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An Algorithm to Train Unrestricted Sequential Discrete Morphological Neural Networks

Diego Marcondes, Mariana Feldman, Junior Barrera

TL;DR

This work tackles the challenge of building interpretable, trainable morphological neural networks capable of representing unrestricted $W$-operators. It introduces unrestricted sequential DMNN (USDMNN) and a hierarchical stochastic lattice descent algorithm (SLDA) to learn both the local windows $W_i$ and the Boolean kernels $f_i$ that define each layer, formalized via the operator class $\mathcal{H}(\mathcal{A})$ and the overparametrized representation $\Theta$. The framework combines a Sequential Morphological Computational Graph (MCG) and a pair of SLDA procedures (one for fixed windows and one for learning windows) to optimize a translation-invariant, locally defined sequence of $W$-operators, with minimization performed in Boolean lattices guided by training and validation errors $L_t$ and $L_v$. An application to boundary recognition on noisy digit images shows USDMNN achieving IoU-based performance comparable to Canonical DMNN (CDMNN), while offering a pathway to training DMNN without strong prior architectural constraints. The results suggest USDMNN can deliver interpretable, morphology-consistent models with competitive performance, and the authors outline efficient implementations and broader comparisons as promising future directions.

Abstract

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 that 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 descent algorithm (SLDA) 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, whose architecture is given by the composition of general W-operators. We illustrate the algorithm in a practical example.

An Algorithm to Train Unrestricted Sequential Discrete Morphological Neural Networks

TL;DR

This work tackles the challenge of building interpretable, trainable morphological neural networks capable of representing unrestricted -operators. It introduces unrestricted sequential DMNN (USDMNN) and a hierarchical stochastic lattice descent algorithm (SLDA) to learn both the local windows and the Boolean kernels that define each layer, formalized via the operator class and the overparametrized representation . The framework combines a Sequential Morphological Computational Graph (MCG) and a pair of SLDA procedures (one for fixed windows and one for learning windows) to optimize a translation-invariant, locally defined sequence of -operators, with minimization performed in Boolean lattices guided by training and validation errors and . An application to boundary recognition on noisy digit images shows USDMNN achieving IoU-based performance comparable to Canonical DMNN (CDMNN), while offering a pathway to training DMNN without strong prior architectural constraints. The results suggest USDMNN can deliver interpretable, morphology-consistent models with competitive performance, and the authors outline efficient implementations and broader comparisons as promising future directions.

Abstract

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 that 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 descent algorithm (SLDA) 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, whose architecture is given by the composition of general W-operators. We illustrate the algorithm in a practical example.
Paper Structure (11 sections, 5 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 11 sections, 5 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. W-operators
  3. Unrestricted Sequential Discrete Morphological Neural Networks
  4. Sequential Morphological Computational graph
  5. Sequential Discrete Morphological Neural Networks Architecture
  6. Representation of USDMNN
  7. Training USDMNN via the stochastic lattice descent algorithm
  8. Training a USDMNN with fixed windows
  9. Learning the windows of USDMNN windows
  10. Application: Boundary recognition of digits with noise
  11. Next steps and future research

Figures (2)

  • Figure 1: Illustration of the deterministic version of the SLDA to (a) learn the USDMNN windows and (b) train a USDMNN with fixed windows. To simplify the illustration, we considered only the possibility of adding a point to a window, and flipping a bit from 0 to 1 of a characteristic function, at each step, even though a point can be erased from a window, and a flipping from 1 to 0 may happen, if they have the least respective error. The training error $L_{t}$ or validation error $L_{v}$ of each point is on top of it.
  • Figure 2: Result obtained after applying, to the validation sample, each layer of the USDMNN trained with a batch size of $10$ that had the least validation error.