The use of Graßmann identities for inversion of a general model in diffuse tomography

Abstract

In this paper we have given a bief description of a general, albeit tiny, problem in diffuse tomography. This is an inverse problem woth 64 variables and a system of only 48 independent non-linear equations. We make 16 linear “identifications” of variables to close the system. Rather than tackle this imposing system directly, we make a change of variables. The transformed system is composed of 48 linear and 16 non-linear equations, which is easily reduced to a system of 8 cubic and 8 quadratic equations in 16 unknowns. The bulk of the paper discusses how one can sumplify these equations. The quadratics may be used to reduce the number of terms in the cubics. Next, Graßmann relations may be used to simplify the coefficients of both the cubics and the quadratics. Finally, we give an algorithm for solving these last 16 equations.

Our next project is to extend our method to a larger problem. At this point, the best option seems to be a recursive algorithm whose “base case” is the 2 x 2 problem described here. The greatest obstacle to inverting the n x n problem will be slℓving highly non-linear consistency conditions analogous to the cubics and quadratics described above. If the two-dimensional n x n problem can be solved with a reasonable computational effort, then a similar method should carry over to the three-dimensional n x n x n problem.

Research partially supported by AFOSR under Contract FDF-49620-92-J-0067-11792 and by the NSF under Grant DMS91-01224.