# Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits

## Abstract

Smith and Grangeat derived a cone-beam inversion formula that can be applied when a nonplanar orbit satisfying the completeness condition is used. Although Grangeat's inversion formula is mathematically different from Smith's, they have similar overall structures to each other. The contribution of this paper is two-fold. First, based on the derivation of Smith, the authors point out that Grangeat's inversion formula and Smith's can be conveniently described using a single formula (the Smith-Grangeat inversion formula) that is in the form of space-variant filtering followed by cone-beam backprojection. Furthermore, the resulting formula is reformulated for data acquisition systems with a planar detector to obtain a new reconstruction algorithm. Second, the authors make two significant modifications to the new algorithm to reduce artifacts and numerical errors encountered in direct implementation of the new algorithm. As for exactness of the new algorithm, the following fact can be stated. The algorithm based on Grangeat's intermediate function is exact for any complete orbit, whereas that based on Smith's intermediate function should be considered as an approximate inverse excepting the special case where almost every plane in 3-D space meets the orbit. The validity of the new algorithm is demonstrated by simulation studies.

- Authors:

- (Univ. of Tsukuba (Japan). Inst. of Information Sciences and Electronics)

- Publication Date:

- OSTI Identifier:
- 7144731

- Resource Type:
- Journal Article

- Journal Name:
- IEEE Transactions on Medical Imaging (Institute of Electrical and Electronics Engineers); (United States)

- Additional Journal Information:
- Journal Volume: 13:1; Journal ID: ISSN 0278-0062

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 62 RADIOLOGY AND NUCLEAR MEDICINE; SINGLE PHOTON EMISSION COMPUTED TOMOGRAPHY; ALGORITHMS; DATA COVARIANCES; IMAGE PROCESSING; THREE-DIMENSIONAL CALCULATIONS; COMPUTERIZED TOMOGRAPHY; DIAGNOSTIC TECHNIQUES; EMISSION COMPUTED TOMOGRAPHY; MATHEMATICAL LOGIC; PROCESSING; TOMOGRAPHY; 550601* - Medicine- Unsealed Radionuclides in Diagnostics

### Citation Formats

```
Kudo, Hiroyuki, and Saito, Tsuneo.
```*Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits*. United States: N. p., 1994.
Web. doi:10.1109/42.276158.

```
Kudo, Hiroyuki, & Saito, Tsuneo.
```*Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits*. United States. doi:10.1109/42.276158.

```
Kudo, Hiroyuki, and Saito, Tsuneo. Tue .
"Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits". United States. doi:10.1109/42.276158.
```

```
@article{osti_7144731,
```

title = {Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits},

author = {Kudo, Hiroyuki and Saito, Tsuneo},

abstractNote = {Smith and Grangeat derived a cone-beam inversion formula that can be applied when a nonplanar orbit satisfying the completeness condition is used. Although Grangeat's inversion formula is mathematically different from Smith's, they have similar overall structures to each other. The contribution of this paper is two-fold. First, based on the derivation of Smith, the authors point out that Grangeat's inversion formula and Smith's can be conveniently described using a single formula (the Smith-Grangeat inversion formula) that is in the form of space-variant filtering followed by cone-beam backprojection. Furthermore, the resulting formula is reformulated for data acquisition systems with a planar detector to obtain a new reconstruction algorithm. Second, the authors make two significant modifications to the new algorithm to reduce artifacts and numerical errors encountered in direct implementation of the new algorithm. As for exactness of the new algorithm, the following fact can be stated. The algorithm based on Grangeat's intermediate function is exact for any complete orbit, whereas that based on Smith's intermediate function should be considered as an approximate inverse excepting the special case where almost every plane in 3-D space meets the orbit. The validity of the new algorithm is demonstrated by simulation studies.},

doi = {10.1109/42.276158},

journal = {IEEE Transactions on Medical Imaging (Institute of Electrical and Electronics Engineers); (United States)},

issn = {0278-0062},

number = ,

volume = 13:1,

place = {United States},

year = {1994},

month = {3}

}