The generalized bilinear model (GBM) has been one of the most representative models for nonlinear unmixing of hyperspectral images (HSI), which can consider the second-order scattering of photons. Recently, robust GBM-low-rank representation (RGBM-LRR) for nonlinear unmixing of HSI has been introduced to capture the spatial correlation of HSI using LRR with nuclear norm minimization (NNM). However, NNM is used to approximate the matrix rank by shrinkage all singular values equally. The singular values have distinct physical significance in many real applications, and NNM may not be able to estimate the matrix rank accurately. To overcome the above issue, a robust GBM with a weighted low-rank representation (RGBM-WLRR) approach is proposed using weighted nuclear norm minimization, which mitigates the penalty on larger singular values by assigning a small weight, so the corresponding shrinkage is small and also takes serious shrinkage on small singular values by assigning larger weights to them. The proposed model is solved using an iterative alternating direction method of the multipliers. A series of experiments with real datasets and a simulated HSI with varying rank and signal-to-noise ratio reveals that RGBM-WLRR performs significantly better than the state-of-the-art algorithms in terms of signal-to-reconstruction error, root-mean-square error, and spectral angle distance.
A four-directional total variation technique is proposed to encapsulate the spatial contextual information for sparse hyperspectral image (HSI) unmixing. Traditional sparse total variation techniques explore gradient information along with the horizontal and vertical directions. As a result, spatial disparity due to high noise levels within the neighboring pixels are not considered while unmixing. Moreover, oversmoothing due to total variation may depreciate the spatial details in the abundance map. In this context, we propose a four-directional regularization technique (Sparse Unmixing with Splitting Augmented Lagrangian: Four-Directional Total Variation, SUnSAL-4DTV) for sparse unmixing. The four-directional total variation scheme is transformed into the fast-Fourier-transform domain to reduce the higher computational requirements. An alternating-direction-method-of-multipliers-based iterative scheme is proposed for solving the large-scale optimization problem. An adaptive scheme is introduced to update the regularization parameters to ensure faster convergence. Extensive numerical simulations were conducted on both simulated and real hyperspectral datasets to demonstrate the robustness of proposed technique. Comparative analysis on noisy (low signal-to-noise-ratio) HSIs shows the robustness of SUnSAL-4DTV over the state-of-the-art algorithms.
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