .
.
. .
.
..
. .
.
. .
. .. . .. ..
. .
. .
. .
. .
.
. .
. .
.
.. . . .. . . ..
. .
. . . .
. .
.
. .
..
.. . .. .. . .. . . .
. .
. .
.
. .
. .
.
. .
...... ....... . ... .... . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
. .
. .
.
. .
. .
.
. .
.. . . .. . . ..
. .
. . . .
. .
.
. .
..
.. . .. .
. .. . . .
. .
. .
.
. .
. .
.
..
..
..
. ..
..
. ....
. ..
..
. ..
..
·
.. ..
..
..
.....
...
..
. ..
...
..
..
~~~~.
lll
ll
l
. .
.
. .
..
.
. . . .
. . ... ..
. .
.
. . . .
. .
. .
..
. ...
. .
..
. .
. .
. .
. .. .
. .
..
.... ..
·
~~~··~~································
m
m
m
.am
m
m
m
m
m
m
m
~
m
. ..
. .. . ..
. ..
. .. . ..
. ..
. ..
.
.
. ..
..
.
. ..
..
.
.
. ..
..
.
.
.
..
..
.
..
..
.
.. ..
.
.. ..
..a.mmmmmmmmmmmmmmmmmmmmmmAAA
..
.
..
..
..
.
..
..
..
.
.. ..
...............
..............
....................................
...................................
mmmm
a.mmm
AAAA&afta&&&&aa~aAaa&&&a~AAA
aaaaaaaaaaaaaaaaa~
&aaaaa&&&*&&a&a~a aAAA
a*aa~a~aAa~a~aAaA&a
aAaA&&a&&&**aA&aa~a
M0 M MM M
.
.
.
. .
. .
.. ..
. .
..
.
..
..
.
..
..
.
..
..
.
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
....
....
. .. .. .. .. .. .
..
.
....
....
... .. ..
....
...
..
..
..
..
..
.. ..
..
..
..
..
..
.
..
..
..
..
.
..
..
..
..
.
..
..
..
..
.
..
.. ..
..
....
....
....
. .. .. .. .. .. .. ....
....
....
....
....
....
....
...
....
....
. .. .. .. .. .. . ....
....
.. .. .. . ....
....
...
....
...
. . ..
. ... ..
. .
. ..
..
.
.........................................
. ..
.
..
..
.aaaaaaaaaa
..
..
.
..
aagaaaa
li
. . .
.
. .
.
. .
. . .
. .
. .
. .
. .
. . . .
. . .. . . . . . . ..
. .
..
. .
. .
. . . .
mmmmmmmmm
m
.
..
&
*&&**.6&
aa m
m
mmmmmAAAAAll
aaaaaaaaaaaamm
.&a11a&a
...........
~ 1 se
m
aa
$
.
..
..
..
..
..
..
..
..
..
..
..
..
.. .. .
....
....
..
..
..
.... .. .. .. .. ..
....
. .. .. .. ..
b
.
·
I~~~I~L+~
~
MEMMEREMEMEMME
. .
A
. M M M M M 0 M M 0 0 wmrwwwwmrwwwwmrmrwwwwwwmrmrwwwmrmrwmrwmrwwwwlwwwww
Figure 5.6: 45' Classical Screen on a Gray Scale Ramp.
. . .
5.2. RECTANGULAR GRIDS
.
.
.
.
.
.
113
.
.
.
.
.
.
.
.
.
...
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
. .
. .
. .
. .
. .
...............
.
. .
.
. .
. .
. .
. .
. . -.
. . .
. .
. .
. .
. .
.
. .
.
.
.
.
.
.
.................
.....
·
.
..
..
..
*
*u69999
Il**f
*
*U
r
f9ffr9frrr
99
..
9
r
..
.
·
.
· d··.
....
..............
.
..
..
.
.
9
uq
p99
r I ~9qqqqqqqqqqlqmllllmmllllu
99~q999l
r! 9
!999 B qugq***z^8**b
i
B *
U l!!ll
99
f
lll
~~i
9
V WV WE
www ww
y
v
WV-...
l.Ufrf
lqww
lllllU
l
l
bwwwwwwwWEW
EWE
EW
E w p * EW
WEEP
wm
EE
m
11
UIUIUUUE
U
ll~l
E
N
***uuz*m..
llluuuuuuuummuuuj
web
EWEEP
m
b
...
U *UEUEEUEU
UU
E
U
U
qqqlqql llqqqlmumu*..uuuu
/S
b/qquuub
*b******b
b *b
b
b
*bb b b *bbbbb
bbb
b bbbbbbbb--
jj
**Oi
b
* *
_
* *
m
**ttteeIeeImmmmmmmmspp
jK-M--Z 0
.
.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
...
..........I~~i
...
. ..
............................
..
.....
.999
r _9991
99rr
VEE
WEE
V
ww VV VVe
VEER VVVVV
.
.........
~~~..
*99
lll·in~~'
I*EUU*YYY
r
* u
..
................
.
. .
..
.-.-.-.-...
. .
. .
.
.
. . ..
............
d···
.
. .
.
. .-- -- -- ·
d
. .
.
. .
. .
.
. . .
.
.
...............................................
..................................................
..............................................
.......................................
I..............................................
.................................................
...............................................
.................................................
..
.
.
...
mm..
M ELML7M-7W7Z
M M w w
wwwwwmmmmm
IMMPME MEIN.
- -
- -
-
- - - wiw-mm
WWWW-M ----
- -
------------------------------------------------
-----
---
-
--
--
-
---------------------
RRNRRRRRRN
----------
Figure 5.6: 45 ° Classical Screen on a Gray Scale Ramp.
(b) 33 levels of gray, M = 4.
--
--
114
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
-.
-.
.
*p *
**m
mm
m
* *
·
-
~~- m.
m
* ,m·····m··im
*· **
6,
·
·
,m
-
$
ma
,i ,
li
u
-
-
aa a
-
!
--
·
--
-
a uau
a
·
I
·m
r
-
-
a***
I
w
r
a
*
·
·
oI
iI m
...
V
m
I
w
·
f
*
*
-
*....--4ma
,m.
I
i*
#
I
·
f
·
---
·
I
*
·
I
-9
a04
6600
O O oo·I
t I I
g
!
9
U
U,
--..
u · *9
j~
4t. ·
mm ol
m
.....
eeeeeeeem..eueeee
""" "" "
'e~ee
'e " 'e'' IIIIIIIIIIawwaw~a
t '"
ME
a
t
maa
NNm
MKK
Iiimimii88mi···········
I
m
Figure 5.6: 450 Classical Screen on a Gray Scale Ramp.
(c) 129 levels of gray, M = 8.
]I
*
-
I*
115
5.2. RECTANGULAR GRIDS
Figure 5.7: 45 ° Classical Screen on a Scanned Picture.
(a) 19 levels of gray, M = 3.
___I__
__·____YIIIII·__·__L__YFI-C--I^
116
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
',.,-...U...-.
U
.
.
.
.E .
.
.
·
,,.
:
.....
,
UU
iji
v,;,.,...
'
:: U!
Figure 5.7: 45 ° Classical Screen on a Scanned Picture.
(b) 33 levels of gray, M = 4.
__
._
5.2. RECTANGULAR GRIDS
117
f
·
P
v ·
W*
P
w ··
.·
v ·
·
-'.
4
''-
*'
H^^^b
&
b
A
·
*
At
·
-
'. .
4b,
-
&
A
*A
v
Figure 5.7: 450 Classical Screen on a Scanned Picture.
(c) 129 levels of gray, M = 8.
__
I
II
___X I__
X_1
IIC_
Il·lll··IPIII-·.ii---
·--- I--.-·I1I--
-L·Il-.^ll-·
--
I
II-I
118
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
bC
,_
t;
o ^
c,) tQ
0
_II
0 ol)_
._
v
c
I1U,
bO
II
(d *-~
o
V
g) ..
-.
WbO
119
5.2. RECTANGULAR GRIDS
-
*[
Ui I .
%%'
Il a
I
m m
!
Li II
I]
I
-
U
I
rol
ki 0 Pi !
I Im ,7 U
rl
_ID
m
0
m a I
"
m '
0
m
aFICK I
;iMMJ PY ;PI
_
AI
0
I
IU
p
m
l
0
~
I,
.
m~ m
Pi
L.._
p
I
aUII Ia
a
I U
I a I
U EUIUI
aI I
c
IPU
dFq%-Mi.. ?I'll dI-qW
Me
m
I .1
I
I
P" 6.
hmP- .W
qlswImq __
m
m
I
u I
0
P-ql
L
J m
..
1
-i
___
5:
M
T
OA
, n
-
n.
. I
U
I
I w
.mmmrN
I
w
al
%-MFL W-WFt
QB·lpi
q
A
Y
M%%~el?
19%Ajo
--m-w
0
- 4.
ICLI
MffMM
m
m
m
m
!
II
'0- l
_
_! *Be.g
-
Pl
Ifil[
@ 1, ·
WEE
m
l
l
r
I JEEM '_,1
mm
I
ALT;
_
I
I
isZBBB
m
m
I)
·
_
_
mm
ii!
a)
bO
_
I
ce
U
I I
I
mo
w
]
i
i
s
s
[
N
-
._
C)
1II
+-' o6
E
a) >to -
t
v
I
I
-
I~~~!
I
-n
l|a
-
·
4lf
n:mdl
.
s
D
L
J
m.
* z | t w~~~D
.
4
10
'
V _
...
d
U),TW
uo
I tR
..
L
l.
|
A.
A.
-
I
I-1
%-%--
I~~~~~~~~~~~~~~~~
he~~~~~~~~~~~~~~
::
a
4
:+
__
m
I
3
II
I
U
m
__
Eu..lml
I El.
m
I
U
II_
_
m
I
*
I
l
m
U
I
ELM
II
.
LO
a)
I-
bO
B.
- -
Ie
*16
I
-
~p.'mw
m.Ri].~.%-...~Elm
L_*
,m
di 0
,mB
.
v
v.
·
.
__
II
.I.E
II
.
120
CHAPTER 5.
CLUSTERED-DOT ORDERED DITHER
0
.
.
.
0
0
.
.
.
0
.
.
.
.
0
.
2S
cycles/unit-length----
Figure 5.9: Composite Fourier Transform of the 45 ° Classical Screen.
(a) Average of 19 patterns, M = 3.
5.2. RECTANGULAR GRIDS
0
121
.
0
0
0
0
0
0
.
0
0
.
.
0
0
0
.
.
.
.
0
0
0
0
.
.
21
28S,
cycles/unit-length--
Figure 5.9: Composite Fourier Transform of the 45 ° Classical Screen.
(b) Average of 33 patterns, M = 4.
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
122
·
S
*
0
0
0
*
0
0
0
0
0
0
0
*
6
0
0
0
a
0
0
S
0
0
0
0
0
0
a
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
S
0
0
0
0
*
*
0
0
0
*
0
0
*
*
*
*
*
*
·
*
*
*
*
*
·
0
0
' cycles/unit-length--
Figure 5.9: Composite Fourier Transform of the 45 ° Classical Screen.
(c) 129 levels of gray, M = 8.
5.2. RECTANGULAR GRIDS
5.2.2
123
Classical Screen at 0 °
Because of the sharp minimum in perceptual sensitivity for spatial frequencies
oriented at 45 ° (or 135 ° ) from horizontal, there is no reason to generate screens
at any other angle for binary displays. Printing color images, however, presents
a new problem.
Color pictures require an overlaying binary image for each of three inks to
represent a reasonable range of the color gamut. In the case of hard copy, a
fourth overlaying image, black, is usually needed to increase the maximum density achievable, eliminate chromatic errors and the need for critical ink balance
across the complete range of neutral grays, and to reduce cost by replacing the
use of the three more expensive inks for one less expensive ink in regions of
neutral gray.
Overlaying identical screens with a fixed phase shift would not be a problem
if the output display was perfectly noiseless. However, even an extremely small
but regular variation in dot position due to the Position Noise, 6(x-
-x)),
in the Physical Reconstruction Function (Figure 2.1, page 34) common in real
printers produce an overwhelming artifact, moire patterns resulting from the
beat frequencies between the periodic screens.
This problem has been well studied and is minimized in practice by simply
orienting the multiple screens at different angles, usually about 15° apart (see
[93, ch. 13]).
Holladay 35] has presented a method for digitally generating
so called "rational screens"-the classical screen at angles which have tangents
that are ratios of relatively small integers. Techniques have also been invented
125,68] to produce such screens at any angle by "dithering" between angles with
rational tangents.
demonstrated at 0° .
In this section, an example of a different screen angle is
124
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
35 30 18 22 31 36 35
2915 10
21 32 29
14
9
5
6 1
13
4
1
2 11 19 13
2
8
3
7242528
34 2
20 14
12 23 26 33 34
35 30 18 22 31 36 35
Figure 5.10: Threshold array for a 0 ° Classical Screen.
Figure 5.10 shows the 36 element threshold array used. Note that the period
is even, and that the border between the dark and light halves of the screen is
as indicated. The effect on a gray scale ramp and scanned picture are shown
in Figure 5.11 and 5.12. The rectangular arrangement of frequency samples
seen in the exposure plot (Figure 5.13) reflect the fact that the spatial period is
even. As in the 45 ° example, energy is symmetrically concentrated at the low
frequency center.
I
125
GRIDS
5.2. RECTANGULAR
. . . : : , : :: ...
...
... .. :.
.... :........
. . . . : ., . .............
p,.........
.......
UPffff
Wf9Y
f
fff
f
IfIfffIfIf9.9.9.9I9
99
99
I
a
99
99
I
99
99 .
.9I99
. .9.9I-+
444444444idid
-*-*yr-*--rrr-
^
^
^
^
^^
1
-------------------e&&&e&essene&s
a~)~~~~~~~~
r
r
i
~
hSS
i
iS i
-ry rvnrvvtw
Fv.r F
^
^^
^
rX
^
.
.
.I..I..I.....
..
.
.
.I..I..I.....
l
n.
.
nEU
1I l li
*ll l i 1 l JS
v
wtwvlwlrvwW
^^
^
Ii
-_:::-......I-Sh
| Si
iSS
SISiiihShihSiS
:~~~~~~~:i
.;r .E.
r!
v
........--_
|
..
..
ismummm
a Umm......l 1111lilll//illliS
aammmmmmmi.m.mmimimeilllllll11111lllbb
a am .....m m m ..... m * 1 1 1 1i e e e e e e e ei 1 1
^
]
~ ~~~ ~~ ~ I" ~l ~ ~
|
i i E U U E U U E E n num
r
^^
r:.
XX[': r :; XX:.
,_r_
l
..
..
^
^
^^
fvVVv
*^--^---
^
b4 P4b4 ,4b4 14 neee004P4Po P4bEI4 04 *eeeeeee000004001404 k4k4pi 0,
II It II 1. II II II-----I
I· II II II II II -------------IBI 1I BIBI I!
I P4 .0P4 I P000004IP4P4 04 04 O04 *eeeeeeeeeee0 10' II0E.4O414 0.
Po P
III 14II II II tl II-I-II II idMI/dBI IIIIIIIIIIIIII
BII lI BI Id II IB BI
b4 r4 b4 P4b4 '1 a....O&O P b4 04 b4 r4b4 kIII4IttItIIt4
0bi klM4kdb4b4 0,
, P4 I P4'I 0000
P
'.
PqP4 P. Pq 0. 04 eeewoweewwew. 0'. I P41404PM0,
l
-----------
i
..
..
.....................................
Fry
:::::~;r
..
..
++
d idddddaddad
da diiiddadd
daddeaddddld
Adfddddddadd
ddddddmddadd
d~dadddddadd
'444444444d~ad
444444444dddd
^^
.I....II....I..I.
..
.
YYYWWYYY99WY++
444f444444d
444444444dadd
4444444444i55
^^^
..
..
.
4,
..
I
........................
-.......................
. .. .. ..
i
iS ii
Si
S
S -..................
Sl S
S
X
X.r
....
.X.X.X.X.X.X.
.X.X.................
i
Sl
L
X:
l
| L
xl
|
l
l
|
l
l
lx
|
l
l
X'
X
-Xxxx
X'|
Figure 5.11: 0 ° Classical Screen on a Gray Scale Ramp.
(37 levels of gray.)
|
126
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
I
p'
" _
. l
._
filte.
4p0.-
h -1.·...
, -i,.,
·--
,
.
·
.
-
L
'
:
!t wl -£^b~w~s
*r+,+p,
' l **-*1 * ,4 *ha 1
4, 4
.
SZ
w^
*,,ab~m,,,
.,*,^^^**,_|^
w , e V +0
t~
'a 4
'
, I fil
n~~~~~~~~~~~~~~~b.
.f
s b n
f~~~~~~~~~~~l~
-
_f
n
g
~ --
,
a*
,..: x^^*|^vf~
"T--rvPF-vwv
'
·
4p
t641
i~~~~~~~~~~~~~~~~~~~~~~6
r+v
'b
.m. . ^++
.il~ tu .... l d.
lppe,
i
,~
~
l,.1
,
·
ql+,d
a A--
*
ipsp. *
d vw
.......
.........
+,,
'I''1'"!"9'"1'9
_,, ...
-
P
4'1
- ~~.
.
*
r, r
'
.
*rP-T
__-s
V
'.'
ff
d^IT
d 41,0u
d.M4
d1,
q*4r~14,-. r,,'-'gsbl11']l'-
.. .. ..
*
.
m
·....
r .
4FW
4
Pm
lr
81q,4
,,
2
4M-v
-t
9
4
r$
1
.-.
. :~
.
P
_
f qr *
q
f~~~~~~p.
k
·
d..M.VW
dlk
~ ~~~ ~ ~ ~ ~ ~~~~~f$,
~~~~~~~~~~~~~~~~~
6 S~~~~~''*,,.::::
+
dq."
,bId.
4 4 4
Q.
md
^++^*
,+-,:,,4,
_
~~~~~~l*_
~~~~~9
i
k4
++s
-P
I
l
-
: qP"1 '2
*a"IF
9 1. , , . 1.
' 4
4
A p.
...
,
|
^**-"
.-1
a
b-4k
**P *. V
~ rs.*,,~lt''6+wTvfrFb
4)'a ,q~~~
.4,
II
-
_
**
+
"
*.
,,
f
FFX7
r
992.s
.......
40.qp
v.41Mpqlt
,14p.
'
~~~~~~~~~~~~~~~~~I
~ ~ ~ ~
T
-P P+
w
v¢-a
$",
,I·4,,·4,·,.
,4i.,iw
l,,41~e
2.;.
:w
..
..
'P
:
~qP* 4
:
:
-
x
*8 _=t
p
mmpl all. So md Qd|*.
:
:
*
Figure 5.12: 0° Classical Screen on a Scanned Picture.
(37 levels of gray.)
;~~~~~~~~~~~W~kaJLl
5.2. RECTANGULAR GRIDS
.
0
.
0
0
127
.
0
0
.
0
0
0
.
0
.
.
.
0
.
.
0
0
.
0
0
I
2It
25S,
cycles/unit-length
.
l
Figure 5.13: Composite Fourier Transform of the 0 Classical Screen.
(Average of 37 patterns.)
128
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
5.2.3
The Spiral and Line Screens
A wide range of special effect clustered-dot screens are available in the graphic
arts industry and all are easily simulated digitally. For example, the ECRM
Autokon Laser Graphics System 201 does this in hardware. In this section, two
such screens are examined.
Figure 5.14 shows the threshold arrays for the spiral and line screens. The
line screen clusters pixels about horizontal lines, while the spiral screen is essentially half of the classical screen, with dark squares growing to fill the plane
without the alternating light squares.
The gray scale ramp and scanned image are displayed in Figures 5.15
and 5.16 for the spiral screen and Figures 5.18 and 5.19 for the line screen.
The exposure plot of the 5 x 5 element spiral screen (Figure 5.17) is quite similar to that of the 0° classical screen. This is the only even period rectangular
threshold array with an odd number of elements on a side in this report. The
horizontal and vertical periods are 5 units. For this reason, no frequency energy
can exist at the edges of the baseband, at fl =
or
f2
=
.
The lack of energy
there is not an issue for a clustered-dot screen, but would be for dispersed-dot
ordered dither (investigated in Chapter 6), the success of which depends on the
concentration of energy at high frequencies.
The exposure plot of the 36 element line screen in Figure 5.20 reveals a strong
concentration along the vertical frequency axis. This is due to the dominant
vertical frequency resulting from the preponderance of horizontal lines. These
vertical frequencies are separated by the inverse of the fundamental vertical
spatial period,
.
129
5.2. RECTANGULAR GRIDS
21 22 23 24 25 21
20
7
8
9 10 20
19
6
1
2 11 19
18
5
4
3 12 18
Spiral Screen
17 16 15 14 13 17
21 22 23 24 25 21
36 34 32 31 33 35 36
24 22 20 19 21 23 24
Line Screen
12 10
8
7
9 11 12
4
2
1
3
6
5
6
18 16 14 13 15 17 18
30 28 26 25 27 29 30
36 34 32 31 33 35 36
Figure 5.14: Threshold arrays for Spiral and Line Screens.
130
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
............
..
-----------...................
................... "I ....
............
................... "I ....
...................
....................
aa
d
~
~
~
a
J
d
dd
ada
d
d
I'll .................
I'll..................
.................
.1,
.1, ..................
.... .................
.1, ..................
m
da
d
........................
m
am
m
m
m
n
m
mm
m
m
mm
.JJ
m
mmmm
.kkhkkdd
m
m
m m
mmm
.......
.......
.......
.......
.......
.......
m mm
m mm
m lm
hm
kkkhkkkkkkkhkkkkhkkkk
kkkkkkkkkkkkhkkkhk
kkk
bkkbkkkhkbkkkkkk
h 6 h
mmmmmmmmm
d mm
aaammm
aamm
mmmm
hkkkkkmmmmmmmm
mmmmmm . ..
mmmmmmm
.. . .u
mmmmm&mm
&mmmmmmm
EEVEEWEEVVVVEWVE
EWE
mmmmmE~u
EEEEEEEE
..
hkkkk
khkkkkkkkkkkkkkkkkk
.kkkmamddmdaammmmmmmmmmmmmmm
6kb
...........
...........
...........
...........
...........
...........
.
U
EEEEEEEE
U
ku EW W
EUEEE
E Wkkkkkk
twwtwwwtrrfww
E[EESESS -SSSWWSS[w.w.w[.mmuummmmmmummummumlmmmm
qqqqqqqqqqqqqqqqqqqqiqqqqqqqqqqqqmqqqmqqqqqqqqqmqqqqqqqqququumuuuuuuuuuuuu~
qqqqqqqqlqqqqqqqqqqq~qqqqqqqqqqqqquqqquqqqqqqqqmuqqqqqqququuuuuu!EEEEuuuulE
qqqqqqqqqqlqqqqqqqqqqqqqqqqqqqqqqqquqqqqqquqqquqqqqqquqquumuuuuuuuuumuuu
qqqq~qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqmqqqqqqqqqqumqququqqqqEEEEEEEEEulEEE
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqmuqqmqqqqqqqlEEullEEEEEEEEE
qqqqqqqqqqqqqqqqqqqlqqqqqqqqqqquqqquqqqqqqqlqqqqqqquqquqqqEEEEEEEEEEIEEEE
qq q q q q q q q q q q q q q.. , mXqm quqq qqqiqmq qqqu uuuu uuuu uu
Iii[[lllllllllllllllnnnnnnnnnmnnn...........u..E.E.mu.EuE-~ummmummmmmmmmmm
||||||||||wll|||""""gg""MUMM
M001-Ax"""
-I-
|||w||||||||||""""""OM-OMMO
--
mm-
-
--
mmm
------------- - - - - - ---
MEN
-- - -- - ---- - - - - - - - - ---------------
nmmm.mmmmm
mm
.mmm mm
mmmm mm
- -S |sr
-------
-- - - - - - - - - - - - - - - -
-mmm-----m-
Figure 5.15: Spiral Screen on a Gray Scale Ramp.
(26 levels of gray.)
___
___MEME
5.2. RECTANGULAR GRIDS
I .. V1
* ·
,wbqg
*
p
*NW=
rir_*
rwwb
Ib..
. *
Ull
N
131
94h 1
b.W|
b* P.
t
i1~
*19 -
ma
~
Ir
~
0,t.^
IILL4maw..wIl.rn
__* W W~~m
,
-rl
ma'
bal q wb
WIb
~ a aw61%
I _i n xJ r
- s
· 4b
.
- -
,
b
o
%
a
=
.r.
....
...
aer
*.a .n
~~EW
r _um-*--qt,--· ' ·.....
i~~~ \,,#·
E3"'_,~BIFI,4B
sbrari
Qr-
. r- .a
*
pn l, · w*
·............
s,,,,,
-
-- -·
s
__-b-,^.---iq,,,*,,*.,,,P,'lffllm|..g'llk
a
...
.
s ^^
¢*_r_rs
iw^.....
.
w
.
q
r.
.:
Ib
* m~"m
w *:. *t.m
-
s
- W *dwt
.
*
ab
a
.........
a
-q-...
d
* I-
....
................
vrq-^-S~|srr--*.--,--,,,....,..,,.,.,.'WE
....
.........
&mwro --- a-,,
1~~~~~~~~~~~~~~~~~
E:
3
-a - ...........*
-l-
~m
.
aa+.
...
. * ..
.
..
| w. Irl
1,E"r :
-1l
NW
~-.
*. :: --r- -..............................
·^^'''a'>i"^^^..................
- ·,, ., -- - *
.. 'p
^ Yi
i
..*
..
.
.!bW
....
i
w
b , m :a
Tor"'-.e OI-,
Ai-L,~~~V
l
WQ_*1hV
-
·
. s
b__pA,*
N
*
'
*rF
.....
u^ - a
r
'.
R
*. · *a
- *W, * *-
d.
w,I
1E
Ea hb.
-__ ir l
Wwi*
' *
.
*
-m---i
C.
.1
r *
-%II..
i'd......
'_·b nu^e.r!Birqla
--
s-->->wz...
··
%%q··r
-
-,;
~-,,,,,.
- · ·......
,.,
"
Figure 5.16: Spiral Screen on a Scanned Picture.
(26 levels of gray.)
.
,
.
..............
lbqklllWm
132
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.
0
.
i
ylsuntlnt
Figure 5.17: Composite Fourier Transform of the Spiral Screen.
(Average of 26 patterns.)
5.2. RECTANGULAR GRIDS
133
................................
......
...........................
.................................-.................................
.............................
.....................................
----------------------------------- -------------------- ------------- -------------------- -----------------------------------
::
:---
...
---------------------------------------------------------------------------------
: --
-:
--
:
-:
_
-
......
. .. ... .
.. . ... .
............
..........................
...................
...................
...................
. .. . . .. .. .
--------------------------------------------------------------------------------------------..............
..............
..............
-------
...........
.......
-- - - - - - -
------------------------------------------------
-----
--------
------
--------------
- -
1
---------------------...........
M .............
............
...............
. . . . . . . . . . . . . . .
. . . . . . . . . . .1
Figure 5.18: Line Screen on a Gray Scale Ramp.
(37 levels of gray.)
IIIII--IY--·-·IIY---111111_-11_----_
_____^__IIXl__il__Ill_··--LIL_---·ll
I_-__II11I--------
134
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
Aph
*
k
!qdI
W..
- =-
.
'.-24
v-4d
-~. I,Ill''l'
-b--
RU
-
.-...
- '-l''--'4
.~.~
eIl
v
....
~ ·
41
-
-
-rrr
I
Y-w--
-r
1-n-
__. 4
l.
l-
. . . .
==
.
--
.
- ..
- - -
bWThW1--
-
Ja.a
-
-
-
- -
ql141111
~~~~~~~41
.- .
-.
- .bU
-.
wy.
--
d- -
-
.9.
1
-
--
Figure 5.19: Line Screen on a Scanned Picture.
(37 levels of gray.)
'fr
5.2. RECTANGULAR GRIDS
.
.
0
0
.
0
0
S
0
0
S
0
135
0
S
0
0
0
0
S
S
a
0
.
1
0
- 2S,S cycles/unit-length
Figure 5.20: Composite Fourier Transform of the Line Screen.
(Average of 37 patterns.)
I II -
136
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
5.2.4
Asymmetric Correction
Thus far, only regular grids have been considered. The problem of correcting a
threshold array for an asymmetric grid is much more complicated for disperseddot ordered dither and well be treated in depth in Chapter 7. The problem
is comparatively simple for clustered-dot ordered dither, and thus only one
representative example will be presented in this section.
Clustered-dot threshold arrays can be thought of as samples of slowly varying
continuous threshold functions. Changing the aspect ratio of the grid requires
sampling the threshold function with a different aspect ratio.
The example to be used here will be the 45° classical screen with M = 8 (examined in section 5.2.1) on a grid with aspect ratio, a =
.
Figure 5.21 displays
the unwanted elongation that will occur if the threshold array of Figure 5.5(c)
is used directly to a scanned image on such a grid.
A corrected threshold array is given in Figure 5.22, along with the results on
the gray scale ramp and scanned picture with a =-
in Figures 5.23 and 5.24.
Figure 5.25 compares the exposure plots due to the (a) uncorrected threshold
array and (b) corrected threshold array. It is first observed that the vertical size
of the baseband has been halved because the line period, L, of the spatial pixel
was doubled. Figure 5.25(a) is simply a squashed version of Figure 5.9(c). The
frequency samples at lower frequency on the vertical frequency axis are evident
in the longer vertical period in the spatial domain picture of Figure 5.21.
5.2. RECTANGULAR GRIDS
137
***
$4'h
4
U
WA4 g;.
*44**94' ,;
,1
* II
Oh 111111
* Eta
*-! 41
i
* it
Figure 5.21: Uncorrected Classical Screen on a Scanned Picture.
Threshold array from Figure 5.5(c), ac=2.
p
138
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
--
35
9 25 43 51 57
1
6 10 26 44 52 58 63 64
19 12 11 18 27 32 36 41 46 53 54 47 38
!
35 42 45 50 49 48 37 34 30 23 20 15 16 17 28 31 35
57 62 61 60 56 40 22 14
8
3
4
64 59 55 39 21 13
7
2
1
38 33
5
9
29 24 19
35
Figure 5.22: Corrected Threshold Array for a grid with a
=
.
__
5.2. RECTANGULAR GRIDS
.
,
·..
.
,
,
A
i
A
*
.
.
OO
0000
a
a
·a
I
·.
,
·,
*a I a U U a *
a
.
I,...k k
,
*
.
,
,
9*
139
a a a U
· ·
·
*~
AWW
Ww
W
****
L
1
W
E
I
I
1
.u
m
**$
·
.
I
I
E
'A WWEALE
J
w
I
I
I
I
1· ·
U
U
.
k
·
I
I
I
.
I
.
.
·
I
I
f
IF
F
P
y
$
!
AL ·
04141·6·
*O***
*44O.44***OOOOO0
.
.
·
.
1
.
I
I
I
F
.
9
I
F
I
·
9
9
B 9B
$444404
¥An in
91
·
¥
,t
eeoc.* ****44*eeeeeeeee.
