Transcript Folie 1 - univie.ac.at

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
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Plan
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What is UNLocboX
Convex optimization: problems of interest
How to write the problem?
Proximal splitting
Algorithms
UNLocboX organization
Solvers
- Proximal operator
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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?
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Matlab convex optimization toolbox
Very general
- http://wiki.epfl.ch/unlocbox
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Why?
In LTS 2 lab of EPFL everyone was rewritting the same code
again and again
- It allows to make reproducible results of experiments
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Very new toolbox
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First public release: august 12
Mistakes?
Evolve quite fast
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New functions will be added
Will take the same structure as LTFAT soon
Convex optimization: problems of
interest
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We want to optimize a sum of convex functions
Mathematical form:
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Example
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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?
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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:
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A would simply be a mask
y the known pixels
f(x) an assumption about the signal
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Example the gradient is sparse, sharp edge => f = TV norm
One way of writing the problem could be
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Proximal splitting
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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:
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Selection of a solver
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3 solvers in the UNLocboX + generalization
Choice depends of the problem
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Forward backward
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Need a Lipschitz continuous gradient
Douglas Rachford
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Form
Function (can we compute the gradient of one function?)
Need only proximal operators
Alternating-direction method of multipliers (ADMM)
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Solves problem of the form
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A bit of matlab – toolbox organisation
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The toolbox is composed of solvers and proximal
operators
All proximal operator takes 3 arguments
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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:
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The measurements
The weight
A structure containing optional parameters
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 results
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Inclusion in the LTFAT toolbox
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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.
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The UNLocBox is a very useful tool for the L1
minimization under constraints.
The UNLocBox can be use to do audio signal
processing.
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Example: Audio in-painting (emerging and promising
field)
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Audio In-painting – A simple example
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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
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A the mask operator and G the Gabor transform
Results: SNR improved from 3.17dB to 8,66dB
Original
Depleted
Reconstructed
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Questions?
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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