Digital Watermarking
7. Image Compression
H. Danyali
hdanyali@ieee.org
Digital Watermarking, Shiraz University of Technology
Raw (uncompressed) Data
Picture size
(pixels)
Bits/pixels
Frames/sec
Bit rate
Common application
600 x 800
24
-
11.2 Mbits
Screen images
1200 x 1600
24
-
46.08 Mbits
2M pixels digital photos
2048 x 2560
24
-
125.83 Mbits
High quality images
96 x 128
24
7.5
2.21 Mbits/sec
Videophone
288 x 352
24
30
72.99 Mbits/sec
Video conference
480 x 720
24
30
248.83 Mbits/sec
Standard TV
1080 x 1920
24
30
1.49 Gbits/sec
High-definition TV
1200 x 1600 x 24 bits/pixel = 4608000 bits
Downloading time (over a 28.8 kbps link) = 26.67 min.
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Image Compression
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Data and Information
• Data and information are not synonymous.
• Data are the means by which information
is conveyed.
• Various amount of data may be used to
represent the same amount of information.
Data Redundancy
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Lossy vs. Lossless Compression
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Some General Concepts
• How Can an Image be Compressed, AT ALL!?
– If images are random matrices, better not to try any compression
– Image pixels are highly correlated  redundant information
• Information, Uncertainty and Redundancy
– Information is uncertainty (to be) resolved
– Redundancy is repetition of the information we have
– Compression is about removing redundancy
• Entropy and Entropy Coding
– Entropy is a statistical measure of uncertainty, or a measure of
the amount of information to be resolved
– Entropy coding: approaching the entropy (no-redundancy) limit
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Some General Concepts
• Bit Rate and Compression Ratio
– Bit rate: bits/pixel, sometimes written as bpp
– Compression ratio (CR):
number of bits to represent the original image
–
CR =
number of bits in compressed bit stream
• Binary, Gray-Scale, Color Image Compression
– Original binary image: 1 bit/pixel
– Original gray-scale image: typically 8bits/pixel
– Original Color image: typically 24bits/pixel
• Lossless, Nearly lossless and Lossy Compression
– Lossless: original image can be exactly reconstructed
– Nearly lossless: reconstructed image nearly (visually) lossless
– Lossy: reconstructed image with loss of quality (but higher CR)
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Data Redundancy
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Coding Redundancy
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Coding Redundancy (cont.)
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Coding Redundancy, An Example
Assume rk : [0,1] represents the gray levels of an image
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A Graphic Representation of Data Compression
l2 ( rk ) increases as pr ( rk ) decreases
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Interpixel / Interframe Redundancy
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Pixel Correlations
Notes:
1- 3 dominant ranges of gray-level value in both histograms.
2- High correlation between pixels separated by 25 and 90
samples in (f) related to the spacing between the matches
in (b). n  1    0.9922 in (a) and 0.9928 in (b)
3- Adjacent pixels of both images are highly correlated.
Interpixel redundancy: also known as spatial redundancy,
geometric redundancy, interaframe redundancy
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Illustration of run-length coding
The binary image can be more efficiently represented by the value and the length of its constant gray-level runs
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Compression by Quantization
• Human visual system (HVS) has limitations; good
example is quantization. Conveys information but
requires much less
IGS: Improved gray-scale quantization recognizes the eye’s inherent sensitivity to edges and breaks
them up by adding to each pixel a pseudo random number (generated from the low order bits of
neighboring pixels) before quantizing.
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Fidelity Criteria
• Objective fidelity criteria
– root-mean-square (rms) error
– mean-square signal-to-noise ratio (SNRms or SNR),
PSNR
Pixel error:
Image error:
– Other criteria (HVS-Based, SSIM)
• Subjective fidelity criteria
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A Subjective Fidelity Criteria
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Image Compression Models
Source encoding: to remove input redundancies
Channel encoding: to increase noise immunity for the source encoder’s output
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Source Encoding Model
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Channel Encoding
• Channel encoding is important when the
channel is noisy or prone to error.
• An example of channel coding: Hamming
encoding
- It is based on appending enough bits to the data to
ensure that some distance between the code words
exist.
- Distance between code words: minimum number of
digits must change in one word so that the other word
results.
- Example: Distance between 010110 and 110011 is 3.
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An Example of Hamming Encoding
•
The 7-bit (4-bit data + 3-bit redundancy):
h1h2 h3h4 h5h6 h7
•
Encoder:
–
4-bit binary data:
b3b2b1b0
h1  b3  b2  b0 (even parity bits for the fields b3b2b0 )
h2  b3  b1  b0 (even parity bits for the fields b3b1b0 )
h4  b2  b1  b0 (even parity bits for the fields b2b1b0 )
h3  b3
h5  b2
h6  b1
h7  b0
–
–
Distance between code words: 3
All single bit error can be detected and corrected.
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Hamming Decoding
•
•
To decode a Hamming encoded result, the channel decoder must check the encoded
value for odd parity over the bit fields in which even parity was established (i.e. C4C2C1 )
A single bit-error is indicated by a nonzero parity word C4C2C1 where:
c1  h1  h3  h5  h7
c2  h2  h3  h6  h7
c4  h4  h5  h6  h7
•
If a nonzero value is found, the decoder simply complements the code word bit position
indicated by the parity word. The final decode binary value is:
h3h5h6h7
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Error-Free (Lossless) Compression
• Variable length coding
– Huffman coding
– Arithmetic coding
• LZW (Lampel-Ziv-Welch) coding
• Bit-plane coding
• Lossless predictive coding
Lossless methods:
- generally consist of two stages:
1- providing an alternative representation (mapping) of to reduce the interpixel redundancy
2- coding the representation to eliminate coding redundancy (symbol coding)
- normally provide compression ratio (CR) of 2 to 10
- applicable to both binary and gray level images
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Huffman Coding
• Most popular technique
• Yields the smallest possible number of code symbol per
source symbol ( the resulting code is optimal).
• First step: source reduction by ordering the symbols
according to their probability and combining the lowest
probability symbol into a single symbol
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Huffman coding (cont.)
• Second step: code assignment procedure
Uniquely decodable by scanning from left-to-right in Fig 8.12
For example: encoded string 010100111100 reveals code words 01010 (a3), 011 (a1),
1 (a2) , 1(a2), 00(a6)
a3a1a2a2a6
- H (z) = Entropy of the source = 2.14 bits/symbol
- Average length of this code is:
Lavg = (0.4)(1) + (0.3)(2) + (0.1)(3) + (0.1)(4) + (0.06)(5) + (0.04)(5) = 2.2 bits/symbol
- Code efficiency = H(z) / Lavg = 0.973
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Arithmetic Coding
• Unlike variable-length code a one-to-one correspondence between
source symbol and code words does not exist. Instead an entire
sequence of source symbol is assigned to a single arithmetic code
word.
• The code word itself defines an interval of real numbers between 0
and 1.
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Lossy Gray-Scale Image
Compression
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Quantization
• Quantization: Widely Used in Lossy Compression
– Represent certain image components with fewer bits (compression)
– With unavoidable distortions (lossy)
• Quantizer Design
– Find the best tradeoff between
maximal compression  minimal distortion
• Scalar Quantization

