@article {2086,
	title = {Robust supervised method for nonlinear spectral unmixing accounting for endmember variability},
	journal = {IEEE Transactions on Geoscience and Remote Sensing},
	volume = {59},
	year = {2021},
	pages = {7434-7448},
	abstract = {Due to the complex interaction of light with mixed materials, reflectance spectra are highly nonlinearly related to the pure material endmember spectra, making it hard to estimate the fractional abundances of the materials. Changing illumination conditions and cross-sensor situations cause spectral variability, further complicating the unmixing procedure. In this work, we propose a supervised approach to unmix mineral powder mixtures, containing endmember variability.

First, the abundances are estimated by calculating the geodesic distances between the mixtures and the endmembers. It is argued and experimentally validated that the estimated geodesic abundances, although not correct, are invariant to external spectral variability.  Then, a supervised approach is applied to learn a mapping from the obtained geodesic abundances to spectra that follow a linear model. To learn this mapping, ground truth fractional abundances of a number of training samples are required. 
Although any nonlinear regression method can be used to learn the mapping, Gaussian process is found to be suitable when a limited number of training samples are available. The trained model is applicable to all manifolds that contain a similar nonlinear behavior as the trained manifold, e.g. when the same mixtures are measured by another sensor. Using the output spectra, a simple inversion of the linear model reveals the true abundances.

Experiments are conducted on simulated and real mineral mixtures. In particular, we developed data sets of homogeneously mixed mineral powder mixtures, acquired by 2 different sensors, an Agrispec spectrometer and a snapscan shortwave infrared hyperspectral camera, under strictly controlled experimental settings. The proposed approach is compared to other supervised approaches and nonlinear mixture models.  },
	doi = {https://doi.org/10.1109/TGRS.2020.3031012},
	author = {Bikram Koirala and Zohreh Zahiri and Alfredo Lamberti and Paul Scheunders}
}
