Sparse acoustical holography from iterated Bayesian focusing

Abstract In a previous work, an attempt was made to give a unified view of some acoustic holographic methods within a Bayesian framework. One advantage of the so-called “Bayesian Focusing” approach is to introduce an aperture function that acts like a lens and thus significantly improves the reconstruction results in terms of spatial resolution, but also of quantification over a larger frequency interval than allowed by conventional methods. This is particularly remarkable when the aperture function is allowed to become very narrow as in the case of sparse sources. The aim of the present paper is to demonstrate that the aperture function – which was previously manually tuned by the user – can be automatically estimated, together with the source distribution, in the same inverse problem. The principle is to use the current estimate of the source distribution to update the aperture function in the next iteration. The resulting algorithm is an iterated version of Bayesian Focusing, which can be formalized as an Expectation-Maximization algorithm with proved convergence. The proof of convergence is based on modeling the aperture function as a random quantity, which assigns the source coefficients with prior probability distribution in the form of a “scale mixture of Gaussians” that enforces sparse solutions. Various types of sparsity enforcing priors can thus be constructed, in a much richer setting than the usual l 1 penalized approach, leading to different updating rules of the aperture function. Some immediate byproducts of iterating Bayesian Focusing are 1) to provide a technique for the automatic setting of the regularization parameter, 2) to apply on the cross-spectral matrix of the measurements, and 3) to easily allow the grouping of frequencies for the broadband analysis of sources that are stationary in space. Experimental results confirm that sparse holography improves the reconstruction of sources not only in terms of localization, but also of quantification and of directivity in a frequency range considerably enlarged as compared to classical methods. These improvements can be achieved even with regular arrays, provided that sparser priors than those leading to the standard l 1 penalization are used.