Uniform scalar
quantization:
8
Non-uniform scalar
quantization:

24
1
2

40
3

248
...
4
Quantization
• Vector Quantization
– Group multiple image components together  form a vector
– Quantize the vector in a higher dimensional space
– More efficient than scalar quantization (in terms of compression)
Vector quantization:
image
Gray-level22
component
Gray-level
1 1
image
component
From Prof. Al Bovik
30
Lossy Image Coding
(Compression) Methods
• Spatial domain methods
• Transform coding
– Adaptive (adaptive to the local image content)
– nonadaptive (fixed for all subimages)
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Ideas on Lossy Image Compression
code each block
independently
• Block-Based Image Compression
Partition
image
MxM
MxM
MxM
MxM
MxM
MxM
MxM
MxM
MxM
From Prof. Al Bovik
• Transform-Domain Compression
– Scalar or vector quantization of transform coefficients (instead of
image pixels)
Transform Coding Scheme
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Discrete Cosine Transform (DCT)
• 2D-DCT:
4C (u )C (v) N 1 N 1
 (2m  1)u
X (u , v) 
x
(
m
,
n
)
cos


N2
2N

m 0 n 0
• Inverse
2D-DCT:
 ( 2 m  1)u
x( m, n )   C (u )C ( v ) X (u, v ) cos
2N

u 0 v 0
N 1 N 1
where
 1

C (u )   2
 1
u  1,, N  1
N
N
2N
From Prof.
Al Bovik
reflected periodic
extension by DFT
  ( 2 n  1)v 
 cos

2N
 

u0
• DFT vs. DCT
periodic extension
by DFT
  (2n  1)v 
 cos

2
N
 

discontinuities:
high frequencies
continuous
2D-DCT
image block
55
55

 51
49
50

 51
 51

 52
36  34 25
25  12 48
40 43 54
49 51 49
50 51 48
52 52 51
51 52 54
51 53 51
66
59
51
51
54
55
55
53
35
38
42
37
36
41
42
42
4
2
3
3
2
4
1
2
37
41
39 
39 
43

