arXiv subjects

Svend Ebert

Publications and source records attributed to Svend Ebert.

3 recordsLinked to original sources

arXiv2012

Generalized splines for Radon transform on compact Lie groups with applications to crystallography

The Radon transform Rf of functions f on SO(3) has recently been applied extensively in texture analysis, i.e. the analysis of preferred crystallographic orientation. In practice one has to determine the orientation probability density function f \in L2(SO(3)) from Rf \in L2(S2\times S2) which is known only on a discrete set of points. Since one has only partial information about Rf the inversion of the Radon transform becomes an ill-posed inverse problem. Motivated by this problem we define a new notion of the Radon transform Rf of functions f on general compact Lie groups and introduce two approximate inversion algorithms which utilize our previously developed generalized variational splines on manifolds. Our new algorithms fit very well to the application of Radon transform on SO(3) to texture analysis.

math.FA↗

arXiv2012

Diffusive wavelets on the Spin group

The first part of this article is devoted to a brief review of the results about representation theory of the spin group Spin(m) from the point of view of Clifford analysis. In the second part we are interested in Clifford-valued functions and wavelets on the sphere. The connection of representations of Spin(m) and the concept of diffusive wavelets leads naturally to investigations of a modified diffusion equation on the sphere, that makes use of the Gamma operator. We will achieve to obtain Clifford-valued diffusion wavelets with respect to a modified diffusion operator. Since we are able to characterize all representations of Spin(m) and even to obtain all eigenvectors of the (by representation) regarded Casimir operator in representation spaces, it seems appropriate to look at functions on Spin(m) directly. Concerning this, our aim shall be to formulate eigenfunctions for the Laplace-Beltrami operator on Spin(m) and give the series expansion of the heat kernel on Spin(m) in terms of eigenfunctions.

math.FA↗

arXiv2010

Diffusive wavelets on groups and homogeneous spaces

The aim of this exposition is to explain basic ideas behind the concept of diffusive wavelets on spheres in the language of representation theory of Lie groups and within the framework of the group Fourier transform given by Peter-Weyl decomposition of $L^2(G)$ for a compact Lie group $G$. After developing a general concept for compact groups and their homogeneous spaces we give concrete examples for tori -which reflect the situation on $R^n$- and for spheres $S^2$ and $S^3$.

math.FA↗