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

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