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Directional Multiscale Modeling of Images

Directional Multiscale Modeling of Images. Duncan Po and Minh N. Do University of Illinois at Urbana-Champaign. Motivation: Image Modeling. A randomly generated image. A “natural” image. A simple and accurate image model is the key in many image processing applications.

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Directional Multiscale Modeling of Images

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  1. Directional Multiscale Modeling of Images Duncan Po and Minh N. Do University of Illinois at Urbana-Champaign

  2. Motivation: Image Modeling A randomly generated image A “natural” image A simple and accurate image model is the key in many image processing applications.

  3. Background: Multiscale Modeling • Initially: Wavelet transform as a good decorrelator • Later: Incorporate dependencies across scale and space

  4. New Image Representations • Idea: Efficiently represent smooth contours by directional and elongated basis elements • Idea: Successive refinement for edges in both location and direction

  5. The Contourlet Transform • Contourlet transform (Do and Vetterli, 2002): extension of the wavelet transform using directional filter banks • Properties: sparse representation for smooth contours, efficient FB algorithms, tree data structure, …

  6. The Contourlet Transform Wavelet Contourlet

  7. Our Goals • Study statistics and properties of contourlet coefficients of natural images. - Good understanding of properties would provide key insights in developing contourlet-based applications. • Based on statistics and properties, develop a suitable model. - Key Idea: Model all three fundamental parameters of visual information: scale, space, and direction. • Apply model to applications. - denoising and texture retrieval.

  8. Marginal Distribution (Peppers) Kurtosis = 24.50 >> 3  non-Gaussian!!

  9. Structure of Contourlet Coefficients Each coefficient (X) has: Parent (PX), Neighbors (NX), Cousins (CX) We refer to all of them as Generalized Neighbors

  10. Joint Statistics: Conditional Distributions (Peppers) Parent Neighbor Cousin

  11. Joint Statistics: Conditional Distributions on Far Neighbors (3 Coefficients away) Peppers Goldhill

  12. Joint Statistics: Conditional Distributions (Peppers) P(X| PX = px) Kurtosis = 3.90 P(X| NX = nx) Kurtosis = 2.90 P(X| CX = cx) Kurtosis = 2.99

  13. Joint Statistics: Quantitative • Use mutual information (Liu and Moulin, 2001) • A measure of how much information X conveys about Y • In estimation, quantifies how easy to estimate X given Y. • In compression, if it takes m bits to encode X, then given Y, it takes only m-I(X,Y) bits to encode X.

  14. Joint Statistics: Quantitative • Histogram estimator for mutual information (Moddemeijer, 1989) • Multiple variables: use sufficient statistics (Liu and Moulin, 2001)

  15. Results: I(X;.)

  16. Lena

  17. Average Mutual Information against Individual Generalized Neighbors

  18. Summary • Contourlet coefficients of natural images exhibit the following properties: • non-Gaussian marginally. • dependent on generalized neighborhood. • conditionally Gaussian conditioned on generalized neighborhood. • parents are (often) the most influential. • Next Step: Develop a simple statistical model that takes into account these properties

  19. Hidden Markov Tree (HMT) Model • Developed for wavelets (Crouse et. al., 1998) Transition Matrix A State S Transition Matrix Coefficient u State 1 State 2

  20. Contourlet HMT Model • Each tree has the following parameters • root state probabilities • state transition probability matrix between subband k in scale j and its parent subband in scale j-1 • Gaussian standard deviations of subband q in scale p

  21. Contourlet HMT Model Wavelets Contourlets

  22. Denoising: Bayesian Estimation

  23. Denoising Results: PSNR

  24. Denoising Results: Zelda Noisy, w = 50 PSNR = 14.61 Wiener2, 5X5 PSNR = 25.78 Original Wavelet Thresholding T = 3w , PSNR = 26.05 Wavelet HMT PSNR = 27.63 Contourlet HMT PSNR = 27.07

  25. Denoising Results: Barbara Noisy, w = 51 PSNR = 14.48 Wiener2, 5X5 PSNR = 22.57 Original Wavelet Thresholding T = 3w , PSNR = 21.96 Wavelet HMT PSNR = 23.71 Contourlet HMT PSNR = 23.74

  26. Contourlet Texture Retrieval System

  27. Texture Retrieval: Use Kullback-Liebler Distance

  28. Retrieval Results • Average retrieval rates: • Wavelet HMT: 90.87% Contourlet HMT: 93.29% • Wavelets retrieve better (>5%) • Contourlets retrieve better (>5%)

  29. Conclusions • Contourlets: new true two dimensional transform allows modeling of all three visual parameters: scale, space, and direction • Statistical measurements show: • Strong intra-subband, inter-scale, and inter-orientation dependencies • Conditioned on their neighborhood, coefficients are approximately Gaussian • Contourlet hidden Markov tree model • Promising results in denoising and retrieval

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