Department of Mathematics and Statistics

Bowling Green State University

Bowling Green, OH, 43403-0221

Abstract

We consider numerical methods for solving problems involving
total variation (TV) regularization for semidefinite
quadratic minimization problems
for image recontruction.
The objective functional (before regularization) is
the norm squared of **(Lu-z)**,
where
**u** is the reconstructed image,
**L** is a compact linear operator, and
**z**
is data containing inexact or partial information about the image.
TV regularization entails adding to
the objective functional
a term which penalizes the total variation
of
**u**; this term
formally appears as (a scalar times)
the
**L1** norm of the gradient of **u**.
The Euler equation for
the regularized objective functional
is a quasilinear elliptic equation of the form

[ L^*L + A(u) ] u = -L^*z

[a(u)](x) = c/sqrt(|grad u(x)|^2 + b^2)

Total variation regularization has enjoyed significant success in
image denoising and deblurring,
laser interferometry,
electrical tomography, and
estimation of permeabilities in porus medua flow models.
Its main advantage is that it
improves the conditioning of the optimization problem
while *not penalizing discontinuities* in the
reconstructed image.
The main difficulty in its use lies in the fact that
the Euler equation is nonlinear with rapidly varying coefficients
and can have a rather large number (e.g., 640-squared) of
degrees of freedom.

In this paper we present results from numerical experiments in which we use a fixed point approach to solving the the Euler equations, using various multilevel preconditioners. Specifically, we shall explore the performance of the "Hierarchical Basis" method, the "wavelet-modified Hierarchical Basis" method, and a conventional multigrid method.

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