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A Transform-based Variational Framework Guy Gilboa Pixel Club, November, 2013

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In a Nutshell Spatial Input Transform Analysis Transform Filtering Spatial Output Fourier inspiration: Fourier Scale Spectral TV Flow TV Scale

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Relations to eigenvalue problems General linear: ( L linear operator) Functional based Linear Nonlinear

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What can a transform-based approach give us? Scale analysis based on the spectrum. New types of filtering – otherwise hard to design: nonlinear LPF, BPF, HPF. Nonlinear spectral theory – relation to eigenfunctions and eigenvalues. Deeper understanding of the regularization, optimal design with respect to data, noise and artifacts.

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Examples of spectral applications today: Eigenfunctions for 3D processing Taken from Zhang et al, Spectral mesh processing, 2010. Taken from L Cai, F Da, Nonrigid deformation recovery.., 2012.

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Image Segmentatoin Eigenvectors of the graph Laplacian [Taken from I. Tziakos et al, Color image segmentation using Laplacian eigenmaps, 2009 ]

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Some Related Studies Andreu, Caselles, Belletini, Novaga et al 2001- 2012– TV flow theory. Steidl et al 2004 – Wavelet – TV relation Brox-Weickert 2006 – scale through TV-flow Luo-Aujol-Gousseau 2009 – local scale measures Benning-Burger 2012 – ground states (nonlinear spectral theory) Szlam-Bresson – Cheeger cuts. Meyer, Vese, Osher, Aujol, Chambolle, G. and many more – structure-texture decomposition. Chambolle-Pock 2011, Goldstein-Osher 2009 – numerics.

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Scale Space – a Natural Way to Define Scale Well talk specifically about total-variation (TV-flow, Andreu et al - 2001): Scale space as a gradient descent:

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TV-Flow: A behavior of a disk in time [Andreu-Caselles et al–2001,2002, Bellettini-Caselles-Novaga-2002, Meyer-2001] Center of disk, first and second time derivatives: t … …

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Spectral TV basic framework Phi(t) definition

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Reconstruction

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Spectral response Spectrum S(t) as a function of time t: t S(t) f

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Spectrum example fS(t)

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Dominant scales

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Eigenvalue problem The nonlinear eigenvalue problem with respect to a functional J(u) is defined by: Well show a connection to the spectral components.

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Solution of eigenfunctions

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What are the TV eigenfunctions? [Giusti-1978], [Finn-1979],[Alter-Caselles- Chambolle-2003].

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Filtering Let H(t) be a real-valued function of t. The filtered spectral response is The filtered spatial response is H(t)

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Filtering, example 1: TV Band-Pass and Band-Stop filters Band-pass Band-stop fS(t)

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Disk band-pass example S(t)

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We have the basic framework Spatial Input Transform Analysis Transform Filtering Spatial Output

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Numerics Many ways to solve. Variational approach was chosen: Currently use Chambolles projection algorithm (some spikes using Split-Bregman, under investigation). In time: 2 nd derivative - central difference 1 st derivative - forward differnce Discrete reconstruction algorithm proved for any regularizing scale-space (Th. 4).

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TV-Flow as a LPF

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Nonlocal TV Reminder: NL-TV (G.-Osher 2008): Gradient Functional

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Spectral NL-TV? The framework can fit in principle many scale-spaces, like NL-TV flow. We can obtain a one-homogeneous regularizer. What is a generalized nonlocal disk? What are possible eigenfunctions? It is expected to be able to process better repetitive textures and structures.

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Sparseness in the TV sense Sparse spectrum – the signal has only a few dominant scales. Or many small ones (here TV energy is large) Can be a large objects Natural images – are not very sparse in general

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Noise Spectrum Various standard deviations: S(t)

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Noise + signal Not additive. Spreads original image spectrum. Needs to be investigated. u f f-u Band-pass filtered

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Spectral Beltrami Flow? Initial trials on Beltrami flow with parameterization such that it is closer to TV Original Beltrami Flow Spectral Beltrami Difference images: Keeps sharp contrast Breaks extremum principle Values along one line (Green channel)

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Segmentation priors Swoboda-Schnorr 2013 – convex segmentation with histogram priors. We can have 2D spectrum with histograms Use it to improve segmentation S(t,h)

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Texture processing Many texture bands We can filter and manipulate certain bands and reconstruct a new image. Generalization of structure-texture decomposition.

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Processing approach Deconstruct the image into bands Identify salient textures Amplify / attenuate / spatial process the bands. Reconstruct image with processed bands

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Color formulation Vectorial TV – all definitions can be generalized in a straightforward manner to vector-valued images. Bresson-Chan (2008) definition and projection algorithm is used for the numerics.

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Orange example

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Orange – close up Original Modes 2,3=0 Modes 2-5=x1.5

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Selected phi(t) modes (1, 5, 15, 40) residual f

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Old man

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Old man – close up Original 2 modes attenuated 7 modes attenuated

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Old Man - First 3 Modes Modes: 1 2 3

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Take Home Messages Introduction of a new TV transform and TV spectrum. Alternative way to understand and visualize scales in the image. Highly selective scale separation, good for processing textures. Can be generalized to other functionals.

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Thanks! Refs. Google Guy Gilboa publications Preliminary ideas are in SSVM 2013 paper. Most material is in CCIT Tech report 803. Up-to-date and organized - submitted journal version – contact me.

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