Blind Unmixing of Hyperspectral Remote Sensing Data: A New Geometrical Method Based on a Two-Source Sparsity Constraint

Blind source separation (or unmixing) methods process a set of mixed signals, which are typically linear memoryless combinations of source signals, so as to estimate these unknown source signals and/or combination coefficients. These methods have been extensively applied to hyperspectral images in the field of remote sensing, because the reflectance spectrum of each image pixel is often a mixture of elementary contributions, due to the limited spatial resolution of hyperspectral remote sensing sensors. Each spatial source signal then corresponds to a pure material, and its value in each pixel is equal to the "abundance fraction" of the corresponding Earth surface covered by that pure material. The mixing coefficients then form the pure material spectra. Various unmixing methods have been designed for this data model and the majority of them are either geometrical or statistical, or even based on sparse regressions. Various such unmixing techniques mainly consider assumptions that are related to the presence or absence of pure pixels (i.e., pixels which contain only one pure material). The case when, for each pure material, the image includes at least one pixel or zone which only contains that material yielded attractive unmixing methods, but corresponds to a stringent sparsity condition. We here aim at relaxing that condition, by only requesting a few tiny pixel zones to contain two pure materials. The proposed linear and geometrical sparse-based, blind (or unsupervised) unmixing method first automatically detects these zones. Each such zone defines a line in the data representation space. These lines are then estimated and clustered. The pairs of cluster centers, corresponding to lines, which have an intersection, yield the spectra of pure materials, forming the columns of the mixing matrix. Finally, the proposed method derives all abundance fractions, i.e., source signals, by using a least squares method with a non-negativity constraint. This method is applied to realistic synthetic images and is shown to outperform various methods from the literature. Indeed, and for the conducted experiments, when considering the pure material spectra extraction, the obtained improvements, for the considered spectral angle mapper performance criterion, vary between 0.02 degrees and 12.43 degrees. For the abundance fractions estimation, the proposed technique is able to achieve a normalized mean square error lower than 0.01%, while the tested literature methods yield errors greater than 0.1%.