eess.IVMar 31, 2026

Polyhedral Unmixing: Bridging Semantic Segmentation with Hyperspectral Unmixing via Polyhedral-Cone Partitioning

Authors: Antoine Bottenmuller, Etienne Decencière, Petr Dokládal

Organizations: CMM, PSL, STIM · Mines Paris, PSL University, Centre for Mathematical Morphology (CMM), 77300 Fontainebleau, France

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

Abstract

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

Figures & tables

Figure 1: Bridge between semantic segmentation and hyperspectral unmixing. From the unmixing data, the dominant-material classification map is directly obtained by taking the arg⁡max⁡\arg\max over the abundances for each pixel. In the opposite way, a model is needed.
Figure 2: Illustration, in the spectral space, of the standard pipeline for blind linear umixing. 2D cross-sections of a 3D space.
Figure 3: Processing chain to determine a polyhedral-cone partition of the spectral space. From left to right: observed data (A.) are pre-processed (B.) and classified into mm classes (C.). Separation hyperplanes are then computed (D.), inducing k≥mk\geq m polyhedral regions. Top row: image domain; bottom row: spectral space representation (3D view + 2D projection plane P(x⃗,y⃗)P(\vec{x},\vec{y}) below, which allows a better visualization of the data and the three polyhedral regions). Each RGB color is associated with one class.
Figure 4: Processing chain for computing an abundance estimate. From left to right: given the spectral partition (D.), signed distances to each cone are computed (E.); a change of basis is then applied in the distance space (F.), and the resulting vectors are projected onto the probability simplex (G.). Top: image ; middle: spectral space ; bottom: distance space . The 2D RGB projection plane P(x⃗,y⃗)P(\vec{x},\vec{y}) shows: polyhedral regions (D.), negative distances in each region (E.), transformed distance vectors (F.) and abundance vectors (G.). Each RGB color is associated with one class. Yellow, magenta and cyan represent separation planes between R-G, R-B and G-B pairs of classes, respectively.
Endm.Endmember Spectral Angle Distance (SAD) (×10−2)(\times 10^{-2})
 VCASISALNMF-QMVEndnetMiSiCNetADMMNetEDAAUNMambaProposed
#1 Soil13.26 ±\pm 9.332.05 ±\pm 0.08.91 ±\pm 0.01.42 ±\pm 0.31.91 ±\pm 0.22.24 ±\pm 0.02.87 ±\pm 0.06.13 ±\pm 0.91.85 ±\pm 0.2
#2 Tree4.39 ±\pm 2.72.96 ±\pm 0.08.44 ±\pm 0.03.48 ±\pm 0.35.76 ±\pm 0.05.32 ±\pm 0.03.45 ±\pm 0.04.48 ±\pm 0.43.41 ±\pm 0.1
#3 Water11.96 ±\pm 0.76.89 ±\pm 0.018.87 ±\pm 0.04.06 ±\pm 0.39.45 ±\pm 0.112.48 ±\pm 0.02.28 ±\pm 0.03.38 ±\pm 0.12.93 ±\pm 0.2
Avg.9.87 ±\pm 4.113.97 ±\pm 0.012.07 ±\pm 0.02.99 ±\pm 0.25.71 ±\pm 0.16.68 ±\pm 0.02.87 ±\pm 0.04.66 ±\pm 0.22.73 ±\pm 0.1
Table 1: The SADs and RMSEs (mean ±\pm std over 10 runs per experiment) on the Samson dataset. The best results are shown in bold; the second best are underlined.
Endm.Endmember Spectral Angle Distance (SAD) (×10−2)(\times 10^{-2})
 VCASISALNMF-QMVEndnetMiSiCNetADMMNetEDAAUNMambaProposed
#1 Soil37.39 ±\pm 9.616.34 ±\pm 0.021.65 ±\pm 0.021.54 ±\pm 2.77.21 ±\pm 0.176.62 ±\pm 0.05.71 ±\pm 0.014.68 ±\pm 0.11.97 ±\pm 0.1