Jwlllilllllin!111~
I
mummm
mm mm
mm m mmmmmmmiii
r
_
_
· II I
I IIIIEIIII·EEE
EIIIII··
-
-
-
[IIIII··III
· · ·
-
-
-
-
-
-In
-
-
II II I
-
-
-
I II · IEIII
· UIUI
· ·
II EIEEII
· ·.. 11..11..11..11..11.,11..I
I I I..11..11..11..11..11~.11~.11;.~~~~
Figure 5.23: Corrected Classical Screen on a Gray Scale Ramp,
(65 levels of gray.)
=
.
140
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
v
q-
MR
UP
is
.
P
'Iii
.
q
-
=
w
.. U'0
a...
a ....
.r
r
J
r
*
·
aar- hp
,-
.
r
, *
l
-
aa a a a
,,4·r4
aa rV
'- a
q,
..
IF , *p
r
t~l~~lk,
·
E~~~ ~~
i.n
-
-
p
-
a* w a
:1J 1AllVl AL
Figure 5.24: Corrected Classical Screen on a Scanned Picture, ca =
(65 levels of gray.)
1
·..
5.2. RECTANGULAR GRIDS
141
*
*
*
S
*
.
.
*
.
S
0
6
U
.
*
.
*
*
|--
.
-
.·
cycles/unit-length---
2S
(a) Uncorrected Threshold Array of Figure 5.5(c).
*
*
*
*
*
*
*
*
*
*
5
.0
.
*
*
*
*
0
0
.
*
*
*
0
.
*
.0
*
*
*
1( cycles/unit-length
5
) C cAr2SFg
(b) Corrected Threshold Array of Figure 5.22·
Figure 5.25: Composite Fourier Transforms with a = .
__ _^·___1_1_1_1__1111Ilyll_
1·-.-
-·---L·^-
-I
142
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
Hexagonal Grids
5.3
Two techniques for rendering clustered dot screens on regular hexagonal grids
will now be presented.
5.3.1
Hexagonal Version of the Classical Screen
The desirable properties of the rectangular classical screen which should be
preserved in developing a hexagonal version are as follows:
1. Clusters should be as symmetrically distributed as possible.
2. There should exist one connected cluster per period.
3. Gray scale symmetry should be exhibited; that is, the pattern generated
for gray level, g, should be the inverse of the pattern generated for 1 - g.
Figure 5.26 illustrates the tri-state ordering scheme proposed for generating
such a hexagonal screen. The base period consists of three hexagons. Thresholds
in the "light" and "dark" hexagons spiral in and out in the same way as in the
light and dark squares of the rectangular classical screen. However, to maintain
the above three properties, the hexagon associated with the middle third has
threshold ordered in the zigzag manner shown.
A 27 element screen generated in this way is shown in Figure 5.27. The period is displayed in the standard hexagonal shape. The outlines show the borders
between the three sub-hexagons. The resulting gray scale ramp and scanned
5.3. HEXAGONAL GRIDS
143
picture are shown in Figures 5.28 and 5.29. These 28 gray level hexagonal patterns can best be compared against the 33 gray level rectangular patterns of
section 5.2.1 for M = 4. The Composite Fourier Transform in Figure 5.30 displays the expected symmetric concentration of energy about the zero frequency
term.
144
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
third
dark third
light third
Figure 5.26: Tri-state Ordering Scheme for the Hexagonal Classical Screen.
1
4
21
27
2 j23
3
20
17
1
9
24
16
26 i11
2
27
18
15
13
0
25
12
< >7
8
1
2 j23
6
5
1
4
19
21
27
Figure 5.27: Threshold array for a 27 element Hexagonal Classical Screen.
5.3. HEXAGONAL GRIDS
{ii~
~ ~~ ~
~~~~~~.
145
.. .. .. .. .. .. .. .. ... . ... . .. ..
~
.. . . . . . . . . . . . .
~~~~~~.
.
a
v
e
~
e w
e
Wve e
v
aM
aMa
~
e
a
*
a
~----~-~-------------~--~-------------------------a e-a e-a w
aa ea eaa -a aab- aa l~ aa
e We
e
e e e-
w
u
we
e
w
a
aa
a
a a-a
~~~
c
w
.*
a
aa
a-
.
aaa
a
~
a
a
~
A
aa
.
.
.
.
.
.
.
.
.*OO
aooo
a
--
~a~a~
a a
~a
A A A
A
.
.*
0
a ~-a
o- -o
a
a a- - *
a a a a a a_~
__~
aeaeaeaeaeoo
a~~
aaa aa a a aa a aa a aa aa
e w
a eea wa
e
a'. *
.
.
.
.
. o
O
O
00
o -o 0-o0 - o.O
0
.
.
.
O
.o
0 0
0
o 0 - O0.0
0
-~0
-
.
.
.
.
O
.
.C
.U
000
E
Uf
W
E
0O 0
U CC UUEW-.0-- C
U0 U-i-W UU
o o o o e0 e0 0e 0e 0e 0e 0E- CE-e Ue Ce U 0
a_~
e O0e Oe O0e O0e O e O 0e 0e0 0e 00e 00e 0 e--E
000
C~~~~~Ue-EU UaC
ae O0
U
a
A
0
A
0 0 0
o oAA~o~o
0
0
0
o
0
0
0
0
0
0
0
0
0
0
0
C UaE
0
0
U
UJ
E
E 4~
EO~~
J
UU UE UCwlb bbb bbbb # * * # G* * # rr ww wwlw
Figure 5.28: Hexagonal Classical Screen on a Gray Scale Ramp.
(28 levels of gray.)
-------1 _ _11~1_·1I-~-__1__1_
--
146
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
Figure 5.29: Hexagonal Classical Screen on a Scanned Picture.
(28 levels of gray.)
147
5.3. HEXAGONAL GRIDS
0
0
0
0
.
0
0
0
0
0
0
0
0
S
a
0
0
.
0
0
.
.
.
.
0
0
.
2
cycles/unit-length
Figure 5.30: Composite Fourier Transform of the Hexagonal Classical Screen.
(Average of 28 patterns.)
_IIIII_^I__LICII___^_I_--i---
II*---·-I
II
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
148
5.3.2
Spiral Screen
The images generated here are simply hexagonal versions of the rectangular
spiral screen of section 5.2.3. Various sizes of such a screen were proposed in
a study by Chao 15, pp. 45-47]. However, the threshold arrays shown in that
work would not tile the plane in a regular hexagonal manner.
A 27 element spiral threshold array is shown in Figure 5.31 along with its
results on the gray scale ramp (Figure 5.32) and scanned picture (Figure 5.33).
Again, the Composite Fourier Transform, shown in Figure 5.34, exhibits the
concentration of energy near the zero frequency term.
A hexagonal spiral screen shares the same shortcomings as in the rectangular
case (see pages 130 and 131) in that it fails to meet one one of the conditions
enumerated on page 142; it does not possess gray scale symmetry.
In light
regions, black pixels form well nucleated clusters, but in dark regions, the white
pixels are not well nucleated. One must keep in mind that the robust clustering
of both white and black pixels is the reason for settling for a clustered-dot
screen. If this this property is not needed, then the visually superior disperseddot patterns presented in the next section for either rectangular or hexagonal
grids are the ones of choice for ordered dither.
__
5.3. HEXAGONAL GRIDS
149
27
25
21
20
26
18
16
15
23
17
6
5
14
24
7
4
23
8
2
3
12
26
20
19
1
13
27
22
9
10
11
22
24
27
25
21
20
Figure 5.31: Threshold array for a Hexagonal Spiral Screen.
(28 levels of gray.)
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
150
.
.
.
.
.
o.o.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. o
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
...
.
i:!''::.::.':i:::!:i .::"
." .
.....
-.- ~~~~~~~~~~~~
i:-:
ww w w
w~
0
0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 00 0 0
0
0
0
0
0
0
0
0
0
0
0
0
4P
-
-
0
0
0
0
0 mU
0
0
0
0
0000
0
-
.
-
.II
-
e
~
U
e
-
-
-
-
.
.
0
00:i
O
00
0
.
0
0
0~~0
0
-
0
0
0
0
W fWV'f
f u
fofs0t0E0t0t0t0t0t0t0t0t0t
a 0
0tl
0Ill 0il0lil
it1
t t tf t E )lt
0t t
0
.0t0t 0
tlt tlt
tltlt
itl
l0l
l0
U U UU U U
U
U
tlti
t
t~~~ttt~~ttt~::lll~~~lll~~~ll
U U U
U U U
U U U
U
U
U 000000000000000.00
0 0.000.0.0.0.0.0.0.
-
0
0
-P
0
0
0
*
-
0
0
0 0
0
0
-
.
.
-
.
.
-
-
.
.
.
.
i' . .:.:.:.:.:.:.:.
.. .:" -. :::
.:"'::-
*
·
0
000
.P
0
0 0 0
0
0
-
t 0. 0
0
U
U
·UO ·
U
Ui
·
U UU
U !U
·
U
·
UUa
00iW
f
!0 0 0IilUil
0I0 0 0
0
lll0
U U
U
U
0.0.0l U U U U
00ltiU
.0.0.0.0.0.0.0.0.9,9
-
WIM0201-0-
BldBaBs
Figure 5.32: Hexagonal Spiral Screen on a Gray Scale Ramp.
(28 levels of gray.)
Uf
U Uff 4
iUU UlU UlU4
U
1 U11 U1lU
5.3. HEXAGONAL GRIDS
Figure 5.33: Hexagonal Spiral Screen on a Scanned Picture.
(28 levels of gray.)
151
CHAPTER 5. CLUSTERED-DOT ORDERED DITHER
152
a
0
.
.
0
0
0
s
.
0
.
0
0
.
0
0
.
0
0
.
.
_2
cce
untln t
Figure 5.34: Composite Fourier Transform of the Hexagonal Spiral Screen.
(Average of 28 patterns.)
II
_I
Chapter 6
Dispersed-Dot Ordered Dither
When the image is to be produced on a device that can faithfully display every binary pixel, the preferred choice of ordered dither is one that generates
dispersed rather than clustered dots. Dispersed-dot threshold arrays yield high
frequency fidelity and illusions of constant gray regions better than do clustereddot arrays of the same resolution and period.
The design of dispersed-dot ordered dither threshold arrays has been studied on square grids by Limb [46], Lippel [47,48], and most notably by Bayer.
Halftoning with a particular homogeneous threshold matrix has become known
as "Bayer's dither" after his famous 1973 proof [11] of optimality with respect
to minimizing low frequency texture. In fact, in a wide sense, "ordered dither"
has come to mean Bayer's dither.
In this chapter, the Method of Recursive Tessellation [861, a technique for
generating optimally homogeneous ordered dither threshold arrays on both
square and hexagonal lattices is introduced and examined in the frequency domain. Not only does this method generalize ordered dither for hexagonal grids,
but for rectangular grids as well in that it applies to both even and odd period
153
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
154
threshold arrays. It will be shown that only odd period arrays should ever be
used for rectangular grids.
6.1
Method of Recursive Tessellation
The Latin word tesselare means "to pave with tiles". In this section, a method
for deriving optimally homogeneous threshold arrays for regularly shaped periods of grid points (tiles) on regular grids is presented.
6.1.1
Tessellating Regular Grids
Rectangular grids are regular for ca = 1 only. Semi-regular hexagonal grids
are regular at three aspect ratios, a =
a = 23
(regular hexagon of the first kind),
(regular hexagon of the second kind), and ca = 2 (square). Two cases
will be worked out in detail, rectangular with ca = 1 and hexagonal with c =
,
hereby referred to as the "rectangular" and "hexagonal" cases. The results can
later be applied to the remaining two cases by rotation. Rotate the hexagonal
result by 90 ° (or 30 ° ) for hexagonal grids of the second kind (
= 23); for
hexagonal grids with a = 2, rotate the rectangular result by 45° .
The fundamental period or tile of the threshold array can have an integral
power of 2 elements for the rectangular case, and an integral power of 3 elements
for the hexagonal case. The power of 2 or 3 is defined as the order,
array.
,7,
of the
Figure 6.1 shows the first 8 orders of rectangular tiles and figure 6.2
shows the first 5 hexagonal orders.
Once a grid and period order is selected, all of two-space is tiled.
I
_
_
_ _ _
IC
6.1. METHOD OF RECURSIVE TESSELLATION
155
Figure 6.1: The first 8 Orders of Rectangular Tiles.
The number of elements unique to each tile is 2 .
.-^-L-YI_·II·-L--X~·I-~1YI^~
-··I~-
IY(-·L·~I .~~·IIICl~el*-Y-·
LI ----
_I_____
1_
11_1_ 1111_-111__111 1
C-_ --
156
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
Figure 6.2: The first 5 Orders of Hexagonal Tiles.
The number of elements unique to each tile is 3 7 .
157
6.1. METHOD OF RECURSIVE TESSELLATION
6.1.2
Generation of Threshold Arrays
The goal is to order the samples from 1 to 2 for the rectangular case or 1 to
3" for the hexagonal case in such a way that as each successive position is numbered (turned on), the total two-dimensional ensemble of "on" positions is as
homogeneously arranged as possible. When used as threshold arrays, the corresponding arrangement of output binary dots will be dispersed as homogeneously
as possible for each gray level to be simulated.
The algorithm for generating the threshold array pivots on the fact that
starting with a tile of a given order,
, the center (prime) point and all the
vertices can act as center points for a re-tiling with tiles of order
- 1. The
vertices of these tiles can further act as center points for another re-tiling of order
q - 2, and so on. All of the vertices at each stage of this recursive tessellation are
numbered before the next tessellation takes place. Breaking down the plane in
this way provides a mechanism for locating the family of points that are exactly
in the center of the voids between points of the higher order families.
The ordering of the family of points within a given stage or "subtessellation",
is governed by an offset vector between the central (prime) point and any one
of the vertices of its circumscribing tile. Figure 6.3 illustrates the process for a
fourth order rectangular array. For an array of order 7, there are 7 stages or
subtessellations. At each stage, i, 2 points are numbered by placing the tail
of an offset vector on the center of each tile to locate one vertex in that tile.
The offset vector can be initially oriented to point at any vertex, but must not
change orientation within a given stage. Each vertex located with the offset
vector is assigned a number equal to that of the center point plus the number
of points assigned in the preceding stage, or 2
- 1.
The same procedure applies to the hexagonal case, except that each stage
-~
-^14
_
1___1___-·~
---
I
I
158
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
2
z
-* I '
I
*
7. *
.
z
.
.
Figure 6.3: The 4 stages of Recursive Tessellation
of a fourth order rectangular tile.
.
.
.
e
6.1. METHOD OF RECURSIVE TESSELLATION
159
requires 2 passes, and the total number of points assigned in each stage is
3
.
The 2 page example illustrated in figure 6.4 shows the method of recursive
tessellation for a hexagonal tile of order 3.
Two passes are required for each
of the 3 stages. Note that on each second pass, the tail of the offset vector is
placed on points assigned in the first pass instead of on tile centers.
Figure 6.5 (pages 162 through 164) shows the resulting rectangular threshold arrays for
ir
= 1 through 8.
These arrays are the same as described by
Bayer [11]. The first 5 orders of hexagonal threshold arrays are shown in figure 6.6 (pages 165 and 166).
Recall that because the array edges are shared
when tiling the plane, two edges of each rectangular tile and three edges of each
hexagonal tile are not unique (figure 3.1, page 63).
As a practical matter, for operating on rectangular shaped images, 2 periods
of these arrays have to be packed together to form rectangularly periodic matrices. The exception, of course, is rectangular threshold arrays with 7 even, which
are already rectangularly periodic. Figure 6.7 shows how this packing is done
for odd ordered rectangular and hexagonal arrays, and figure 6.8 illustrates this
for even ordered hexagonal arrays.
Note that rectangular tessellation of these
packed matrices preserves the periodicity of the originally shaped tiles.
----.- 1.__11_1_--11-I__l_-l·I1IUIIII1
.--1 __------II·II·Y ......._ ·- -I
--
I·
160
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
Figure 6.4: (a) The first 2 stages of Recursive Tessellation
of a third order hexagonal tile.
Two passes are required for each stage.
6.1. METHOD OF RECURSIVE TESSELLATION
Figure 6.4: (b) The third stage of Recursive Tessellation
of a third order hexagonal tile.
__
161
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
162
2
2
1
2
2
2
3 2
4
1
4
2
3
2
-2
2 16
3
13
2
6
11
7
10
4 14
1
15
4
12
8
9
5
12
2
16
3
13
2
2 60
16
56
3 57
34
18
48
32
10
50
42
2
2
3
6
4
7
1
8
4
5
3
r/=3
10
2
2
2
16
3
32
13 19
11
2 29
18 10
6
26
22 14 20
7 27
1
30
15
17
9
4 31
5
25
10 23
12
5
4
21 13
2
8
12
28
24 16
3
2
13 53
2
45 29 34
35
19
6
64 11
51
26
38
22 43
27
39 23 42
4 58
14
54
1
59
15 55
36
20
46
30 33
17
47
31 36
12
52
5
63 12
44
28
2
60
8 62
9
49
40
24
41
25
16
56
3 57
Figure 6.5: Rectangular Threshold Arrays.
The edges of each array are shared in the tiling.
(Continued on next two pages.)
7
61 10
4
37 21 44
13 53
2
6.1. METHOD OF RECURSIVE TESSELLATION
163
2
60 66
16 124
56 80
3 120
57 67
13 121
53
2 117
34
77
45 83
29 109
93
19
10 125
42
51 75
64
23 103
87
55 79
4 119
70
54 78
58 68
33 94
97
46 84
62 72
12
76
44
28 108
40
92
24 104
16
41 88
25 105
37 89
52
8 116
9 126
49 73
36
20 100
30 110
21 101
44 85
4
14 122
5 113
63 69
12 127
38 90
1 118
17
42
26 106
22 102
47 81
31 111
36 95
6 114
59 65
15 123
10
50 74
43 86
27 107
39 91
98
48 82
99 11 128
7 115
61 71
18
32 112
35 96
34
2
60
56
3
57
13
53
2
Figure 6.5: Rectangular Threshold Arrays (continued).
I
_
_I·_I
lyl^ll-_l--l
*U1----_I^YILI
.*·XLYIYII-·-II--LIDILYIII)·I-·
I-I--I
_... ------X
__·A-^
-.
_lII··n·-dl
~·-^-·1s·1-Pll
^~~-
CI
I
I
I
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
164
2 236
130
60 220
16 232
66188 124 144
34194
162
18252
98 146
10 226
138
26242
170 106 154
4 234
132
6 240
38 198
22256
14230
11 227
43 203
1235
20 250
30 246
164 100 148
12 228
84 174 110 158
52 212
8 238
76 180 116 136
44 204
28 244
172 108 156
2 236
62222
24 254
92 168 104 152
60 220
16 232
45 205
29 245
7 237
34
93 162
61 221
10
71 189 125 138
39199
23 253
91 167 103 151
59 219
15231
42
87 170
55215
4
17 251
31 247
97145
47 207
81 175 111 159
49 209
5 239
73 177 113 133
41201
25 241
88 169 105 153
56 216
77 181 117 130
79 183 119 132
9225
72 190 126 137
40 200
27243
2
65 187 123 143
33 193
94 161
53 213
83 173 109 157
51 211
86171 107 155
54214
13 229
75179 115 135
78 182 118 129
46 206
19 249
99 147
70 192 128 139
90 166 102 150
58 218
35195
96 163
64 224
57 217
67 185 121 141
68 186 122 142
36 196
140
32 248
82 176 112 160
50 210
3 233
80 184 120 131
48 208
74 178 114 134
42 202
56 216
3 233
95164
63223
12
69 191 127 140
37 197
21 255
89 165 101 149
57 217
36
13 229
85 172
53 213
Figure 6.5: Rectangular Threshold Arrays (continued).
44
2
6.1. METHOD OF RECURSIVE TESSELLATION
3
2
1
2
1
3
165
2
3
3
5
2
1
7
2
18
11
8
25
15
q
3
1
21
12
3
2
11
20
76
25
2
18
3
22
13
4
24
9
26
17
3
2
7
19
10
9
15
6
23
6
4
2
25
16
14
5
22
7
27
3
3
20
2
8
22
3
18
78
47
20
3
57
30
25
50
15
69
42
20
40
13
36
73
46
19
79
52
31
80
53
9
11
71
4
64
37
60
33
6
26
51
24
55
2
62
44
17
77
10
35
75
28
1
70
8
68
15
66
48
21
61
43
59
41
23
74
16
67
81
34
39
12
32
14
65
7
58
5
54
27
38
11
72
45
56
29
2
49
63
3
22
18
2
= 4
?777=4
Figure 6.6: Hexagonal Threshold Arrays.
The edges of each array are shared in the tiling.
(Continued on next page.)
_I__^11I_1_^I__·_·_XP111··--·-··IPIYI
L
_
-·-II_-
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
166
2
150
208
18
154
198
22
144
202
3
225
49
84
229
111
103
130
165
76
157
192
25
106
219
133
126
184
72
153
211
12
93
120
214
140
228
96
177
62
123
204
~
2
156
197
24
143
129
170
75
116
102
221
48
89
162
194
21
201
35
241
=n
5
174
135
167
81
113
8
147
15
86
108
218
54
234
59
187
66
137
159
191
27
207
32
238
39
110
5
164
78
180
56
235
45
231
29
99
83
105
224
51
132
83 164
11
92
240
38
119
216
95
122
210
98
125
11
179
206
4
152
213
109
158
190
26
107
217
53
134
85 166
233
31
131
163
77
55 136
186
71
82
104
223
50
230
28
237
44
203
155
196
23
142
128
169
74
176
1
149
17
227
61
183
68
88
101
220
47
34 115
243
41
200
7
146
14
20
58 139
189
65
173
80
161
112 193
92 173
182
67
3
148
209
30
97
16
236
43
124
212
239
37
118
145
215
13
94
121
20 101 182
114
15
195
241
42
19 100
141
199
222
90
117
175
171
73
123
181
69
46 127
226
63
168
79 160
172
36
242
40
87
9
219
52 133
232
60
188
64
205
33
91
25
138
6
151
10
111
178
57
185
70
229
150
208
18
154
198
22
144
202
3
Figure 6.6: Hexagonal Threshold Arrays (continued).
2
167
6.1. METHOD OF RECURSIVE TESSELLATION
=
II'
I----
-
-
-
-
-
-
-
--- I
-
-
-
-
I
I
--.1
I
*I
· I
I
I
· I
LL
L
L
i- -
-
-
--
-- I
. r- - - - zi
I
\ ,
L .
*
.
*
·
*
......
Figure 6.7: Packing two odd ordered periods for rectangular storage.
I____
Illl_·I___I_I__PI1III_··P
1-·111_III
11
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
168
S
IS
S
S
S
S
.T21.
S
S
.
.
I
I
*1
I
. . . . . . . . I
.......... ...... I
. . . . . . . .1
I
L
I
'I
*'
.
.
.
r[
I
.
.
.
.'
.
.
*I
.
I
*.
I
II
*:
Figure 6.8: Packing two even ordered periods for rectangular storage.
.
169
6.2. EXAMPLES FOR REGULAR GRIDS
Examples for Regular Grids
6.2
To insure a fair comparison, each of the images shown in this section has an
equal number of samples per unit area. This is guaranteed by the condition,
SL, = ShLh. The rectangular examples are shown with a, = Sr/Lr = 1, and
the hexagonal examples with ah
=
Sh/Lh
=
.
These contraints leave one
degree of freedom, the actual number of samples per unit area, which has been
set at 4550 pixels/square-inch, or roughly 67 lpi for the rectangular case.
Figure 6.9 (pages 170 through 177) shows the result of halftoning a wrapped
one-dimensional gray scale ramp on a rectangular grid with each of the first 8
orders. As usual, the beginning of the ramp is marked with a black line. In each
case, 2
'
+ 1 gray levels (including white) are simulated. Similarly, the first 5
hexagonal orders are shown in figure 6.10 (pages 178 through 182). Figure 6.11
(pages 184 through 188) and figure 6.13 (pages 191 through 194) illustrate the
result of halftoning a scanned image.
Two examples of rectangular ordered
dither of the synthesize image is shown in Figure 6.12.
These examples demonstrate the tradeoff between different threshold array
orders. Note that "zeroth order" halftoning would correspond to using a single
fixed threshold as in Figure 1.1 (page 19). For small orders, spurious contours
result from the insufficient number of gray levels. Images produced with large
order threshold arrays suffer from the appearance of low frequency patterns
in areas of uniform gray. The choice of optimum order, ri, depends on the
resolution in cycles/degree with which the image is to be displayed.
170
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
Figure 6.9: Rectangular Ordered Dither of a Gray Scale Ramp.
(a) 3 (or 21 + 1) levels of gray, a = 1, ?7, = 1.
6.2. EXAMPLES FOR REGULAR GRIDS
...............................................................................................................
········································
Figure 6.9: Rectangular Ordered Dither of a Gray Scale Ramp.
(b) 5 (or 22 + 1) levels of gray, a = 1, r/,= 2.
171
172
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
..................................................................................
..................................................................................
..................................................................................
..................................................................................
..................................................................................
..................................................................................
..................................................................................
..................................................................................
..................................................................................
..................................................................................
..................................................................................
..................................................................................
..................................................................................
..................................................................................
..................................................................................
..................................................................................
.....................
.....................
.....................
..............
"""""""""""""""""""" """"""""
----------------------------------------------------------------------------------------------
-
*r
Figure 6.9: Rectangular Ordered Dither of a Gray Scale Ramp.
(c) 9 (or 2 + 1) levels of gray, a = 1, ?7,= 3.
A
6.2. EXAMPLES FOR REGULAR GRIDS
173
.....................................
......................................................
................................................
... .... .... ... .... .... .
.. .... ...
.....................................................
......................................................
................................................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.. .... ...
.......................................................
......................................................
................................................
... .....
... .... .
.. .... ...
..........................................................
......................................................
...............................................
... .... .
... .... ... .... .
.......................................................
~:i*-:
.
I
_':,:",:
...........................................
.%|
~o-:
A:-o:.
:-:-:-
:*
*
s X
*
:.:.....
4
...........
" ....
"""
.....
::xxxxxx
~~~~~~~~~~xxxxxx
~~~~~~~~~~~xxxxxx
: :::::::::::: :::::xxxxxx
:::: :::::::
: :::: : xxxxxx
:::: ::::::
:: ::::xxxxxx
.
.
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX>xxxxxxxxxxxxX
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX~
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXxxxxxxx~xxx~cx
XXXXXXXXXXXXxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx XX
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx x
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxX
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
XXXXXXXXXXXXXXXXXXXXXXXXXXXX.XXXXXXXXXXXXXXXX
·
igf~
~~~~xxxx
=--------
xxxxxx
Xxx
xxxxxxxxxx
---
xxxx
~
'''
'''''
' ''''''''''--''''"''"-_~-'_-"'"
Figure 6.9: Rectangular Ordered Dither of a Gray Scale Ramp.
(d) 17 (or 24 + 1) levels of gray, a = 1, rq = 4.
''''''-'-'"-----
174
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
.
.
.
.
.*o.....o.............
........................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
................
.. ... ... ... ..
.................
. . . . . . .
..........
. . . .
......................
...........................
. . . . . . .
..........
. . . .
.................
...........................
. . . . . . ...................................
. .............
..........................
o*o
oo..........o@o...o
. .. . :.: .. :..::..::.:::.:::.:::.:::.:::.::.:.:.::..:::::::::::::::::::::::::::::::::::::::::::::::::::::::
.......
....
* --. . .
-:::-:::-::~~~~~~~~~~~~~~~~.x.-:.x..x.......
........................................
~~~~~~~~~~**x
--:::::::::::::::::::x:
-*
:-.::| . -v. -v..v..v--v--v--
x::x::xxxxxxxxxxxxxxx
::x::x::xxxxxxxxxxx
x::x::xxxxxxxxxxxxxxxxxxxxxxxxx
::x::x::xxxxxxxxxxxxxxxxxxxxxxxx
x::x::xxxxxxxxxxxxxxxxxxxxxxxxx
::x::x::xxxxxxxxxxxxxxxxxx
x
x::x::xxxxxxxxxxxxxxxxxxxx
::x::x::xxxxxxxxxxxxxxxxxxxxxxxxx
xx
x::x::xxxxxxx:,xxxxxxxxxxcxxxxxx
~~ ~ ~ ~ ~ ~ ~
X::
X X X X X X X X X X Xm
X::~
X Xo~
:X:X:X~X~
XX~
X X~
X0o
. X~
XoX~
Xa--X
~
x..
- XX
---
XXX XXX
x
XX XX
X>XX
X
Figure 6.9: Rectangular Ordered Dither of a Gray Scale Ramp.
(e) 33 (or 2 + 1) levels of gray, ca = 1, r, = 5.
A
6.2. EXAMPLES FOR REGULAR GRIDS
.......
~
°°°O°°H.
. .
'.'......;
ZZ''N';'...:..
~~~~~~~~~~~~~~~~~~....
~.;.;.;.~.~;
; ;
Z .Z]' ..ZZ' .' . ~~
.......~~~;
. . . . . .. .
. . .........................................
. .
~~~~~~~~~~~~~~~~~~~~~~~::.::.::.::.::.:::::::.......::::::.:..::::::.::::::::x~~
~ ~. ...
! ~ ~ ~ ! ~. ~ ....~ ~. ~
. .. .. . . ... .. ...~...........................................
~
. . . . . . . . . .....................
......................... . .. . .. . . .. ........................................
:x
. . . ..
---- :.:.:....
........
........
· ·
· o
o~
......
175
.
·
o.
·... o
.
.
.
...,.
..
...... .
.
.
.
.o
.H ~..~
. . . . . . . . . . . . . .
... .
:.
.......
..
.......
. .........
................
~o.
.°
.o.
°,o.......
o..o
~oo~
........... °....
.
~~~~~~~~.. .-.-..........
.~.o
. .
. .
~~...........
oo................
.
. .
. ..
H........·
. . .............---
: . . . ... ... . ....
.................. - --
..
. :. -..
.. .
.
x
.... ...... .....................................................................
..
.........
:::-:::-
.
...
..... ...
:::-:::-:::-:::-:::-
... ......
.......
... ........
. - . -. -. -. -. - . -
-
-
-
.
-
-
-
........................
-
:::-::- - --:::' - - --- -- -- - -
.
.
.
.
.
.
.
.
.
.
.
.
.
x::x::x::x::xxxxxxxxxxxxxx~~
XXXXN
x
x
..........
N N
Wlwsw
-----------
Figure 6.9: Rectangular Ordered Dither of a Gray Scale Ramp.
(f) 65 (or 26 + 1) levels of gray, ca = 1, ?7r = 6.
.`---"-~II-P."-~l'-1^-·^11·-·111--^
-·I_-·-·I----·----
-I-sl_
.
.
.
-- -- -- XXX
-- -- --:: --X::X::X::X::X::x-X--X--X
-X -X --X- -X-- X -X--X- -XXXX X Xl
.~~~~~~~~~~~~~~~~~~X
X - X - --- -- - -X--X---- -- :--XX--XX XX
176
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
.
.
.
.
. .. . . .
.
.
. . . ....
.
. .
..
. .
. .
.
·
.
.
.
.
.
.
. .
.*
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
....................................
..........................................
.
o,.°
...
. . ................................
.
.
.
.
.
.
.
..
· oo,%,
......................
,% %,%
........................
%°%
%,%
**...
°
..........................
