Inspired by the recent low dimension manifold model (LDMM), which achieves high performance in imaging inpainting,we consider regularization on image patches from their low dimensional embedding. We provide a framework for patch-based signal decomposition by frames generated from the embedding coordinates and from patch bases in a convolution form. For any given embedding of patches (or more generally a set of data points in high dimensional space), the energy concentration of the coefficients of such frames characterizes the optimal patch basis, with respect to which the linear reconstruction from the embedding of the image has minimum error. This leads to a weighted $\ell_2$-regularization on coefficients in the decomposition of an image with respect to this optimal frame; we use this to extend the original LDMM to a re-weighted LDMM that achieves better inpainting results. Our framework can be generalized to the setting of a union of local embedding, and we show that the state-of-the-art denoising algorithm BM3D fits in our framework.

该研究论文提出了一种图像表示方案，其中等价于使用局部和非局部patches 刻画构建紧密的帧图像，并称其为卷积 framelets。分析该表示方案得出了高性能的patch-based image inpainting 算法的新解释，并提出了对其的改进。此外，还建立了卷积 framelet 系数的能量集中性质，为一些成像处理算法提供了自然的框架。

两个基地的故事:用卷积框架施加局部-非局部正则化于图像块