Morphology on Categorical Distributions

Mathematical morphology (MM) is an indispensable tool for post-processing. Several extensions of MM to categorical images, such as multi-class segmentations, have been proposed. However, none provide satisfactory definitions for morphology on probabilistic representations of categorical images. The categorical distribution is a natural choice for representing uncertainty about categorical images. Extending MM to categorical distributions is problematic because categories are inherently unordered. Without ranking categories, we cannot use the standard framework based on supremum and infimum. Ranking categories is impractical and problematic. Instead, we consider the probabilistic representation and operations that emphasize a single category. In this work, we review and compare previous approaches. We propose two approaches for morphology on categorical distributions: operating on Dirichlet distributions over the parameters of the distributions and operating directly on the distributions. We propose a “protected” variant of the latter and demonstrate the proposed approaches by fixing misclassifications and modeling annotator bias.