Course Description

Exploring inverse problems with non-smooth optimization and sparse representations

by Prof. Jalal Fadili from the GREYC (ENSICAEN, Caen, France), on March 29th 2012

Abstract

The course is split in three lectures (1h30 each) which will focus on inverse problems in signal and image processing (but not only), and on modern tools to solve them. These problems are formalized in a unified and versatile framework, by building upon theoretical tools from mathematical statistics, computational harmonic analysis, and non-smooth convex optimization.  The lectures will tackle in detail the main three facets of inverse problems : direct problem modeling ; prior models ; and the inversion method.  Modeling the direct problem allows to establish a degradation equation as faithfully as possible to the measurement scenario that produced the observations (i.e. degradation/measurement operator, noise).  As the degradation operator is typically ill-behaved since it models an acquisition process that encounters loss of information, and also because of the presence of noise, the problem is ill-posed in the sense of Hadamard. The classical approach is to incorporate prior knowledge on the typical structure of the original object in order to reduce the space of candidate solutions. This prior information ranges from the uniform smoothness assumption to more complex knowledge of the geometrical structures. In these lectures, we will focus on priors entailed by sparse overcomplete/redundant representations. We will also shed light on these prior models from a stochastic point of view.  Taking into account the direct model (to build the data fidelity) and the prior model on signals and images, solving the inverse problem can be cast as the minimization of an energy functional. In these lectures, we will focus on convex but non-necessarily smooth objectives. To solve these optimization problems, we will invoke tools from convex analysis and maximal monotone operator splitting. We will derive proximal splitting algorithms in their primal or dual versions. These algorithms take the form of fast iterative algorithms that achieve full splitting by exploiting the structure of the functional to be minimized, e.g. composite sum of simple and/or smooth functions, composition with linear operators, etc..  The general framework will be illustrated on comprehensive suite of experiments on several inverse problems such as denoising, deconvolution, sparse component separation, inpainting, compressed sensing, and more generally in sparse recovery problems. We will also show that the framework has implications beyond inverse problems in signal and image processing, such as in machine learning and optimal control.

Short biography of Jalal Fadili

Jalal M. Fadili graduated from the Ecole Nationale Supérieure d'Ingénieurs (ENSI) de Caen, Caen, France, and received the M.Sc. and Ph.D. degrees in signal and image processing from the University of Caen. He was a Research Associate with the University of Cambridge (MacDonnel--Pew Fellow), Cambridge, U.K., from 1999 to 2000. He has been an Associate Professor of signal and image processing since September 2001 at ENSI. He also received an Habilitation from  University of Caen in 2010.  He was a visitor at several universities (QUT--Australia, Stanford University, CalTech, MIT, EPFL).  He is the co-­author of a book entitled "Sparse Signal and Image Processing: Wavelets, Curvelets, Morphological Diversity" Cambridge University Press.  His research interests include statistical approaches in signal and image processing, inverse problems, computational harmonic analysis, optimization and sparse representations.  His areas of application include medical and astronomical imaging.
Schedule

The course program is structured as follows:

• 9h15 - 10h45: first part
• (Tea/Coffee break)
• 11h15 - 12h45: second part
• (Lunch break -- on your own)
• 14h30 - 16h: third part

Course location

Auditoire MERC 04,  Address : Place Louis Pasteur, 3 - Louvain-la-Neuve.   Building number "9", in cell "D8" on the map http://uclouvain.be/cps/ucl/doc/adpi/documents/PLAN_2007recto.pdf

Venue

http://www.uclouvain.be/en-acces-lln.html 

External visitors are welcome to park on the (free) Redime parking 

http://www.uclouvain.be/9913.html 

or see cell "D9" on  http://uclouvain.be/cps/ucl/doc/adpi/documents/PLAN_2007recto.pdf

Organizer and contact person:

L. Jacques