Digital Halftoning
Sasan Gooran
Department of Science and Technology, Linköping University, Campus Norrköping, Sweden
Contents
1. Introduction ............................................................................................................................................... 2
1.1
Digital Image ................................................................................................................................... 2
1.2
Why Halftoning? ............................................................................................................................ 2
1.3
Short History.................................................................................................................................... 3
2. General Concepts ..................................................................................................................................... 3
2.1
Line Screen Ruling and Print Resolution ................................................................................ 3
2.2
AM and FM Halftoning ................................................................................................................ 5
2.3
The Rule of Thumb ........................................................................................................................ 6
3. Dot Gain ..................................................................................................................................................... 7
3.1
Physical Dot Gain .......................................................................................................................... 7
3.2
Optical Dot Gain............................................................................................................................. 7
3.3
Dot Gain and Halftoning .............................................................................................................. 7
3.4
Dot Gain Models ............................................................................................................................ 7
3.5
Dot Gain Compensation ............................................................................................................... 8
4. Halftoning Methods ................................................................................................................................ 9
4.1
Table Halftoning ........................................................................................................................... 10
4.2
Threshold halftoning ................................................................................................................... 12
4.2.1
Ordered Dithering ............................................................................................................... 13
4.3
Error Diffusion .............................................................................................................................. 14
4.4
Iterative Halftoning ...................................................................................................................... 16
4.5
Hybrid AM-FM Halftoning ....................................................................................................... 16
4.6
Multi-level Halftoning ................................................................................................................ 17
5. Color Halftoning .................................................................................................................................... 17
5.1
Color................................................................................................................................................. 18
5.2
Color Reproduction ..................................................................................................................... 18
5.3
Neugebauer and Demichel Equations .................................................................................... 19
5.4
AM Color Halftoning .................................................................................................................. 20
5.5
FM Color Halftoning................................................................................................................... 22
6. Quality Aspects of Halftoning ........................................................................................................... 23
7. Summary and Discussion .................................................................................................................... 24
8. References ................................................................................................................................................ 25
1. Introduction
The main focus of this chapter is on digital halftoning. Halftoning is one of the most important parts of the image
reproduction process for devices with a limited number of colors. Before describing why halftoning is needed
and point out its importance in image reproduction a short description and definition of digital images is given in
this section. This is followed by the general definition of halftoning and a short history. Section 2 provides a
brief description of a number of fundamental and important concepts related to halftoning. In Section 3, Dot
Gain, which is closely related to halftoning and has a great impact on print quality, is discussed. A number of
basic and advanced halftoning methods are described and discussed in Section 4. In Section 5 color halftoning
and related concepts are introduced and described. Section 6 provides a short discussion on the quality aspects of
halftoning. A summary of this chapter together with a short discussion is finally given in Section 7.
1.1
Digital Image
In order to store the information from the original photographs in computer they have to be converted to digital
format. In other words, the photographs should be digitized, which is normally done by scanning them. If you
have taken a picture by a digital camera of course you don’t need to scan it because the image is already saved in
digital format. When a scanner scans a photograph or when a digital camera takes a picture the information
(color) is measured at discrete points (pixels) of the original. The number of these sample points per an inch is
called the scanning resolution and is denoted by ppi (pixels per inch). In the grayscale photographs the gray tone
of each sample point is normally stored using eight bits (or one byte). Thus, the digital grayscale image contains
256 levels of gray shadows, varying from white (0 or 255) to black (255 or 0). In the case of color photography
the color of each sample point is represented by its three primary colors, namely Red (R), Green (G) and Blue
(B). In this case each sample point needs 3 * 8 bits (3 bytes) to be stored in computer. The bit depth is thus 24
bits or 3 bytes and the digital color image contains 256 3 (about 16.8 million) colors. It is obvious that the higher
the resolution the more detailed the stored image and consequently the bigger the image file. Therefore it is
important to keep resolution as low as possible. The choice of resolution mostly depends upon what you will be
doing with the digital image and in what medium it will be reproduced. For example if your digital image is
eventually going to be reproduced by computer screens, which normally have a reproduction resolution at 72
pixels per inch, it is unnecessary to scan your original photograph with a ppi higher than 72 at 100% scale. If
your digital image is supposed to be printed the choice of ppi depends upon a number of factors that will be
discussed in Section 2.
1.2
Why Halftoning?
The digitized photographs, or the digital images, will eventually be reproduced by a device or a number of
devices. Most of the image reproduction devices, particularly the printing devices, are restricted to few colors
while the digital image mostly consists of millions of colors. In the grayscale case, as been discussed earlier, the
digital image consists of 256 different shadows of gray, while the black and white printers normally use only one
colored ink, i.e. black. These 256 levels of gray should somehow be represented by the black color and the white
substrate. In order to do that the original continuous tone digital image, which in the following will be called the
original image, is transformed into a binary image consisting of 1’s and 0’s, i.e. a bitmap. A 1 at a pixel means
that a black dot should be printed (or shown) at that particular position and a 0 means that the corresponding
position should remain empty. This transformation from a continuous tone image to a bitmap representation is
called Halftoning, also referred to as Screening. The most straightforward way of halftoning is to represent the
average color of different parts of the original image by a so-called halftone cell (or halftone screen). The
fractional area of the halftone cell that is covered by the ink should represent the average color of the
corresponding area in the original image. If the dots are small enough, the eye cannot detect the dot patterns,
2
instead it integrates the black halftone dots and the non-printed areas as varying shades of gray. This is illustrated
in Figure 1.1. Since the halftone dots in the image to the left are small enough they are hardly detected by the
eye at a viewing distance of about 50 cm in this example. On the other hand, the dots in the image in Figure
1.1b, which is an enlargement of a part of the image in Figure 1.1a, are easily detected by the eye at the same
distance.
a
b
Fig. 1.1. The image is halftoned. The dot patterns in the image to the left are hardly detected by the
eye from a distance of 50 cm while they are easily detected in the image to the right.
In the color case this transformation is performed for a number of color channels, normally for the color channels
that the reproduction device uses. More details about color halftoning are given in Section 5.
1.3
Short History
The history of halftoning technology goes back to the middle of 19 th century. The first halftoning technologies
were so called optical. The binary halftone was obtained by projecting the negative of the original image through
a mesh screen [34]. Bright light, as it passes through the mesh screen, would form a large and round spot on the
plate. Dark light would on the other hand form a small spot. Then, a plate was made. A letterpress plate, for
example, is raised where the image is black, and etched out elsewhere. Finally the plate was on the printing
press, where it comes in contact with the ink and selectively transfers it to paper. Digital halftoning, which is the
main topic of this chapter, started around the 1920s and was used to display images on bi-level devices and
reduce the transmission bandwidth [28]. Some of the optical halftoning technologies are applied directly to
digital halftoning, such as ordered dithering (clustered dot), which is described in Section 4.
