A Linear Scale-Space Theory for Continuous Nonlocal Evolutions

Most scale-space evolutions are described in terms of partial differential equations. In recent years, however, nonlocal processes have become an important research topic in image analysis. The goal of our paper is to establish well-posedness and scale-space properties for a class of nonlocal evolutions. They are given by linear integro-differential equations with measures. In analogy to Weickert’s diffusion theory (1998), we prove existence and uniqueness, preservation of the average grey value, a maximum–minimum principle, image simplification properties in terms of Lyapunov functionals, and we establish convergence to a constant steady state. We show that our nonlocal scale-space theory covers nonlocal variants of linear diffusion. Moreover, by choosing specific discrete measures, the classical semidiscrete diffusion framework is identified as a special case of our continuous theory. Last but not least, we introduce two modifications of bilateral filtering. In contrast to previous bilateral filters, our variants create nonlocal scale-spaces that preserve the average grey value and that can be highly robust under noise. While these filters are linear, they can achieve a similar performance as nonlinear and even anisotropic diffusion equations.