Ralf Hielscher

Ronny Bergmann, Raymond H. Chan, Ralf Hielscher et al.

The paper addresses the generalization of the half-quadratic minimization method for the restoration of images having values in a complete Riemannian manifold. We recall the half-quadratic minimization method using the notation of the c-transform and adapt the algorithm to our special variational setting. We prove the convergence of the method for Hadamard spaces. Extensive numerical examples for images with values on spheres, in the rotation group SO(3) and in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm. In particular, the method with SO(3)-valued data shows promising results for the restoration of images obtained from Electron Backscattered Diffraction which are of interest in material science.

Ralf Hielscher, Daniel Potts, Michael Quellmalz

In spherical surface wave tomography, one measures the integrals of a function defined on the sphere along great circle arcs. This forms a generalization of the Funk--Radon transform, which assigns to a function its integrals along full great circles. We show a singular value decomposition (SVD) for the surface wave tomography provided we have full data. Since the inversion problem is overdetermined, we consider some special cases in which we only know the integrals along certain arcs. For the case of great circle arcs with fixed opening angle, we also obtain an SVD that implies the injectivity, generalizing a previous result for half circles in [Groemer, On a spherical integral transform and sections of star bodies, Monatsh. Math., 126(2):117--124, 1998]. Furthermore, we derive a numerical algorithm based on the SVD and illustrate its merchantability by numerical tests.