Variational And Partial Differential Equation Models For Image Processing With Numerical Simulation

Image processing has interdisciplinary characteristics,which is connected to pattern recognition,automatic control,and computer vision.At the current stage,there are mainly model-driven traditional methods and data-driven learning methods in the field of image processing.This thesis focuses on some key issues in image restoration,image smoothing,and image decomposition with traditional methods based on variational regularization and partial differential equations(PDEs).The specific research contents are as follows:The anisotropic diffusion equation plays an important role in the field of image processing based on PDEs.In this study,an anisotropic diffusion equation with a source term is proposed for restoration of degraded document images with bleed-through,in which the diffusion term is used to selectively smooth the bleed-through texts,and the source term is responsible for restoring the polluted background in the image.Meanwhile,a parallel-series splitting algorithm is designed to solve the proposed model numerically,by combining the finite difference,series splitting algorithm and parallel splitting algorithm.Experiments have confirmed the effectiveness of the model for the restoration of bleed-through degraded document images.Guided image filtering(GIF)is an edge-preserving smoothing operator with low time complexity;however,it does not preserve sharp edges,and therefore exhibits halo artifacts caused by the edge blurring.To address this problem,a robust double-weighted guided image filtering(RDWGIF)is presented in this paper,which includes a data fidelity term based on a mollifier from Sobolev space theory and a regularization based on robust edge-aware weighting(REAW).To illustrate its effectiveness in preserving sharp edges and reducing halo artifacts,the proposed RDWGIF is applied to edge-aware smoothing,image denoising,single image detail enhancement,tone mapping of high dynamic range images,and texture removal smoothing,and compared with previous weighted versions of the GIF.Most of the existing variational Retinex models are proposed by combining the data fidelity term with different regularization terms of the illumination and reflectance components;however,they do not consider the collaborative relationship between illumination and reflection components.Based on this observation,a variational Retinex framework model is proposed by considering the cooperative mechanism of illumination and reflection components for the estimation of illumination and reflection components.To numerically solve this framework model,it is decomposed into two sub-problems regarding to the illumination component and the reflection component,and then their Euler-Lagrangian equations are derived.In order to reduce the computational cost,following the idea behind parallel splitting,the Euler-Lagrange equations are discretized by combining the central difference scheme,and the approximate solution is obtained.Finally,the framework model is applied to low-light image enhancement and tone-mapping of high dynamic range(HDR)images.