Back to Search
Start Over
Time-invariant radon transform by generalized Fourier slice theorem
- Source :
- Inverse Problems & Imaging. 11:501-519
- Publication Year :
- 2017
- Publisher :
- American Institute of Mathematical Sciences (AIMS), 2017.
-
Abstract
- Time-invariant Radon transforms play an important role in many fields of imaging sciences, whereby a function is transformed linearly by integrating it along specific paths, e.g. straight lines, parabolas, etc. In the case of linear Radon transform, the Fourier slice theorem establishes a simple analytic relationship between the 2-D Fourier representation of the function and the 1-D Fourier representation of its Radon transform. However, the theorem can not be utilized for computing the Radon integral along paths other than straight lines. We generalize the Fourier slice theorem to make it applicable to general time-invariant Radon transforms. Specifically, we derive an analytic expression that connects the 1-D Fourier coefficients of the function to the 2-D Fourier coefficients of its general Radon transform. For discrete data, the model coefficients are defined over the data coefficients on non-Cartesian points. It is shown numerically that a simple linear interpolation provide satisfactory results and in this case implementations of both the inverse operator and its adjoint are fast in the sense that they run in \begin{document}$O(N \;\text{log}\; N)$\end{document} flops, where \begin{document}$N$\end{document} is the maximum number of samples in the data space or model space. These two canonical operators are utilized for efficient implementation of the sparse Radon transform via the split Bregman iterative method. We provide numerical examples showing high-performance of this method for noise attenuation and wavefield separation in seismic data.
- Subjects :
- Control and Optimization
Radon transform
Mathematical analysis
Fourier inversion theorem
020206 networking & telecommunications
02 engineering and technology
010502 geochemistry & geophysics
01 natural sciences
Fractional Fourier transform
Parseval's theorem
symbols.namesake
Discrete Fourier transform (general)
Fourier transform
Modeling and Simulation
Projection-slice theorem
0202 electrical engineering, electronic engineering, information engineering
symbols
Discrete Mathematics and Combinatorics
Pharmacology (medical)
Convolution theorem
Analysis
0105 earth and related environmental sciences
Mathematics
Subjects
Details
- ISSN :
- 19308345
- Volume :
- 11
- Database :
- OpenAIRE
- Journal :
- Inverse Problems & Imaging
- Accession number :
- edsair.doi...........36a938553e94ede32fa9e6ae4a6c2382