Robust blind unmixing: A geometric approach to overcoming basis variation

2026-10-02

AI Summary

This study addresses the challenge of basis variability in signal unmixing caused by noise and physical processes, where traditional parametric models suffer from limited generalizability due to domain-specific assumptions. To overcome this, we propose a novel geometric unmixing framework grounded in metric spaces that eliminates parametric constraints by formulating data generation as a metric space problem. The approach leverages optimal transport distances to precisely capture the geometric relationships underlying basis transformations, driving optimization through within-class variance minimization while achieving computational efficiency for one-dimensional cases. Experimental evaluations on Gaussian mixture models, simulated powder X-ray diffraction, and laboratory hyperspectral imaging datasets demonstrate that the proposed method significantly enhances both unmixing accuracy and generalizability in complex scenarios.

Abstract

Signal separation problems are common in science. A prominent example of this occurs during the use of diffraction or spectroscopy to identify the individual components of a mixture by measuring it. In the simplest case, the measured signal is a linear combination of basis patterns corresponding to the constituent parts. The unmixing problem is to infer all or some of these basis patterns and abundances of components from measurements of distinct mixtures. One of the core challenges of this task is the variation of the basis from mixture to mixture due to noise and the exact physics of the measurement process. This is usually addressed with tailored model-based and parametric methods that are then limited in use to specific application domains by the nature of the assumptions made. We propose a novel geometric approach to unmixing problems which views the generation of data during measurement through a metric space lens, thereby shifting the focus from parametrised models to a general relationship between basis transformations and the corresponding geometry. We take advantage of the optimal transport distances to capture commonly occurring basis variations, and use minimisation of in-class variance of candidate solutions to drive the optimisation. We pay special attention to the one-dimensional case due to its practical importance and availability of efficient distance and transport map routines. The effectiveness of our approach is demonstrated on a range of unmixing tasks using random Gaussian mixture models, simulated powder X-ray diffraction, and laboratory hyperspectral imaging datasets.

Problem

• blind unmixing
• signal separation
• basis variation
• robust unmixing

Innovation

• Blind unmixing
• Geometric approach
• Optimal transport
• Basis variation
• Metric space