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From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images Alfred M. Bruckstein (Technion), David L. Donoho (Stanford), Michael Elad (Technion) SIAM REVIEW 2009. Presented by: Mingyuan Zhou Duke University, ECE June 11, 2009. Outline. Introduction
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From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and ImagesAlfred M. Bruckstein (Technion), David L. Donoho (Stanford), Michael Elad (Technion) SIAM REVIEW 2009 Presented by: Mingyuan Zhou Duke University, ECE June 11, 2009
Outline • Introduction • The sparsest solution of Ax = b • Variations on P0 • Sparsity-seeking methods in signal processing • Processing of sparsely generated signals • Applications in image processing
Introduction • Under-determined linear system equation • L2 norm • L0 norm How can uniqueness of a solution be claimed? How to verify a candidate solution? How to efficiently solve the problem (the exhaustive search is a NP-hard problem)? What kind of approximations will work and how accurate can those be?
Current achievements • Conditions under which has a unique solution • Conditions under which has the unique solution as • Conditions under which the solution can be found by some “pursuit” algorithm • Less restrictive notions of sparsity, impact of noise, the behavior of approximate solutions, and the properties of problem instances defined by ensembles of random matrices…
The signal processing perspective • JPEG, DCT • JPEG-2000, DWT • The sparsity of representation under given basis is key to many important signal and image processing problems: Image compression, Image denoising, image deblurring, speech compression, audio compression… Measuring sparsity
Uniqueness • Uniqueness via the Spark • Uniqueness via the Mutual Coherence
Pursuit Algorithms: Practice • Greedy Algorithms • Convex Relaxation Techniques
Pursuit Algorithms: Performance • Greedy Algorithms • Convex Relaxation Techniques
Variations on P0 Uncertainty Principles and Sparsity
From Exact to Approximate Solutions • Relaxed constraint: • Stability: • Pursuit algorithms: OMP Iteratively reweighted least squares (IRLS) Iterative thresholding Stepwise algorithms: LARS and Homotopy
Beyond Coherence Arguments • Empirical evidence: The column of A is drawn at random from a Gaussian distribution, , Without noise With noise
The sparsest solution of Ax = b: A summary • Uniqueness • Solvability • Approximate solutions • Beyond coherence
Sparsity-Seeking Methods in Signal Processing • Non-Gaussian Prior • Combined representation
Processing of Sparsely Generated Signals • Applications Analysis Compression Denoising Inverse problems Compressive sensing Morphological Component Analysis
The quest for a dictionary Reconstructed dictionaries Dictionaries learned from training data Dictionaries learned from data under test Learning Methods: MOD, K-SVD, BPFA
Applications in Image Processing • Compression of Facial Images