Spherical Harmonics Decomposition in inverse acoustic methods involving spherical arrays

Abstract

Inverse methods for acoustic source mapping have gathered the attention of the beamforming community during the last years. Indeed, they provide higher accuracy in both source localization and strength estimation with respect to Conventional Beamforming (CB). One of the main drawbacks of the current formulations is the need of a regularization strategy for tackling the ill-posedness of the problem. Very often, Tikhonov regularization is exploited to face this issue, but different methods for estimating the regularization factor associated to the Tikhonov formulation may lead to different regularization levels and, therefore, to different results. This paper presents a way to face this problem when dealing with spherical arrays. The new approach proposed by the authors exploits Spherical Harmonics Decomposition (SHD) of complex pressure data at microphone locations. SHD performs a spatial filtering that reduces the effect of noise and causes an intrinsic stabilization of the numerical problem associated to the inverse problem formulation. When the source-receiver propagation model is appropriate to describe the acoustic environment in which the test takes place and noise is not spoiling excessively measurement data, the SHD approach is sufficient to obtain a regularized solution. If these conditions are not satisfied, SHD can be exploited as a pre-processing step in a twofold procedure also involving classical Tikhonov regularization. In this paper the SHD approach is tested in the Generalized Inverse Beamforming (GIBF) formulation. Classical Tikhonov approach, in which the regularization factor is estimated using the Generalized Cross-Validation (GCV), L-curve functions and Bayesian regularization, is presented as a way to enhance data processed by SHD. A sensitivity analysis of the approach to measurement noise and source-receiver relative positions is presented on simulated data. Results on experimental data are presented and discussed for both a simplified test case and an application to a real car cabin.