In this paper, we make use of the fact that the matrix $\mathit{u}$ is (approximately) low-rank in image inpainting, and the corresponding gradient transform matrices ${\mathcal{D}}_x\mathit{u},{\mathcal{D}}_y\mathit{u}$ are sparse in image reconstruction and restoration. Therefore we consider that these gradient matrices ${\mathcal{D}}_x\mathit{u},{\mathcal{D}}_y\mathit{u}$ also are (approximately) low-rank, and also verify it by numerical test and theoretical analysis. We propose a model called compressive total variation (CTV) to characterize the sparsity and low-rank prior knowledge of an image. In order to solve the proposed model, we design a concrete algorithm with provably convergence, which is based on inertial proximal ADMM. The performance of the proposed model is tested for magnetic resonance imaging (MRI) reconstruction, image denoising and image deblurring. The proposed method not only recovers edges of the image but also preserves fine details of the image. And our model is much better than the existing regularization models based on the TGV, Shearlet-TGV, ${\ell }_1-\alpha {\ell }_2$TV and BM3D in test for images with piecewise constant regions. And it visibly improves the performances of TV, ${\ell }_1-\alpha {\ell }_2$TV and TGV, and is comparable to Shearlet-TGV in test for natural images.

在本文中，我们利用矩阵 $\mathit{ü}$ 在图像修复中是（大约）低等级的，并且相应的梯度变换矩阵 ${\mathcal{d}}_X\mathit{ü}，{\mathcal{d}}_{ÿ}\mathit{ü}$在图像重建和还原方面稀疏。因此，我们认为这些梯度矩阵${\mathcal{d}}_X\mathit{ü}，{\mathcal{d}}_{ÿ}\mathit{ü}$也是（大约）低等级的，并通过数值测试和理论分析进行验证。我们提出了一个称为压缩总变化（CTV）的模型来表征图像的稀疏性和低阶先验知识。为了解决该模型，我们设计了一种基于惯性近端ADMM的具有可证明收敛性的具体算法。测试了所提出模型的性能，以进行磁共振成像（MRI）重建，图像去噪和图像去模糊处理。所提出的方法不仅恢复图像的边缘，而且保留图像的精细细节。而且我们的模型比基于TGV，Shearlet-TGV的现有正则化模型要好得多，${\ell }_{\mathrm{1个}}-\alpha {\ell }_2$TV和BM3D正在测试具有分段恒定区域的图像。明显改善了电视的性能，${\ell }_{\mathrm{1个}}-\alpha {\ell }_2$TV和TGV，在自然图像测试中可与Shearlet-TGV媲美。