2. General Concepts
In this section we introduce a number of important concepts that are quite fundamental for the understanding of
halftoning. The concepts such as, print resolution, line screen ruling, halftone cell, AM (Amplitude Modulated)
and FM (Frequency Modulated) halftoning and their advantages and disadvantages are discussed here.
2.1
Line Screen Ruling and Print Resolution
As mentioned earlier in Section 1 one of the most straightforward ways of halftoning is to represent different
areas in the original image by a halftone screen (halftone cell). The fractional area covered by the ink represents
the gray tone of the corresponding area in the original image. Each halftone cell itself consists of a number of
smaller dots, microdots. Two 8 x 8 halftone cells are illustrated in Figure 2.1. The halftone dot in Figure 2.1a is 2
x 2 and thus represents the gray tone of 4/64. The gray tone represented by the halftone cell to the right is 44/64.
3
Therefore, totally it is possible to represent 65 (8 2+1) different gray tones by 8 x 8 halftone cells. The number of
halftone cells per inch is called line screen ruling or screen frequency and is denoted by lpi, lines per inch. It is
obvious that the higher the lpi the smaller the halftone cell and consequently the halftone dot and the more
difficult for the eye to detect the halftone dots. Studies have shown that the halftone dots are not detected by the
eye from the normal viewing distance at screen frequencies above 200 lpi [34, 36].
The number of the micro dots per inch is called the print resolution and is denoted by dpi (dots per inch).
Halftone dots
a
b
Micro dots
(The small squares)
Figure 2.1. Two halftone cells (halftone sreens). a) The gray tone is 4/64. b) The gray tone is
44/64.
From the definition above it follows that the size of the halftone cell, and consequently the number of
represented gray tones, is determined by the ratio dpi/lpi. This can be summarized in the following equation,
2
 dpi 
  1
Number of gray tones  
lpi


Equation 2.1
As can be observed from Equation 2.1, a higher lpi will lead to a decrease of the number of gray tones when the
dpi is constant. Choosing an appropriate lpi is therefore a trade off between the number of gray tones and the fine
details, see Figure 2.2. In all these images the print resolution is kept constant at 300 dpi. In Figure 2.2a the
screen frequency is 25 lpi, which is quite low, and the number of gray tones is 145. As can be seen in this image
the halftone dots are easily detected and therefore most of details are lost. In Figure 2.2b the screen frequency is
50, thereby 37 levels of gray. In Figure 2.2c lpi is 100 and the number of gray levels is only 10. Since in the
image in Figure 2.2c quite few gray tones are represented the image is not perceived that well, because the
transition from one gray tone to another is quite visible. The best image in this case would probably be the image
in the middle.
a
b
c
Figure 2.2.The print resolution is kept constant at 300 dpi. The screen frequency is in a, 25 lpi, in
b, 50 lpi and in c, 100 lpi.
4
2.2
AM and FM Halftoning
The halftoning methods can mainly be divided into two main types, namely AM (Amplitude Modulated) and FM
(Frequency Modulated). In the AM methods the size of the halftone dots vary, while their spatial frequency is
constant. This means that the size of the halftone dot becomes bigger as the tone gets darker. In the FM methods,
on the other hand, the dot size is constant while the frequency (the number of micro dots) varies. Something
worth observing here is that the terms AM and FM halftoning are sometimes incorrectly replaced by
conventional and stochastic halftoning, respectively. However, in this book we rather use the terms AM and FM
halftoning as defined above. This means a conventional halftoning, where we use halftone cells, could be either
AM or FM. Figure 2.3 shows four different 8 x 8 halftone cells. The two halftone cells in each column represent
the same gray tone, while the upper halftone cells are constructed using AM and the lower using FM.
Figure 2.3. Four 8 x 8 halftone cells are shown. The two halftone cells in each column represent
the same gray tone while the upper halftone cells are built as AM and the lower ones as FM.
When using halftone cells to build the halftoned images, it doesn’t matter whether AM or FM is used, the final
image will include periodically repeated structures (halftone cells). The FM techniques, however, generally don’t
necessarily need to use halftone cells to build the final image. Actually, most of the known FM methods, which
are wrongly called stochastic, such as Error Diffusion, don’t use any halftone cells to build the final image and
therefore the term lpi is not used in this case. The only term that is used is the print resolution dpi. Figure 2.4
illustrates two images halftoned by AM and FM methods. In the image in Figure 2.4a an AM method is used, the
screen frequency is 50 lpi and the print resolution 300 dpi. The image in Figure 2.4b is halftoned by the wellknown Error Diffusion, which will be described in Section 4. The print resolution is 300 dpi. Images shown in
Figures 2.4c and d are enlargements of a part of images shown in Figures 2.4a and b, respectively, printed at 75
dpi. As can be seen the AM-halftoned image has a periodical structure and the FM one, in this example, doesn’t
possess any ordered structures. When the tones become darker, in the AM-halftoned image the size of the dots
becomes bigger. In the FM-halftoned image, on the other hand, the size of the smallest micro dots, decided by
the print resolution (here 300 dpi), are constant and when the tones become darker the number of these micro
dots increases. AM halftoning has for long been the most used halftoning technique in the printing industry. The
most important reason has been the inability of the printing devices to produce the small single microdots.
However, since the beginning of the 1990s, when FM halftoning was used as an alternative to AM for low-cost
inkjet printers [34], FM has started competing with AM technologies. These days, they are widely used not only
in inkjet printers but also in the newspaper prints. The AM and FM halftoning have however their own
advantages and disadvantages. The FM techniques are superior when it comes to reproducing the details,
especially when the screen frequency cannot be as high as one would like because of the mechanical limitations
of the print press. For example, a typical screen frequency for newspaper print is 65-85 lpi and 600 dpi laser
printers print at 85 lpi [45]. The AM methods, however, are better for the areas where the tones vary slowly. The
FM methods generally give a “noisy” impression in these regions. The advantages and drawbacks of these
methods have made researchers carry out studies in combining these two methods for different applications and
purposes. This will be discussed more in Section 4.
5
a
b
c
d
Figure 2.4. The image is halftoned by AM and FM techniques. In a, the screen frequency is 50 lpi
and the print resolution is 300 dpi. In b, the Error Diffusion method is used and the print resolution
is 300 dpi.
2.3
The Rule of Thumb
In Section 1 we had a short discussion on how to choose the scanning resolution ppi (pixels per inch). We
mentioned that if the digital image is supposed to be displayed by for example a computer screen it is
unnecessary to scan the image with a ppi higher than the resolution of that computer screen, if the reproduced
image is supposed to be the same size as the original. However, when it comes to print one should consider the
screen frequency of the print before deciding the scanning resolution. There is a rule of thumb that says that the
scanning resolution (ppi) should be about 1.5 to 2 times the screen frequency (lpi) used in the press at 100%
scale [45]. The scaling factor should also be multiplied by this factor if the printed image is not supposed to be
the same size as the original photograph. This rule of thumb can be summarized in the following equation.
ppi 
printed size
 2  lpi
original size
Equation 2.2
Let us give an example. Assume that you have a photograph that is 10 x 15 cm and you want to print it with a
laser printer at 85 lpi and make it 20 x 30 cm. Then, to achieve the best possible quality you need to scan your
photograph with a ppi about 4*85 =340. Notice that the rule of thumb also suggests that each halftone cell
should represent a 2 x 2 pixel area in the original image.
