Multilevel Image Reconstruction with Natural Pixels

             Van Emden Henson (1)               Mark A. Limber (2)
           Stephen F. McCormick (3)           Bruce T. Robinson (4)

                                   Abstract

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.


(1) Department of Mathematics 
    Naval Postgraduate School  
    email: vhenson@nps.navy.mil 

(2) Auto-trol Technology Corporation, Denver, CO.
    email: marlim@Auto-trol.COM  

(3) Program in Applied Mathematics
    University of Colorado
    email: stevem@newton.colorado.edu

(4) Accurate Information Systems
    Eatontown, NJ. 
    email: brobinso@Accurate.COM