An application of the finite element method to maximum entropy tomographic image reconstruction

Abstract

A new approach to maximum entropy tomographic image reconstruction is presented. It is shown that by using a finite-dimensional subspace, one can obtain an approximation to the solution of a maximum entropy optimization problem, set inL ₂ D. An example of an appropriate finite element subspace for a two-dimensional parallel beam projection geometry is examined. Particular attention is paid to the case where the x-ray projection data are sparse. In the current work, this means that the number of projections is small (in practice, perhaps only 5–20). A priori information in the form of known maximum and minimum densities of the materials being scanned is built into the model. A penalty function, added to the entropy term, is used to control the residual error in meeting the projection measurements. The power of the technique is illustrated by a sparse data reconstruction and the resulting image is compared to those obtained by a conventional method.