用于高光谱解混的乘法变量误差模型
Multiplicative Errors-in-Variables Models for Hyperspectral Unmixing
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中文总结 AI 辅助
针对高光谱解混中加性误差模型置信区间覆盖缺陷，提出分层贝叶斯乘法EIV模型，其置信区间随信号幅度比例缩放，在多数据集上覆盖效果优于标准模型且重构误差相当或更优。
中文摘要 AI 辅助
从高光谱图像中估计端元光谱及其对应丰度是遥感领域的一个基础逆问题。标准线性混合模型依赖加性同方差误差，该模型错误设定了光学测量中固有的均值-方差关系，这类测量的噪声强度可能与信号幅度成正比，导致置信区间的覆盖效果较差。为解决这些局限性，我们提出了一种分层贝叶斯乘法变量误差（EIV）模型。受光传播物理特性的启发，我们的公式通过多元对数正态设定捕捉依赖于信号的噪声，该设定自然地适应了光谱波段之间的相关性。EIV框架将端元特征视为随机变量，并允许按类别进行方差缩放，从而产生灵活且基于物理的数据生成过程。我们证明，乘法模型下丰度向量的最大似然估计（MLE）具有恒定的渐近多元变异系数，这意味着校准良好的置信区间应与信号幅度成比例缩放。相比之下，当真实噪声为乘法型时，加性模型会产生宽度恒定的区间，导致低丰度时覆盖过度，高丰度时覆盖不足。通过对模型参数的后验分布，可实现完整分层模型的实际实施与推断。在多个真实和模拟数据集上的实验表明，与标准加性模型相比，所提模型导出的置信区间在各丰度水平上实现了更好的覆盖，区间宽度与信号幅度成比例缩放，同时达到了相当或更优的信号重构误差。
英文摘要
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.
