mgaws1_5056 - Sparse Representations of Signals: Theory and...

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Click to edit Master subtitle style Sparse representations for Signals – Theory and Sparse Representations of Signals:    Theory  and Applications * Michael Elad The CS Department The Technion – Israel Institute of technology Haifa 32000, Israel IPAM – MGA Program September 20th, 2004 * Joint work with:  Alfred M. Bruckstein   – CS, Technion                             David L. Donoho   – Statistics, Stanford Vladimir Temlyakov   – Math, University of South Carolina Jean-Luc Starck   – CEA - Service d’Astrophysique, CEA-Saclay, France
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Sparse representations for Signals – Theory and 22 Collaborators Freddy Bruckstein   CS Technion Vladimir Temlyakov  Math, USC Jean Luc Starck   CEA-Saclay, France Dave Donoho Statistics, Stanford
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Sparse representations for Signals – Theory and 33 Agenda 1. Introduction Sparse & overcomplete representations, pursuit algorithms 2. Success of BP/MP as Forward Transforms Uniqueness, equivalence of BP and MP 3. Success of BP/MP for Inverse Problems  Uniqueness, stability of BP and MP 4. Applications Image separation and inpainting 
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Sparse representations for Signals – Theory and 44    Problem Setting – Linear Algebra Our dream – solve an linear system of                            equations of the form known α Φ = x L N L>N, Ф is full rank, and  Columns are  normalized Φ where
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Sparse representations for Signals – Theory and 55    Can We Solve This?  Our assumption for today: the sparsest possible solution is preferred  Generally NO * Unless additional information is introduced. *
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Sparse representations for Signals – Theory and 66    Great … But,  Why look at this problem at all? What is it good for? Why  sparseness?   Is now the problem well defined now? does it lead to a unique  solution? How shall we numerically solve this problem?  These and related          questions will be discussed in  today’s talk
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Sparse representations for Signals – Theory and 77    Addressing the First Question  We will use the linear relation   as the core idea for modeling signals α Φ = x
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Sparse representations for Signals – Theory and 88    Signals’ Origin in  Sparse-Land   Random  Signal  Generator x     We shall assume that our signals of interest           emerge from a random  generator machine   M M
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Sparse representations for Signals – Theory and 99    Signals’ Origin in  Sparse-Land   = α sparse Instead of defining       over the signals directly,  we define it over “their  representations ”  α : § Draw the number of none-zeros (s)   in  α with probability P(s), §
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This note was uploaded on 05/28/2010 for the course EE EE564 taught by Professor Runyiyu during the Spring '10 term at Eastern Mediterranean University.

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mgaws1_5056 - Sparse Representations of Signals: Theory and...

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