{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–20. Sign up to view the full content.

View Full Document Right Arrow Icon
Sparse approximations in image processing Anna C. Gilbert Department of Mathemtics University of Michigan
Background image of page 1

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

View Full Document Right Arrow Icon
Basic image compression: transform coding
Background image of page 2
Tutorial on sparse approximations This tutorial is about the sparse approximation of images, signals, data, etc.
Background image of page 3

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

View Full Document Right Arrow Icon
Tutorial on sparse approximations This tutorial is about the sparse approximation of images, signals, data, etc. Compress data accurately concisely efficiently (encoding and decoding)
Background image of page 4
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
Background image of page 5

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

View Full Document Right Arrow Icon
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.
Background image of page 6
Orthogonal basis Φ: Transform coding Φ x = c Φ c x Compute orthogonal transform Φ T x = c
Background image of page 7

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

View Full Document Right Arrow Icon
Orthogonal basis Φ: Transform coding Φ x = c Φ c x Compute orthogonal transform Φ T x = c Threshold small coefficients Θ( c )
Background image of page 8
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
Background image of page 9

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

View Full Document Right Arrow Icon
Nonlinear encoding = c x Φ x = Ψ a c'
Background image of page 10
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
Background image of page 11

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

View Full Document Right Arrow Icon
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
Background image of page 12
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
Background image of page 13

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

View Full Document Right Arrow Icon
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
Background image of page 14
Redundancy If one orthonormal basis is good, surely two (or more) are better...
Background image of page 15

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

View Full Document Right Arrow Icon
Redundancy If one orthonormal basis is good, surely two (or more) are better... ...especially for images
Background image of page 16
Background image of page 17

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

View Full Document Right Arrow Icon
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
Background image of page 18
Matrix representation Form a matrix Φ = ϕ 1 ϕ 2 . . . ϕ N so that Φ c = X c ϕ .
Background image of page 19

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

View Full Document Right Arrow Icon
Image of page 20
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}