### Abstract

All of exact cone-beam reconstruction algorithms^{6-8} for so called "long-object problem" do not use equi-angular sampling but use equi-space sampling. However, cylindrical detectors (equi-angular sampling in xy and equi-spatial in z) have advantage in their compact design. Therefore, toward the long-object problem with equi-angular sampling, the purpose of this study is to develop a cone-beam reconstruction algorithm using equi-angular sampling for short-object problem. A novel implementation of Grangeat's algorithm^{9} using equi-angular sampling has been developed for short-object problem with and without detector truncation. First, both cone-beam projection g^{ψ} (θ,φ) and the first derivative of plane integral (3D Radon transform) p̃^{ψ} (θ,φ) are described using spherical harmonics with equi-angular sampling. Then, using Grangeat's formula, relationship between coefficients of spherical harmonics for g^{ψ} (θ,φ) and p̃^{ψ} (θ,φ) are found. Finally, a method has been developed to obtain p̃^{ψ} (θ,φ) from cone-beam projection data in which the object is partially scanned. Images are reconstructed using the 3D Radon backprojection with rebinning. Computer simulations were performed in order to verify this approach: Isolated (axially bounded) objects were scanned both with circular and helical orbits. When the orbit of the cone vertex does not satisfy Tuy's data sufficiency conditions, strong oblique shadows and blurring in the axial direction were shown in reconstructed coronal images. On the other hand, if the trajectory satisfies Tuy's data sufficiency condition, the proposed algorithm provides an exact reconstruction. In conclusion, a novel implementation of the Grangeat's algorithm for cone-beam image reconstruction using equi-angular sampling has been developed.

Original language | English (US) |
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Pages (from-to) | 867-876 |

Number of pages | 10 |

Journal | Proceedings of SPIE-The International Society for Optical Engineering |

Volume | 4322 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2001 |

Externally published | Yes |

### Keywords

- Cone-beam
- Equi-angular sampling
- Image reconstruction
- Short-object problem
- Spherical harmonics

### ASJC Scopus subject areas

- Electronic, Optical and Magnetic Materials
- Condensed Matter Physics
- Computer Science Applications
- Applied Mathematics
- Electrical and Electronic Engineering

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## Cite this

*Proceedings of SPIE-The International Society for Optical Engineering*,

*4322*(2), 867-876. https://doi.org/10.1117/12.431166