Anisotropic Diffusion Equation In Image Processing And Analysis

Anisotropic Diffusion Equation in Image Processing and AnalysisImage is acquired by using various observation systems in different forms and means of observation and access to the objective world, it can affect on the human eyes directly or indirectly, generates the perception entity. Image is an important means of information visualization. It can be divided into types of gray-scale image and color images by color type, it can be divided into static and dynamic image by movement type, too. In this paper, we consider static gray image. Image processing is the process which makes image signal convert into digital signal, and use the computer to deal with it. Image processing is a basic tool used to reconstruct the relative order, geometry, topology, patterns, and dynamics of the three-dimensional world from two-dimensional images. Image enhancement, image restoration and image segmentation arc main issues in image processing and analysis. In recent years, partial differential equations (PDE) theory turn into a new type of tool in image processing gradually, which attracted attention of many people. Traditional image denoising methods, such as Gaussian filtering, median filtering etc, mainly filter component of the image high-frequency. It experiences a process from linear to nonlinear, and from the isotropic diffusion to the anisotropic diffusion. As a great deal of findings succession of mathematical physics equations, in many image processing and computer vision application areas, at present, the research of the image partial differential is gradually rich and mature. It has been rapidly developed into a kind of method which is rigorous in theory and effective in practice. The idea that the method of partial differential equations (PDE) is used in image processing can be traced back to the work of Gabor and Jain followed. But the substantial and original achievement in this field should own to the individual effort of Koenderink and Witkin who connected the convolution of original image with Gaussian kernel with solution of PDE, they introduce the definite mathematical form of the image multiple scale.Image transformation can be denoted with partial differential equations as followsu(x,y,t) : R²Ã—[0, (?))→R , u(x,y,t) is changing with t. It denotes one image. u₀ is initial conditions, F is a linear operator. u is the solution of this equation, and gives image state of time l.. Witkin pointed out that basing on the gaussian kernel in the space of different scales, in essence, convolution model solves a linear partial differential equations - heat conduction equation (isotropic diffusion)whose initial value contains noise image. Although the heat conduction equation model can remove noise, this model may cause the border drift, fuzzy images, and so on, for it has the same speed of proliferation at every point in the equation. In 1990, Perona and Malik proposed the nonlinear anisotropic diffusion, is called P-M model.diffusion coefficient c(s) is a smooth non-increasing function in [0,+∞), s.t. lim c(s) = 0. But c(s)≠0, c(s) usually takes the following two formss→+∞c(s)=e^-(s/k)s≥0,orc(s) = (1 + (s/k)²)^-1 s≥0The P-M model has some drawbacks: it diffuses not only in the vertical directionproliferation, but also in the gradient, and associates with the gradient assignment; At the same time, P-M model is not well-posedness.There are various anisotropic diffusion models, as the Perona and Malik anisotropic diffusion can not preserve edge details well, many new anisotropic diffusion models are proposed. This chapter introduces these different forms. such as anisotropic diffusion model in complex number field, Alvarez, Lions and Morel model, F. Catte model, You Yu-Li and M. Kaveh model. In 2001, Guy Gilboa proposed anisotropic diffusion model in complex number field. For the anisotropic diffusion equation. multi-dimensional diffusion equation can be expressed as follows wherec(Im(u)) = e^iθ/(l + {Im(u)/kθ)²),k is parameter.The total variation models were firstly given by Rudin, S. Osher, E. Fatemi in. It is called TV (Total Variation) modelTV model wiped off image noise, and obtained very good denoising effects on the base of the image noise information, the solution to this model allows the existence of discontinuity, therefore, it is better than other models to maintain the border. Total variation model is a variational model, it is consistent with the image restoration method based on the PDE, and even more, it can be regarded as a special form of the anisotropic diffusion equation. Total variation model can be expressed ass.t.andGenerally we consider the similar total variation modelwhere R is a linear degradation operator.σis variance, TV model can preserve edges very well, in TV model. its effects can be improved greatly by selecting the fidelity of the norm form. as well as the size of regularization parameter. It is very important for partial differential equations. the paper also briefly introduced the discrete format of anisotropic diffusion equation.The well-posedness of mathematical model refers to the existence of solutions, the only, and initial stability. For the equation, we only want to obtain stable solution of the equation, hut most partial differential equations can not obtain analytical solutions, generally we obtain the solution of the equation through numerical solution methods, it is very necessary to discuss the existence and uniqueness of weak solution of the equation. To permit the well-posedness of anisotropic diffusion in complex number field, it is similar to proof method of selective smooth model by F.catt(?): L^P(0,T, H^k(Ω))denote, for (0,T) and any t, u{·,t)∈H^k(Ω),k G , L^p(0,T,H^k(Ω)is a normed space,Firstly, we use the classic Schauder fixed-point theory to prove the existence of the weak solution of its equation. Defineusing the parabolic type theory of partial differential equations, (3.1.4)has one and only one solution U(ω) in W(0.T).U(ω) s.t.We can prove existence of weak solution of the equation. Then Gronwall Lemma can get the uniqueness of the solution.G. Aubert, L. Vese [48] proved the existence and uniqueness of variational model in V = {f∈L²(Ω),â–½f∈L¹(Ω)²}. They used the method which introduces half of the second of auxiliary variables. We denote by BV(Ω) the set of all functions of locally bounded variation on ft, variational problem in image processing and statistics already is quite common, the choice of function space in the variational problems mainly depends on the actual application. BV(Ω) is very suitable space for image processing. The solutions existence of variational model in BV can be discussed by variational methods. To consider the BV solution of the equation, we should remark that sequence bounded in V are also bounded in BV(Ω). Therefore, they are compact for the BV weakly star topology. In this case, it is classical to compute the relaxed energy. where c = (?) ((φ(s))/s)Anisotropic diffusion can be applied not only to image restoration, but also to image enhancement, image segmentation, image classification, and other fields. Finally, we summarize the full text, and prospect further research directions.