Morphology on categorical distributions

TL;DR

This paper develops morphological operations tailored for categorical distributions in multi-class segmentation, addressing the challenge of applying morphology to uncertain, probabilistic data.

Contribution

It introduces a set of requirements and novel operators for morphology on categorical distributions, bridging classic morphology with probabilistic uncertainty.

Findings

Operators respect probabilistic constraints

Applied to modeling annotator bias in brain tumor segmentation

Used for vesicle instance segmentation with improved results

Abstract

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.