#2 Tree23.52 ±\pm 4.13.64 ±\pm 0.025.25 ±\pm 0.010.59 ±\pm 3.12.18 ±\pm 0.019.36 ±\pm 0.06.81 ±\pm 0.07.63 ±\pm 0.02.05 ±\pm 0.1
#3 Water29.08 ±\pm 6.69.71 ±\pm 0.025.37 ±\pm 0.07.17 ±\pm 1.77.14 ±\pm 0.024.06 ±\pm 0.04.44 ±\pm 0.03.55 ±\pm 0.16.45 ±\pm 0.2
#4 Road52.93 ±\pm 8.750.21 ±\pm 0.079.69 ±\pm 0.09.41 ±\pm 0.234.14 ±\pm 0.031.33 ±\pm 0.02.82 ±\pm 0.010.66 ±\pm 0.65.35 ±\pm 0.4
Avg.35.73 ±\pm 2.719.97 ±\pm 0.037.99 ±\pm 0.012.18 ±\pm 1.912.67 ±\pm 0.037.84 ±\pm 0.04.94 ±\pm 0.09.13 ±\pm 0.23.96 ±\pm 0.1
Table 2: The SADs and RMSEs (mean ±\pm std over 10 runs per experiment) on the Jasper Ridge dataset. The best results are shown in bold; the second best are underlined.
Endm.Endmember Spectral Angle Distance (SAD) (×10−2)(\times 10^{-2})
 VCASISALNMF-QMVEndnetMiSiCNetADMMNetEDAAUNMambaProposed
#1 Asph.54.30 ±\pm >> 989.21 ±\pm 0.042.81 ±\pm 0.06.44 ±\pm 0.912.98 ±\pm 0.0>> 99 ±\pm 0.04.34 ±\pm 0.018.44 ±\pm 0.23.50 ±\pm 0.2
#2 Grass46.65 ±\pm 8.431.42 ±\pm 0.037.27 ±\pm 0.09.46 ±\pm 1.118.36 ±\pm 0.036.17 ±\pm 0.019.74 ±\pm 0.016.27 ±\pm 0.35.08 ±\pm 0.1
#3 Tree34.54 ±\pm 3.322.06 ±\pm 0.039.96 ±\pm 0.013.15 ±\pm 0.616.62 ±\pm 0.026.35 ±\pm 0.019.73 ±\pm 0.08.50 ±\pm 0.07.48 ±\pm 0.2
#4 Roof71.74 ±\pm >> 929.53 ±\pm 0.085.19 ±\pm 0.036.90 ±\pm 2.523.53 ±\pm 0.040.85 ±\pm 0.017.79 ±\pm 0.06.92 ±\pm 0.16.59 ±\pm 0.6
#5 Metal64.28 ±\pm 8.744.79 ±\pm 0.049.18 ±\pm 0.011.11 ±\pm 1.215.61 ±\pm 0.045.09 ±\pm 0.044.70 ±\pm 0.022.66 ±\pm 0.211.58 ±\pm 0.8
Table 3: The SADs and RMSEs (mean ±\pm std over 10 runs per experiment) on the Urban-6 dataset. The best results are shown in bold; the second best are underlined.
Data MethodGround Truthk-MeansGMMSpectral Cl.SDSTEGFSC
Samson\begin{array}[]{c}\text{Acc.: }100.00\\ \text{SAD: }2.79\\ \text{RMSE: }3.43\end{array}\begin{array}[]{c}\text{Acc.: }97.28\\ \text{SAD: }2.90\\ \text{RMSE: }3.63\end{array}\begin{array}[]{c}\text{Acc.: }93.03\\ \text{SAD: }2.73\\ \text{RMSE: }3.52\end{array}\begin{array}[]{c}\text{Acc.: }96.03\\ \text{SAD: }2.87\\ \text{RMSE: }3.56\end{array}\begin{array}[]{c}\text{Acc.: }96.04\\ \text{SAD: }2.94\\ \text{RMSE: }3.68\end{array}\begin{array}[]{c}\text{Acc.: }94.54\\ \text{SAD: }2.77\\ \text{RMSE: }3.55\end{array}
Jasper Ridge\begin{array}[]{c}\text{Acc.: }100.00\\ \text{SAD: }4.41\\ \text{RMSE: }7.87\end{array}\begin{array}[]{c}\text{Acc.: }72.73\\ \text{SAD: }10.52\\ \text{RMSE: }16.42\end{array}\begin{array}[]{c}\text{Acc.: }75.39\\ \text{SAD: }8.37\\ \text{RMSE: }12.88\end{array}\begin{array}[]{c}\text{Acc.: }77.87\\ \text{SAD: }6.75\\ \text{RMSE: }12.52\end{array}\begin{array}[]{c}\text{Acc.: }83.76\\ \text{SAD: }5.85\\ \text{RMSE: }8.28\end{array}\begin{array}[]{c}\text{Acc.: }88.56\\ \text{SAD: }5.55\\ \text{RMSE: }8.08\end{array}
Urban-6\begin{array}[]{c}\text{Acc.: }100.00\\ \text{SAD: }6.82\\ \text{RMSE: }11.07\end{array}\begin{array}[]{c}\text{Acc.: }68.09\\ \text{SAD: }18.24\\ \text{RMSE: }19.39\end{array}\begin{array}[]{c}\text{Acc.: }71.73\\ \text{SAD: }15.55\\ \text{RMSE: }18.70\end{array}\begin{array}[]{c}\text{Acc.: }76.65\\ \text{SAD: }23.12\\ \text{RMSE: }17.82\end{array}\begin{array}[]{c}\text{Acc.: }67.89\\ \text{SAD: }17.87\\ \text{RMSE: }19.72\end{array}\begin{array}[]{c}\text{Acc.: }77.37\\ \text{SAD: }7.57\\ \text{RMSE: }12.08\end{array}
