A New TV Model For Oscillatory Patterns

Image processing is widely used in astronomy, medicine, computer vision and many other fields. But the image will receive the noise pollution frequently. If people directly carry on other processing to the image with noise, such as edge detection, image segmentation, feature extraction, pattern recognition and so on, then the result is clearly unsatisfactory. For further processing, carring on the de-noising to the image is reliable guarantee. The image de-noising belonging to the technical category of the image restores is an ancient and challenging task.With the variety of large-scale digital images and digital TV appearance, people need one kind of image restoration method which the condition necessary to be simple, real-time stronger, and effect better. However, most algorithms have not yet attained a desirable level of applicability.For a long time, according to the actual characteristics of the image, the statistical characteristics of noise, and the frequency spectrum distributed rule, people has developed many de-noising methods. Generally speaking, the de-nosing method can be used to remove the noise, but, it will often lose the high-frequency characteristic of the target image, causing the edge and texture fuzziness. In other words, it has contradiction between noise reduction and edge,texture protection in the denoising process. Moreover, the algorithm complexity direct impact on the effectiveness of its practical applications, and is also the factor which we need to consider.This article mainly divides into two parts, respectively be the discussion of the image denoising method and the statement of a new TV model for the oscillation texture. In the first part, this article mainly carries on the discussion to several typical de-noising methods, including the dual-tree complex wavelet combine with the hidden Markov model denoising, Curvelet transform threshold denoising, wave stoms transform threshold denoising, PDE model combine with split Bregman method denoising. This article carries on the theoretical analysis to the above question, including the construction of the basic elements and the discussion of the details of the algorithms. Then we achieve the results through numerical implementation and sum up the advantages and disadvantages and associated research tendency of each kind of the method. At the end, the future trend of image de-noising is pointed out, though, in personal opinion. In the second part, we propose a new TV model for the oscillation texture. The solution method is given by using split Bregman iteration. Then we give the proof of the astringency. Compared with the traditional TV model, the new model has a good denoising effect and can maintain the texture better.In the chapter one of this paper, we outline the significance of image de-noising, as well as domestic and foreign scholars for research to image denoising. We classify and discuss all kinds of denoising method, including linear filtering and nonlinear filtering algorithm, and wavelet based denoising algorithm, as well as PDE based image denosing algorithm.In the chapter two, we first summarize the mathematical essence of wavelet transform denoising. The wavelet-based denoising approach can be sum up as follows:1) uses the appropriate wavelet function space, in order to achieve the best image representation.2) formulates the appropriate metrics, namely, carries on corresponding processing to the wavelet coefficient, in order to achieve the best denoising results.Then we respectively discusse the dual-tree complex wavelet transform denoising, Curvelet transform denoising and wave atoms transform denoising, form the construction of one-dimensional wavelet bases to the construction of two-dimensional wavelet bases, then, to the discretization, the fast algorithm and the numerical implementation. Separable wavelet, dual-tree complex wavelet, curvelet and wave atoms have basically reflected the various stages of the development of the wavelet basis. DT-CWT has multi-scale, approximate shift-invariant and multi-directional nature, which makes them easily maintain the edge features in image denoising, compared with may separate wavelet transform. But there is limitation that the direction is insufficient. Curvelet has the directional parameters, achieve the heterosexual relationship between scales, and its approaching rates superior to the traditional wavelet transform. However, for more complex images, such as the texture image, curvelet transform can not reach optimal. The wave atoms transform has given the vibration function most optimal sparse representation, therefore achieves good results in maintaining the texture. Although the wavelet transform develops gradually in the image characteristic representation, there is big redundancy in the de-noising applications, and the computing time is long. Wavelet transform denoising is difficult to achieve optimal. If we use wavelet threshold denoising, the denoisied image will has seriously pseudo-Gibbs phenomena. Because of the multi-directional of the basis, the processed image appears scratches, and this phenomena greatly reduce the de-noising effect. If we use the complex model to carry on coefficient processing, the operation complexity will be increased, but it can maintain the texture better, moreover the computer operating speed enhanced unceasingly has provided the feasibility for this reason.In the chapter three, we first summarize the PDE based denoising methods. We divide it into axiomatic approach based on partial differential equation, geometric partial differential equation method and minimization of energy function method based on variation method. Then we focused on the variational-based methods, including the model proposed and development, as well as solving methods. We have theoretically analyzed the advantages and disadvantages of such methods.After again, we introduces splits Bregman method, and solve the ROE denoising model with it. Finally, adoption of numerical experimentation, we prove that RDE model denoising can maintain image edge well, but for the image texture features, it falls short. Using split Bregman method to solve the ROE denoising model demonstrated efficient.In the chapter four, we propose a novel model for the oscillation texture image——TV model for oscillatory patterns.The fifth chapter is a summary of the above three chapters and some future prospects: Whichμis the weight determined to the experience,Du is the local direction variation, namely, the partial difference direction is the texture derivative direction. Then we give the solution method by using the split Bregman iteration.And eventually solution procedure translates into the following iteration step: Then we give the proof of the astringency.Theorem 4.1 Let "u be the unique solution of the minimization problem(4.1). ForThe split Bregman iteration greatly increased the speed of the solution. The effectiveness is guaranteed, moreover, the nature of the model itself had decided that the model is simple and effective in image denoising application of oscillation texture image.The fourth chapter is a summary of the above three chapters and some future prospects. Considering the advantages and disadvantages of the total variation-based model and the wavelet transform denoising, the TV-synthesized wavelet transform will be one of focus of the future research. If split the Bregman iteration and the wavelet fast algorithm unifies, that will definitely raise the operating speed greatly.In the near future, with the increase in computer performance, high-precision denoising model and the algorithm will have better prospects.