Also notice that when it comes to FM halftoning the scanning resolution should be equal to the printer’s or
image-setter’s print resolution (dpi) at 100% scale.
6
3. Dot Gain
The printed dots normally appear bigger than they are in the bitmap. This is partly because that the dots become
physically bigger due to for example the ink spread and other distortions produced by the printer or print press.
This is what we call the physical (mechanical) dot gain. Another reason why the printed dots appear bigger than
their real physical size is the diffusion of the light in the paper or substrate. This is called the optical dot gain. In
this section these concepts are briefly described.
3.1
Physical Dot Gain
Due to many different factors the printers or the print presses are not able to produce the dots exactly the same
size as they are in the bitmap. Normally, they become bigger. One of the most important reasons is that the
viscosity of the ink is not high enough to avoid some spreading. The pressure from the print cylinder also plays a
significant role; the bigger the pressure the bigger the mechanical dot gain.
3.2
Optical Dot Gain
Unlike the physical dot gain the optical dot gain, also called the Yule-Nielsen effect, has actually nothing to do
with the mechanical processes of printing. It is simply due to the light scattering in the paper or the substrate.
Some of the photons that enter the non-inked areas of the print are scattered in the paper and absorbed by the ink
on their way back from the paper. Figure 3.1 shows a simple illustration of three different paths a photon can
travel when it enters the surface of a print that is partly covered by the ink and partly blank. Arrow C illustrates
the reason for optical dot gain [58].
A
C
B
Ink
Paper
Figure 3.1. Three different paths that a photon can take when it enters a print. Arrow C illustrates
the reason for optical dot gain.
3.3
Dot Gain and Halftoning
Due to the different arrangements of the micro dots in AM and FM halftoning the impact of dot gain on these
techniques is also different. It is quite obvious that the effect of the gain of each simple micro dot has greater
impact on the total dot gain when the dots are dispersed. This fact is actually valid for both physical and optical
dot gain. Since the microdots are clustered in AM halftoning and dispersed in FM halftoning, FM halftoning
suffers more from dot gain than AM in the same printing conditions, see Figure 2.4. The image shown in Figure
2.4b, which is FM halftoned, looks darker than the image in Figure 2.4a, which is AM halftoned, although the
print resolution is 300 dpi in both cases.
3.4
Dot Gain Models
Because of dot gain’s great impact on the print quality many studies have been carried out in the last decade to
investigate dot gain and find an appropriate and useful model for it [2, 3, 25, 26, 33, 44, 55, 56].
7
When the optical dot gain is neglected, only arrows A and B in Figure 3.1 describe the paths a photon can take.
Hence the total reflectance of the print can easily be approximated by Murray-Davies formula [21, 40],
Rtot  aRink  (1  a) R pap
Equation 3.1
where Rtot, Rink and Rpap denote the total reflectance of the print, the reflectance of the full tone ink and the
reflectance of the bare paper, respectively. a denotes the fractional area of the print surface covered by the ink.
Note that a in Equation 3.1 corresponds to the real physical dot size in print, i.e. after the physical dot gain.
When the optical dot gain is not negligible then Equation 3.1 is not valid anymore. One of the first models for
optical dot gain was introduced by Yule and Nielsen. Their model is actually an extension to the Murray-Davies
model and is as follows [59],
1/ n
1/ n
/n
Rtot
 aRink
 (1  a) R1pap
Equation 3.2
where Rtot, Rink and Rpap and a are defined as before and factor n is actually a fitting factor, which is determined
by experiment. Something worth mentioning is that although Equation 3.2 is used to only model the optical dot
gain, it is sometimes utilized to model both physical and optical dot gain [21].
There have been many other studies and investigations on optical dot gain [2, 25, 39, 44, 56] during recent years.
Recently, a unified dot gain model has been proposed that incorporate both physical and optical dot gain [57].
3.5
Dot Gain Compensation
Since due to dot gain each dot appears bigger than its correspondence in the bitmap, the printed image will be
darker than it actually is, see Figure 3.2a.
a
b
Figure 3.2 The image is halftoned by a FM method. In a, the original image has not been
compensated for dot gain. In b, the original image has been compensated for dot gain prior to
halftoning.
In order to make the printed halftoned image represent the correct gray-tones, as they are in the original, the
effect of dot gain has to be compensated. This compensation is normally performed on the original image prior
to halftoning by using an experimental dot gain curve or alternatively an appropriate dot gain model. The dot
gain curve is normally found by experiments. We measured a number of halftone patches with different nominal
dot coverage. The patches were then halftoned by a FM method and printed out at 300 dpi using a laser printer
(HP LaserJet 4050N) [21]. By using a spectrophotometer we measured the spectra of all these patches. The
Murray-Davies formula was then used to find the effective dot coverage,
8
a eff 
R pap  Rmea
R pap  Rink
Equation 3.3
where Rink and Rpap are as before and Rmea denotes the measured spectra. By using for example the least square
errors method and Equation 3.3 one can calculate the effective dot coverage, aeff. Notice that since the
spectrophotometer is an optical equipment both physical and optical dot gain are included in our measurement
and consequently aeff represents both of them. Figure 3.3 shows aeff versus the commanded dot coverage for the
laser printer mentioned above and the utilized FM halftoning method at 300 dpi. The straight line in this figure
shows the ideal case, i.e. when there is no dot gain. Now by using the experimental curve shown in Figure 3.3
one can compensate the original image for dot gain prior to halftoning. Assume that we want to print a halftone
patch with a gray tone corresponding to 30% (or 0.3) coverage. From the curve, follow the arrows in Figure 3.3,
we can see that in order to have 30% coverage in print we need to have about 15% coverage in the digital
halftone patch. The original image can be compensated for dot gain using the same strategy for each pixel value.
The compensated image can then be halftoned with the halftoning method for which the measurements were
carried out. In the image shown in Figure 3.2b the original image was first compensated for dot gain using the
curve in Figure 3.3 and then FM-halftoned. Notice that in order for the printed image to have the correct gray
tone the halftoned image should be printed with the same laser printer and at 300 dpi. It has to be mentioned that
since the dot compensation is a non-linear transformation the compensated image, and consequently its halftoned
version, might lose some details of the original image [21].
Figure 3.3 The effective dot coverage vs. the commanded dot coverage for a laser printer and FM
halftoning at 300 dpi. The ideal case is when there is no dot gain (the straight line).
Some authors have suggested that considering the effect of dot gain within the halftoning process, if the
halftoning method allows for that, can increase the quality of the final printed image compared with the case
where the compensation is performed prior to halftoning [5, 21].