Table 4: Evaluation of the method on different clustering algorithms (Avg. values, ×10−2\times 10^{-2} ).
Table 5: Evaluation of the method robustness to uniform random noise (Avg. values, ×10−2\times 10^{-2} ).
Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: The Samson dataset ( 5(a) ) with its ground-truth (GT) abundance map ( 5(b) ) and endmembers ( 5(c) ). In subfigures ( 5(b) ) and ( 5(c) ), red is associated with the soil class, green with trees, and blue with water.
Figure 6: The Jasper Ridge dataset ( 6(a) ) with its ground-truth (GT) abundance map ( 6(b) ) and endmembers ( 6(c) ). In subfigures ( 6(b) ) and ( 6(c) ), red is associated with the soil class, green with trees, blue with water and black with roads.
Figure 7: The Urban-6 dataset ( 7(a) ) with its ground-truth (GT) abundance map ( 7(b) ) and endmembers ( 7(c) ). In subfigures ( 7(b) ) and ( 7(c) ), red is associated with asphalt, green with grass, blue with trees, yellow with roofs, magenta with metal and cyan with dirt.
Figure 8: Visualizations of the endmember estimates on Samson given by the nine tested blind unmixing algorithms (columns). The ground-truth endmembers of the three materials (rows) are provided in the associated cells as light-gray line plots.
Figure 9: Visualizations of the material abundance estimates on Samson given by the nine tested blind unmixing algorithms (columns). The ground-truth abundance maps of the three materials (rows) are provided in the left column as references.
Figure 10: Visualizations of the endmember estimates on Jasper Ridge given by the nine tested blind unmixing algorithms (columns). The ground-truth endmembers of the four materials (rows) are provided in the associated cells as light-gray line plots.
Figure 11: Visualizations of the material abundance estimates on Jasper Ridge given by the nine tested blind unmixing algorithms (columns). The ground-truth abundance maps of the four materials (rows) are provided in the left column as references.
Figure 14: Visualizations of the classification maps given by the five tested clustering algorithms (columns) over the three hyperspectral datasets (rows). The ground-truth classification maps are provided in the left column as references.

Explore similar work

CardsList

  1. Interpretable Hyperspectral Unmixing Framework with Fixed Endmember Prior and Structured Residual Refinement

    Sep 8, 2026Ziyi Guan, Jianping Zhang, Qian LiuHyperspectral Imaging

    School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China

  2. Agentic Multimodal Models for Environmental Hyperspectral Unmixing

    Sep 1, 2026Michał Cholewa, Luca Ciampi, Nicola Messina +2Hyperspectral ImagingMultimodal Agents

    IITiS-PAS, Gliwice, Poland · ISTI-CNR, Pisa, Italy

  3. Hyperspectral Unmixing Hierarchies

    Apr 18, 2026Joseph L. Garrett, P. S. Vishnu, Pauliina Salmi +4Non-Negative Matrix FactorizationRemote Sensing

    Department of Engineering Cybernetics, Norwegian University of Science and Technology · Department of Biology, Norwegian University of Science and Technology · Department of Automatic Control and Systems Engineering, University Politehnica Bucharest +1