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E. Fatemizadeh, Sharif University of Technology, 2011

1

Digital Image Processing

Image Compression

1

• Goal:

– Reducing the amount of data required to represent a digital image.

• Transmission

• Archiving

– Mathematical Definition: • Transforming a 2-D pixel array into a statistically uncorrelated data

set.

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Image Compression

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• Two Main Category:

– Lossless: TIFF (LZW,ZIP)

– Lossy: JPEG and JPEG2000

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Image Compression

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• Example: Original vs. JPEG2000

I am 1,085,496 Byte I am 81,503 Byte

13.31

0.925

R

D

C

R

Best Compressor: 510,576 bytes

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Image Compression

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• Fundamentals:

– Raw image: A set of n1bits

– Compressed image: A set of n2 bits.

– Compression ratio:

– Relative Data Redundancy in A:

– Example: n1= 100KB and n2 = 10Kb, then C0 = 10 , and RD = 90%

• Special cases: 1) n1>>n2 → CR ≈ ∞, RD ≈ 1

• 2) n1 ≈ n2 → CR ≈ 1, RD ≈ 0

• 3) n1 << n2 → CR ≈ 0, RD ≈ -∞

1

2

R

nC

n

11D

R

RC

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Image Compression

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• Three basic data redundancies:

– Coding redundancy

– Spatial/Temporal (Inter-Pixel) redundancy,

– Irrelevant/Psycho-Visual redundancy.

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Image Compression

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• Coding redundancy:

– Type of coding (# of bits for each gray level)

– Image histogram: • rk: Represents the gray levels of an image

• pr(rk): Probability of occurrence of rk

• l(rk): Number of bits used to represent each rk.

• Lavg: Average # of bits required to represent each pixel:

( ) 0,1,2,..., 1kr k

np r k L

n

1

0

( ) ( )L

avg k r k

k

L l r p r

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Image Compression

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• Example (1): Variable Length Coding

1

2

= 0.25(2)+0.47(1)+0.25(3)+0.03(3)=1.81 bits

Code 2: 256 256 8 8 14.42 1 0.774 77.4%

256 256 1.81 1.81 4.42

avg

R D

L

nC R

n

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Image Compression

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• Example (2): Spatial Redundancy

– Uniform intensity

– Highly correlated in horizontal direction

– No correlation in vertical direction

– Coding using run-length pairs:

1 1 2 2, , , ,g RL g RL

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Image Compression

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• Irrelevant/Psychovisual Redundancy:

– Human perception of images: • Normally does NOT involve quantitative analysis of every pixel

value.

– Quantization eliminates psychovisual redundancy

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Image Compression

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• Image Compression using Psychovisual Redundancy:

– A Simple method: Discard lower order bits (4)

– IGS (Improved Gray Scale) Quantization: • Eye sensitivity to edges.

• Add a pseudo random number (lower order bits of neighbors) to central pixel and then discard lower order bits.

169 ÷ 16 = 10.56

167 ÷ 16 = 10.43

11

10

11 × 16 = 176

10 × 16 = 160

Rounded

values

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Image Compression

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Pixel Gray Level SUM IGC

i-1 N/A 0000 0000 N/A

i 0110 1100 0110 1100 0110

i+1 1000 1011 1001 0111 1001

i+2 1000 0111 1000 1110 1000

i+3 1111 0100 1111 0100 1111

• IGS Example:

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Image Compression

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• Sample Results: Original Simple Discard IGS

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Image Compression

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• Image information Measure:

– Self Information

1log log : m-ary units

2 for bit expression ( 0.5 1)

mI E P EP E

m P E I E

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Image Compression

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• Information Channel:

– A Source of statistically independent random events

– Discrete possible events with associated probability:

– Average information per source output (Entropy):

– For “zero-memory” (intensity source) imaging system: • Using Histogram

• This lower band of bits-length

1 2, , , Ja a a

1 2, , , JP a P a P a

1

logJ

j j

j

H P a P a

1

2

0

log /L

r k k k

k

H p r p r bits pixel

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Image Compression

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• Example:

1.6614 bits/pixelH

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Image Compression

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• Shannon First Theorem (Noiseless Coding Theorem):

– For a n-symbol group with Lavg, n average number of code symbol:

,

limavg n

n

LH

n

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Image Compression

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• Fidelity Criteria:

– Objective (Quantitative)

• A number (vector) is calculated

– Subjective (Qualitative)

• Judgment of different subject is used in limited level.

