Irregularity Index for Vector-Valued Morphological Operators

Abstract

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.

Discovering Irregularities from Computer Networks by Topological Mapping

“…The review found anomalies in the topologies of the various variations of bridge graphs or networks, i.e., Gr (Ps, v), Gr (Cs, v) and Gr (Ks, v), honeycomb network HCn, hexagonal network H(m, n), Sierpinski networks, i.e., Sn, S(n, k) networks. Figures 2,4,6,8,10,12,and 14 give the graphical representation of the ISO index for the above-mentioned graphs of networks. Irregularity Sombor Index found lower bounds, upper bounds, and irregularities of all mentioned networks well prediction quality of best characteristic.…”

“…The nearby irregularity index, which can be taken care of significantly more rapidly by amounting to expected gains of neighborhood windows, yields a lower bound for the general irregularity index. Computational primers with customary pictures outline the possibility of the proposed irregularity indexes [12].…”

Shortest Length Total Orders Do Not Minimize Irregularity in Vector-Valued Mathematical Morphology

“…The irregularity introduced by a morphological operator can be measured using the so-called irregularity index [9]. Briefly, the irregularity index is given by the gap between the generalized sum of pixel-wise distance and the Wasserstein metric.…”

“…Because morphological operators only take into account the partial order ≤, they can introduce irregularities if the image I have pixel values x, y, z ∈ V such that x ≤ y ≤ z but d(x, z) ≤ d(x, y). Accordingly, despite x is closer to z than to y, the inequalities x ≤ y ≤ z suggest w is farther from x than y [9]. Because morphological operators are defined using extrema operators, they do not consider the metric of V, resulting in the irregularity issue.…”

“…Because morphological operators are defined using extrema operators, they do not consider the metric of V, resulting in the irregularity issue. At this point, we would like to remark that the irregularity introduced by a morphological operator can be quantitatively measured using the so-called irregularity index [9].…”