UNLocBox: Matlab convex optimization toolbox http://wiki. epfl 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 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

What is UNLocboX? Matlab convex optimization toolbox Why? 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

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

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).

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

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:

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

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.

Image in-painting results

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)

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

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