### Abstract

Standard, or global, tomography involves the reconstruction of a function f from line integrals. Local tomography, in this paper, involves the reconstruction of a related function, Lf = α(Λf + μΛ^{-1}f), where Λ is the square root of the positive Laplacian, -Δ. This article is a sequel to the article "Local Tomography" [SIAM J. Appl. Math., 52 (1992), pp. 459-484, 1193-1198] by Faridani, Hitman, and Smith. The principal new results are (1) good bounds for Λf and Λ^{-1}f outside the support of f, particularly when f has 0 moments up to some order; (2) identification and reduction of global effects in local tomography, i.e., identification and reduction of the dependence of Lf (x) on the values of f at points at an intermediate distance from x; (3) an algorithm for computing approximate density jumps from Λf when f is a linear combination of characteristic functions and a smooth background. Several examples are given: some from real x-ray data, some from mathematical phantoms. They include three-dimensional 7-micron resolution reconstructions from microtomographic scans.

Original language | English (US) |
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Pages (from-to) | 1095-1127 |

Number of pages | 33 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 57 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1997 |

### Keywords

- Local
- Microscopic tomography
- Three-dimensional computed tomography
- Tomography
- Two-dimensional computed tomography

### ASJC Scopus subject areas

- Applied Mathematics

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## Cite this

*SIAM Journal on Applied Mathematics*,

*57*(4), 1095-1127. https://doi.org/10.1137/S0036139995286357