Sparsity and image processing

Transcription

1 Sparsity and image processing Aurélie Boisbunon INRIA-SAM, AYIN March 6,

2 Why sparsity? Main advantages Dimensionality reduction Fast computation Better interpretability Image processing pattern recognition denoising / deblurring compression super-resolution source separation A. Boisbunon (AYIN) Sparsity & Image Proc. March 6, /

3 Context and objectives Linear regression x D ε x = D α + ε (vectorized) image dictionary noise Assumption α is a sparse vector/matrix Dictionary D = {φ j } J j= Fixed: Fourier basis, Wavelets Learned Source: [Hastie et al., 8] Source: [Donoho et al., 99] A. Boisbunon (AYIN) Sparsity & Image Proc. March 6, /

4 Sparse optimization problem { } min x Dα α + pen(α) goodness of fit / distortion rate Goodness of fit Measures how close two images are Original Salt & pepper Gaussian Negative S&P Gauss Neg Goodness of fit x A. Boisbunon (AYIN) Sparsity & Image Proc. March 6, /

5 Sparse optimization problem { } min x Dα + pen(α) α penalty / regularization Penalty Special case: non-differentiable in zero sparse solution ˆα β β l β β l /Lasso... β β Reweighted-l... β β MCP MCP = Minimax Concave Penalty [Zhang, ] with belonging to subgradient of pen A. Boisbunon (AYIN) Sparsity & Image Proc. March 6, /

6 Sparse optimization problem { } min x Dα + pen(α) α penalty / regularization Penalty Special case: non-differentiable in zero sparse solution ˆα 6 l l LS LS lasso Lasso LS LS l /hard threshold l /Soft threshold MCP = Minimax Concave Penalty [Zhang, ] with belonging to subgradient of pen adalasso Adaptive LS 6 6 LS 6 Reweighted-l FS FS LS LS MCP A. Boisbunon (AYIN) Sparsity & Image Proc. March 6, 6 /

7 Matching/Basis pursuit Algorithm Start: α =, J = Repeat. Find vector φ j most correlated with residual arg max φ t j (x D (J) α (J) ). Add it to the active set J J {j}. Update the coefficients α (J) until stopping rule. A. Boisbunon (AYIN) Sparsity & Image Proc. March 6, 7 /

8 Matching/Basis pursuit β LS β LAR β Ada β MCP 6 step l /matching p. 6 step l /Basis p. 6 step Reweighted-l 6 8 step MCP A. Boisbunon (AYIN) Sparsity & Image Proc. March 6, 8 /

9 Applications Compression Original l Reweighted-l [Candes et al., 8] A. Boisbunon (AYIN) Sparsity & Image Proc. March 6, 9 /

10 Applications Denoising/Deblurring Original Noisy l (FISTA) [Beck and Teboulle, 9] A. Boisbunon (AYIN) Sparsity & Image Proc. March 6, /

11 Dictionary learning Optimization problem { min x Dα + pen(α) } α, D Algorithm Start: α =, D. Extract patches from image. Repeat Solve optimization problem for α with D fixed Solve optimization problem for D with α fixed until stopping rule. Source: [Bach et al., ] A. Boisbunon (AYIN) Sparsity & Image Proc. March 6, /

12 Dictionary learning Applications Inpainting Texture recognition 6 [Mairal et al., 9] 6 [Mairal et al., 8] A. Boisbunon (AYIN) Sparsity & Image Proc. March 6, /

13 Thank you! A. Boisbunon (AYIN) Sparsity & Image Proc. March 6, /

14 References Bach, F., Jenatton, R., Mairal, J., and Obozinski, G. (). Optimization for Machine Learning, chapter Convex optimization with sparsity-inducing norms, pages 9. MIT Press. Beck, A. and Teboulle, M. (9). A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences, ():8. Candes, E. J., Wakin, M. B., and Boyd, S. P. (8). Enhancing sparsity by reweighted l minimization. Journal of Fourier analysis and applications, (-6): Donoho, D. L., Buckheit, J. B., Chen, S., Johnstone, I., and Scargle, J. D. (99). About wavelab. Hastie, T., Tibshirani, R., and Friedman, J. (8). The Elements of Statistical Learning: Data Mining, Inference and Prediction (nd Edition), volume. Springer Series in Statistics. Mairal, J., Bach, F., Ponce, J., and Sapiro, G. (9). Online dictionary learning for sparse coding. In Proceedings of the 6th Annual International Conference on Machine Learning, pages ACM. Mairal, J., Ponce, J., Sapiro, G., Zisserman, A., and Bach, F. R. (8). Supervised dictionary learning. In Advances in Neural Information Processing Systems, pages. Zhang, C. (). Nearly unbiased variable selection under minimax concave penalty. Annals of Statistics, 8():89 9. A. Boisbunon (AYIN) Sparsity & Image Proc. March 6, /