Inverse Fourier Transform in the Gamma Coordinate System
This paper provides auxiliary results for our general scheme of computed tomography. In 3D parallel-beam geometry, we first demonstrate that the inverse Fourier transform in different coordinate systems l...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2011-01-01
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| Series: | International Journal of Biomedical Imaging |
| Online Access: | http://dx.doi.org/10.1155/2011/285130 |
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| Summary: | This paper provides
auxiliary results for our general scheme of
computed tomography. In 3D
parallel-beam geometry, we first demonstrate
that the inverse Fourier transform in different
coordinate systems leads to different
reconstruction formulas and explain why the
Radon formula cannot directly work with
truncated projection data. Also, we introduce a
gamma coordinate system, analyze its properties, compute the Jacobian of the coordinate transform, and define weight functions for the inverse Fourier transform assuming a simple scanning model. Then, we generate Orlov's theorem and a weighted Radon formula from the inverse Fourier transform in the new system. Furthermore, we present the motion equation of the frequency plane and the conditions for sharp points of the instantaneous rotation axis. Our analysis on the motion of the frequency plane is related to the Frenet-Serret theorem in the differential geometry. |
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| ISSN: | 1687-4188 1687-4196 |