A Fast Algorithm For Forth-order Vector-valued Image Denoising

Image processing techniques based on PDEs have been widely used in image restoration now. Image denoising and image deblurring are two major problems in image restoration. The basic idea of image denoising based on variational PDE is to construct a regularization term which is suitable for image denosing, then gain the corresponding variational PDE model according to the variational principle, and finally use numerical methods to solve the model, from which we get the denoising image and achieve the purpose of image denoising.In1992, Rudin, Osher and Fatemi presented the total variation model, which can be utilized to obtain the denoised image with removing noise while preserving the edge of an image. However, in many practical applications. TV solutions are piecewise constant, and they will lead to staircasing effects, which will develop "false edges" that doesn’t exist actually. So researchers proposed high-order mod-els, e.g., the LLT model presented by Lysaker, Lundervold and Tai. to overcome the above shortcomings. It has been proved that LLT model can not only preserve edges well, but also weaken the staircasing effects from the TV model. In addi-tion, variational techniques for gray-scale image denoising have been extensively studied, however, little research has been done for vector-valued images (eg. col-or images) denoising. So in the dissertation we will consider vector-valued image denoising models.With the LLT model as a starting point, combining the above ideas, we present a new model with high-order coupling term—coupling multi-channel LLT-based model, and compare it with existing multi-channel models. In the algorithm ap-plication aspect, the dissertation studies a class of fast algorithms—augmented Lagrangian method to remove noise. The thesis consists of five chapters in the following:In the first chapter, the basic concepts of digital image processing and the basic knowledge of PDE-based image processing are briefly presented, and the background and content of this dissertation are also summarized.The second chapter is devoted to introducing some basic mathematical knowl-edge, including bounded variation function space, variational method, convex anal-ysis and related quantitative criteria, which are theoretical foundation of image denoising.In the third chapter, we first introduce three low-order vector-valued TV models, then present three kinds of forth order multi-channel coupling models, and simply analyze the similarities and differences between the six models.In the fourth chapter. we first introduce the general idea of augmented La-grangian method for solving the TV model, and then apply the method to the coupling multi-channel LLT model, finally we prove the convergence of the algo-rithm.In the fifth chapter, the numerical implementation process of the augmented Lagrangian method is derived in detail, discretization of the corresponding for-mulas is obtained, and the numerical experiments are implemented. Experimental results show that the augmented Lagrangian method is fast with good vision effect.Finally, we conclude the dissertation and point out the subjects of future researches.