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# Wavelet and Variational PDE Based Image Modeling Theory, Algorithm and Application

Author: LiMin
Tutor: FengXiangChu
School: Xi'an University of Electronic Science and Technology
Course: Applied Mathematics
Keywords: Wavelet transform Variational PDE approach Wavelet shrinkage Edge detection Image decomposition Image zoom Image restoration Multiscale image representation Iterative regularization Inverse scale space Besov space
CLC: TP391.41
Type: PhD thesis
Year: 2008
Downloads: 634
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### Abstract

 Wavelets and variational PDE theory are two basic mathematical tools in image processing and the low-level vision process. This dissertation focuses on combination of wavelets and variational PDE for the major problems of image restoration, decomposition, magnification, multiscale image representation and edge detection. The main work can be summarized as follows:1. The paper first gives a new model for image restoration which is based on variational model with an adaptive fidelity term (G-model) in BV space and“pure”anisotropic diffusion equation. Due to the introduction of“pure”anisotropic diffusion term, this model can not only removal noise but also enhance edges and keep the locality of them. And it can also keep textures and large-scale fine features that are not characterized by edges. Because of these favorable characteristics, not only the processed images look much more clear and smooth, but also significant details are kept, resulting in appealing vision. Secondly, the paper discusses combination of variational model with H ?1 norm (OSV model) in BV space and wavelets. On the one hand, a total variation (TV) denoising model can be obtained by replacing H ?1 -norm with norms of wavelet coefficients in OSV model. Since this method leads to combining geometric information of image with wavelet transform, one does not solve the associated PDE systems. On the other hand, in terms of the property of the residual v of being of zero mean in L2 (Ω), the paper gives the wavelet Galerkin algorithm of OSV model. And experimental results verify the efficiency of the algorithm.2. The paper deals with the applications of variational methods in image decomposition. Following the ideas of Y. Meyer in a total variation minimization framework, we discuss the generalization of theoretic models (BV, G ) and (BV, E ) in Besov spaces. And instead of solving PDE, we propose wavelet based treatments of the generalized models in terms of the equivalence between Besov semi-norm and the norms of wavelet coefficients. Concretely speaking, there are the following three models. Firstly, a new class of variational models based on Besov spaces and negative Hilbert-Sobolev spaces is given. By the description of equivalence, the associated minimizers of this model can be expressed through applying different shrinkage functions that depend on the wavelet scale to each wavelet coefficient. Secondly, we propose the variant of Besov smoothness term in the above model. Thus, an improved model is produced. To this model, we give a projection algorithm based on wavelet. Here the associated minimizers of the improved model can be interpreted as the global shrinkage functions. Thirdly, a class of image decomposition models using Besov spaces and homogeneous Besov space is proposed. And we obtain the successive iterative scheme and the rigorous convergence analysis for this model.3. The three iterative regularization methods are given. Firstly, we develop a new iterative regularization procedure for VO model, which is based on the use of generalized Bregman distance. And we give the simple numerical algorithm for new iterative regularization sequence. Experimental results show that the new method improves significantly the quality of restoration when VO standard model is used to real textured image denoising. Secondly, we present a new class of iterative regularization methods in Besov spaces B1α,1 (Ω)(α> 0). By incorporating translation invariant wavelet transform, minimizers of the new methods can be understood as the alternative to translation invariant wavelet shrinkage with weight that is dependent on wavelet decomposition scale、Besov smoothness order、scale parameter and iterate degrees. In addition, by idea borrowing from total variation based image restoration, we generalize the iterative regularization methods to a new class of nonlinear inverse scale spaces with weight. Finally, a new model for multiscale image representation using hierarchical ( )BV ,H ?1 decompositions is given. Due to the introduction of monotone scale parameter, this method offers a hierarchical, adaptive representation for the different features of general images in the intermediate spaces lying between the larger H ?1 (Ω) and the smaller BV (Ω). The numerical results show an excellent decomposition effect.4. Inspired from idea of the AAFC image decomposition model, by introduction of the half-quadratic regularization, the paper proposes a coupled image decomposition and edge detection model with variational formulation. This model can split an image into three features of a structure component, an oscillating component (both textures and noise) and the edges. And we obtain the successive iterative algorithm based on a projection and two coupled PDE systems. In addition, we give the associated numerical discretization of the model. Experimental results demonstrate its superiority.5. A new algorithm for image zooming combining variation and wavelet transform is proposed. We find the zoomed image by minimizing the variational functional which uses the Besov norm to measure the regularity of an image. Thus, the variational functional can be minimized by the wavelet projection scheme. And experimental results verify the efficiency of the zooming algorithm.

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CLC: > Industrial Technology > Automation technology,computer technology > Computing technology,computer technology > Computer applications > Information processing (information processing) > Pattern Recognition and devices > Image recognition device
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