PDEs for Morphological Scale-Spaces and Eikonal Applications

In contrast to the all-discrete approaches that dominated image processing in the recent past, in computer vision since the mid 1980’s there have been proposed continuous models for several vision tasks based on partial differential equations (PDEs). The discrete part of such approaches comes only in the corresponding difference equations (numerical algorithms) that approximate the solution of these PDEs. Motivations for using PDEs in image analysis and vision include better and more intuitive mathematical modeling, connections with physics, better approximation to the Euclidean geometry of the problem, and existence of efficient numerical algorithms for solving them. One major effort [31] was the use of PDEs in problems of shape from shading and optical flow. However, the most well known vision problem modeled via PDEs is that of multiscale image analysis, i.e. scale-spaces. While many such continuous approaches have been linear, the majority and the most useful ones are nonlinear. Several classes of nonlinear PDEs used in image analysis and vision are based on or related to morphological operations. Morphological image processing has been based traditionally on modeling images as sets or as points in a complete lattice of functions and viewing morphological image transformations as set or lattice operators. Thus, the two classic approaches to analyze or design the deterministic systems of mathematical morphology have been (i) geometry by viewing them as image set transformations in Euclidean spaces and (ii) algebra to analyze their properties using set or lattice theory. Geometry was used mainly for intuitive understanding, and algebra was restricted to the space domain. Despite their limitations, these approaches have produced a powerful and self-contained broad collection of nonlinear image analysis concepts, operators and algorithms, which have found a broad range of applications in image processing and computer vision; see [73, 49, 28] and Chapter 3.3 of this book for surveys and references. In parallel to these directions, there is a growing part of morphological image processing that is based on ideas from differential calculus and dynamical systems. It combines some early ideas on morpholological signal gradients and some recent ideas on using differential equations to model nonlinear multiscale processes or distance propagation in images. In this chapter we present a unified view of the various interrelated ideas in this area, using occasionally some systems analysis tools in both the space and the slope transform domain. Among the few early connections between morphology and calculus were the morphological gradients. Specifically, given a function f : Rm → R, with m = 1, 2, its isotropic morphological sup-derivative at a point x is defined by