41
45

41
DC component
2D-DCT
low frequency
high frequency
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low
frequency
 313
 38

  20
  10
 6

 2
 4

 3
high
frequency
56  27 18 78  60 27  27 
 27 13 44 32  1  24  10 
 17 10 33 21  6
 16  9 
8
9 17
9  10  13 1 
1
6 4
3 7
5 5 

3
0 3
7 4
0 3 
4
1  2  9 0
2 4 

1
0  4  2 1
3 1 
DCT block
JPEG Compression
– Partition the image into 8x8 blocks, for each block
183
183

179
177
178

179
179

180
160
153
168
177
178
180
179
179
94
116
171
179
179
180
180
181
153
176
182
177
176
179
182
179
194
187
179
179
182
183
183
181
163
166
170
165
164
169
170
170
132
130
131
131
130
132
129
130
165
169
167 
167 
171

169
173

169
- 128
55
55

 51
49
50

 51
 51

 52
DCT
 313
 38

  20
  10
 6

 2
 4

 3
56  27 18 78  60 27  27 
 27 13 44 32  1  24  10 
 17 10 33 21  6
 16  9 
8
9 17
9  10  13 1 
1
6 4
3 7
5 5 

3
0 3
7 4
0 3 
4
1  2  9 0
2 4 

1
0  4  2 1
3 1 
scalar
quantization
36  34 25
25  12 48
40 43 54
49 51 49
50 51 48
52 52 51
51 52 54
51 53 51
 20
 3

 1
 1
 0

 0
 0

 0
66
59
51
51
54
55
55
53
35
38
42
37
36
41
42
42
5 3
2 1
1 1
0
0
0
0
0
0
0
0
0
0
zig-zag scan
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4
2
3
3
2
4
1
2
37
41
39 
39 
43

41
45

41
1 3
2 1
1
1
1
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0

0
0

0
JPEG Compression
– Adjust Quantization Step to Achieve Tradeoff between CR and distortion
Original:
100KB
JPEG: 9KB
JPEG: 5KB
– Artifacts:
Inside blocks: blurring (why?); Across blocks: blocking (why?)
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Wavelet and JPEG2000 Compression
• Wavelet Transform
Energy Compaction
Lower Entropy
Wavelet and JPEG2000 Compression
• Bitplane Coding
– Scan bitplanes
from MSB to LSB
– Progressive (scalable)
JPEG (64:1)
sign
..
...
0
...
0
...
0
...
0
...
1
..
..
...
s s s s s s s s s s s s s s s s s s s s s
msb 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2
1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3
1 1 1 1 1 0 0 0 0 0 0 0 0 0
4
1 1 1 1 1 1 1 0 0
5
1 1
6
7
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ...
. . . . . . . . . . . . . . . . . . . . . . .
JPEG2000 (64:1)
Wavelet Image Coding
Examples:
• EZW (1993)
• SPIHT (1996)
• JPEG2000
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Mutiresolution Wavelet Decomposition
Multi level wavelet decomposition of an image
2D-DWT
LL1
Input image
LL2 HL2
HL1
HL1
LH2 HH2
2D-DWT
LH1
HH1
LH1
HH1
0
0
63
127
255
0
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127
127
255
255
0
63
127
255
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Bitplanes and Self-Similarity Across Scales
• Most of the image’s energy is concentrated in the highest level of the
subband pyramid.
• There is a spatial self-similarity between subbands in different levels of the
pyramid.
• Coding of a subband in higher level of the pyramid is started with higher
bitplane during a bitplane coding process.
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Bitplane coding
Sign bit
37
0
18 -20 14
0
1
0
9
0
-6
1
Bitplane level 5
1
0
0
0
0
0
Bitplane level 4
0
1
1
0
0
0
Bitplane level 3
0
0
0
1
1
0
Bitplane level 2
1
0
1
1
0
1
Bitplane level 1
0
1
0
1
0
1
Bitplane level 0
1
0
0
0
1
0
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Set Partitioning in Hierarchical Trees (SPIHT)
Introduced by A. Said and W. A. Pearlman, in 1996
http://www.cipr.rpi.edu/research/SPIHT/
Key Ideas
• Multi-pass zero-tree coding.
• Ordered bit plane transmission.
• Exploitation of self-similarity
across wavelet scales.
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SPIHT Definitions
•
Sets




