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Bach Mairal Ponce Sapiro

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32 Pages
English

Description

Dictionary Learning ICCV 2009 Bach, Mairal, Ponce, Sapiro

  • random vector

  • learning sparsity

  • energy minimization

  • prototype signal

  • vector ?

  • being learned

  • given measurements

  • contains very


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Dictionary Learning
ICCV 2009
Bach, Mairal, Ponce, SapiroRestoration by Energy Minimization
Restoration/representation algorithms are often related to the minimization
of an energy function of the form
1 2
f x   x  y  Pr x 
22
y : Given measurements Relation to
Prior or regularization
measurementsx : Unknown to be recovered
 Bayesian type of approach
 What is the prior? What is the image model? Thomas Bayes
1702 - 1761
Learning Sparsity 2The Sparseland Model for Images
 Every column in
D (dictionary) is
a prototype signal M K (Atom).
N
 The vector N 
contains very few
(say L) non-zeros.
A sparse A fixed Dictionary x
& random
vectorD α
Learning Sparsity 3What Should the Dictionary D Be?
1 2 0
  argmin D   y s.t.   L ˆx  D ˆ ˆ
022
D should be chosen such that it sparsifies the representations
Learn D :
One approach to choose D is from a
Multiscale Learningknown set of transforms (Steerable
wavelet, Curvelet, Contourlets,
Color Image Examples
Bandlets, …)
Task / sensing adapted
Internal structure
Learning Sparsity 4What is being learned?
Learning Sparsity 5Learning D to reconstruct
X D A
P 2 0
Min D   x s.t. j,   L jj j Field & Olshausen (‘96)2 0 Engan et. al. (‘99)D,A j 1 Lewicki & Sejnowski (‘00)
Each example has a Each example is Cotter et. al. (‘03)
Gribonval et. al. (‘04)sparse representation with a linear combination
Aharon, Elad, & Bruckstein (‘04)
no more than L atomsof atoms from D Aharon, Elad, & Bruckstein (‘05)
Ng et al. (‘07)
Mairal, Sapiro, Elad (‘08)
Learning Sparsity 6The K –SVD Algorithm – General
Aharon, Elad, & Bruckstein (`04)
Initialize DD
Sparse Coding
Orthogonal Matching TPursuit (or L1)
X
Dictionary
Update
Column-by-Column by
SVD computation over
the relevant examples
Learning Sparsity 7Non-uniform noise
Learning Sparsity 8Show me the pictures
Learning Sparsity 9Change the Metric in the OMP
Learning Sparsity 10