Hielscher, Ralf ; Potts, Daniel ; Prestin, Jürgen ; Schaeben, Helmut ; Schmalz, Matthias : The Radon Transform on SO(3): A Fourier Slice Theorem and Numerical Inversion

Author(s):

Hielscher, Ralf
Potts, Daniel
Prestin, Jürgen
Schaeben, Helmut
Schmalz, Matthias

Title:

The Radon Transform on SO(3): A Fourier Slice Theorem and Numerical Inversion

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 20, 2007

Mathematics Subject Classification:

44A12			[ Radon transform ]
92C55			[ Biomedical imaging and signal processing ]
65R10			[ Integral transforms ]
65T40			[ Trigonometric approximation and interpolation ]
65T50			[ Discrete and fast Fourier transforms ]

Abstract:

The inversion of the one--dimensional Radon transform on the rotation group SO(3) is an ill posed inverse problem which applies to X--ray tomography with polycrystalline materials. This communication presents a novel approach to the numerical inversion of the one--dimensional Radon transform on SO(3). Based on a Fourier slice theorem the discrete inverse Radon transform of a function sampled on the product space $\mathbb S^2 \times \mathbb S^2$ of two two--dimensional spheres is determined as the solution of a minimization problem, which is iteratively solved using fast Fourier techniques for $\mathbb S^2$ and SO(3). The favorable complexity and stability of the algorithm based on these techniques has been confirmed with numerical tests.

Keywords:

Radon transform, fast spherical Fourier transform, trigonometric approximation and interpolation, rotation group, ill posed inverse problem

Language:

English

Publication time:

10 / 2007