Spatially adaptive hyperspectral unmixing through endmembers analytical localization based on sums of anisotropic 2D Gaussians

Spectral unmixing provides, for each pixel in the image, an estimated vector of fractional abundances that correspond to pure signatures, known as endmembers (EMs). Standard unmixing techniques rely only on spectral information and each pixel is solved as an individual entity. Recent studies show that incorporating the image's spatial information enhances accuracy of the unmixing results. Spatial information may allow better selection of relevant EMs, for each pixel, rather than utilizing all potential EMs in solving the spectral mixing problem. To implement this approach, we developed a new method for spatially adaptive spectral unmixing called Gaussian-based spatially adaptive unmixing (GBSAU). GBSAU fits for each EM, a surface by a series of spatial anisotropic 2D Gaussians whose sum represents the EM's fraction distribution over the whole image. These analytical surfaces facilitate a sparse solution for the unmixing process by spatial localization of EMs, which is then used to determine an adaptive subset of actual EMs for each pixel. The performance of our novel method was compared with that of the state-of-art spatially adaptive unimximg method, sparse unmixing via variable splitting augmented Lagrangian and total variation (SUnSAL-TV) as well as with two ordinary non-spatial methods, sparse unmixing by variable splitting and augmented Lagrangian (SUnSAL) and vectorized code projected gradient descent unmixing (VPGDU). The comparison was carried out on both simulated and real hyperspectral images. The results obtained with GBSAU indicated a significant improvement in the overall accuracy of the unmixing process compared with both spatially adaptive and ordinary methods. Quantitatively, GBSAU reduces the average mean absolute error (MAE) of the results by ∼15%, for cases with SNR = 30 db and 20 db, and by ∼30% for cases with SNR = 10 db and 5 db.