Spectral matrix completion by Cyclic Projection and application to sound source reconstruction from non-synchronous measurements
Résumé
A fundamental limitation of the inverse acoustic problem is determined by the size of the array and the microphone density. A solution to achieve large array and/or high microphone density is to scan the object of interest by moving sequentially a small prototype array, which is referred to as sequential measurements. In comparison to a large array and/or high microphone density array that can acquire simultaneously all the information of the spectral matrix, in particular all cross-spectra, sequential measurements can only acquire a block diagonal spectral matrix, while the cross-spectra between the sequential measurements remain unknown due to the missing phase relationships between consecutive positions. Nevertheless, these unknown cross-spectra are necessary for sound source reconstruction. The objective of this paper is to propose an algorithm to recover the missing elements of the spectral matrix in the case where the acoustical field is highly coherent. This issue is shown to boil down to a matrix completion problem subject to given constraint of hermitian symmetry, measurements fitting, reduced rank and spatial continuity of the sound field. A Cyclic Projection (CP) algorithm is proposed in this work to find an optimal solution at the intersection between three predefined sets. The proposed method is analyzed through numerical simulations of diverse setups and is also validated experimentally.