Diffuse Scattering and Diffuse Optical Tomography on Graphs

Abstract

We formulate and analyze difference equations on graphs analogous to time- independent diffusion equations arising in the study of diffuse scattering in con- tinuous media and consider the associated inverse problem, which we call discrete diffuse optical tomography.

For the forward problem we show how to construct solutions in the presence of weak scatterers from the solution to the homogeneous (background problem) using Born series, providing necessary conditions for convergence and demonstrating the process through numerous examples. In addition, we outline a method for finding Green’s functions for Cayley graphs for both abelian and non-abelian groups. Finally, we conclude our discussion of the forward problem by considering the effects of sparsity on our method and results, outlining the simplifications that can be made provided that the scatterers are weak and well-separated.

For the inverse problem, we present an algorithm for solving inverse problems on graphs analogous to those arising in diffuse optical tomography for continuous media. In particular, we formulate and analyze a discrete version of the inverse Born series, proving estimates characterizing the domain of convergence, approximation errors, and stability of our approach. We also present a modification which allows additional information on the structure of the potential to be incorporated, facilitating recovery for a broader class of problems.