### 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 language | English (US) |
---|---|

Title of host publication | Proceedings of SPIE - The International Society for Optical Engineering |

Editors | U. Bonse |

Pages | 301-309 |

Number of pages | 9 |

Volume | 5535 |

DOIs | |

State | Published - 2004 |

Externally published | Yes |

Event | Developments in X-Ray Tomography IV - Denver, CO, United States Duration: Aug 4 2004 → Aug 6 2004 |

### Other

Other | Developments in X-Ray Tomography IV |
---|---|

Country | United States |

City | Denver, CO |

Period | 8/4/04 → 8/6/04 |

### Fingerprint

### 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

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

TY - GEN

T1 - A generalized projection-slice theorem for the divergent beam projections

AU - Chen, Guang Hong

AU - Leng, Shuai

AU - Mistretta, Charles A.

PY - 2004

Y1 - 2004

N2 - 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.

AB - 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.

KW - Cone-beam projections

KW - Fan beam projections

KW - Image reconstruction

KW - Micro-CT

KW - X-ray computed tomography (CT)

UR - http://www.scopus.com/inward/record.url?scp=15844407853&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=15844407853&partnerID=8YFLogxK

U2 - 10.1117/12.560165

DO - 10.1117/12.560165

M3 - Conference contribution

AN - SCOPUS:15844407853

VL - 5535

SP - 301

EP - 309

BT - Proceedings of SPIE - The International Society for Optical Engineering

A2 - Bonse, U.

ER -