| The image is between the humanity exchanges one of most direct-viewing ways. Especially in current information technology rapid development time, image infor-mation as an important communication medium are widely used in multimedia, digital health, artificial intelligence, aerospace and geological remote sensing, and other fields. Generally speaking, the digital image processing technology take the digital image as an object, including image acquisition, image analysis and image understanding three main areas. Image analysis is between image acquisition and image understanding between one of the links, it directly affects computer vision, robot vision, and machine vision. In image processing, image restoration and image segmentation is important problem and has been extensively studied, they all belong to the image analysis of research areas.Image processing research has a long history, goes back to the 1920s, most of the image processing methods are based on a dimensional signal processing technologies, such as filtering technology, statistical theory, etc. With remote sensing, medical, and other fields growing demands, as well as physical technology breakthrough, peo-ple are increasingly concerned about the nature of image processing, and attempt to use strict mathematical theory to the current image processing method for colla-tion and improvement. Currently, the image processing technology has three main tools:Stochastic Models, Wavelet Analysis Theory and Partial Differential Equa-tions Methods. The first two have a long history:Stochastic Models are based on Bayes and markov random field and are to work directly on the images; Wavelet Analysis Theory is based on Fourier analysis, transform the image information to the frequency domain information, and then are used for image processing. Since the 1990s, image processing methods based on PDEs (partial differential equations) developed rather quickly, becoming a very attractive study areas. At present, in im-age restoration, image segmentation, image reconstruction, image recognition, image analysis and so on, image processing methods based on PDEs has been widely ap-plied. In PDEs models for image processing, image information is quantified as a continuous two-dimensional function, and can be seen as a field or physical state. PDEs methods work directly on the image, is closer to reality the objective world and easy to analysis models. In the mathematical theory, there are lots of the the-ories on PDEs, and then the models can be deeply analyzed. From this stage of development, we can not know which is the best in these three tools, And these three tools is kept in touch with each other and cross-promotion.In this article, the first problem we are concerning is PDEs methods for image restoration and decomposition and the corresponding theoretical analysis. In image processing, image restoration has always been one of the most important fundamen-tal research subjects. In the image acquisition, copy, scan, transmit, display, and so on,the images will be inevitably degraded, such as blur and noise. However, in many application areas, people need a clear, high-quality images, so image restora-tion (such as denoising, deblurring, etc) is of great significance. In theory, noise is unpredictable random error, and subject to some probability distribution. Assume that u:Ω(?) R2→R is the intensity of the ideal image, n(x) is the noise,f is the intensity of the observed image. The idea image u and the observed image f arerelated byf= u+n.Generally speaking, the noise n(x) satisfies:where, the first constrain indicates that the noise has zero mean, and the second one uses a priori information that the standard deviation of the noise n(x) isη. For ease of analysis, we assume that the image signal and noise are.independent of each other. The problem is to recover the restoration image, u, from the observed, noisy image, u0, where the two are related by u0= u+n(χ). The image restoration and decomposition models based on PDEs have the two important framework:the variational methods based on the optimal problem for energy functionals and the diffusion equation methods based on fluid diffusion theory. Image decomposition is the extension of image noise, Image denoising model basically can also be used as image decomposition model. Next, we present some image denoising models based on the variational method, image decomposition models based on the variational method, and image restoration models based on diffusion equations.First, we introduce image denoising models based on the variational method and some classical denoising models. The famous classical and general method for removing Gaussian noise firs consists in solving the following constrained minimiza-tion problem:λ> 0 is a weight parameter. The first term in E(u) is a smoothing term for deniosing. The second term in E(u) measures the fidelity to the initial data. In 1977, in the model proposed by Tikhonov and Arsenin [1],(?)(s)= s2. And the solution space iswhereΩis a bounded open domain of-RN with the appropriate smooth boundary, n denotes the unit outward normal to the boundary (?)Ω, and then the constrained minimization problem has unique solution in W. However, the above-mentioned (?) punish the gradient too much and W1,2(Ω) is a smoother space, and therefore the recovery image is too smooth to preserve the image edge information. The real image maybe not be smooth, so someone try to find a better function (?), for example, (?)(s)=|s|p,1< p< 2, and then the corresponding solution space is W1,p(Ω). In 1992,Rudin, Osher and Fatemi [2] propose the famous total variation method, the ROF model. In the TV model,(?)