Brendan J. Babb, Frank Moore, and Pat Marshall University of Alaska, Anchorage and AFIT CIISP 2007

1. 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

2. 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.

3. Overview • Why I might care? • Wavelet image compression and quantization • Evolving wavelet like transforms • Results • Future Research • Questions

4. Applications • JPEG 2000 • FBI Fingerprints database – 200 million cards – 2000 Terabytes of data • Web • Digital Cameras • Video • MP3s

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

6. Multiresolution Analysis

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

8. Original “Zelda” Image

9. Quantization 64

10. 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

11. Information Entropy

12. Genetic Algorithms • Optimization techniques inspired by Darwinian evolution

13. 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

14. 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

15. Genetic Operators • Initial Population • Fitness • Selection • Mutation • Cross-over

16. 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

17. 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)

18. Resulting images • 23% MSE improvement for the same filesize for Fruits.bmp that generalizes • 40% MSE improvement for Zelda image

19. Original “Zelda” Image

20. Quantization 64

21. Evolved 40%

22. Original “Zelda” Image

23. Test Images (Partial)

24. 1 Level Runs Run #1 Run #2 Run #3

25. Error Difference for D4

26. Error Difference for Evolved

27. Multiresolution Analysis

28. MRA3 Same at each level

29. MRA 3 different at each level

30. Evolved Coeffs SetMRA LevelValues (% 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%)

31. 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

32. 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

33. Fingerprint Image

34. IE 110% - 60%

35. Original “Zelda” Image

36. 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

37. Questions

38. 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