Inversion of spherical Radon transform in the class of discrete random functions

The paper deals with the problem of reconstructing the probabilistic distributions of random functions from distribution of spherical projections that describe the images in certain types of tomographic experiments, including optoacoustic tomography, thermoacoustic tomography, and radiolocation. The problems of this kind arise when the object under study may randomly change its structure during the registration of the projection data and the time within which its structure changes radically is considerably smaller than the time of registration of a required number of projections. In such cases, the conventional tomographic approach cannot be used directly. The authors assume that a random object may have at most countable set of structural states which are described by integrable functions with compact support. For such discrete class of random functions, the uniqueness of solution is proved and the reconstruction method is developed which is based on the properties of the so-called moments of projections. It is shown that the developed method is stable and gives adequate results when the projection data are corrupted by noise.