Study On Non-local Variational And Partial Differential Equation Methods In Image Restoration

Image restoration while preserving image feature such as edge, detail and texture is a key problem in image processing and computer vision. In the past two decades, image processing based on Variational Equations and Partial Differential Equations (PDEs) has made great progress and attracted the attention of many researchers. The first PDE which is used to restore image is Heat Equation. In 1980s, Koenderink and Witkin respectively defined and developed scale-space theory and appled PDEs to image processing. The ROF model proposed by Rudin etc and PM model proposed by Perona etc achieved great success in image restoration. Though image edge feature can be preserved by Variations and PDEs, above methods which belong to a calss of geometry-driven methods can not preserve texture and detail. The NL-means method put forward by Buwades etc can avoid PDEs'flaw to a certain extent, but the method, which essentially is an adaptive non-linear weighting filter based on image self-similarity, will blur image edge feature for less similar pixels to other edges and corners in image. Therefore, to overcome flaws of above image restoration methods, we propose two new image restoration methods followed by the non-local idea in this paper.1,Non-local PDEs method and numerical algorithms for image restoration. First, we introduce the scale-space theory briefly. Then, we introduce selectively the ROF model, the PM model and the NL-means model. Afterwords, some Non-local operators are introduced. Since the PDEs method cannot preserve texture and the NL-means method blurs edge of image, two new Non-local PDEs models based on non-local operators are proposed, which can preserve image feature effectively, and corresponding discrete numerical methods are provided in the paper. Proposed models are carried out on some real images to verify the effect of the algorithms. At the same time, for the NL-means method, a fast algorithm is put forward which can reduce computing time significantly.2,Adaptive mixed image restoration method based on image decomposition. Based on image decomposition, we propose a new image processing strategy that is an adaptive mixed method for image restoration. First, this method decomposes a given image as the sum of two components:geometric structure and oscillating pattern according to the Meyer's theory. Then, a coupled bidirectional diffusion equation is used to restore the structure part, and a non-local means filter is used to remove noise in the oscillating part.Experimental results show that, as adaptive image restoration methods proposed methods can effectively restore image features including image edge, texture and detail.The key techniques presented in the paper are expected to be applied to such fields as medical image processing, visual surveillance, digital TV and aerospace engineering etc widely. As a cross-disciplinary research related to information sciences and mathematics, our work enriches applications of variational and PDE methods in image processing, which has both important theoretical value and broad application prospect.