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Image Compression

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• Objective Criteria:

– :input image

– : approximated image

1 1

0 0

1 21 1 2

0 0

1 1 2

0 0

1 1 2

0 0

ˆ, , ,

1 ˆ , ,

1 ˆ , ,

ˆ ,

ˆ , ,

M N

abs

x y

M N

rms

x y

M N

x y

ms M N

x y

e x y f x y f x y

E f x y f x yMN

E f x y f x yMN

f x y

SNR

f x y f x y

ˆ ,f x y

,f x y

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Image Compression

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• Objective Criteria:

– PSNR: Peak SNR

2

10 1 1 2

0 0

10log1 ˆ , ,

: Maximum level of images (255 for 8bits)

M N

x y

LPSNR

f x y f x yMN

L

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Image Compression

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• Subjective Test

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Image Compression

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• Example:

– Original and three approximation. Erms=5.17

Erms=15.67

Erms=14.17

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Image Compression

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• Image Compression Model:

– Encoder: Create a set of symbols from image date.

• Source Encoder: Reduce input redundancy.

• Channel Encoder*: Increase noise immunity (noise robustness)

– Decoder: Construct output image from transmitted symbols

• Source Decoder: Reverse of Source encoder

• Channel Decoder*: Error detection/correction.

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Image Compression

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• Source Encoder-Decoder:

– Mapper: Map input data to suitable format for interpixel redundancies reduction (i.e., Run-Length coding)

– Quantizer: Reduce accuracy of mapper output according to fidelity criteria (i.e., Psychovisual Redundancy)

– Symbol Encoder: Create Fix or variable length symbols (code redundancy)

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Image Compression

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• Channel Encoder-Decoder*:

– Add controlled redundancy to source encoded data

– Increase noise robustness.

– Hamming (7-4) as an example:

3 2 1 0

4 2 1 4 2 1

1 3 2 1

7 6 5 3 0 1

1 1 3 5 7

2 3 1 0 2 2 3 6 7

4

2 3

4 5 6 74 2 1 0

4 bit binary input data:

7 bit binary output data:

Parity bits, Error check:

b b b b

h h h h h h

h b b b c h h h h

h b b b c h h h h

h h h h b b

c h h h hh b

b b

b b

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Image Compression

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• Image Compression Standards

• See Tables 8.3 and 8.4 for more details

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Image Compression

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• Error-Free Compression:

– ZIP, RAR, TIFF(LZW/ZIP)

• Compression Ratio: 2-10

– Steps:

• Alternative representation to reduce interpixel redundancy

• Coding the representation to reduce coding redundancy

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Image Compression

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• Variable-Length Coding:

– Variable-Length Coding

• Assign variable bit length to symbols based on its probability:

• Low probability (high information) more bits

• High Probability (low information) less bits

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Image Compression

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• Huffman Coding:

– Uses frequencies (Probability) of symbols in a string to build a variable rate prefix code.

– Each symbol is mapped to a binary string.

– More frequent symbols have shorter codes.

– No code is a prefix of another. (Uniquely decodable)

– It is optimum!

– CCITT, JBIG2, JPEG MPEG (1, 2, 4), H261-4

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Image Compression

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• Huffman Coding:

– Example:

• We have three symbols a, b, c, and c.

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Image Compression

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• Huffman Coding:

– A source string: aabddcaa

• Fixed Length Coding: 16 bits (ordinary coding)

– 00 00 01 11 11 10 00 00

• Variable length coding: 14bits (Huffman coding)

– 0 0 100 11 11 101 0 0

• Uniquely Decodable:

0 0 1 0 0 1 1 1 1 1 0 1 0 0

a a b d d c a a

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Image Compression

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• Huffman Coding:

– Consider an information source:

– Assume symbols are ranked decreasing with respect to occurrence probabilities:

– We seek for optimum code, the lengths of codewords assigned to the source symbols should be:

1 2, , , mS s s s

1 2 1m mp s p s p s p s

1 2 1m ml l l l

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Image Compression

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• Huffman Coding: – Huffman (1951) Derived the following rules:

• Two least probable source symbols have equal-length codewords.

• These two codewords are identical except for the last bits, with binary 0 and 1, respectively.

• These source can be combined to form a new symbol. – Its occurrence probability is

– Its codeword is common prefix of order lm-1of the two codewords assigned to Sm and Sm-1 , respectively.

1 2 1m ml l l l

1m mP s P s

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Image Compression

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• Huffman Coding: – The new set of source symbols thus generated is referred to as the first

auxiliary source alphabet, which is one source symbol less than the original source alphabet.

– In the first auxiliary source alphabet, we can rearrange the source symbols according to a nonincreasing order of their occurrence probabilities.

– The same procedure can be applied to this newly created source alphabet.

– The second auxiliary source alphabet will again have one source symbol less than the first auxiliary source alphabet.

– This is repeated until we form a single source symbol with a probability of 1.

– Start from the source symbol in the last auxiliary source alphabet and trace back to each source symbol in original source alphabet to find the codewords.

