An explanation for the logarithmic connection between linear and morphological system theory

Abstract

Dorst/van den Boomgaard and Maragos introduced the slope transform as the morphological eqivalent of the Fourier transform. It formed the basis of a morphological system theory that bears an almost logarithmic relation to linear system theory. This surprising logarithmic connection, however, has not been understood so far. Our article provides an explanation by revealing that morphology in essence is linear system theory in specific algebras. While linear system theory uses the standard plus-prod algebra, morphological system theory is based on the max-plus algebra and the min-plus algebra. We identify the nonlinear operations of erosion and dilation as linear convolutions in the latter algebras. For the subsequent theoretical analysis, it is advantageous to focus on two concepts from convex analysis: We consider the conjugacy operation and the multivariate Laplace transform instead of the closely related slope and Fourier transforms. While the Laplace transform maps convolution into multiplication, the conjugacy operation turns erosion into addition. This logarithmic connection triggers us to consider the logarithmic Laplace transform. The logarithmic Laplace transform in the plus-prod algebra corresponds to the conjugacy operation in the max-plus algebra. Its conjugate is given by the so-called Cramer transform. Originating from the theory of large deviations in stochastics, the Cramer transform maps Gaussians to quadratic functions and relates standard convolution to erosion. This fundamental transform constitutes the direct link between linear and morphological system theory. Many numerical examples are presented that illustrate the convexifying and smoothing properties of the Cramer transform.