Hyperspectral Unmixing Under Endmember Variability: A Variational Inference Framework

TL;DR

This work addresses hyperspectral unmixing under endmember variability ($HU\text{-}EV$) and outliers by formulating a marginalized maximum likelihood ($MML$) objective and solving it with a lightweight variational inference ($VI$) framework. It introduces a patch-wise endmember model, where endmembers are static within each image patch, and derives the HELEN algorithm to perform inference under both Beta and Gaussian priors for endmembers. The proposed Beta- and Gauss-variants of HELEN deliver accurate, outlier-robust endmember and abundance estimates with efficient, sampling-free updates, and demonstrate superior performance against multiple baselines on synthetic, semi-real, and real HSIs. The approach balances model flexibility and computational efficiency, enabling scalable HU-EV analysis and providing spatially resolved endmember maps useful for subsequent hyperspectral interpretation.

Abstract

This work proposes a variational inference (VI) framework for hyperspectral unmixing in the presence of endmember variability (HU-EV). An EV-accounted noisy linear mixture model (LMM) is considered, and the presence of outliers is also incorporated into the model. Following the marginalized maximum likelihood (MML) principle, a VI algorithmic structure is designed for probabilistic inference for HU-EV. Specifically, a patch-wise static endmember assumption is employed to exploit spatial smoothness and to try to overcome the ill-posed nature of the HU-EV problem. The design facilitates lightweight, continuous optimization-based updates under a variety of endmember priors. Some of the priors, such as the Beta prior, were previously used under computationally heavy, sampling-based probabilistic HU-EV methods. The effectiveness of the proposed framework is demonstrated through synthetic, semi-real, and real-data experiments.

Figures (12)

• Figure 1: Illustration of EV in the Samson dataset. The spectra of the pixels that only contain "soil" are shown on the right.
• Figure 2: Segmentation of an image into blocks. $|\mathcal{I}_5|$ denotes the number of pixels of 5-th block.
• Figure 3: Illustration of spatial smoothness of the endmembers in the presence of EV. The two manually cropped blocks (with size $5\times 5)$ for the water in the Samson and Moffett images are shown in the left column. The spectra are shown in the right column.
• Figure 4: (Synthetic data experiment) The ground-truth and the algorithm-estimated ${\bm A}_t$ in three selected pixels located at $(10,10), (30,30)$ and $(50,50)$.
• Figure 5: (Synthetic data experiment) The ground-truth and the HELEN-estimated ${\bm A}_t$ in three selected pixels located at $(10,10), (30,30)$ and $(50,50)$. The MSEs of HELEN that use Gaussian and Beta prior are marked in green and red. The runtime is also shown.

Theorems & Definitions (2)

• Remark 1
• Remark 2