New Numerical Discrete Algorithms For Tangential Diffusion In Image Restoration

With the high-speed development in computing speed and communication technique, digital image processing has developed fast to a popular subject. It has an important impact on people’s daily life and high-tech area. As a major branch of image processing, image restoration has been a hot topic for many researchers. Some modern restoration methods which are based on Partial Differential Equations (PDEs) have developed fastest and have become some important methods in image analysis and computer vision. Image restoration methods based on PDEs were first derived from Gaussian filter, and then Perona and Malik changed linear diffusion model to non-linear diffusion model with an initiative by introducing the gradient operator to detect edges. After that, Catte,Scherzer proposed regularization methods; Whitaker,Pizer and Weickert also developed the model. In the image restoration based on PDEs, non-linear diffusion process can be separated to two parts which are diffusion along the tangential direction and diffusion along the normal direction. The two parts are AuTT and BuNN. In those areas near image edges, more tangential smoothing and less normal diffusion are needed to remove noise on edges and protect edge information. So the tangential diffusion uTT is a very important item in analyzing and numerical modeling.In the traditional uTT discrete method, only simple normal differentiate discretization is taken. There is no fast algorithm for the method. Small time step must be taken for the scheme stability because of its explicit characteristic. But in the real resolving process, small time step is not easily controlled and the method needs a lot of iterations. The method is time-consuming and low-SNR. Based on shortcomings of the traditional method, a new fast stable reliable uTT numerical discrete algorithm is proposed. It overcomes claws of traditional method and can better help analyze the diffusion along the tangential direction.In the new nTT numerical discrete algorithm, the equivalent form of uTT is first deduced from some real theories. And then, the alternant different method and the AOS scheme (additive operator splitting) proposed by Weickert are used to do the new discretization. At last, Thomas fast algorithm is taken to do fast resolution. The new discrete method is extant of the semi-implicit scheme and is unconditionally stable. It can take big time steps and can equally treat every coordinate axis. Some high-dimensional problems can be easily dealt with by using the new method. The new discrete method has a good rotational invariance and lower computational complexity. When doing the space discretization, alternative difference scheme is used to form triple diagonal system which can be fast solved by Thomas algorithm. The difference scheme has a good approximate order.Numerical experiments are done to give results that new discrete method has obvious advantage in speed and is12times faster than the traditional method when two methods realize similar effects. And in protecting image character, traditional method loses some important character information with removing noise. While, new discrete method can protect image details and image texture well when removing a lot of noise.New uTT numerical discrete algorithm is fast and stable. Besides it can help better analyze the diffusion along the tangential direction and be better applied in the areas of image restoration based on PDEs, it also can be used in some other areas of image processing which are related to the second tangential derivative uTT discretization. In the computational physics, it also can be used to do some analysis. All in all, new algorithm has much theoretical value and application prospect.