An Explanation for the Logarithmic Connection between Linear and Morphological Systems

Abstract

Since the introduction of the slope transform by Dorst/van den Boomgaard and Maragos as the morphological equivalent of the Fourier transform, people have been surprised about the almost logarithmic relation between linear and morphological system theory. This article gives an explanation by revealing that morphology in essence is linear system theory in a specific algebra. While classical linear system theory uses the standard (+,×)-algebra, the morphological system theory is based on the idempotent (max, +)-algebra and the (min, +)- algebra. We identify the nonlinear operations of erosion and dilation as linear convolutions *e and *d induced by these idempotent algebras. The slope transform in the (max, +)-algebra, however, corresponds to the logarithmic multivariate Laplace transform in the (+, ×)-algebra. We study relevant properties of this transform and its links to convex analysis. This leads to the definition of the so-called Cramer transform as the Legendre-Fenchel transform of the logarithmic Laplace transform. Originally known from the theory of large deviations in stochastics, the Cramer transform maps standard convolution to *e-convolution, and it maps Gaussians to quadratic functions. The article is a step towards the unification of linear and morphological system theories on the basis of a general linear system theory in an appropriate algebra.