| Title: |
Crystallographic and geodesic Radon transforms on SO(3): motivation, generalization, discretization |
| Authors: |
Bernstein, Swanhild, Pesenson, Isaac Z. |
| Publication Year: |
2012 |
| Collection: |
Mathematics |
| Subject Terms: |
Mathematics - Functional Analysis |
| More Details: |
In this paper we consider the so-called crystallographic Radon transform (or crystallographic $X$-ray transform) and totally geodesic Radon transform on the group of rotations SO(3). As we show both of these transforms naturally appear in texture analysis, i.e. the analysis of preferred crystallographic orientation. Although we discuss only applications to texture analysis both transforms have other applications as well. In section 2 we start with motivations and applications. In sections 3 and 4 we develop a general framework on compact Lie groups. In section 5 we give a detailed analysis of the totally geodesic Radon transform on SO(3). In section \ref{relations} we compare crystallographic Radon transform on SO(3) and Funk transform on $S^{3}$. In section \ref{1} we show non-invertibility of the crystallographic transform. In section 8 we describe an exact reconstruction formula for bandlimited functions, which uses only a finite number of samples of their Radon transform. Some auxiliary results for this section are collected in Appendix.
Comment: Will appear in a volume dedicated to 85-th birthday of S. Helgason "Geometric Analysis and Integral Geometry", Contemporary Mathematics 2013; Volume: 598 ISBN-10: 0-8218-8738-6 ISBN-13: 978-0-8218-8738-7 |
| Document Type: |
Working Paper |
| Access URL: |
http://arxiv.org/abs/1210.2652 |
| Accession Number: |
edsarx.1210.2652 |
| Database: |
arXiv |
