Discrete Morphological Neural Networks

TL;DR

The paper addresses learning and automatic design of binary image operators within Mathematical Morphology by introducing Discrete Morphological Neural Networks (DMNN), which encode operator design as a Morphological Computational Graph and learn parameters via lattice-based optimization. It develops two training schemes, the lattice gradient descent (LGDA) and its stochastic variant (SLGDA), to efficiently search the discrete operator space while preserving interpretability. Theoretical results show that DMNN are universal representers of translation-invariant, locally defined set operators and that canonical (CDMNN) architectures can be trained to approximate target operators with data. A proof-of-concept demonstrates boundary recognition of noisy digits, highlighting the practical potential and interpretability advantages of DMNN over black-box neural models. The work positions DMNN as a principled bridge between classical MM design and modern data-driven learning, with extensions to broader lattices and grayscale domains on the horizon.

Abstract

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.

Figures (13)

• Figure 1: The lattice isomorphisms between representations of W-operators.
• Figure 2: Trivial DMNN representation of $\psi \in \Omega$.
• Figure 3: (a) Sup-generating and (b) inf-generating DMNN representation of a $\psi \in \Psi_{W}$ with $\boldsymbol{B}_{W}(\psi) = \{\mathscr{I}_{1},\mathscr{I}_{2},\mathscr{I}_{3}\}$and $\boldsymbol{B}_{W}(\psi) = \{\mathscr{I}_{1}^{\star},\mathscr{I}_{2}^{\star},\mathscr{I}_{3}^{\star}\}$.
• Figure 4: Three DMNN representations of a same alternate-sequential filter. The operators $\gamma\phi_{B_{1}}$ and $\gamma\phi_{B_{2}}$ denote, respectively, the compositions $\phi_{B_{1}}\gamma_{B_{1}}$ and $\phi_{B_{2}}\gamma_{B_{2}}$.
• Figure 5: A DMNN representation of the composition of an ASF with structuring element $A \in \mathcal{P}(W)$ and a W-operator $\psi \in \Psi_{W}$ with $\boldsymbol{B}_{W}(\psi) = \{\mathscr{I}_{1},\mathscr{I}_{2},\mathscr{I}_{3}\}$.

Theorems & Definitions (28)

• Proposition 3.1
• Corollary 3.2
• Proposition 4.1
• Corollary 4.2
• Proposition 4.3
• Proposition 5.1
• proof
• Remark 5.2
• Definition 5.3
• Example 5.4: Trivial DMNN