J. Weickert,
Nonlinear diffusion scale-spaces,
J. Sporring, M. Nielsen, L. Florack, P. Johansen (Eds.),
Gaussian scale-space theory, Kluwer, Dordrecht, 221-234, 1997.
In order to reduce difficulties that are linked to the
Gaussian scale-space paradigm such as blurring of interesting features
or the correspondence problem, we study modifications which
renounce the linearity assumption, but keep the divergence formulation.
After having briefly discussed ill-posedness aspects
of the first nonlinear model in this direction, we turn to
a regularized version, for which well-posedness results and a
maximum--minimum principle holds. Since nonlinear diffusion filters
can be edge-enhancing by acting locally like a backward heat equation,
the question arises in which sense they can be regarded a smoothing,
simplifying scale-space transformations. We shall see that
Lyapunov functionals are a natural framework to describe global
smoothing properties of nonlinear diffusion filters.
Prerequesites are given under which these results carry over to the
semidiscrete and discrete setting. We illustrate how these
requirements are satisfied by numerical schemes. Finally it is
indicated how these results can be generalized to more flexible
filter classes.
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