Spectral unmixing aims to determine the relative amount (so-called abundances) of raw materials (so-called endmembers) in hyperspectral images (HSI). Libraries of endmember spectra are often given. Since the linear mixing model assigns one spectrum to each raw material, the endmember variability is not considered. Computationally costly algorithms exist to still derive precise abundances. In the method proposed in this work, we use only the pseudoinverse of the matrix of the endmember spectra to estimate the abundances. As can be shown, this approach circumvents the necessity of acquiring a HSI and is less computationally costly. To become robust against model deviations, we iteratively estimate the abundances by modifying the matrix of the endmember spectra used to derive the pseudoinverse. The values to modify each endmember spectrum are derived involving the singular value decomposition and the grade of violation of physical constraints to the abundances. Unlike existing algorithms, we account for the endmember variability and force simultaneously to meet physical constraints. Evaluations of samples for material mixtures, such as mixtures of color powders and quartz sands, show that more accurate abundance estimates result. A physical interpretation of these estimates is enabled in most cases.
Using appropriately designed spectral filters allows to optically determine material abundances. While an infinite number of possibilities exist for determining spectral filters, we take advantage of using neural networks to derive spectral filters leading to precise estimations. To overcome some drawbacks that regularly influence the determination of material abundances using hyperspectral data, we incorporate the spectral variability of the raw materials into the training of the considered neural networks. As a main result, we successfully classify quantized material abundances optically. Thus, the main part of the high computational load, which belongs to the use of neural networks, is avoided. In addition, the derived material abundances become invariant against spatially varying illumination intensity as a remarkable benefit in comparison with spectral filters based on the Moore-Penrose pseudoinverse, for instance.