O. Scherzer, J. Weickert,
Relations between regularization and diffusion filtering,
Technical Report DIKU-98/23, Dept. of Computer Science,
University of Copenhagen, Denmark, 1998. Revised version published
in Journal of Mathematical Imaging and Vision, Vol. 12, 43-63, 2000.
Regularization may be regarded as diffusion filtering with an
implicit time discretization where one single step is used. Thus, iterated
regularization with small regularization parameters approximates
a diffusion process. The goal of this paper is to analyse relations
between noniterated and iterated regularization and diffusion
filtering in image processing.
In the linear setting, we show that with iterated Tikhonov regularization
noise can be better handled than with noniterated.
In the nonlinear framework, two filtering strategies are considered:
total variation regularization and the diffusion filter of Perona and
Malik.
It is established that the Perona-Malik equation decreases the total
variation during its evolution.
While noniterated and iterated total variation regularization is
well-posed, one cannot expect to find a minimizing sequence which
converges to a minimizer of the corresponding energy functional for
the Perona-Malik filter.
To address this shortcoming, a novel regularization of the Perona-Malik
process is presented which allows to construct a weakly lower
semi-continuous energy functional.
In analogy to recently established results for a well-posed class of
regularized Perona-Malik filters, we introduce Lyapunov functionals
and convergence results for regularization methods.
Experiments on real-world images illustrate that iterated linear
regularization performs better than noniterated, while
no significant differences between noniterated and iterated total
variation regularization have been observed.
The
full technical report is available online as well.
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