An Introduction to Morphological Associative Memories in Complete Lattices and Inf-Semilattices

In the mid 1990's, the morphological associative memory (MAM) was introduced as a distributive associative memory model. Since then several extensions of MAMs as well as applications in different domains have appeared in the literature. Just like other morphological neural network models, a MAM performs an elementary operation of mathematical morphology, possibly followed by an activation function, at every node. Generally speaking, a common trait of all distributive MAM models is their foundation in mathematical morphology on complete lattices. Morphological operators in the complete lattice framework come in dual pairs such as dilation/erosion, opening/closing, etc.. Therefore, MAM models also have two versions (denoted using the symbols W and M) that are tolerant to different types of noise in the input patterns. To overcome this drawback for MAM models, we resort to the more recent theory of mathematical morphology on inf-semilattices whose elementary operators are self-dual. This paper represents a first attempt at formulating an associative memory (AM) in this framework.