Sparse l1 Regularisation of Matrix Valued Models for Acoustic Source Characterisation

TL;DR

This paper introduces a convex optimization approach with l1 regularization for reconstructing sparse acoustic source matrices from microphone array data, addressing practical measurement issues.

Contribution

It develops a novel convex model with structured matrix constraints and combines splitting algorithms and matrix differential theory for efficient source reconstruction.

Findings

Effective in reconstructing sparse sound sources

Handles measurement inaccuracies and data corruption

Achieves near-perfect solutions with post-processing

Abstract

We present a strategy for the recovery of a sparse solution of a common problem in acoustic engineering, which is the reconstruction of sound source levels and locations applying microphone array measurements. The considered task bears similarities to the basis pursuit formalism but also relies on additional model assumptions that are challenging from a mathematical point of view. Our approach reformulates the original task as a convex optimisation model. The sought solution shall be a matrix with a certain desired structure. We enforce this structure through additional constraints. By combining popular splitting algorithms and matrix differential theory in a novel framework we obtain a numerically efficient strategy. Besides a thorough theoretical consideration we also provide an experimental setup that certifies the usability of our strategy. Finally, we also address practical issues, such as the handling of inaccuracies in the measurement and corruption of the given data. We provide a post processing step that is capable of yielding an almost perfect solution in such circumstances.