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With the advent of deep learning, there have been attempts to insertmathematical morphology (MM) operators into convolutional neural networks(CNN), and the most successful endeavor to date has been the morphologicalneural networks (MNN). Although MNN have performed better than CNN in solvingsome problems, they inherit their black-box nature. Furthermore, in the case ofbinary images, they are approximations, which loose the Boolean latticestructure of MM operators and, thus, it is not possible to represent a specificclass of W-operators with desired properties. In a recent work, we proposed theDiscrete Morphological Neural Networks (DMNN) for binary image transformationto represent specific classes of W-operators and estimate them via machinelearning. We also proposed a stochastic lattice gradient descent algorithm(SLGDA) to learn the parameters of Canonical Discrete Morphological NeuralNetworks (CDMNN), whose architecture is composed only of operators that can bedecomposed 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 generalW-operators. We consider the representation of a W-operator by itscharacteristic Boolean function, and then learn it via a SLGDA in the Booleanlattice of functions. Although both the CDMNN and USDMNN have the Booleanlattice structure, USDMNN are not as dependent on prior information about theproblem at hand, and may be more suitable in instances in which thepractitioner does not have strong domain knowledge. We illustrate the algorithmin a practical example.

An Algorithm to Train Unrestricted Sequential Discrete Morphological Neural Networks