Image Restoration in Optical Acoustic Tomography

We consider the optical-acoustic tomography problem. In the general case, the problem is to reconstruct a real-valued function with a compact support in the $n$-dimensional Euclidean space via its spherical integrals, i.e., integrals over all $(n-1)$-dimensional spheres centered at points on some $(n-1)$-dimensional hypersurface. We deal with the cases $n=2$ and $n=3$, which are of the most practical interest from the standpoint of possible medical applications. We suggest a new effective method of reconstruction, develop restoration algorithms, and investigate the quality of the algorithms for these cases. The main result of the paper is construction of explicit approximate reconstruction formulas; from the mathematical standpoint, these formulas give the parametrix for the optical-acoustic tomography problem. The formulas constructed is a background for the restoration algorithms. We performed mathematical experiments to investigate the quality of the restoration algorithms using the generally accepted tomography quality criteria. The results obtain lead to the general conclusion: the quality of the restoration algorithms developed for optical-acoustic tomography is only slightly lower then the quality of the convolution and back projection algorithm used in Radon tomography, which is a standard de facto.