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Study of Partial Differential Equations and Wavelet in Image Restoration and Feature Extraction

Author: SunXiaoLi
Tutor: SongGuoXiang
School: Xi'an University of Electronic Science and Technology
Course: Applied Mathematics
Keywords: Partial Differential Equations Wavelet Image Denoising Image Inpainting Variational Morphology Structure tensor Diffusion tensor Direction of diffusion equation
CLC: TP391.41
Type: PhD thesis
Year: 2007
Downloads: 732
Quote: 3
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Abstract


Over the past ten years,Partial Differential Equations(PDEs),as a new tool,have been widely used in image processing field.The combination of PDEs and image processing injects new vitality for many research areas based on image processing. Wavelet also plays a more and more important role in image processing and related fields,there are successful examples in image restoration,image segmentation,target recognition and so on.In this dissertation,the applications of PDEs and wavelet in image denoising、image inpainting and fractal feature extraction of IC defect outlines are discussed.The main work can be summarized as follows:1.An adaptive anisotropic diffusion equation based on morphology is proposed. Based on the character that morphological operator can estimate the intensity of noise, a new operator is defined,which used to estimate the threshold of gradient.The type of pixels and the two regular coefficients in tangent direction and normal direction can be determined by the threshold.Then the two regular coefficients in different pixels vary adaptively on their types.In addition,inspired by the variational model,a fidelity term is added into the diffusion equation.Experiments show that the new diffusion equation performs better for removing noise while preserving edges.2.A secondary image filtering model based on noise-texture operator is proposed. Since details and textures always will be compromised in the process of filtering,a local energy operator and a noise-texture operator are proposed.Combined with Weickert’s coherence-enhancing model,the textures and details filtered by mistake during the process of filtering can be extracted as much as possible.After refilling detailed information into the filtered image,final filtered image can be obtained. Experiment results show that our new method can be a follow-up process of any other filtering methods,and the results will be obviously improved after the secondary filtering process.3.Through the following three improvements of Weickert’s tensor diffusion equation,an adaptive tensor diffusion equation model is proposed.Firstly,when determining the orientation of edges,the large-scale Gaussian kernel is always inaccurate,so an edge-orientation operator based on nonlinear wavelet threshold is presented.Secondly,combining edge-enhancing model and coherence-enhancing model,we introduce a weighted function and reset the two eigenvalues of diffusion tensor.Finally we add an adaptive fidelity term into the tensor diffusion equation, which can adaptively vary with the time scale and the spatial position.Experiments show that the new model has a more extensive scope of application,and the details and edges will be retained better during the process of denoising.4.The correlation between directed diffusion equation and wavelet transform is studied.Combining the soft wavelet shrinkage model,an improved directed diffusion equation model is proposed.Firstly,we find that directed diffusion equation can realize the gradual variations of wavelet decomposition and reconstruction in two contiguous levels.The two methods in the realization process are different,so they are not substitutable.By substituting anisotropic diffusion operator for isotropic Laplace operator in directed diffusion equation,adding two different coefficients in the two diffusion terms,and choosing the denoised image by soft wavelet shrinkage as the initial approximation image,an improved directed diffusion equation model is proposed.Experiments show that the new model overcomes the disadvantage that the edges will be rapidly blurred during the diffusion in directed diffusion equation.The denoising result is better.5.Two image inpainting models are proposed:a 3rdPDE image denoising and inpainting model based on Taylor expansion and a wavelet image inpainting model based on curvature driven.When inpainting,the image information should be transmitted and extended along the isophote direction.Therefore,3rdTaylor expansion is made along the isophote direction in this paper,by omitting senior infinitesimal,the new 3rdPDE image inpainting model is proposed.In particular,we testified that the new model is morphologically invariant.At the same time,different smooth processes are added inside and outside of the inpainting domain.In our new model,the denoising process and inpainting processe can be performed simultaneously.Based on the disadvantage that small bars can not be connected successfully by total variation wavelet inpainting model,a curvature-driven term is added into the energy functional, and the new regular criterion is defined.Then a wavelet inpainting model based on curvature driven is proposed.The new regular criterion requires that the length of edge should be shortest and the curvature should be continuous.The Euler-Lagrange equation of the new model is derived by variational method,and the discrete format is given.Experiments show the effectiveness of these two new models.6.The fractal feature extraction of IC defect outlines based on wavelet is studied. Projecting the defect outline in x direction and y direction,we verify that the two sequences are both with the fractal feature.The fractal dimensions of these two sequences are estimated by wavelet method.In addition,based on the inaccuracy of single fractal dimension in characterizing the outline,we propose that the outlines have multifractal features,and the multifractal spectrum of one typical real defect outline of the two sequences is estimated by the method of wavelet transform modulus maximum(WIMM).The new parameters for the precise characterization of IC defect outline are proposed.

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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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