Regularization on Image Patches: a linear reconstruction from manifold embedding

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.