Recently, mapping a signal/image into a low rank Hankel/Toeplitz matrix has become an emerging alternative to the traditional sparse regularization, due to its ability to alleviate the basis mismatch between the true support in the continuous domain and the discrete grid. In this paper, we introduce a novel structured low rank matrix framework to restore piecewise smooth functions. Inspired by the total generalized variation to use sparse higher order derivatives, we derive that the Fourier samples of higher order derivatives satisfy an annihilation relation, resulting in a low rank multi-fold Hankel matrix. We further observe that the SVD of a low rank Hankel matrix corresponds to a tight wavelet frame system which can represent the image with sparse coefficients. Based on this observation, we also propose a wavelet frame analysis approach based continuous domain regularization model for the piecewise smooth image restoration. Finally, numerical results on image restoration tasks are presented as a proof-of-concept study to demonstrate that the proposed approach is compared favorably against several popular discrete regularization approaches and structured low rank matrix approaches.

最近，将信号/图像映射到低秩 Hankel/Toeplitz 矩阵已成为传统稀疏正则化的新兴替代方案，因为它能够缓解连续域中的真实支持与离散网格之间的基础失配。在本文中，我们引入了一种新颖的结构化低秩矩阵框架来恢复分段平滑函数。受总广义变异使用稀疏高阶导数的启发，我们推导出高阶导数的傅立叶样本满足湮灭关系，从而产生低秩的多重汉克尔矩阵。我们进一步观察到，低秩 Hankel 矩阵的 SVD 对应于可以用稀疏系数表示图像的紧小波框架系统。基于这一观察，我们还提出了一种基于连续域正则化模型的小波帧分析方法，用于分段平滑图像恢复。最后，图像恢复任务的数值结果作为概念验证研究提出，以证明所提出的方法与几种流行的离散正则化方法和结构化低秩矩阵方法相比具有优势。