Morphology on categorical distributions

The categorical distribution is a natural representation of uncertainty in multi-class segmentations. In the two-class case the categorical distribution reduces to the Bernoulli distribution, for which grayscale morphology provides a range of useful operations. In the general case, applying morphological operations on uncertain multi-class segmentations is not straightforward as an image of categorical distributions is not a complete lattice. Although morphology on color images has received wide attention, this is not so for color-coded or categorical images and even less so for images of categorical distributions. In this work, we establish a set of requirements for morphology on categorical distributions by combining classic morphology with a probabilistic view. We then define operators respecting these requirements, introduce protected operations on categorical distributions and illustrate the utility of these operators on two example tasks: modeling annotator bias in brain tumor segmentations and segmenting vesicle instances from the predictions of a multi-class U-Net.