°, .. °% ,%°% ,%,%
°% °% ,%---
. . . . . . . . . . . . . . . . .-----.
. . . .-...-.
. . . ..... . ... .. . ... ... . ......
:-.-.
........
.!.!!.!.
. .......
.......
..!..
. . ....
... . --.-.-!.
:-:-:-. -:-. -:- . :- . -:.-. -:--!- - - - - . - . -. - - - .- .
xxxxmmmmmmmx
...
....
....
-
.
-
..
- - - --------------
-
----
Of
Figure 6.9: Rectangular Ordered Dither of a Gray Scale Ramp.
(g) 129 (or 2 + 1) levels of gray, a = 1, r/r = 7.
_YI
6.2. EXAMPLES FOR REGULAR GRIDS
·
.
.
. . ..
.
..
0o 0 0 0.
.
.
·
.... .............
. . . . . . . . . . ....
.......
.
......
.....................
..
o. .
.
·
.
.o ..o.
.o
.
.
. .
.
.
.
.
.
.
.
.
...
.
.
.
.
.
.
.
. .
...
ooo...o.
.
.
.
.
.
177
.
.
.
...
.
.................................
..............
...........
..ooo
.....
o..ooo...
..
ooo. ......
o.........o..
..
.
o.·o.......
. .... o.oo..
. . ..
....
..........
.
:
.. ... ....
O.....
..................
~ :::::::::::::::x:::::::
~
~ ~ ~ ~
,,,,... ,..,,,,,@ ...........
.........
0..O.
.......
.....
00..00.0....0...
....
. ... . . .......... ..
....
..........................
.
.
. .
.
. .
.
... . . o.. . . . o. . . . . . . . . . . . . . . .....................
.0 ..... . . . ...
..
0o. o0..0o....o.0
.
.
.
.
00.
.. 0.
. 0
o
0.......
. . . . . . . ... ...................
0 o..o.0...o
..
o ... 0.o.o.0
·
· ·
·0.0..'.
·
· . .' . '. . ' .. ·0'0 ' 0 ................................
o 0 . ............
......
.
.
.
.........
0 '
O.'.O. .
.O.0.0...'0
. . . ..
.. . ..
. . . . ..........................
.... .
. ... ......
.....
...............
*.. ..0. . .....................................................
.. ... . . ......
. ...
.....
.......
.....
..........
000'000.
....
'
.0...0........0.........................................................0...00.0'.....0.....
········································
. ..................
...
O.o..
..........................................
..
. .
.............
:'l.,
o..........
.,,,,,,.,,,
. . . ...
. .
. .
.
0 0 . . 0 O 0 00..
O.0.
o 0
0.0
.. 0.00
.'0
0 '
.O..H
........
..............
.
.
.........
00........................00'000'0
....
.......**,
0.
.0.00..
. .........
O .. '.
, ,o,,,,,,,,,,,,,,,,,,,,
::X:::::::::::xx:::::::::::::::::::::::::::::x::::::::::::::::
::xxxxx
::..x::x::x::
x:x :
x::x::::
K:.........-.::.::: :: ::::-::::::-v-X-vv-uv-vv
:::::::::::::::::::::::x::x::x::x::::
:::::::::::::::::::::::::::x::
~~K::::::::::::::x::x::::::::--:::::::::
:
::::::::
::::
xx
:::::::::::::::::.....
:::
::::::::
***o... : ..
*....*...
I I
Ix
I1
kXXXXMXX
x
ex
Cx
X
x
cx
0 ex
ex
x X3
KxxKN
K~ .01x
Kxx
axz
ex.$xKxxq
ex
....
I
c
: : :: *..*
::
.. .......
KK X
x:::::::::::::::::::::::
...
" -''~"
-.
' '."'v.
-v
SV f
M...........V...
Figure 6.9: Rectangular Ordered Dither of a Gray Scale Ramp.
(h) 257 (or 28 + 1) levels of gray, a = 1, tr/ = 8.
---
--
---·
e -----vv
:- --v'
,. xx--
178
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
l·l·L
Figure 6.10: Hexagonal Ordered Dither of a Gray Scale Ramp.
(a) 4 (or 31 + 1) levels of gray, c =
2, 7h = 1-
__
6.2. EXAMPLES FOR REGULAR GRIDS
,
,
,
,
,
179
,,.................................................................
.................................
.. .........................................................................................................
. . . . . . . . . . . . . . . . . . . . . . . . . . . ..................
. . . ............
. . ..... . .
.......... ........ ... ....
... ....
... .... ... .... ... ....
... ...
.... .......................
.. .. .....................
. . . . . . . . . . . .. .. .. .. ...........
.........
.................
.. . . ...
. ... .... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .....................
..
.
.
..
.
............
. ... .......
...............................................................................................................
,,,,,,,,,,,,,,,,,....··.....,,........... .....
.....
........................................ ~ · ,,...........·....
~
oo,,o,,o,......,~............~,.... .......
~,...........,o~............·.....,......,~,...........................
...........................................................................................
":
!::
!1
!:::::
*:::::::::::::::
~~**~~~~,~o,·
,...................~o*'
~o~,~,*~ ~'o',~*
:":::::::
,.,..,.,,.,..,.
.. .. .. .
.. .- . .
.
... .- . .
........
....... ,,--XZ@-.-.-.-.-.-
..
"--,.eo,
:1 -..
.
, ;:;:;
g....
°:::5_'...
j:
M:..:
.-.
5_'
1
I
F
:.:.:.:R.:.:..:
r
I
I
I
I
-1:
:'~:
Figure 6.10: Hexagonal Ordered Dither of a Gray Scale Ramp.
(b) 10 (or 3 + 1) levels of gray, = , .7h = 2.
_~--lli~l
_I
_-_-
_
i-~
l
l~-I
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
180
..
~~~~~~
~
..
........................
. ....~~~
..............................
. ~.~~
. ... . .. ...... . .
. .
. . . . . . . . .
''''' ''''' ''''' ''''' ''''' ''''''''''''''''''''''''''''''''''''''''
w¢; ;
.
.
.
.
.
.
..
.
.
.
;
.
.
.
.
. . ..
.
;-z;;
.
.
;
.
..
.
;-;
..
.
. .
.
.
..
.
..
. . ..
;-- ;
.
. . ..
..
.
;-S-wRD;---S
.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- ....................................
; :: :.::
*.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-:-.-.-:-. .-:-.-.-:-.-.-:-.-.-:-.-.-:-.-.-:-.-. :-.-.-:-.-.-:-.-.-:-.-.-:-.-.-:-:-.-:-:-.-:-:-.-:-:-.-:-p.
-.-.-.-.-.-.-.-............ -.-.--.
>--.-.-.-.-.--.
>--.-.- -.
>--.-.- -.......................................... --.-.- -.-.--.-.- -............... ---.--.' -.
!-.--.,
- - -.-.-.-.-.-.-.-.-.- >--.
- ->.-.-.-.-.-:: : -.- -.-.-.-.-.-.- -.- -.- q-.-.
:- -.'.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- -.-.-.-.- '.-.- -.-.- -.-.-.-.- -.-.- - ..---.-.-.-.- -.-.- -.-.- - -.--.- .- --.-.-.- '
,---------,-------------------------------------------------'-----------------------:----:----:--*:--- - - ------P.. -!..P-.'
- -.
P.!.-.P.--X.Z.:...s-.-.s.-.-.!.-.-.@.-.-.@.@.- -.-.!.- -.-.- -.-.- -.-.- -.
-.
!.- -.
!.- -.
'.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
I
D4
i,
I 1
._l^a . l
3so
_S
__
_X
---.-
-
I ---;'."---.;.: _-.-:
'.: '; .:-:-S:;:-:-!;!.!S
. ;
. .
;·
·········
···· ···
w:
i
1
-.
ir
n-eM
i
il
I
wI
1.
I
q
DD
:i
.-C.w
.:.:z
:,C:.
.-C-
..
%,M4.. -40
.4
:.1!:4
:qe'
:h'
4,.
.
--------------------------w,.,,
,=s
Os
-- - - -- - - -- - - -- - - -- _ ,
.
-- - - = - -- _ - -- . -
Figure 6.10: Hexagonal Ordered Dither of a Gray Scale Ramp.
(c) 28 (or 33 + 1) levels of gray, a =2, t7 h =3.
"`
.
6.2. EXAMPLES FOR REGULAR GRIDS
181
....:~...:..:..........:.......:....::....:..:.. . . ......... :..:......:..:...:..:..........
:..:.::...:....: ..:.... .::.:.:....:
.. . .::.
~~.:::
.:. . .i i:. ....
:.:::....:..
..........
:....
.::.....:.
....: .:~~
]..:.:....ii.:..
.
.4.~.4_. .4
.4
.'.=4
....
. ....
....
....1
9,
.9
4
4
14
I
4
Ye:
Ci
Ci
.4
e
4
. :...:.....- ::......::...:
..
~~~ ~ ~ ~ ~ ~ ~ ~ ~~
i i i i i i i i i i i i E+: -C
.n'
. . . . . : ~
I
.4
4.9
01 C..-C-C-C-C M
C-C-C--C
RN
2
-1
A
C_4
.4..
- -- -- --- - --- - -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- - -- -- -- -- -- -
-M-1
C-0C-P
go- 4-1
4-0-9-F
I--
Figure 6.10: Hexagonal Ordered Dither of a Gray Scale Ramp.
(d) 82 (or 3 + 1) levels of gray, a~ =
2s, 77h
= 4.
..
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
182
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
.
.
. .
-..................................................................................................
........................................
........................................................................
"°`.~..~.~...-.......... ............ ~......~?
.~.-..
.?.i..~.i...............
-%... ..-
.......
..:
:
.............
.......
.:
......
.......
......
.......
......
.......
.............
.......
......
.......
......
.......
.......
................................
..
~·..~.
..........................................................................................................................
......................................................................
DIi D
r·
j::·
il.
i:
,:i:~:
M 1 i,
A:
I
.
-,
...
il.
i';
i.
.4"
.
.
.
.
..
--
.
.
.
.
......
:~J
. - - - -- - - - - - - - - ---
.......
......
.......
......
..................
ii ,I
i:
:i
::.w.-C.:
..
"".:ii"i'"'"i:"'':'.:?"i
o...o...~.........·........:-......................................:..............:.....o:..................
-
:$ ,I
iil
;:~. D:.
I
:-
-
i:CC i
i::-':i.i.~
Z;:~
- - ------
--- -- -- -- -- -- -- -- -- -- --- -- -- -- -- -- -- -- -- -- --
Figure 6.10: Hexagonal Ordered Dither of a Gray Scale Ramp.
(e) 244 (or 3 + 1) levels of gray, a = , r/h = 5-
I
.
.
183
6.2. EXAMPLES FOR REGULAR GRIDS
In any case, the dominance of the basic period seen in clustered-dot ordered
dither is gone,
The images of Figure 6.11(e) (page 188) and Figure 612(b)
(page 190) for a, = 7 and the classical screens of Figure 5.7(c) (page 117) and
Figure 5.8(b) (page 119) each have a period of exactly the same size (Z = 128)
and shape!
Upon comparing rectangular and hexagonal images produced with comparable number of gray levels, the images on the hexagonal grid possess patterns
that are less disturbing than those on the rectangular grid. As explained in
section 5.1.1, human vision is more acute for horizontal and vertical orientations than for screens oriented at a 45 ° angle. The preponderance of horizontal
and vertical patterns seen in rectangular ordered dither is not present in the
hexagonal case, which probably explains its more pleasing appearance.
As was mentioned at the onset of this section, these results can be applied
to the two other regular grids, hexagonal grids with a = 23
and a = 2. While
rotating the rectangular results by 45° for application on a hexagonal grid with
a = 2 is straightforward, it may not be clear why the hexagonal results in this
section produced on a hexagonal grid of the first kind need to be rotated 90 °
for application on a hexagonal grid of the second kind (a = 2).
Figure 6.14
(page 195) illustrates why this rotation is necessary.
It should be pointed out that while the idea of applying ordered dither to
hexagonal grids is new, using a homogeneous distribution of dots to create the
illusion of gray scale is not a modern one. Figure 6.15 shows the detail of a binary
image produced in 1844. It was a 125 thread per inch silk weaving generated
on the punchcard operated Jacquard loom [421. It shows a close resemblance to
the patterns produced by rectangular ordered dither.
_I
_t
_
~I(I
I____
__
-*
C---
_
184
CHAPTER 6, DISPERSED-DOT ORDERED DITHER
Figure 6.11: Rectangular Ordered Dither of a Scanned Picture.
(a) 3 (or 21 + 1) levels of gray, a = 1, r = 1.
185
6.2. EX AMPLES FOR REGULAR GRIDS
Figure 6.11: Rectangular Ordered Dither of a Scanned Picture.
(b) 5 (or 22 + 1) levels of gray, ca = 1, 7r,= 2.
__1_1__1^______l____li
11
.1-111·111111111
186
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
'V
...1.
P,
4.Ne.
"
'- M.1,
....
...
V
'ph
Figure 6.11: Rectangular Ordered Dither of a Scanned Picture.
(c) 9 (or 23 + 1) levels of gray, a = 1, tr/ = 3.
6.2. EXAMPLES FOR REGULAR GRIDS
Figure 6.11: Rectangular Ordered Dither of a Scanned Picture.
(d) 17 (or 24 + 1) levels of gray, a = 1, 7r = 4.
187
188
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
Figure 6.11: Rectangular Ordered Dither of a Scanned Picture.
(e) 129 (or 27 + 1) levels of gray, a = 1, f7r = 7.
6.2. EXAMPLES FOR REGULAR GRIDS
189
bo
mmmprl--
--
Ci2)
44
....
0
....
....
.....
,CCCC.
xxxxx"
x--,cx-xxxxx
.:Xxx..
:xxxxx
xx
xc
x
x
xx
bo
ca)
._
bo
q,=C
c
_I
-
IF
---------------
+
190
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
al
bO
oJ2-4 bI
._
o
c
4Q
o[Q
Q>
O
m,
6.2. EA AMFLBS FPOR KPIGULAR GRILS
trCsr
-a
d
191
Figure 6.13: Hexagonal Ordered Dither of a Scanned Picture.
(a) 4 (or 31 + 1) levels of gray, a = 3, U7h = 1.
_1·__
.111_-_1111_1
192
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
Figure 6.13: Hexagonal Ordered Dither of a Scanned Picture.
(b) 10 (or 32 + 1) levels of gray, a = 7, 7h = 2.
6.2. EXAMPLES FOR REGULAR GRIDS
193
Figure 6.13: Hexagonal Ordered Dither of a Scanned Picture.
(c) 28 (or 33 + 1) levels of gray, a = 2, 7h = 3.
_-
_~~I
_I
I
__IYX
_
__
IL
.
-
194
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
.. .. .. . .. :.- .. :........
*.:::::.::.::.::.:.. ..-.. ;'..:':
.......
. .-*.o
.o , .o *o ,
;/ o: -''o-','°,'*o'................-.
o,o.
....................
.....
......
.
. .......
... .
...
-°o'o -'o°- -'o', *'.....
..
...... i.:.................
.
o ',- ,:,
-.
:.
.....
.........
....
· :;.------ Ot-- :..~.~i~-';'':,........
.....
·4
.--------- -~-*,:*.-'*' ...
Figure 6.13: Hexagonal Ordered Dither of a Scanned Picture.
(e) 244 (or 35 + 1) levels of gray, ac = , r7h = 5.
195
6.2. EXAMPLES FOR REGULAR GRIDS
3
2
5
7
8
2
1
3
6
4
9
3
2
Second order threshold array
for a regular hexagonal grid of the First Kind (a =
2
5
7
3
3
8
1
6
).
2
4
9
3
2
Incorrect use of the above data
on a regular hexagonal grid of the Second Kind (a = 2V3).
2
3
4
8
3
9
1
5
2
6
7
2
3
Correct translation of threshold values
for a hexagonal grid of the Second Kind.
Figure 6.14: Importance of Hexagonal Kind.
196
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
We
*^
-ms
'*
ar
N%,
:
tJ v ew
*t
-..
p
soo
a--Wjp
a
v· V
*R~
a| - se
a-% a
J* mnZa
ae
a
a
a
a · _ S5
*''
-
C|w=
a
a
4m m mNjbiev_
mAm
.
eSU %aa %
Par~
%
m a a Im mmmn
%"
%" %'
%,
%r!
a
s
·
pan - a
4
mm
V a
I
I
q,%8
%',',,
e-
,.
W_ t
em
.a a at IL a It a
Jae*
'a
.m,
*, *:Z
St
UiU3SuJ *
a
M
-
a
3
*~~~~~~~~~
.%
. su.
· u. %U%3.. s:4
U533~~
d dldr'l
S..%.~,%.~
a
a
Figure 6.15: Detail from a 1844 silk weaving by the Jacquard loom [42].
197
6.3. EXPOSURE PLOTS
6.3
Exposure Plots
Figure 6.16, pages 198 through 205, displays exposure plots of Cr(f) for r = 1
through 8 on a rectangular spatial grid (a = 1), along with the corresponding
values of I[k] for the first quadrant (Cr,(f) possesses 4 fold symmetry). At
the bottom of each exposure plot, a scale defining the actual dimensions (in
cycles/unit-length) of the plot is terms of the original sample period, S, is
provided.
Similarly, figure 6.17, pages 206 through 210, displays exposure plots for
7h =
(a =
1 to 5 where the sample grid was a regular hexagonal grid of the first kind
). The numerical values of I[k] for the first sextant are shown at the
top of each plot completely specifying the Composite Fourier Transform since
it has 6 fold symmetry.
Consistent with the fair comparison approach of section 6.2, the number of
spatial samples per unit-area are held constant, and the exposure plots are all
generated with the same dimensional scale. Note that rectangular and hexagonal
images with the same spatial pixel area have basebands of equal area.
The rectangular and hexagonal sample periods, Sr and Sh, used to define the
scales in each exposure plot are constrained by the equal pixel area condition,
SrLr = ShLh, and the aspect ratios used, a, = 1 and ah =
related as
Sr
Sh
V
2
2. They are thus
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
198
0.33
1.00
a
*'
*a
1 cycles/unit-length-Figure 6.16: Composite Fourier Transform of Rectangular Ordered Dither.
(a) Average of 3 patterns, r/, = 1.
(First quadrant of IE[k] shown at top.)
6.3. EXPOSURE PLOTS
199
0.40
0.80
2.00
0.40
*
*
**
*
*S
2cycles/unit-length---j
I-
Figure 6.16: Composite Fourier Transform of Rectangular Ordered Dither.
(b) Average of 5 patterns, tr = 2.
(First quadrant of Ir[k] shown at top.)
200
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
1.78
0.89
0.44
0.89
4.00
0
*
0
*
0
*
0
*
*
0
-
2S
cycles/unit-length -
Figure 6.16: Composite Fourier Transform of Rectangular Ordered Dither.
(c) Average of 9 patterns, r7, = 3.
(First quadrant of IE[k] shown at top.)
6.3. EXPOSURE PLOTS
201
1.88
0.47
3.76
0.47
0.94
0.47
8.00
0.47
1.88
.
0
.
.
.
.
a
.
.
.
.
.
.
.
.
I
yce/u i-lnt
Figure 6.16: Composite Fourier Transform of Rectangular Ordered Dither.
(d) Average of 17 patterns, rr = 4.
(First quadrant of Is[k] shown at top.)
_
qll
I
_Igll
__II_
I-L-·l_--__llIILYL·-_----__I_.
).
I
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
202
3.88
0.97
7.76
0.48
0.48
0.97
1.94
0.97
0.48
0.48
16.00
0.97
3.88
.
a
.
0
.
0
0
a
.
0
0
.
0
0
.
0
0
.
0
0
0
0
*
*
|I2
i
cycles/unit-length___
Figure 6.16: Composite Fourier Transform of Rectangular Ordered Dither.
(e) Average of 33 patterns, r7, = 5.
(First quadrant of Ir[k] shown at top.)
__
_I
6.3. EXPOSURE PLOTS
203
7.88
0.49
1.97
0.49
15.75
0.49
0.98
0.49
0.98
0.49
1.97
0.49
3.94
0.49
1.97
0.49
0.98
0.49
0.98
0.49
32.00
0.49
1.97
0.49
7.88
0
.
.
0
0
0
0
U
.
.
0
.
0
0
0
S
.
0
0
.
0
S
0
S
.
0
.
0
*
0
0
.
.
I
i-
0
.
2S1
.
0
cycles/unit-length---
Figure 6.16: Composite Fourier Transform of Rectangular Ordered Dither.
(f) Average of 65 patterns, r = 6.
(First quadrant of IE[k] shown at top.)
LLI-C
I~I^~~-..-··ILI
l~--·
1_1__4I_
--·- · CI111)-l-~~~·
__ __·__1__····~~_
l·I IIYI~LII~~---------. --X_~~_-.III·~·U·YIIY~~s
_____1_11
__·_II
II_
I
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
204
15.88
0.99
0.50
0.99
3.97
0.50
1.98
0.50
3.97
0.99
0.50
0.99
0.50
1.98
0.50
64.00
1.98
7.94
0.50
0.99
0.50
15.88
0
0
.
0
*
.
0
.
0
*
0
0
.
0
0
0
0.99
0.50
0
0
0
3.97
0.50
0.99
0
.
0
0.99
1.98
3.97
0
0.99
0.50
0.50
0.99
31.75
0.50
0.50
0.99
0.50
0.99
0.50
0
0
.
0
0
0
*
*
*
U
- 2S-S, cycles/unit-length---Figure 6.16: Composite Fourier Transform of Rectangular Ordered Dither.
(g) Average of 129 patterns, 7r/ = 7.
(First quadrant of Ir[k] shown at top.)
I
6.3. EXPOSURE PLOTS
0.50
1.00
0.50
1.00
0.50
1.00
0.50
1.00
0.50
31.88
0.50
1.99
0.50
7.97
0.50
1.99
0.50
128.00
1.99
0.50
3.98
0.50
1.99
0.50
3.98
0.50
1.99
0 *
.
205
0.50 7.97
1.00 0.50
0.50 1.99
1.00 0.50
0.50 15.94
1.00 0.50
0.50 1.99
1.00 0.50
0.50 7.97
S
1.99
0.50
3.98
0.50
1.99
0.50
3.98
0.50
1.99
0.50
1.00
0.50
1.00
0.50
1.00
0.50
1.00
0.50
a
0.50 63.75
1.00 0.50
0.50 1.99
1.00 0.50
0.50 7.97
1.00 0.50
0.50 1.99
1.00 0.50
0.50 31.88
0.e~.@0
S
.
*
0~
.
.
0
S.
*0
0
.
0
*
*
S.
.
0
.
*
.
.
.
*
0
a
0* -
S
*
.
.
*.
a.
a.
.
*
0
.
.
.
S
*
.
0
a
a
-
cycles/unit-length----
Figure 6.16: Composite Fourier Transform of Rectangular Ordered Dither.
(h) Average of 257 patterns, r7, = 8.
(First quadrant of IE[k] shown at top.)
.-l·_ll--I__Y1--^--···P·----------
206
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
0.50
0.50
1.50
|Ol n0
0
I
F -_~-235h cycles/unit-length
---
I
Figure 6.17: Composite Fourier Transform of Hexagonal Ordered Dither.
(a) Average of 4 patterns, Tlh
=
1.
(First sextant of I[k] shown at top.)
____
__I__
_I
6.3. EXPOSURE PLOTS
207
1.73
1.73
0.60
4.50
*
@
*
0
0
*
*
0
*
0
|
[
-3
-s
cycles/unit-length
5~~h
Figure 6.17: Composite Fourier Transform of Hexagonal Ordered Dither.
(b) Average of 10 patterns, r7h = 2.
(First sextant of I[k] shown at top.)
_I_
__I__
sXLIIII__--II.·-I------^ILI1-
-141 _-CI._.-LIII-· -
_ _1_1__
II-·---^--·_IIII1__I
·IIXI-·----·---
·- L------C-·-^IIIIII-
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
208
5.53
0.64
0.64
5.53
1.85
0.64
0.64
0.64
0.64
13.50
.
.
.
0
.
0
0
0
.
0
.
0
.
0
0
0
.
.
.
.
1
0
0
0
0
0
.
.
0
0
0
0
ls/ni-lngh
Figure 6.17: Composite Fourier Transform of Hexagonal Ordered Dither.
(c) Average of 28 patterns, /h = 3.
(First sextant of It[k] shown at top.)
209
6.3. EXPOSURE PLOTS
1.90
1.90
17.00
17.00
0.66
0.66
0.66
0.66
0.66
1.90
1.90
5.67
0.66
0.66
0.66
1.90
1.90
0.66
40.50
.
0
0
.
.
.
.
.
.
0
.
.
0
.
0
0
.
0
.
.
0
0
a
0
.
0
0
.
.
0
.
.
.
.
0
.
0
.
.
.
.
t-
32
cycles/unit-lengthl
35h~
Figure 6.17: Composite Fourier Transform of Hexagonal Ordered Dither.
(d) Average of 82 patterns, h = 4.
(First sextant of IE[k] shown at top.)
____1_11
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
210
51.41 0.66 0.66 5.72 0.66 0.66 5.72 0.66 0.66 51.41
0.66 1.91 0.66 0.66 1.91 0.66 0.66 1.91 0.66
0.66 0.66 1.91 0.66 0.66 1.91 0.66 0.66
5.72 0.66 0.66 17.14 0.66 0.66 5.72
0.66 1.91 0.66 0.66 1.91 0.66
0.66 0.66 1.91 0.66 0.66
5.72 0.66 0.66 5.72
0.66 1.91 0.66
0.66 0.66
121.50
0
.
'
.
.
.
.
0
.
.
0
S
*
0
*
.
.
.
S
*
.
.
.
.
.
*
*
S
.
*
*
*
*
0
.
*
*
*
*
*
S
S
.
*
*
.
S
.
.
*
.
.
.
0
*
.
.
.
. .
.
*
.
.
*
*
.
.
.
.
@
0
*
.
*
S
*
S
.
@
@
.
.
*
*
@
.
.
.
0
*
.
.
*
.
0
0~~
6
S
.
S
S
|
-
32
cycles/unit-length
3 Sh
Figure 6.17: Composite Fourier Transform of Hexagonal Ordered Dither.
(e) Average of 244 patterns, r/h = 5.
(First sextant of I[lkI shown at top.)
O
6.3. EXPOSURE PLOTS
6.3.1
211
Analysis
The exposure plots of figures 6.16 and 6.17 beautifully illustrate the results of
the Method of Recursive Tessellation.
For 7 = 1, the only location of energy in the frequency domain besides the
zero frequency term is at the "corners" of the baseband. This corresponds to
the highest frequency which the grid is capable of accommodating; look ahead
to Figure 8.2, page 271, for an illustration of this. For the rectangular grid,
it is the checkerboard pattern. Hexagonal grids have two "highest frequency"
patterns which are negatives of one another. The patterns occur when either 1
or 2 of the 3 elements of a first order threshold array are "on".
For every order,
, these high frequency corners are second only to the zero
frequency term in magnitude. This is exactly what one would expect from a
good dispersed-dot ordered dither algorithm.
As
i/,
and thus the number of elements in the threshold period, increases,
the additional frequency terms in the Fourier Transform assume positions in
between those of smaller Il, with proportionally less energy associated with
them. The size of the baseband, however, always remains fixed.
Observe that families of frequency coefficients of the same magnitude are
arranged exactly as the families of threshold points within the stages of the
Method of Recursive Tessellation! Families of larger dots correspond to earlier
stages.
Also, the gray-level-number vs. low-frequency-texture tradeoff demonstrated
in the examples of section 6.2 can be explained in the frequency domain. As
the number of gray levels increases, so does the number of coefficients.
As
coefficients get closer to the zero frequency location, textures of lower frequency
appear in the image.
212
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
On high spatial resolution displays, it is the set of frequency terms closest
to the zero frequency center that determine whether or not any patterns will
be perceived. Redrawing the data given by Taylor [84], shown in Figure 5.3
(page 107), in polar form would reveal a somewhat square shaped frequency
threshold plot with the cusps oriented along the horizontal and vertical axes.
Any non-zero frequency components inside this perceptual mask would be seen
as a periodic pattern; components outside this area would be spatially integrated
to yield a sensation of a solid average gray value.
So, in observing the exposure plots for rectangular grids (Figure 6.16) for
any even ordered period, rl, the same susceptibility of the human visual system
to perceive low frequency periods exists for the next higher odd ordered period,
7 + 1, which can display twice as many gray levels! Therefore, from this examination in the frequency domain it can be argued that only odd rectangular
ordered dither arrays should used.
The superior radial symmetry of hexagonal grids is also evident in the frequency domain. Energy is distributed more isotropically. While the corners
of the rectangular Fourier Transforms are slightly further from the zero frequency term than the corners of the hexagonal Fourier Transforms, a fact of
great importance in Chapter 8, the hexagonal transforms have higher frequency
components along both the horizontal and vertical directions, where high frequencies are most needed due to the visual system's acute sensitivity there. The
superposition of a square and hexagon of equal area in Figure 6.18 demonstrates
these geometric characteristics.
__
6.3. EXPOSURE PLOTS
213
Figure 6.18: Superposition of Square and Regular Hexagonal Basebands
of equal area corresponding to equal sample densities.
Hexagonal grids support higher horizontal and vertical frequencies, but the
highest frequencies are accommodated on rectangular grids.
-^11_1111·1_.
11 1 --·-·--^1-1.111 ·Il·XI
II
214
CHAPTER 6. DISPERSED-DOT ORDERED DITHER
Chapter 7
Ordered Dither on Asymmetric
Grids
The general solution for generating ordered dither threshold arrays for regular
grids, both rectangular and hexagonal, is provided by the method of recursive
tessellation as demonstrated in the last chapter. Figure 7.1 illustrates the failure
of such threshold arrays to perform well on an asymmetric grid. In this example,
both the scanned and synthesized image have been scaled to fit on a rectangular
grid with aspect ratio,
a
=
.
The development of compensated threshold
arrays for halftoning by ordered dither on asymmetric grids of any aspect ratio
is the subject of this chapter.
The physical mechanisms and constraints which fix the sample period, S,
and line period, L, in real display devices are usually very different. It may be
the case that it is easy to increase the resolution in one direction and difficult or
impossible to do so in the other. Some theoretical questions come to mind. If
a resolution increase is available in only one direction, should that increase be
used at the expense of grid symmetry? And if so, to what extent should such a
215
_yl
111_1-1--·111^11---^-·-1
CI
I
II
-
--P··----
216
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
IIIIIII IIIIII
I 1111111
IIIII I A
1116
w 111111 I
I I I I 11I
11I IIA II II 111
111111111
Illy IIII
I
NQmmW~Ijpyifn C t1111
l
htl IIII~il1111111III
IllI
!1lilI
IIIIIIII
MMINUMlmr
lIllmnln
I~
I il!!111111iH
HI
§Ilii
rmlE
1111
UI z 1 N0Y1
| |11
l
s~~~~~~ I!
ffllI
12 !
l~
e1ll111|
IIIIII
IIIiil
fieiil
lh hnliih HM2II UY
31mlIIU
nIII~11
11
lll111 I|IllllllllllllllllllllIIIII1ii1111111~111
~~""'l'~'' "" """" I' %%"''" %fj~4vull
h ' '
'""" f
'"""' '~'"
'~~~'~~
" "'
' _~
"'" '
Y\ YSiIil'
~n
t r lyf~ | '~YYh
qw
Ei °~
'"' i
iEGb
Figure 7.1: Failure of uncompensated ordered dither on an asymmetric grid.
