An Algorithm to Train Unrestricted Sequential Discrete Morphological Neural Networks

With the advent of deep learning, there have been attempts to insert mathematical morphology (MM) operators into convolutional neural networks (CNN), and the most successful endeavor to date has been the morphological neural networks (MNN). Although MNN have performed better than CNN in solving some problems, they inherit their black-box nature. Furthermore, in the case of binary images, they are approximations, which loose the Boolean lattice structure of MM operators and, thus, it is not possible to represent a specific class of W-operators with desired properties. In a recent work, we proposed the Discrete Morphological Neural Networks (DMNN) for binary image transformation to represent specific classes of W-operators and estimate them via machine learning. We also proposed a stochastic lattice gradient descent algorithm (SLGDA) to learn the parameters of Canonical Discrete Morphological Neural Networks (CDMNN), whose architecture is composed only of operators that can be decomposed as the supremum, infimum, and complement of erosions and dilations. In this paper, we propose an algorithm to learn unrestricted sequential DMNN (USDMNN), whose architecture is given by the composition of general W-operators. We consider the representation of a W-operator by its characteristic Boolean function, and then learn it via a SLGDA in the Boolean lattice of functions. Although both the CDMNN and USDMNN have the Boolean lattice structure, USDMNN are not as dependent on prior information about the problem at hand, and may be more suitable in instances in which the practitioner does not have strong domain knowledge. We illustrate the algorithm in a practical example.