Something worth mentioning here is that some studies have shown that although the optical dot gain is
considered as an un-wanted phenomenon it can actually increase the color gamut (the volume of the color that
can be reproduced by a device) in print [25, 56].
4. Halftoning Methods
Now when we have introduced halftoning and pointed out its importance in image reproduction it is time to
briefly describe a number of fundamental and also advanced halftoning methods. In this section we concentrate
9
on halftoning methods for grayscale images. Color halftoning is discussed later in Section 5. It is assumed that
the original images are scaled between 0 and 1, which represent white and black respectively.
4.1
Table Halftoning
Probably one of the simplest halftoning methods is table halftoning. In this method different areas in the original
image are replaced by their corresponding tables (halftone cells). If we want to follow the rule of thumb
discussed earlier at least each 2 x 2 pixel area in the original image should be replaced by a table. The size of this
table is decided by screen frequency and the print resolution. Recall from the previous sections that the size of
the halftone cell is decided by the ratio of dpi and lpi. For simplicity assume that each 2 x 2 pixel area is going to
be replaced by a 3 x 3 table. The mean of the pixel values in each 2 x 2 pixel area in the original image is
calculated. Since the table is 3 x 3 then only ten different gray tones can be represented by this table, see Figure
4.1. One of these ten possible candidates that has the closest mean to each 2 x 2 pixel area in the original image
is placed in the corresponding position in the final image. Figure 4.1 illustrates how this method works. In this
illustration only the table representation for the first (follow arrow A) and the last (follow arrow B) 2 x 2 pixel
areas are shown. The same is done for other 2 x 2 pixel areas in the original image.
Ten different gray
tones (candidates)
Original image
Table halftoned image
A
B
A
B
A
Calculate the mean for this 2 x 2
area and choose the candidate that
has the closet mean
B
Figure 4.1 Table halftoning. Each 2 x2 pixel area in the original image is replaced by a 3 x 3 table
(halftone cell). The table with the closest mean is chosen for each 2 x 2 region.
Observe that if the pixel values in the original image are scaled between 0 and 1, a black microdot in the table is
1 and an empty dot 0. Figure 4.2 shows an image halftoned by a simple table halftoning. In this image a 3 x 3
table has replaced each 3 x 3 region in the original image. Hence, only 10 levels of gray. The tables were
arranged exactly as shown in Figure 4.1. An enlargement of a part of the image is also shown in this figure.
Notice that the arrangement of the black micro dots in the tables doesn’t necessarily need to be clustered, it could
also be dispersed. The two tables in Figure 4.3 both represent the same gray tone 4/9, while the one to the left is
arranged as a clustered dot and the one to the left as a dispersed dot. Another important observation is that the
shape of the halftone dots can also be changed by the arrangement of the black microdots in the screen table.
Since normally the screen tables are much bigger than 3 x 3, it is not so difficult to arrange the black micro dots
to get a circular, elliptical or square dot. To illustrate this we arranged the black micro dots in order to get line
and circular raster, see Figure 4.4. In both images each 4 x 4 pixel area in the original image has been replaced
by a 4 x 4 screen table. Hence, 17 levels of gray. In the image shown in Figure 4.4a the black micro dots have
been arranged to build a circular dot.
10
a
b
Figure 4.2 The image has been halftoned by table halftoning. The tables are 3 x 3, hence 10 levels
of gray. An enlargement is shown in b.
a
b
Figure 4.3 Two screen tables both representing the gray tone 4/9. In a, the black micro dots are
clustered. In b, the black micro dots are dispersed.
Observe that since a 4 x 4 screen table is quite small it is not possible to build “perfect” circular dots. In the
image shown in Figure 4.4b the black microdots have been arranged to make a line raster. To show the
differences more clearly enlargements of a part of both images are also shown in this figure.
a
b
c
d
Figure 4.4 Table halftoning with 4 x 4 screen tables. a) The black microdots are arranged to make
circular dots. b) The black microdots are arranged to make line raster. c) An enlargement of a part
of the image shown in a. d) An enlargement of a part of the image shown in b.
11
4.2
Threshold Halftoning
This technique is quite similar to table halftoning and also results in images with similar quality. However, in
this technique instead of having a pre-decided set of screen tables, a threshold matrix is used. Depending on the
content of the original image the result will vary due to the form of the threshold matrix. This technique can
simply be described by following equation,
1 if
b(i, j )  
0 if
g (i, j )  t (i, j )
Equation 4.1
g (i, j )  t (i, j )
where b, g and t denote the final halftoned image, the original image and the threshold matrix respectively. The
pixel value at each position (i, j) in g is compared with the corresponding position in the threshold matrix t. If
this is bigger or equal than the threshold, then a 1 (black dot) is set at the corresponding position in the halftoned
image b. Otherwise, a 0 (white dot) is set there.
7
8
9
10
1
2
3
4
6
1
2
11
5
6
7
8
5
4
3
12
9
10
11
12
16
15
14
13
13
14
15
16
a
b
Figure 4.5 Two 4 x 4 threshold matrices representing 17 levels of gray. In a, the threshold values
are arranged to build a spiral dot form. In b, the threshold values are arranged to build line raster.
a
b
c
d
Figure 4.6 The image is halftoned by threshold halftoning. a) The threshold matrix shown in
Figure 4.5a is used. b) The threshold matrix shown in Figure 4.5b is used. c) An enlargement of a
part of the image shown in a. d) An enlargement of a part of the image shown in b.
12
Observe that, in Equation 4.1 it is assumed that the threshold matrix is the same size as the original image. If the
threshold matrix is smaller than the original image, which normally is the case, it should be periodically repeated
in both directions to be the same size as the original image. Like in the table halftoning case the size of the
threshold matrix and the arrangement of the threshold values have a great impact on the final image. The two
threshold matrices shown in Figure 4.5 both represent 17 levels of gray. The threshold values in the matrix
shown in Figure 4.5a are arranged to build a spiral form halftone dot. The values in the other matrix are arranged
to build a line raster. Note that since the original image is scaled between 0 and 1, the threshold values in both
matrices should be divided by 17. Other shapes of halftone dots are also possible to produce by a proper design
of threshold matrices [28]. Images shown in Figure 4.6a and b are thresholded using the threshold matrices
shown in Figure 4.5a and b, respectively. To show the differences more clearly enlargements of a part of these
images are also shown.
4.2.1 Ordered Dithering
One of the well-known classes of halftoning techniques that is based on threshold halftoning is Ordered
Dithering. As mentioned earlier the design of the threshold matrix has a great impact on the characteristics of the
final halftoned image.
