Numerical inversion of the spherical Radon transform and the cosine transform using the approximate inverse with a special class of locally supported mollifiers

摘要

The focus of this paper is on the numerical inversion of two integral transforms, namely the spherical Radon and the cosine transform. Both transforms are frequently used in integral geometry and related fields, and their numerical inversion is needed in several applications. To derive fast regularization schemes, the method of the approximate inverse is utilized. We introduce a new family of mollifiers and calculate the corresponding reconstruction kernels analytically for dimension d = 3 and even dimensions d ≥ 4. Numerical results for the three-dimensional case are presented showing that the new class of mollifiers clearly improves the quality of the reconstruction in comparison to the Gaussian mollifier. Moreover, the regularization theory for the method is extended to a framework for arbitrary dimension d ≥ 3 (the special case d = 3 was already considered in [21]).