Summability Methods for the Inversion of the Spherical Mean Operator

Image reconstruction from circular or spherical mean data is a central problem in thermoacoustic and photoacoustic tomography. To recover the image from such data a summability-type approximate reconstruction method is presented which in the limit leads to exact reconstruction. Fast and efficient numerical algorithms for approximative reconstruction for various data acquisition surfaces are developed. These numerical schemes apply Fourier techniques for certain types of convolution integrals and discretization based on polar respectively spherical grids. A detailed discussion of the reconstruction error is provided and, moreover, all results are supported by numerical considerations. In addition a detailed analysis of the degree of approximation of the presented method is carried out. Furthermore, new exact inversion formulas for elliptical acquisition surfaces in space dimension two and three are obtained by certain limit processes.