1. Digital Image Processing & Pattern Analysis (CSCE 563) Image Compression
Prof. Amr Goneid
Department of Computer Science & Engineering
The American University in Cairo

2. Image Compression Definitions Image Compression Models
Prefix (Huffman) Coding
Run-Length Coding
Bit-Plane Coding
Constant Area Coding
Lossy Compression
Standards
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3. 1. Definitions Symbol: A one-to-one representation of a single entity.
Alphabet:
A finite set of symbols.
Message:
A sequence of symbols.
Encoding:
Translating symbols to a string of bits.
Decoding:
The reverse.
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4. Definitions (continued)
Coding Redundancy
Interpixel Redundancy
Psychovisual Redundancy
Lossless Compression
Lossy Compression
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5. 2. Image Compression Models
General Model
f(x,y)
Source
Encoder
Channel
Encoder
Channel
g(x,y)
Source
Decoder
Channel
Decoder
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6. Source Encoder/Decoder
f(x,y)
Mapper
Quantizer
Symbol
Encoder
g(x,y)
Inverse
Mapper
De-
Quantizer
Symbol
Decoder
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7. Source Encoder/Decoder
Mapper: e.g. FFT, DCT
Quantizer: makes use of psychovisual redundancy (e.g. scalar or vector quantization)
Symbol Encoder: e.g. Huffman, Run-length,
Also called redundancy encoder
Symbol Decoder: the reverse
Inverse Mapper: e.g. IFFT, IDCT
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8. 3. Prefix (Huffman) Coding
Terminology
Coding Efficiency & Redundancy
Variable Length Coding
Building The Optimal Merge Tree
File Compression
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9. (a) Terminology Binary Merge Tree:
A binary tree with external nodes representing entities and internal nodes representing merges of these entities.
Optimal Binary Merge Tree:
The sum of paths from root to external nodes is optimal (e.g.minimum)
Fixed and Variable Length Coding
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10. Example: Coding Tree for 4-Symbol Alphabet (a,b,c,d)
Encoding:
a 00
b 01
c 10
d 11
Decoding:
b c a d a
This is fixed length coding
abcd 1 ab cd 1 1 a b c d
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11. (b) Coding Efficiency & Redundancy
di =Length of path from root to symbol (i) = no. of bits representing that symbol.
Pi = probability of occurrence of symbol (i) in message.
n = size of alphabet.
< L > = Average Symbol Length = 1 i  n Pi di bits/symbol (bps)
For fixed length coding, di = d = constant, < L > = d (bps)
Is this optimal (minimum) ? Not necessarily.
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12. Coding Efficiency & Redundancy
The minimum length (in bits) is called the Entropy defined as:
H = <L>min = 1 i  n Pi log2 (1/ Pi) bps
When the probabilities Pi of n symbols are all equal, we have fixed length coding. In this case:
H = log2 n
(e.g. the 256 ASCII characters require 8 bits per symbol)
When the probabilities in a given message are not equal, we could have H < log2 n
(e.g. the message “aaaabbcc” in a 4-symbol alphabet has H = 1.5 bps)
In this case, if we are using fixed codes of < L > = 2 bps, there will be a coding redundancy
R = 1 – H / < L > 0  R  1
The coding efficiency is:
 = 1 – R = H / < L >    1
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13. Example Symbol(i) code pi -log pi -pi log pi a 00 0.5 1 0.5
4- Symbol Alphabet (a,b,c,d)
Message : abbcaada
Symbol(i) code pi log pi -pi log pi a b
c 10 0, d
Entropy H = = 1.75 (bps), < L > = 2 (bps)
 = H / < L > = 1.75 / 2 = 0.875
R = 0.125
This is not optimal
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14. (c) Variable Length Coding (Huffman Coding)
Only when all probabilities are equal that fixed length coding is optimal
To achieve optimality, use optimal binary merge trees to code symbols of unequal probabilities.
Huffman Coding: More frequent symbols occur nearer to the root
( shorter code lengths), less frequent symbols occur at deeper levels (longer code lengths).