14
5
6
9
19
28
27
24
12
4
1
7
21
29
32
26
13
3
2
8
20
30
31
25
16
10
11
15
17
23
22
18
19
28
27
24
14
5
6
9
21
29
32
26
12
4
1
7
20
30
31
25
13
3
2
8
17
23
22
18
16
10
11
15
2 29 7 27
18 10 23 15
6 26 4 31
22 14 20 12
1 30 8 28
17 9
24 16
5 25 3 32
21 13 19 11
a
b
Figure 4.7. Two threshold matrices representing 33 levels of gray. a) Clustered dot ordered
dithering. b) Dispersed dot order dithering.
a
1
17
5
21
2
18
6
22
30
9
25
13
29
10
26
14
8
24
3
19
7
23
4
20
28
16
32
11
27
15
31
12
b
c
d
Figure 4.8 The image is halftoned by ordered dithering. a) Clustered dot ordered dithering,
threshold matrix shown in Figure 4.7a. b) Dispersed dot ordered dithering, threshold matrix shown
in Figure 4.7b. c) Enlargement of a part of the image in a. d) Enlargement of a part of the image in
b.
13
The ordered dithering techniques are divided into two parts, clustered dot and dispersed dot [51]. In the clustered
dot dithering the threshold matrices are arranged in a way that the final halftone dot is a cluster of black
microdots. In the dispersed dot dithering the black microdots are dispersed. In Figure 4.7 two threshold matrices
according to clustered dot and dispersed dot dithering are shown. As can be seen in the matrix in Figure 4.7a the
threshold values grow coherently and the halftone dot will thus be clustered. In the other threshold matrix the
threshold values grow separately, hence the halftone dot will be dispersed. Since half of these threshold values
are repeated, they only represent 33 levels of gray, although both threshold matrices are 8 x 8. The reason for
them being repeated is to make a raster angle of 45 degrees [51]. Notice that the threshold values in both
threshold matrices shown in Figure 4.7 should be divided by 33. To know more about how the threshold
matrices should be designed to make different dot shapes and different raster angles see [28]. Figure 4.8 shows
images being halftoned using the threshold matrices shown in Figure 4.7. Although both images are printed at
300 dpi one can see that the dispersed dot image appears darker. The reason, as mentioned earlier, is that dot
gain is bigger for FM halftoning than AM. Enlargements of a part of these images are also shown in this figure.
4.3
Error Diffusion
Although the FM halftoning methods started being commercially used in the beginning of the 1990s the first real
FM halftoning was actually invented in the mid 1970s by Floyd and Steinberg [16]. Unlike the ordered dithering
methods, which operate point-by point, i.e. the value of each output point depends only on the value of the
corresponding input position, error diffusion operates on a neighborhood of currently processed pixel. Error
diffusion is graphically illustrated in Figure 4.9.
g
+
b
Threshold
+
-
+
+
+
”error”
Error Filter
Figure 4.9. Error Diffusion halftoning method.
In this figure g and b denote the original and the halftoned image, respectively. This algorithm goes through all
pixels in the original image, normally starting from the pixel up to the left and then goes through all pixels from
left to right and up down. The value of each pixel in g is compared with a threshold, which in the non-modified
error diffusion is normally 0.5. Depending on whether the pixel value is bigger or smaller than the threshold a 1
(black dot) or a 0 (empty dot) is set at the corresponding position in b. Since the pixel value in g, which is a real
number between 0 and 1, has been replaced by 0 or 1 in b an “error” has been occurred. The “error” is the
difference between the pixel value in g and b at that position. This error is then diffused to a number of nonprocessed pixels. To what pixels and how this error is diffused is decided by an error filter. In Figure 4.10 three
different error filters are shown. The filter shown in Figure 4.10a is the one Floyd and Steinberg introduced.
(*1/48)
(*1/16)
(*1/42)
7
5
8
4
7
3
5
3
5
7
5
3
2
4
8
4
2
1
3
5
3
1
1
2
4
2
1
1
a
b
c
Figure 4.10 Three error filters. a) Floyd and Steinberg. b) Jarvis, Judice and Ninke. c) Stucki.
14
When using this filter the error occurred at the position (i, j) is weighted by 7/16 and added to the pixel value at
(i+1, j). The same error is weighted by 1/16 and added to the pixel at (i+1, j+1) and so on. After the error has
been diffused the pixel value of the next position is compared to the threshold and the same process continues
until all pixels have been met. Figure 4.11 shows the test image being halftoned by error diffusion. In Figure
4.11a the error filter is one-dimensional, that means the error is diffused to only one pixel to the right of the
currently processed position. In Figure 4.11b the error filter is the Floyd and Steinberg filter shown in Figure
4.10a.
a
b
c
Figure 4.11 The image is halftoned by error diffusion. a) one dimensional error diffusion. The
error is diffused to only one pixel. b) The error filter is the filter shown in Figure 4.10a.The
threshold is fixed at 0.5. c) The error filter is the filter shown in Figure 4.10a.The threshold is a
random number between 0.25 and 0.75.
Since the error in Figure 4.11a has been diffused only in one direction and only to one pixel the result is not
satisfying and the visible vertical structures are disturbing. The disturbing structures are much less evident in
Figure 4.11b. The details are reproduced very well in this image. Although the non-modified error diffusion is
quite simple and results in fairly good images it has a few shortcomings. One of them is the correlated artifacts,
best seen in the mid-tones of the image shown in Figures 4.11a and b. Another problem is the directional
hysteresis, which is best seen in the highlights and shadows of an image [51]. Due to the simplicity and
efficiency of error diffusion there have been many studies on this method during the recent decades and many
different modifications have been suggested [12, 13, 30, 31, 32, 52]. One of the simplest modifications to reduce
the mentioned artifacts is to add some noise to the original image before halftoning. Equally, one can change the
threshold from being fixed at 0.5 to a random number between 0 and 1. Figure 4.11c shows the test image being
halftoned by error diffusion with Floyd and Steinberg filter and a random threshold varying from 0.25 and 0.75.
As can be seen the result gives a noisy impression but the correlated artifacts are much less visible. Another
simple modification is to use a bigger error filter. Jarvis et al. and Stucki have proposed bigger error filters,
shown in Figure 4.10b and c, respectively [27, 49]. The test image being halftoned by these filters are shown in
Figure 4.12.
a
b
Figure 4.12. The image is halftoned by error diffusion. a) The error filter is the filter shown in
Figure 4.10b.The threshold is fixed at 0.5. b) The error filter is the filter shown in Figure
4.10c.The threshold is fixed at 0.5.
15
Note that in order for the final image to have the same mean as the original the sum of the weights in the error
filter should always be one. Although using a bigger filter result in sharper image [19, 20, 32] and reduces some
of the artifacts directional hysteresis in the very dark and light regions increases [51]. It has also been shown that
error diffusion techniques, in general, have a tendency of high-pass filtering (edge enhancing) the original [19,
20, 32].