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Image Compression

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• Huffman Coding:

– Source Reduction sequence.

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Image Compression

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g0.05

a b c d e f0.05 0.1 0.2 0.3 0.2 0.1

g0.05

a b c d e f0.05 0.1 0.2 0.3 0.2 0.1

0.1

g0.05

a b c d e f0.05 0.1 0.2 0.3 0.2 0.1

0.1 0.3

0.2 0.6

0.4

1.0

0

0

0

0

0

0

1

1

1

1

1

1

• More Illustration: Symbol Probability Codeword

a 0.05 0000

b 0.05 0001

c 0.1 001

d 0.2 01

e 0.3 10

f 0.2 110

g 0.1 111

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Image Compression

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2.2 bits/symbolavgL

• Example:

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Image Compression

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• Example:

– 8bits monochrome • Estimated Entropy: 7.3838 bits/pixel

– Huffman’s Coding: • MATLAB: 7.428 bits/pixel, CR = 1.077, RD = 0.0715

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Image Compression

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• Near Optimal

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Image Compression

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• Huffman drawbacks: – For large number symbols, construction of

optimal Huffman code is nontrivial.

– Huffman is block coding: • That is, some codeword having an integral number of

bits is assigned to a source symbol.

• A message may be encoded by cascading the relevant codewords. It is the block-based approach that is responsible for the limitations of Huffman codes.

• Computationally not efficient for a message with all possible symbols.

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Image Compression

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• Golomb Coding (1):

– Non-negative integer inputs

– Exponentially decaying probability distribution

– Inputs may be near optimal codes (simple than Huffman)

– JPEG-LS and AVS

– Notation: : Largest integer less than or equal to ,

:Smallest integer greater than or equal to ,

x x Floor

x x Ceil

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Image Compression

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– Goal: • Golomb Coding of n (Non-Negative) with respect to m>0:Gm(n)

– Combination of unary coding quotient and binary representation of remainder “n mod m”

– Unary coding of integer q is: q “1s” followed by a “0”

n m

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Image Compression

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– Steps: • Form unary coding of

• With compute truncated remainder:

• Concatenate the results of two steps

n m

2log , 2 , modkk m c m r n m

truncated to -1 bits 0

truncated to bits o.w.

r k r cr

r c k

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Image Compression

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• Golomb Coding Example (1):

– Goal: G4(9)

– Step #1:

– Step #2:

– Step #3: Concatenate 110 and 01: 11001

Unary Coding9 4 0112

2

2

truncated to =2

log 4 2, 2 4 0

9mod 4 1(0 010 ) 01k

k c

r r

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Image Compression

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• Golomb Coding Example (2):

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Image Compression

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– Easy (binary shift operation) for: • Golomb-Rice or Rice Codes

– Optimality condition:

• Non-negative geometrically distribution:

• With:

2 , 0, modkm c r r n m

1 , 0 1nP n

2

2

log 1

log 1m

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Image Compression

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• Usability (1): – Intensity image distribution are NOT geometrical.

– Difference images are.

• How handle negative value: – Map to Odd and even non-negative integer:

2 0

2 1 0

n nM n

n n

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Image Compression

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• Usability (2):

– Geometrically distribution(s)

– Difference image histogram

– Mapped histogram

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Image Compression

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• Use Golomb Coding in image:

– Consider P(n-μ) instead of P(n), μ: mean intensity

– Map the negative image.

– Code using Gm(n)

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Image Compression

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• Golomb Image Coding Example (1):

Map

pin

g

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Image Compression

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• Golomb Image Coding Results (1):

– G1(n)

• C=4.5

• 88% of theoretical compression ratio (4.5/5.1)

• 96% of Huffman coding compression (Both in MATLAB) – Less computational efforts

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Image Compression

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• Golomb Image Coding Example (2):

– G1(n) • C=0.0922!!!

• Data Expansion

– Golomb is model based

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Image Compression

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• Golomb Exponential Coding of order k:

– Step #1: Find non-negative integer i such that:

• And form unary coding of i.

– Step #2: Truncate the following binary representation to k+I (LSB)

– Step #3: Concatenate results of step #1 and #2

– k=0 is known as Elias-gamma code

1

0 0

2 2i i

j k j k

j j

n

1

0

2i

j k

j

n

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Image Compression

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• Example:

– Let’d compute: • Step #1:

• Step #2:

• Step #3: 1110001

0

exp 8G

Unary Coding

20 log 9 3 1110k i

3 1Unary Coding Trunc. to 3+0

0

8 2 8 1 2 4 1 0 000 101j k

j

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Image Compression

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• Arithmetic Coding:

– Stream-based (A string of source symbols is encoded as a string of code symbols):

– Free of the integer number of bits/symbol restriction and more efficient.