Example:
a (50%) (0) , b (25%) (10) , c (12.5%) (110) , d (12.5%) (111)
< L > = ( 1* * * * 0.125) = 1.75 (bps)
 = H / < L > = 1.75 / 1.75 = , R = 0.0
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15. (d) Building The Optimal Merge Tree
Merge Table
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16. Optimal Merge Tree for the Example
Symbols exist at leaves, i.e., no symbol code is the prefix of another symbol code.
abcd 1 a
bcd 1 cd b 1 c d
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17. Example: 8 x 8 image Red 19 Black 17 Green 16 Blue 5 Cyan 4 Magenta 2
Yellow 1
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18. Huffman code for the example
red 00
black 01
green 10
blue 111
cyan 1100
magenta 11010
yellow 11011
< L > = 2.34 bps
Compression ratio = 0.293
for 256 colors
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19. (e) File Compression
Variable Length Codes can be used to compress files. Symbols are initially coded using ASCII (8-bit) fixed length codes.
Steps:
1. Determine Probabilities of symbols in file.
2. Build Merge Tree (or Table)
3. Assign variable length codes to symbols.
4. Encode symbols using new codes.
The Compression Ratio = < L > / 8
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20. 4. Run-Length Coding
Each row is coded as a sequence of lengths describing successive runs of black and white pixels.
Assume that the first run of each row is always white (which could have zero length)
The 2D version is the standard FAX coding
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21. RL Coding Example Run-Length for 2 rows Row 1 : 0 7 2 3 1 4 3
Symbols can be coded using Huffman code
Inefficient for high rate of 0 , 1 transitions
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22. 5. Bit-Plane Coding Bit-Plane decomposition For example for
8-bit gray level 1 7
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23. Example of Bit Plane Decomposition
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24. Bit-Plane Coding Decompose image into bit planes
Each binary plane is coded using binary coding technique (e.g Run-Length)
Small changes in gray level produce frequent 0,1 transitions
e.g. pixel(1) = 127 =
pixel(2) = 128 =
Solution: Code planes using Gray-Code
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25. Gray Code Successive code words differ only by one bit
e.g. pixel(1) = 127 =
pixel(2) = 128 =
g(i) = a(i)  a(i+1)
i = 0 ,1 , .. ,6
g(7) = a(7)
a7……..a0
g7 …... g0
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26. 6. Constant Area Coding
Four 3x3 blocks in binary bit plane, normally coded in 36 bits.
Block Coding scheme:
white 0, black 10
mixed 11
Resulting Code (14 bits):
11( ) 10
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27. 7. Lossy Compression Q Q Transform Coding T(g) = FFT or DCT Symbol
Encoder
f(NxN)
QTZR
T(g) G
g(nxn) G’
(N/n)2
subimages Q
f’(NxN)
Merge
T-1(G’)
De-
QTZR
Symbol
Decoder g’ G’
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28. Fidelity Criteria Root-Mean-Square Error
erms = [(1/N2) x y [f ’(x,y) – f(x,y)]2]1/2
Signal-to-Noise Ratio
x y f ’(x,y)2
SNR = ___________________
x y [f ’(x,y) – f(x,y)]2
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29. DCT Coding Example(1)
Image (I) 57x62 DCT(I)
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30. Zigzag Mapping of 8x8 Block; ; ; ; ; ; ; DC
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31. Example(1) continued
|DCT(I)| Highest 25%
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32. DCT Coding Example(2)
100 % Highest 43 %
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33. Example(2) continued
Highest 10 % Highest 1 %
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34. 8. Standards
CCITT
Committee of the International Telephone and Telegraph
Run-length with Variable Length Coding
JPEG (Joint Photographic Experts Group)
DCT on 8x8 pixel regions. Variable length encoding.
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35. JPEG
Five Steps:
1. Transform image to luminance/chrominance color space (YCbCr).
2. Reduce the color components (optional).
3. Partition image into 8 x 8 pixel blocks and perform the DCT on each block.
4. Quantize resulting DCT coefficients.
5. Entropy code the reduced coefficients.
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36. JPEG DC value defines the overall background brightness
64 HARMONIC FREQUENCY VALUES DC = BACKGROUND BRIGHTNESS V DC
QUANTIZATION, SELECT HIGHEST VALUES, ZERO OUT LESSER VALUES V
RUN-LENGTH CODE FOR ZERO VALUES AND HUFFMAN CODING
VARIABLE BIT LENGTH CODE FOR EACH COLOR BLOCK
DC value defines the overall background brightness
63 harmonic frequency values define the block pattern
Applying a quantization threshold for lossy compression
High threshold = high compression ratio = low image quality
Zero out all frequency values which are below the threshold
Run length coding defines a string of zeroes by a single number
Huffman coding uses variable bit length output code for each block
Prof. Amr Goneid, AUC