Methods for spectrally unmixing heterogeneous pixels in hyperspectral datasets using Bregman algorithms

The pixels of any image are, by definition, of finite spatial dimension. In most remote sensing applications, and in particular remote sensing of the surface of the Earth, any pixel in an image can contain more than one constituent material. Recent advances in hyperspectral imaging technology allow for the production of remote imaging products of very high spectral resolution, allowing materials of sub-pixel spatial dimension to be detected and identified, a process termed spectral unmixing. The L1 linear mixing model (L1 LMM) has been shown to produce accurate results on hyperspectral data. Processing the high dimensionality data for this model poses challenges in terms of computer resources, however. Recently developed high-speed algorithms can be applied that mitigate these issues. By applying a recently developed split Bregman algo- rithm employing an augmented Lagrangian method which prefers non-negativity and sparsity to two AVIRIS datasets (224 spectral bands) and one HYDICE dataset (210 spectral bands), it is shown that this algorithm has very fast convergence and is very efficient at solving the L1 LMM problem. Tikhonov regularization is applied to the spectral bands of the test datasets and its effect on abundance vector production and rate of convergence is observed and analysed. Finally, it is shown that applying a total variation (TV) minimization using Bregman techniques to the unmixed pixels results in regularized fraction planes.