Multiscale Spatial Graph-Regularized Hierarchical Sparse Unmixing Based on the Framelet Transform

Hyperspectral unmixing (HU) is dedicated to disassemble mixed pixels into a group of pure spectral signatures (endmembers) and their respective fractional abundances. By utilizing available spectral libraries, sparse unmixing (SU) methods can estimate the fractional abundances associated with the endmembers. Many SU algorithms have focused on how to effectively integrate spatial information to improve unmixing accuracy. However, under imaging conditions with low signal-to-noise ratio (SNR) or complex noise interference, the exploitation of spatial information may be difficult, which restricts the robustness of SU algorithms. To address this problem, we propose a new multiscale spatial graph-regularized hierarchical SU (MSGHSU) algorithm based on the framelet transform. Our algorithm introduces the framelet transform into the unmixing framework, leveraging its multiscale decomposition property to separate effective information from noise in the image, providing a high-quality data foundation for the unmixing task. Meanwhile, MSGHSU introduces a superpixel-based adaptive spatial graph regularization strategy able to model the similarity between pixels within superpixel spatial clusters by constructing a graph to enhance the spatial correlation. In addition, a hierarchical graph regularization method is proposed that dynamically updates the graph by reconstructing pixel data layer-by-layer, reducing the impact of noise on the graph regularization process and better capturing spatial details. Finally, our method introduces two weights to constrain the abundance matrix, one is based on the row sparsity of the estimated abundances, enhancing endmember sparsity, and the other is based on the global spatial prior information, improving spatial uniformity and smoothness. For optimization and solution, the alternating direction method of multipliers (ADMM) is employed to ensure efficient model solving. Our experiments with simulated and real data confirm that MSGHSU outperforms existing unmixing methods, demonstrating its effectiveness in addressing the mixed pixel problem in hyperspectral images (HSIs).