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.

用离散形态神经网络（DMNN）进行二进制图像转换，表示具有特定特性的 W 算子，并通过机器学习来估计它们，同时提出了一种用于学习无限制顺序 DMNN（USDMNN）的算法。

训练无限制的顺序离散形态神经网络的算法

An Algorithm to Train Unrestricted Sequential Discrete Morphological  Neural Networks

A classical approach to designing binary image operators is Mathematical
Morphology (MM). We propose the Discrete Morphological Neural Networks (DMNN)
for binary image analysis to represent W-operators and estimate them via
machine learning. A DMNN architecture, which is represented by a Morphological
Computational Graph, is designed as in the classical heuristic design of
morphological operators, in which the designer should combine a set of MM
operators and Boolean operations based on prior information and theoretical
knowledge. Then, once the architecture is fixed, instead of adjusting its
parameters (i.e., structural elements or maximal intervals) by hand, we propose
a lattice gradient descent algorithm (LGDA) to train these parameters based on
a sample of input and output images under the usual machine learning approach.
We also propose a stochastic version of the LGDA that is more efficient, is
scalable and can obtain small error in practical problems. The class
represented by a DMNN can be quite general or specialized according to expected
properties of the target operator, i.e., prior information, and the semantic
expressed by algebraic properties of classes of operators is a differential
relative to other methods. The main contribution of this paper is the merger of
the two main paradigms for designing morphological operators: classical
heuristic design and automatic design via machine learning. Thus, conciliating
classical heuristic morphological operator design with machine learning. We
apply the DMNN to recognize the boundary of digits with noise, and we discuss
many topics for future research.

提出了离散形态神经网络（DMNN）用于二值图像分析以表示 W 算子并通过机器学习进行估计的经典方法。