– Arithmetic coding may reach the theoretical bound of coding efficiency specified in the noiseless source coding theorem for any information source.

– JBIG1, JBIG2, JPEG-2000, H264, MPEG-4 AVC

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Image Compression

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• Arithmetic Coding, Basic idea:

– With n bits, a code interval of [0,1] divided to 2n parts with

length of 2-n.

– Vise versa A=2-n can be coded using –log2A.

000 001 010 011 100 101 110 111

00 01 10 11

0 1

0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1

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Image Compression

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• Arithmetic Coding-Basic Idea:

– Represent the entire input stream as a small interval between the range [0,1].

– 1. Divide interval ([0,1]) to subintervals according to probability distribution.

– The first symbol interval is taken and again divide to sub interval with length of relative to probabilities.

– Now take second symbol interval and continue …

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Image Compression

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• Arithmetic Encoding By Example:

– Stream is baca

Si Pi Range

a 0.5 [0.0-0.5)

b 0.25 [0.5-0.75)

c 0.25 [0.75-1.0)

Range

[0.5-0.625)

[0.625-0.6875)

[0.6875-0.75)

Range

[0.5-0.5625)

[0.5625-0.59375)

[0.59375-0.625)

Range

[0.59375-0.609375)

[0.609375-0.6171875)

[0.6171875-0.625)

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Image Compression

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• Arithmetic Decoding By Example:

– Code is 0.59375

• 0.5<0.59375<0.75 b (b section of main interval)

• 0.5<0.59375<0.625 a (a section of subinterval b)

• 0.59375<0.59375<0.625 c (c section of subinterval a)

• 0.59375<0.59375<0.609375 a (a section of suninterval c)

– Loop. For all the symbols.

• Range = high_range of the symbol - low_range of the symbol

• Number = number - low_range of the symbol

• Number = number / range

• Stop for Number = 0

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Image Compression

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• Implementation:

– Change [0,1] to [0000H,FFFFH]

– Change Probabilities value

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Image Compression

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• Arithmetic Coding

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Image Compression

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• Lempel-Ziv-Welch (LZW) Coding

– Reduce Spatial-Coding redundancy

– Assign fixed-length codewords to variable-length sequences of source symbols.

– No priori knowledge about intensity probability

– GIF, TIFF, PNG, PDF

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Image Compression

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• Introductory Example:

• First Order Entropy:

21 21 21 95 169 243 243 243

21 21 21 95 169 243 243 243

21 21 21 95 169 243 243 243

21 21 21 95 169 243 243 243

Gray level Count Probability

21 12 0.375

95 4 0.125

169 4 0.125

243 12 0.375

1.81 bits/pixel

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Image Compression

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• Introductory Example: – Second Order Entropy:

1.25 bits/pixel

Gray level

pair Count Probability

(21, 21) 8 0.250

(21, 95) 4 0.125

(95, 169) 4 0.125

(169, 243) 4 0.125

(243, 243) 8 0.25

(243, 21) 4 0.125

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Image Compression

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• Meta AlgorithmWikipedia:

– Initialize the dictionary to contain all strings of length one.

– Find the longest string W in the dictionary that matches the current input.

– Emit the dictionary index for W to output and remove W from the input.

– Add W followed by the next symbol in the input to the dictionary.

– Go to Step 2.

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Image Compression

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• ExampleWikipedia:

– String to be code: TOBEORNOTTOBEORTOBEORNOT#

– 1) Initial Dictionary:

Symbol Code Symbol Code Symbol Code

# 0 I 9 R 18

A 1 J 10 S 19

B 2 K 11 U 20

C 3 L 12 V 21

D 4 M 13 W 22

E 5 N 14 X 23

F 6 O 15 Y 24

G 7 P 16 Z 25

H 8 Q 17 # 26

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Image Compression

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• Example: TOBEORNOTTOBEORTOBEORNOT# Input

Code Output Sequence

New Dictionary Entry

Full Conjecture

20 T 27: T?

15 O 27: TO 28: O?

2 B 28: OB 29: B?

5 E 29: BE 30: E?

15 O 30: EO 31: O?

18 R 31: OR 32: R?

14 N 32: RN 33: N?

15 O 33: NO 34: O?

20 T 34: OT 35: T?

27 TO 35: TT 36: TO?

29 BE 36: TOB 37: BE?

31 OR 37: BEO 38: OR?

36 TOB 38: ORT 39: TOB?

30 EO 39: TOBE 40: EO?

32 RN 40: EOR 41: RN?

34 OT 41: RNO 42: OT?

0 #

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Image Compression

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0. Initialize a dictionary by all possible gray values (0-255) 1. Input current pixel 2. If the current pixel combined with previous pixels form one of existing dictionary entries Then 2.1 Move to the next pixel and repeat Step 1 Else

