Compressive total variation for image reconstruction and restoration

In this paper, we make use of the fact that the matrix u is (approximately) low-rank in image inpainting, and the corresponding gradient transform matrices D_xu,D_yu are sparse in image reconstruction and restoration. Therefore we consider that these gradient matrices D_xu,D_yu 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, ℓ₁−αℓ₂TV and BM3D in test for images with piecewise constant regions. And it visibly improves the performances of TV, ℓ₁−αℓ₂TV and TGV, and is comparable to Shearlet-TGV in test for natural images.