### 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) |
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Pages (from-to) | 459-484 |

Number of pages | 26 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 52 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1992 |

### ASJC Scopus subject areas

- Applied Mathematics

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## Cite this

*SIAM Journal on Applied Mathematics*,

*52*(2), 459-484. https://doi.org/10.1137/0152026