On singularities and instability of reconstruction in thermoacoustic tomography

TL;DR

This paper investigates the stability of reconstructing initial wave conditions in thermoacoustic tomography, demonstrating that incomplete geometric coverage leads to instability in the reconstruction process.

Contribution

It establishes a theoretical link between geometric properties of the wave equation and the stability of the reconstruction in TAT, highlighting conditions for instability.

Findings

Reconstruction is not Hölder stable if some rays do not intersect the observation surface.

Geometric conditions critically affect the stability of TAT reconstructions.

The study provides a mathematical foundation for understanding limitations in TAT imaging.

Abstract

We consider the problem of thermoacoustic tomography (TAT), in which one needs to reconstruct the initial value of a solution of the wave equation from its value on an observation surface. We show that if some geometric rays for the equation do not intersect the observation surface, the reconstruction in TAT is not H\"{o}lder stable.