Angelica Lourenço Oliveira

Angelica Lourenço Oliveira, Marcos Eduardo Valle

This paper presents a hybrid morphological neural network for regression tasks called linear dilation-erosion regressor ($\ell$-DER). An $\ell$-DER is given by a convex combination of the composition of linear and morphological operators. They yield continuous piecewise linear functions and, thus, are universal approximators. Besides introducing the $\ell$-DER model, we formulate their training as a difference of convex (DC) programming problem. Precisely, an $\ell$-DER is trained by minimizing the least-squares using the convex-concave procedure (CCP). Computational experiments using several regression tasks confirm the efficacy of the proposed regressor, outperforming other hybrid morphological models and state-of-the-art approaches such as the multilayer perceptron network and the radial-basis support vector regressor.

Angelica Lourenço Oliveira, Marcos Eduardo Valle

Mathematical morphology (MM) is a theory of non-linear operators used for the processing and analysis of images. Morphological neural networks (MNNs) are neural networks whose neurons compute morphological operators. Dilations and erosions are the elementary operators of MM. From an algebraic point of view, a dilation and an erosion are operators that commute respectively with the supremum and infimum operations. In this paper, we present the \textit{linear dilation-erosion perceptron} ($\ell$-DEP), which is given by applying linear transformations before computing a dilation and an erosion. The decision function of the $\ell$-DEP model is defined by adding a dilation and an erosion. Furthermore, training a $\ell$-DEP can be formulated as a convex-concave optimization problem. We compare the performance of the $\ell$-DEP model with other machine learning techniques using several classification problems. The computational experiments support the potential application of the proposed $\ell$-DEP model for binary classification tasks.

Angelica Lourenço Oliveira, Marcos Eduardo Valle

In this work, we briefly revise the reduced dilation-erosion perceptron (r-DEP) models for binary classification tasks. Then, we present the so-called linear dilation-erosion perceptron (l-DEP), in which a linear transformation is applied before the application of the morphological operators. Furthermore, we propose to train the l-DEP classifier by minimizing a regularized hinge-loss function subject to concave-convex restrictions. A simple example is given for illustrative purposes.