4.4
Iterative Halftoning
The need for having cheap, fast and high quality print has increased during the last decade of 20 th century. The
fast increase of the computer has opened the door for the researchers to be able to introduce high quality and
complex but still applicable halftoning methods. During 1990 th and beginning of the 2000th there have been
many new iterative halftoning methods proposed in literature [1, 19, 23, 43, 46]. Unlike the ordered dithering
methods, which are point-by-point, and unlike error diffusion, which operates on a neighborhood of currently
processed position, the iterative methods normally operate over the entire original image. These methods
normally start with an initial binary image and iteratively decrease the difference between the binary image and
the original image. The difference is normally defined as the visual difference between images. Therefore,
different models of human eye are used in these methods [37, 41, 50]. These methods, as mentioned, result in
high quality halftoned images, but are still time demanding and so far they are only applicable in digital print.
Figure 4.13. The test image is halftoned by an iterative halftoning method, IMCDP. An
enlargement of the image is also shown.
Since the computer power has not yet reached its highest limit, these methods are probably going to be more and
more popular in future. Figure 4.13 shows the test image being halftoned by an iterative method called “Iterative
Method Controlling the Dot Placement” (IMCDP) [19, 23]. An enlargement of a part of this image is also
shown. Compare this image with those halftoned by error diffusion shown in 4.11 and 4.12. As can be seen, the
result is more pleasing than the results from error diffusion. Halftoning this image, which is 512 x 512, took less
than 1 second using Pentium (R) 4 CPU, 3.00 GHZ and 1.00 GB RAM [11].
4.5
Hybrid AM-FM Halftoning
We have discussed and illustrated how different halftoning technologies perform. We specially discussed AM
and FM halftoning and presented their characteristics, advantages and disadvantages. The idea of using the
strength of both techniques in one halftoned image has been the topic of many researches. Since the FM
techniques are normally better in reproducing the details and the AM in the homogenous parts one of the ideas
has been to use a FM technique for the details of the original image and an AM for the rest of it [18, 22].
Although the idea seemed to be promising and test runs showed an improvement in the image quality compared
to AM, to our knowledge this idea has not yet been used in practice.
Another more important, and necessary application of hybrid halftoning is for the printing methods, such as
Flexography, that are unable to produce the dots smaller than a specific critical size. Since the method is unable
16
to produce the dots small enough, by only using an AM method the very highlights that correspond to the gray
tones brighter than the critical dot size will remain empty. The same problem occurs in the shadow areas as well.
There have been many researches carried out in studying and finding efficient methods to combine AM and FM
halftoning to overcome this problem [6, 7, 8, 9, 24]. The main idea behind the proposed methods is to use a FM
method in the highlights (and shadows) with the smallest producible dot and using the AM method in the rest of
image. There have been many ideas on how to do this. Some suggest the so called FM classic halftoning which
is not actually a mixture of AM and FM halftoning. Instead, it is AM halftoning, where part of the highlight dots
is left away depending on the dot percentage [6]. Another method to combine AM and FM is to use AM
halftoning until the certain critical size is reached. The FM halftoning is then made of screen dots with this
critical size. To avoid a clear transition between AM and FM parts both of them are mixed in an intermediate
density range [6]. In SambaFlex halftoning method, which is another hybrid method, there is a gradual transition
between FM and AM screens. In the highlights the halftone dots do not follow any AM pattern. At 25% dots
come nearer to the AM theoretical center. You see the dots “dancing” around the AM dot centers. Finally, at
50%, the dots are fully on the AM grid [6]. There are also methods that are similar to hybrid halftoning described
above without having any mixture of AM and FM in the intermediate density range [24].
There have also been researches done to find halftoning methods that produce dot clusters which vary in both
their size and frequency depending on the tone [34]. These methods have the benefits of both AM and FM. In the
highlights and shadows they behave almost like a FM method and in the mid-tones they suffer less from dot gain
since the dot size is varying as well.
4.6
Multi-level Halftoning
All the halftoning methods discussed so far assumed that the print or reproduction device is bi-level. There are
however devices in xerographic systems and inkjet printers, that are capable of rendering more than two and
normally less than 16 levels [28]. The number of levels they can produce is still much less than 256, which is the
number of gray levels in the original image. Therefore, the original image should still be halftoned (quantized) to
a fewer number of levels. The halftoning techniques doing this transformation are called multi-level halftoning.
Many different multi-level halftoning methods have been proposed in literature [10, 15, 17, 28, 38, 48]. Some of
these methods are based on ordered dithering [28, 48]. In these techniques a threshold matrix is still used but the
comparison between the pixel value in the original image and the threshold value is different. In the bi-level
threshold halftoning, as discussed earlier, either a 1 or a 0 was set in the output. In multi-level case the closest
output gray level depending on the threshold matrix is chosen. Some of the multi-level techniques are based on
error diffusion [10, 17]. These methods are quite similar to the ordinary error diffusion. The difference is that
instead of comparing each pixel value with a fixed threshold, the closest output gray level is chosen. The error is
then calculated and diffused as before.
5. Color Halftoning
Since almost all the print materials are colored these days it is very important to study color halftoning. Although
at the first glance one could expect that the extension to color halftoning only means repeating the same process
a number of times, in practice other problems will also occur. In order to be able to study color halftoning we
have to have a wide knowledge on color in general. In this chapter, however, we don’t go too deep into the
details of color science and only concentrate on those parts that are useful for understanding color halftoning.
17
5.1
Color
All scientists seem to agree that color doesn’t exist in the outside world and it only exists in our eyes and brain.
The electromagnetic radiations with wavelength between 380 and 780 nm are detected, modulated and
interpreted by human eyes’ three color-receptors (cones) and the brain. The spectral power distribution of the
radiation describes the color we see. For example, a spectral color distribution that is mostly concentrated at
longer wavelength (say 700 nm) is perceived as reddish. In order to be able to define a color, we need to know
the sensitivity functions of eye’s three receptors. In the 1930 th the CIE (Commission Internationale de
l’Éclairage) carried out a number of experiments to find these three sensitivity functions. By knowing them the
tri-stimulus values are calculated as [14, 47, 54],
X  k  R( ) I ( ) x( )d
Equation 5.1
Y  k  R( ) I ( ) y ( )d
Equation 5.2
Z  k  R( ) I ( ) z ( )d
Equation 5.3



where X, Y and Z are the tri-stimulus values (color coordinates). R and I represent the reflectance of the object
and the intensity of the light source, respectively.
x , y and z are the three sensitivity functions of the eye and 
is the wavelength. The factor k is a normalization factor and is multiplied by all three integrals to get Y = 100 for
the white point of the light source. The white point is calculated by assigning R ( )  1 . Therefore, k is
calculated as,
k
100
 I ( ) y( )d
Equation 5.4

5.2
Color Reproduction
To reproduce a color, a device needs to mix a number of primary (basic) colors. A computer screen, or a TV,
normally uses Red, Green and Blue as primary colors and by mixing them with different intensity can reproduce
other colors. When all three colors are present, white color is perceived, and when none of them is present, black
is perceived. This is called additive color mixing and is illustrated in Figure 5.1a. To reproduce a color in print
the primary colors are represented by inks. The three colors that are normally used in print are Cyan, Magenta
and Yellow. The printing inks are actually color filters, which absorb some parts of the incoming light and
transmit the rest. When there is no ink, all the incoming light is reflected back (assuming that the paper or the
substrate is white), hence white is perceived. When all three colors are printed on each other then all the
incoming light is absorbed and black is perceived. This is called subtractive color mixing and is illustrated in
Figure 5.1b.
a
b
Figure 5.1. a) Additive color mixing. b) Subtractive color mixing.
