Research On Image Restoration And Segmentation Models Based On High-order Variational Partial Differential Equations

With the development of big data and the popularization of smart devices,images are playing an important role in the fields of digital medicine,industrial inspection,multimedia,and criminal investigation.The research on image processing technology is becoming more and more important and urgent.In particular,image restoration and image segmentation are two important research topics in the field of image processing.In the field of image processing,variational partial differential equation methods have been widely studied due to their strong local adaptability and high flexibility.Low-order variational and partial differential equation models often cause the “staircase” effect,resulting in false edges during image processing.In this dissertation,the geometric properties of highorder variation and partial differential equations are employed to eliminate the “staircase”effect.For image restoration and image segmentation,high-order variational and partial differential equation models are proposed.The main research contents are as follows:For image denoising,in order to preserve edges in the image,an adaptive high-order variational model is proposed by introducing a gradient-based weighting function.Furthermore,based on the Euler-Lagrange equation corresponding to the new variational model,a class of fourth-order nonlinear anisotropic diffusion equation models are proposed.The new diffusion coefficients depend not only on the first-order derivative describing the edge of the image,but also on the second-order derivative describing the geometric features of the image,so the new model can effectively preserve the edge and geometric details while removing noise.For numerical implementation,since the convergence condition of the finite difference explicit scheme of the fourth-order equation is relatively strict,the fast explicit diffusion scheme and the additive operator split implicit scheme are employed to improve the computational efficiency.The experimental results show that the new model has better performance in edge and geometric feature preservation and image denoising.For image restoration,this dissertation proposes a nonlocal adaptive biharmonic regularization term in the variational nonlocal framework by taking advantage of the similarity of image patches.Based on the new regularization term,variational nonlocal models for image denoising,image deblurring,and image inpainting are proposed,respectively.By the high-order property,the new model can preserve the geometric features of the image.On the other hand,by the nonlocal property,the new model can preserve the image texture details.Based on the method of contraction mapping,this dissertation proves the existence of the solution,and further analyzes the uniqueness and mean invariance of the solution.For the problem of imbalance of similarity patches,a weight matrix normalization method is proposed,and a finite difference explicit scheme and semi-implicit scheme for the Euler-Lagrange equation corresponding to the variational model are proposed.Simulation experiments show that the new models have the advantages of preserving texture and geometric details and preserving image consistency in image denoising,image deblurring,and image inpainting tasks.For the problem that the BM3 D method deeply depends on the noise level parameter and has artificial and bias effects,this dissertation proposes a non-parametric single image denoising method based on the BM3 D method and partial differential equations.On the one hand,a partial differential equation filter is proposed,and the discontinuity maintaining,mean invariance,convergence,and local continuity of its solution are proved.Based on these theoretical properties,a class of noise level estimators based on partial differential equation filtering are proposed.On the other hand,combining the BM3 D method with the partial differential equation filtering,a stable BM3 D method is proposed to avoid the artificial effect and bias effect.Furthermore,based on the partial differential equation filtering-based noise level estimator and the stable BM3 D method,a non-parameter single image denoising method is proposed.Comparison experiments with other blind denoising methods show that the proposed method can effectively remove noise for real images and preserve detailed features.For image segmentation,the surface diffusion current and nonequilibrium diffusion current in the molecular beam epitaxy equation are employed as the regularization terms in the level set evolution equation,and a high-order level set regularization variational model based on the molecular beam epitaxy equation is proposed.The nonequilibrium term of the new model constrains the magnitude of the gradient of the function to avoid it being too large or too small,thereby avoiding re-initialization.On the other hand,the surface diffusion current term can control the smoothness of the segmentation contour,thereby overcoming the influence of noise on the segmentation results.The level set evolution equation is obtained by calculating the Euler-Lagrange equation corresponding to the variational model,the finite difference semi-implicit scheme of the evolution equation is designed,and the Fourier transform is applied to calculate the numerical solution of the new model.Simulation results show that,while avoiding re-initialization,the proposed model can lead to smooth segmentation curves and retain fine segmentation targets for noisy images with intensity inhomogeneity.