(a) Scanned Picture, tr = 5, a(r = (vertical resolution decrease).
I
217
o._
-
- o
d0
cd:
o
O
oa) in
I-
,_
._
szI
o Cd
1
C11
o
z
4
1I1III1
_sm·.IY..---·I·PIl*/C-·--*-ll
.-----
--
218
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
unidirectional increase be used?
As one might expect, a resolution increase, symmetric or otherwise, will
always enhance the display of digital imagery. Figure 7.2 shows the scanned
image of Figure 7.1(a) halftoned in the same way on the same grid with aCr,
=
but with the input subsampled and the binary output replicated six times;
symmetry was achieved at the expense of resolution.
In some cases, a unidirectional increase in resolution can enhance the quality
of a display in the same way as a symmetric increase with an equal number of
samples per unit area. As early as 1940, it was found that in the case of m-ary
images, altering the aspect ratio within the range .4 < a < 2.5 had little effect
on perceived sharpness 91.
In this chapter, the solution to providing symmetrically ordered dither patterns for any unidirectional resolution increase will be given, with examples on
rectangular grids of various resolutions. As will be seen, the solution for rectangular grids hinges on solving the asymmetry problem for hexagonal grids. For
this reason, the hexagonal solution is presented first.
As discussed in section 1.3, the resolution (in pixels per unit area) with which
images on asymmetric grids are shown is lower than that for symmetric grids
by an amount proportional to the degree of asymmetry.
r.
--
219
ch~~a
ini iii I
U.
II
i[]hU
Figure 7.2: Sacrificing Resolution for Symmetry with a, =
Ordered Dither on a Scanned Image with tr = 5.
___l·X______··_l_CU·_II
X-C--I-
-
II
I
220
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
7.1
Hexagonal Grid Solution
The trick to be employed here is based on applying the threshold arrays generated in Chapter 6 to symmetric subsamples of the asymmetric grid.
In generating the hexagonal threshold arrays by the method of recursive
tessellation (Figure 6.4, page 160), the centers of the even and odd period tiles
are on "super grids" of only two aspect ratios. One is the aspect ratio of the
grid itself,
Cah,
or
, and the other is 3 h, or 2v3, that of a regular hexagonal
grid of the second kind. These two super grid shapes can be seen in several of
the patterns in Figure 6.10, particularly on page 180.
The two super grid aspect ratios of ah and 3 ah will be observed when these
patterns appear on any hexagonal grid of the first kind (ah < 2). For hexagonal
ordered dither patterns on grids of the second kind (ah > 2), the two super grid
aspect ratios will be ah and cah/3.
Recalling that Covering Efficiency introduced in Section 2.1.2.1 was defined
as the ratio of pixel shape to a circumscribing circle and plotted in Figure 2.5,
page 46, such a metric also describes the radial symmetry of any grid as a
function of aspect ratio. A close up of Figure 2.5 for hexagonal grids is shown
in Figure 7.3. On this curve, complement pairs of super grid aspect ratios are
shown. The range
< ah
< 2 is defined as the Principal Aspect Ratio Range
for hexagonal grids of the first kind, and 2 <
ah
< 6 as the Principal Aspect
Ratio Range for hexagonal grids of the second kind.
All aspect ratios in the principal range of a hexagonal grid on one kind have a
221
7.1. HEXAGONAL GRID SOLUTION
1.0
rj
v
4)
._,
0
· ,,
t~q
0
bO
0H
0)
2
I
.5
1i1
Aspect Ratio, a
Figure 7.3: Complement Aspect Ratios Pairs
found in Hexagonal Ordered Dither Patterns
on grids with aspect ratios in the principal range.
E(a) is a measure of symmetry.
--·-·
1-r__ ---rrurrs-l·4-----
-·111 --
222
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
complement aspect ratio in the principal range of a hexagonal grid of the other
kind.
metric,
For grids in that range, no compensation is necessary; the symmetry
E(ah),
is very high.
When the aspect ratio is in the range
<
h < 2, a
compensated threshold
array is generated by a method of one-dimensional ternary subsampling and
replication which effectively multiplies the aspect ratios of the two super grids
by 3, thus getting them into the principal range. By ternary subsampling is
meant subsampling an asymmetric hexagonal grid by 3 in the closer-packed
direction in the manner shown in Figure 7.4. Note that the subsampled grid,
with an aspect ratio of 3 ah, is much more symmetric than the original grid.
Generation of the compensated threshold array begins by assigning the val-
ues of a regular hexagonal ordered dither array of any order, Oh, to these samples.
ternary replication is the method by which the in between samples are assigned.
Very similar to the technique of assigning values described in section 6.1.2, the
offset vector in this case always points along the direction of subsampling.
An example of this is shown in Figure 7.5. A hexagonal threshold array of
order h = 1 shown in (a) is valid for any grid in the principal range,
<
ah
2.
In (b), an array compensated with one ternary replication is valid for grids with
an aspect ratio in the range
<
h <
2. The two passes in this example add 3
to previously assigned threshold values and assign the result to the location at
the immediate right.
As one might expect, this procedure can be repeated for grids of the even
smaller aspect ratio,
27
22
< ah _<
2
9.
Figure 7.5(c) shows such a threshold array. In
each of these cases, the values of the original regular threshold array are circled.
A generalization of this method for any aspect ratio will now be formally stated.
7.1. HEXAGONAL GRID SOLUTION
O
0
-*0
223
0
0
.00 .
0.
.0-
0-
. 0. .
'0.0 .
0
.0
0- 0 0
*
O
0
0- '0
.0
'0
- -0
. 'O- -O0-
.
'0)-
.0..0- .
Figure 7.4: Ternary Subsampling of an Asymmetric Hexagonal Grid.
( = 1 in this case.)
--^·_IILI_--
·--·1II*1---1^1I*IPI-CIILI^L-.
.1
-IX_·
I-_-I
--L-·I^·-*IIC_-
_
__I
224
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
3
2
2
1
3
3
2
(a) rh = 1, , = 0, for the range
0
0
0
12 21
6
15
24
0
8
14
5
18
20
0
1, for the range 2(3)2 < ah < 2(3).
20
9
1
(c)
=
11
(
0
,c
2.
®
5
= 1,
a
0
9
(b) r7 h
<
8
5
6
2
23
14
23
8
26
17
1O 19
27
5
8
4
17
13
22
7
16
25
26
/h = 1, c = 2, for the range 2(3) 3 < ah < 2(3)2.
Figure 7.5: Compensating Threshold Arrays by Ternary Replication.
225
7.1. HEXAGONAL GRID SOLUTION
7.1.1
Generalized Procedure
In general, for asymmetric hexagonal grids, a new hexagonal grid is created by
repeated ternary subsampling in the closer-packed direction until the resulting
grid has an aspect ratio in the principal range. In this way, the pair of complementary aspect ratios have the highest possible "symmetry values", E(cah),
(Figure 7.3).
The rule determining the number of ternary subdivisions and replications, c,
is as follows. For asymmetric hexagonal grids of the first kind with an aspect
ratio in the range
2 (3
< ahe < 2
)(7.1)
a subsampling factor of (3)C is required in the S direction, and for asymmetric
hexagonal grids of the second kind with an aspect ratio in the range
2(3)K < ah < 2(3)'
+l
(7.2)
a subsampling factor of (3)K is required on the L direction.
The procedure for assigning values to the array begins by assigning the
subsamples the values of a threshold array of any order,
nh.
Note that for grids
with aspect ratios in the Principal Range, c = 0 and no subsampling is required;
the ordered dither threshold arrays can be used without further compensation.
For s; 5 0, the in between samples are assigned with c ternary replications.
The offset vector (see section 6.1.2) is oriented along the closer-packed direction
with length equal to ( 3 )
-i
units, for i = 1 to . At each step, i, the grid point
pointed to by the offset vector is assigned a value equal to that at the tail of the
vector plus (3)7h +i - 1. The resulting number of elements in the threshold array
is Z = (3)' h+
--.......--_I-I·-_
II--J-^~---~--lX-l--t_
_~____1~.__.^_·
-·-·ILIC-
II~1~
__ly·__·IIXI11___··--1-1··_11--·111
1II1
--
226
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
7.2
Rectangular Grid Solution
The aspect ratio immunity argument (section 2.1.2.1) in favor of hexagonal
grids, which showed that hexagonal grids were more symmetric over an order of
magnitude in aspect ratio than a perfectly square rectangular grid, can be exploited in the rectangular case by selecting every other grid point in a hexagonal
fashion. Besides enjoying the attributes of hexagonal grids, dividing a rectangular grid in this way preserves the very high frequency checkerboard pattern
at middle gray, a most important feature of rectangular grids.
Remarkably then, the solution for generating compensated ordered dither
threshold arrays for asymmetric rectangular grids is a simple extrapolation of
the hexagonal solution. Figure 7.6 shows how an asymmetric rectangular grid
with aspect ratio ar is subsampled to form a hexagonal grid of aspect ratio
th =
acr/2. The values of the rectangular threshold array are determined by
first assigning to the hexagonal subsamples a corrected hexagonal threshold
array of order, r/h, with c appropriate for
ah =
a,/2. The in between samples
are given the value of the neighbor in the tighter packed direction plus
(3)7h+r.
The resulting compensated rectangular ordered dither threshold array has Z =
2
(3) "7h+' elements.
In terms of the asymmetric aspect ratio of the rectangular grid, the value of
227
7.2. RECTANGULAR GRID SOLUTION
00.0.00.00
o-0.0-0.0-0.0-0
o0-0o0o0oo0oo
.o.0oo0Q0Qoo
00000.00
.0000.0.0.0
00-00-000
o 0-0o0.0.0.0
00-00-000
Figure 7.6: Hexagonal Subsampling of an Asymmetric Rectangular Grid.
(ic = 0 in this case.)
_I_LI11__1·_l___l__-·IIIPIIIICI-IILII
-----·- · PII-
228
,;
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
is determined as that which satisfies
(1)K+1 < ar < ()
for ar <_
1
with subsampling along the S direction
or
(3)
<
r. < (3)K +'
for ar > 1
(7.3)
(7.4)
with subsampling along the L direction.
The examples in Figure 7.7 were generated from the compensated hexagonal
threshold arrays in Figure 7.5 for the three aspect ratios shown. The circled
values in Figure 7.7 highlight those hexagonal arrays used.
All of the example images in the following sections were generated on rectangular grids with small aspect ratios. Examples of grids with large aspect ratios
are not shown since the effects are the same through transposition.
_
7.2. RECTANGULAR GRID SOLUTION
229
0
In
0
O
O
VI
v
0
c'
·'l
~o._
cd
0
~LO
n
-Im
VI
D
V
O
D
I
0O0
Cr
(U
VI
V
co
C
-4
oo
b(
®
0
n
ce) :d
UQ
cd
® A
_0 )
4-
40
~606° ©
©
10®0
r.
_ ( ®~~~~~~c
E) -4
0
~~~~II0 ~
®(0~~~~~
®
vQ
Ln
1-4
Ln
0
II
II
11
14
-4
_
._
d fq
-n
O
co
9..
0
®
~®(
®D
-
©
-4
0
®4,e
0
0
_..___._II__--
___
Il.l-·_._^__LI--
WIX^IIII--··__-·-PIII
I-L----LLI-
^-·I
·
-·---C-
230
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
38
45
40
©
38
®
47 030
43
®
Q34
54
49 032 0
0
39
42
41
48 028
0 52 0 35
0
43
50
0
44 8
®
033
37
53
36
031 0
51
Q
42
46 0293 38
45
40
38
Figure 7.8: Compensated Threshold Array for
rlh = 3,
K
< Ca, < 1.
= 0, Z = 54.
Circled values correspond to the original threshold array.
7.3
Examples for
,r=22
The proper value of c for the range into which the aspect ratio a,
=
2
falls
is 0. The compensated threshold array to be used is shown in Figure 7.8 in
the usual odd rectangular period form. The hexagonally subsampled locations
are circled and correspond to a hexagonal threshold array with order rh
=
3.
Notice how these circled values comprise an almost regular hexagonal grid. The
total number of elements is Z = 2(3)3 + ° = 54.
To examine the effect of this compensated threshold array, the gray scale
ramp (Figure 7.9) and scanned picture (Figure 7.10) compare the results of
halftoning with
7.3. EXAMPLES FOR aR
= 1
231
(a) an uncompensated rectangular ordered dither threshold array of order
f7r
=
5 and Z
=
32,
(b) the same as above but first replicating pixels in the S direction, sacrificing
resolution for symmetry, and
(c) the compensated threshold array of Figure 7.8.
The horizontal banding evident in (a) is somewhat eliminated at the great expense of forfeiting horizontal resolution in (b). The images in (c) use that extra
resolution dexterously to form radially symmetric patterns.
Considerable insight is gained by studying the exposure plots in Figure 7.11.
Of primary interest are those frequency components closest to the zero frequency
term for they contribute the most to the visibility of low frequency patterns.
Figure 7.11(a) is the same as that for the regular grid case on page 202
shrunk in the vertical direction, while (b) is shrunk by a factor of two in both
directions. The horizontal banding in the spatial domain is evidenced by the
strong vertically oriented frequency components indicated by arrows in (a).
While the low frequency neighbors are symmetrically arranged in (b), all are
lower in frequency. The success of the compensated array (c) is supported in the
frequency domain by the radially symmetric arrangement of frequency samples.
___
-·-1I-·I1I^)·-I.
Y--·IIIICIII III 1___11 --
I
232
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
.
,
.. .
. . . . . ....................................
~ ~~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~...
........................................
. .
. .
.....
.
.
. I........
. .
. . . . . ... . .....
. . .... .. .. . .... ........................
. . ......
. . .I... .. .....
. ................................
.................................................
.......................
. . . . . . . . . ...................................................................
.. . . . . . . . . . . . . . . . . . . . . . . . . .......................................
..
.. . . . .. . .. . . .. . . .. . .. . . .. . .. . . .. ................................
I
........
.......................
....................................
..........................
...................
.....
...
.l
...........................................
.................................................
::x:x:xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
x:',x::xxxxxxxxxxxxxxxxxxxxxxxxx~
::xax:xxxxxxxxxxxxxxxxxxxxxxxx~
xx:xxxxxxxxxxxxxxxxxxxxxxxxx
"Y""Y""YYYYYYYYYYYYYyyyyyyyyyyy
~~~~~~~~~x
..............
X"~
~~~~~~ §l~~~~~~~x.x.xx...xx..xx
"444444V444444444:44
~-';~~;.
4xxnlx..xhx..hhhxnxnnx.,n..w
Y:4 4, 4444
,
4~ 4 ~444444444
44...........
:'''-'--::~
44'"'-~-:lf 1fx1 f f Ih"In'.....n'n---h"n''<:.
:::::::
xxx..x..x..x..x..x........................
x..x..x..x..x..x..x..x..x....x...x.
~...
:~~ ~ ~ : ........ "" ~;-"-~<:.....:::::::....
Figure 7.9: Comparison of Gray Scale Ramp for a =
(a) Uncompensated Rectangular Ordered Dither, 7r = 5
7.3. EXAMPLES FOR aR
I*.
=
233
1
..............
.
.
.':::.:":::.::
.........
* .
.
.
.. . .. .*... . *.. .*. ..*..... .*.. *
.. *
. . .. . ..
.* . .* ... *. .... .*
. ..
...................
. . . . .. .. ..
* . * . .*.. * .***
.
. .. .. .. .. . . .. .. .. .. .. .. . . .. . . .. . ..
....
.
*. ......
. . .s .. s*
. | . l". ....
.*
. *
. . *******.
. .. . . . . . . . . . . . . .
********....*....**...**
... . . . . . . . . . . . .*****.*
::::::::::::::::::::::::::::::
.
.
.
.
.
. . .. . .. .. . .* * * * * * **
. .. . .. . ... . . .. . ........
. ...
. . . . . . . . . . . . .
..
..
. .
* *
... *.* *****
.
. .
.
................................................................
. ..
.
...............
* iillifm|
....
l
l
...
llllll
...
...
...
:,.:.· :..:...:...:.................vvv
..
.. ..........
ll
lilllllll-Xll
ll
·
.::
::cXXXXXXXX~XXXX XX
::XXXXXXXcXXXX
V MVMV VVW
V V V VMJ9v4
V Vr
I
Figure 7.9: Comparison of Gray Scale Ramp for ar =
(b) qr = 5, sacrificing resolution for symmetry.
.~_
I_
,~~II-YII
3~--
-I
I
234
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
.
.~~~~~~
.
~~~.
.
.
.
.
.
!
..
..
..
I~hr
I cerzto
..
.
.
..
. .
....
I
II
.
I
...,,
.
.
..
.
~
. .
!
............
.
. .
II
..
.
,,,.,a,,,,,,.,,.,,,, ~',.,
,,.
.
a~maaI~ainainaaaa..
I
I
,.,..,, ,
:::::
: :
'
i~lllll
I
I~
.:.:.::'
a.1
.,.
{<{jes~fl{{jeS
.
..
..
.
. . .I
..
I............................ I..
. . .I. .. .. .. .l... .. .II...
. . . . . . . . . . . . . .........
:!Airbus!,~~~~~~~~~~~~~~
,,a,
.
iitgi§X|20l#Si021000
I I~
I~
I
II
I III
I
....
.,,:.:::::::
.
.
.l
.
I
.le
I
I
...
.....
. '11 :: :<,1.,
,
MI
,
.ll
.le...
I III
ll
I~M
..
I
11:.',
a::. a:.:::::::::::::::<:s:
I
. . !'''''''''''
.
/ !' wwI1ff~~~~~f
'
t
w
Figure 7.9: Comparison of Gray Scale Ramp for ar (c) Compensated Threshold Array, r/h = 3,
= 0.
I
1
.
--
7.3. EXAMPLES FOR aR =
'I~~~~~~~~~~~
1
235
NX ....
~~~~~.........'..?
Ki
,* ,K~,,,
p~~~~~~~~
..................
'U
~~~~......
.....
,,, ...
. . ..........
...
........
~~~~~~~~~~~~.
~......),......
~~~
~~~ ~~~~~~~~~~~~~~~~~I
XN~~~~~~~~~~~~
-JF~
Rl~~~~~~~~~
I
IIIIII
I
IIIIIII
IIIII
II
'X'X
IIIII
Figure 7.10: Comparison of Scanned Picture for a 7 =' 1
(a) Uncompensated Rectangular Ordered Dither, rq,= 5.
I___ ___I_·UII^IIII__L___-
_
-1-·1-·--·11111-
1-·1
____._
-
236
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
--.
....... ..
~
z.".6M~
"'X"~'
_ _r"~ . "".
.. .. . . . . . . . . .
,W
_,
. .
'
..
C ,.rr
~
.
. "...........
~r .4 .x. w"".
...
.............
.. ..'.
.· :*::rrl,:..
..
:x.
.......-... *.|.........~
.
~.:.--
@.............
.-. .. { 'H.':'.
,.
...'x..'..'
'
*
~c ' -:........
'~*~|*|
~~~~~
v~~.......
*........."
·
aa
·0 · mm
·00-
"|"|^H
-|
mirai-ra
^
x
Z- w " ...................................
?
_..-',-r.
·"· · mm
, *
~
.................................
s
lU *
~.2,,' " ·.'.........................
_l · g .. m~~~~~~~~':
'- : ·. · : ".imi.
: " *@lm
4.
: *-1 @.ww
Figure 7.10: Comparison of Scanned Picture for ar = 2'
(b) r7r = 5, sacrificing resolution for symmetry.
__
-
237
FOR
=EXAMPLES
7.3. EXAMPLES FOR aR 7.3.
a!.
I
,
Ii
Figure 7.10: Comparison of Scanned Picture for a,-=
(c) Compensated Threshold Array, 7h = 3, Kc= 0.
_..L_I^---
----IIII_·ill
^-I~~~L-L-I·~~Y·-l·-~~--*
-.--IY__C·II.
238
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
0
*
*
.
.
0
0
/
.
a
0
0
0
0
0
0
.
0
/7
0
0
.
F-
21
25r
cycles/unit-length--
(a) Uncompensated case, 7r, = 5.
*
a
*
*
0
0
4
0
·
0
*
0
*
*
0
0
0
*
0
0
*
*
~0
*
0
0
0
0
0
*
*
0
0
*
a
*
0
(b) Horizontal resolution forfeited.
Figure 7.11: Comparison of Exposure Plots for a =
2'
7.3. EXAMPLES FOR aR
*
*
= I
239
0
0
0
0
0
.
a
0
.
0
0
0
t
0
*
0
0
*
0
0
0
0
i
.
(c) Compensated Case, h
cycles/unit-length l
= 3, K =
Figure 7.11: Comparison of Exposure Plots for
r
0.
=
(continued).
__IYUIII_1I____IIWICICI_-II
·---- ---·
240
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
7.4
Examples for
ar-
The number of ternary subdivisions, c, for the value of the rectangular aspect
ratio,
ar =
,
is 1. To maintain a number of gray levels equal to that of the
last section, a hexagonal threshold array of order 7h = 2 is used. Figure 7.12
shows the compensated threshold array to be used in this section. The total
number of elements is Z = 2(3)3+1 = 54; the circled values are from the original
hexagonal threshold array.
An aspect ratio of l corresponds to a display which has the ability to resolve
6 times as many pixels in the S direction than in the L direction. This time 4
approaches are compared: the uncompensated rectangular ordered dither with
= 5, the case of sacrificing resolution for grid symmetry, the threshold array
used in the last section for compensate for grids in the range 1 < ar < 1 with
rh = 3 and
7-h =
= 0, and finally the correctly compensated threshold array with
2 and I = 1.
All four cases on the gray scale ramp are shown in Figure 7.13. The horizontal banding in (a) has become even more pronounced this extreme aspect
ratio, as does the coarseness of (b). Horizontal banding even begins to appear
in (c). In Figure 7.13(d), the asymmetries have been correctly compensated.
We've already seen the first two cases at the beginning of this chapter for
the scanned picture in Figure 7.1(a) (page 216) and Figure 7.2 (page 216). The
remaining two are illustrated in Figure 7.14. The uncorrected synthesize image
seen in Figure 7.1(b) is also shown with proper threshold array in Figure 7.15.
----
7.4. EXAMPLES FOR aR
= 1
241
40
45 27 54
38 20 27 ®32
40 22 49
D30 12 39
14 41 23 50
735
17 44
34 16 43 25 52 ® 28 10 37 19 46 031
39 21 48
D33 15 42 24 51
44 26 53
13 40
36 18 45
29 11 38
40
4~~~~~~~
1
Figure 7.12: Compensated Threshold Array for < a, < .
Tlh= 2,
= 1, Z = 54.
Circled values correspond to the original threshold array.
242
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
II
l
ll
~~~~~~~iiJl
11111I1111
Il
ll
l 11111111111111111111111111
ll l l
111ll~lIl~Ill~ll
lll11111111111111111111
IllIlI l II IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
11
ll llI Illl ll ll llIll
IIII l Il l l lIII
111111111
IIIl
111111
I l l l mmmmmJmmml
I I I mmml
hlhl
i I I il illllilJ alJllllllllllllllllllllllllllllllll~lll
I
mlilh
mmhlimhmhmhlh
1ll
lIIIIIIIIIUIIIIIII 1111"1IIIIHIIUIIIIIIIIIIII l iiii 1IIIIIIIUIIMIIIIIIIIIIIIIIIS1
uhWE M WWI
11
Yh~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
b
~
~~h
hb
~~h~~hh~~hh~~~h~~hhhlb~~~~~lb~~~
~~~~ih
~~~~
Ih 1~~~~99
71D~~~~~
iml~~~~~bm~~~~
9
h h h h h h h A ^ h h h~~~~~~~~~~~~~h
I IUIU i i 1 ll
1N
flMM
lM
lDlh
lllAhWUU
1 YYYYYYYYYYYYYYYYYYYnnrmmrnYYYYYY Illil
rmmrmil l
IIlIIIIIIlal
Imnm InnImIIII~~~
WINNN
Figure 7.13: Comparison of Gray Scale Ramp for ar =
1
6.
(a) Uncompensated Rectangular Ordered Dither, r, = 5.
7.4. EXAMPLES FOR caR
I*
~
U U
U
U|
U
U
MU M
*U
*
IIIIIIIIIIIlI OEM
U
~*
~
~~~~~~~~~~~~~
*
*
U *
·
*
*
243
= 6
U
U
E
~*
*
*
*
*
*
*
*
*
*
SE E
EEEME
||U|||U|UU|U|U
U
E EE E U U E EE E U U E U
EE
U
E
U
°11111fillfi
IEIIIIIIIIII
OEMIIIEEE
vmx
dONEONEME
IIIFIIIIIIIIIIII~iI
.
. -
.
.-
-
.-
E
ON O ME
·
F_
.
.
.
.
.
·
·
. --...
.
-
·
.
-
Figure 7.13: Comparison of Gray Scale Ramp for ca. =
(b) 7 = 5, sacrificing resolution for symmetry.
.
U
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
244
I
I
I
II
I
I II
1111111111111111111111
I
I
I
II
II
IIIIII
II
II IIIIIIIIIIIIIIIIIIIIIIIIIIII
I
I
I
II II
I II IIIIIIIIIIIIIIIIIIIIIIIIIIIIII
JIII
IIIIIIIIIIIIIII
I
ill1 l 1I I1 I1
11111111111111,!II,111111!11111111,111111111111111111111
iilil
I II I I [I IIIl I I 1 I I I II 1
1 1 1 i11
llllllllilllelttlileletllilelt
tlsl~
I
sItttllltlflslilftli~lttll ltlslsl~llll
<<
II< <<:<<<<<<<<<<<<<<<1<sS<e1 14:sirss
lslellelelltlelltl~I
ii
s~
liisssiiitiitis
t11Sslasti$R$lSaaalWi8agifS§92
:g
14~a4If
II
I
B I XI$
II l1(
IIRS W1
Ie
III11IIII III II IIIIIII
U UU
1 Y UU
8'
ml~
F T U1
Tml
LJJUbJ JIJLIUI~
II
.n!
l
1LU
I YUJ
11
T# W
l I I1I
I
I'I
I
In
11111111111111
~~~~~~111
|1ENNMEIME11
N1||1|E NEENE| 1|EN1
11IME UENr
Figure 7.13: Comparison of Gray Scale Ramp for aCr = 1
(c) Threshold Array, h = 3, = 0 compensated for the range < ar < 1.
7.4. EXAMPLES FOR aR = 6
I
I
I
I
I
I
I1
I
I I I III
I I
II
I
II I
II I
I
I
I
I I II I I
I
I
I I I I I II IIIIII
I
I
I I
I I I
I III
III IIIiI II III
1I1I1IIIII II I IIII IIII
IIIIiii I II I IIII IIII I
11 II IIIIIIIIIIIIIIIIIIIIIIIIIIII
I I I II I 111 1 1
I
I
I
I
I
I
I
II
245
II I I I III :I I I I
1 1 II I
1 1 1 1 1:,1 1:1
II IIIII
II II IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII II II III
I I
III
II III I
I
I I[
I 11 I
U........
Tt g~i UygiMg~l U# .................
, ,N
R:,:,,::,,',,,,,',ingiRaYS
,,,gi ,,,,
i~ , .....
.,.
. .,.,', ' ..........
. . . ,,,,,'
.....
,.........
3 ...... . . ..... . ............
~ ~
I
1111 I
t l l g S
§S§SaIlietltllllli~iaiaif~l§
t S
l §
l l~~~I
1 g
I1 I 1 1
1 11
~~
I
W
Iml1 mlYI
Al
I.IU
I.W,11W1.1
ll AI
IU N
P)1
11
U a~P EIX
] Di
III) 11 la
xalY IIII
IIfl OiI
EB
:1
II II IIIII IIIII11
IIIIIIII
II11III
II
I
IIII I! I!
II
NI I11
wINI1CW
M!1VN N11111111111
m1
IIIII!1111111111
BlaJI Dill
aIIIleaa
I )NEMUMmY~m
II
III1 II III
11111111111111111111
1E1111E1111
1111E1111E11
Figure 7.13: Comparison of Gray Scale Ramp for aTr =. 1
(d) Correctly Compensated Threshold Array, h = 2, K;= 1.
I
II
246
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
As in the last section, the exposure plots provide much insight. All four
cases conveniently fit on the one page in Figure 7.16. The plots in (a) and (b)
are even more compressed than those for ar
in Figure 7.11, which follows
=
the even greater low frequency artifacts in the spatial domain.
Recall that the threshold array with ?lh
=
3 and Kc= 0 produced a Com-
posite Fourier Transform with symmetrically arranged frequency components
around the zero frequency center for the case of ar
=
.
In Figure 7.16(c) the
location of the closest frequency components to the zero frequency center are
also symmetrically arranged for a,
=
Ir, however, the horizontal banding is due
to the greater concentration of energy in the vertical components indicated with
arrows.
The exposure plot for the correctly compensated case in Figure 7.16(d) reflects the symmetry seen in the spatial domain. The frequency components are
in the same locations as in case (c), however, the energy in those surrounding
the zero frequency term is uniformly distributed.
7.4. EXAMPLES FOR aR =
IIIIII 1111
I
247
6
IIIIIIII
I
11A
III
I
I
I
II
I
:I,
I
I III
II
II
t
TAW
Figure 7.14: Comparison of Scanned Picture for ar =
(a) Threshold Array, h = 3, = 0 compensated for the range
.
<
1.
248
CATR7
fill
ORDERED DITHER ON ASYMMETRIC GRIDS
111HIMINIMINIIIII lill Itul [I I
I 11
I
11
I
II
Ilill
III
t', l F
"'¥¥v
a'
'
I'
~: M~~,f,.~ :l .,?~~, t1!',~
,i',S C,
I
I.Ev9
II·hI
III II~~~~~~~
i
_ll _1la
&:,.:',.
I I' __~~~~~~~~~~~~n
I I mmoniii,,,,
~~~~~~~~II
_
1
Figure 7.14: Comparison of Scanned Picture for a,. = .
(b) Correctly Compensated Threshold Array, r7h = 2, K = 1.
7.4. EXAMPLES FOR
aR =
249
1
II
WM---
-
-_
-e
lit .
II
0
cd
-e
-7-
-
-
)E
-
o
-1
-
z_-
-
-
-
-
-
-
250
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
0
0
0
/e
0
.
0
0
0
?
0
(a) Uncompensated case, %, = 5.
0
.
*
.
0 .
0.0
6
.
. . ..
.
*
U
. ..0 . . ·
.
.
.
.
..
(b) Horizontal resolution forfeited.
0
/·
.
0
.
.
.
.
0
0
(c)
77h
= 3, K = O,
. 7
.
compensated for the range
.
< ar
1.
.
*
0
0
a
0
0
.
.
0
0
.
2
21
(d) Correctly Compensated Case,
cycles/unit-length
h =
2,
K=
1.
Figure 7.16: Comparison of Exposure Plots for Ca.= .
