Inversion of the Weighted Spherical Mean

TL;DR

This paper derives an inversion formula for the weighted spherical mean, extending classical results to generalized translations, which is significant for advanced image reconstruction techniques.

Contribution

It introduces a generalized inversion formula for weighted spherical means involving generalized translations, broadening the scope of classical spherical mean inversion methods.

Findings

Derived an explicit inversion formula for weighted spherical means.

Extended classical spherical mean inversion to generalized translation operators.

Applicable to radially symmetric functions and advanced image reconstruction scenarios.

Abstract

The paper contains the inversion formula for the weighted spherical mean. The interest to reconstruction a function by its integral by sphere grews tremendously in the last six decades, stimulated by the spectrum of new problems and methods of image reconstruction. We consider a generalization of the classical spherical mean and its inverse in the case when generalized translation acts to function instead of regular. As a particular case this problem includes action of spherical means on radially symmetric functions.