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

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 propose a robustness step to estimate the statistical parameters of these heavy-tailed elliptical distributions. In addition, an edge bias, which contains prior boundary information, is incorporated into both the data fidelity term and the perimeter term of the Mumford-Shah functional. Numerical experiments show that, to the best of our knowledge, our method achieves the highest overall accuracy among fully unsupervised methods on three of the four standard benchmarks (Indian Pines, Kennedy Space Center, Pavia University and Salinas) evaluated on the full scenes. It also improves upon our previous Mumford-Shah based model.