Ralf Hielscher: Numerical Inversion of the Funk transform on the Rotation Group

Author(s):

Ralf Hielscher

Title:

Ralf Hielscher: Numerical Inversion of the Funk transform on the Rotation Group

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 13, 2013

Mathematics Subject Classification:

44A12	[radon transform]
53C65	[Integral geometry]

Abstract:

The reconstruction of a function on the rotation group from mean values along all geodesics is an overdetermined problem, i.e., it is sufficient to know the mean values for a three dimensional subset of all geodesics on the rotation group. In this paper we give a Fourier slice theorem for the restricted problem. Based on the Fourier slice theorem and fast Fourier transforms on the rotation group and the sphere we introduce a fast algorithm for the forward transform. Analyzing the inverse problem we come up with an exact inversion formula for bandlimited functions on the rotation group. Unfortunately, this inversion formula turns out to be extremely ill conditioned. Therefore, we introduce an iterative approach which makes use of regularization and the fast algorithm for the forward transform. Numerical experiments indicate the applicability of our algorithms.

Keywords:

Funk transform, Fourier transform, inversion formula, fast algorithms

Language:

English

Publication time:

07/2013