Department of Mathematics

Naval Postgraduate School

Monterrey, CA

Auto-trol Technology Corporation

Denver, CO.

Program in Applied Mathematics

Campus Box 526

University of Colorado at Boulder

Boulder, CO 80309-0526

Accurate Information Systems

Eatontown, NJ

The sampled Radon transform of a 2D function can be represented as a
continuous linear map A:L_{2}(W)--> R^{N}, where
(Au)_{j} = < u, p_{j}> and p_{j} is the characteristic
function of a strip through W approximating the set of line integrals in the
sample. The image reconstruction problem is: given a vector b in
R^{N}, find an image (or density function) u(x,y) such that Au=b. In
general there are infinitely many solutions; we seek the solution with minimal
2-norm, which leads to a matrix equation Bw=b, where B is a square dense
matrix with several convenient properties. We analyze the use of Gauss-Seidel
iteration applied to the problem, observing that while the iteration formally
converges, there exists a near null space into which the error vectors
migrate, after which the iteration stalls. The null space and near null space
of B are characterized in order to develop a multilevel scheme. Based on the
principles of the Multilevel Projection Method (PML), this scheme leads to
somewhat improved performance. Its primary utility, however, is that it
facilitates the development of a PML-based method for spotlight tomography,
that is, local grid refinement over a portion of the image in which features
of interest can be resolved at finer scale than is possible globally.

Contributed