### Abstract

Sufficiency conditions for cone-beam data are well known for the case of continuous data collection along a cone-vertex curve with continuous detectors. These continuous conditions are inadequate for real-world data where discrete vertex geometries and discrete detector arrays are used. The authors present a theoretical formulation of cone-beam tomography with arbitrary discrete arrays of detectors and vertices. The theory models the imaging system as a linear continuous-to-discrete mapping and represents the continuous object exactly as a Fourier series. The reconstruction problem is posed as the estimation of some subset of the Fourier coefficients. The main goal of the theory is to determine which Fourier coefficients can be reliably determined from the data delivered by a specific discrete design. A Fourier component will be well determined by the data if it satisfies two conditions: it makes a strong contribution to the data, and this contribution is relatively independent of the contribution of other Fourier components. To make these considerations precise, the authors introduce a concept called the cross-talk matrix. A diagonal element of this matrix measures the strength of a Fourier component in the data, while an off-diagonal element quantifies the dependence or aliasing of two different components. One reasonable approach to system design is to attempt to make the diagonal elements of this matrix large and the off-diagonal elements small for some set of Fourier components. If this goal can be achieved, simple linear reconstruction algorithms are available for estimating the Fourier coefficients. To illustrate the usefulness of this approach, numerical results on the cross-talk matrix are presented for different discrete geometries derived from a continuous helical vertex orbit, and simulated images reconstructed with two linear algorithms are presented.

Original language | English (US) |
---|---|

Article number | 012 |

Pages (from-to) | 451-476 |

Number of pages | 26 |

Journal | Physics in Medicine and Biology |

Volume | 39 |

Issue number | 3 |

DOIs | |

State | Published - 1994 |

### Fingerprint

### ASJC Scopus subject areas

- Radiological and Ultrasound Technology
- Radiology Nuclear Medicine and imaging
- Physics and Astronomy (miscellaneous)
- Biomedical Engineering

### Cite this

*Physics in Medicine and Biology*,

*39*(3), 451-476. [012]. https://doi.org/10.1088/0031-9155/39/3/012

**Cone-beam tomography with discrete data sets.** / Barrett, Harrison H; Gifford, H.

Research output: Contribution to journal › Article

*Physics in Medicine and Biology*, vol. 39, no. 3, 012, pp. 451-476. https://doi.org/10.1088/0031-9155/39/3/012

}

TY - JOUR

T1 - Cone-beam tomography with discrete data sets

AU - Barrett, Harrison H

AU - Gifford, H.

PY - 1994

Y1 - 1994

N2 - Sufficiency conditions for cone-beam data are well known for the case of continuous data collection along a cone-vertex curve with continuous detectors. These continuous conditions are inadequate for real-world data where discrete vertex geometries and discrete detector arrays are used. The authors present a theoretical formulation of cone-beam tomography with arbitrary discrete arrays of detectors and vertices. The theory models the imaging system as a linear continuous-to-discrete mapping and represents the continuous object exactly as a Fourier series. The reconstruction problem is posed as the estimation of some subset of the Fourier coefficients. The main goal of the theory is to determine which Fourier coefficients can be reliably determined from the data delivered by a specific discrete design. A Fourier component will be well determined by the data if it satisfies two conditions: it makes a strong contribution to the data, and this contribution is relatively independent of the contribution of other Fourier components. To make these considerations precise, the authors introduce a concept called the cross-talk matrix. A diagonal element of this matrix measures the strength of a Fourier component in the data, while an off-diagonal element quantifies the dependence or aliasing of two different components. One reasonable approach to system design is to attempt to make the diagonal elements of this matrix large and the off-diagonal elements small for some set of Fourier components. If this goal can be achieved, simple linear reconstruction algorithms are available for estimating the Fourier coefficients. To illustrate the usefulness of this approach, numerical results on the cross-talk matrix are presented for different discrete geometries derived from a continuous helical vertex orbit, and simulated images reconstructed with two linear algorithms are presented.

AB - Sufficiency conditions for cone-beam data are well known for the case of continuous data collection along a cone-vertex curve with continuous detectors. These continuous conditions are inadequate for real-world data where discrete vertex geometries and discrete detector arrays are used. The authors present a theoretical formulation of cone-beam tomography with arbitrary discrete arrays of detectors and vertices. The theory models the imaging system as a linear continuous-to-discrete mapping and represents the continuous object exactly as a Fourier series. The reconstruction problem is posed as the estimation of some subset of the Fourier coefficients. The main goal of the theory is to determine which Fourier coefficients can be reliably determined from the data delivered by a specific discrete design. A Fourier component will be well determined by the data if it satisfies two conditions: it makes a strong contribution to the data, and this contribution is relatively independent of the contribution of other Fourier components. To make these considerations precise, the authors introduce a concept called the cross-talk matrix. A diagonal element of this matrix measures the strength of a Fourier component in the data, while an off-diagonal element quantifies the dependence or aliasing of two different components. One reasonable approach to system design is to attempt to make the diagonal elements of this matrix large and the off-diagonal elements small for some set of Fourier components. If this goal can be achieved, simple linear reconstruction algorithms are available for estimating the Fourier coefficients. To illustrate the usefulness of this approach, numerical results on the cross-talk matrix are presented for different discrete geometries derived from a continuous helical vertex orbit, and simulated images reconstructed with two linear algorithms are presented.

UR - http://www.scopus.com/inward/record.url?scp=0028260061&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028260061&partnerID=8YFLogxK

U2 - 10.1088/0031-9155/39/3/012

DO - 10.1088/0031-9155/39/3/012

M3 - Article

C2 - 15559983

AN - SCOPUS:0028260061

VL - 39

SP - 451

EP - 476

JO - Physics in Medicine and Biology

JF - Physics in Medicine and Biology

SN - 0031-9155

IS - 3

M1 - 012

ER -