Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

Abstract

It is well-known that there are striking analogies between linear shift-invariant systems and morphological systems for image analysis. So far, however, the relations between both system theories are mainly understood on a pure convolution / erosion level. A formal connection on the level of differential or pseudodifferential equations and their induced scale-spaces is still missing. The goal of our paper is to close this gap. We present a simple and fairly general dictionary that allows to translate any linear shift-invariant evolution equation into its morphological counterpart and vice versa. It is based on a scale-space representation by means of the symbol of its (pseudo)differential operator. Introducing a novel transformation, the Cramer---Fourier transform, puts us in a position to relate the symbol to the structuring function of a morphological scale-space of Hamilton---Jacobi type. As an application of our general theory, we derive the morphological counterparts of many linear shift-invariant scale-spaces, such as the Poisson scale-space, $$\alpha $$ź-scale-spaces, summed $$\alpha $$ź-scale-spaces, relativistic scale-spaces, and their anisotropic variants. Our findings are illustrated by experiments.