Research Director at the National Center for Scientific Research (CNRS),
Ali Mohammad-Djafari received the B.Sc. degree in electrical engineering
from Polytechnic of Teheran, in 1975, the diploma degree (M.Sc.) from
Ecole Supérieure d'Electricit(SUPELEC), Gif sur Yvette, France, in 1977,
the "Docteur-Ingénieur" (Ph.D.) degree and "Doctorat d'Etat" in Physics,
from the University of Paris Sud 11 (UPS), Orsay, France, respectively in 1981 and 1987.
He was Assistant Professor at UPS for two years (1981-1983). Since 1984, he has a
permanent position at "Centre National de la Recherche Scientifique (CNRS)" and works
at "Laboratoire des signaux et systèmes (L2S)" at SUPELEC. He was a visiting Associate
Professor at University of Notre Dame, Indiana, USA during 1997-1998. From 1998 to
2002, he has been at the head of Signal and Image Processing division at this laboratory.
Presently, he is "Directeur de recherche" and his main scientific interests are in developing
new probabilistic methods based on Bayesian inference, Information Theory and
Maximum Entropy approaches for Inverse Problems in general in all aspects of data
processing, and more specifically in imaging and vision: image reconstruction, signal and
image deconvolution, blind source separation, sources localization, data fusion, multi and
hyper spectral image segmentation. The main application domains of his interests are
Computed Tomography (X rays, PET, SPECT, MRI, microwave, ultrasound and eddy
current imaging) either for medical imaging or for non destructive testing (NDT) in industry,
multivariate and multi dimensional data, signal and image processing, data mining,
clustering, classification and machine learning methods for biological or medical
He has supervised more than 20 Ph.D. Thesis, more than 10 Post-doc research activities
and more than 50 M.Sc. Student research projects. In 2013, he was supervising 6 Ph.D.
Thesis where four graduated successfully. He has more than 40 full journal papers and
more than 200 papers in national and international conferences. He has organized or coorganized
about 10 international workshops and conferences. He has been expert for a
great number of French national and international projects. Since 1988 he has many
teaching activities in M.Sc. and Ph.D. Level in SUPELEC, University of Paris sud, ENSTA
and Ecole centrale de Paris.
He also participated and managed many industrial contracts with many French national
industries such as EDF, RENAULT and THALES and great research institutions such as
CEA, INSERM, INRIA as well as the regional (such as Digiteo), national (such as ANR)
and European projects (such as ERASYSBIO).
For an overview and acces to more details of his activities and publications, please see his
http://djafari.free.fr for general,
for news and
http://publicationslist.org/djafarie for the list of publications.
Bayesian Approach to Inverse Problems of Acoustic Source Localization
Laboratoire des signaux et System
UMR 8506 CNRS-SUPELEC-UNI PARIS SUD
In this tutorial, first the problem of source localization is considered as an inverse problem of spatially varying deconvolution and different methods of regularization are investigated. In particular the Least Squares (LS) methods subject to the minimum L0 or L1-norm are presented. Then, to push further the limitations of the deterministic methods which need, for example the prior knowledge of the number of sources or the regularization parameter, the Bayesian estimation approach is presented.
An important step in the Bayesian approach is the assignment of the prior laws to noise and to the desired solution of the inverse problem. When this step is done appropriately, the expression of the posterior law can be obtained and used to do inference about the unknowns. Two point estimators are commonly used: the Maximum A Posteriori (MAP) and the Expected A Posteriori (EAP). In this tutorial, two classes of heavy tailed prior laws are considered: Generalized Gaussian (GG) and the Student-t or Cauchy distributions. The corresponding Bayesian methods are implemented and compared to the classical Beam-Forming (BF), CLEAN, Minimum Norm (MN) LS or Quadratic Regularization (Tikhonov) or Sparsity enforcing based methods.
Between the advantages of the Bayesian framework, we can mention the possibility to account for the approximation errors. As an example, we will see how the near-field propagation model which is a spatially variant Point Spread Function (PSF) convolution can be approximated by a far field model which is a fixed PSF convolution which can be computed via Fast Fourier Transform (FFT). However, this approximation generate errors on the predicted data which is spatially variant and this has to be accounted for during the inversion method. We will see how the Bayesian framework can handle this with a non stationary Gaussian
model for these errors where we need to estimate the spatially variant variances simultaneously. Inverse Gamma priors are assigned to these unknown variances which are conjugate priors for the Gaussian probability laws.
At the end, we give some details of a Bayesian method which accounts for the spatial variability of the observation predicting errors via a non stationarity Gaussian probability model and for the sparsity of the acoustic sources spatial distribution. This last property is modeled via a Student-t probability law which can be modeled in a hierarchical way through its Infinite Gaussian Mixture (IGM) equivalence.
Finally, to propose a non supervised inversion and source localization method
a Variational Bayesian Approximation (VBA) computational method is presented
and its performances are compared to more classical deterministic or Bayesian