(s)=|s|, and mathematically this is reasonable, since it is natural to study solutions of this problem in the space of functions of bounded variation, BV(Ω), allowing for discontinuities which are necessary for edge reconstruction. The TV model has been studied extensively and has proved to be an invaluable tool for preserving edges in image restoration problem. In 1997, Chambolle and Lions [3] detailed theoretical analysis for the TV model, and propose the corresponding numerical calculation method. In 2003, Rudin, Lions and Osher [4] summary theoretical results for the total variation model, as well as the related theory of partial differential equations. When (?)(s)= (1+s2)1/2, the corresponding model is based on the minimal surface functional, and the functional has similar properties with the TV model and well-posedness [13]. When (?)(s) is the non-convex function, such as(?)(s)= log(1+s2)[9], using the model, we can have better results for image restoration. However,the existence and uniqueness of the solutions for the model still are a open problem, the convergence of the numerical scheme for the model had not been proved. The TV model is well posed, but TV-based denoising favors the piecewise constant solutions. Sometimes this also causes a staircasing effect in which noisy smooth regions are processed into piecewise constant regions [3], [5], and [38]. Not only'blocky'solutions fail to satisfy the ubiquitous'eyeball norm' but they can also develop'false edges' which can mislead a human or computer into identifying erroneous features not present in the true image. Some authors consider another regularizing term to remove the noise [5], which is as followswhere (?) p(s)→2, (?) p(s)→1, and p is monotonically decreasing. This model should reap the benefits of both isotropic and TV-based diffusion, as well as a com-bination of the two. However, it is difficult to study mathematically since the lower semi-continuity of the functional is not readily evident. In [6], Chen, Levine, and Rao modify the model, and propose a functional with variable exponent,1≤p(x)≤2, which is a combination of Total Variation based regularization and Gaussian smooth-ing.Secondly, we introduce some image decomposition model based on the varia-tional method. If we decompose the image into two components:the smoothed version of the original image f, denoted by u, which represents the cartoon in-formation of the original image, and the componentυ= f - u, which represents the texture or noise information, then the above model is actually a new energy functional for image restoration and image decomposition into cartoon plus texture, where the first term is a regularizing term to remove the noise, and the second term is a fidelity term. Though the ROF model performs very well for image denoising and edge protecting, it may also destroy small details, such as textures, see [7]. To overcome this drawback, Meyer [7] proposed a new minimization method by intro-ducing a weaker norm which is more appropriate to represent textured or oscillatory patterns. Subsequently, Osher et al. [8] presented the following model which com- bined the norm for oscillatory functions proposed by Meyer [7] (involving the H -1 norm) with the total variation minimization from the ROF model [2] whereλ> 0 is a weight parameter. Osher, Sole and Vese[8] is in turn a particular case of an earlier work by Osher and Vese [24]. Inspired by Meyer's G-norm, some authors have proposed various numerical models for image decomposition and mod-eling texture with Meyer's G-norm and F-norm, and also with Besov norms[25]-[27].Finally, we present and analysis some denoising models based on the diffusion equations noise model, which is from the fluid diffusion physical phenomena. Early in the 1960s, the oldest and most investigated equation proposed by Gabor in the processing is probably the parabolic linear heat equation with the initial data f. Alvarez et al. [10] rigorously establish the connection between scale space analysis and PDEs. In [9], Perona and Malik developed an anisotropic diffusion scheme for image denosing. The basic idea of this nonlinear smoothing scheme was to smooth the image while preserving the edges in it. This was done by using the equationwhere C(s) is a smooth non-increasing function such that C(s)→1, as s→0 and C(s)→0, as s→∞. In fact, the PM diffusion equation is a forward-backward diffusion and the well-posedness of the equation is still a open problem. It is worth mentioning that, the TV model can be seen as the Perona-Malik diffusion when c(|▽υ|)= (?). In [11], A simple adjustment with practical applications, is to include a short range mollifier in the nonlinear diffusion. Existence and uniqueness of solutions to this modified Perona-Malik equation has been proved by Catte et al. for initial dataυ0∈L2(Ω). Inspired by the PM model, many researchers proposed new models which is to work on the nonlinear anisotropic diffusion equations, without thinking of any energy functional. All models are nonlinear anisotropic diffusion filters with diffusivities being adapted to the local image structure. Weickert [12] propose the model which is a nonlinear anisotropic diffusion filter with diffusion tensors being adapted to the local image structure. The author modify the diffusion tensor matrix eigenvectors and eigenvalues to control the diffusion direction and diffusion strength. The models based on PDEs can easily extend to the color image case. In the color image case, the geometry properties of surfaces [13] is involved. In