Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries

We propose three fast algorithms for solving the inverse problem of the thermoacoustic tomography corresponding to certain acquisition geometries. Two of these methods are designed to process the measurements done with point-like detectors placed on a circle (in 2D) or a sphere (in 3D) surrounding the object of interest. The third inversion algorithm works with the data measured by the integrating line detectors arranged in a cylindrical assembly rotating around the object. The number of operations required by these techniques is equal to O(n^3 log n) and O(n^3 log^2 n) for the 3D techniques (assuming the reconstruction grid with n^3 nodes) and to O(n^2 log n) for the 2D problem with n-by-n discretizetion grid. Numerical simulations show that our methods are at least two orders of magnitude faster than the existing algorithms, without any sacrifice in accuracy or stability. The results of reconstructions from real measurements done by the integrating line detectors are also presented, to demonstrate the practicality of our methods.