Recall of patterns using binary and gray-scale autoassociative morphological memories

Abstract

Morphological associative memories (MAM's) belong to a class of artificial neural networks that perform the operations erosion or dilation of mathematical morphology at each node. Therefore we speak of morphological neural networks. Alternatively, the total input effect on a morphological neuron can be expressed in terms of lattice induced matrix operations in the mathematical theory of minimax algebra. Neural models of associative memories are usually concerned with the storage and the retrieval of binary or bipolar patterns. Thus far, the emphasis in research on morphological associative memory systems has been on binary models, although a number of notable features of autoassociative morphological memories (AMM's) such as optimal absolute storage capacity and one-step convergence have been shown to hold in the general, gray-scale setting. In previous papers, we gained valuable insight into the storage and recall phases of AMM's by analyzing their fixed points and basins of attraction. We have shown in particular that the fixed points of binary AMM's correspond to the lattice polynomials in the original patterns. This paper extends these results in the following ways. In the first place, we provide an exact characterization of the fixed points of gray-scale AMM's in terms of combinations of the original patterns. Secondly, we present an exact expression for the fixed point attractor that represents the output of either a binary or a gray-scale AMM upon presentation of a certain input. The results of this paper are confirmed in several experiments using binary patterns and gray-scale images.