2.2 Output the dictionary location of the currently recognized sequence (which is not include the current pixel) 2.3 Create a new dictionary entry by appending the currently recognized sequence in 2.2 with the current pixel 2.4 Move to the next pixel and repeat Step 1

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Image Compression

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• LZW for Image coding:

– Initialize a dictionary by all possible gray values (0-255)

– (1) Input current pixel

– (2) If the (current pixel concatenate previous pixels) exists in dictionary, then

• (2-1) Move to the next pixel and repeat Step 1

– Else • (2-2) Output the dictionary location of the currently recognized

sequence (which is not include the current pixel)

• (2-3) Create a new dictionary entry by appending the currently recognized sequence in (2-2) with the current pixel

• (2-4) Move to the next pixel and repeat Step 1

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Image Compression

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Input

39

39

126

126

39

39

126

126

39

39

126

126

Dictionary

0 0

1 1

… …

255 255

256 39-39

257 39-126

258 126-126

259 126-39

260 39-39-126

261 126-126-39

262 39-39-126-126

Encoded

Output (9 bits)

39

39

126

126

256

258

260

Sequences

39

39

126

126

39

39-39

126

126-126

39

39-39

39-39-126

126

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Image Compression

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• Binary image compression:

– A very simple approach: 8 pixels to one byte (code redundancy)

– Constant Area Coding (CAC):

• Image is divided to blocks (p×q): – White, Black, and Mixed

• Coding: – Most Probable (W/B/M) will code by one bit (0)

– Two others will code by two bits(10/11)

– Mixed code is prefix of its data

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Image Compression

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• Binary image compression:

– White Block Skipping (WBS):

• 0 for whole white blocks

• 1 is used as prefix for mixed or whole black (less probable)

– Symbol-based (Token-based) Coding:

• Image modeled as a collection of frequently occurred sub-images (symbols or tokens)

• (xi,yi,ti):(xi,yi) are position of symbols and ti position of symbol in dictionary

– JBIG2 (Text Halftone, Generic region), Page: 561-2

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Image Compression

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• Symbol-based (Token-based) Coding:

– Image modeled as a collection of frequently occurred sub-images (symbols or tokens)

– (xi,yi,ti):(xi,yi) are position of symbols and ti position of symbol in dictionary

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Image Compression

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• One dimensional Run Length Coding (RLE):

– Basic idea:

• Code each cluster of white (1) and Black (0) pixel by its length (Or cluster of constant gray level).

• Two Approach: – Specify value of first run of each row.

– Assume each row begin with white (its length may be zero)

– We may use variable length coding for each length of black or white.

– Facsimile (FAX) and BMP

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Image Compression

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• Run-Length Coding Implementation:

– One-dimensional CCITT

– Two-Dimensional CCITT • Coding Black-to-white and white-to-black transition

– Read pages 555-559

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Image Compression

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• Bit-Plane Coding: – Ordinary Coding:

– We could have 8 Binary images, but its drawback: • Small changes in gray level can have significant effect on bit-plane

images: 127=01111111 and 128=10000000, all eight images are different!

– Gray Code:

– 127=11000000 and 128=01000000

1 2 1 0

1 2 1 02 2 2 2m m

m mA a a a a

1

1 1

, 0,1, , 2i i i

m m

g a a i m

g a

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Image Compression

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• Bit-Plane Coding Example

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Image Compression

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Gray Binary

Bit 7

Bit 6

Bit 5

Bit 4

Gray Images are less complex

• Example:

– Bit-Planes (7-4)

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Image Compression

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• Example:

– Bit-Planes (3-0)

Gray Binary

Bit 3

Bit 2

Bit 1

Bit 0

Gray Images are less complex

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Image Compression

79

• JBIG2 Results:

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Image Compression

80

• Block Transform Coding:

– A Reversible- Linear transform maps the image to a set of coefficients.

– These coefficients then quantized and coded. (Local/Global)

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Image Compression

81

• Transform Selection:

– Compression is done during quantization NOT transforming.

– Type of transformation relate to Compression ratio.

