Aitchison Geometry on the Simplex for Uncertainty Quantification in Bayesian Hyperspectral Image Unmixing

Most algorithms for hyperspectral image unmixing produce point estimates of fractional abundances of the materials to be separated. However, in the absence of reliable ground truth, the ability to perform abundance uncertainty quantification (UQ) should be an important feature of algorithms, e.g. to evaluate how hard the unmixing problem is and how much the results should be trusted. The usual modeling assumptions in Bayesian models for unmixing rely heavily on the Euclidean geometry of the simplex and typically disregard spatial information. In addition, to our knowledge, abundance UQ is close to nonexistent. In this paper, we propose to leverage Aitchinson geometry from the compositional data analysis literature to provide practitioners with alternative tools for modeling prior abundance distributions. In particular we show how to design simplex-valued Gaussian Process priors using this geometry. Then we link Aitchinson geometry to constrained sampling algorithms in the literature, and propose UQ diagnostics that comply with the constraints on abundance vectors. We illustrate these concepts on real and simulated data.