Kernel-based methods for inversion of the Radon transform on SO(3) and their applications to texture analysis

Authors:
K. G. van den Boogaart;R. Hielscher;J. Prestin;H. Schaeben
Affiliations:
Department of Mathematics and Computer Science, Ernst-Moritz-Arndt-University Greifswald, Greifswald, Germany;Institute of Geology, Freiberg University of Mining and Technology, Freiberg, Germany;Institute of Mathematics, University of Lübeck, Lübeck, Germany;Institute of Geology, Freiberg University of Mining and Technology, Freiberg, Germany
Venue:
Journal of Computational and Applied Mathematics - Special issue: Special functions in harmonic analysis and applications
Year:
2007

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Abstract

Texture analysis is used here as short term for analysis of crystallographic preferred orientation. Its major mathematical objective is the determination of a reasonable orientation probability density function and corresponding crystallographic axes probability density functions from experimentally accessible diffracted radiation intensity data. Since the spherical axes probability density function is modelled by the one-dimensional Radon transform for SO(3), the problem is its numerical inversion. To this end, the Radon transform is characterized as an isometry between appropriate Sobolev spaces. The mathematical foundations as well as first numerical results with zonal basis functions are presented.