Spectral unmixing is one of the main research topics in hyperspectral imaging. It can be formulated as a source separation problem, whose goal is to recover the spectral signatures of the materials present in the observed scene (called endmembers) as well as their relative proportions (called fractional abundances), and this for every pixel in the image. A linear mixture model (LMM) is often used for its simplicity and ease of use, but it implicitly assumes that a single spectrum can be completely representative of a material. However, in many scenarios, this assumption does not hold, since many factors, such as illumination conditions and intrinsic variability of the endmembers, induce modifications on the spectral signatures of the materials. In this paper, we propose an algorithm to unmix hyperspectral data using a recently proposed extended LMM. The proposed approach allows a pixelwise spatially coherent local variation of the endmembers, leading to scaled versions of reference endmembers. We also show that the classic nonnegative least squares, as well as other approaches to tackle spectral variability can be interpreted in the framework of this model. The results of the proposed algorithm on two different synthetic datasets, including one simulating the effect of topography on the measured reflectance through physical modelling, and on two real data sets, show that the proposed technique outperforms other methods aimed at addressing spectral variability, and can provide an accurate estimation of endmember variability along the scene because of the scaling factors estimation.

The Linear Mixing Model is often used to perform Hyperspectral Unmixing because of its simplicity, but it assumes that a single spectral signature can be completely representative of an endmember. However, in many scenarios, this assumption does not hold since many factors such as illumination conditions and intrinsic variability of the endmembers have consequences on the spectral signatures of the materials. In this paper, we propose a simple yet flexible algorithm to unmix hyperspectral data using a recently proposed Extended Linear Mixing Model. This model allows a pixelwise variation of the endmembers, which leads to consider scaled versions of reference endmember spectra. The results on synthetic data show that the proposed technique outperforms other methods aimed at tackling spectral variability, and provides an accurate estimation of endmember variability along the observed scene thanks to the scaling factors estimation, provided the abundance of the corresponding material is sufficient.

The spectral signatures of the materials contained in hyperspectral images (HI), also called endmembers (EM), can be significantly affected by variations in atmospheric, illumination or environmental conditions typically occurring within an HI. Traditional spectral unmixing (SU) algorithms neglect the spectral variability of the endmembers, what propagates significant mismodeling errors throughout the whole unmixing process and compromises the quality of its results. Therefore, large efforts have been recently dedicated to mitigate the effects of spectral variability in SU. This resulted in the development of algorithms that incorporate different strategies to allow the EMs to vary within an HI, using, for instance, sets of spectral signatures known a priori, Bayesian, parametric, or local EM models. Each of these approaches has different characteristics and underlying motivations. This paper presents a comprehensive literature review contextualizing both classic and recent approaches to solve this problem. We give a detailed evaluation of the sources of spectral variability and their effect in HI spectra. Furthermore, we propose a new taxonomy that organizes existing work according to a practitioner's point of view, based on the necessary amount of supervision and on the computational cost they require. We also review methods used to construct spectral libraries (which are required by many SU techniques) based on the observed HI, as well as algorithms for library augmentation and reduction. Finally, we conclude the paper with some discussions and an outline of possible future directions for the field.