A Tale of Two Bases: Local-Nonlocal Regularization on Image Patches with Convolution Framelets

We propose an image representation scheme combining the local and nonlocal characterization of patches in an image. Our representation scheme can be shown to be equivalent to a tight frame constructed from convolving local bases (e.g., wavelet frames, discrete cosine transforms, etc.) with nonlocal bases (e.g., spectral basis induced by nonlinear dimension reduction on patches), and we call the resulting frame elements convolution framelets. Insight gained from analyzing the proposed representation leads to a novel interpretation of a recent high-performance patch-based image processing algorithm using the point integral method (PIM) and the low dimensional manifold model (LDMM) [S. Osher, Z. Shi, and W. Zhu, Low Dimensional Manifold Model for Image Processing, Tech. Rep., CAM report 16-04, UCLA, Los Angeles, CA, 2016]. In particular, we show that LDMM is a weighted `2-regularization on the coefficients obtained by decomposing images into linear combinations of convolution framelets; based on this understanding, we extend the original LDMM to a reweighted version that yields further improved results. In addition, we establish the energy concentration property of convolution framelet coefficients for the setting where the local basis is constructed from a given nonlocal basis via a linear reconstruction framework; a generalization of this framework to unions of local embeddings can provide a natural setting for interpreting BM3D, one of the state-of-the-art image denoising algorithms.