Ralf Hielscher, Michael Quellmalz: Reconstructing a Function on the Sphere from Its Means Along Vertical Slices

Author(s):

Ralf Hielscher
Michael Quellmalz

Title:

Ralf Hielscher, Michael Quellmalz: Reconstructing a Function on the Sphere from Its Means Along Vertical Slices

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 10, 2015

Mathematics Subject Classification:

    65T40 []
    45Q05 []
    65N21 []
    44A12 []
    62G05 []

Abstract:

We present a novel algorithm for the inversion of the vertical slice transform, i.e. the transform that associates to a function on the two-dimensional unit sphere all integrals along circles that are parallel to one fixed direction. Our approach makes use of the singular value decomposition and resembles the mollifier approach by applying numerical integration with a reconstruction kernel via a quadrature rule. Considering the inversion problem as a statistical inverse problem, we find a family of asymptotically optimal mollifiers that minimize the maximum risk of the mean integrated error for functions within a Sobolev ball. By using fast spherical Fourier transforms and the fast Legendre transform, our algorithm can be implemented with almost linear complexity. In numerical experiments, we compare our algorithm with other approaches and illustrate our theoretical findings.

Keywords:

Radon transform, spherical means, statistical inverse problems, minimax risk, asymptotic bounds, spherical harmonics, fast algorithms

Language:

English

Publication time:

06/2015