MiSiSUn: Minimum Simplex Semisupervised Unmixing

Semantic segmentation and hyperspectral unmixing are two central problems in spectral image analysis. The former assigns each pixel a discrete label corresponding to its material class, whereas the latter estimates pure material spectra, called endmembers, and, for each pixel, a vector representing material abundances in the observed scene. Despite their complementarity, these two problems are usually addressed independently. This paper aims to bridge these two lines of work by formally showing that, under the linear mixing model, pixel classification by dominant materials induces polyhedral-cone regions in the spectral space. We leverage this fundamental property to propose a direct segmentation-to-unmixing pipeline that performs blind hyperspectral unmixing from any semantic segmentation by constructing a polyhedral-cone partition of the space that best fits the labeled pixels. Signed distances from pixels to the estimated regions are then computed, linearly transformed via a change of basis in the distance space, and projected onto the probability simplex, yielding an initial abundance estimate. This estimate is used to extract endmembers and recover final abundances via matrix pseudo-inversion. Because the segmentation method can be freely chosen, the user gains explicit control over the unmixing process, while the rest of the pipeline remains essentially deterministic and lightweight. Beyond improving interpretability, experiments on three real datasets demonstrate the effectiveness of the proposed approach when associated with appropriate clustering algorithms, and show consistent improvements over recent deep and non-deep state-of-the-art methods. The code is available at: https://github.com/antoine-bottenmuller/polyhedral-unmixing

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.

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 $ρ> 0.90$.