Looking into a Pixel by Nonlinear Unmixing -- A Generative Approach

The Linear Mixing Model (LMM) dominates spectral unmixing for its simplicity, but fails under multiple scattering; existing nonlinear models compensate by applying a fixed regime uniformly across entire scenes. We propose Physics-Guided Regime Unmixing (PGRU), which estimates a pixel-wise scalar $ξ_i \in [0,1]$ from observable physical features to activate nonlinear mixing only where justified. Residuals from the Generalized Bilinear Model (GBM), the Post-Nonlinear Mixing Model (PPNM), and Hapke are combined via learned attention, yielding interpretable regime maps. Experiments on Samson, Jasper Ridge, and Urban show consistent improvements over baselines, with physical coherence $ρ> 0.90$.

Unmixing reveals the spatial distribution and spectral details of different constituents, called endmembers, in a hyperspectral image. Because unmixing has limited ground truth requirements, can accommodate mixed pixels, and is closely tied to light propagation, it is a uniquely powerful tool for analyzing hyperspectral images. However, spectral variability inhibits unmixing performance, the proper way to determine the number of endmembers is ambiguous, and the clarity of the endmembers degrades as more are included. Hierarchical structure is a possible solution to all three problems. Here, hierarchical unmixing is defined by imposing a hierarchical abundance sum constraint on Deep Nonnegative Matrix Factorization. Binary Linear Unmixing Tactile Hierarchies (BLUTHs) solve the hierarchical unmixing problem with a simple network architecture. Sparsity modulation unmixing growth tailors the topology of a BLUTH to each scene. The structure imposed by BLUTHs allows endmembers with varying levels of spectral contrast to be revealed, mitigating the challenge of spectral variability. The performance of BLUTHs exceeds state-of-the-art unmixing algorithms on laboratory scenes, particularly with regard to abundance estimation, while their performance remains competitive on remote sensing scenes. In addition, ocean color unmixing by BLUTHs is demonstrated on hyperspectral scenes from the HYPSO and PACE satellites.

Hyperspectral unmixing (HU) is crucial for analyzing hyperspectral imagery, yet achieving accurate unmixing remains challenging. While traditional methods struggle to effectively model complex spectral-spatial features, deep learning approaches often lack physical interpretability. Unrolling-based methods, despite offering network interpretability, inadequately exploit spectral-spatial information and incur high memory costs and numerical precision issues during backpropagation. To address these limitations, we propose DEQ-Unmix, which reformulates abundance estimation as a deep equilibrium model, enabling efficient constant-memory training via implicit differentiation. It replaces the gradient operator of the data reconstruction term with a trainable convolutional network to capture spectral-spatial information. By leveraging implicit differentiation, DEQ-Unmix enables efficient and constant-memory backpropagation. Experiments on synthetic and two real-world datasets demonstrate that DEQ-Unmix achieves superior unmixing performance while maintaining constant memory cost.