Since sparse unmixing has emerged as a promising approach to hyperspectral unmixing, some spatial-contextual information in the hyperspectral images has been exploited to improve the performance of the unmixing recently. The total variation (TV) has been widely used to promote the spatial homogeneity as well as the smoothness between adjacent pixels. However, the computation task for hyperspectral sparse unmixing with a TV regularization term is heavy. Besides, the convergence of the primal alternating direction method of multipliers (ADMM) for the hyperspectral sparse unmixing with a TV regularization term has not been explained in detail. In this paper, we design an efficient and convergent dual symmetric Gauss-Seidel ADMM (sGS-ADMM) for hyperspectral sparse unmixing with a TV regularization term. We also present the global convergence and local linear convergence rate analysis for this algorithm. As demonstrated in numerical experiments, our algorithm can obviously improve the efficiency of the unmixing compared with the state-of-the-art algorithm. More importantly, we can obtain images with higher quality.

由于稀疏分解已经成为高光谱分解的一种有前途的方法，因此近来利用高光谱图像中的一些空间上下文信息来改善分解的性能。总变化量（TV）已被广泛用于促进空间均匀性以及相邻像素之间的平滑度。但是，将高光谱稀疏分解与TV正则化项的计算任务繁重。此外，还没有详细解释用于高光谱稀疏解开的乘数的原始交替方向乘数法（ADMM）与TV正则化项的收敛性。在本文中，我们设计了一种高效且收敛的双对称高斯-赛德尔ADMM（sGS-ADMM），用于具有电视正则项的高光谱稀疏解混。我们还介绍了该算法的全局收敛性和局部线性收敛率分析。如数值实验所示，与最新算法相比，我们的算法可以明显提高解混效率。更重要的是，我们可以获得更高质量的图像。