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The normal compositional model (NCM) has been extensively used in hyperspectral unmixing. However, previous research has mostly focused on the estimation of endmembers and/or their variability, based on the assumption that the pixels are independent random variables. In this paper, we show that this assumption does not hold if all the pixels are generated by a fixed endmember set. This introduces another concept, endmember uncertainty, which is related to whether the pixels fit into the endmember simplex. To further develop this idea, we derive the NCM from the ground up without the pixel independence assumption, along with: 1) using different noise levels at different wavelengths and 2) using a spatial and sparsity promoting prior for the abundances. The resulting new formulation is called the spatial compositional model (SCM) to better differentiate it from the NCM. The SCM maximum a posteriori objective leads to an optimization problem featuring Noise-weighted least-squares minimization for unmixing. The problem is solved by projected gradient descent, resulting in an algorithm that estimates endmembers, abundances, noise variances, and endmember uncertainty simultaneously. We compared SCM with current state-of-the-art algorithms on synthetic and real images. The results show that SCM can, in the main, provide more accurate endmembers and abundances. Moreover, the estimated uncertainty can serve as a prediction of endmember error under certain conditions.

A Spatial Compositional Model for Linear Unmixing and Endmember Uncertainty Estimation