Localization, Stability, and Resolution of Topological Derivative Based Imaging Functionals in Elasticity

TL;DR

This paper rigorously analyzes topological derivative imaging in elasticity, revealing limitations due to wave coupling and introducing a new functional that improves localization and resolution to the diffraction limit.

Contribution

It introduces a novel imaging functional based on weighted Helmholtz decomposition that ensures accurate localization and diffraction-limited resolution in elastic inclusion detection.

Findings

Original topological derivative functional may not locate inclusions accurately.

Coupling of pressure and shear waves affects imaging resolution.

The new functional achieves maximum at the inclusion and reaches the diffraction limit.

Abstract

The focus of this work is on rigorous mathematical analysis of the topological derivative based detection algorithms for the localization of an elastic inclusion of vanishing characteristic size. A filtered quadratic misfit is considered and the performance of the topological derivative imaging functional resulting therefrom is analyzed. Our analysis reveals that the imaging functional may not attain its maximum at the location of the inclusion. Moreover, the resolution of the image is below the diffraction limit. Both phenomena are due to the coupling of pressure and shear waves propagating with different wave speeds and polarization directions. A novel imaging functional based on the weighted Helmholtz decomposition of the topological derivative is, therefore, introduced. It is thereby substantiated that the maximum of the imaging functional is attained at the location of the inclusion and the resolution is enhanced and it proves to be the diffraction limit. Finally, we investigate the stability of the proposed imaging functionals with respect to measurement and medium noises.