_
_
l
251
7.5. CROSSOVER POINTS
7.5
Crossover Points
Equations (7.1) and (7.2) on page 225 for hexagonal grids and equations (7.3)
and (7.4) on page 228 for rectangular grids precisely define the ranges over
which particular values of nawill yield the most symmetric threshold arrays.
This section addresses the nature of compensated ordered dither patterns on
grids with aspect ratio values on the boundaries of those ranges.
7.5.1
Examples for ar = 1
The ranges in the above equations include regular hexagonal and rectangular
grids. As was seen in section 7.1, hexagonal grids with aspect ratios ah =
or
ah = 2V3 use the regular ordered dither threshold arrays unchanged. However,
equation (7.3) suggests a threshold array formed from a hexagonal array with
= 0 for use on regular rectangular grid (a, = 1).
Using the threshold array from Figure 7.8 (page 230), the gray scale ramp,
scanned image, and Composite Fourier Transform are displayed in Figure 7.17.
This output should be compared to that produced by ordered dither for regular rectangular grids in section 6.2. While some of the patterns produced in
Figure 7.17(a) are radially symmetric, many suffer from some vertical banding.
If the threshold array were generated from a hexagonal grid of the second kind
(on the other side of the a,
=
1 crossover point), the banding would be in the
horizontal direction.
____IL
·_··1_1
I___
_II_·
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
252
I.
.
.
i*
~*4*c.~*:*~*<~c
<
< .4..
.
.
.
.
.
.
.
..
.
.
.
..
.
.
.
. ..
. .
. .
..
..
..
..
.
. .
.
.. . . . . . .
.
. . . °. . .. . . .. .. ..
. ...
. . . ...
..........
. .'. .........
. '. ..'....
.
.
. .
. .
. . . . . .
'
~~~~~~~~.... ~.... ~ ~ ~cc~~.<$~
~
4p
.................................................
cc4
*
.
.
.
!
.
N
'. * .~Pi~~~~~~~~~~~~~~~~~~~~~~~
*. ~.~. *, ~.
.~.-.*. .- *. .-ii~ii~
*.. .- *.-....----....
:: :~: ~: :
: - . :- . Cc. - ~
.
...
.
.C C
.
*. .- *. .- *. .--*~~,
.~~. .
CC
: Ac.s.
. C C:
....................................................................
Sc
...
:C: CC
: cc:4.:. : CC C
.
:
ccc...
..>C
n'!hib
l4N
SL! N
L
s SI
Figure 7.17: The Compensated Array of Figure 7.8 on an
(a) Gray Scale Ramp.
Ca,
= 1 Grid.
7.5. CROSSOVER POINTS
253
Figure 7.17: The Compensated Array of Figure 7.8 on an a,. = 1 Grid.
(b) Scanned Image.
·^
·L---X-I·-·
--_
I
254
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
0
0
0
0
0
0
0
0
*
0
0
0
0
0
·
0
·
0
0
a
0
0
0
0
0
0
0
0
0
a
i-
1 cycles/unit-length
Figure 7.17: The Compensated Array of Figure 7.8 on an a = 1 Grid.
(c) Composite Fourier Transform.
__
___
7.5. CROSSOVER POINTS
255
In section 6.2 it was pointed out that that since hexagonal grids with ah = 2
are actually regular rectangular grids, rectangular ordered dither arrays can be
applied if rotated 45°. The hexagonally subsampled half of a regular rectangular
grid will have an aspect ratio of 2. If a rectangular ordered dither array for any
?7r
is rotated 45 ° and applied to this hexagonal subset with the in between
elements determined as in section 7.2, the result is a rectangular ordered dither
array of order
7r
+ 1! So all of the threshold arrays in Chapter 6 are a special
case of the generalized techniques of this chapter.
--
Iylll·_-·---.
_ -····-·--II..
IIUI.·-LII----_I1II^-·I-I---.---·.
I__--I..
256
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
7.5.2
Examples for Car =
1
The observation above is a surprising theoretical discovery for ar = 1. Rotating
rectangular ordered dither arrays can be applied to other crossover points as
well. Figure 2.5 on page 46 revealed that a hexagonal grid with aspect ratio
ah =
2 is exactly as symmetric as a regular rectangular grid with ar = 1, at
least in terms of covering efficiency, E(ca). So it appears that either a hexagonal
ordered dither array or a rectangular array rotated by 45° can be successfully
applied to hexagonally subsampled grids with an effective aspect ratio of ah
In this section, the rectangular crossover point at aCr
=
=
2.
will be explored.
Figure 7.18 illustrates how a threshold array at this aspect ratio can be generated. The underlined values are the hexagonal subsamples of the rectangular
gird, and the circled values correspond to one ternary subsampling of the hexag-
onal subsamples. These circled subsamples have an aspect ratio of
ah
=
2; the
values assigned are those of a rectangular ordered dither threshold array with
77,r= 3 rotated 45 ° . The in between values are then assigned in the usual way.
The value of t, = 3 was chosen so as to produce a threshold array size (Z = 48)
as close as possible to that of the other two arrays to be compared.
The effect of halftoning with
(a) the threshold array of Figure 7.8 (page 230) with r7h
Z = 54 compensated for the range
<
< 1,
(b) the threshold array of Figure 7.12 (page 241) with
Z = 54 compensated for the range
3, / = 0, and
77h =
2, c = 1, and
< a, < , and
(c) the threshold array of Figure 7.18 with 7, = 3,
I: =
1, and Z = 48
is compared in Figure 7.19 for the gray scale ramp, and Figure 7.20 for the
scanned picture.
7.5. CROSSOVER POINTS
257
026 10 34 18
42
40 2 48
14 38 22
027
30
11 35 19 43 025
37 21 45 (731
026
28
15 39 23
10 34 18 42
Q28
12 36 20 44 0
46
32
16 40
9 33 17 41
47
29
12 36 20
0
13 37
44
Figure 7.18: Threshold Array designed for et, =
3.'
Circled values correspond to the rectangular array with tr, = 3, rotated 45 ° .
Underlined values are the result of one ternary replication.
~~ ~ ~ ~ ~ ~
-
_
__-·
258
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
I
I
I
is
I
II
I
ISI
111
Ii
I
I
5 sim
I
~~~~I
1 I
mI I
icc
I
lii
mh I IcIIIIsI
I I III
ii
i
II
I
5
Ii
I II
5
I
I
I
ll
lII mII
l
lil l
I
lll
II
I
sIIII
I
I
IliilI
I
I
II
I IIIIIIIIIIIIIIIIIIIIIII
I
I
5 ill~li
IIII
IIII
iiIIlII
Ii
I
I Il II s
II
mil
m
l l
i
IIII
lcIiiilII
II
I
I 55%l 5 I1ll
llI
I I
1ll
,,,,,,,,,,,,,,,,,,1,
,, ,,,,,,,,~,,,,
,,,,,,,.,
, ,<
,,,,,,, ,,,,,,,,,
, ,,, ,,,, ,,,,.,,,,,
,,,,,,,,,,,,,,,,,,,,,,,,,,,,.,,,,,.,,,,,,,,.,,,
,.,.,,, :::,, ,,,,,
,,sm,
,,,,,
is,,,,,m,,,esimmi,,,,,,,,,,i,,,i,,,,111111,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::',,,:,
II..............................
~~I
~~~ ~~~,,.,,,,.,.,,,.,,,,,,,,,
~~~ ~~~~~~~~~~~~~~~~~
, ,,,,.,,,,,,,,,,,,,,,,,s,,,,,,,,,,,.,,,,,,,,,,,,,,,,,,,,:,
'"m
is''''',,"'':",,,,
miii isilmmisml
,,,,,,:,
miimmimmi
umm
miimm,,.,, i
mu
, ii ,' ,~ ~ ~~~ym mm5 5 i i %mmj,m
m~mc~i~i
lm~mi~st~
mm~mi~me,
~
5~i~m,,
'"
,,,',:,,I,','m
'"
ii:
a
mi~~~~u~~
m~
m,,,i,,
m
m
~~
~
5',,s,<,,5,s'
, 5,,ss~s5'<,<<"~~~~~
s@ ,, <s' ',,<<,",<
>'ss,",:',"
i:"''sss4$~~>$~,56
i¶' 't[ t
~~
cii
mliii
11111111
fltS~ilii
5
llli0glli 51
ii
ii
i~~~~~~~~
55
111111111111111111
I i00eilI
iS
i
i
ii I'~
IIII5 siiiI
iS
ii sIii
I
II
im Ii
t
mm
ii
memm
mic~~~lsilmill
s sem s~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
m imi i a
s
I ma ai m
i II m I mm i
5
m [ mm,[a I I
------
-----
mm
~,'~ I
..............
5 I ll
,
,Illillc
,s
~ sm
m
i
_is
e
I
..............
I
- -----
======mm
Figure 7.19: Comparison of Gray Scale Ramp for car = .
(a) Threshold Array, 7r1h = 3, c = 0 compensated for the range I < ca, < 1.
7.5. CROSSOVER POINTS
259
~~~~~~~~~I
~~~~~~~~~~~~~~~~~~~~~~~~I
~ ~ ~~~I
~~~~~
~
I
I
I
I
I I
I
I
I
I
I
I
3
I
I
I
I
I I
S~
~~~
~~~~~ ~~~
I
SI
SI
I
II
I
SI IS
II
saII
I
!
I
I VI,,.
S~
Si
S I I
SI
SI
ISII
ii
IIII
SIIIIIIS
I
II ii
IIII
111551
iIIIIIIISI~i
SI
I
31 II
, i , ,iI SI J,
i
I
I
, , ,, , , ,JI, , , , , , ,
I 3i31
II
I
3ii I
I
1
II
1111111111111~~~~~~~~~~~~~~~~~i
~~~~~II
II
II
I
I
I
I
I
I
II
1I15111
SI
I
I IS I
II
1
II 5
I S
aii
I
S I
IiiII
I
5
I I I I
ai
ii
I
I
I
I I
S
I
153II3II31I
1
1
1
,
IIIII
II
1.
5
1
I
I
II I
I I
3I
I
,
D
9D
l
I
II
IyI
i
,
i
l
I
*Sy
1 1
, ,, '
II II III
l
i
I
I
I
I
~II~
1
mE"
R RR
i
I I I II S I I I
I
I
i
i1
IIII
I I
I,, , , , I, ,
I
...............
.............................
W l Wm W R
M
iiiiiiiiiiiiiiiiiiniiiiiiiiiiiiiiiiiii
- - - - -MM-=
I
I
I
I
I
I
I
I
IIIIIIIIIII
5
1
IS
I
I
1
II
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~II
~~~
i
I I
I
111111
1551511
I SI I I II I I I I I I
S I I I I
S I I I I I
II I I I
II
IS
SI i II s IS
IISIIISISIIISIIIIIIIIISSIIIII~~II
ilI
i
i
l l
I II I
ii
I Si
I IISI
S
I
S
I
I
I
I
I
II
I
I
I
i
I
IS
I
I
1
1
I
II
SI
I
I
D
RD
|
-
W
W
W
l
W
l W
^ii!11111111
9WIiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
=--
-- -
-------
Figure 7.19: Comparison of Gray Scale Ramp for a =
(b) Threshold Array, r/h = 2, ,c = 1 compensated for the range 1 <a ar <1
ly
_____LI lll-_l·L·1PII
-^__1114------1-1--
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
260
I
I
INN i
,'
II
i
I'
I
I
i~~~~~au
I II
I II
I II
[1
~~~~~~~~~~I
I
I
I
II
I.~~~~~~~~~,
I
I
I
I
I
I
I
I
[I
I
I
I
I
~~~~I
!
I
I
I
I
~~~~I
I
I
[I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
II
.,,,, I~~~~~~~11111IIIIII
, ,,l , ,,,,',,'|,,,,, , , ,,,,
... ,..,..,..,..,..,..,..,..,l,
II
I
II
I
IiiI
I I I 11
1
I
I
III
II
~
1
61
1
II
l
31
.
le
I
I
I
I
I
I
'
Il
l I l l Il l Il Iiil I
I I IiIII
,..,,,,,,
,,,,,,,,.,, ,,
1
I I
1
'"'',"'
:,.,,
''''",,"'"'','
','
'
I
II
I
, , I ~:".,,,..'
.
IlIi1
i
II I
i
'::
IIII
{','g'~ 'e''
I I
t ~~
r0161IIUPel
I
I1
a Ig1IIIIIg3II
Prg681
IBE
stlerla 3
.
j81 le
lI
I
l
l
ll
l
II
I
I
I
I
I
aiaiaii
I
I
I
I
I
I
I
I
I
I
II
I
I
I
I
I
I
I
l
ll
l
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I-[111[1[il'lie|tiii~l|iiitigilliglllgt~liiliii
IJul.1Ul J U IJIULIlIIIUIIII ILJIIUI IIIUI ILl IIILI II Ill II
1
&I6
IIIIYii I!I01bI
!i0 Y La~ld6
111Mil
EI I Ia WI
IIIII
T)HI ilfil HI
Mil
niI[]IDIIrI
nIIII:
ff1IiOiH
I11I I11
I H
II[!1
H1 il 111ii Mfill
Il[II11I
I[ IlI IU1
-t 1I HII
I il nuWiaiJu U1
nUII
UI
I110I
II11
II01Ul
II01tM
IIItK Lo It
JuflE
M1
6bl~d bl 6I
I
IIII
.,, .,l,,I,,',,%',1:%',,,,,,,l,,,,,,,,,,,,,,e,,,,,,...
"
III
~~~~~~II
I
I
,,,,
,,,,
' <.,~,< l.''''i'tt'|'l'~'~'1'''''
,',~, ,',, ', '~
l1|<el'lee~
I
I
ii*i
I
I!
I
I
I
I
]
III
II
I
-
I
N"
~~~~~MO
. . . . . . . . . . . . ...
~~~~~~~~~~III
I
I
IIlllllllllllllllllllilllllllllllllllnlllllllllll~llllllll~llftlttl
-IIIIII
W'M'hxrW mIIIIIIIIIII
t xnls wX1 :|| zmx In mn gm~ntX~l~wlsslX~sXX~sX..
Z"'"'"'"'...
At
..............
- - - - - - -
---
=--=
......
-----
0...........
=---
- - - - - - - -
=--
Figure 7.19: Comparison of Gray Scale Ramp for aCr
(c) Threshold Array of Figure 7.18.
=
.
7.5. CROSSOVER POINTS
Figure 7.20: Comparison of Scanned Picture for ar =.
(a) Threshold Array, 7h = 3, = 0 compensated for the range
261
1
< er < 1.
262
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
.P
1-1
Figure 7.20: Comparison of Scanned Picture for ar =
(b) Threshold Array, rih = 2, tc = 1 compensated for the range
6
-
1 <
< a,
4.
v
7.5. CROSSOVER POINTS
263
f1111%
I
IY
i
I) I]1
I
I
I
i~1
II.
/I 1:
I
I
W
I
l, ,
PY'W
%All
Y '"', 'n'm 'iI, '
I
I I
I
[11. II
IiI
II I I I I I
-.:::
1:.: .:(.
I ,,.
IW
,%..[..
, ,,
I1II
.. :
I
I IIIIIII11
,
I
I
, ,,t, I ,/,-,,
I
m
I
.. :~. ~,,,
,.I~u
t!~~~,,~~.,
,.!A'~:~............
Figure 7.20: Comparison of Scanned Picture for ar
(c) Threshold Array of Figure 7.18.
=
!~i
.
264
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
Once again, much can be learned by looking at the Composite Fourier Transforms for this case in Figure 7.21. The slight horizontal and vertical artifacts
observed in the spatial domain for cases (a) and (b) are of equal but opposite
magnitude at the crossover point. Their associated Exposure Plots reveal this
asymmetry with the guilty frequency components highlighted with arrows. The
Exposure Plot in Figure 7.21(c) does not show any sign of such asymmetry, however, it possesses horizontal and vertical frequency components that are lower
in frequency than those found in (a) or (b). It appears that the optimum choice
is a toss-up at such crossover points.
265
7.5. CROSSOVER POINTS
0
0
*
I
a
.
.
0
*
0
.
*
0
0
0
0
v
0
0
.
*
.
S
0
* *
.
(a) Threshold Array, rih
3,
.
·
*
*
0
1
= 0 compensated for the range
*
S
*
@~
·
0
*
< a
0
1.
0
lb
*
0
S
(b) Threshold Array, 7Oh
vI*
*
S
*
0
*
0
*
0
0
0
S
0
-
S
.
0
0
0
0
2, c = 1 compensated for thie range
=
*
0
0
*
*
*
0*
*
*
*
0
S
0
*
0
-+*
0
*
*
'
0
*
0
0
0
0
1~~ *
*
*0
0
*
0
0
0
0
*
*
0
*
*
*
<ar <
1
' cycles/unit-length
Array of Figure
2reshold 7.18.S
(c) TI ireshold Array of Figure 7.18.
Figure 7.21: Comparison of Exposure Plots for car
1=
0
S
266
CHAPTER 7. ORDERED DITHER ON ASYMMETRIC GRIDS
7.6
Summary
This concludes the investigation of the point process of ordered dither.
Chapter 5 reviewed the relatively simple problem of generating clustered-dot
halftone patterns on both rectangular and hexagonal grids. The more involved
problem of dispersed-dot ordered dither has been generalized for all semi-regular
girds.
Chapter 6 developed the basic theory with the Method of Recursive
Tessellation to generate threshold arrays for regular grids, both rectangular
and hexagonal, with odd or even periods. Here in Chapter 7, a general rule
for creating correctly compensated threshold arrays for any asymmetric aspect
ratio has been established.
The use of exposure plots to represent Composite Fourier Transforms of
all possible patterns produced by a given threshold array on a particular grid
adds depth to our understanding of the ordered dither process. It reveals the
low frequency clustering of energy for clustered dot patterns, the high frequency
distribution for dispersed-dot patterns, and culprits of asymmetry in cases where
such artifacts exist.
Chapter 8
Dithering with Blue Noise
The nature of various types of noise is often described by a color name. The most
well known example is "white noise", so named because its power spectrum is
fiat across all frequencies, much like the visible frequencies in white light. "Pink
noise" is used to describe low frequency white noise, the power spectrum of
which is flat out to some finite high frequency limit. There is even the curious
case of "brown noise", named for the spectrum associated with Brownian motion
1281. Introduced in this chapter is "blue noise", the high frequency complement
of pink noise, which will be shown to be important in the generation of good
digital halftones.
Unlike any of the halftoning techniques presented thus far in this report,
schemes which generate blue noise are neighborhood operations. As in Chapter 4, which explored dithering with white noise, blue noise patterns are aperiodic and radially symmetric. Although white noise patterns do not suffer from
the correlated periodicity of ordered dither, the fact that they possess energy
at very low frequencies results in a grainy appearance.
Blue noise patterns
enjoy the benefits of aperiodic, uncorrelated structure without low frequency
267
I_^_1_1__1 _.IL_·--X-III-I
--X--
·-_-X-----
268
CHAPTER 8. DITHERING WITH BLUE NOISE
graininess.
The important algorithm known as error diffusion is closely examined and
with some variation is found to be a good blue noise generator. This algorithm
is examined for both rectangular and hexagonal grids.
After the success of
hexagonal grids for the case of ordered dither, it is surprising to learn, as will
be theoretically established, that a rectangular grid is the superior choice for
dithering with blue noise.
8.1
Principal Wavelength
Consider the problem of rendering a fixed gray level, g, with binary pixels on
regular rectangular or hexagonal grids. A goal is to distribute the binary pixels
as homogeneously as possible. These pixels would be separated by an average
distance in two dimensions. This distance is called the Principal Wavelength,
and would have the value
Ag
vl/
Ivl/
where
Ivl
IvlII
-g
g
g > ~~~~~1
(8.1)
S, and v and v 2 are the spatial sampling vectors
= Iv 2 =
defined in section 2.1.1.
I
v
= Iv2 1 because we are considering regular grids
only.
Since the distribution is assumed to be homogeneous, the corresponding
power spectrum would be radially symmetric. The principal wavelength would
be manifested as the Principal Frequency,
fg =
V9-g•
9< ~2(8.2)
1-guI
g>
AS
lul
2
269
8.1. PRINCIPAL WAVELENGTH
where ul = lulJ = ju2 1, and ul and u 2 are the frequency bandpass replication
vectors. Recalling that for a regular rectangular grid, L = 5, and for a regular
hexagonal grid of the first kind, L =
2
S. Upon combining these with the
relations of Equations (3.17) and (3.19) (page 68),
i
Iul=
for rectangular grids.
s
(8.3)
2
l S_
for hexagonal grids.
We can now plot fg as a function of g in Figure 8.1 for (a) rectangular and
(b) hexagonal grids. These plots reveal an amazing shortcoming of hexagonal
grids; they cannot support a principal frequency for
< g<
Figure 8.2(b) il-
lustrates the highest frequency which can exist on a hexagonal grid, f/S-1 =-3
for either a pattern for g =
or its complement, g =
.
On a rectangular grid
(Figure 8.2(a)), the checkerboard pattern for gray level, g =
fr/-S'
=
1
and frequency,
' can most definitely be supported. These spatial patterns corre-
spond to the high frequency corners of the frequency baseband.
For gray levels in the range
<
<
on a hexagonal grid, pixels must be
grouped together resulting in frequencies lower than fr/S- ' =
2
3'
A principal
frequency exists for all gray levels on rectangular grids.
A well formed binary dither pattern rendering of a fixed gray level should
consist of an isotropic field of binary pixels with an average separation of A.
This average separation should vary in an uncorrelated manner, but the wavelengths of this variation must not be significantly longer that Ag. The failure of
dithering with white noise was due to the presence of long wavelengths.
Figure 8.3 depicts these well formed dither pattern characteristics in the frequency domain. The radially averaged power spectrum (defined in section 3.2.2)
of a fixed gray level, g, has 3 important features. First, its peak should be at
I__II__IPI^_Y__IULUI___U____-YLIY·IPI
270
CHAPTER 8. DITHERING WITH BLUE NOISE
1
_
a)
4
*_
I._
1
2
_
t
0
1
2
0
1
Gray Level, g
(a) Regular Rectangular Grid.
2
3
1---Z
1
Q
g w -
n
wCr
:z
4
1-4
14.
0
(n
:t
CZ
9
p,
:
U
r.
$-4
114
0
0
1
3
1
2
2
3
Gray Level, g
(b) Regular Hexagonal Grid.
Figure 8.1: Principle Frequency, fg, as a function of g.
1
271
8.1. PRINCIPAL WAVELENGTH
0
0
0
0
1
Rectangular9(a)
Grid, fg/S21
1
(a) Rectangular Grid, f1S- = V72-
0
0
0
0
0
0
0
0
O
0
0
0
0
0
0
0
-.
...
*
O
..
*
*0
* .0
*0
1
.
.
10
*.
O0"
.
2
3=
g=3
(b) Hexagonal Grid, f/S
-1 -
2
3'
Figure 8.2: Patterns with the highest possible Spatial Frequency
(Corresponding to the corners of the spectral baseband).
II-Il·IY···
I___I_.IIX
-II---
272
CHAPTER 8. DITHERING WITH BLUE NOISE
I
0
0
Radial Frequency, fr (units of S -
1)
Figure 8.3: Spectral Characteristics of a Well Formed Dither Pattern.
1. Low frequency cutoff at principal frequency.
2. Sharp transition region.
3. Flat high frequency "Blue Noise" region.
273
8.1. PRINCIPAL WAVELENGTH
the principal frequency for that gray level, fg. This frequency marks a sharp
transition region below which little of no energy exists. And finally, the uncorrelated high frequency fluctuations are characterized by high frequency white
noise, or "blue noise".
The visually pleasing nature of some of the patterns generated by the error
diffusion algorithm to be presented in the next section can be attributed to
a spectral signature as just described. However, several shortcomings exist in
this algorithm. In section 8.3, error diffusion enhanced with certain stochastic
perturbations is found to be a good blue noise generator.
The presence of significant low frequency energy is responsible for the visibility of disturbing artifacts in halftoning patterns. For dispersed-dot ordered
dither, half of the total possible gray level patterns available for a given threshold array have low frequency components which correspond to wavelengths of
the size of the threshold period. For good blue noise processing, the lowest
frequency is essentially f.
The negative feedback of error diffusion acts as a
low frequency inhibitor.
_·X____I
L·lllllll-ll__lPI--·-·-·
1·_-*II-·_-------II-----
CHAPTER 8. DITHERING WITH BLUE NOISE
274
J[n]-
Threshold
I[n]
Figure 8.4: Point Processes.
8.2
The Error Diffusion Algorithm
Up to this point, all of the halftone processes considered could be modeled as
shown in Figure 8.4. All were point processes, that is, the output depends only
on the current input pixel. Pixels of the continuous-tone or m-ary digital image,
J[n], are simply compared with a threshold to determine the state of the output
pixels, I[n].
The error diffusion algorithm, first introduced by Floyd and Steinberg in
1975 [22,231, requires neighborhood operations and is thus more computationally
intensive. It is currently the most popular neighborhood halftoning process and
has received considerable attention in spite of some shortcomings. A generic
form of this algorithm is graphically illustrated in Figure 8.5.
The threshold in this case is fixed at
where the input, J[n], varies as usual
from g = 0 (white) to g = 1 (black). The resulting binary output value of 0
or 1 is compared with the original gray level value. The difference is suitably
called the error" for location n. The signal consisting of past error values is
passed through an error filter, e[n], to produce a correction factor to be added
275
8.2. THE ERROR DIFFUSIONALGORITHM
I[n]
J[n]
Figure 8.5: The Error Diffusion Algorithm.
to future input values. Errors are thus "diffused" over a weighted neighborhood
determined by en].
Figure 8.6 summarizes error filter impulse responses promoted in the literature. Note that in all cases the values are deterministic and sum to 1 so that
errors are neither amplified nor reduced. The first three listed are designed for
rectangular grids and will be examined in both the spatial and frequency domain in this section. The error filter in (d) is intended for use on a hexagonal
grid; it will be similarly addressed in section 8.4.
-_
II·L-I-L_--.
-r~11
-11
CHAPTER 8. DITHERING WITH BLUE NOISE
276
7
1
1A
3 5
(a) Floyd and Steinberg (1975) 122,23].
(1 x
1
(48
3 5
35753
*~ 7 5
5 3
1 3 5 3 1
(b) Jarvis, Judice and Ninke (1976) [39].
(
x)
42
2 4 8 4 2
1 2 4 2 1
(c) Stucki (1981) [83].
*
200
1
12
12
16
12
26
12
5
32
30
26
12
5
(d) Stevenson and Arce (1985) [80].
Figure 8.6: Error Filters reported in the Literature.
(a), (b) and (c) are for rectangular grids.
(d) is for hexagonal grids.
("." represents the origin.)
*
277
8.2. THE ERROR DIFFUSIONALGORITHM
8.2.1
The Floyd and Steinberg Filter
The original error filter or set of "weights" suggested by Floyd and Steinberg
is shown in Figure 8.6(a). They argued that a filter with four elements was the
smallest number that could produce "good" results. The values were chosen to
particularly assure the checkerboard pattern at middle gray.
Figure 8.7 shows the effect of error diffusion with the Floyd and Steinberg
filter for (a) the gray scale ramp, (b) scanned picture, and (c) the synthesized
image. The reason for the popularity of this algorithm is clear; several gray
levels are represented by pleasingly isotropic, structureless distributions of dots.
However, some shortcomings are also apparent:
1. Correlated artifacts in many of the gray level patterns. This can best be
seen in the gray scale ramp.
2. Directional hysteresis due to the raster order of processing. This artifact
is most apparent in very light and very dark patterns.
Note the light
regions of the sky in the scanned picture, the highlight in the egg shaped
object and the dark squares of the synthesized image.
3. Transient behavior near edges of boundaries. A good example of this is
in the rendering of the fixed background at the top and left edges of the
synthesized image.
The radially averaged power spectrum, P(f,), and anisotropy measure,
8 2 (fr)/P,(fr) (described in section 3.2.2) are plotted for selected gray levels
in Figure 8.8 on pages 281 through 287. In each case here and throughout this
chapter, the principal frequency, fg, is marked with a small diamond on the
frequency axis of the power spectrum.
-.-- -·l._.____-PI
-·l-.-LI
----
278
CHAPTER 8. DITHERING WITH BLUE NOISE
:~~~~.
a..,
o.·~~~.
·~
o.,
·
.......
o
o
Figure 8.7: Error Diffusion with the Floyd and Steinberg Filter.
(a) Gray Scale Ramp, aC = 1.
279
8.2. THE ERROR DIFFUSIONALGORITHM
Figure 8.7: Error Diffusion with the Floyd and Steinberg Filter.
(b) Scanned Picture, ar = 1.
____ILI
__II_I
--^ILS-I
280
CHAPTER 8. DITHERING WITH BLUE NOISE
Q)
;-s
P._
._U)
qJ
O
o
- 4
bOt~
._2 w
"-,
0
s-,
ic1
oS..
00
._
*
8.2. THE ERROR DIFFUSIONALGORITHM
m
IV
281
+10
1-
0
0
0
ii---
---
i~~~~~~~~~~~~~~~~
.eq
-10
Celvzh
t5
C-
l
2
72
Radial Frequency, f, (units of S-1)
Figure 8.8: Radial Spectra for the Floyd and Steinberg Filter.
(a)
, fg/S-1 .1768, C2 - .0303
-
~
-
~
He
~
w
~
s
282
CHAPTER 8. DITHERING WITH BLUE NOISE
+10
m
-10
C
- 10
2.8
ellb.,,
,b
-1
I.-
-"S
L
i
Radial Frequency, f, (units of S-1 )
Figure 8.8: Radial Spectra for the Floyd and Steinberg Filter.
.0586
f/s- 1 = .25, or2
(b)
72
283
8.2. THE ERROR DIFFUSIONALGORITHM
r·
0
1
·
I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+10
04
0m
0
0
wu
-10
v
I
o,, ig
4.88
2.88
8.81
1
72;
2
Radial Frequency, f, (units of S- ')
Figure 8.8: Radial Spectra for the Floyd and Steinberg Filter.
1
(c) =(c)
fg/S-'
8 ,~~~~
g '194 .3495, o'g .1094
"~1
II
II'------'I
_·IIU--r----_l-_-
.
----
---
284
CHAPTER 8. DITHERING WITH BLUE NOISE
I
m +10
Po
0o
e
0
-10
.
--
I
1
2
Radial Frequency, f (units of S - l)
Figure 8.8: Radial Spectra for the Floyd and Steinberg Filter.
(d)g = ,fg/S-' = .5, ' = .1875
~
4~
_
1
8.2. THE ERROR DIFFUSION ALGORITHM
285
+10
0
1-
0
e$
0
._1
-10
-
I
i
72'
Radial Frequency, f (units of
S - 1)
Figure 8.8: Radial Spectra for the Floyd and Steinberg Filter.
(e) = 2_ , fg/S-l .7071, ' 2U = .25
_
____ -.
I11I--_I.-.------LIIII_
-X -
-1-111--1
286
CHAPTER 8. DITHERING WITH BLUE NOISE
I-
m
+10
o
0
'o
rt~
ax
.1.
-10
l
2
Radial Frequency, f, (units of S- l)
Figure 8.8: Radial Spectra for the Floyd and Steinberg Filter.