– Separable Transform:

– Symmetric Transform:

1 1

0 0

1 1

0 0

, , , ; ,

, , , ; ,

n n

x y

n n

u v

T u v g x y r x y u v

g x y T u v s x y u v

1 2, ; , , ,r x y u v r x u r y v

1 1, ; , , ,r x y u v r x u r y v

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Image Compression

82

• Transform Coding:

– DFT (M=N):

• Fast

• Complex

• Gibbs Phenomenon

2

2

2

, ; ,

1, ; ,

j ux vy n

j ux vy n

r x y u v e

s x y u v en

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Image Compression

83

• DFT Basis Function:

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Image Compression

84

• Walsh-Hadamard Transform (WHT):

1

0

0 1

1 1 2

2 2 3

1 1 0

1, ; , , ; , 1

2 , : kth bit (righ to left) of binary representation of

m

i i i i

i

b x p u b u p v

m

k

m

m m

m m

m

r x y u v s x y u vn

n b z z

p u b u

p u b u b u

p u b u b u

p u b u b u

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Image Compression

85

• Walsh-Hadamard Basis Function:

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Image Compression

86

• Discrete Cosine Transform (DCT):

, ; , , ; ,

2 1 2 1cos cos

2 2

10

21,2, , 1

r x y u v s x y u v

x u y vu v

n n

un

u

u nn

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Image Compression

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• DCT Basis Function:

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Image Compression

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• Example:

– n=8,

– Truncating 50% of coefficients

DFT (2.32) WHT (1.78) DCT (1.13)

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Image Compression

89

• Transform Coding Details:

– Consider a sub-image (block-wise processing):

1 1

0 0

1 1

0 0

, 1, 1

, 0,0

, , , ; ,

,

Matrix Formulation:

, ; ,

u v

uv

u v

i j n n

uv i j

g x y T u v s x y u v

T u v

s h i j u v

G S

S

n n

n n

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Image Compression

90


– Transform Masking Function:

1 1

0 0

0 , in a predefined region,

1 o.w.

ˆ , , Compressed Imageuv

u v

T u vu v

u v T u v

G Sn n

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Image Compression

91


– Error Analysis:

22

21 1 1 1

0 0 0 0

21 1

0 0

1 12

, 20 0

ˆ

, , ,

, 1 ,

Orthonormality of Transform Coeffs1 , :

N 0,

ms

uv uv

u v u v

uv

u v

T u vu v

e E

E T u v u v T u v

E T u v u v

u v

G G

S S

S

G

n n n n

n n

n n

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Image Compression

92

• Transform Compression: – How to select Transform:

• Who collect more information in less coefficients.

– Best Transform: • KL (Karhune-Loeve)

– Data Dependent!

– Most used transform: • DCT:

– Single Chip Implementation

– Packing the most information into the fewest coefficients.

– Minimum Blocking Artifact.

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Image Compression

93

• Periodicity Implicit DFT and DCT

– Gibbs phenomenon causes erroneous boundary • Blocking artifacts

DFT

DCT

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Image Compression

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• Block size effect:

– Small Blocks: • Faster

• More blocking effect (correlation between neighboring pixels)

• Low compression ratio

– Large Block: • Slower

• Better compression in “flat” regions.

• Lower Blocking Effect

– Power of 2, for fast implementation.

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Image Compression

95

• Block-Size and Transform Effects:

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Image Compression

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• Block Size effect for DCT:

– Reconstruct using 25% of the DCT coefficients

Original 2×2 4×4 8×8

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Image Compression

97

• Bit Allocation:

– Bio Allocation: Truncation-Quantization-Coding

– How to select retained coefficients. • Maximum Magnitude (Threshold Coding)

• Maximum Variance (Zonal Coding)

– Threshold Coding: • Keeping P-largest coefficients.

– Zonal Coding: • DCT Coeffs. are considered as RV’s , all statistics computed using

ensemble of transformed subimages.

• Keeping P-largest variances (same -P (u,v)’s for all subimages.)

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Image Compression

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• Threshold vs. Zonal Coding:

– Keep coefficients 8 out of 64

RMS=4.5

RMS=6.5

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Image Compression

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• Zonal Coding Implementation:

– Ensemble average over MN/n2 subimages transforms and select non zero χ(u,v) , Zonal Masking. (High variances)

• 1 or 0 for each position in transform domain, Zonal Mask

• Number of bits used to code each coefficients, Zonal bit allocation

– Quantization of the retained coefficients: • Fixed Length (Uniformly Quantized)

• Optimal Quantizer:

– A model for DCT coefficients:

» Rayleigh for DC component. ( A positive RV’s)

» Laplacian or Gaussian for others (CLT theorem)

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Image Compression

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Z-Coding # of bits

• Zonal Coding Example:

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Image Compression

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• Threshold Coding Implementation:

– An Adaptive mask selection of non-zero χ(u,v)

– Location of maximum magnitude vary from one sub-image to another.

• Re-order 2D DCT to a 1D (Run Length Coded Sequence) in a zigzag arrangement.

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Image Compression

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• Why zigzag arrangement:

– This is done so that the coefficients are in order of increasing frequency.

– This improves the compression of run-length encoding.

