Adaptive Partial Differential Equation And Image Denoising

Image denoising is fundamental problem in both image processing and computer vision with numerous applications.Image denoising aims to preserving important struc-ture features including edges and corners while removing the noise in the image.The method based on partial differential equations have achieved fast development because of their strong local adaptability,high flexibility and broader mathematical theory.This thesis mainly uses partial differential equation to study the problem of image denoising.We propose three new models for noise removal.Moreover,we prove the existence of a weak solution of the proposed models.In charter 2,in order to overcome the drawback of TV model,we propose an adap-tive second order partial differential equation for noise removal,which combines TV and p-Laplacian(1<p?2).Utilizing the edge indicator,we can adaptively control the diffusion model,which alternates between the TV and the p-Laplacian(1<p ?2)in accordance with the image feature.The main advantage of the proposed model is able to alleviate the staircase effect in smooth regions and preserves edges while removing the noise.The existence of a weak solution of the proposed model is proved.Experimental results show that the proposed method achieved good denoising result.In charter 3,to overcome the weakness of second order methods such as Peron-a-Malik model for image denoising,various high order models have been proposed and studied.However,there is not too much analysis of these equations to be found in the literature.In this paper,we propose an adaptive fourth-order partial differential equation,which joints a fourth-order term and a second-order term.The model takes advantage of the fourth-order model's better image avoiding staircase effect and the second-order model's better edge preserving effect.By introducing a functional framework and k-bounded partial variation(BPVk)space,we prove the existence of a weak solution of the proposed model.Experimental results show that the proposed model can alleviate the staircase effect and preserve edges accurately.In charter 4,using the fractional derivative defined in Fourier transform domain,an adaptive fractional-order partial differential equation(PDE)has been proposed which joints a fractional-order equation and a total variation(TV)equation.The two diffusion can be adaptively selected according to the edge detector function.When the pixels locate at the edges,the TV equation is selected to filter the image,which can preserve the edges.When the pixels belong to the flat regions,the fractional-order equation is adopted to smooth the image,which can eliminate the staircase effect.The existence of a weak solution of the proposed equation is proved.A set of experiments demonstrates that the proposed method outperforms the state-of-the-art methods cited in the paper in both the qualitative and quantitative evaluations...