Research On Fractional-order PDE Based Image Structure-Preserving Denoising Algorithms

Usually, an image is complex, which contains a lot of local structure features, such as curved edges, straight edges, texture details, corners, ramp region and the flat region, etc. It is very essential to maintain these rich, complex and extremely important structure features in the process of image noise reduction. To keep these various structure features, it should be used different differential geometric variables to distinguish and characterize them, respectively, and then establish appropriate mathematical models for preserving different structure features.In the image processing, the characteristic of fractional calculus theory is superior to one of integer order calculus. In this dissertation, with the image richly contains complex structures as the object, we study on the variational and nonlinear diffusion equation based method and the model for image denoising, and explore the different image representation method for the different structure characteristics by using the fractional calculus theory and differential geometry theory, etc. Based on above researching, a framework with a new variational and nonlinear diffusion equation is proposed by using calculus of variations for image denoising. The new theory framework can broaden and enrich the application of the nonlinear diffusion equation and variational method in the image processing, and can be further extended to the deep-level field of the image subsequent processing. The developed theory framework has theoretical significance and application prospects. The main works and contributions of this dissertation include:(1) A class of fractional order anisotropic diffusion with adaptive p-Laplace equation is proposed for image denoising in this dissertation, which is based on fractional order derivative and fractional order diffusion factor in the frequency domain. Since it is not enough to characterize the complex image structure features only by adopting the image gradient(first-order differential geometric variable), we introduce the abundant information contained curvature(second-order differential geometric variables) to the diffusion factor, and extend the diffusion factor to the fractional order. The designed diffusion factor can be adaptively changed according to the fractional order gradient and the fractional curvature of the isolux lines, and adaptively controls diffusion intensity and direction. Numerical results show that the proposed model can effectively improve the denoising ability and maintain the important local features, especially the edges with large bend, small scale texture details and ramp structures, and also can achieve a good tradeoff between staircase effect and speckle artifacts, so that the denoised image looks more natural.(2) A class of fractional order mean curvature driven fractional order Telegraph-diffusion equations for image denoising is proposed in this dissertation, which is based on the Grümwald-Letnikov fractional order derivative and mean curvature. In addition, the analysis of the existence and uniqueness of the solution of the proposed model is given. Firstly, we introduce the fractional derivative to the telegraph-diffusion equation, which makes a “natural interpolation” between the second order telegraph- diffusion equations and fourth order ones. Secondly, the mean curvature which can characterize more image local structure information is generalized to be a fractional order version. Compared with the integer-order mean curvature, besides can effectively retain curved edges, corners and grayscale intensity contrasts, the fractional-order version can well keep the scale and the image strcture with large bend. Finally, we consider the obtained fractional order mean curvature as a conduction term and the Laplacian kernel as a controlling function. As in the the high frequency regions, the diffusion velocity of Laplacian kernel function is between that of two common controlling functions of PM model, while is smaller than them as in the flat regions. By carrying out above designs, we obtain a new diffusion coefficient to control a new telegraph-diffusion process. Numerical experiments show that the proposed model is significant in improving the denoising effect, and can effectively keep the image curved edges, corners and contrasts, while avoids speckle artifacts and staircase effect, especially for images with rich-texture and large bended structures, it can well maintain these features.(3) The image edge indicator based on the image gradient cannot effectively distinguish edges and ramps, one based on the mean curvature cannot distinguish edges and isolated noise in a noisy image and one based on the Gaussian curvature does not work effectively when the image includes a high level of noise. To impove the proformnece of above mentioned indicators, a new edge indicator named difference curvature is proposed in this dissertation, which can better depict an image and can effectively remove noise while can effectively distinguish edges from ramps and flat areas. For this purpose, based on the difference curvature and fractional derivative in the frequency domain, two classes of image denoising models are established, which called as difference curvature based fractional order nonlinear diffusion model and difference curvature based fractional order adaptive total variation model, respectively.Firstly, an image structure-preserving denoising model has been established based on difference curvature driven fractional nonlinear diffusion. In this dissertation, the fractional derivative in the frequency domain to anisotropic diffusion equation is introduced, which makes a “natural interpolation” between a PM equation and a fourth order anisotropic diffusion equation. And then we consider difference curvature as conduction term and the Laplacian kernel as a controlling function. Compared with the two common controlling functions of PM model, Laplacian kernel function can better maintain the image edges(details explained in(2)). Numerical experiments show that the proposed model can improve the PSNR effectively, and can effectively distinguish edges from ramps and flat areas, while restrains staircase effect and speckle artifacts.Secondly, a fractional order adaptive total variation denoising model based on the difference curvature has been established. In this dissertation, a function of difference curvature is proposed to be as the regularization term with a variable exponent which possesses adaptive diffusion characteristic, and another function of difference curvature is proposed to be as regular parameter which can adjust regularization term and fidelity term. At image edges, the variable exponent in the regularization term can be adaptively chose, that leads to the image edges can been well keep; in flat and ramp regions, and also isolated noise, the fractional total variation in the regularization term can well suppress noise and restrain staircase effect; For the small scale textures, it can be well kept as employing the fractional derivative and adaptive regular parameter. Numerical experimenttal resules show that the proposed model can effectively keep edges and ramps, while enhances textures.(4) In order to better depict the important visual geometric structure in an image,a class of fractional telegraph-diffusion equations based on fractional structure tensor for image structure-preserving denoising has been proposed. In addition, the analysis of the existence and uniqueness of the solution of the proposed model has been carried out theoretically. Fractional-order tensor diffusion can not only inherit the characteristics of fractional calculus, but also accesse more additional image structure information by using the characteristics of structure tensor. Therefore, it can fine distinguish image important structures such as flow-like structures, corners and T shape structures(structures with different direction), and also can detail with weak structure, small scale structure and fractal structure. In this dissertation, a fractional telegraph-diffusion equation is proposed, which forms a “natural interpolation” between the second order telegraph- diffusion equations and the fourth order ones. It can be viewed as a damped wave equation, enable better to preservate edges than the other diffusion based methods, and achieve a good tradeoff between staircase effect and speckle artifacts. Fractional-order structure tensor is employed to replace the usual scalar diffusion function or integer order structure tensor can obtain the anisotropic diffusion in a real sense or can deal with better complex fractal textures where gray-level does not change evidently. Numerical experimental resules show that the proposed model is significant in improving the denoising effect, while can achieve a better performance of structure features preservation and enhancement.