Effect of Linearization on Normalized Compression Distance

1. Effect of Linearization on Normalized Compression Distance Jonathan Mortensen Julia Wu DePaul University July 2009

2. Introduction • Kolmogorov Complexity is an emerging similarity metric • Transformation Distance • Universal Similarity Measure • Does not require feature identification and selection • How can it be applied to images? • CBIR, Classification • Investigate its effectiveness • Discovered some fundamentals have been overlooked thus far

3. Outline • Background • Kolmogorov Complexity and Complearn • Research Topics • Spatial Transformations • Intensity Transformations • Image Groupings • Conclusion • Future Work

4. Background • Li (2004): successful clustering of phylogeny trees, music, text files • 1D to 2D data? • Tran (2007): NCD not a good predictor of visual indistinguishability • Only one photograph used, one type of linearization (row-by-row) • Gondra (2008): CBIR using NCD produced statistically significant measures against H0 of random retrieval and other similarity measures • Test set of hundreds of images, inconsistent methods of compression and concatenation, linearization unclear

5. K(x) – The length of the shortest program or string x* to produce x K(x|y) - The shortest binary string to convert output x given input y E(x,y)=max{K(x|y),K(y|x)} Normalized Information Distance: Kolmogorov Complexity

6. Kolmogorov Complexity • Universal, in that it captures all other semi-computable normalized distance measures • Therefore also semi-computable • Compression losslessly simplifies strings, and therefore is used as an approximation, C(x) “The human brain is incapable of creating anything which is really complex.”--Kolmogorov,  A.N., Statistical Science, 6, p314, 1990

7. CompLearn • Open Source package which implements K-Complexity • Developed by Rudi Cilibrasi, Anna Lissa Cruz, Steven de Rooij, and Maarten Keijzer • Uses basic linux compression tools to develop the comparison map

9. Images from “Google Similar Images”

10. Initial Questions • Linearization Methods and Alternatives • How to Preserve a 2D signal • Linearization’s affect NCD on spatial transformations and intensity shifts • Do additional feature images lower NCD? • CBIR: Can K-Complexity be used with feature vectors or image semantics

11. Spatial Transformations • Applied 4 types of linearization to 800 images (original and 7 transformations) • Found that each linearization type produced distinctly different NCDs • Certain linearizations result in lower NCDs for certain transformations

12. Linearization Methods Row Major Column Major Hilbert-Peano SPC: Images transformed to 128x128 SCPO: Images transformed to 35% of original size

13. Spatial Transformations Original Image Down Shift Left Shift 180 rotation 90 rotation 270 rotation Reflection Y Axis Reflection X Axis

14. Intensity Transformations • Additive Constant • Three types of noise • Gaussian • Speckle • Salt and Pepper • Least Significant Bit (LSB) Steganography • Contrast Windowing

15. Additive Constant Image 937.jpg +32 and +64 respectively • P = Intensity + Constant • +4, +8, +12… +100 • 16 bit • 255 (+4)-> 259 • Truncation • 255 (+4)-> 255 • Wrap • 255 (+4)-> 4

16. Additive Constant

17. Various Noise • Gaussian (Statistical) • Speckle (Multiplicative) • Salt and Pepper (Drop-off) 0.32 and 0.64 Variance/Noise Density Respectively

18. Noise Cont: • Gaussian and Speckle Noise don’t compress well • Gaussian and Salt Pepper experience some posterior decay

19. Least Significant Bit Steganography • Hide4PGP • “Scrambles” message • Changes pixel bit to most similar color with opposite bit assignment • Spreads secret data over entire file • True Grayscale: Changes two bits per pixel Image with No Text Image hiding “Gettysburg Address”

20. LSB Steganography

21. Hamming Distance

23. Contrast Windowing • Computed Tomography image enhancement that increases contrast in certain structures • Brief Medical Exploration

24. Contrast Windowing Lung Window (-200 HU, width 2000 HU) Bone Window (300 HU, width 1500 HU) Patient 5: Original Image top left Soft Tissue Window (50 HU, width 350 HU)

25. P1 P3 P5

26. Cross Dicom Comparison

27. Conclusion: "How Many" vs "How Little" • NCD for Ordinal Comparisons • Numerical Redundancy Selective Entire Picture Gaussian Speckle Noise Salt and Pepper Noise Steganography Additive Constants Contrast Windowing Larger NCD SmallerNCD

28. Feature Image Comparison and Grouping • Feature Image: Pixel based values derived from the original image • 3 Main Types of Linearization • Avg NCD inter > Avg NCD intra • The greater inter - intra, the better NCD finds groupings

29. Feature Image Linearization • Image-At-Once – row-order one feature image at a time • Row Concatenation – Appends all images, then performs row-order linearization • Pixel Order – Selects value from same pixel of each feature image in row-order fashion • Gray Row-Major – Grayscales an image and follows row-order on intensities

31. Data Set and Methods • Corel Image Database with 10 predefined groupings • Linearized by 5 methods • NCDs were found within a group and then to the left and to the right

33. Results • Nearly every linearization produced statistically different NCDs • Intra Group was always less than Inter Group • Gray provided the greatest difference Inter-Intra • Thought this was due to filesize • Triple Concat’ed Gray creating equal filesize: Found an even greater difference

37. Conclusion • NCD is a good model for predefined human groupings and linearization has little impact on this • Gray-Triple Row-Major may be the best form of linearization • Direction of concatenation does not matter • Defined a methodology for any number of feature images

38. Conclusion • Compressor Errors • Numerical Redundancy • Ordinal Variables vs Nominal Variables • EX: 195 195 195 195 <=> 198 198 198 198 • NCD = 0.100000 • 199 199 199 199 <=> 202 202 202 202 • NCD = 0.128205 • NCD needs refinement • 2D image as a 1D string?

39. Future Work • Image Scaling and Normalization • Additional Feature Images • New Forms of Image concatenation • Investigate Compressors (Numeric?)

40. References • A. Itani and D. Manohar. Self-Describing Context-Based pixel ordering. Lecture notes in computer science, pages 124{134, 2002. • M. Li, X. Chen, X. Li, B. Ma, and P. M.B Vitnyi. The similarity metric. IEEE.Transactions on Information Theory, 50:12, 2004. • R. Dafner, D. Cohen-Or, and Y. Matias. Context-based space lling curves. In Computer Graphics Forum, volume 19, pages 209{218. Blackwell Publishers Ltd, 2000. • R. Cilibrasi, Anna L. Cruz, Steven de Rooij, and Maarten Keijzer. CompLearn home. http://www.complearn.org/. • R. Cilibrasi, P. Vitanyi, and R. de Wolf. Algorithmic clustering of music. Arxiv preprint cs.SD/0303025, 2003. • N. Tran. The normalized compression distance and image distinguishability. Proceedings of SPIE, 6492:64921D, 2007. • I. Gondra and D. R. Heisterkamp. Content-based image retrieval with the normalized information distance. Computer Vision and Image Understanding, 111(2):219{228, 2008.

41. Questions