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Image Compression

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T-Coding Zigzag tracing

• Threshold Coding Example:

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Image Compression

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• Thresholding:

– A single global threshold for all subimages.

• Different CR from image to image

– A different threshold for each subimages.

• N-Largest coding (same number of images are discarded)

• Same CR for all images.

– A variable threshold as a function of location.

• Different CR from image to image.

• Combined Quantization and Thresholding.

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Image Compression

105

• Location Variant Thresholding:

,ˆ ,,

, : Elements of Transform Normalization

ˆ, , ,

:

,

:

inv

Compression

Decompression

T u vT u v round

Z u v

Z u v

T u v Z u v T u v f x y

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Image Compression

106

Curve Normalization Matrix

,ˆ ,,

T u vT u v round

Z u v

ˆ , , ,2 2

,ˆ , 0, ,2

c cT u v k kc T u v kc

Z u vT u v T u v

• Thresholding-Quantization:

– Left for specific Z(u,v)=c

– Map of Z(u,v)

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Image Compression

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• Quantization Effect:

Z(u,v) 2Z(u,v) 4Z(u,v)

8Z(u,v) 16Z(u,v) 32Z(u,v)

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Image Compression

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• JPEG – Based on the Text Book

– Three different coding systems: • Lossy baseline coding system (most application)

• Greater Compression/Progressive reconstruction (Web)

• Lossles independent coding (reversible compression)

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Image Compression

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• Baseline system (Sequential baseline system):

– Input and Output precision: 8 bits

– Quantized DCT coefficients precision: 11 bits

– Performs in three steps: DCT, Quantization, VL coding

• Preprocessing:

– Level shifting (2k-1, k bit depth)

• Coding:

– Different VLS coding for nonzero AC and DC components.

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Image Compression

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• Sample subimages (8*8)

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Image Compression

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• Level shifting (-128)

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Image Compression

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• Forward DCT Transoform:

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Image Compression

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• Normalized:

,ˆ ,,

ˆ 0,0 26

T u vT u v round

Z u v

T

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Image Compression

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• Zig-Zag Scanning:

• Two different Coding for DC and AC

– See Tables A-3, A-4, A-5 for codebook

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Image Compression

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• Quantized Coefficients:

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Image Compression

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• Denormalization:

ˆ, , ,

0,0 416

0,0 415

T u v Z u v T u v

T

T

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Image Compression

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• Inverse DCT Transform:

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Image Compression

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• Level Shifting (+128):

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Image Compression

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• Difference Image:

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Image Compression

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• Example:

1:25

1:52

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Image Compression

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• JPEG (Joint Photographic Expert Group)

– Color Models

– Image Subsampling

– Coding (DC and AC)

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Image Compression

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• YUV Color model:

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Image Compression

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• YIQ Color Model:

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Image Compression

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• Chromatic components subsampling:

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Image Compression

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• Chrominance Decimation of eye:

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Image Compression

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• Compression Block Diagram

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Image Compression

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• Decompression Block Diagram

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Image Compression

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• DCT on Image block

– Each image is divided into 8×8 blocks. The 2D DCT is applied to each block image f(i, j), with output being the DCT coefficients F(u,v) for each block.

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Image Compression

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• Quantization:

– F(u,v) represents a DCT coefficient, Q(u,v) is a “quantization matrix” entry, and F(u,v) represents the quantized DCT coefficients which JPEG will use in the succeeding entropy coding.

• The quantization step is the main source for loss in JPEG compression.

• The entries of Q(u,v) tend to have larger values towards the lower right corner. This aims to introduce more loss at the higher spatial frequencies

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Image Compression

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• Quantization Table:

– Default Q(u,v) values obtained from psychophysical studies with the goal of maximizing the compression ratio while minimizing perceptual losses in JPEG images.

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Image Compression

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• DCT + Quantization:

Encoder

Decoder

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Image Compression

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• DPCM: Differential Pulse Code Modulation

– Used for DC components compression: • DC coefficient is unlikely to change drastically within a short

distance so we expect DPCM codes to have small magnitude and variance.

– Entropy coding of DC components: • Huffman

150 155 149 152 144 150 5 6 3 8

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Image Compression

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• RLC: Run Length Coding – Used for AC components compression:

• RLC is effective if the information source has the property that symbols tend to form continuous groups. In this case, the quantized DCT coefficients tend to form continuous groups of 0s.

– Used in Conjunction with zig-zag scan

– Entropy coding of DC components: • Huffman

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Image Compression

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• RLC: Run Length Coding

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Image Compression

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• Lossless Predictive Coding:

– Code only new information in each pixel.

– New Information: Difference between two successive pixel.

– en is coded with a variable-length coding (symbol Encoder)

: New incoming pixel value.

