鲁棒截断L1-L2全变分稀疏恢复模型

信号获取过程中，除了有高斯噪声外，还有具有脉冲性质的稀疏噪声，常用的鲁棒稀疏信号恢复模型能够在稀疏噪声环境下恢复出原始的稀疏信号。但是，许多实际应用问题需要考虑原始信号的结构稀疏性，如梯度稀疏。为了从稀疏噪声和高斯噪声共存的环境下恢复出结构稀疏的原始高维信号，文中基于截断L₁-L₂全变分、3维截断L₁-L₂全变分和鲁棒压缩感知，提出了两个非凸非光滑优化模型，用于解决高斯噪声和稀疏噪声混合影响下的结构稀疏信号恢复问题，并采用含有外推的邻近交替线性极小化算法求解这两个优化模型，使用含外推的邻近凸差算法求解子问题，在势函数具有Kurdyka-Lojasiewicz(KL)性质的条件下，给出了含外推交替极小化算法和含外推邻近凸差算法的收敛性分析。数值实验测试了高斯噪声灰度图像、混合噪声彩色图像、混合噪声灰度视频等，采用图像峰值信噪比(PSNR)作为评价准则。实验结果表明，文中模型能够更好地恢复出原始的结构稀疏信号，且在同一噪声环境下文中模型恢复的信号具有更优的PSNR值。

Robust Truncated L1-L2 Total Variation Sparse Restoration Models

In addition to Gaussian noise, there is sparse noise with impulsive properties in the signal acquisition process. The common robust sparse signal recovery models can recover the original sparse signal under sparse noise environment. However, in many practical applications, the structural sparsity of the original signal, for example, gradient sparsity needs to be considered. In order to recover the sparse structure of the original high-dimensional signal from the coexistence of sparse noise and Gaussian noise, this paper proposed two nonconvex and nonsmooth optimization models based on truncated L₁-L₂ total variation (TV) and 3D truncated L₁-L₂ TV, respectively. These optimization models were solved by the proximal alternating linearized minimization algorithm with extrapolation, and the sub-problems involved were solved by the proximal convex difference algorithm with extrapolation. Under the assumption that the potential function has Kurdyka-Lojasiewicz (KL) property, the convergence analysis of these algorithms was given. The numerical experiments test grey images with Gaussian noise, color images with mixed noise, grey video with mixed noise and so on. The peak signal-to-noise ratio (PSNR) was used as the evaluation criterion for recovered quality. The experimental results show that the new models can correctly recover the original structured sparse signal, and have better PSNR values in the same noisy environment.