•
O (i,j): set of coordinates of all offspring of node (i,j).
D (i,j): set of coordinates of all descendants of node (i,j).
H : set of coordinates of all spatial orientation tree roots.
L (i,j): D (i,j) - O (i,j).
Lists
 LIS: List of Insignificant Sets
• Type A: Entries are elements D (i,j)
• Type B: Entries are elements L (i,j)
 LIP: List of Insignificant Pixels
 LSP: List of Significant Pixels
•
Significance test
1,

S n (T )  
0,

max {| ci , j |}  2 n
( i , j )T
otherwise
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SPIHT Algorithm
1. Initialization
• Output n  log 2 (max (i , j ) {| ci , j |}) 
• Set the LSP as an empty list
• Add (i, j )  H to the LIP and only
those with descendants to the
LIS as type A.
2. Sorting Pass
• Sort each entry in the LIP.
• Sort each entry in the LIS type A.
• Sort each entry in the LIS type B.
3. Refinement Pass
• Output the nth most significant bits
for all entries in the LSP except
those added in the last sorting pass.
4. Quantization Step Update
• Decrement n by 1 and go to Step 2.
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LSP
LIP
(0,0)
(0,1)
(1,0)
(1,1)
LIP
S. P.
LIS
LIS
S. P.
(0,1)A
(1,0)A
(1,1)A
SPIHT Sorting Pass
LSP
LIP S. P.
LIP
(0,0)
(0,1)
(1,0)
(1,1)
10
10
11
0
LIS S. P. (type A)
(0,0)
(0,1)
(1,0)
10
11
0
0
(0,4)
(0,5)
(0,4)
(0,5)
(1,4)
(1,5)
1
1
0
If leaves
exist
(1,4)
(1,5)
1 1
1
11
LIS
(0,1)A
(1,0)A
(1,1)A
LIS S. P. (type B)
(0,1)B
(1,0)B
1
0
(0,4)
(0,5)
(1,4)
(1,5)
Add as type A,
remove (0,1)
1
1
Output bitstream
Header Bitplane n
Bitplane n-1
1010110 … 11011001 … 1 … 0 … 10 …
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Bitplane 0
SPIHT Properties
• Provides good image quality (high PSNR).
• Produces a fully embedded bit stream (optimized
for progressive image transmission).
• Can code to exact bit-rate or distortion.
• Can be used for lossless compression.
• Fast coding/decoding (nearly symmetric).
• Has wide application, completely adaptive.
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Demo
Bpp = 0.31
SPIHT PSNR = 35.12 dB
(Show a demo by using VCDemo software)
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JPEG PSNR = 31.8 dB (quality factor 15%)
Original Lena and Barbara Images
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JPEG (0.25bpp, CR: 32:1)
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SPIHT (0.25bpp, CR: 32:1)
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Comparison (JPEG and SPIHT at CR :32:1)
Top: JPEG
Bottom: SPIHT
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Failure of Logarithmic Wavelet Transform,
An Example
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Wavelet Packet
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Wavelet Packet (Cont.)
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Wavelet Packet (Cont.)
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Binary Image Compression
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Run Length Coding
• Run Length
– The length of consecutively identical symbols
• Run length Coding Example
what's
stored: '1'
row m
7
5
8
3 1
• When Does it Work?
– Images containing many runs of 1’s and 0’s
• When Does it Not Work?
what's
stored: '1'1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
row m
Run Length Coding
CCITT test image No. 1
Size: 17282376
1 bit/pixel (bpp)
original: 513216 bytes
compressed: 37588 bytes
CR = 13.65
Run Length Coding
• Decoding Example
A binary image is encoded using run length code row by row, with
“0” represents white, and “1” represents black. The code is given by
Row 1: “0”, 16
Row 2: “0”, 16
Row 3: “0”, 7, 2, 7
Row 4: “0”, 4, 8, 4
Row 5: “0”, 3, 2, 6, 3, 2
Row 6: “0”, 2, 2, 8, 2, 2
Row 7: “0”, 2, 1, 10, 1, 2
Row 8: “1”, 3, 10, 3
Row 9: “1”, 3, 10, 3
Row 10: “0”, 2, 1, 10, 1, 2
Row 11: “0”, 2, 2, 8, 2, 2
Row 12: “0”, 3, 2, 6, 3, 2
Row 13: “0”, 4, 8, 4
Row 14: “0”, 7, 2, 7
Row 15: “0”, 16
Row 16: “0”, 16
decode
Decode the image
Run Length Coding
