Efficient and accurate computation of spherical mean values at scattered center points

Torsten Görner,Ralf Hielscher,Stefan Kunis

doi:10.3934/ipi.2012.6.645

Abstract

Spherical means are a widespread model in modern imaging modalities like photoacoustic tomography. Besides direct inversion methods for specific geometries, iterative methods are often used as reconstruction scheme such that each iteration asks for the efficient and accurate computation of spherical means. We consider a spectral discretization via trigonometric polynomials such that the computation can be done via nonequispaced fast Fourier transforms. Moreover, a recently developed sparse fast Fourier transform is used in the three dimensional case and gives optimal arithmetic complexity. All theoretical results are illustrated by numerical experiments.

Highlights

In analogy to the classical Radon transform, we consider the spherical mean value operator M that assigns to each function f : Rd → R its mean values Mf (y, r) =f (y + rξ) dσ(ξ) ωd−1 Sd−1 along the spheres with center point y ∈ Rd and radius r > 0, where σ denotes the surface measure on the sphere and ωd−1 = σ(Sd−1)
Acoustic tomography [26, 18, 4] are based upon the spherical mean value operator
The inverse problem is of interest, i.e. given the spherical means

Summary

Introduction

In analogy to the classical Radon transform, we consider the spherical mean value operator M that assigns to each function f : Rd → R its mean values. Our idea is to restrict the spherical mean value operator to periodic functions and to discretize these by trigonometric interpolation. We show that in the three dimensional case the discrete spherical mean value operator can be identified with a four-dimensional sparse Fourier transform with nodes and frequencies restricted to some three dimensional submanifolds. For this setting, the sparse Fourier transform [28, 19] applies and has the numerical complexity O(N 3 log N ), where N is the number of discretization points in each dimension. It should be noted that compactly supported functions fit in our framework

Inverse Problems and Imaging

Then the spherical mean value operator has the singular value decomposition