Bayesian vision of inverse problems in acoustics

The inverse problem in acoustics is known to be inherently ill-posed due to several reasons such as the lack of information in the measurements, disturbances and noise, and possible uncertainties in the propagation laws. Contrary to the deterministic approach, a probabilistic formalism is ideally suited to modeling these various sources of errors - be they stochastic or purely epistemic - and to control their effect in the inversion process. In this context, the Bayesian framework is particularly attractive since it expresses the inverse probability (amplitude probability of the source field given the measurements) in terms of the direct problem (amplitude probability of the measurements given the source field) and any information that is a priori available on the source characteristics before the experiment is made (the so-called "prior"). This formalism has several advantages; among other things, a) it contains an implicit mechanism of regularization (of which Tikhonov regularization happens to be a particular case) together with automatic tuning of the regularization parameters, b) it returns probabilistic results from which confidence intervals can be easily set up, c) it makes possible the propagation of the various sources of errors or uncertainties to the final result. These concepts will be reviewed and some other recent results that the Bayesian approach has made possible in the field of acoustic imaging will be presented. These include the definition of an optimal basis for source reconstruction, a solution to blindly separate sound sources of distinct physical origins, the reconstruction of missing phase relationships between non-synchronous (e.g. sequential) measurements, and the possibility to account for modeling errors. Illustrations will be given on different academic and industrial cases.