Evolving Multiresolution Analysis Transforms for

Improved Image Compression and Reconstruction under

Quantization

Brendan J. Babb, Frank Moore, and Pat Marshall

University of Alaska, Anchorage and AFIT

CIISP 2007

Results

We were able to improve image quality on average by 23% over a known wavelet transform with quantization using a Genetic Algorithm to evolve forward and reverse transforms.

For 3 level MRA the improvement is 11% over the standard wavelet.

Overview

Why I might care? Wavelet image compression and

quantization Evolving wavelet like transforms Results Future Research Questions

Applications

JPEG 2000 FBI Fingerprints database – 200

million cards – 2000 Terabytes of data

Web Digital Cameras Video MP3s

Wavelet Compression

Forward Wavelet Transfor

m

Inverse Wavelet Transfor

m

Quantizer

Dequantizer

Encoder

Decoder

Compressor

Decompressor 10011…

Original Image

Lossy Image

Multiresolution Analysis

Quantization

Quantization of 64 Y value is 300 300/64 = 4.6875 = 4 Dequantization multiplies 4 * 64 = 256 17 times smaller file size

Original “Zelda” Image

Quantization 64

Mean Squared Error (MSE) The common method for comparing the

quality of a reproduced image is Mean Squared Error

The average of the square of the difference between the desired response and the actual system output (the error)

Must consider file size

Information Entropy

n

iii xxnxEntropy

12log)(

n

iii yx

nyxMSE

1

2)(1

),(

Genetic Algorithms

Optimization techniques inspired by Darwinian evolution

Previous Research

Dr. Moore published papers on 1-D signals and images, evolving the Inverse transform

90% improvement on 1-D and 5 – 9 % improvement on images over Wavelets

Specifics

Matlab code modified from Michael Peterson’s code based on Dr. Moore’s code.

Forward and Reverse at the same time Start with a population of real coefficients

from a known Wavelet Daubechies 4 ( 8 forward and 8 reverse) MR Levels 1 through 3 Parallel operation on Supercomputer

Genetic Operators

Initial Population Fitness Selection Mutation Cross-over

Fitness Function

Restrain File Size A * MSE ratio + B * File Size ratio Good MSE but bigger files or vice

versa Penalize for bigger file size or bigger

MSE with if statement combinations

GA Parameters

• Population size: 500 to 10000• Generations: 500 to 2000• Elite Survival Count: 2• Parental Selection: stochastic uniform• Crossover: Heuristic• Mutation: varies by experiment• Population initialization: Random factor times the

original Wavelet• Crossover to Mutation ratio: 0.7 (unless noted)

Resulting images

23% MSE improvement for the same filesize for Fruits.bmp that generalizes

40% MSE improvement for Zelda image

Original “Zelda” Image

Quantization 64

Evolved 40%

Original “Zelda” Image

Test Images (Partial)

1 Level Runs

image IE % Size SE % SE imprv image IE % Size SE % SE imprv image IE % Size SE % SE imprv

airplane 95.34 72 28 airplane 96.26 72.7 27.3 Airplane 99.98 57.86 42.14

baboon 94.38 93.2 6.8 baboon 98.8 85.07 14.93 baboon 105.88 68.6 31.4

barb 97.85 77.12 22.88 barb 100.47 77.72 22.28 barb 105.56 66.09 33.91

boat 98.03 79.28 20.72 boat 99.06 77.34 22.66 boat 105.39 61.73 38.27

couple 96.45 81.61 18.39 couple 100 77.67 22.33 couple 105.35 62.55 37.45

fruits 98.06 96.38 3.62 Fruits 100 74.82 25.18 fruits 105.24 64.61 35.39

goldhill 98.82 72.91 27.09 goldhill 100.97 73.27 26.73 goldhill 105.58 61.93 38.07

lenna 99.11 70.26 29.74 lenna 100.05 76.75 23.25 lenna 104.47 56.6 43.4

park 97.04 81.64 18.36 park 100.76 86.72 13.28 park 104.87 65.17 34.83

peppers 99.61 68.79 31.21 peppers 101.05 69.02 30.98 peppers 105.72 56.49 43.51

susie 97.57 72.55 27.45 susie 100.02 74.45 25.55 susie 104.12 57.4 42.6

Zelda 100 60.22 39.78 zelda 101.51 67.95 32.05 zelda 106.19 57.48 42.52

                       

