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Derivative-based regularization of inverse problems in acoustic holography

Authors Pagavino, M.
Year 2019
Thesis Type Master's thesis
Topic Audio Signal Processing
Keywords Akustische Holographie
Abstract The visualization of the sound field close to the source is often helpful to understand the vibroacoustic origin. This gave rise to the development of several acoustic imaging techniques that can be used to model the measured sound field radiated by an arbitrary source. One of these models is the equivalent source method (ESM). It models the local sound field by superimposing distributed elementary sources of different strengths. From spatially discrete sound pressure measurements, the strength of these sources can be determined through solving a linear inverse problem. Due to the underdetermined and ill-posed nature of the inverse problem, the introduction of some form of regularization is a prerequisite for obtaining a meaningful solution. Imposing additional constraints on the solution to enforce expected spatial structures can provide suitable regularization. Inverse problems with constraints typically minimize some norm functional acting on the spatial domain. Sparsity promotion through Compressive Sensing, based on L1-norm minimization, has received increasing attention in recent years due to its ability of providing solutions that are valid beyond the spatial sampling limit. However, typical vibroacoustic source phenomena are not necessarily spatially sparse themselves, as they frequently contain spatially distributed patterns as well. This thesis regards regularization methods that impose sparsity on first- and second-order spatial derivatives. This promotes piecewise constant or linear solutions with minimum curvature as a more probable spatial constraint. Such regularizers are heavily used in various fields of image processing. They were only recently introduced in acoustics, where they have consistently proven to effectively model common structures. In this thesis, I propose to adapt the Schatten-norms of the Hessian as regularizers, which to the best of my knowledge has not been considered for acoustic holography yet. What is more, a fused approach is considered where additional sparsity is imposed on the spatial domain, suitable for the characterization of sparse and extended sources. A proximal splitting algorithm is adopted to solve the minimization problem, which allows an efficient implementation of the proposed regularizers. This work provides the fundamental understanding of derivative-based regularization and reveals its characteristics and abilities. The proposed methods are investigated and verified by numerical simulations and by using measurements obtained from an experimental setup. The required theory behind the algorithm is examined and a detailed exposition of its use is provided.
Supervisors Zotter, F., Höldrich, R.