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SIAM.Imaging.2008

# SIAM.Imaging.2008 - Sparse approximations in image...

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Sparse approximations in image processing Anna C. Gilbert Department of Mathemtics University of Michigan

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Basic image compression: transform coding
Tutorial on sparse approximations This tutorial is about the sparse approximation of images, signals, data, etc.

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Tutorial on sparse approximations This tutorial is about the sparse approximation of images, signals, data, etc. Compress data accurately concisely efficiently (encoding and decoding)
Tutorial on sparse approximations This tutorial is about the sparse approximation of images, signals, data, etc. Compress data accurately concisely efficiently (encoding and decoding) Focus on algorithms mathematical approximation theory

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Tutorial on sparse approximations This tutorial is about the sparse approximation of images, signals, data, etc. Compress data accurately concisely efficiently (encoding and decoding) Focus on algorithms mathematical approximation theory Not focus on Compressed Sensing (see Ron DeVore’s talk) image models, codecs, etc.
Orthogonal basis Φ: Transform coding Φ x = c Φ c x Compute orthogonal transform Φ T x = c

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Orthogonal basis Φ: Transform coding Φ x = c Φ c x Compute orthogonal transform Φ T x = c Threshold small coefficients Θ( c )
Orthogonal basis Φ: Transform coding Φ x = c Φ c x Compute orthogonal transform Φ T x = c Threshold small coefficients Θ( c ) Reconstruct approximate image Φ(Θ( c )) x

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Nonlinear encoding = c x Φ x = Ψ a c'
Nonlinear encoding = c x Φ x = Ψ a c' = c y Φ y = Ω c' Position of nonzeros depends on signal Different matrices Φ T , Ω T for 2 different signals

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Linear decoding c x Φ c Φ y Given the vector of coefficients and the nonzero positions, recover (approximate) signal via linear combination of coefficients and basis vectors Φ(Θ( c )) x
Linear decoding c x Φ c Φ y Given the vector of coefficients and the nonzero positions, recover (approximate) signal via linear combination of coefficients and basis vectors Φ(Θ( c )) x Decoding procedure not signal dependent

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Linear decoding c x Φ c Φ y Given the vector of coefficients and the nonzero positions, recover (approximate) signal via linear combination of coefficients and basis vectors Φ(Θ( c )) x Decoding procedure not signal dependent Matrix Φ same for all signals
Redundancy If one orthonormal basis is good, surely two (or more) are better...

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Redundancy If one orthonormal basis is good, surely two (or more) are better... ...especially for images

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Dictionary Definition A dictionary D in R d is a collection { ϕ } N =1 R d of unit-norm vectors: k ϕ k 2 = 1 for all . Elements are called atoms If span { ϕ } = R d , the dictionary is complete If { ϕ } are linearly dependent, the dictionary is redundant
Matrix representation Form a matrix Φ = ϕ 1 ϕ 2 . . . ϕ N so that Φ c = X c ϕ .

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