Ralf Hielscher, Michael Quellmalz: Optimal Mollifiers for Spherical Deconvolution

Author(s):

Ralf Hielscher
Michael Quellmalz

Title:

Ralf Hielscher, Michael Quellmalz: Optimal Mollifiers for Spherical Deconvolution

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 04, 2015

Mathematics Subject Classification:

    65T40 []
    45Q05 []
    65N21 []
    44A12 []
    44A35 []
    62G05 []

Abstract:

This paper deals with the inversion of the spherical Funk--Radon transform, and, more generally, with the inversion of spherical convolution operators from the point of view of statistical inverse problems. This means we consider discrete data perturbed by white noise and aim at estimators with optimal mean square error for functions out of a Sobolev ball. To this end we analyze a specific class of estimators built upon the spherical hyperinterpolation operator, spherical designs and the mollifier approach. Eventually, we determine optimal mollifier functions with respect to the noise level, the number of data points and the smoothness of the original function. We complete this paper by providing a fast algorithm for the numerical computation of the estimator, which is based on the fast spherical Fourier transform, and by illustrating our theoretical results with numerical experiments.

Keywords:

Spherical Radon transform, spherical deconvolution, statistical inverse problems, minimax risk, asymptotic bounds, fast algorithms

Language:

English

Publication time:

02/2015