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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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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