Thermoacoustic tomography with detectors on an open curve: an efficient reconstruction algorithm

Practical applications of thermoacoustic tomography require numerical inversion of the spherical mean Radon transform with the centers of integration spheres occupying an open surface. A solution of this problem is needed (both in 2D and in 3D) because frequently the region of interest cannot be completely surrounded by the detectors, as happens, for example, in breast imaging. We present an efficient numerical algorithm for solving this problem in 2D (similar methods are applicable in the 3D case). Our method is based on the numerical approximation of plane waves by certain single-layer potentials related to the acquisition geometry. After the densities of these potentials have been precomputed, each subsequent image reconstruction has the complexity of the regular filtration backprojection algorithm for the classical Radon transform. The performance of the method is demonstrated in several numerical examples: one can see that the algorithm produces very accurate reconstructions if the data are accurate and sufficiently well sampled; on the other hand, it is sufficiently stable with respect to noise in the data.