A discrete Funk transform on the Cubed Sphere

Abstract

Computing accurately Funk transforms from discrete values is crucial in some applications, such as Q-Ball Imaging in medicine. This paper deals with a discrete Funk transform devoted to such a computation. The studied transform is based on a spectral method applied on a least squares fitting, with the special feature that regularization is not performed. We investigate several mathematical and numerical aspects in this context, including stability and pseudoinversion. As a specific instance, we introduce a simple framework based on the equiangular Cubed Sphere to guarantee the stability. Various numerical experiments attest to the accuracy and the convergence of the approach, in particular for synthetic Gaussian signals from Q-Ball Imaging.