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 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

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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

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Convex optimization: problems of interest

We want to optimize a sum of convex functions Mathematical form:

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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).

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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

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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 proximity operator of a lower semi-continuous convex function ff 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:

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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

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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.

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Image in-painting results10

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)

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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 8,66dBOriginal Depleted Reconstructed

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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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