Document Type

Article

Publication Date

2009

Keywords

Boltzmann equation, Riemannian metric, Transport equation

Abstract

Optical tomography is the use of near-infrared light to determine the optical absorption and scattering properties of a medium M ⊂ Rn. If the refractive index is constant throughout the medium, the steady-state case is modeled by the stationary linear transport equation in terms of the Euclidean metric and photons which do not get absorbed or scatter travel along straight lines. In this expository article we consider the case of variable refractive index where the dynamics are modeled by writing the transport equation in terms of a Riemannian metric; in the absence of interaction, photons follow the geodesics of this metric. The data one has is the measurement of the out-going flux of photons leaving the body at the boundary. This may be knowledge of both the locations and directions of the exiting photons (fully angularly resolved measurements) or some kind of average over direction (angularly averaged measurements). We discuss the results known for both types of measurements in all spatial dimensions.

Publication Title

Cubo

Volume

11

Issue

5

First Page

71

Last Page

97

Required Publisher's Statement

CUBO: A mathematical journal published by the Universidad de La Frontera Universidade Federal de Pernambuco Volume 11/No ¯ 05 – DECEMBER 2009

Subjects - Topical (LCSH)

Optical tomography; Transport theory; Integral geometry; Inverse scattering transform

Genre/Form

articles

Type

Text

Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

Creative Commons License

Creative Commons Attribution-Noncommercial 3.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 License

Language

English

Format

application/pdf

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Included in

Analysis Commons

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