ece-c490 winter 2004 image processing architecture

Image Processing Architecture, © 2001-2004 Oleh Tretiak Lecture 4

ECE-C490 Winter 2004 Image Processing Architecture

Lecture 4, 1/20/2004

Principles of Lossy CompressionOleh Tretiak

Drexel University


Compression: Input is a sequence of discrete symbols Main theoretical tool: Entropy

Technology: Coding Methods Huffman codes

Variable length code words, high probability symbols get short code words

Ideally, length of code word should be -log2pi

Arithmetic codes Many symbols translate to many code bits Each input symbol produces -log2pi

Other codes: Golomb-Rice

Review: Lossless Compression

H =− pi log2 pii∑


Successful preprocessing can reduce per/symbol entropy Example: Run-length encoding Example: Prediction on the basis of past values

Review: Coder Structures

SignalPreprocessorEntropy CoderData

High entropy Low entropy


Review: Coder Design Strategies Fixed encoder

Example: ITU-T, Rec. T.4 (Group 3), Rec. T.6 (Group 4) Excellent compression on text documents of fixed resolution Simple system

Adaptive encoder Example: JBIG

Adaptive predictor on basis of context Arithmetic coding with p estimated from documents Superior performance over a large range of documents

ContextpixelsPixelbeing

predicted


Review: Coder Designs Example: Lossless JPEG

Prediction on basis of context 8 prediction rules, choose the best one

Selection Value Prediction

0 No prediction1 a2 b3 c4 a+b-c5 a+((b-c)/2)6 b+((a-c)/2)7 (a+b)/2

abcx

Prediction context

Available Prediction Rules


This Lecture Lossy Compression

Tradeoff between quality and number of bits Bit reduction Resolution reduction Quantizers

Rate distortion theory


Review: Lossless Compression of Images

Can get quite good compression of graphics images Lossless compression of gray-scale images (JPEG) is ~ 2 What to do?

Keep trying Give up Take another approach

Lossy Compression


Lossy Compression Ideas Tradeoff between bits and quality

Name of the game: Large reduction in bits at virtually no loss in quality

Theoretical tool Shannon Rate-Distortion Theory


Quality - Rate Tradeoff Given: 512x512 picture, 8 bits per pixel

Bit reduction Fewer bits per pixel Fewer pixels Both

Issues: How do we measure compression?

Bits/pixel — does not work when we change number of pixel Total bits — valid, but hard to interpret

How do we measure quality? RMS noise Peak signal to noise ratio (PSR) in dB Subjective quality


Reduced bits/pixel Keep b most significant bits Compression = 8/b Distortion: o(i, j) original picture, r(i, j) reduced picture,

we define the noise as RMS error:

Peak Signal to Noise Ratio (PSNR) = 20log10(255/)

n(i , j ) =r(i, j) –o(i, j)

s =

1HV

(n(i, j ))2

j =1

V

∑i =1

H

∑

Bits 8 7 6 5 4 3 2 1Compression 1.00 1.14 1.33 1.60 2.00 2.67 4.00 8.00Error 0 0.7 1.22 2.35 12.5 22.5 35.5 51.5PSNR, db 51 46 41 26 21 17 14


Reducing bits per pixel.Upper row, left to right:8, 4, 3 bits per pixel. Lower row, left to right:2, 1 bits per pixel. With 5, 6, and 7 bits per pixel the image is almost the same as 8 bits.


Loss vs. Bit Rate

0

10

20

30

40

50

60

0 2 4 6 8

bits/pixel


Resolution Reduction Start with 512x512 8 bit image Sample over a WxW point grid, keep 8 bits per pixel r(i, j) is interpolated to a 512 x512 grid Compression = (512/W)2

n(i, j) = r(i, j) - o(i, j) Compute RMS Noise and PSNR


512 382

256 181

128

Resolution reduction. Upper left is full resolution (512x512 pixels). Lower right image has 1/8 of the pixels, Compression = 8.


Difference Between Compressed and Original

Difference between 128x128 (16 fold compression) and original image


Quantitative PerformanceResolution 512 362 256 181 128Compression 1 2 4 8 16RMS error 0 2.7 4.08 7.29 7.75PSNR 40 36 30.9 30.4

PSNR

0

5

10

15

20

25

30

35

40

45

0 100 200 300 400Resolution


Comparison, Bit and Pixel Reduction

0

10

20

30

40

50

60

0 1000000 2000000

Total bits in image

SubsampleDrop bits


Conclusion and Questions For this image, resolution reduction is a much more effective

compression method than bit reduction 8-fold reduction in bits at 30 dB PSNR with subsamplng 2.67-fold reduction bits at 29 dB PSNR with bit reduction

What is the best combination of sampling and bits? Can entropy coding help? Is there a theory that can guide us?


