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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| _version_ | 1832564753445158912 |
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| author | Yuchuan Wei Hengyong Yu Ge Wang |
| author_facet | Yuchuan Wei Hengyong Yu Ge Wang |
| author_sort | Yuchuan Wei |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-697f7bf0cd2840caaab0db2b8a1b5328 |
| institution | Kabale University |
| issn | 1687-4188 1687-4196 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Biomedical Imaging |
| spelling | doaj-art-697f7bf0cd2840caaab0db2b8a1b53282025-02-03T01:10:20ZengWileyInternational Journal of Biomedical Imaging1687-41881687-41962011-01-01201110.1155/2011/285130285130Inverse Fourier Transform in the Gamma Coordinate SystemYuchuan Wei0Hengyong Yu1Ge Wang2Department of Radiation Oncology, Wake Forest University School of Medicine, Winston-Salem, NC 27157, USABiomedical Imaging Division, VT-WFU School of Biomedical Engineering and Sciences, Wake Forest University Health Sciences, Winston-Salem, NC 27157, USABiomedical Imaging Division, VT-WFU School of Biomedical Engineering and Sciences, Wake Forest University Health Sciences, Winston-Salem, NC 27157, USAThis 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.http://dx.doi.org/10.1155/2011/285130 |
| spellingShingle | Yuchuan Wei Hengyong Yu Ge Wang Inverse Fourier Transform in the Gamma Coordinate System International Journal of Biomedical Imaging |
| title | Inverse Fourier Transform in the Gamma Coordinate System |
| title_full | Inverse Fourier Transform in the Gamma Coordinate System |
| title_fullStr | Inverse Fourier Transform in the Gamma Coordinate System |
| title_full_unstemmed | Inverse Fourier Transform in the Gamma Coordinate System |
| title_short | Inverse Fourier Transform in the Gamma Coordinate System |
| title_sort | inverse fourier transform in the gamma coordinate system |
| url | http://dx.doi.org/10.1155/2011/285130 |
| work_keys_str_mv | AT yuchuanwei inversefouriertransforminthegammacoordinatesystem AT hengyongyu inversefouriertransforminthegammacoordinatesystem AT gewang inversefouriertransforminthegammacoordinatesystem |