(f)
=,
f/S
-
= 5, o2 = .1875
72'
8.2. THE ERROR DIFFUSION ALGORITHM
I
287
I
+10
0
5
0
-6qa
e
0
.a"l
¢
-10
kllll
v11
I
bC
t
I
7i'
2
Radial Frequency, f, (units of S -
l)
Figure 8.8: Radial Spectra for the Floyd and Steinberg Filter.
(g) g , f/S.3536, o 20~; .1094
·-- ·-·----- · ---·-----
----·--·-
·---- ··---
·-------
----
-·-·-
288
CHAPTER 8. DITHERING WITH BLUE NOISE
The lack of symmetry in these patterns is strongly acknowledged in the
Anisotropy measure, especially for (d) g =
and (e)
=
.
Recall that the
"background noise limit" due to the spectral estimation method is -10 dB
(equation (3.31)) indicated by a reference line at that level. Any measure greater
than 0 dB at any frequency indicates an especially anisotropic pattern; at such
a level the sample variance is greater than the square of the average.
289
8.2. THE ERROR DIFFUSION ALGORITHM
8.2.2
Filters with 12 Weights
In 1976 Jarvis, Judice and Ninke 39] reported an error filter with the 12 elements
shown in Figure 8.6(b). Although their technique was identical to error diffusion,
they called it the "Minimum Average Error" algorithm.
Output from error
diffusion with this filter is shown in Figure 8.9. The larger filter size does reduce
some of the artifacts seen with the 4 element filter of Floyd and Steinberg, but
directional hysteresis in the very dark and light regions has increased, and pixels
are clustered together more in the middle gray regions.
It also sharpens the picture more. The amount of sharpening may or may
not be to the degree desired. In fact, Stucki [831 argues that it is objectionable
an uses an additional filter to inhibit sharpening.
Sharpening should not be
compounded with halftoning. If the computational overhead of a neighborhood
process can be afforded, as is the case when error diffusion is used, a separate
presharpening step can be employed (see Figure 1.2). This will be demonstrated
in section 9.1.1 where the degree of sharpening can be precisely controlled.
The 12 element error filter used by Stucki is shown in Figure 8.6(c). For
computational efficiency, the selected values are all powers of 2. The effect of
error diffusion with this error filter is displayed in Figure 8.10, and is found to be
quite similar to the images of Figure 8.9. In each case the directional nature of
the inherent sharpening is particularly evident in the synthesized image under
the glass-shaped object. In that image, the background which has a fixed value
just under g =
, reveals high sensitivity to the slight difference between the
two 12 weight filters.
The patterns generated for fixed gray levels with the Jarvis, et al. filter
is examined in the frequency domain in Figure 8.11 (pages 297 through 301.
While all gray levels shown still suffer some anisotropy, g = 32 ' 16'1
I and
_b_
IYI_)_
8
I~_l-1-1~1-
show
C-
l
1
_
CHAPTER 8. DITHERING WITH BLUE NOISE
290
...
'''''''''"..
..~~~~~~~~~~~~.
''':..---..
~ ~ ~ ~~~~~~
~ ~ ~ ~ ~ ~ ~ ~......
*..
.
... ................
.y...:....:....:......
"":::::v-;..
....
'''
"''''
..
..
. . .'.....
..
...
.....
:'...*.:-*
-. % -.. %%
... ~o..
s.%.'..·
''-.........
.e..?..
-..
, ...
.
...................
'
I
.
.'.".'....-..::..:-.:...5:...
..- -...
~.~.----,-~
.......
j~,.....i
,'1
...
,.,
, . .,
....
.
9101
·.
$·.~..
_.,,j.
i·q.
·.
ry-fc
II r
,
......
r-
I
~.,--
rr'rrr
...
rtr.n
·..
MalP
M
M
))Wmi
E
i X
i YE
S
Abut
RU
S
|: | eE S
- w ' n
|
t si19ei i
8
Figure 8.9: Error Diffusion with the Jarvis, et al. Filter.
(a) Gray Scale Ramp, a, = 1.
--
b-~~
]rzJ'.xC
wr'cum
Ifr=zzznx
..................
FlqsmZiml
---- i
'·'"'.
?
... ..................................................
...........
,-....o
.
-... oo
'"'...:......':"";-'...%
:
~ "i'; ".';.~;
.:...- .;.
o
8.2. THE ERROR DIFFUSIONALGORITHM
Figure 8.9: Error Diffusion with the Jarvis, et al. Filter.
(b) Scanned Picture, err = 1.
291
292
CHAPTER 8. DITHERING WITH BLUE NOISE
.:
I>
.' ..
~:
_
~>
a; I11
b
;~-4
0o.!
._
8.2. THE ERROR DIFFUSIONALGORITHM
........
....
~~
..o
. o O O o
..
·
ooo
oo
.. ·..
..
·
. o
o.
o *
.·
·
o * o......o
o . . .
o
o·
oOo
o o.
oo o*
..
.
...
...
293
o o o *....o
. O o o.......
· = · o·
. o
.oo·
o.
..
·.
. ....
..
.......
......
X.....
o . . . o o o.
. ..
,o
oo
o o
o.....o.
o .
,o=
o.o.....
. .. ,,.e
..
,,....,,
.,,.
,
...*
....
. ,
%·.
oo
.
o o
~
B.V.
"'...·';"......-.';-·..2:..""
·.·oo
o .~*o.
~o
..
·..
·
:
.o~o·oo*.o
.
.
o
.
.
'
. o.~ ooOo~oo
A
.
-
,
" " "··'::"...·"'.."..
.o ooo..,. ,,.o..o ..
.
-
.
,
Figure 8.10: Error Diffusion with the Stucki Filter.
(a) Gray Scale Ramp, ca, = 1.
,
,
.
294
CHAPTER 8. DITHERING WITH BLUE NOISE
Figure 8.10: Error Diffusion with the Stucki Filter.
(b) Scanned Picture, a = 1.
8.2. THE ERROR DIFFUSIONALGORITHM
295
1)
U2
0)
*-bO
4
:(
n -IE
o
.
296
CHAPTER 8. DITHERING WITH BLUE NOISE
a stronger concentration of energy at the principal frequency, fg, than those of
Figure 8.8. The clustering of pixels at g =
frequencies
lower than f2g
at frequencies lower than f.
I
-
and
4at
results in significant energy
297
8.2. THE ERROR DIFFUSIONALGORITHM
:. ,,
:'
~ ~~~~~
*
-o
.
. .. . ,,
.
fff'DE''''''X
*'~~~~~~~~~o
'
'. '
'...:
. .
.
.
.
'''''''fS·
. ,
.o
.
.
.','.,...
* ,**.
o *......
.
. *..........
:.
' · :
,a
0·
:
.
.
.
.o
.- .
*
*
.-
:
,...
e
.'
.
'
'::
I
1-
Pq
.....
~~.
*, *,
.
.
.
.
+10
0
-
e
54
0
0
an
*_
- AAA I
-10
I
I
-
2.r
eq
U..,
1
2
Radial Frequency, f, (units of S-')
Figure 8.11: Radial Spectra for the Jarvis, et al. Filter.
(a)
i fg/S- 1 .1768, r .0303
72'
CHAPTER 8. DITHERING WITH BLUE NOISE
298
.............
.. .... ................
.'..v..
'-'',
.... '..'' . ::..........
"
'
:
::::
.....
....
:.....
:..,.:
:.':
i:':. '': :.':'i.".': :'.' .':.'.:.''.
...... .... ....
. . . .... . . ......
i.....**.
.. :::.:.:.:
: : .. ::.:..::.
.::::
+10
,
1-
0
0
sa
0
-o10
-10
2,1
I*
Radial Frequency, f (units of S -')
Figure 8.11: Radial Spectra for the Jarvis, et al. Filter.
(b)
=
fg/S-1 = .25, o2 ; .0586
or1
299
8.2. THE ERROR DIFFUSION ALGORITHM
+10
0
m.
0
0
eP~
-10
3.0
2.9
1.8
8.-8
Radial Frequency, fr (units of S - ')
Figure 8.11: Radial Spectra for the Jarvis, et al. Filter.
(C) g8,
fg/S
-1
.3495,
Ue2
.1094
1_1_·
^1_1_11______111·*_yllll·---
11 --I ·
300
CHAPTER 8. DITHERING WITH BLUE NOISE
I
m
+10
10
0$4
0
-1_
I
-10
i .k
jr -, .
,nr
I
..... AA '
. VVI
3.88
2.88
1.88
2
Radial Frequency, fr (units of S-')
Figure 8.1 .1: Radial Spectra for the Jarvis, et al. Filter.
(d)
I fg/S-I = .5, 2 = .1875
g_=4 1_J
l
72'
8.2. THE ERROR DIFFUSION ALGORITHM
301
+10
0
0
e
c-;--
zci~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
._
-10
0b
2.
-W
-
_S-~
1.
8-.
i
72;
Radial Frequency, f, (units of
S-
Figure 8.11: Radial Spectra for the Jarvis, et al. Filter.
(e)
l fg/S-1
.7071,
=.25
-
')
CHAPTER 8. DITHERING WITH BLUE NOISE
302
8.3
Error Diffusion with Perturbation
The idea of dithering" or perturbing a method in image processing to defeat
visual artifacts of a regular and deterministic nature has been used before. Randomizing sampling grids 181 is one method used to reduce the aliasing effects of
undersampled images. Allebach improved the classical screen by randomizing
the centers of dot clusters [2,41 which eliminated the occurrence of morie patterns. And in the color printing industry, arbitrary or "irrational" clustered-dot
screen angles have been digitally produced by employing random perturbations
[25,681.
Proposed here are several modifications to the basic error diffusion algorithm
graphically depicted in Figure 8.5. They are categorized in the following four
areas.
Choice of Error Filter
An enormous number of choices are available for this, the deterministic part of
the algorithm. An error filter can consist of weights of any
1. number,
2. position, and
3. value.
For computational efficiency, as small a filter as possible is preferred.
303
8.3. ERROR DIFFUSION WITH PERTURBATION
Threshold Perturbation
In 1983, Billotet-Hoffman and Bryngdahl [12] suggested using an ordered dither
threshold array in place of the fixed threshold used in error diffusion. However,
the resulting halftoned output differs little from conventional ordered dither. A
modification to this idea would be to perturb a fixed threshold within a given
maximum percentage with ordered dither and/or white noise.
So, additional parameters include:
1. Choice of period size of ordered dither.
2. Magnitude of ordered dither perturbation.
3. Magnitude of white noise perturbation.
Raster Direction
The directional artifacts seen in the examples of error diffusion are due largely to
the traditional raster order of processing. Many choices of space filling curves
to define the order of processing are possible. Although they did not call it
error diffusion, Witten and Neal [91] demonstrated fairly good results by essentially using an error filter with one deterministic weight and processing all of
the two-dimensional image data on a Peano curve (a type of fractal). While
this particular approach imposes heavy demands on memory, the idea of using
nonstandard raster ordering should be tried.
One idea that breaks up the directionality of a normal raster without the expense of a full two-dimensional buffer is to process along a serpentine raster (see
Figure 8.12). Neighborhood operations in image processing hardware or software buffer image data in full lines. So the choice of serpentine raster processing
does not require any memory increase over a normal raster.
__ -.
---lla-·l··UliP·CII-
1--
_111_-1_11111--
---- ·-
304
CHAPTER 8. DITHERING WITH BLUE NOISE
-
i
~~~-
(a) Normal Raster.
II.__
I
J.0
I
I
-
.
fI
I
l
I-
_·
i
r~~~~~~~~~~~~~~~~~~~~0
>
I
(b) Serpentine Raster.
Figure 8.12: Two Processing Path Options.
8.3. ERROR DIFFUSION WITH PERTURBATION
305
Stochastic Filter Perturbations
Along with threshold perturbations, random noise can be added to the elements
or weights of the error filter.
This idea was proposed by Schreiber [701 and
demonstrated by Woo [921, but only on the 12 element filter of Jarvis, et al..
The magnitude or range of additive noise can be adjusted for each element.
The sum of all of the weights in the resulting stochastic filter should still be
unity at all times. This condition can be met by pairing weights of comparable
value.
For each pair of weights a scaled random value is added to one and
subtracted from the other.
A random value, X, is generated by the method described in Chapter 4, with
the adjusted uniform probability density function,
P(X) =
ai
2
0
for
-ai < X < ai
(8.4)
otherwise
for each pair of weights, i, in the filter. Then at every image location for each
pair of weights, aiX is added to the first weight, and subtracted from the second.
The ai's, expressed as a percentage of the smaller weight in the pair, are yet
other adjustable parameters.
The error filter can also be perturbed by randomizing the positions of the
weights.
The number of adjustable parameters available to modify the basic error
diffusion algorithm is tremendously large. Much is to be learned about the effect
of each parameter used independently and in combination with others. Over
a hundred combinations of these parameters were experimented with in this
investigation; the examples that follow are a carefully selected representative
1___·_1
--·111-11111
·11··111
..
1_ ··IIIIl-^·--LI-llIIII_·l__·YII1-
11-----
L-. Y11 l~llL· 1111
-II_U
.--_l_-·-i^·il-----(·-^_-rm_-.·IC
-1_1(
--
306
sample.
CHAPTER 8. DITHERING WITH BLUE NOISE
307
8.3. ERROR DIFFUSION WITH PERTURBATION
8.3.1
One Weight
The most computationally inexpensive form of the error diffusion algorithm is
one implemented with an error filter with one weight. In this case all of the
error is diffused to only one location; no multiplication is necessary.
If the one weight is fixed to some predetermined location, the resulting patterns fail in a big way. This is demonstrated in Figure 8.13 where the position
of the weight was fixed diagonally adjacent to the origin. Failure is at least as
great for any other location choice, or when a serpentine raster is used.
However, when the position of the one weight is randomly determined over
some finite set of candidate positions, a much more acceptable result emerges.
The images of Figure 8.14 where produced with the position of the one weight
selected with equal probability between only two candidate locations, immediately below and preceding the filter origin.
This example represents a broad class of parameter combinations. Effectively
the same output results for any local neighborhood of candidate locations, as
well as for a broad range of probability mass functions governing the selection of
a location. Several combinations were tried without change in output, including
the 4 Floyd and Steinberg locations using the values of their weights as the
probability mass function for position selection. Even when 2 or 3 weights were
randomly selected, no significant difference was seen.
The radially averaged power spectrum is displayed in Figure 8.15. The
spectra for the various gray levels all reveal the desirable properties of
1. very low anisotropy,
2. flat blue noise region, and
3. cutoff at fg.
·- ILI
--.
·---L·l·-L
^-LIIIII*I-·Y-Y·LII··CIIIIIIIIX-IYI·
.·
_1 _111__^___1__1____11_-_--^C.I·l*ll
1- ---
-
CHAPTER 8. DITHERING WITH BLUE NOISE
308
If
Fr
Vt
Halml
_
_
__
------
Figure 8.13: Failure of One Deterministic Weight Location.
_
309
8.3. ERROR DIFFUSION WITH PERTURBATION
I~~~~~~
:l
· ~~~~~~~~~~~~J L..A
..
' o
,..
'...'
....-..-.......-....
A
'
~~~~~~~~~~~~~~~~~~~~~~~
°
i
..
,oo
: I
I
-
I
I
o
I
.
I
oo·
o
..
ON
.
o
o°
o
OO
· ·
'.o
-Ck
·-
-W
~
i·I
o
ileiII
,...-...-.. ...-..-... ;
o
o
o·
·
iIi~
·i
I
i
I
,
,,
oo
o
o
o
.,
·
·
·
o.o
, . · ·
o "o
;
o~o
o,
o
, o,
...
' oo'o
o~o~
oo.-
Figure 8.14: Effect of One Randomly Positioned Weight.
(a) Gray Scale Ramp, ar = 1.
-
~ ~ ~ ~ ~
-I
I
I
X-·
310
CHAPTER 8. DITHERING WITH BLUE NOISE
Figure 8.14: Effect of One Randomly Positioned Weight.
(b) Scanned Picture, a, = 1.
I
8.3. ERROR DIFFUSION WITH PERTURBATION
311
The only feature that the spectra fall short of is that of a sharp transition region.
This is most pronounced at
=
.
The low frequency leakage is responsible
for the grainy texture; the associated patterns can be called "light-blue noise".
The suppression of frequencies below fg is sufficient, however, to produce a vast
improvement over the white noise images of Chapter 4.
312
CHAPTER 8. DITHERING WITH BLUE NOISE
*.:~~~~~~~~~~~
.
..
;
.
: . . ..
.:
... '
.
:2':
,·'.
'.:.:
.o.:
~~~... ..
: . ...
.':.:': . ::
.: ' ...
..
.
: ... ;.:
:.
:...
.
.. ::
.. :
.. ..
:'
..
-
:
...
.
;.
::
.
.:
.: ....
.
..
.
..::
:.
.:.:.
.:.
.::
I
1-
I
+10
0
0
0
eq
ax
*
.1 A.
-10
- L--
-
l
F{
2
Radial Frequency, f, (units of S-')
Figure 8.15: Radial Spectra for One Randomly Positioned Weight.
(a)
, fg/S-1
.1768, a2 .0303
g~=~2~~~~~~~~~
-
---
---
--
--
---
-- -
-------------
----
8.3. ERROR DIFFUSION WITH PERTURBATION
I
1-
M
X
313
+10
0
0
-4a
0
-10.
- 10
-W
40"
m-.
' I VI
-w6vvk*A&%A0--
V2q.Pk%Aom- .-
I 11 -45
I
-A- v-
I
A M
b
4
Radial Frequency, f, (units of S-')
Figure 8.15: Radial Spectra for One Randomly Positioned Weight.
(b)
-1------ -
, fgL/S-' = .25, a
I-"~'11`
'~
YiU~
_
IU-
LII
.0586
-S·-~·^~·*~··l·L-illl--·----~L~pl~ll
i~-·
I·---
314
CHAPTER 8. DITHERING WITH BLUE NOISE
;·.
·.:.
.... : .,~'.
::'....':
:t'
:.:.'.'.;:.
·. ,:'.^....: * ;;.::;.
··...
:~,..,:.
...-.....
,.... ,·.
.
..
,....t
..
'.
-%
%'2 ....
·..
A..~~~.
'.:::::'"':'
l
:-
+10
0
I"
+a
0
.
00
,._...._i~~~
.
10
-10
Radial Frequency, f, (units of S - ')
Figure 8.15: Radial Spectra for One Randomly Positioned Weight.
(c)
-- , f~S'
fg/S 1 345
.3495,' C2 .1094
315
8.3. ERROR DIFFUSION WITH PERTURBATION
+10
0
0
M
IV
0
-10
-_
Radial Frequency, f (units of S-')
Figure 8.15: Radial Spectra for One Randomly Positioned Weight.
(d)
__I._
___I··ll__. ·_II
1g 49, fg/S-l = .5, a
-
....-··U··llll----I
---...
= .1875
_I·
I·C_1
II
____·IC
·l-----^IIY·l--
--··-------- ·-- ·- ·----
316
CHAPTER 8. DITHERING WITH BLUE NOISE
+10
M
0
-1
0
k
0
e-,4
.
-10
fg
2
Radial Frequency, f, (units of S -')
Figure 8.15: Radial Spectra for One Randomly Positioned Weight.
(e) g = ,
/S-'
.7071, a2 = .25
__
l
72'i
8.3. ERROR DIFFUSION WITH PERTURBATION
I
1__
m
0
0
I-,
317
+10
0
r"o
-10
HVI %I
.v
M
-)% - QF,-Iyn AAUWN
v
" -V.-,w
I . IV
-
v4u,-AAw-v -,0v0W
A^
A
-
eq I
C4b
b
C-
2
l
72
Radial Frequency, f, (units of S-')
Figure 8.15: Radial Spectra for One Randomly Positioned Weight.
-1
(f) g_8
=,
; .3536, 29 : .1094
, ff/S
jg
I
318
CHAPTER 8. DITHERING WITH BLUE NOISE
11 X
*
Figure 8.16: Deterministic Part of a Two Weight Error Filter.
8.3.2
Two Weights
On this section, four variations on the error filter shown in Figure 8.16 will be
considered.
As might be expected, using this filter unperturbed yields unacceptable results as seen in Figure 8.17. The strong diagonal texture patterns result in an
extremely anisotropic power spectrum evidenced by Figure 8.18.
The use of a serpentine raster corrects the directionality of these textures
but still leaves many undesirable patterns. This is shown in Figure 8.19.
Figure 8.20 shows the result of adding the perturbation of 100% randomness
to the two weights.
This is a particularly interesting case because it simply
requires the selection of a random number distributed between 0 and 1 for one
weight, and its two's complement for the other.
The very random nature of this approach, while making the patterns more
isotropic, passed too much low frequency content.
A compromise is seen in
Figure 8.21 where the weights were perturbed with 50% noise. The well behaved
Radially Averaged Power Spectra results in Figure 8.22 (pages 327 through 331).
8.3. ERROR DIFFUSION WITH PERTURBATION
fly~~~~~~~~~~~e
I
,/""',
,{.;/.S'..,z'.'/dS~~~~e,
,
... o
.'
.
. , ..
...
.
,: .
...
·
319
~...
....-
.''°°''''°e,
ed.
'.
·
~~~~~~~~~
/
Figure 8.17: Effect of Two Deterministic Weights.
·I·_LC·
I·
C__I IrC __I I_
I__
320
CHAPTER 8. DITHERING WITH BLUE NOISE
-
m
+10
IV
SC6
0o
b4
0
To
.a
-10
A
-
b
-1
l
2
Radial Frequency,
fr
72-
(units of S - ')
Figure 8.18: Radial Spectra for Two Deterministic Weights.
= , f/S-'
.3495, 2 .1094
I i '1
321
8.3. ERROR DIFFUSION WITH PERTURBATION
I
.'.. '..'."': ':
.
':.'.":
x
'.:".'::' .'.
.:'::
,'.',,,,.'.'
.'..:'w
'.. ,',':'.":.'':'''''''
........
,......,..,,=,,_,,.,j:
~~~~~~~~~~~~~~~~..
J..
'#
':'::.':'.''.:':''.;:;".%'.
. o ~%
,,,,.,:
Nj4['... . ,..,....,...
......
%
.,o.
x.`~`
Z~
A~b
M7411
Figure 8.19: Effect of 2 Deterministic Weights on a Serpentine Raster.
(a) Gray Scale Ramp, a, = 1.
___11__
___
IY___I
111
_I_··I____
__
322
CHAPTER 8. DITHERING WITH BLUE NOISE
Figure 8.19: Effect of 2 Deterministic Weights on a Serpentine
Raster.
(b) Scanned Picture, ar = 1.
__
______
_
323
8.3. ERROR DIFFUSION WITH PERTURBATION
· "'.
...
? .
'.'.·..
.'.·.' .
-------- --------- ---- --------
.',.''''......9t0t"'
'.'''''.''''"''
......
· . .....
- .... ,-.,...·
..:'. . .,-..,....
: .....
.
;;..
·.. ...........
..........
;
" ,
.
.
.
..
~~~~'"'"'':"...
Ef~~~~~~A
,
..
.,·..%°..o..
·.
...
' . .. '
~...........·"...............
· ... · ........-. · ........ ,... .. ·
..... I ....... .......... ... .......
.
- ------
'.
..
.. '..Fi.......... ' . · ~...-%
· . . v:.
.. ~......· ... ..*..
;.-.... ' .
..t .%....., ,. .... ,....'.....,.i..
·... :...
------ ---------- -------------------------------------------
Figure 8.20: Effect of Two 100%1 Random Weights on a Serpentine Raster.
_LI___
____P----ll_--^l-·-
·I II^--
324
CHAPTER 8. DITHERING WITH BLUE NOISE
Figure 8.20: Effect of Two 100% Random Weights on a Serpentine Raster.
(b) Scanned Picture, r = 1.
8.3. ERROR DIFFUSION WITH PERTURBATION
325
*'..*-..:..: .*.*
.:' .: .-.
.:.:.
'.:-:::.::
. - .:. -' ..: . .. . ..
·
.'..'... ..
: ..-.
.
-.::
:.,'....'.
..·
:..:.'..'...:'..-.:.
~~
~ ~~ ~~ ~ ~ ~ ~
5,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.~
;~~~~~~~~
·
·
·
~~~~~~~~~~~~~~~~~~
· '::...~::'..-.-..................
- ... .....: Z.
Ct~~~~~~~~~~~~~~~~~
.
·
,..~...,..
.....
Figure 8.21: Effect of Two 50% Random Weights on a Serpentine Raster.
(a) Gray Scale Ramp, ar = 1.
II
_··· ·11111111
.·--·1-·-1111111-Y-_.·llll
1 -^--L^I_---
326
CHAPTER 8. DITHERING WITH BLUE NOISE
Figure 8.21: Effect of Two 50% Random Weights on a Serpentine Raster.
(b) Scanned Picture, ar, = 1.
8. 3. ERR OR DIFFUSION WITH PERT URBATION
-
.
.
......
:.
. .. .
.
,' ',·
~ ~~.
~~~~~~~~~~.
* .
:
.
·.
..
.
.'
,:
:.:
.
:
',,.''.'.'.:
'o,
:':
..
.
..
.
.
'
.,~~.'
.
.:,"....
.~~~~~.·.
.
:~~~~~~~~~~~~~~~~~~~
..
.
.:
.
.
.
*
.
,'
'
.. '
.
:
.
:.
*.
'.
.·
.
·
327
.:
:
.
:
.
..
.
.::o...'.
.
.·
.:
.:
,'
:
''
:
::
+10
oo
0
:.4
-10
.F-.
-
.
i~~~~~
.
A
1
2
Radial Frequency, f, (units of S - ')
Figure 8.22: Radial Spectra for Two 50% Random Weights.
(a) g=d , fg/S - 1 .1768, 2
0303
721
328
CHAPTER 8. DITHERING WITH BLUE NOISE
I
m, +10
Xo
00~
.'4
. KA
-
-00AOft -1-AkA
0
,4
oo
- 10
- 10
WV
I"
m P,&AOA
- I- -
2.88
,·
-
1.88
8.82
Radial Frequency, f, (units of S-')
Figure 8.22: Radial Spectra for Two 50% Random Weights.
(b)
, f/S-' = .25, 2
.0586
vIm
8.3. ERROR DIFFUSION WITH PERTURBATION
........
....
329
::·..
....
:........-..:
:.....
.: ........
:....
I
M
P
+10
1-
0
-4
0
0
- 10
hMnM
-v
-AAgnA.,
v
',A'o,,-A=k4
. ---A
VW
I
cb
b
C4 1.88
8.8%
1
2
7i
Radial Frequency, f (units of S -1 )
Figure 8.22: Radial Spectra for Two 50% Random Weights.
(c)-i3 , f/S
fg/S-- 11
(c
g=L
8
t
.3495, Og
U2 ~ .1094
.3495,
___
CHAPTER 8. DITHERING WITH BLUE NOISE
330
·
·
i~~~~~~~~~~~~~~~~~
+10
0
0
eml
+a
To
0
Wo
Uk
-10
M'(W
-A
I
2.88
9.9%
2
· w
Radial Frequency, f, (units of S - l )
Figure 8.22: Radial Spectra for Two 50% Random Weights.
(d)
, f9/S-1 = .5, o, = .1875
8.3. ERROR DIFFUSION WITH PERTURBATION
331
+10
0
I--
0Se
. -
-10
50
40
30
20
10
n
-
1
2
a
2
Radial Frequency, f, (units of S-')
Figure 8.22: Radial Spectra for Two 50% Random Weights.
(e) 9 =Ifg/- ~ .7071, ag2 = .25
_____lii·l·_
_·I
_I
332
CHAPTER 8. DITHERING WITH BLUE NOISE
Four Weights
8.3.3
Although better than that which can be achieved with one weight, the transition
regions for the two weight case just examined were still not as steep as desired.
Experiments with various choices of three weights did not produce a significant
improvement.
In trying several combinations of deterministic values in a four element error filter, none proved better than the famed filter of Floyd and Steinberg. In
this section, two variations on this basic filter are presented, both processed
with serpentine rasters. Figure 8.23 shows the results of error diffusion holding the filter values constant but perturbing the threshold by 30% with white
noise. The serpentine raster used in processing is responsible for much of the
directional artifact elimination. The noisy threshold breaks up most remaining
stable texture patterns yielding good radial symmetry at the expense of adding
some low frequency energy.
In Figure 8.24, instead of perturbing the threshold, noise was added to the
weights. For this purpose, the two larger weights (
weights
(
and
) and the two smaller
and 1 ) were paired together. To prevent weights with negative
values, the maximum noise amplitude (100%) is the value of the smaller weight
in each pair. 50% noise was added to each pair in this case.
The radially averaged power spectra in Figure 8.25 (pages 338 through 344
reveal a good blue noise process. Along with low anisotropy and flat high frequency regions, the extra number of weights provided additional low frequency
inhibition and steeper transition regions.
Here, as in many of the other cases considered, the most disturbing patterns
evolve near g
=
.
Unless the power spectra contain impulses at the corners
of the frequency baseband, perturbations to this perfect spatial checkerboard
333
8.3. ERROR DIFFUSION WITH PERTURBATION
· .
·
·...
.
.....
O.'.·
·
~~~~~~~~~~~~~ ~ ~.
·
' .... o°.o
%
.
°.
.
.
°
.
.A.o·...o..·..o..o·oo.'o.**:*...'.°o...."'
*.P9
kne3
OooO****o~o .*.......o~.-***..**:....
-.-°%°.....%..° *:o°%*°**...*
o .*· o-....%.-**
o o.: o......... %..oloo~o
o.o. .o.. oooo ·. ·o°%..o**o ..**......o***..
Figure 8.23: Floyd and Steinberg Filter with a 30% Random Threshold
Processed on a Serpentine Raster.
(a) Gray Scale Ramp, cr = 1.
-r-----··1--111-·i
-·--- 1. ...
--
-·-1111
334
CHAPTER 8. DITHERING WITH BLUE NOISE
Figure 8.23: Floyd and Steinberg Filter with a 30% Random Threshold
Processed on a Serpentine Raster.
(b) Scanned Picture, ar = 1.
_________111_1_______
335
8.3. ERROR DIFFUSION WITH PERTURBATION
;'
.~:~~
.
......
" ;' '";"
>?""
"
'.'"
..
'".
...'....';;;
.....
':'.....
::
::!:"~'::!::i
,.
- .;
.
.
**..*:::
-~'.'.0":.'-"Z.'.:.;
.'....-'.'0;5--.0'/;-'.'
'
-..'.'.'.';;,:.~'--':."
; :.';.::
.''.;.":':
::'.';."':.''.::..'F:~:'}:::';' '';.
: ; '....-::..;..;-:..; :-..:: :...-;~;~.:-.-..;'..;...:.:
A\. *:
:-
~:
:
:::~:
:;~
::::~~::~::~::;
Figure 8.24: Floyd and Steinberg Filter with 50% Random Weights
Processed on a Serpentine Raster.
(a) Gray Scale Ramp, ar = 1.
------
-·II-
336
CHAPTER 8. DITHERING WITH BLUE NOISE
Figure 8.24: Floyd and Steinberg Filter with 50% Random Weights
Processed on a Serpentine Raster.
(b) Scanned Picture, a = 1.
