  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 descent algorithm (LDA) 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 LDA 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. As a proof-of-concept, we
apply the DMNN to recognize the boundary of digits with noise, and we discuss
many topics for future research.
