Note on analytic recovery of transition probabilities in three dimensional diffuse tomography

摘要

The word ''tomography'' refers to imaging an object by slices. X rays, for example, have high energy and travel straight through the body. Data analysis is linear and yields a scalar valued function. The oxymoron diffuse tomography refers to low energy imaging in which the paths of the radiant energy are not necessarily straight and are unknown. Data analysis in diffuse tomography is highly nonlinear and yields a vector valued function. Problems in diffuse tomography are highly nonlinear because low energy is used. Clinical applications such as neonatal imaging and annual mammograms are not amenable to high energy techniques which might overexpose the patient to harmful radiation. Experimentalists in the medical arena are presently working with near infrared radiation; mathematicians have done preliminary mathematical analysis of diffuse tomographic methods. An analytic algorithm for recovering Markov transition probabilities from boundary value data for the smallest nontrivial problem in three dimensions is outlined in this paper.