Sparse_and_BP - Sparse Representations and the Basis...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Click to edit Master subtitle style Sparse representation and the Basis Pursuit Sparse Representations and the  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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 
Background image of page 4
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 
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 6
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. Observation: Lack of universality - Different bases good for different  purposes.  § Sound = harmonic music (Fourier) + click noise (Wavelet), § Image = lines (Ridgelets) + points (Wavelets).
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 63

Sparse_and_BP - Sparse Representations and the Basis...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online