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UNLocBox: Matlab convex optimization toolbox wiki.epfl.ch/unlocbox

UNLocBox: Matlab convex optimization toolbox http://wiki.epfl.ch/unlocbox. Presentation by Nathanaël Perraudin Authors: Perraudin Nathanaël, Shuman David Vandergheynst Pierre and Puy Gilles LTS2 - EPFL. Plan. What is UNLocboX Convex optimization: problems of interest

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UNLocBox: Matlab convex optimization toolbox wiki.epfl.ch/unlocbox

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  1. UNLocBox:Matlab convex optimization toolboxhttp://wiki.epfl.ch/unlocbox Presentation by Nathanaël Perraudin Authors: Perraudin Nathanaël, Shuman David Vandergheynst Pierre and Puy Gilles LTS2 - EPFL

  2. Plan • What is UNLocboX • Convex optimization: problems of interest • How to write the problem? • Proximal splitting • Algorithms • UNLocboX organization • Solvers • Proximal operator • A small image in-painting example • Inclusion into the LTFAT toolbox • Use of the UNLocboX through an sound in-painting problem

  3. What is UNLocboX? • Matlab convex optimization toolbox • Very general • http://wiki.epfl.ch/unlocbox • Why? • In LTS 2 lab of EPFL everyone was rewritting the same code again and again • It allows to make reproducible results of experiments • Very new toolbox • First public release: august 12 • Mistakes? • Evolve quite fast • New functions will be added • Will take the same structure as LTFAT soon

  4. Convex optimization: problems of interest • We want to optimize a sum of convex functions • Mathematical form:

  5. Example • Usually a signal contain structure and this sometimes implies that it minimizes some mathematical functions. • Example: On image, the Fourier transform is mainly composed of low frequencies. The gradient is usually sparse (Lot of coefficients are close to zero, few are big).

  6. How to write the problem? • One way to write the problem is: • With this formulation the signal should be close to the measurement and satisfy also the prior assumption. • Suppose we want to recover missing pixel on a image: • A would simply be a mask • y the known pixels • f(x) an assumption about the signal • Example the gradient is sparse, sharp edge => f = TV norm • One way of writing the problem could be

  7. Proximal splitting • The problem is solved by minimizing iteratively each term of the sum. • We separate the problem into small problems. This is called proximal splitting. • The term proximal refers to their use of proximity operators, which are generalizations of convex projection operators. • The proximity operator of a lower semi-continuous convex function f is defined by: • In the toolbox, the main proximal operator are already implemented. • In our image in-painting problem the proximal operator we need to define is:

  8. Selection of a solver • 3 solvers in the UNLocboX + generalization • Choice depends of the problem • Form • Function (can we compute the gradient of one function?) • Forward backward • Need a Lipschitz continuous gradient • Douglas Rachford • Need only proximal operators • Alternating-direction method of multipliers (ADMM) • Solves problem of the form

  9. A bit of matlab – toolbox organisation • The toolbox is composed of solvers and proximal operators • All proximal operator takes 3 arguments • The measurements • The weight • A structure containing optional parameters • The solvers have various structures but take usually the starting point the functions and optional parameter • In matlab, each function is represented by a structure containing two fields: • f.norm : evaluation of the function • f.prox or f.grad: gradient or proximal operator of the function • This structure allows a quick implementation. • This structure allows to solve a big range of problem.

  10. Image in-painting results

  11. Inclusion in the LTFAT toolbox • The LTFAT toolbox provides a set of frame and frame operator that could be used with the UNLocBox. • Project of including wavelet in the LTFAT toolbox. • The UNLocBox is a very useful tool for the L1 minimization under constraints. • The UNLocBox can be use to do audio signal processing. • Example: Audio in-painting (emerging and promising field)

  12. Audio In-painting – A simple example • Suppose we have a audio signal with some samples have been lost. • We know that the Gabor transform of audio signal is usually smooth and localized. Using this information we can try to recover the original audio signal. • The problem would be • A the mask operator and G the Gabor transform • Results: SNR improved from 3.17dB to 8,66dB Original Depleted Reconstructed

  13. Questions? • Thank you for your attention • Any question? • Thanks to Pierre Vandergheynst and Peter L. Soendergaard for helping me to do this presentation. • More information on: http://wiki.epfl.ch/unlocbox

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