addition, the convection term in PDEs is introduced, and can speed up calculation speed [14]-[15]. In practice, the ill-posedness results in a mild instability in the PM problem. Regions of high gradients develop a "staircase" instability that involves dynamic coarsening of the steps as time evolves. To make the images more pleasing to the eye, it would be useful to reduce this effect. To this end, many researchers consider higher-order version of the Perona-Malik equations, examples of which.are given in [16]-[18].In this article, the second problem we are concerning is image segmentation models based on PDEs and the level-sets method. Image segmentation is one of the most fundamental problems in the fields of image processing and computer vision. The aim is to find a partition of an image into its constituent parts. However, this definition is rather unsatisfactory and ambiguous. As we will see, the main difficulty is that one need to manipulate objects of different kinds:functions, domains in R2, and curves. In the last century in the late 1980s, partial differential equation method is first applied to the image segmentation. Kass, Witkin and Terzopoulos[19] first propose the snake and the geodesic active contours models. Mumford and Shah [20] propose the variational segmentation method for the region detection. Since the image edges is one-dimensional, in early segmentation algorithms it is difficult to indicate the edge information for images. At that time, using theΓ-convergence or spline curves and other techniques, the researchers obtain the image segmentation results. However, these methods are not only difficult to understand, and solve complex, and do not have the expected effect.1988年, Osher and Sethian [21] introduce the level-sets method which allows for cusps, corners, and automatic topological changes. The curve is represented implicitly via a Lipschitz function, by the zero level-set of the Lipschitz function, and the evolution of the curve is given by the zero-level curve from the evolution of the function. Since then, the PDEs method is widely used to image segmentation. The iconic work can be see the geodesic active contours models by Caselles, Kimmel and Sapiro[22] and the active contours without edges models by Chan and Vese [23]. At present, the CV model has many improvements and development and has attracted the attention of many scholars.In this paper we studies image processing methods based on PDEs, especially in image restoration, decomposition and segmentation applications. Since in im-age processing the models based on PDEs (system) which is from the research and analysis of the image structure, is usually the new type diffusion equation (system), so the existence and uniqueness of solutions for the new PDEs(system) is impor-tant practical significance for the numerical calculation. In this paper the following problems are studies:the first chapter is the theoretical research and numerical implementation for the image restoration and decomposition model based on new partial differential systems; the second chapter is the theoretical research and nu-merical implementation for the image restoration model based on new anisotropic diffusion equations; the third chapter is the introduction and analysis of two new fast segmentation algorithms. every problem in this paper can be divided into the following sections:first, we introduce a new model, usually based on some diffu-sion equation (system); then, we have demonstrated the existence, uniqueness and boundedness of solutions for the new models; finally, we structure the numerical scheme which is applied to image processing with respect to other known models. Since the new equation (system) structure is different, we applied the regularization method, the Galerkin method, and the fixed-point method to obtain the existence of solutions for the new models, respectively. Then, experimental results illustrate the effectiveness of the new models with respect to other known models.In the first chapter, the two image restoration and decomposition models based on reaction-diffusion PDEs (system) is proposed. In the first part of this chapter, inspired by Osher, Sole and Vese[8], we propose the following model based on the reaction-diffusion partial differential system: where f is the original image,Ωis a bounded domain of RN with appropriately smooth boundary, n the unit outward normal toΩ,1≤N< 4, T> 0,λ> 0, and as far as we know, there is few literature about PDE system devoted to image processing. Different from the models as before, in the new system, the two equations interact with each other:first we obtainωfrom (1) to adjust the fidelity betweenυand f in (2), then we use (2) which is from the classical TV model to obtain the smoother image u. In other words, the denoising equation (2) has a different fidelity termωwhich is also derived from an evolutionary equation, thus we can select an appropriate scale to achieve an appropriate fidelity betweenυand f and obtain a better result for noise removal and edge or texture preservation. Furthermore, we can proveυis the element of BV space theoretically, which allows for discontinuities along curves, therefore edges and contours are kept in the image u. In addition, in theory, the problem (2) can be seen as a TV flow equation with nonlinear source. In [28,29,30], Andreu et al. studied the TV flow ut= div(Du/|Du|) with L1-data using the Crandall-Liggett semigroup theory. The main difficult for this class of equations is due to the fact that (2) has a strongly degenerate term, i.e. when|Du|= 0. To overcome it, we employ the regularization method. We firstly prove the existence and uniqueness for weak solutions of the approximate p-Laplace equations. Then some prior estimate are given. Last by a standard limiting process as p→1+, we recover the well-posedness of the problem (1)-(6).In the second part of this chapter, Analysising the models in [6] and [8], we combine the two models and propose the following model based on the reaction-diffusion partial differential system: where Gσ(x) is the Gaussian kernel, namelyT> 0, A> 0,f is the original image,κ1and k2 are positive constants,Ωis a bounded domain of RN with the smooth boundary, and n denotes the unit outward normal to the boundary (?)