Asymptotic Expansions for Higher Order Elliptic Equations with an Application to Quantitative Photoacoustic Tomography

TL;DR

This paper develops asymptotic expansions for higher order elliptic equations with small inclusions and applies them to improve inverse problem solutions in quantitative photoacoustic tomography.

Contribution

It introduces new asymptotic expansions for complex elliptic equations and a topological derivative algorithm for image reconstruction and edge detection.

Findings

Effective numerical demonstration in photoacoustic tomography

Enhanced edge detection capabilities

Improved inverse problem solutions

Abstract

In this paper, we derive new asymptotic expansions for the solutions of higher order elliptic equations in the presence of small inclusions. As a byproduct, we derive a topological derivative based algorithm for the reconstruction of piecewise smooth functions. This algorithm can be used for edge detection in imaging, topological optimization, and for inverse problems, such as Quantitative Photoacoustic Tomography, for which we demonstrate the effectiveness of our asymptotic expansion method numerically.