Nonlocal Operators with Applications to Image Processing

Abstract

We propose the use of nonlocal operators to define new types of flows and functionals for image processing and elsewhere. A main advantage over classical PDE-based algorithms is the ability to handle better textures and repetitive structures. This topic can be viewed as an extension of spectral graph theory and the diffusion geometry framework to functional analysis and PDE-like evolutions. Some possible application and numerical examples are given, as is a general framework for approximating Hamilton-Jacobi equations on arbitrary grids in high demensions, e.g., for control theory.

An Improved Image Denoising Model Based on Nonlocal Means Filter

“…which is different from the nonlocal gradient vector field given in [20,29,30]. Here one views as a metric.…”

“…The three standard test images and USC-SIPI images database are widely used in the image processing literature, for example, [2,11,20,36,40]. All images in this experiment are of size 256 × 256.…”

“…This is an inverse problem, and in general there is no solution. Nevertheless, there are many useful methods to deal with this problem, for example, the median filter, the Wiener filter, the bilateral filter, variational methods [1][2][3][4][5], sparse representation based methods [6], wavelet-based methods [7], the convolutional neural network based methods [8][9][10], nonlocal self-similarity based methods [11][12][13][14][15][16][17][18][19][20], and collaborative filtering methods [12][13][14][15][21][22][23].…”

“…Motivated by the work of Kindermann, Osher, and Jones [28], Gilboa and Osher [20,29] wanted to combine the TV model and the nonlocal means filter into a nonlocal TV model, so that the corresponding 1 model is just a regularization of the nonlocal means filter. They realized this idea successfully, by introducing suitable nonlocal operators, which was motivated by the work of Zhou-schölkopf [30,31].…”

“…They realized this idea successfully, by introducing suitable nonlocal operators, which was motivated by the work of Zhou-schölkopf [30,31]. Having nonlocal operators, it is natural to propose a nonlocal version of the ROF model, the NLTV model [20], and other some nonlocal models. The nonlocal models they proposed are very useful in image processing.…”