I
8.3. ERROR DIFFUSION WITH PERTURBATION
337
o
cd
._
:.
b:OCn "
Ld
.°.n ' Q
U U
~n
c
C
0o
0
-
bO
____1111
11-·-_4-_1_
I^l·-_IIL·l··ll·--I
-----1--I ._
338
CHAPTER 8. DITHERING WITH BLUE NOISE
I
+ 10
0
0
k¢
-6-
0
a
-10
. IV
17 -All
v----TIvw
wtoov -OXV-v,
, -- -MUP
AAA1
I
-
- -2.8
to"
l
2
V2'
Radial Frequency, f, (units of S-')
Figure 8.25: Radial Spectra for the F&S Filter with 50% Random Weights
Processed on a Serpentine Raster.
(a)
, fg/S 1
.1768, 02 - .0303
g-~~3~~~~~~~~~
339
8.3. ERROR DIFFUSION WITH PERTURBATION
+10
10
54
.eW
0
0 - -
-10
e
-
We'veV
I
C..
-b
2
L2
Radial Frequency, f (units of S l )
Figure 8.25: Radial Spectra for the F&S Filter with 50% Random Weights
Processed on a Serpentine Raster.
(b)
= .25, o
.0586
g[, --fg/S-'
t~{~~~~~~
I-
340
CHAPTER 8. DITHERING WITH BLUE NOISE
I
+10
"0
0o
0
0o
k_
-10
WY
A
.. , U4
IL
9=1A
.
I
VW
. 1\ -
1.99
e.9%
-b
,__
&. L -"
C4^
2
l
727
Radial Frequency, f (units of S - ')
Figure 8.25: Radial Spectra for the F&S Filter with 50% Random Weights
Processed on a Serpentine Raster.
.3495, a 2 ~ .1094
(c) -lg 1'[, IS-'
(c)
---
8.3. ERROR DIFFUSION WITH PERTURBATION
341
_
m
+10
Po
I~~
0
0
.
0
-10
-0 v -.
q~~~~~~~~~~~~~
·
·
v
8_
.vv
.I
n
i
3.9
b
2.9
- 1
.98
i-"s
898.%
Radial Frequency, f (units of S - ')
Figure 8.25: Radial Spectra for the F&S Filter with 50% Random Weights
Processed on a Serpentine Raster.
(d) [g = , f/S-1 = .5, 2 = 1875
4~1
-11^'-----"I '-'---1~
11_1_11_1
342
CHAPTER 8. DITHERING WITH BLUE NOISE
+10
0
o0
0
Mo
To
-10
50
40
-
b
30
Q;
20
10
fo
0
J
2
Radial Frequency, f (units of S'
V2
l)
Figure 8.25: Radial Spectra for the F&S Filter with 50% Random Weights
Processed on a Serpentine Raster.
(e)
HE,
2LI
J fg/S9
.7071, o 2 = .25
343
8.3. ERROR DIFFUSION WITH PERTURBATION
m
+10
0
so
0
00
._.
-10
,4
a%
1
2
72'
Radial Frequency, f, (units of S - )
Figure 8.25: Radial Spectra for the F&S Filter with 50% Random Weights
Processed on a Serpentine Raster.
(f) g[ =
fg/S - ' = .5,
= 4~lg
= 1875
__11
1__1__11___1_)_1
L__ll
344
CHAPTER 8. DITHERING WITH BLUE NOISE
I
m
+10
10
0
-
To
0
-10
A.,.
wTv-
v-
00--Ada A
A6_-%Cn9h4
4p PC46, . pM-r x
vqAIW
I
2.88
C4 M
.a88
8.8%
l
2
72;
Radial Frequency, f (units of S - ')
Figure 8.25: Radial Spectra for the F&S Filter with 50% Random Weights
Processed on a Serpentine Raster.
(g)
= ,
fg/S - ' .3536, Uo2 .1094
(g
g~1
--
8.3. ERROR DIFFUSION WITH PERTURBATION
345
pattern, no matter how slight, become very visible. The fixed background in
the synthesized image of Figure 8.24(c), which is near g =
i
an extreme
example. Rendering gray levels near this value is perhaps the greatest weakness
of the error diffusion algorithm and variations on it.
--.I__1I11I---I
_X-
I_
I-
CHAPTER 8. DITHERING WITH BLUE NOISE
346
8.3.4
Asymmetric Robustness
Figure 8.26 demonstrates the ability of error diffusion to perform reasonably
well on grids with asymmetric aspect ratios. In the coarse example shown here,
ar =
1. This should be compared with the images of Chapter 7 with the same
aspect ratio. As in the other asymmetric examples with a =
the resolution in
pixels per unit area of the images shown is reduced by a factor of 6; the width of
the pixels in the horizontal direction is fixed at the minimum that will survive
reproduction.
While uncompensated ordered dither clearly fails at this aspect ratio, when
properly compensated for ar =
6'
it suffers less anisotropicity than uncompen-
sated error diffusion. The power spectrum of gray levels produced on asymmetric grids by error diffusion will certainly no longer be radially symmetric;
however, because spectral energy is not concentrated at points as in the ordered
dither case, the contortions of asymmetry are more diluted.
Even though error diffusion is somewhat robust in this regard, it could benefit from some compensation. As of yet, the nature of compensated error filters
for asymmetry remains an open problem.
347
8.3. ERROR DIFFUSION WITH PERTURBATION
I II
II
11 1
1
1111111
II III III II I
I . II
IIIIIIll
~~~~~~I
I
1
Figure 8.26: Asymmetric Robustness of Error Diffusion.
Floyd and Steinberg Filter with 50% Random Weights
Processed on a Serpentine Raster.
Scale
(a) Gray
(a) Scae
GrayRamp,
Ramp, a,
ar = (vertical resolution decrease).
__
eI~-^I^I
X---·
l
-^~-
348
CHAPTER 8. DITHERING WITH BLUE NOISE
Figure 8.26: Asymmetric Robustness of Error Diffusion.
Floyd and Steinberg Filter with 50% Random Weights
Processed on a Serpentine Raster.
(b) Scanned
Picture, Picture,
a = 6 (vertical resolution decrease).
(b) Scanned
----
-----------
I
__
349
8.4. HEXAGONAL CASE
8.4
Hexagonal Case
In section 8.1 it was argued that hexagonal grids were inferior to rectangular
grids as far as generating blue noise. This is based on the inability of a hexagonal
grid to support a principal wavelength,
., for
< g <
(see Figure 8.1(b),
page 270). In spite of this deficiency, the other features of hexagonal grids,
particularly its superior covering efficiency (Figure 2.5), are still reason to devote
attention.
The only reported attempt at performing error diffusion on a hexagonal grid
was by Stevenson and Arce [80X, whose error filter was given Figure 8.6(d) on
page 276. They stated that this is the filter which gave the "highest image
quality", but admitted that no optimization was done. The effect of hexagonal
error diffusion with this filter is shown in Figure 8.27. The gray scale ramp
reveals many disturbing texture patterns.
A close look at the Radially Averaged Power Spectrum in Figure 8.28 reflects
the large measure of anisotropy in 7 selected patterns. Perhaps the reason for
the bizarre shape of many of the anisotropy plots is due to the stable texture
patterns that begin to "grow" in regions of constant gray producing localized
spikes in the power spectra.
__II_)^
___I··
___ll·IIL__LIII___I
CHAPTER 8. DITHERING WITH BLUE NOISE
350
...
. . . .
_.
.--.oo........-...o- o. : :o...... :..:......
... :
:::::-:,::::.......
·...............
..........
.... ........- .;.
~ ~ ~ ~~
~'
;
. %
........
; ;
;":"'-...
:"":...
:
."
:.......: :.:-'-''::
. ...........
.':...
::3
;:.."...........................................
' oo..
·
.o o
.-.....
...........
;.....
;.'
~~~~~~~~~~~~~~~~~..........
' %
oo -oo
;';"" :'.
........
.
..
;;...........
....
....... ooo... o ·..
... ' ;; .. ".%%-*" ..:;:.'':....
.'...........
~
.............
..
_
.
.
....
;-
:..~.,=
.-.- "
==o
oo.~
-:o~
-. Tam-*o
°
q-'b
,~'~- ('L~F.~';%.
~~~~
~.
";% "''-1W1
-'-
...............
..
-,,.
·. '
.
t)....
,
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~........
:, .......
q :;:'.'
-,·.'Ffw'.)¢-.-x
.'
,
.. ; : · "~'
............
%;-:;";%;;.%%'.%%%'"
-
-
::;:";":::;:
.
)I..D
..............
~
.~.Y.
. .
,%
.~ ,
· o :%~,'~'~
Figure 8.27: Error Diffusion with the Stevenson and Arce Filter.
(a) Gray Scale Ramp, ah = 2
x
351
8.4. HEXAGONAL CASE
Figure 8.27: Error Diffusion with the Stevenson and Arce Filter.
(b) Scanned Picture, ah = a3.
IC··_ ·I^ ._-·t-_IXII·II---PL --I I-^-------
352
CHAPTER 8. DITHERING WITH BLUE NOISE
· ..
*..
.
.. .
.·
.. ...
..
'. o
.'·
.
'··
·
..
. '
.....
.o.
.·
...
.......... ...........
:
.
"' " :.
-.
.
'...
',
...
·
.',''
.
..
· ..
. ·
.'.
.
.
.
.
...
.
.......
.....
..
.
'
.''',
.
'
.
..
',','-'
.
· ·
.
...
.
.-~.
.
..
I-
m
+10
0o
o
0
-10
b
-
2
3
Radial Frequency, fr (units of S- 1 )
Figure 8.28: Radial Spectra for the Stevenson and Arce Filter.
(a)
,g~-~~~~~~~~~~~
fg/S - 1 .2041, Ua2 .0303
353
8.4. HEXAGONAL CASE
*::;... :
·
p
o0
1
0
<
':72
"
oo
. ..... .......... :'.""
:
o°.·
o
·..
.
,.,.oo..·
.
+10
0
- 10
eb
'-z
__
2
_3
_
Radial Frequency, fr (units of S-1)
Figure 8.28: Radial Spectra for the Stevenson and Arce Filter.
-1 .2887, o 2 ; .0586
f
fgS,]
(b)[ =
Lil1i'
_·
_r I1^___I_·_
_
II
__
354
CHAPTER 8. DITHERING WITH BLUE NOISE
..
:
...
..
..
..
.,......., .
::oo...:::
:
..
. .. .
..
.
.
.......
:,
.
.;.
...
~~~~~
.
.
.........................................................
..
..
....................
............
. ................
......
.
-~~~~~~~~~~~~~~~~~~~~.:
... 2. .;.'...°..".
.. . i....................................
'
-...
."..:-......:-......-.......:::-.............................................
· . ,...-.:'
;::,.'.-. ;:'4 .
; .,,"~.
"·-"-'....',
... .' . -...
.. ............................................
°" ..
- .~ .~'o..,~
°.° ,o'%o. ··"..........
.o °-°%
..
. .
-.
...
:;;:~~~~~~~~~~~~~~~~~~~.:
;. ..... ;:..:.·.;:..,
;.1.:
........................... :.;
.:
'
.
.. ....... . ,.";.. - .:..
...
...
..........
....,.- '...~#.
.. · ,,.';
-..
'°
".......
."..,r
1..'.~ .........
...................
: .o
..
..
..........
. . .... ...........................
.................
.....
'-".'-';'",'."
::.--.--....-........
...;--';./';-;'
,:.;-': ."....
;.;......
.....·o4°."
-....;...
o-.. .'.'":.'
.--.-..
.;""..o~,
"." o".°..
o. · ·......
· ~.o. ·°%°
·
o
o . ".%
r
o~o
·
·
o.
o
.o
.. .....
. .................
.
· ·
...
·..
........................-..... .4.
-.....
...........................
".'-
-. .............................-
.
..
... ..o . . .
¢ o . o ..
................
;"."~.,;
....;'.-;
..,.;%
....· ......
.... ,'.; ... :....:....-. o.· · ;...:
'. ,.,:
.-....-~
...
: .....
::...:..-.:
::::. ......-........................
..........
.......... ...................
..
o-P.
.;·-.-...
.......~~~~~~...·
·. ;;.o
... .- .
· -%~~~~~~~~~~~~~
,,·.o
I
._
m
+10
o
0
0
.-10
eq C3%
2
3
73'
Radial Frequency, f (units of S'
l)
Figure 8.28: Radial Spectra for the Stevenson and Arce Filter.
(c)C) = 8 , fg/S .4082, a 2g .1094
8.4. HEXAGONAL CASE
m
355
+10
10
0
$e
-6--
0
an
*0
-10
g
3
Radial Frequency, f (units of S-')
3
Figure 8.28: Radial Spectra for the Stevenson and Arce Filter.
(d)g , f/S - = 2,-3,
C
.1111
1II
.Y-LIIII*Y·-·D·I·IYL-.-.)LI*L^
I-··- -IL_*
_ I...
.-
_·____ _L_-~ -
_
356
CHAPTER 8. DITHERING WITH BLUE NOISE
74X 610111F,
YJ
, "' 1:5 W. - !
m
+10
Po
.4a
0
an
-10
-a
4
.-
2
3
Radial Frequency,
fr
(units of S - 1)
Figure 8.28: Radial Spectra for the Stevenson and Arce Filter.
(e)
=
(fg is not supported), a2
= .25
8.4. HEXAGONAL CASE
5
l
g
357
_g__
C[
iS
I
C 5
|W
__W
I
E__
I
_
{
____________
Ell
_____...
x
__w___
a
______B__-fi
1
.
I
l
I
;
:
.
t
+1 0
0
-6.;o10
0
0
B
|I
!5
%
7
_
E
IE
=
_..____.
sr
_
w_
Rns
_v
M___________
IE
a
___
s___
z
!s
m
w__>
E
i
___Z_
n
a
E
.
Z
1
|
R
w
_
=__
_
n_
'
X
Y
E
W]i
n__
a
g
c
W
BE
R
,
,
,
,,
m
S
ea
v
4
s
__
z
__
.
_
_s
_.____
|
D
-
=
_
B
su=
.
-
...
Y
*
-
- 10
N
f9
7
Radial Frequency, f (units of S
-1
3
)
Figure 8.28: Radial Spectra for the Stevenson and Arce Filter.
(f)(f)
2,
3 ,/ -f/S1
=
,g--
1____1141__1_111_11--_1111_
lt·-·i^l----_-3--·I·LI·-LII-I
2
=
.1111
____·_____11_11___-I---I
-·11·---
CHAPTER 8. DITHERING WITH BLUE NOISE
358
m0 +10
0
0
Go
..4
-10
2
v/i
3
Radial Frequency, f, (units of S-')
Figure 8.28: Radial Spectra for the Stevenson and Arce Filter.
.1094
fg/S-1' .4082, o2
(g)g =,
W -
j
8
I__
·
359
8.4. HEXAGONAL CASE
(1 X
t2X
*
1
1
(a) Two Weights.
7
16x)1635
3
5
1
(b) Variant of Floyd and Steinberg Filter.
Figure 8.29: Two Hexagonal Error Filters to be Perturbed.
("i" represents the origin.)
8.4.1
Error Diffusion with Perturbation
It turns out that the most successful methods of halftoning by error diffusion on
rectangular grids works quite well on hexagonal grids after adjusting the error
filters slightly. Figure 8.29 displays the deterministic part of the two and four
element filter that will be demonstrated in this section.
Parallel to the methods used for Figures 8.21 and 8.24, these filters will
have 50% noise added to their weights, and processing will be on a serpentine
raster. The result of error diffusion with the two element filter is illustrated in
Figure 8.30; the result of the four element filter is shown in Figure 8.31.
Both render gray levels in a much more isotropic manner than that of Figure 8.27. The large 12 element filter did sharpen the picture more, however.
Again, as was argued in the rectangular case, sharpening should not be compounded with halftoning.
In the next chapter, an example of how precisely
controlled sharpening can be achieved in a separate operation for both rectan-
I-. -III~IYPIY~--------
-~...11--1_
__I
·-
_1
1_
1- 1~-
360
CHAPTER 8. DITHERING WITH BLUE NOISE
K
:Y
q
.:....·
.':'.::·-:.'~:.-....,,~.:.~.-,..·'...::'.:...:~:?-,-..:-...:
.. -'-:',','::..:-.- .-.
'~~·.~ :...~-.·
..... :,',..:%
:-.':..:~~~',:~;;:. :,·,M',.-?.:,.
:;:.--.:.-:-....:
.,--.
:::.. :-:, .'~'.'-,-,~.~.,
',.~.,'.. ~%-¢.';t4'.--.·:·;;'.0
.. 'c;.:,;:·::.i:
;c:
I
·..~:'~~.?.~
:~.':~:?~ :.:~:::...~~~:~.....~:~.~..:..:.;::.;.:...~N.~~:EV~::i::/.?n:.
,: .,;;,~--:;..~!;~
.' ·. .;/-,,,.::,:,,~~,~:q,,,.,,,:, :::::~; : ~ .:q..~':.~:'::::;-.
%.:F'
~~$~ ;~~ :~:::~: ',c' i' .:b' ~~~
; )f.f, ' 4
Figure 8.30: Effect of Two 50% Random Weights on a Hexagonal Grid
Processed on a Serpentine Raster.
(a) Gray Scale Ramp, Cth = v/i.
2.
8.4. HEXAGONAL CASE
361
Figure 8.30: Effect of Two 50% Random Weights on a Hexagonal Grid
Processed on a Serpentine Raster.
22
(b) Scanned Picture, ah
-I--·li--^
__1_
----
·
-.
-IIC
K~
I
_YL-I·-·--_l-·-
)-_
I
362
CHAPTER 8. DITHERING WITH BLUE NOISE
I.~
~ ' ~~~.
. ,' .
·
o
"
· ' .'''
.', .......
'.-...-.'.'"'
.'.''..'-..',..
· o Wa':I
~~~~~~~~~I
·
o
V.....
.
.
.
,..',.'X
..
·
...
o.,
o
.et
.-.: -..'':.' .:--:
' ' "...'-.
-,' :'.'.':...::.:
:' -''
.-'.:..'.:'.':-':::-:.'"..:.'.'-'..y::::.-:- :.~ .' -:' -$-;.: :.": ::-:v--':-- --
Figure 8.31: Effect of Four 50% Random Weights on a Hexagonal Grid
Processed on a Serpentine Raster.
(a) Gray Scale Ramp, ch = 2.
363
8.4. HEXAGONAL CASE
Figure 8.31: Effect of Four 50% Random Weights on a Hexagonal Grid
Processed on a Serpentine Raster.
(b) Scanned Picture, ah = 2 .
___II_·LL^a.lll1_-_ll-
---YII.--UL--.
·--- I
11 _1_·1_1·1__1_·_^___I__IIICU__II
··-
-·-X-----
364
CHAPTER 8. DITHERING WITH BLUE NOISE
gular and hexagonal girds.
The Radially Averaged Power Spectrum of both the two and four weight
stochastic hexagonal filters are well behaved with the four weight case having
sharper transition regions. So this case is included in Figure 8.32. Notice how
well the peak of these spectra follow the principal frequency (marked as usual
with a diamond on the frequency axis). The important exception is for g =__1
which is in the region where the hexagonal grid cannot support a principal
frequency.
Two other particularly interesting cases are those for g =
and
- .
The principal frequency for these gray values is at the high frequency limit for a
hexagonal grid, f =
.
These cases are similar to the g =
1
case for rectangular
grids.
In balance, anisotropy is generally low for the stochastic hexagonal filter,
not nearly as wild as in Figure 8.28. Outside of the range 12
I<g
are also good blue noise generators.
<
,
such filters
8.4. HEXAGONAL CASE
365
I
1-
+10
To
m
1.0
0
0
·i-,
-10
401V-4-049
I
k
T
2
3
Radial Frequency, f (units of S - 1 )
Figure 8.32: Radial Spectra for the Stochastic Hexagonal Filter.
(a)
,L~~~~~~23i'~
f/S~ ~- 1~ .2041, r 2 .0303
III_---_I-·ILPIIY----I
-^-l--ll)-yllll--
·Il)ll·-··IIIIIIIY·
11·--114--
^-
L--
II·(·C·
-
-
-l
I-
CHAPTER 8. DITHERING WITH BLUE NOISE
366
· :--."'".':-:.'.:...
m
.- ' · .' .- -.
+10
04
0o
:4
-6.
0
0o
-10
- 10
-=&
A'!:%PM
-- A*.Ao- --Atcs-
--,mAPA*
-
-- av -
-
C-
2
73'
Radial Frequency, fr (units of S
i3
1)
Figure 8.32: Radial Spectra for the Stochastic Hexagonal Filter.
.0586
, f/S-'
.2887, , 2
(b)
8.4. HEXAGONAL CASE
m
To
To
0
U,
·-la,
367
+10
0
-10
, AO , "-/'
vy v
-A-.-
I
2.M
3.8%,
~3
Radial Frequency, f, (units of S-')
I
3
3
Figure 8.32: Radial Spectra for the Stochastic Hexagonal Filter.
(c) [
,= fg/S' z .4082, a 2 .1094
g_[_~ 9
~~~~~~~~~~~~~~~-
-·--------
368
CHAPTER 8. DITHERING WITH BLUE NOISE
+10
0
I-
-
0
e._
-10
cl
h
t
C-
73'
Radial Frequency, f, (units of S-')
Figure 8.32: Radial Spectra for the Stochastic Hexagonal Filter.
1
(d)[g=
2 .1111
_ _ _ _ _~,
_3 fg/S_ = §,, Y3
9
3
8.4. HEXAGONAL CASE
4
IV
1-
369
+10
0
4a
0o
0
.e4
-10
eq
-
b
I-
4~
73
r
3
Radial Frequency, f (units of S - 1)
Figure 8.32: Radial Spectra for the Stochastic Hexagonal Filter.
(e)
= , (fg is not supported), a o = .25
(e)
9 ~1
__I___II____III^__
·II(·-·IIIYI1(I^L_
· _IIX-- --^ ·-P--·l -·---·-C
370
1-
o
la04
0
CHAPTER 8. DITHERING WITH BLUE NOISE
+10
0
4
o
eno4
Z_
-10
PA-T-.A
v %j
73'
Radial Frequency, f (units of S)
Figure 8.32: Radial Spectra for the Stochastic Hexagonal Filter.
(f)g
=_ 3_ fg/S 1 ' (Tg2,2 .1111
__
,
3
371
8.4. HEXAGONAL CASE
·
1
0
7
1
·
+10
04
;e
4a
0
-10
.
.
Wc--
73'
i3
Radial Frequency, fr (units of S-')
Figure 8.32: Radial Spectra for the Stochastic Hexagonal Filter.
1
.1094
.4082, a2
,g_~~~~~~~~~~~~
= /Sfg
(g)
----·1C________I_... .. _Il-^----Y1
__
372
d
CHAPTER 8. DITHERING WITH BLUE NOISE
I
Chapter 9
Concluding Remarks
For completeness, this chapter reviews some of the other halftoning related
neighborhood process. In particular, the effect of presharpening an image with
a digital Laplacian is demonstrated for both rectangular and hexagonal grids.
Patterns from each of the major classes of halftoning are compared, with blue
noise found to exhibit a grid defiance property. The details on extrapolating
halftone methods for use on multi-bit displays are explained, and suggestions
for future research is presented.
At the end of this chapter, a summary of the major points of this text is
provided with recommendations.
9.1
Other Neighborhood Processes
Besides the popular method of error diffusion, other neighborhood processes
have been presented that use a local average to modify the halftone process.
Jarvis's "Constrained Average" [381 algorithm is one such scheme. The success
of the algorithm depends heavily on the digitization noise inherent in the picture.
373
__II. ..-....·.--LIIIIII·L··l*··*lpl·lll
-
-I.._--l·-Y·--^·----·
·------- -ll---L----·------1--III
CHAPTER 9. CONCLUDING REMARKS
374
The threshold at any point is a function of the local average, dependent on the
probability density function of the noise. If the noise level is too low, artificial
"noise" must be added. Lippel [491 later complained that this algorithm was
nothing more than a standard edge enhancing technique and that the value of
its halftoning ability depended on the nature of the added noise.
Roetling [64] modified the classical screen by modulating the amplitude of
the screen as a function of the local average; he called the algorithm ARIES,
for Alias Reducing Image Enhancing Screen.
White [90] of IBM suggested separating the low and high frequencies and
processing each separately. The low frequencies are represented by a clustered
dot within an 8 pixel cell, which allows only 9 gray levels. A 3 by 3 Laplacian operator generates the high frequencies. An improvement on this idea was
reported [8] that used a "nonlinear Laplacian" to reduce moire patterns when
screening scanned classical halftones, and a larger (18 pixel) cell to better represent the low frequencies with less contouring. Steve Silverberg [771 experimented
with this algorithm on a color ink-jet printer.
9.1.1
Sharpening
As explained in section 1.2, the virtues of a halftoning scheme should be decoupled from its ability to sharpen. The improved output perceived from a method
that intrinsically sharpens can misleadingly outweigh other shortcomings in its
ability to render gray levels accurately and without algorithmic artifacts.
Sharpening does improve, or at least defeats, unsharpening degradations
that halftoning imparts. The proper degree of sharpening is a subjective quality
and can easily be controlled independently of halftoning. Sharpening can be
combined with interpolation, that is, in the retrospective resampler of of the
9.1. OTHER NEIGHBORHOOD PROCESSES
375
image rendering system of Figure 1.2 (page 24) for the case of digital enlarging.
The sharpened Gaussian [73,74] is often used for such a dual purpose.
When sharpening is not combined with resampling, a separate high pass
filter operation is needed. Perhaps the most popular high pass filter used for
the purpose of sharpening is the digital Laplacian. The Laplacian, V 2 , is an
operator which produces the second spatial derivative,
V 2 J(x) =
a 2 J(X)
(92 J(x)
(x
+
(-
(9-.1)
When applied to an image, it produces large amplitudes at edge locations, and
zero in constant or uniformly varying regions (regions where the zeroth or first
derivative is zero).
This operator can be described as convolution with the filter V 2 6(x). The
discrete-space analog to this on a rectangular grid is the 5 element filter shown
in Figure 9.1(a), although the 9 element variety achieves a similar effect. A
hexagonal version is shown in Figure 9.1(b).
Denoting a digital Laplacian filter as T[n], sharpening is achieved by subtracting the Laplacian filtered image from the original image,
Jsharp[nl = J[n] - ,3[n] * J[n].
(9.2)
The amount of sharpening is controlled by the value of /3 > 0. A tradeoff must
be made between the accentuation of edge detail and amplification of noise.
An example of the effect of presharpening in this way on a rectangular
grid with
d
= 2.0 is displayed in Figure 9.2. It should be compared to that on
page 336 which was identically halftoned without presharpening. The hexagonal
example in Figure 9.3 is the prefiltered version of that shown on page 194.
Adaptive sharpening techniques exist which are not as sensitive to noise but
are, as one might expect, more compute intensive.
---_11_-·11. --.11111·rm··1·-LI
ll---P-
---_·-1111
il--·1
-)--IIIII·IW-^tC-__---)--·II
11I
---L
376
CHAPTER 9. CONCLUDING REMARKS
4
4
1
1
4
8
-1
1
4
or
1
8
8
-1
1
1
1
1
8
8
8
1
$
1
4
1
(a) Rectangular.
1
6
1
6
-1
6
1
6
16
1
6
(b) Hexagonal.
Figure 9.1: Digital "Laplacian" Filters,
_
___
__
I_
__
__
i[n].
9.1. OTHER NEIGHBORHOOD PROCESSES
Figure 9.2: Sharpening with a Rectangular Laplacian Filter.
Laplacian amplitude, s3 = 2.0.
Halftone by error diffusion. Compare Figure 8.24(b) (page 336).
377
378
CHAPTER 9. CONCLUDING REMARKS
Figure 9.3: Sharpening with a Hexagonal Laplacian Filter.
Laplacian amplitude, = 2.0.
Halftone by ordered dither. Compare Figure 6.13(e) (page 194).
9.2. BLUE NOISE IS PLEASANT
9.2
379
Blue Noise is Pleasant
Figure 9.4 collectively compares greatly enlarged portions of the four major
classes of dither patterns arranged in order of increasing correlation (decreasing
1 and thus
entropy). All patterns are representations of a fixed gray level, g = 6,
all have roughly the same number of black pixels.
While white noise appears too random or "noisy", ordered dither appears
too "structured". The purpose of a dither pattern is to represent a continuoustone level. It therefor should not have any form or structure of its own; a pattern
succeeds when it is innocuous. Blue noise is visually pleasant because it does
not clash with the structure of an image by adding one of its own or degrade it
by being too "noisy" or uncorrelated.
Blue noise even defies the structure of the underlying grid. Even though
the dots in Figure 9.4(b) are perfect squares, each precisely aligned to a given
position on a rectangular grid, the collective ensemble tends to destroy this
rigorous alignment creating a grid defiance illusion.
For many years, noise with 1/f power spectrum distributions have been
known to exist in electrical systems. But recently, remarkable discoveries have
repeatedly confirmed the existence of a 1/f power spectrum in almost every
aspect of nature [28,51,89] including such things as variations in sunspots, wobbling of the earths axis, and flood levels of the River Nile. Evidence of 1/f fluctuations in human biological systems [56] has also been found; a 1/f spectrum
was found in electroencephalogram (brain wave) measurements when subjects
CHAPTER 9. CONCLUDING REMARKS
380
* *
*
*
*
_
*
"u
*
l
N*
* ..
*
**
v
_
U
.
.
%'
*I.
*:
1-
.
*
*:ie**n***m
e
.
.Me
*
it
e
mEml
,'1|'
'
*r
*
U
(a) White Noise.
·
0~~~~~~
in
order
(b)*EU
Noise.
~ UArranged
U Blue
I*ofI decreasing
U I*
*
E*
**entropy.
**~~~
*
*
* .UU· U
U
U * 'U.. U
U
U
U
U
U
U]
*~ ~ U
U
U2
*
*U U U·
U
U
U
U
I
I
*~ ~U·
*
UU
U
U
I IU ·
U
II U
U
UiI
U
U
U
·. *
**
*
*
(c *Dipre-o
U
E*
.M
* ~M
U·
*
*
· EE*
~~
*
·~
'U
UI
*
~
I
m· U
'
.
U~~~~
**
Oree
*U
*
mm*.
*
ml U
Diter
*
11'
U
*
*
.
EU
.
Ii
*
U
l
I
II
I
*
.
U
·
·
( dClustered-dot
Clu
)(b e()(d)
e - o Ordered
r e e Dither.
Di t he r.
Arrnge insorers-of dereasn enrop.
Figure 9.4: Comparison of Halftone Patterns for g =9
·
9.2. BLUE NOISE IS PLEASANT
381
were exposed to "pleasing" stimuli.
A study by Richard Voss [87,881 has found that practically all forms of music
possess 1/f noise. Experiments with stochastic music composition revealed that
listeners found 1/f music far more interesting than white (1/f
0
) music, described
as "too random", or brown (1/f2) music, described as "too correlated".
Blue noise can be described as the "pleasing" complement of 1/f noise. The
dominance of low frequencies in 1/f phenomenon is responsible for its interesting
and natural structure. Blue noise, by contrast, is not "interesting"; nor is it
annoying. Being devoid of low frequencies and localized concentrations of spikes
in the frequency domain, it has no structure and thus does not interfere with
the interesting features of that which it is representing.