avg 97.69 77.16 22.84 avg 99.91 76.12 23.88 avg 104.86 61.38 38.62

Run #1 Run #2 Run #3

Error Difference for D4

Error Difference for Evolved

Multiresolution Analysis

MRA3 Same at each level

Trained on Zelda SAME coeffs at each level MRA 3

image 512 IE % MSE % MSEI %

airplane 100.06 92.32 7.68

baboon 101 88.01 11.99

barb 100.8 91.86 8.14

boat 100.41 91.94 8.06

couple 100.45 90.67 9.33

fruits 100.29 95.92 4.08

goldhill 100.49 90.06 9.94

lenna 99.94 92.86 7.14

park 100.18 92.24 7.76

peppers 100.08 94.41 5.59

susie 100.06 91.37 8.63

zelda 99.99 89.82 10.18

100.31 91.79 8.21

Trained on Fruits SAME coeffs at each level MRA 3

image 512 IE % MSE % MSEI %

airplane 100 92.14 7.86

baboon 99.95 90.28 9.72

barb 99.95 92.77 7.23

boat 100.08 92.18 7.82

couple 99.99 91.81 8.19

fruits 99.95 93.9 6.1

goldhill 100.06 91.99 8.01

lenna 99.94 92.86 7.14

park 100.03 92.6 7.4

peppers 100.08 93.44 6.56

susie 99.84 92.78 7.22

zelda 100.12 91.98 8.02

100.00 92.39 7.61

MRA 3 different at each level

Trained on Zelda DIFFERENT coeffs at each level MRA 3

image 512 IE % MSE % MSEI %

airplane 100.17 89.51 10.49

baboon 100.89 93.59 6.41

barb 100.61 106.97 -6.97

boat 100.31 88.45 11.55

couple 100.43 88.34 11.66

fruits 100.43 88.34 11.66

goldhill 100.34 88.24 11.76

lenna 100.23 88.49 11.51

park 100.47 90.13 9.87

peppers 100.16 97.58 2.42

susie 100.25 93.66 6.34

zelda 100 87.79 12.21

100.36 91.76 8.24

Trained on Fruits DIFFERENT coeffs at each level MRA 3

Image 512 IE % MSE % MSEI %

airplane 99.98 87.38 12.62

baboon 100.07 88.14 11.86

barb 100.04 97.56 2.44

boat 100.09 86.99 13.01

couple 99.97 86.87 13.13

fruits 100.43 88.34 11.66

goldhill 100.02 89.1 10.9

lenna 99.9 88.89 11.11

park 100 88.09 11.91

peppers 100.02 93 7

susie 99.84 91.01 8.99

zelda 100.16 89.96 10.04

100.04 89.61 10.39

Evolved Coeffs

Set MRA Level Values (% Change Relative to D4 Wavelet)h1 (Lo_D) 1 -0.1278, 0.2274, 0.8456, 0.4664 (-1.24%, +1.47%, +1.09%, -3.44%)

2 -0.1274, 0.2289, 0.8446, 0.4661 (-1.55%, +2.14%, +0.97%, -3.50%)3 -0.1278, 0.2281, 0.8455, 0.4670 (-1.24%, +1.78%, +1.08%, -3.31%)

g1 (Hi_D) 1 0.4791, 0.8474, -0.2347, -0.1278 (-0.81%, +1.30%, +4.73%, -1.24%) 2 -0.4894, 0.8447, -0.2317, -0.1279 (+1.33%, +0.98%, +3.39%, -1.16%) 3 -0.4901, 0.8462, -0.2291, -0.1288 (+1.47%, +1.16%, +2.23%, -0.46%)

h2 (Lo_R) 1 0.4811, 0.8152, 0.2274, -0.1069 (-0.39%, -2.55%, +1.47%, -17.39%) 2 0.4805, 0.8159, 0.2279, -0.1093 (-0.52%, -2.46%, +1.70%, -15.53%) 3 0.4820, 0.8172, 0.2278, -0.1097 (-0.21%, -2.31%, +1.65%, -15.22%)

g2 (Hi_R) 1 -0.2008, 0.0274, 0.5960, -0.1472 (+55.18%, -87.78%, -28.75%, -69.52%) 2 -0.1618, -0.1105, 0.6870, -0.3201 (+25.04%, -50.69%, -17.87%, -33.73%)

3 -0.1572, -0.1495, 0.7861, -0.4033 (+21.48%, -33.29%, -6.03%, -16.50%)

Summary

Forward and Inverse Transforms evolved from Wavelets have better image quality than the Wavelet under quantization and multiple levels

Improves image quality with the same amount of file size

Training images exist which generalize well across other images

Recent Research

Increased Information Entropy results in 60% improvement for Zelda

Evolving for fingerprint images results in 16% improvement over FBI standard for 80 images (Humie)

Training over 4 images and using Differential Evolution

Evolved Fingerprint wavelet does poorly on standard test images

Fingerprint Image

IE 110% - 60%

Original “Zelda” Image

Future Research

Evolving different shape wavelets Mathematically analyze Use of different operators and

techniques What makes a good representative

training image Improve on JPEG 2000 wavelets Custom wavelets for other

applications

Questions

Fitness Logic

If (SE ratio > 1) and (IE ratio > 1) then fitness = (SE ratio)^4 +(IE ratio)^4 else if (SE ratio > 1) then fitness = (SE ratio)^4 + IE ratio else if (IE ratio > 1) then fitness = SE ratio + (IE ratio)^4 else fitness = (SE ratio)^2 + IE fitness = fitness *1000