Digital Image Processing & Pattern Analysis (CSCE 563) Image Compression Prof. Amr Goneid Department...

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Digital Image Processing&

Pattern Analysis(CSCE 563)

Image Compression

Digital Image Processing&

Pattern Analysis(CSCE 563)

Image Compression

Prof. Amr Goneid

Department of Computer Science & EngineeringThe American University in Cairo

Prof. Amr Goneid, AUC 2

Image Compression

Definitions

Image Compression Models

Prefix (Huffman) Coding

Run-Length Coding

Bit-Plane Coding

Constant Area Coding

Lossy Compression

Standards

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.

Definitions (continued)

Coding Redundancy

Interpixel Redundancy

Psychovisual Redundancy

Lossless Compression

Lossy Compression

2. Image Compression Models

General ModelGeneral Model

Source Encoder

ChannelEncoder

Channel

ChannelDecoder

SourceDecoder

f(x,y)

g(x,y)g(x,y)

Source Encoder/Decoder

QuantizerSymbol

EncoderMapper

De-Quantizer

InverseMapper

g(x,y)

f(x,y)

Symbol Decoder

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

3. Prefix (Huffman) Coding

Terminology

Coding Efficiency & Redundancy

Variable Length Coding

Building The Optimal Merge Tree

File Compression

(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

Example: Coding Tree for 4-Symbol Alphabet (a,b,c,d)

a b c d

Encoding:

Decoding:

0110001100

b c a d a

This is fixed length coding

(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.

Coding Efficiency & RedundancyThe minimum length (in bits) is called the Entropy defined as:

H = <L>min = 1 i n Pi log2 (1/ Pi) bpsWhen 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 1The coding efficiency is: = 1 – R = H / < L > 0 1

Example4- Symbol Alphabet (a,b,c,d)

Message : abbcaada

Symbol(i) code pi -log pi -pi log pi

a 00 0.5 1 0.5

b 01 0.25 2 0.5

c 10 0,125 3 0.375

d 11 0.125 3 0.375

Entropy H = 0.5 + 0.5 + 0.375 + 0.375 = 1.75 (bps), < L > = 2 (bps)

= H / < L > = 1.75 / 2 = 0.875

R = 0.125

This is not optimal

(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* 0.5 + 2 * 0.25 + 3 * 0.125 + 3 * 0.125) = 1.75 (bps)

= H / < L > = 1.75 / 1.75 = 1.00 , R = 0.0

(d) Building The Optimal Merge Tree

Merge Tablesymbol prob symbol prob symbol prob symbol prob code

a 0.5 a 0.5 a o.5 abcd 1 0b 0.25 b 0.25 bcd 0.5 10c 0.125 cd 0.25 110d 0.125 111

Optimal Merge Tree for the Example

Symbols exist at leaves, i.e., no symbol code is the prefix of another symbol code.

Red 19

Black 17

Green 16

Blue 5

Cyan 4

Magenta 2

Yellow 1

Example: 8 x 8 image

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

(e) File CompressionVariable 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

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

RL Coding Example

Run-Length for 2 rows

Row 1 : 0 7 2 3 1 4 3Row 2 : 5 5 4 6

Symbols can be coded using Huffman codeInefficient for high rate of 0 , 1 transitions

5. Bit-Plane Coding

Bit-Plane decomposition

For example for

8-bit gray level

Example ofBit Plane Decomposition

Bit-Plane Coding

Decompose image into bit planesEach binary plane is coded using binary coding technique (e.g Run-Length)Small changes in gray level produce frequent 0,1 transitionse.g. pixel(1) = 127 = 0 1 1 1 1 1 1 1

pixel(2) = 128 = 1 0 0 0 0 0 0 0Solution: Code planes using Gray-Code

Gray Code

Successive code words differ only by one bit

e.g. pixel(1) = 127 = 0 1 0 0 0 0 0 0

pixel(2) = 128 = 1 1 0 0 0 0 0 0

g(i) = a(i) a(i+1) i = 0 ,1 , .. ,6 g(7) = a(7)

g7 …... g0a7……..a0

6. Constant Area Coding

Four 3x3 blocks in binary bit plane, normally coded in 36 bits.Block Coding scheme:white 0, black 10mixed 11Resulting Code (14 bits):011(100010001) 100

7. Lossy Compression

Transform Coding

T(g) = FFT or DCT

SymbolEncoder

Symbol Decoder

T-1(G’)Mergef’(NxN)

g(nxn)f(NxN) T(g) G QTZR

(N/n)2

subimages

De-QTZR

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

DCT Coding Example(1)

Image (I) 57x62 DCT(I)

Zigzag Mapping of 8x8 Block

1 2 6 7 15 16 28 29;

3 5 8 14 17 27 30 43;

4 9 13 18 26 31 42 44;

10 12 19 25 32 41 45 54;

11 20 24 33 40 46 53 55;

21 23 34 39 47 52 56 61;

22 35 38 48 51 57 60 62;

36 37 49 50 58 59 63 64

Example(1) continued

|DCT(I)| Highest 25%

DCT Coding Example(2)

100 % Highest 43 %

Example(2) continued

Highest 10 % Highest 1 %

8. Standards

CCITTCommittee of the International Telephone and TelegraphRun-length with Variable Length Coding

JPEG (Joint Photographic Experts Group)DCT on 8x8 pixel regions. Variable length encoding.

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.

RUN-LENGTH CODE FOR ZERO VALUES AND HUFFMAN CODINGVARIABLE BIT LENGTH CODE FOR EACH COLOR BLOCK

64 HARMONIC FREQUENCY VALUES DC = BACKGROUND BRIGHTNESS

V V V V V V V V V V V V V V V V V V V V V V V V V V V V V

DCQUANTIZATION, SELECT HIGHEST VALUES, ZERO OUT LESSER VALUES

V 0 V V 0 V 0 V V V0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

DC value defines the overall background brightness63 harmonic frequency values define the block patternApplying a quantization threshold for lossy compressionHigh threshold = high compression ratio = low image qualityZero out all frequency values which are below the thresholdRun length coding defines a string of zeroes by a single numberHuffman coding uses variable bit length output code for each block

Digital Image Processing & Pattern Analysis (CSCE 563) Image Compression Prof. Amr Goneid Department...

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