Research On Partial Differential Equation-based Methods For Image Processing

Image is an important method to obtaining and conveying information.It is widely used in human life,and is critical for scientific research and social production.Partial Differential Equation(PDE) is an important mathematical analysis method,and its property is determined by the diffusion directions and diffusion items in the equation.In this paper,the usage of PDE in image processing is deeply analyzed and compared.The PDE anisotropiccally diffuses in image domain and the diffusion procedure is constrained by the local geometric information.The diffusion items and directions could be computed by the geometric properties in image directly.So PDE smoothes image while preserving the edge information.In this thesis,we focus on the research on the PDE models used in image processing.We give the profound research on functional theory,Markov random filed theory,Wavelet transform analysis,pattern recognition etc..We analyze the validity of PDE models and prove the relationship between PDE and other modern image processing methods.Our objective is to explore the interdependency and complementarity between PDE and the image processing methods based on the global information.We explore the effective PDE models and the composite models used in image denoising,image magnification,image inpainting,and the pre-processing step for face recognition under complex conditions.The anisotropic diffusion property, smoothing and locality preserving property of PDE are analyzed and proved from various aspects and feature levels.The main creative work in this thesis is summarized as follows:1.The composite image denoising models.In this paper,the relationship between PDE and the Wavelet shrinkage denoising method,and the relationship between PDE and the Gaussian-Markov model are proved based on the Functional theory and the Markov random field theory,respectively.Then two composite denoising models are proposed.The composite models could denoise while preserving the edge and textured information in image.Also it decreases the blocky effects caused by PDE. There is no Gibbs phenomenon in the denoised results;2.The two-layer constrained image magnification models.There are severe zigzag effects and ringing effects in the interpolated image processed by the conventional magnification models.In the two-layer constrained models,the processing procedures are executed under the constraint of the geometric information in the original image.So the processed image is smoothed and preserves the geometric property;there is less zigzag and ringing effect.The two-layer constrained model has two forms:A PDE model which forward diffuses along two orthogonal directions.It preserves the linear structure and removes the zigzag and ringing effects.The diffusion items and directions in this PDE model are the constrained result of the geometric properties in the original image;A magnification model which is based on both local and global image information.This model uses a bi-directional diffused PDE to preserve the geometric property.Then the zigzag and ringing effects are weakened while preserving the linear structure.The textured and noisy information in the image is processed by the non-local means algorithm.This model could be directly used to magnify the natural image,texture image and noisy image;3.A PDE inpainting model which is used to restore the image with a small target region and containing no textured information.The gradient information is diffused into the target region by minimizing the total variation energy function.The pixel's gray value is diffused along two orthogonal directions based on the updated geometric vectors.The diffusion PDEs are all morphological invariant.The novel model could smoothly restore the image while preserving the geometric property;4.An exemplar-based model which is used to restore the natural image with the large target region and containing the textured information.The exemplar is the basic processing unit and its processing order is determined by a cross isophote diffused PDE.This PDE is morphological invariant,and it is diffused under the constraint of the edge width.The exemplar along the linear structure has a higher processing priority.So the model could preserve the geometric property of image better.The similarity comparison of exemplar is implemented in the CIELAB space and PDE is used to remove the seams between exemplars.The model is further extended for more complex restoring tasks;5.A pre-processing model for face recognition under complex conditions.The model uses PDE and the Lambertian surface model to extract the illumination invariant small-scaled image features.Also it uses PDE and the region-based histogram equalization method to extract the enhanced large-scaled image features.Last,two scaled image features are fused at the feature-level.The face sample processed by this model is robust to the lighting conditions.This model achieves good performances in the face databases with the large variations of expression,occlusion, etc.In this paper,the diffusion items and diffusion directions of PDE are analyzed based on the mathematical theory and image processing theory,respectively.The difference and connection between PDE and other modern image processing methods are proved.Different PDEs are used in low-level,middle-level and pre-processing for high-level image processing fields.They are combined with other modern image processing methods and achieve good performances in different image processing tasks.