18
These two color systems, however, are based on the characteristics of the device. The same R, G and B value
could be perceived as different colors when reproduced by different devices. The same is valid for C, M and Y
values. That is why these color systems are called device dependent. Although C, M and Y printed on top of
each other will theoretically reproduce black, in practice it produces dark brown. In order to produce the pure
black and avoid printing three primary colors on top of each other, which causes another problems, normally a
fourth primary color, black (K), is used in printers and print presses.
One of the biggest issues in color reproduction is to reproduce the color as accurate as possible. Because of the
reproduction devices’ properties and limitations of course all the colors in the original image cannot be
reproduced. The volume of the colors, which can be reproduced by a device, is called the color gamut of the
device. In order to reproduce the colors correctly, or as accurate as possible, the color reproduction should be
traced from the start, for example scanning, to the end, i.e. printing. This is done by Color Management Systems
(CMS) and ICC-profiles. To do so, we of course need color systems that are independent of the reproduction
devices. Therefore, different device independent color systems have been proposed and applied in practice. One
of the most commonly used color system is Lab color system, which is based on the tri-stimulus values defined
in Equations 5.1-5.4 [14, 47, 54].
5.3
Neugebauer and Demichel Equations
As discussed earlier a device reproduces different colors by using a number of primary colors. The question is
now how to calculate (approximate) the resulting color when these primary colors are mixed. When several
small different colored areas are averaged together by the human eye a simple model, Neugebauer’s equations,
can be used to find the resulting color [42],
 X ave 
X i 
Y   a Y 
i i 
 ave  
i
 Z ave 
 Z i 
Equation 5.5
where ai is the fractional area covered by the ink color (Xi, Yi, Zi) and
a
i
1
Equation 5.6
i
For example when we print with four inks (CMYK) it is possible to create 16 different colors (of which 9 is in
effect black), so the sum in Equation 5.5 will have 16 terms. When two different colored inks are printed on top
of each other, we have a typical subtractive color mixing, as discussed earlier. For example, the blue color (cyan
on magenta or vice versa) will have the spectrum,
Rblue ( )  R p ( )C ( ) M ( )
Equation 5.7
where C() and M() are the transmittance characteristics of cyan and magenta inks and Rp() is the reflectance
of the paper. By using Equations 5.1-5.4, the tristimulus values for blue can be calculated. The same can be done
for all 16 possible colors when printing with four inks. These tristimulus values can then be used in
Neugebauer’s equations, Equation 5.5. It has to be pointed out here that Equation 5.5 is a simple model that is
not directly applicable to all practical print situations. First of all, in these equations the effect of optical dot gain
is ignored. Secondly, as in the Murray-Davies formula, it is assumed that ai denotes the real physical coverage of
ink i. Thirdly, it is also assumed that the inks are not penetrating into each other, or paper. In many practical
cases, there is always some ink penetration and optical dot gain cannot be ignored. In these cases this simple
model does not apply and more complicated models should be used [2, 25, 44, 56]. One simple way of
19
modifying Neugebauer’s model to account for optical dot gain is to follow the Yule-Nielsen modification,
Equation 3.2, and do the same kind of modification and get the so called n-modified Neugebauer equations [58],
1/ n
 X ave

 X i1 / n 
 1/ n 
 1/ n 
 Yave    ai  Yi 
i
1/ n 
 Z ave
 Z i1 / n 




Equation 5.8
Although the models that present the print in practice are more complicated these simple models can still be used
to approximate the resulting color in some printing situations. The question that arises now is how to know the
fractional area, ai, covered by each color. Let c, m, y and k be the fractional coverage for cyan, magenta, yellow
and black, respectively. The fractional coverage for each color can be approximated by Demichel’s equations
assuming a semi-stochastic overlap behavior [59]. Let us give two examples. First, assume that we want to find
out the coverage for pure cyan. By pure cyan we mean that we only have cyan and no other ink on/under that.
That means, only cyan and no magenta, no yellow and no black. Then, the coverage for pure cyan can be
calculated by using the probability concept. Since it is assumed that the colored dots are printed independently,
the possibility of only having cyan can be approximated by,
ac  c(1  m)(1  y)(1  k )
Equation 5.9
Using the same logic the fractional area for blue (cyan and magenta, i.e. both are present) can be approximated
by,
acm  cm(1  y)(1  k )
Equation 5.10
The fractional areas for all 16 colors can be approximated by similar equations. Observe that the assumption of
semi-stochastic overlap behavior is fulfilled when the color channels are FM-halftoned and independent of each
other. Also, when using AM-halftoning with different screen angles for different channels Demichel’s equations
can be used to approximate the average color with good acceptance.
5.4
AM Color Halftoning
The concept of AM halftoning for gray scale images has already been introduced in previous sections. Since a
color image is divided into its color channels, which most commonly are Cyan, Magenta and Yellow (and
Black), it is halftoned by halftoning its color channels. In Figure 5.2 the test image is AM halftoned at 50 lpi and
300 dpi. An enlargement of a part of this image is also shown. As can be seen the color channels, here C,
a
b
Figure 5.2. The test image is AM-halftoned using 50 lpi and 300 dpi. The screen angle for the four
channels (C, M, Y and K) is 45 degrees. An enlargement of a part of the image is shown in b.
20
M, Y and K, have been halftoned using the same screen angle, 45 degrees. As long as the print press is stable
and there is no miss-registration, i.e. the printed dots are printed at the same position they are supposed to, using
the same screen angle will not introduce any problem. However, in practice there are always some kind of missregistration and consequently using the same screen angle can cause color shift and unwanted Moiré pattern.
Figure 5.3 illustrates the problems. The image shown in Figure 5.3a is the image that is supposed to be printed. If
we now have some positioning error, that means the printed dots (cyan dots in this example) are shifted in
position, then it could lead to the case shown in Figure 5.3b, which is reproduced as a different color. Figure 5.3c
illustrates another problem, the visible and disturbing Moiré pattern, which is because that some screen angle
error has occurred.
a
b
c
Figure 5.3. The positioning and angle error can cause color shift and unwanted Moiré pattern. a)
Correct registration. b) Miss-registration in position. c) Miss-registration in angle.
