From Proper Scoring Rules and Edge Biases to Mumford-Shah Models: Robust Unsupervised Hyperspectral Image Segmentation

Fabian Conrad, Benjamin Berkels

Hyperspectral images provide a rich representation of the underlying spectrum for each pixel, allowing for a pixel-wise classification/segmentation into different classes. As the availability of labeled hyperspectral images is limited and the acquisition of labeled training data is time-consuming, unsupervised segmentation methods are important in hyperspectral image segmentation. Spectral variability, mixed pixels, noise as well as high correlation between neighboring spectral bands make this task challenging. We present a feature-agnostic, unsupervised hyperspectral segmentation framework based on the piecewise-constant Mumford-Shah functional. Our framework turns a strictly proper scoring rule and a family of parametric probability distributions into an indicator function for the Mumford-Shah functional, and we naturally generalize some previous Mumford-Shah based models for image segmentation this way. Our focus will be on heavy-tailed elliptical distributions, the negative log-likelihood score and the score based on the beta-divergence to deal with the heavy outliers in hyperspectral imaging. Using the interplay between the calculus of variations and statistics, we also…