Ω. In [8], we change (?)|▽υ|p(χ) by |▽υ|, and obtain the following model:The corresponding Euler-Lagrange is as followsNote that our model is the evolutionary form of the steady system (15). The main distinguish between the problem (7)-(12)and the problem (1)-(6) is the principal part of the diffusion equation onυ, and in the problem (7)-(12), the diffusion is p(χ)-Laplace. From (13) and (14), g(x) and p(x) depend on the location, x, in the original image. This way the speed and direction of diffusion at each location depends on the local behavior. The term|▽u| has larger value on the edges than in the homogeneous region, while g(x) is just the opposite. Therefore, the diffusion applied by equation (7) may cause little smoothing near the edges. The main feature of equation (7) is the variable character of nonlinearity which causes a gap between the homogeneous regions and nonhomogeneous regions (edges). So it ensures TV based diffusion (as p→1) along edges and Gaussian smoothing (as p→2) in homogeneous regions. On the other hand, as far as we know, there is few literature about PDE system devoted to image processing. Different from the models as before, in the new system, the two equations interact with each other:first we use (7) which is from the mixed diffusion model to obtain the smoother imageυ. then we obtainωfrom (8) to adjust the fidelity betweenυand f in (7). In other words, the denoising equation (7) has a different fidelity termωwhich is also derived from an evolutionary equation, thus we can select an appropriate scale to achieve an appropriate fidelity betweenυand f and obtain a better result for noise removal and edge or texture preservation. Furthermore, the new mode is to work directly on the equations, without thinking of any energy. Starting from the initial image f(χ) and by running the system (7)-(12) we construct a family of functions (i.e. the smoother images){υ(t,x)}t>o representing successive versions of f(x). we can choose the best image for the appropriate t and observe the whole evolution process. Finally, using the Galerkin method, we can proveυis the element of belong to Orlicz-Sobolev spaces W1,p(χ)(QT) (the rigorous definition is given in Section 2 below) theoretically, which allows for better regularity, and therefore achieve a good denoising result.In the second chapter, we propose the following model where where, Ga(x) is the Gaussian kernel,f is the original image,κ1> 0,κ2> 0,σ1,σ2 and T> 0 are fixed constants, c is a constant dependent ofκ1 andσ1,Ωis a bounded open domain of RN with the appropriate smooth boundary, n denotes the unit outward normal to the boundary (?)Ω. LetwhereIt is clearly that the model can be rewrite as followingOn the one side, we detailedly analysis some properties of the function C(s), and show that the edge detection function C(s) is likely to that of the original Perona-Malik diffusion, while the new diffusion equation has not backward diffusion. On the other side, letting g(s)= 1, as s→+∞, p(s)→1, the divergence principal part of the proposed diffusion equation is likely to the divergence operator of the mean curvature diffusion equation[13]; while, as s= 0, p(s)≈2, the divergence principal part is the diffusion term of the heat equation. Because of C(s), the new model has a hybrid diffusion type combined the mean curvature diffusion with the heat diffusion, and g(s) enhances the efficacy of C(s). In theory, using the fixed point theorem, we verify the existence and uniqueness of the solutions for the problem (16)-(18). And then we continue to verify the stability, boundedness, and the behavior of the solutions as t→∞, and these properties influence the results of image restoration. Next, we propose a convergent iterative scheme and verify the convergence of the scheme. Finally, experimental results illustrate the effectiveness of the new models with respect to other known models.In the third chapter, using the CV model, we propose two fast image segmen-tation algorithm. The first model is primarily to improve the speed of level-sets segmentation methods. we develop a fast level set method for image segmentation without solving the Euler-Lagrange equation of the underlying variational problem proposed by Chan and Vese. Compared with the original CV model, our approach is a new practical fast way to solve the active contour without edge problem and avoids the process of complicated, expensive reinitialization. Since the calculation cost of the second step is O(N), even for the larger images, it is also very efficient. The second model is mainly to improve the color image segmentation results. From the idea of some the color image denoising models, we propose the color image seg-mentation model based on a convex energy functional, and using the principle of duality, we obtain the corresponding algorithm. From the numerical experiments, we can see the superiority of our model.
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