A semidiscrete nonlinear scale-space theory and its relation to the Perona-Malik paradox

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 pro..