CHAPTER 9. CONCLUDING REMARKS
382
9.3
Beyond Binary Displays
As stated in the Introduction, halftoning or dithering can be used to redistribute
coarse quantization noise whenever the number of gray levels to be displayed
exceeds that which can be accommodated by the display device.
of binary displays has been the focus of this report.
The case
The results are easily
extrapolated to m-ary or multi-bit displays.
For systems of more than one bit, dither is not as critical as in binary
displays. Thus not much attention has been paid to it. The old idea of random
dither 29,62] works fairly well and is widely used in practice. Nonetheless, just
as dithering with white noise proved to be inferior to other halftoning techniques
in the binary case, dithering m-ary displays can also be improved.
For ordered dither, the threshold value was assumed to vary from 0 to 1 with
the relative magnitudes governed by some threshold array. This process could
be equivalently described by the relation,
I[n] = int{J[n] + D[n]}
(9.3)
where Dfn] is a dither signal comprised of the ordered dither threshold values,
int{} represents integer truncation, and as usual, JnJ is the continuous-tone
image, and I[n] is the halftoned image equal to either 0 or 1.
If the display could render 4 true levels (2 bits), dithering could be expressed
as
I(2 )[n] = -int{3J[nJ + D[n])
3
___
(9.4)
9.3. BEYOND BINARY DISPLAYS
where the quantized output, I( 2 )[n]
383
[0, ,
,
1. In general, ordered dither for
a K bit display is
1
__
I(K)[n]
2
i:
1mnt{(2
(9.5)
- 1)J[n + D[nJ}.
This method assures that an input image, J[n], with a uniformly distributed
histogram will result with the same discrete histogram as quantizing with fixed
thresholds.
The output levels of a fixed K-bit quantizer would included 0 and 1 with the
remaining
2 K-1
levels equispaced between them. The
2K -
1 threshold levels
are equispaced between the output values. Such a quantizer would replace the
fixed threshold of a K-bit error diffusion algorithm. The other aspects of the
algorithm, along with all of the permutations of Chapter 8 remain unaltered. It
is interesting to note that as early as 1969, an m-ary error distribution algorithm
similar to error diffusion was demonstrated on a multiple exposure microfilm
recorder [75].
The most efficient way to distribute a finite set of gray levels in a m-ary display device is not uniformly in reflectance or luminance, but in uniform steps of
perceived brightness, which is quasilogarithmically related to luminance. If the
Physical Reconstruction Function of the device has such properly separated gray
levels, the compensation should be handled in the Tone Scale Adjust portion of
the display preprocessor (Figure 1.2, page 24) and not in the quantizer.
Color
These same dithering techniques are also easily extended to multi-spectral displays. By far, the most studied case is that of four color printing with the
classical screen as discussed in section 5.2.2. For dispersed-dot ordered dither,
_____ ^1111111l-11-11
--.·-_--1
--
CHAPTER 9. CONCLUDING REMARKS
384
the common practice is to simply halftone each component color independently
and overprint. The problem of morie patterns due to beat frequencies between
overprinted patterns is not appreciable in this case.
Using error diffusion on component colors has also proved to be free of beat
frequency concerns, however, output is sometimes described as "muddy". Perhaps experiments with overlaying combinations of dispersed-dot ordered dither
and blue noise processes would yield improvements.
In general, quantizing and displaying color images while faithfully preserving chromaticity is a complex and delicate problem. The fundamentals of color
hard copy are presented by Yule [931, Heckbert [331 reviews some color quantization issues, and Engeldrum [211 discusses the variables that contribute to
color gamuts of dot formed printing systems.
However, most of these issues are often ignored in the implementation of
computer displays. A common application involves dithering each component
of a source image coded with the additive primaries-red, green, and blueresulting in the 3 bits (bRbGbB). A color hard copy device can typically display
the three subtractive primaries-cyan, magenta, and yellow-with each pixel
represented by the bits (bcbMby). The simple transformation
bc
=
bR
bM
-
bG
by
=
bB
where b refers to the logical complement of b, is used to produce the data to be
printed.
I
I
9.4. SUGGESTIONS FOR FUTURE RESEARCH
9.4
385
Suggestions for Future Research
Several interesting open problems remain to be addressed on the topic of digital
halftoning. A few suggestions are presented in this section.
Many research opportunities exist in the investigation of blue noise processes.
A considerable improvement in the error diffusion algorithm was realized by
introducing the simple perturbation of processing on a serpentine rather than
a normal raster.
Perhaps experimenting with other space filling curves will
uncover even more improvement. Practical candidate curves would be those
that transverse N line buffers at a time, for various values of N.
Adjusting the blue noise techniques described in this work for use on asymmetric grids needs to be considered. The new metrics of radially averaged power
spectrum and anisotropy were developed for symmetric grids only. They should
be generalized for all semi-regular grids. It is possible that other algorithms
besides variations of error diffusion can be found that prove to be good blue
noise generators.
The quality of the images resulting from the techniques presenting in this
work should be quantified through formal perceptual experiments. The tradeoffs
between many parameters can be made more meaningful in this way. For example, such experiments would contribute to the establishment of guidelines for
selecting the threshold period size for a given resolution display using ordered
dither, or the optimum degree of sharpening required for a given halftoning
technique and Physical Reconstruction Function.
_1__..___'_.__
-..
386
CHAPTER 9. CONCLUDING REMARKS
In the case of m-ary displays, an important practical concern is the tradeoff
between spatial and amplitude resolution and its effects on image quality. Experiments should be performed on such displays with various halftoning methods, holding the total number of image bits constant.
The process of halftoning on color displays also needs to be carefully investigated. Such displays should be modeled with an expanded Physical Reconstruction Function with a well defined color "Tone Map". It was already suggested
that halftoning component colors with combinations of ordered dither and blue
noise techniques should be tried to minimize intercolor interference. The effect
of such combinations on chromaticity and perceived quality should be studied.
9.5. SUMMARY
9.5
387
Summary
A comprehensive investigation of new and old techniques available for digital
halftoning has been presented. The corresponding digitally produced examples
can serve as a catalog for evaluating the tradeoffs between methods.
Halftoning is but one component in a total image rendering system. The production of high quality images requires a preprocessor tailored to the peculiarities of a target Physical Reconstruction Function. Before halftoning, an image
must be appropriately resampled, tone scale adjusted and optionally sharpened.
A clustered-dot technique should only be used if the Physical Reconstruction
Function (which might include a reproduction process) cannot accommodate
isolated black or white pixels. Otherwise, a dispersed-dot method should always
be used.
In this text nonstandard grid geometries were considered. The "aspect ratio
immunity" argument revealed that semi-regular hexagonal grids outperform the
best covering efficiency of rectangular grids over an order of magnitude of aspect
ratio. Dot overlap decreases with improved covering efficiency, so tone scale can
be more precisely controlled on hexagonal grids.
The advantages of hexagonal displays can be realized in most cases without
converting all storage formats in an imaging system to hexagonal form; some
classes of images such as text are better rendered on rectangular grids [61]. For
the case where a rectangularly sampled continuous-tone image is to be produced
on binary displays of lower resolution, subsampling could be performed in a
__1111
_1_·
X _
I_
·_
__I_
CHAPTER 9. CONCLUDING REMARKS
388
hexagonal fashion before halftoning. As was pointed out earlier, the hardware
modification needed to convert a rectangular display to a semi-regular hexagonal
one is quite simple; an alternate line offset of one half sample period is all that
is required.
The point process of ordered dither was generalized for both rectangular
and hexagonal grids by the Method of Recursive Tessellation. In the case of
asymmetric grids, the "immunity" of hexagonal grids can be used even in the
case of rectangular grids. A set of guidelines defining the number of ternary
replications required in the generation of asymmetric threshold arrays to create patterns with maximum symmetry as a function of aspect ratio has been
presented.
The concept of dithering with blue noise along with its "grid defiance" property was introduced. Conventional methods of error diffusion with previously
reported error filters were closely examined and found to be fair blue noise generators. Experiments with a broad array of perturbations found that excellent
blue noise patterns could be achieved with error filters of four or fewer weights
when noise was added and processed on a serpentine raster. While more computationally expensive than ordered dither, a blue noise technique should be
used when ever the resolution of a display device is so low that every pixel is
clearly visible. Being devoid of low frequencies, it does not interfere with the
structure of the image it is rendering.
However, dithering with blue noise on hexagonal grids was inferior to rectangular grids as predicted from theoretical arguments in the frequency domain.
This is perhaps a surprising result, considering the success of ordered dither on
such grids.
The nature of the patterns resulting from halftoning uniform planes of con-
389
9.5. SUMMARY
stant gray level were evaluated in the frequency domain for rectangular and
hexagonal grids. New metrics for analyzing the frequency content of both periodic and aperiodic patterns were developed. For the periodic patterns resulting
from ordered dither, insight was gained by displaying Composite Fourier Transforms in the form of "exposure plots". Radially average power spectra combined
with a measure of anisotropy were used to evaluate aperiodic patterns.
This collection of halftoning schemes should prove useful in the implementation of image networks that employ a wide variety of practical displays.
__
_··L_1
I
I
_
I~I-
·
-
390
CHAPTER 9. CONCLUDING REMARKS
Glossary of Principal Symbols
The following notational conventions are followed in this report:
* Bold lower case symbols are vectors.
* Bold upper case symbols are matrices.
* All other (script) symbols are scalars.
* Arguments of discrete-space signals are enclosed in square brackets,
l.
* Arguments of continuous-space signals are enclosed in parentheses, ().
This alphabetically arranged glossary presents Greek symbols first, followed
by English symbols. The number following each description is the page number
in this text where the symbol is defined or illustrated.
aIt
~
Aspect ratio. It equals the Sample Period divided by the
(41)
Line Period, S/L.
df~
~
Magnitude of a Laplacian to be subtracted from an image for
the purpose of sharpening. It is proportional to the degree
(375)
of sharpening.
6( O
Impulse function. Has the property of being equal to zero
everywhere except where the argument is zero, and has unit
"area", f
6(x)dx = 1. Bracewell 13, ch. 5 has a good
discussion of this function.
391
1_1_
1_1^1___1__·_11_1_1_I·
1 C--
--
GLOSSARY OF PRINCIPAL SYMBOLS
392
A^
~
Width of the annuli over which the Power Spectrum Estimate, P(f), is averaged to form the Radially Averaged Spec(76,78)
trum, Pr(fr).
r17 ~
Order or size of an ordered dither threshold matrix which
was generated by the method of recursive tessellation. Rectangular and hexagonal matrices have 27 and 3 elements,
(154)
respectively.
~
Number of ternary subdivisions and replications necessary
to compensate an ordered dither threshold array for use on
(225)
an asymmetric grid.
Ktc
Ag
r2
Principal radial wavelength in a homogeneously distributed
field of binary pixels representing the constantgray level, g.
(268)
Variance of an individual output binary pixel resulting from
(75,81)
halftoning homogeneous region of gray level g.
I[n]
Digital Laplacian filter used for presharpening an image
(375)
prior to halftoning.
b(x)
Global Background. The linear but shift-varying global
background luminance contribution in the Physical Recon(34,35)
struction Function.
c(x)
Output of the Discrete-toContinuous-space image.
(34,37,65)
Continuous Space Converter.
C(f)
Continuous-space Fourier Transform of image c(x).
CE(f)
Composite Fourier Transform. The average of the Fourier
transform magnitudes of all possible gray level renderings
produced by thresholding with a particular ordered dither
(73)
matrix.
d(x)
Dot function. The linear, shift invariant function which
models the visible output pixel in the Physical Reconstruc(34,47)
tion Function.
(65)
393
GLOSSARY OF PRINCIPAL SYMBOLS
e[n]
Error Filter. Governs how past quantization errors are negatively distributed or "diffused" into the yet to be quantized
(275)
image in the error diffusion algorithm.
E(a)
Efficiency of a periodic sampling grid as a function of aspect
ratio. Defined as the ratio of pixel area (see pages 39 and 38
for definition of pixel shape) to the area of a circumscribing
(44,46)
circle.
f
Continuous-space frequency vector, f =
fg
Principal radial frequency in a field of homogeneously dis(268)
tributed binary pixels representing gray level g.
Ifr.
[ ]
(65)
.
Radial frequency. The scalar distance in frequency units
from zero frequency in a two dimensional Fourier Transform.
(80,78)
g
Gray level. It has a continuous value in the range 0 (white)
(56)
to 1 (black) inclusive.
I[n]
Quantized discrete-space image. The output from a halfton(24,23,34)
ing process.
I[n;g]
The binary output image resulting from halftoning an image
(73)
consisting of a fixed gray level, J[n] = g.
I[k]
Discrete Fourier Transform (DFT) of I[n].
I[k; g]
The DFT of the binary output image resulting from halftoning an image consisting of a fixed gray level, J[n] = g. (73)
IE[k]
Composite DFT. The average of the Discrete Fourier Transform magnitudes of all possible gray level renderings produced by thresholding with a particular ordered dither ma(73)
trix, that is, the average of I[k; g] for all g.
I(x)
Visible Image. Output of the Physical Reconstruction Func(24,23)
tion.
J[n]
Continuous amplitude, discrete-space image.
halftoning process.
-I_
(67)
II·_
Input to a
(24,23,34)
--- *l----L-^l
I
._
-
I·-
GLOSSARY OF PRINCIPAL SYMBOLS
394
Discrete-space frequency index vector, k = [ k i
k
k2'
kl and k2 are integers.
]
(67)
L
Line period.
(38,39)
£
Luminance. That photometric quantity which is perceived,
having units of lumens/area/steradian.
(56)
n
Discrete-space spatial index vector, n =
[n ]
rt 2
nland n 2 are integers.
Nr (fr)
Number of discrete frequency samples in an annulus about
radial frequency f.
(80,83)
P1,P2
Spatial period replication vectors.
P
Spatial period replication matrix, [PI SP2 .
P(f)
Power Spectrum. In this report, only the Power Spectra
of binary output of aperiodic halftone processes on a single
input gray level are considered.
(75)
P(f)
Power Spectrum Estimate.
Pr(fr)
Radially Averaged Power Spectrum. Sample mean of the
frequency samples of P(f) in the annulus, Ilf I- fl < A//2,
about f.
(80)
ql,
Frequency sampling vectors.
q2
(62,70)
(66)
(76)
(66,70)
Q
Frequency sampling matrix, [ql q 2 j].
R
Reflectance. The ratio of reflected to incident radiant power.
(66)
(56)
S 2(fr)
Sample variance of the frequency samples of P(f) in the
annulus, If - fi < A/2, about f7 .
(80)
s2 (f)/Pr2 (fr)
Anisotropy of P(f).
(80)
S
Sample period.
(38,39)
GLOSSARY OF PRINCIPAL SYMBOLS
395
U1 , U2
Frequency bandpass replication vectors.
U
Frequency bandpass replication matrix, [l
V1,v
2
(65,70)
U 2 ].
(65)
Spatial sampling vectors.
(37,70)
Tv2].
V
Spatial sampling matrix, [v1
var{P(f)}
Ensemble variance of the spectral estimate.
w (x)
Dot noise. The visual noise local to a single pixel in the
physical reconstruction function.
(34,35)
x
Continuous-space spatial vector, x
(37)
=
[ 1]
(76)
(39)
.
Z2
z
Number of elements in an ordered dither threshold array.
(71)
____I
__
-_
I
1_
1
__
396
GLOSSARY OF PRINCIPAL SYMBOLS
References
[1] Adobe Systems (1985)
Postscript Language Reference Manual.
Reading, MA: Addison-Wesley.
[2] Allebach, J.P. and B. Liu (1976)
"Random quasi-periodic halftone process",
J. Opt. Soc. Am., vol. 66, pp. 909-917.
[3] Allebach, J.P. and B. Liu (1977)
"Analysis of halftone dot profile and aliasing in the discrete binary
representation on images",
J. Opt. Soc. Am., vol. 67, pp. 1147-1154.
[4] Allebach, J.P. (1978)
"Random nucleated halftone screening",
Photogr. Sci. Eng., vol. 22, no. 2, pp. 89-91.
[5] Allebach, J.P. (1979)
"Aliasing and quantization in the efficient display of images",
J. Opt. Soc. Am., vol. 69, pp. 869-877.
[6] Allebach, J.P. (1980)
"Binary display of images when spot size exceeds step size",
Applied Optics, vol. 19, pp. 2513-2519.
[7] Allebach, J.P. (1981)
"Visual model-based algorithms for halftoning images",
Proc. SPIE, vol. 310, pp. 151-158.
[8] Anastassiou, D. and K.S. Pennington (1982)
"Digital Halftoning of Images",
IBM J. Res. Develop., vol. 26, pp. 687-697.
[9] Baldwin, M.W. (1940)
"The subjective sharpness of simulated television images",
Proc. IRE, Oct., pp. 458-468.
397
Y_·__I-III*-·-----^---·I111-
1(-
I
398
REFERENCES
[10] Bartlett, M.S. (1955)
An Introduction to Stochastic Processes with Special Reference to
Methods and Applications.
New York: Cambridge University Press, pp. 274-284.
[11] Bayer, B.E. (1973)
"An optimum method for two level rendition of continuous-tone pictures",
Proc. IEEE Int. Conf. Commun., Conference Record, pp. (26-11)(26-15).
[12] Billotet-Hoffman, C. and O. Bryngdahl (1983)
"On the error diffusion technique for electronic halftoning",
Proc. SID, vol. 24, pp. 253-258.
[13] Bracewell, R.N. (1978)
The Fourier Transform and Its Application.
New York: McGraw-Hill.
[14] Campbell, F.W., J.J. Kulikowski, J. Levinson (1966)
"The effect of orientation on the visual resolution of gratings",
J. Physiology London, vol. 187, pp. 427-436.
[151 Chao, Y. (1982)
"An investigation into the coding of halftone pictures",
M.I.T. Ph.D. Thesis.
[16] Clapper, F.R. and J.A. Yule (1953)
"The effect of multiple internal reflections on the densities of halftone prints on paper",
J. Opt. Soc. of Am., vol. 43, no. 7, pp. 600-603.
[17] Cornsweet, T.N. (1970)
Visual Perception.
New York: Academic Press.
[18] Dippe, M.A. and E.H. Wold (1985)
"Antialiasing through stochastic sampling",
Computer Graphics (AMC SIGGRPAPH'85 Conf. Proc.), vol. 19,
no. 3, pp. 69-78.
[19] Dudgeon, D.E. and R.M. Mersereau (1984)
Multidimensional Digital Signal Processing.
Englewood Cliffs, NJ: Prentice-Hall, pp. 39-41.
[20] ECRM (Tewksbury, MA)
"Optional screens", Autokon User's Manual, ch. 9,
and "Autokon 8400 Theory of Operation".
399
REFERENCES
[21] Engeldrum, P.G. (1985)
"Computing color gamuts of ink-jet printing systems",
SID Int. Sym. Digest of Tech. Papers, pp. 385-385.
[22] Floyd, R.W., and L. Steinberg (1975)
"Adaptive algorithm for spatial grey scale",
SID Int. Sym. Digest of Tech. Papers, pp. 36-37.
[23] Floyd, R.W., and L. Steinberg (1976)
"An adaptive algorithm for spatial greyscale",
Proc. SID, vol. 17/2, pp. 75-77.
[24] Freund, J.E. (1971)
Mathematical Statistics.
Englewood, NJ: Prentice-Hall.
[25] Gall, W., K. Wellendorf, and K. Kiel (1985)
"Production of screen printing blocks",
U.S. Patent 4499489.
[26] Garcia, A. (1986)
"Efficient rendering of synthetic images",
M.I.T. Ph.D. Thesis.
[27] Gard, R.L. (1976)
"Digital picture processing techniques for the publishing industry",
Computer Graphics and Image Proc., vol. 5, pp. 151-171.
[28] Gardner, M. (1978)
"White and brown music, fractal curves and one-over-f fluctuations",
Scientific Am., Apr., pp. 16-32.
[29] Goodall, W.M. (1951)
"Television by pulse code modulation",
Bell Sys. Tech. Journal, vol. 30, pp. 33-49.
[30] Graham, C.H., ed. (1965)
Vision and Visual Perception.
New York: John Wiley & Sons.
[31] Hamill, P. (1977)
"Line printer modification for better grey level pictures",
Computer Graphics and Image Proc., vol. 6, pp. 485-491.
[32] Hecht, E. and A. Zajac (1974)
Optics.
Reading, MA: Addison-Wesley, pp. 361-363.
-
e
_____III__P__L_1IIl__l_*I-I-
400
REFERENCES
[33] Heckbert, P.S. (1982)
"Color image quantization for frame buffer display",
Computer Graphics (AMC SIGGRPAPH'82 Conf. Proc.), vol. 16,
no. 3, pp. 297-307.
[34] Higgins, G.C. and K. Stultz (1948)
"Visual acuity as measured with various orientations of a parallelline test object,"
J. Opt. Soc. Am., vol. 38, no. 9, pp. 756-758.
[35] Holladay, T.M. (1980)
"An optimum algorithm for halftone generation for displays and hard
copies",
Proc. SID, vol. 21, no. 2, pp. 185-192.
[36] Holman, L.A. (1926)
The Graphic Process, a Series of Actual Prints.
Boston: Charles E. Goodspeed & Co.
[37] Jaeger, C.W., H. McManus, and D. Titterington (1984)
"The influence of ink/media interactions on copy quality in ink-jet
printing",
Proc. SID, vol. 25/1, pp. 65-70.
[38] Jarvis, J.F., and C.S. Roberts (1976)
"A new technique for displaying continuous-tone images on a bilevel
display",
IEEE Tran. on Commun., vol. COM-24, pp. 891-898.
[39] Jarvis, J.F., C.N. Judice, and W.H. Ninke (1976)
"A survey of techniques for the display of continuous-tone pictures
on bilevel displays",
Computer Graphics and Image Processing, vol. 5, pp. 13-40.
[40] Judice, C.N., J.F. Jarvis, and W.H. Ninke (1974)
"Using ordered dither to display continuous-tone pictures on an AC
plasma panel",
Proc. SID, vol. 15/4, pp. 161-169.
[41] Kermisch, D. and P.G. Roetling (1975)
"Fourier spectrum of halftone images",
J. Opt. Soc. Am., vol. 65, pp. 716-723.
[42] Knowlton, K. and L. Harmon (1972)
"Computer-produced greyscales",
Computer Graphics and Image Proc., vol. 1, pp. 1-20.
r*
REFERENCES
401
[431 Knuth, D.E. (1981)
The Art of Computer Programming, vol. 2.
Reading, MA: Addison-Wesley, Ch. 3, "Random Numbers".
[44] Kuhn, L. and R.A. Myers (1979)
"Ink-jet printing",
Scientific American, vol. 240, no. 4, pp. 162-178, April.
[45] Lewis, W.G.B. and W.A. Pite, ed. (1902)
Plate 26: "photo of Eltham Place, Kent",
The Architectural Association Sketchbook, 3rd series, vol. VI, London:
C.F. Kell and Son.
[46] Limb J.O. (1969)
"Design of dither waveforms for quantized visual signals",
Bell Sys. Tech. J., Sep., pp. 2555-2582.
[47] Lippel, B. and M. Kurland (1971)
"The effect of dither on luminance quantization of pictures",
IEEE Trans. Comm., vol. COM-19, no. 6, pp. 879-888.
[48] Lippel, B. (1976)
"Two and three-dimensional ordered dither in bi-level picture displays",
Proc. SID, vol. 17/2, pp. 115-121.
[49] Lippel, B. (1978)
"Comments on 'A new technique for displaying continuous-tone images on a bilevel display' ",
IEEE Trans. Commun., vol. COM-26, pp. 309-310.
[501 Lowel, G (1916)
photo: "Summer house on road north of Florence", Smaller Italian
Villas and Farmhouses.
New York: The Architectural Book Publishing Co., p. 76.
[51] Mandelbrot, B. and R. Voss (1983)
"Why is nature fractal and when should noises be scaling?",
Noise in Physical Systems and 1/f Noise.
New York: North Holland Physics Publishing, pp. 31-39.
[52] Marks, R.J. (1986)
"Multidimensional-signal sample dependency at Nyquist densities",
J. Opt. Soc. Am. A, vol. 3, no. 2, pp. 268-273.
[53] Mersereau, R.M. (1978)
"Two-dimensional signal processing from hexagonal rasters",
IEEE InternationalConf. on Acoustics, Speech, and Signal Processing, pp. 739-742.
__1__I________·_ICIII___LI__I____
.IYPIII··L·_I1^-*111-·1
XL- I
.-I-C.L-··---·XL·------
----·1II^.^----
I1
I
402
REFERENCES
[54] Mersereau, R.M. (1979)
"The processing of hexagonally sampled two-dimensional signals",
Proc. IEEE, vol. 67, no. 6, pp. 930-949.
[55] M.I.T. Museum (c. 1929)
Photo: "Building 10", contact print from 5x7 glass negative.
[56] Musha, T. (1981)
"1/f fluctuations in biological systems",
Proc. 3rd symposium on 1/f fluctuations, pp. 143-146.
[57] Oppenheim, A.V. and R.W. Schafer (1975)
Digital Signal Processing.
New York: Printice-Hall, pp. 548-549.
[58] Papoulis, A. (1984)
Probability, Random Variables, and Stochastic Processes.
New York: McGraw-Hill, p. 178.
[59] Perry, B. and M.L. Mendelsohn (1964)
"Picture generation with a standard line printer",
Comm. of the ACM, vol. 7, no. 5, pp. 311-313.
[60] Petersen, D.P. and D. Middleton (1962)
"Sampling and reconstruction of wave-number-limited functions in
N-dimensional Euclidean spaces",
Information and control, vol. 5, pp. 279-323.
[61] Ratzel, J.N. (1980)
"The discrete representation of spatially continuous images",
M.I.T. Ph.D. Thesis.
[621 Roberts, L.G. (1962)
"Picture coding using pseudo-random noise",
IRE Trans. Infor. Theory, vol. IT-8, pp. 145-154.
[63] Robinson, A.H. (1973)
"Multidimensional Fourier transforms and image processing with finite scanning apertures",
Applied Optics, vol. 12, no. 10, pp. 2344-2352.
[64] Roetling, P.G. (1976)
"Halftone method for edge enhancement and moire suppression",
J. Opt. Soc. Am., vol. 66, pp. 985-989.
[65] Roetling, P.G. (1976)
"Visual performance and image coding",
Proc. SID, vol. 17/2, pp. 111-114.
.........-
REFERENCES
403
[66] Roetling, P.G. (1977)
"Binary approximation of continuous-tone images",
Photographic Science and Engineering, vol. 21, pp. 60-65.
[67] Roetling, P.G. and T.M. Holladay (1979)
"Tone reproduction and screen design for pictorial electrographic
printing",
J. Applied Photographic Eng., vol. 5, no. 4, 179-182.
[68] Rosenfeld, G. (1984)
"Screened image reproduction",
U.S. Patent 4456924.
[69] Schreiber, W.F. (1976)
"Laser scanning for the graphic arts",
Proc. SPIE, vol. 84, pp. 21-26.
[70] Schreiber, W.F. (1981)
"The representation of continuous tone images on binary recording
material",
Unpublished paper.
[71] Schreiber, W.F. (1983)
"An electronic process camera",
Tech. Ass. of the Graphic Arts Proc., May.
[72] Schreiber, W.F. (1983)
RLE Progress Report No. 125, MIT, p. 6.
[73] Schreiber, W.F. and D.E. Troxel (1985)
"Transformation between continuous and discrete representation of
images: a perceptual approach",
IEEE Trans. PAMI, vol. PAMI-7, no. 2, pp. 178-186.
[74] Schreiber, W.F. (to be published)
Fundamentalsof Electronic Imaging Systems: Some Aspects of Image
Processing.
New York: Springer-Verlag.
[75] Schroeder, M.R. (1969)
"Images from computers",
IEEE Spectrum, vol. 6, 66-78.
[76] Shaw, R., P.D. Burns and J.C. Dainty (1981)
"Particulate model for halftone noise in electrophotography I. Theory, and II. Experimental verification",
Proc. SPIE, vol. 310, pp. 137-150.
__II
___·___1I__XL_YI1___C^·-Lsl^l·II·-Y-·II
_I__
^^__^1_1_
··__
C^_
404
REFERENCES
[77] Silverberg, S.J. (1983)
"An adaptive dither algorithm for hardcopy devices",
M.I.T. M.S. Thesis.
[78] Sonnenberg, H. (1982)
"Laser-scanning parameters and latitudes in laser xerography",
Applied Optics, vol. 21, pp. 1745-1751.
[79] Sonnenberg, H. (1983)
"Designing Scanners for Laser Printers",
Lasers & Applications, April, pp. 67-70.
[80] Stevenson, R.L. and G.R. Arce (1985)
"Binary display of hexagonally sampled continuous-tone images",
J. Opt. Soc. Am. A, vol. 2, no. 7, pp. 1009-1013.
[81] Stoffel, J.C. and J.F. Moreland (1981)
"A survey of electronic techniques for pictorial reproduction",
IEEE Tran. Commun., vol. 29, 1898-1925.
[82] Stucki, P. (1979)
"Image processing for document reproduction",
in Advances in Digital Image Processing.
New York: Plenum Press, pp. 177-218.
[83] Stucki, P. (1981)
"MECCA -a multiple-error correcting computation algorithm for
bilevel image hardcopy reproduction",
Research Report RZ1060, IBM Research Laboratory, Zurich,
Switzerland.
[84] Taylor, M.M. (1963)
"Visual discrimination and orientation",
J. Opt. Soc. Am., vol. 53, June, pp. 763-765.
[85] Ulichney, R.A. and D. Troxel (1982)
"Scaling Binary Images with the Telescoping Template",
IEEE Trans. on Pattern Analysis and Machine Intelligence,
vol. PAMI-4, no. 4, pp. 331-335.
[86] Ulichney, R.A. (1985)
"Generalized ordered dither",
M.I.T., ATRP-T-51.
also Digital Equipment Corporation, DEC-TR-412.
[87] Voss, R.F. and J. Clarke (1975)
"'1/f noise' in music and speech",
Nature, vol. 258, no. 5533, Nov. 27, pp. 317-318.
405
REFERENCES
[881 Voss, R.F. and J. Clarke (1978)
" '1/f noise' in music: music from 1/f noise",
J. Acoustic Soc. Am., vol. 63, no. 1, pp. 258-263.
[89] Voss, R.F. (1979)
"1/f (flicker) noise: a brief review",
Proc. of the 33rd Annual Symposium on Frequency Control, May 30June 1, 1979, pp. 40-46.
[901 White, J.M. (1980)
"Recent advances in thresholding techniques for facsimile",
J. Applied Photographic Eng., vol. 6, pp. 49-57.
[91] Witten, I.H. and M. Neal (1982)
"Using peano curves for bilevel display of continuous-tone images",
IEEE CG&A, May, pp. 47-52.
[92] Woo, B. (1984)
"A survey of halftoning algorithms and investigation of the error
diffusion technique",
M.I.T. S.B. Thesis.
[93] Yule, J.A.C. (1967)
Principles of Color Reproduction,
New York: John Wiley & Sons.
_1_(·-~~~~~·II~~l~·P--·
__~~~__11
_ 11____
1 1~~~__
___1_
-..
____II
__1
_1__1_ ^__II
__