Approaches to Multivalued Mathematical Morphology Based on Uncertain Reduced Orderings

Abstract

Mathematical morphology (MM) is a powerful non-linear theory that can be used for signal and image processing and analysis. Although MM can be very well defined on complete lattices, which are partially ordered sets with well defined extrema operations, there is no natural ordering for multivalued images such as hyper-spectral and color images. Thus, a great deal of effort has been devoted to ordering schemes for multivalued MM. In a reduced ordering, in particular, elements are ranked according to the so-called ordering mapping. Despite successful applications, morphological operators based on reduced orderings are usually too reliant on the ordering mapping. In many practical situations, however, the ordering mapping may be subject to uncertainties such as measurement errors or the arbitrariness in the choice of the mapping. In view of this remark, in this paper we present two approaches to multivalued MM based on an uncertain reduced ordering. The new operators are formulated as the solution of an optimization problem which, apart from the uncertainty, can circumvent the false value problem and deal with irregularity issues.

Irregularity Index for Vector-Valued Morphological Operators

“…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 (%)…”

“…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].…”

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

“…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 [13].…”

Reduced Dilation-Erosion Perceptron for Binary Classification

“…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].…”

“…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].…”