Theory Theory tells us that the data we encode should be statistically

independent In a picture, adjacent pixels are correlated.

Shannon’s rate-distortion theory provides a way of finding the tradeoff between rate (total bits per image) and distortion (rms. error)

Image should be transformed (preprocessed) so that statistically independent samples are encoded

Correlated images (with low detail) can be compressed more than uncorrelated images.


Typical Rate-Distortion Functions

Rate is in bits per image, or bits per time Distortion is 1/(rms. error)

0.0001

0.001

0.01

0.1

10 5 10

Rate (bits)

IndependentCorrelated


Information Theory of Continuous Sources

Quantizers Decorrelation and Bit Allocation Rate-Distortion Theory DCT (Discrete Cosine Transform)


Quantizer: The Basics Given: 8 bit gray value samples, 0 ≤ data ≤ 255 Task: Keep 4 bits per sample Method: Shift 4 bits to right (divide by 24 = 64)

This is called quantization Quantization step S = 64 All values betwee 0 and 63 are mapped to 0, values from 64 to 127

are mapped to 1, etc Restoring data: Multiply by 24 = 64 and add 32

This is called dequantization What used to be between 0 and 63 is now at 32, data values

between 64 and 1


Example of quantization

0

20

40

60

80

100

120

140

160

Quantizer inputDequantizer output

Quantization with S = 20. Note that the dequantizer output is typically not equal to the input!


Quantizer Performance Questions:

How much error does the quantizer introduce (Distortion)? How many bits are required for the quantized values (Rate)?

Rate: 1. No compression. If there are N possible quantizer output values, then it takes

ceiling(log2N) bits per sample. 2(a). Compression. Compute the histogram of the quantizer output. Design Huffman

code for the histogram. Find the average lentgth. 2(b). Find the entropy of the quantizer distribution 2(c). Preprocess quantizer output, ....

Distortion:Let x be the input to the quantizer, x* the de-quantized value. Quantization noise n = x* - x. Quantization noise power is equal to D = Average(n2).


Quantizer: practical lossy encoding

Quantizer Symbols

x — input to quantizer, q — output of quantizer,S — quantizer step

Quantizer:q = round(x/S)

Dequantizer characteristicx* = Sq

Typical noise power added by quantizer-dequantizer combination: D = S2/12

noise standard deviation = sqrt(D) = 0.287SExample: S = 8, D = 82/12 = 5.3,

rms. quatization noise = sqrt(D) = 2.3 If input is 8 bits, max input is 255. There are 255/8

~ 32 quantizer output values PSNR = 20 log10(255/2.3) = 40.8 dB

Δxq

Δx*q x* =Dq Quantizer characteristic

Dequantizer characteristic


Rate-Distortion Theory

Probabilistic theory, developed by Claude Shannon in 1949 Input is a sequence of independent random variables, output

are lossy (distorted) reproduction Joint probability density function p(x, y) Distortion: D =E[(X–Y)2] (mean squared error, noise power),

Rate-distortion function

X i YiEncoderDecoderBits/sample

I(X ,Y ) = log2∫∫p(x,y)

p(x)p(y)dxdy

R(D) = minover p(y|x)with D fixed

I(X,Y )


Rate-Distortion Theorem When long sequences (blocks) are encoded, it is possible to

construct a coder-decoder pair that achieves the specified distortion whenever bits per sample are R(D) +

Formula: X ~ Gaussian random variable, Q = E[X2] ~ signal power

D = E[(X–Y)2 ] ~ noise power

p(x ) =1

2πQ(exp −x2 / 2Q )

R =12

log2QD

D ≤Q

0 D >Q

⎧ ⎨ ⎪

⎩ ⎪


Rate-Distortion Function for Gaussian Distribution

‘Theoretical’ plot ‘Practical’ plot

Gaussian Rate-Distortion Function

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.5 1 1.5 2

D/Q

R - bits per sample

Gaussian Rate-Distortion Function

0

1

2

3

4

5

-10 0 10 20

SNR = 10log10(Q/D)

R (bits)


Rate - Distortion, in words

Q - actual signal power (size) D - specified distortion power D > Q (specified distortion bigger than signal)

R = 0, send nothing, actual distortion = received - sent distortion power = Q

D < Q (specified distortion less than signal Coded bits per sample = 0.5 log2(Q/D)

This formula applies for coding statistically independent symbols!

R =12

log2QD

D ≤Q

0 D >Q

⎧ ⎨ ⎪

⎩ ⎪

Image Processing Architecture, © 2001-2004 Oleh Tretiak Page 34Lecture 4

Basic Solution If the total distortion is small,

Make Di = D/n

Ri = 0.5 log2 (Qi /Di) = 0.5 log2(nQi /D)

This holds as long as Qi ≥ D/n for all i

ece-c490 winter 2004 image processing architecture

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