Abstract
Dilation and erosion are two elementary operations from mathematical morphology, a non-linear lattice computing methodology widely used for image processing and analysis. The dilation-erosion perceptron (DEP) is a morphological neural network obtained by a convex combination of a dilation and an erosion followed by the application of a hard-limiter function for binary classification tasks. A DEP classifier can be trained using a convex-concave procedure along with the minimization of the hinge loss function. As a lattice computing model, the DEP classifier assumes the feature and class spaces are partially ordered sets. In many practical situations, however, there is no natural ordering for the feature patterns. Using concepts from multi-valued mathematical morphology, this paper introduces the reduced dilation-erosion (r-DEP) classifier. An r-DEP classifier is obtained by endowing the feature space with an appropriate reduced ordering. Such reduced ordering can be determined using two approaches: one based on an ensemble of support vector classifiers (SVCs) with different kernels and the other based on a bagging of similar SVCs trained using different samples of the training set. Using several binary classification datasets from the OpenML repository, the ensemble and bagging r-DEP classifiers yielded mean higher balanced accuracy scores than the linear, polynomial, and radial basis function (RBF) SVCs as well as their ensemble and a bagging of RBF SVCs.
Highlights
Cyber-physical systems (CPS) is a broad interdisciplinary area which combines computational and physical devices in an integrated manner [1,2,3,4]
Morphological neural network (MNN) refer to the broad class of neural networks whose neurons perform an operation from Mathematical morphology (MM) possibly followed by the application of an activation function [34]
In analogy to the previous example, we evaluated the performance of the r-dilation-erosion perceptron (DEP) classifier on the double-moon problem presented in Example 5
Summary
Cyber-physical systems (CPS) is a broad interdisciplinary area which combines computational and physical devices in an integrated manner [1,2,3,4]. Complete lattice provides an appropriate mathematical background for binary and gray-scale MMs, defining morphological operators for multi-valued images is not straightforward because there is no universal ordering for vector-valued spaces [26,27]. Pessoa and Maragos used pulse functions to circumvent the non-differentiability of lattice-based operations on a steepest descent method designed for training a hybrid morphological/rank/linear network [39]. Based on the ideas of Pessoa and Maragos, Araújo proposed a hybrid morphological/linear network called dilation-erosion perceptron (DEP), which is trained using a steepest descent method [40]. We present a brief review on the basic concepts from lattice theory and MM, including the supervised reduced ordering-based approach to multi-valued MM.
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