UNLocBox: Matlab convex optimization toolbox Presentation by Nathanaël Perraudin Authors:...
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Transcript of UNLocBox: Matlab convex optimization toolbox Presentation by Nathanaël Perraudin Authors:...
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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