Energy-based geometric regularization in three-dimensional inverse obstacle scattering

by Jannik Rönsch
Doctoral thesis
Date of Examination:2024-09-23
Date of issue:2024-12-13
Advisor:Prof. Dr. Max Wardetzky
Referee:Prof. Dr. Max Wardetzky
Referee:Prof. Dr. Thorsten Hohage
Persistent Address: http://dx.doi.org/10.53846/goediss-10943

 

 

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Abstract

English

The classic "inverse obstacle scattering problem" in three-dimensions names the challenge of reconstructing 3D-objects from (noisy) far-fied measurements of scattered waves. Like any interesting inverse problem, this problem is ill-posed in the sense of Hadamard, meaning that the inversion does not depend continuously on the measured data. This yields the necessity to employ a regularization method in the reconstruction process. In this thesis, we suggest to use the tangent-point energy, a geometric surface energy, to stabilize the reconstruction as it naturally enforces suitable regularity properties on the surfaces. We motivate this choice by proving this energy to be a suitable regularizer in the sense of generalized Tikhonov reglarization. To do so, we present a correspondence between surfaces which are locally graphs of Sobolev-Slobodeckij functions and Sobolev-Slobodeckij embeddings of smooth manifolds and make use of the state-of-the-art analysis of the tangent-point energy. Afterwards, we use this energy and a suitable pre-conditioning Sobolev-type metric on the energy space to construct a Gauss-Newton-like optimization algorithm of iterative kind. We perform regularity analysis on the parametrized formulation of the tangent-point energy and on this metric, yielding the well-definedness of the proposed algorithm and its parametrization-independence. Finally, we present a sketch of the implementation of our algorithm and show its versatility and robustness with the reconstruction of various obstacles.
Keywords: inverse obstacle scattering; tangent-point energy; Gauss-Newton type methods; iterative regularization; generalized Tikhonov regularization; Sobolev-Slobodeckij surfaces; space of surfaces; Sobolev-Slobodeckij functions; acoustic inverse scattering
 
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