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

2026-03-25

AI Summary

This work addresses the lack of uncertainty quantification in existing hyperspectral unmixing methods and the common oversight of the intrinsic geometry of the abundance simplex in most Bayesian models, which typically rely on Euclidean assumptions and neglect spatial structure. To overcome these limitations, we introduce the Aitchison geometry into hyperspectral unmixing for the first time, formulating a Gaussian process prior directly on the simplex. By integrating this prior with constrained Markov chain Monte Carlo sampling, our approach enforces the non-negativity and sum-to-one constraints inherent to abundance fractions while enabling principled Bayesian uncertainty quantification. Experiments on both real and synthetic datasets demonstrate that the proposed method effectively characterizes abundance uncertainty, substantially enhancing the reliability and interpretability of unmixing results.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Data transparency and provenance

Abstract

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.

Problem

• hyperspectral image unmixing
• uncertainty quantification
• abundance estimation
• simplex geometry
• Bayesian modeling

Innovation

• Aitchison geometry
• uncertainty quantification
• Bayesian hyperspectral unmixing
• compositional data
• Gaussian process on simplex