### Abstract

Tomography produces the reconstruction of a function f from a large number of line integrals of f. Conventional tomography is a global procedure in that the standard convolution formulas for reconstruction at a single point require the integrals over all lines within some plane containing the point. Local tomography, as introduced initially, produced the reconstruction of the related function Λf, where Λ is the square root of -Δ, the positive Laplace operator. The reconstruction of Λf is local in that reconstruction at a point requires integrals only over lines passing infinitesimally close to the point, and Λf has the same smooth regions and boundaries as f. However, Λf is cupped in regions where f is constant. Λ
^{-1}f, also amendable to local reconstruction, is smooth everywhere and contains a counter-up. This article provides a detailed study of the actions of Λ and Λ
^{-1}, and shows several examples of what can be achieved with a linear combination. It includes the results of x-ray experiments in which the line integrals are obtained from attenuation measurements on two-dimensional image intensifiers and fluorescent screens, instead of the usual linear detector arrays.

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

Pages (from-to) | 459-484 |

Number of pages | 26 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 52 |

Issue number | 2 |

State | Published - Apr 1992 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*52*(2), 459-484.

**Local tomography.** / Faridani, Adel; Ritman, Erik L.; Smith, Kennan T.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 52, no. 2, pp. 459-484.

}

TY - JOUR

T1 - Local tomography

AU - Faridani, Adel

AU - Ritman, Erik L.

AU - Smith, Kennan T.

PY - 1992/4

Y1 - 1992/4

N2 - Tomography produces the reconstruction of a function f from a large number of line integrals of f. Conventional tomography is a global procedure in that the standard convolution formulas for reconstruction at a single point require the integrals over all lines within some plane containing the point. Local tomography, as introduced initially, produced the reconstruction of the related function Λf, where Λ is the square root of -Δ, the positive Laplace operator. The reconstruction of Λf is local in that reconstruction at a point requires integrals only over lines passing infinitesimally close to the point, and Λf has the same smooth regions and boundaries as f. However, Λf is cupped in regions where f is constant. Λ -1f, also amendable to local reconstruction, is smooth everywhere and contains a counter-up. This article provides a detailed study of the actions of Λ and Λ -1, and shows several examples of what can be achieved with a linear combination. It includes the results of x-ray experiments in which the line integrals are obtained from attenuation measurements on two-dimensional image intensifiers and fluorescent screens, instead of the usual linear detector arrays.

AB - Tomography produces the reconstruction of a function f from a large number of line integrals of f. Conventional tomography is a global procedure in that the standard convolution formulas for reconstruction at a single point require the integrals over all lines within some plane containing the point. Local tomography, as introduced initially, produced the reconstruction of the related function Λf, where Λ is the square root of -Δ, the positive Laplace operator. The reconstruction of Λf is local in that reconstruction at a point requires integrals only over lines passing infinitesimally close to the point, and Λf has the same smooth regions and boundaries as f. However, Λf is cupped in regions where f is constant. Λ -1f, also amendable to local reconstruction, is smooth everywhere and contains a counter-up. This article provides a detailed study of the actions of Λ and Λ -1, and shows several examples of what can be achieved with a linear combination. It includes the results of x-ray experiments in which the line integrals are obtained from attenuation measurements on two-dimensional image intensifiers and fluorescent screens, instead of the usual linear detector arrays.

UR - http://www.scopus.com/inward/record.url?scp=0026846049&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0026846049&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0026846049

VL - 52

SP - 459

EP - 484

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 2

ER -