Evolving Multiresolution Analysis Transforms for Improved Image Compression

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 ¢ MP 3 s ¢

Wavelet Compression Compressor Original Image Forward Wavelet Transform Quantizer Decompressor Encoder 10011… Lossy Image Inverse Wavelet Transform Dequantizer Decoder

Multiresolution Analysis

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

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 Run #1 image Run #2 IE % Size SE % SE imprv image Run #3 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

Error Difference for D 4

Error Difference for Evolved

Multiresolution Analysis

MRA 3 Same at each level Trained on Fruits SAME coeffs at each level MRA 3 image 512 IE % MSEI % Trained on Zelda SAME coeffs at each level MRA 3 image 512 IE % MSEI % airplane 100 92. 14 7. 86 airplane 100. 06 92. 32 7. 68 baboon 99. 95 90. 28 9. 72 baboon 101 88. 01 11. 99 barb 99. 95 92. 77 7. 23 barb 100. 8 91. 86 8. 14 boat 100. 08 92. 18 7. 82 boat 100. 41 91. 94 8. 06 couple 99. 99 91. 81 8. 19 couple 100. 45 90. 67 9. 33 fruits 99. 95 93. 9 6. 1 fruits 100. 29 95. 92 4. 08 100. 06 91. 99 8. 01 goldhill 100. 49 90. 06 9. 94 lenna 99. 94 92. 86 7. 14 park 100. 03 92. 6 7. 4 park 100. 18 92. 24 7. 76 peppers 100. 08 93. 44 6. 56 peppers 100. 08 94. 41 5. 59 susie 99. 84 92. 78 7. 22 susie 100. 06 91. 37 8. 63 zelda 100. 12 91. 98 8. 02 zelda 99. 99 89. 82 10. 18 100. 00 92. 39 7. 61 100. 31 91. 79 8. 21 goldhill

MRA 3 different at each level Trained on Fruits DIFFERENT coeffs at each level MRA 3 Image 512 IE % MSE % Trained on Zelda DIFFERENT coeffs at each level MRA 3 MSEI % image 512 IE % MSEI % airplane 99. 98 87. 38 12. 62 airplane 100. 17 89. 51 10. 49 baboon 100. 07 88. 14 11. 86 baboon 100. 89 93. 59 6. 41 barb 100. 04 97. 56 2. 44 barb 100. 61 106. 97 -6. 97 boat 100. 09 86. 99 13. 01 boat 100. 31 88. 45 11. 55 99. 97 86. 87 13. 13 couple 100. 43 88. 34 11. 66 fruits 100. 43 88. 34 11. 66 goldhill 100. 02 89. 1 10. 9 goldhill 100. 34 88. 24 11. 76 lenna 99. 9 88. 89 11. 11 lenna 100. 23 88. 49 11. 51 park 100 88. 09 11. 91 park 100. 47 90. 13 9. 87 100. 02 93 7 peppers 100. 16 97. 58 2. 42 susie 99. 84 91. 01 8. 99 susie 100. 25 93. 66 6. 34 zelda 100. 16 89. 96 10. 04 zelda 100 87. 79 12. 21 100. 04 89. 61 10. 39 100. 36 91. 76 8. 24 couple peppers

Evolved Coeffs Set h 1 (Lo_D) 1 2 3 g 1 (Hi_D) 1 2 3 h 2 (Lo_R) 1 2 3 g 2 (Hi_R) 1 2 3 MRA Level Values (% Change Relative to D 4 Wavelet) -0. 1278, 0. 2274, 0. 8456, 0. 4664 (-1. 24%, +1. 47%, +1. 09%, -3. 44%) -0. 1274, 0. 2289, 0. 8446, 0. 4661 (-1. 55%, +2. 14%, +0. 97%, -3. 50%) -0. 1278, 0. 2281, 0. 8455, 0. 4670 (-1. 24%, +1. 78%, +1. 08%, -3. 31%) 0. 4791, 0. 8474, -0. 2347, -0. 1278 (-0. 81%, +1. 30%, +4. 73%, -1. 24%) -0. 4894, 0. 8447, -0. 2317, -0. 1279 (+1. 33%, +0. 98%, +3. 39%, -1. 16%) -0. 4901, 0. 8462, -0. 2291, -0. 1288 (+1. 47%, +1. 16%, +2. 23%, -0. 46%) 0. 4811, 0. 8152, 0. 2274, -0. 1069 (-0. 39%, -2. 55%, +1. 47%, -17. 39%) 0. 4805, 0. 8159, 0. 2279, -0. 1093 (-0. 52%, -2. 46%, +1. 70%, -15. 53%) 0. 4820, 0. 8172, 0. 2278, -0. 1097 (-0. 21%, -2. 31%, +1. 65%, -15. 22%) -0. 2008, 0. 0274, 0. 5960, -0. 1472 (+55. 18%, -87. 78%, -28. 75%, -69. 52%) -0. 1618, -0. 1105, 0. 6870, -0. 3201 (+25. 04%, -50. 69%, -17. 87%, -33. 73%) -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