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
