Nonlinear Scale-Space

Abstract

We propose a generalisation of scale-space theory for scalar images. The starting point is the assumption that the conventional Gaussian model constitutes an unbiased (that is, task independent), multiscale image representation. A generalised scale-space is then considered as a conventional scale-space “in disguise”, representing the data in a format that is more convenient for specific applications. Although formally equivalent to the conventional representation (at least locally), a generalised representation may be more apt for a dedicated task. In particular, it may potentially solve the so-called “localisation problem” of linear scale-space. Several models based on nonlinear diffusion have emerged by the desire to deal with this problem. The proposed theory provides a unifying framework for a variety of such models that can be related to conventional scale-space in a one-to-one way.

Our defining constraint for a generalised scale-space is the requirement of equivalence: it should formally correspond to a transformation of linear scale-space. The key idea is a metric transform that preserves the intrinsic properties of the spatial domain. This allows one to regard a generalised scale-space as a strategy for reading out a single data representation. The equivalence constraint is shown to yidld a particular class of (linear or nonlinear) diffusion equations. Conventional, linear scale-space is a convenient representative of the equivalence class for “universal” purposes. The emphasis is on nonlinear scale-spaces, although the principle of equivalence can be used within the linear context as well. Examples are included to illustrate the theory both in the linear as well as in the nonlinear sector.