Explicit inversion formulas for a model in diffuse tomography

Abstract

Optical, or diffuse tomography, refers to the use of low energy probes to obtain images of highly scattering media. This represents an emerging area for the application of new mathematical tools to important problems in medical imaging. It is our contention that it poses important mathematical challenges dealing with the solution of large systems of nonlinear equations. In this paper we present a case for which a complete solution is possible; however the consideration of any larger system presents at the moment a number of unresolved difficulties. A variety of mathematical models of the underlying physics are reviewed in [1] including the ones considered in [4,7–10], where we have discussed both the “direct”, as well as the more interesting “inverse problem” for a discrete model. The time evolution of the system is governed by a Markov chain with discrete state space and discrete time. The one step transition probability matrix is denoted by P . Some of the states are “incoming”, some are “outgoing”, and the rest are “hidden” states. The first two types correspond to sources and detectors respectively, and the hidden ones represent the state of a photon while it moves in the interior of the object.