ˆ : Prediction of based on a limited range history

n

n n

f

f f

ˆn n ne f f

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Image Compression

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• Prediction:

– An optimal (local/global) and adaptive predictor:

– A nearest integer operator is needed

1-D

1 1

ˆ ˆ , ,m m

n i i

i i

f round f n i f x y round f x y i

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Image Compression

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• Coder and Decoder:

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Image Compression

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• Lossless Predictive Coding:

– For 2D images:

– Example:

• Zero difference is code with 128

• Negative/Positive error are coded with dark light region.

ˆ , ,m m

i

i m j m

f x y round f x i y j

ˆ , 1nf round f x y

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Image Compression

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• Example:

– Prediction residual

– α=1

– CR = 8/3.99 = 2

– Residual PDF

ˆ, , ,e x y f x y f x y

2

1

2

e

e

e

e

P e e

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Image Compression

140

• Example (Temporal Redundancy):

ˆ , , , , 1 , 1

ˆ, , , , , , 1

f x y t round f x y t

e x y t f x y t f x y t

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Image Compression

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• Motion Compensated Prediction Residuals

– Self study (589-596)

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Image Compression

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• Lossy Compression:

– JPEG, JPEG2000, GIF

– 1:10 to 1:50

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Image Compression

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• Lossy Predictive Coding:

– Map prediction error into a limited range of outputs (control compression ratio)

ˆn n nf e f

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Image Compression

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• Delta Modulation:

1ˆ 1

0

O.W.

Singel bit code (1bit/pixel)

n n

n

n

f f

ee

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Image Compression

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• Optimal Prediction:

22

1

2

2

1

, 1

2 2

11

ˆ ˆ ˆ,

ˆ

: Image Autocorrelation Matrix, ,

n n n n n n n n n

m

n i n i

i

m

n n i n i

i

m

n i n j i j

mm

n n k e i n n iki

E e E f f f e f e f f

f f

E e E f f

i j E f f

k E f f E f f

-1α = R r

R R

r

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Image Compression

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• Optimal Prediction:

– Computation for image by image is hard.

– Global coefficient is computed by assuming a simple image model.

2

1 2 3 4

1 2 3 4

m

i

i=1

, ,

ˆ , , 1 1, 1 1, 1, 1

, , , 0

and : Horizontal and Vertical correlation coefficient

Dynamic Range Const.: 1

i j

v h

h h v v

h v

E f x y f x i y i

f x y f x y f x y f x y f x y

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Image Compression

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• Results of Prediction Error Coding:

ˆ , 0.97 , 1

ˆ , 0.5 , 1 0.5 1,

ˆ , 0.75 , 1 0.75 1, 0.5 1, 1

0.97 , 1ˆ ,0.97 1, O.W.

1, 1, 1

, 1 1, 1

f x y f x y

f x y f x y f x y

f x y f x y f x y f x y

f x y h vf x y

f x y

h f x y f x y

v f x y f x y

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Image Compression

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• Example:

– Prediction Errors

rms = 11.1 rms = 9.8

rms = 9.1 rms = 9.7

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Image Compression

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• Optimal Quantizer:

– t=q(s)

– q(s)=-q(-s)

– Map s in (si,si+1] to ti

– What is best si and ti

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Image Compression

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• Lloyd-Max Quantizer

– For even p(s), E{(si-ti)2}will minimize with these

conditions:

1

, 1, 2, ,2

0 0

11, 2, , 1

2 2

2

,

i

i

s

i

s

ii

i i i i

Ls t p s ds i

i

t ti Ls i

Li

s s t t

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Image Compression

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• Lloyd-Max Quantizer

– L=2, 4, 8

– Laplacian pdf (σ=1)

– ti-ti-1= si-si-1=θ

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Image Compression

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• Wavelet Coding

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Image Compression

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• Wavelet Selection: Haar Daubechies

Symlet Cohen

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Image Compression

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• Wavelet Selection:

– Truncating the transforms below 1.5:

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Image Compression

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• Decomposition Level Selection:

– More scales More computational efforts

– An Experiments:

• Biorthogonal wavelet;

• Fixed global threshold (25);

• Truncate only details

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Image Compression

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• Jpeg 2000 (607-614)

– DC Level shift

– Convert using color video transform (RGB YCbCr)

– Convert the whole image to tile components

– Using different filter for lossy or loss-free compression

– Quantized the coefficients

– Arithmetic Coding

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Image Compression

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• Example:

– C=25, 52, 75, 105.

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Image Compression

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• Image Watermarking

– Skipped

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Image Compression

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• Matlab Command

– Huffmandeco, huffmandict, huffmanenco

– Arithdeco, arithenco

– Dpcmdeco, dpcmenco

– Lloyds, quantiz

– wcompress (wavelet image compression)

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