Mathematical Challenges Arising in Thermoacoustic Tomography with Line Detectors

Abstract

Thermoacoustic computed tomography (thermoacoustic CT) has the potential to become a mayor non-invasive medical imaging method. In this paper we derive a general mathematical framework of a novel measuring setup introduced in [P. Burgholzer, C. Hofer, G. Paltauf, M. Haltmeier, and O. Scherzer, "Thermoacoustic tomography with integrating area and line detectors", IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 52 (2005)], that uses line shaped detectors instead of the usual point like ones. We show that the three dimensional thermoacoustic imaging problem reduces to the mathematical problem of reconstructing the initial data of the two dimensional wave equation from boundary measurements of its solution. We derive and analyze an analytic reconstruction formula which allows for fast numerical implementation.