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

Although much effort has been spent in the recent decade to establish a theoretical foundation of certain partial differential equations (PDEs) as scale-spaces, it is almost never taken into account that, in practice, images are sampled on a fixed pixel grid1. For nonlinear PDE-based filters, usually straightforward finite difference discretizations are applied in the hope that they reflect the nice properties of the continuous equations. Since scale-spaces cannot perform better than their numerical discretizations, however, it would be desirable to have a genuinely discrete nonlinear framework which reflects the discrete nature of digital images. In this paper we discuss a semidiscrete scale-space framework for nonlinear diffusion filtering. It keeps the scale-space idea of having a continuous time parameter, while taking into account the spatial discretization on a fixed pixel grid. It leads to nonlinear systems of coupled ordinary differential equations. Conditions are established under which one can prove existence of a stable unique solution which preserves the average grey level. An interpretation as a smoothing scale-space transformation is introduced which is based on an extremum principle and the existence of a large class of Lyapunov functionals comprising for instance p-norms, even central moments and the entropy. They guarantee that the process is not only simplifying and information-reducing, but also converges to a constant image as the scale parameter t tends to infinity.