Multiplicative Errors-in-Variables Models for Hyperspectral Unmixing

2026-07-30

AI Summary

This study addresses the limitation of conventional hyperspectral unmixing models, which assume additive homoscedastic noise and fail to capture the physically realistic scenario where noise intensity scales with signal magnitude, often resulting in undercovered confidence intervals. To overcome this, the authors propose the first hierarchical Bayesian error-in-variables model incorporating a multiplicative error structure, modeling signal-dependent noise via a multivariate log-normal distribution and treating endmembers as random variables to better align with optical measurement physics. The approach explicitly accounts for inter-band noise correlations and reveals that abundance estimators exhibit a constant asymptotic coefficient of variation. Experiments on both synthetic and real datasets demonstrate that the proposed model yields more accurate confidence interval coverage, produces interval widths that appropriately scale with signal strength, and achieves reconstruction errors comparable to or lower than existing methods.

Abstract

Estimating endmember spectra and their corresponding abundances from hyperspectral images is a fundamental inverse problem in remote sensing. The standard linear mixing model relies on additive homoscedastic errors, which misspecifies the mean-variance relationship inherent to optical measurements for which the noise intensity might be proportional to the signal magnitude. This leads to confidence intervals with poor coverage. To address these limitations, we propose a hierarchical Bayesian multiplicative Errors-in-Variables (EIV) model. Motivated by the physics of light propagation, our formulation captures signal-dependent noise through a multivariate log-normal specification that naturally accommodates dependence across spectral bands. The EIV framework treats endmember signatures as random and allows for class-specific variance scaling, yielding a flexible and physically grounded data-generating process. We establish that the MLE of the abundance vector under the multiplicative model has a constant asymptotic multivariate coefficient of variation, implying that well-calibrated confidence intervals should scale proportionally with signal magnitude. The additive model, by contrast, produces intervals of constant width regardless of signal magnitude, leading to overcoverage at low abundance and undercoverage at high abundance when the true noise is multiplicative. Practical implementation and inference for the full hierarchical model is obtained from posterior distributions over the model parameters. Experiments on several real and simulated datasets demonstrate that confidence intervals derived from the proposed model achieve improved coverage across abundance levels compared to those from a standard additive model, with widths that scale proportionally with signal magnitude, while achieving comparable or superior signal reconstruction error.

Problem

• hyperspectral unmixing
• multiplicative noise
• errors-in-variables
• signal-dependent noise
• confidence interval coverage

Innovation

• Multiplicative Errors-in-Variables
• Hyperspectral Unmixing
• Signal-dependent noise
• Hierarchical Bayesian model
• Confidence interval calibration