J. Weickert, B. Benhamouda,
A semidiscrete nonlinear scale-space theory and its relation
to the Perona-Malik paradox,
F. Solina, W.G. Kropatsch, R. Klette, R. Bajcsy (Eds.), Advances
in computer vision, Springer, Wien, 1-10, 1997.
We discuss a semidiscrete framework for nonlinear diffusion scale-spaces,
where the image is sampled on a finite grid and the scale parameter is
continuous. This leads to a system of nonlinear ordinary differential
equations.
We investigate conditions under which one can guarantee
well-posedness properties, an extremum principle, average grey
level invariance, smoothing Lyapunov functionals, and convergence
to a constant steady-state. These properties are in analogy to
previously established results for the continuous setting.
Interestingly, this semidiscrete framework helps to
explain the so-called Perona--Malik paradox: The Perona--Malik
equation is a forward--backward diffusion equation which is
widely-used in image processing since it combines intraregional
smoothing with edge enhancement.
Although its continuous formulation is regarded to be ill-posed,
it turns out that a spatial discretization is sufficient to create
a well-posed semidiscrete diffusion scale-space.
We also prove that an explicit 1-D Perona--Malik scheme is
monotonicity preserving, which explains that staircasing is
essentially the only practically appearing instability.
The full paper is available
online as well.
Return to
Joachim Weickert's publication list.