1. UNLocBox: Matlab convex optimization toolbox http://wiki. epfl
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 Why?
Very general
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.
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