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Sparse_and_BP

# Sparse_and_BP - Sparse Representations and the Basis...

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Click to edit Master subtitle style Sparse representation and the Basis Pursuit Sparse Representations and the  Basis Pursuit Algorithm* Michael Elad The Computer Science Department –  Scientific Computing & Computational mathematics (SCCM) program Stanford University November 2002 * Joint work with: Alfred M. Bruckstein – CS, Technion     David L. Donoho – Statistics, Stanford   Peyman Milanfar – EE, UCSC

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Sparse representation and the Basis Pursuit 22 Collaborators Dave Donoho Statistics Department  Stanford Freddy Bruckstein Computer Science    Department – Technion Peyman Milanfar EE - University of  California Santa-Cruz
Sparse representation and the Basis Pursuit 33 General Basis Pursuit algorithm  [Chen, Donoho and Saunders, 1995] § Effective for finding sparse over-complete representations, § Effective for non-linear filtering of signals. Our work (in progress) – better understanding BP and  deploying it in  signal/image processing and computer vision applications.  We believe that over-completeness has an important role!  Today we discuss: § Understanding the BP: why successful? conditions?   § Deploying the BP: through its relation to Bayesian (PDE) filtering.

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Sparse representation and the Basis Pursuit 44 Agenda Understanding  the BP  1. Introduction Previous and current work 2.  Two Ortho-Bases Uncertainty �  Uniqueness n  Equivalence 3.  Arbitrary dictionary Uniqueness   Equivalence   4.  Basis Pursuit for Inverse Problems Basis Pursuit Denoising P  Bayesian (PDE) methods 5.  Discussion Using the BP for  denoising
Sparse representation and the Basis Pursuit 55 { } { } α = = α - 1 T s : Backward s T : Forward Define the forward and backward transforms by (assume  one-to-one mapping) s  – Signal (in the signal space CN)  – Representation (in the transform domain CL, L* N) Transforms T in signal and image processing used for coding, analysis,  speed-up processing, feature extraction, filtering, …    Transforms

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Sparse representation and the Basis Pursuit 66 = L N s Atoms from a  Dictionary General transforms Special interest - linear  transforms (inverse)             Linear α Φ = s    The Linear Transforms Square In square linear transforms,      is an N-by-N & non- singular. Unitary
Sparse representation and the Basis Pursuit 77 Many available square linear transforms – sinusoids, wavelets, packets,  ridgelets, curvelets, …  Successful transform – one which leads to sparse representations.

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Sparse_and_BP - Sparse Representations and the Basis...

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