J. Weickert,
Scale-space properties of nonlinear diffusion filtering with
a diffusion tensor,
Report No. 110, Laboratory of Technomathematics,
University of Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern,
Germany, 1994.
In spite of its lack of theoretical justification, nonlinear diffusion
filtering has become a powerful image enhancement tool in recent years.
The goal of the present paper is to provide a mathematical foundation for
continuous nonlinear diffusion filtering as a scale-space transformation
which is flexible enough to simplify images without loosing the capability
of enhancing edges.
By studying the Lyapunov functionals, it is shown that nonlinear
diffusion reduces $L^p$ norms and central moments and increases the
entropy of images.
The proposed anisotropic class utilizes a diffusion tensor which may be
adapted to the image structure. It permits existence, uniqueness and regularity
results, the solution depends continuously on the initial image, and it
satisfies an extremum principle.
All considerations include linear and certain nonlinear isotropic models
and apply to m-dimensional vector-valued images. The results are
juxtaposed to linear and morphological scale-spaces.
The full paper is available
online as well.
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