A fast method for the solution of some tomography problems

Tomography is an imaging technique, commonly used in medical applications, but also in geophysics and astrophysics. This technique allows estimation of internal characteristics of an object apart from measurements performed outside the object. Its main interest lies in the fact that the estimation process does not require any intrusive operations. From a methodological point of view, tomography may be divided into two main stages: first a forward model has to be provided (i.e. a model of the observed physical phenomena generally based on one or several partial differential equations); then an inverse model based on the forward model is derived which consists in reconstructing physical characteristics of the object. In general such an inverse model is a large-scale ill-posed problem. This paper is devoted to a new approach based on the derivation of a low complexity forward model. The inverse problem is then solved very efficiently with reduced computation time compared to classical approaches based on the finite element method (FEM). The here-proposed approach is applicable to problems governed by elliptic partial differential equations. An example of 2D bioluminescence tomography illustrates the effectiveness of the approach.