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

Abstract

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.

Irregularity Index for Vector-Valued Morphological Operators

“…Our motivation is to formulate measures to study the irregularity implied by a morphological operator on vector-valued images. We believe this is the first work proposing a framework based on the Wasserstein metric to score the irregularity effect considering pairs of input/output images besides our conference paper [30]. Indeed, this paper extends [30] by presenting more efficient estimators for the irregularity measure using local windows and entropic regularized optimal transport methods [19,26].…”

“…We believe this is the first work proposing a framework based on the Wasserstein metric to score the irregularity effect considering pairs of input/output images besides our conference paper [30]. Indeed, this paper extends [30] by presenting more efficient estimators for the irregularity measure using local windows and entropic regularized optimal transport methods [19,26]. This paper also presents extensive computational experiments to show the effectiveness of the proposed irregularity measures.…”

“…Remark 1 We would like to point out that we previously used the geometric mean to aggregate the quotient of W p (I, J|W ) by D p (I, J|W ) [30]. However, the geometric mean does not scale well for moderate and large size images because 0 < W p (I,J|W )…”