Bayesian formalism of mapping relationship based nearfield acoustical holography

Mapping relationship based NAH (MRS-based NAH) consists in constructing analytically a mapping relationship between modes on surfaces of boundary and hologram surface based on the fundamental solutions for Helmholtz equation (FS) in spherical coordinates. Therefore, the source reconstruction is converted to obtain the coefficients of participant modes on holograms. However, the number of FS is a crucial parameter to be chosen for the successful reconstruction, thus the purpose of current paper is to solve this parameter choosing problem. In this paper, MRS-based NAH with a fixed and large enough number of basis is firstly reformulated in a Bayesian formalism by endowing the physical quantities of interest with probability density functions (PDF). This formalish provides a careful description of measurement noise via the likelihood function; and at the same time it makes possible the incorporating of physical admissible fluctuations for the reconstructed source through the definition of a prior PDF, a mechanism which intrinsically gives rise to a regularization technique. Therefore, the reconstruction of MRS-based NAH is converted to obtain the coefficients of participant modes on holograms by solving an inverse problem of under-determined equation under the Bayesian formalism, which eliminates the necessary to choose the number of FS in traditional MRS-based NAH. The proposed method has been elaborately investigated by the simulation and the results are given in diverse scenarios.