Approaches to Multivalued Mathematical Morphology Based on Uncertain Reduced Orderings

Sangalli, Mateus; Valle, Marcos Eduardo

doi:10.1007/978-3-030-20867-7_18

Cited by 11 publications

(6 citation statements)

References 19 publications

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“…Thus, the variables z ij satisfy the constraints of the linear programming problem (11). Because W p (I, J) p is the minimum value of (11), from (25) we conclude that Local Irregularity Index (%)…”

Section: The Local Irregularity Indexmentioning

confidence: 90%

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Irregularity Index for Vector-Valued Morphological Operators

et al. 2022

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Mathematical morphology is a valuable theory of nonlinear operators widely used for image processing and analysis. Although initially conceived for binary images, mathematical morphology has been successfully extended to vector-valued images using several approaches. Vector-valued morphological operators based on total orders are particularly promising because they circumvent the problem of false colors. On the downside, they often introduce irregularities in the output image. This paper proposes measuring the irregularity of a vector-valued morphological operator by the relative gap between the generalized sum of pixel-wise distances and the Wasserstein metric. Apart from introducing a measure of the irregularity, referred to as the irregularity index, this paper also addresses its computational implementation. Precisely, we distinguish between the ideal global and the practical local irregularity indexes. The local irregularity index, which can be computed more quickly by aggregating values of local windows, yields a lower bound for the global irregularity index. Computational experiments with natural images illustrate the effectiveness of the proposed irregularity indexes.Keywords mathematical morphology • vector-valued images • total order • irregularity issue • optimal transport.

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Section: The Local Irregularity Indexmentioning

confidence: 90%

“…The irregularity index can also be used for the design of efficient morphological operators. For example, it can be used as the objective function for the design of vector-valued morphological operators based on uncertain reduced orderings [25].…”

Section: Discussionmentioning

confidence: 99%

Irregularity Index for Vector-Valued Morphological Operators

et al. 2022

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Section: Discussionmentioning

confidence: 99%

Measuring the Irregularity of Vector-Valued Morphological Operators Using Wasserstein Metric

Valle

Francisco

Granero

et al. 2021

Lecture Notes in Computer Science

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Mathematical morphology is a useful theory of nonlinear operators widely used for image processing and analysis. Despite the successful application of morphological operators for binary and gray-scale images, extending them to vector-valued images is not straightforward because there are no unambiguous orderings for vectors. Among the many approaches to multivalued mathematical morphology, those based on total orders are particularly promising. Morphological operators based on total orders do not produce the so-called false-colors. On the downside, they often introduce irregularities in the output image. Although the irregularity issue has a rigorous mathematical formulation, we are not aware of an efficient method to quantify it. In this paper, we propose to quantify the irregularity of a vector-valued morphological operator using the Wasserstein metric. The Wasserstein metric yields the minimal transport cost for transforming the input into the output image. We illustrate by examples how to quantify the irregularity of vector-valued morphological operators using the Wasserstein metric.

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“…Furthermore, the component-wise order given by (1) results the so-called "false color" problem in multi-valued MM [52]. As a consequence, a great deal of effort has been devoted to finding appropriate ordering schemes for vector-valued data [26,27,29,31,53]. Among the many approaches to multi-valued MM, those based on reduced orderings are particularly interesting and computationally cheap [32,33,53].…”

Section: Multi-valued Mathematical Morphologymentioning

confidence: 99%

“…Although complete lattice provide 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]. In fact, the development of appropriate ordering scheme for multi-valued MM is an active area of research [28][29][30][31][32]. In this paper, we make use of the suprevised reduced ordering proposed by Velasco-Forero and Angulo [33].…”

Section: Introductionmentioning

confidence: 99%

Reduced Dilation-Erosion Perceptron for Binary Classification

Valle

2020

Mathematics

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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 in 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.

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Approaches to Multivalued Mathematical Morphology Based on Uncertain Reduced Orderings

Cited by 11 publications

References 19 publications

Irregularity Index for Vector-Valued Morphological Operators

Irregularity Index for Vector-Valued Morphological Operators

Measuring the Irregularity of Vector-Valued Morphological Operators Using Wasserstein Metric

Reduced Dilation-Erosion Perceptron for Binary Classification

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