• Decoding Example
A binary image is encoded using run length code row by row, with
“0” represents white, and “1” represents black. The code is given by
Row 1: “0”, 16
Row 2: “0”, 16
Row 3: “0”, 7, 2, 7
Row 4: “0”, 4, 8, 4
Row 5: “0”, 3, 2, 6, 3, 2
Row 6: “0”, 2, 2, 8, 2, 2
Row 7: “0”, 2, 1, 10, 1, 2
Row 8: “1”, 3, 10, 3
Row 9: “1”, 3, 10, 3
Row 10: “0”, 2, 1, 10, 1, 2
Row 11: “0”, 2, 2, 8, 2, 2
Row 12: “0”, 3, 2, 6, 3, 2
Row 13: “0”, 4, 8, 4
Row 14: “0”, 7, 2, 7
Row 15: “0”, 16
Row 16: “0”, 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
decode
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
10
11
12
13
14
15
16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Chain Coding
Assume the image contains only single-pixel-wide contours,
like this, not this
contour image
After the initial point position,
code direction only (3bits/step)
region image
From Prof. Al Bovik
n
2
3
1
0
4
7
6
= initial point
Code Stream:
5
(3, 2), 1, 0, 1, 1, 1, 1, 3, 3, 3, 4, 4, 5, 4
m
initial point position
chain code
Digital Watermarking, Shiraz University of Technology
65
Chain Coding
• Decoding Example
The chain code for a 8x8 binary image is given by:
column row
2
3
(1, 6), 7, 7, 0, 1, 1, 3, 3, 3, 1, 1, 0, 7, 7
decode
Decode the image
1
0
4
5
6
7
Chain Coding
• Decoding Example
The chain code for a 8x8 binary image is given by:
column row
2
3
(1, 6), 7, 7, 0, 1, 1, 3, 3, 3, 1, 1, 0, 7, 7
decode
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8
0
4
5
1 2 3 4 5 6 7 8
1
6
7
Lossless Gray-Scale Image
Compression
Digital Watermarking, Shiraz University of Technology
Variable Word Length Coding
• Intuitive Idea
– Assign short words to gray levels that occur frequently
– Assign long words to gray levels that occur infrequently
• How Much Can Be Compressed?
– Theoretical limit: entropy of the histogram
– Practical algorithms (approach entropy):
Huffman coding, arithmetic coding
K 1
  p (k ) log 2 p (k )
k 0
Maximum entropy:
Uniform distribution
H I (k)
typically in-between
0
K-1
gray level k
Minimum entropy:
Impulse (delta) distribution
Variable Word Length Coding: Example
• A 4x4 4bits/pixel original image is given by
Default Code Book
0:
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
2
8
6
6
6
8
8
8
8
8 10 10
9 10 10 14
encode
0010 1000 0110 0110
0110 1000 1000 1000
1000 1000 1010 1010
1001 1010 1010 1110
Bit rate = 4bits/pixel
Total # of bits used to
represent the image:
4x16 = 64 bits
Variable Word Length Coding: Example
• Encode the original image with a CODE BOOK given left
Huffman Code Book
0:
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
0000000
0000001
0001
0000010
0000011
0000100
01
0000101
10
00100
11
0000110
0000111
001010
0011
001011
2
8
6
6
6
8
8
8
8
8 10 10
Total # of bits used to
represent the image:
4+2+2+2+2+2+2+2+2+
2+2+2+5+2+2+4
= 39 bits
9 10 10 14
encode
0001
10
01
01
01
10
10
10
10
10
11
11
00100
11
11
0011
Bit rate = 39/16
= 2.4375 bits/pixel
CR = 64/39 = 1.6410
Predictive Coding
• Intuitive Idea
– Image pixels are highly correlated (dependent)
– Predict the image pixels to be coded from those already coded
• Differential Pulse-Code Modulation (DPCM)
– Simplest form: code the difference between pixels
Original pixels:
82, 83, 86, 88, 56, 55, 56, 60, 58, 55, 50, ……
DPCM:
82, 1, 3, 2, -32, -1, 1, 4, -2, -3, -5, ……
– Key features: Invertible, and lower entropy (why?)
H I (k)
H D(k)
0
K-1
high entropy image
1-K
K-1
reduced entropy image
image histogram (high entropy) DPCM histogram (low entropy)
From Prof.
Al Bovik
Advanced Predictive Coding
• Higher Order (Pattern) Prediction
– Use both 1D and 2D patterns for prediction
1D Causal:
1D Non-causal:
2D Causal:
2D Non-Causal:
• Apply Image Transforms before Predictive Coding
– Decouple dependencies between image pixels
• Use Advanced Statistical Image Models
– Better understanding of “the nature” of image structures implies
potentials of better prediction
Digital Watermarking, Shiraz University of Technology