Fast Multipole Techniques for a Mumford-Shah Model in X-Ray Tomography

Abstract

This paper presents a numerical approach for the fast evaluation of convolution operations which occur in the problem of inversion and segmentation of X-ray tomography data via the minimization of a Mumford-Shah functional. This approach is based on ideas of the fast multipole method. The problem which we want to solve is the problem of inversion and segmentation of X-ray tomography data, using a piecewise smooth Mumford-Shah model. The numerical framework that we use is a recently developed finite difference approximation for the determination of a piecewise smooth density function as the solution of a variational problem on a variable domain. This approach achieves simultaneously the reconstruction and segmentation directly from the raw tomography data. This work focuses on the application of ideas from the fast multipole technique. The numerical analysis predicts and the numerical experiments which are presented confirm a significant speed-up of the computational time.