A generalized projection-slice theorem for the divergent beam projections

Guang Hong Chen, Shuai Leng, Charles A. Mistretta

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

Abstract

The general goal of this paper is to extend the parallel-beam projection-slice theorem to the divergent fan-beam and cone-beam projections without rebinning the divergent fan-beam and cone-beam projections into parallel-beam projections directly. The basic idea is to establish a novel link between the local Fourier transform of the projection data and the Fourier transform of the image object. Analogous to the two- and three-dimensional parallel-beam cases, the measured projection data are backprojected along the projection direction and then a local Fourier transform is taken for the backprojected data array. However, due to the loss of the shift-invariance of the image object in a single view of the divergent-beam projections, the measured projection data is weighted by a distance dependent weight w(r) before the local Fourier transform is performed. The variable r in the weighting fonction w(r) is the distance from the backprojected point to the X-ray source position. It is shown that a special choice of the weighting function, w(r) = 1/r, will facilitate the calculations and a simple relation can be established between the Fourier transform of the image function and the local Fourier transform of the 1/r - weighted backprojection data array. Unlike the parallel-beam cases, a one-to-one correspondence does not exist for a local Fourier transform of the backprojected data array and a single line in the 2D case or a single slice in the 3D case of the Fourier transform of the image function. However, the Fourier space of the image object can be built up after the local Fourier transforms of the 1/r - weighted backprojection data arrays are shifted and added up in a laboratory frame. Thus the established relations Eq. (19) and Eq. (21) between the Fourier space of the image object and the Fourier transforms of the backprojected data arrays can be viewed as a generalized projection-slice theorem for divergent fan-beam and cone-beam projections. Once the Fourier space of the image function is built up, an inverse Fourier transform could be performed to reconstruct tomographic images from the divergent beam projections.

Original languageEnglish (US)
Title of host publicationProceedings of SPIE - The International Society for Optical Engineering
EditorsU. Bonse
Pages301-309
Number of pages9
Volume5535
DOIs
StatePublished - 2004
Externally publishedYes
EventDevelopments in X-Ray Tomography IV - Denver, CO, United States
Duration: Aug 4 2004Aug 6 2004

Other

OtherDevelopments in X-Ray Tomography IV
CountryUnited States
CityDenver, CO
Period8/4/048/6/04

Fingerprint

Fourier transforms
theorems
projection
Fans
fans
Cones
cones
Inverse transforms
weighting functions
Invariance
invariance
X rays
shift

Keywords

  • Cone-beam projections
  • Fan beam projections
  • Image reconstruction
  • Micro-CT
  • X-ray computed tomography (CT)

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Condensed Matter Physics

Cite this

Chen, G. H., Leng, S., & Mistretta, C. A. (2004). A generalized projection-slice theorem for the divergent beam projections. In U. Bonse (Ed.), Proceedings of SPIE - The International Society for Optical Engineering (Vol. 5535, pp. 301-309). [33] https://doi.org/10.1117/12.560165

A generalized projection-slice theorem for the divergent beam projections. / Chen, Guang Hong; Leng, Shuai; Mistretta, Charles A.

Proceedings of SPIE - The International Society for Optical Engineering. ed. / U. Bonse. Vol. 5535 2004. p. 301-309 33.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Chen, GH, Leng, S & Mistretta, CA 2004, A generalized projection-slice theorem for the divergent beam projections. in U Bonse (ed.), Proceedings of SPIE - The International Society for Optical Engineering. vol. 5535, 33, pp. 301-309, Developments in X-Ray Tomography IV, Denver, CO, United States, 8/4/04. https://doi.org/10.1117/12.560165
Chen GH, Leng S, Mistretta CA. A generalized projection-slice theorem for the divergent beam projections. In Bonse U, editor, Proceedings of SPIE - The International Society for Optical Engineering. Vol. 5535. 2004. p. 301-309. 33 https://doi.org/10.1117/12.560165
Chen, Guang Hong ; Leng, Shuai ; Mistretta, Charles A. / A generalized projection-slice theorem for the divergent beam projections. Proceedings of SPIE - The International Society for Optical Engineering. editor / U. Bonse. Vol. 5535 2004. pp. 301-309
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