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Remote sensing hyperspectral sensors, with high spectral resolution, allow precise classification of endmembers present in imaged areas. These sensors have a limited spatial resolution, which results in mixed pixels. The mixture is usually assumed to be linear and blind linear spectral unmixing (LSU) methods are used to unmix all observed pixel spectra. Most blind LSU approaches assume that each endmember is represented by a unique spectrum in all image pixels. But, in many practical applications, this assumption is not valid and more complex models are needed to describe other phenomena, e.g. when each endmember needs to be represented by slightly different spectra in all image pixels. This spectral variability must be handled by replacing the concept of endmembers by classes of endmembers, to avoid errors when processing the considered data. In this paper, a new linear mixing model is firstly introduced in order to handle the spectral variability. In the proposed model, the endmember spectra are additively tuned. Then, an algorithm, based on pixel-by-pixel nonnegative matrix factorization, is proposed for unmixing the considered data. That algorithm, which derives, for each class of endmembers, slightly different estimated spectra in all pixels, optimizes a cost function and uses additional constraints that are related to the introduced linear mixing model. Experiments, based on realistic synthetic data, are conducted to evaluate the performance of the proposed algorithm. The obtained results are compared to those of three approaches from the literature. These test results show that the proposed approach outperforms all other tested methods.
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