In order to reduce the effect of miss-registration, in practice four different angles are used for Cyan, Magenta,
Yellow and Black. Since the human eye is less sensitive for raster at 45 degrees the strongest color, Black, is
halftoned at this angle. The weakest color, Yellow, is printed at 0 degrees, where the human eye is most
sensitive. The other two colors are placed in between with the same distance to Yellow and Black. Therefore
Cyan and Magenta are normally halftoned at 15 and 75 degrees, respectively. Figure 5.4 shows the test image
being AM-halftoned with different screen angles being used for C, M, Y and K channel. An enlargement of a
part of this image is also shown.
a
b
Figure 5.4. The test image is AM-halftoned using 50 lpi and 300 dpi. The screen angle for the four
channels (C, M, Y and K) is 15, 75, 0 and 45 degrees, respectively. An enlargement of a part of the
image is shown in b.
A simple example on how to change the angle when halftoning has already been shown in previous sections. A
more detailed description on how to screen an image with an arbitrary angle can be found in [28].
Although using different angles for different channels reduce the effect of miss-registration it introduces a new
type of patterns, Rosette patterns, which are quite visible at lower screen frequencies. Two different types of
21
Rosette patterns that may occur are shown in Figure 5.5. In Figure 5.5a there is a dot at the center of the rosette
while in the image shown in Figure 5.5b the center is empty.
a
b
Figure 5.5. Two different types of Rosette patterns that may occur are shown. a) there is a dot at
the center of the rosette. b) the center is empty.
Since the problems caused by miss-registration when using the same screen angle are more severe than the
presence of the rosette patterns, the screen angles discussed above are commonly used for the four color
channels in printing industry.
5.5
FM Color Halftoning
Like AM halftoning of color images the FM halftoning is also applied to the color channels that the image is
divided into. Normally the color channels are halftoned independently. Figure 5.6 shows three FM-halftoned
color images where the channels (C, M and Y) are halftoned independently.
a
b
c
d
e
f
Figure 5.6. The test image has been halftoned a) by dispersed dot ordered dithering with 33 tone
levels in each channel. b) by non-modified error diffusion using Floyd and Steinberg’s error filter.
c) by the iterative halftoning method, IMCDP . d) Enlargement of a part of the image shown in a.
e) Enlargement of a part of the image shown in b. f) Enlargement of a part of the image shown in c.
22
In Figure 5.6a the image has been halftoned by dispersed ordered dithering with 33 tone levels for each channel.
The image in Figure 5.6b is halftoned by non-modified error diffusion using Floyd and Steinberg’s error filter. In
Figure 5.6c the image has been halftoned by the iterative halftoning method IMCDP. Enlargement of a part of
these images are also shown. In recent years researchers have been looking at the possibility of halftoning
different color channels in a dependent manner in order to reduce the color noise [23, 29, 35]. It has also been
shown that dependent color halftoning not only increases the print quality but also reduces the amount of ink
needed to print the same image [23]. In Figure 5.7 we show two examples. The color channels in the image
shown in Figure 5.7a have been halftoned independently by IMCDP. The image shown in Figure 5.7b has been
halftoned by an extended version of IMCDP where the Cyan and Magenta channels are halftoned dependently.
As can be seen the image shown in Figure 5.7b is much homogenous and has less color noise. Notice that no
color matching calibration has been performed when making these images and therefore the colors in these two
images might appear different [23]. How much they differ in color depends on the characteristics of the
reproduction device. Enlargements of a part of these images are also shown in this figure.
a
b
c
d
Figure 5.7. The image has been halftoned by IMCDP. a) The C, M and Y channels have been
halftoned independently. b) The C and M channels have been halftoned dependently. c)
Enlargement of a part of the image shown in a. d) Enlargement of a part of the image shown in b.
6. Quality Aspects of Halftoning
There is no doubt that halftoning is one of the most important parts of the image reproduction process, especially
in printing. The print quality is therefore very much dependent upon the quality of halftoning. Judging the
quality of a halftoning method is however not so simple because there are other factors that also affect the print
result. For example, a certain FM halftoning method might result in better print quality when the image is printed
by ink-jet but the same technique could give rise to poor print quality when using a certain print press, just
because the printing method might not be able to print the distinct FM dots. In this case one would prefer AM
halftoning. Thus, studying the quality of a halftoning method is rather difficult. However, it is always useful to
have tools to study the characteristics of halftoning methods. During recent decades researchers have tried to find
different tools that objectively measure the quality of halftoning [1, 4, 19, 20, 34, 51]. In many of these tools the
digital halftoned image is compared with the digital original image. The quality is then based on the similarity
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between these two images. However, this is not so simple that it might seem to be because the “similarity”
between two or more images is not simple to be objectively defined. However, in many of the works the
similarity is based on a model of human visual system [1, 5]. If the printer and the other involved factors are
possible to be modeled they can also be used in the tool. If we now concentrate on FM halftoning we can define
a number of aspects or demands that a good FM halftoning should fulfill. One of the demands is that the final
image is structure-less. Look at the images halftoned by non-modified error diffusion in Figures 4.11a and b.
Although the images are quite smooth but the structures in the mid-tones are disturbing. Another demand is, as
mentioned earlier, that the halftoned image is as similar as possible to the original image. This similarity aspect
should however be connected to the human eye. Since the human eye acts as a low-pass filter the difference
between these two images should be studied in the frequency (for example Fourier) domain. It is desirable to
have no difference in lower frequencies and the noise (the differences) shifted to higher frequencies, i.e. the blue
noise characteristics [51, 52]. However, this doesn’t always correlate with what we see. For example, the image
halftoned by non-modified error diffusion (Figure 4.11b) gives smaller difference compared to the one halftoned
by the modified error diffusion (Figure 4.11c) but due to the fact that the latter image is more structure-less,
though it is a bit noisy, it might be preferred over the image in Figure 4.11b [19]. Another important behavior of
FM halftoning methods that should be studied is how they operate in the highlights (and shadows). It is desirable
that the distinct FM dots are placed homogenously in these regions. Figure 6.1 illustrates this demand. The
images shown in Figures 6.1a, b and c have been halftoned by non-modified error diffusion, modified error
diffusion (the simple modification discussed in Section 4.3) and IMCDP. As can be seen non-modified error
diffusion operates very poorly in the highlight regions.
a
b
c
Figure 6.1. A constant image with 2% coverage has been halftoned by a) non-modified error
diffusion. b) modified error diffusion (simple modification discussed in Section 4.3). c) IMCDP
7. Summary and Discussion
In this chapter the most fundamental and basic concepts related to halftoning have been discussed and briefly
described. A number of renowned halftoning methods have been described and their results have been shown.
Going into more details of the definitions and methods presented in this chapter is however out of the scope of
this chapter. The interested reader is therefore referred to the references listed below. The aim has however been
to give a brief overview on halftoning and make the reader familiar with it and its related concepts. The FM
halftoning methods presented in this chapter have been programmed by the authors and were run in Matlab. For
a number of AM-halftoned images Adobe PhotoShop was used. Although the print quality has dramatically been
increased during recent years the researches on halftoning have not yet ended and there will still be more studies
carried out on this subject for years to come.
24
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