Partial Differential Equation Models With Numerical Implementations For Image Restoration And Segmentation

Images are the most important way for us to get information from the outer world.Due to various objective factors or improper processing,the images may be degraded images with noise,blur,or low contrast.This could influence the quality of the subsequent processing(image segmentation,recognition,and understanding).Therefore,we need to restore the “real” image as much as possible,such as denoising and deblurring the image.However,image restoration is mathematically an inverse problem that is ill-posed.Thus,image restoration has always been a challenging topic in image processing.Image segmentation is a basic task in the field of image analysis.The results of segmentation would influence the subsequent processing steps,such as image analysis and understanding.Since there is no universal criterion,the judgment of segmentation results is usually uncertain,image segmentation has also been a challenging problem in image processing.This dissertation mainly studies the restoration(denoising,deblurring)and segmentation(binarization)of images under the framework of partial differential equations(PDE).The main ideas behind this paper is the uses of shock filter and diffusion and exponential smoothing in time series analysis.The shock filter is used for deblurring of images and enhancing of low-frequency components,while the role of edge-preserving diffusion is to smooth out false details(such as noise)and retain the object edges.The research contents are as follows:(1)Adaptive shock-diffusion model with numerical method for image restorationWe propose a PDE model with regularized shock filter and diffusion operators for the restoration of blurred images with noise,in which the adaptability is achieved by an edge detection operator.In this study,different from Gaussian regularization,we use a mollifier(instead of Gaussian function)as the convolution kernel function,while calculating the corresponding convolution mask for the numerical implementation.In order to solve our model efficiently,we propose a hybrid numerical algorithm combining finite difference and exponential smoothing.Experiments show that our model has a better restoration effect on blurred images with noise,compared with the related PDE model.(2)Regularization of Perona-Malik equation for image denoisingPerona-Malik anisotropic diffusion equation(PM model)plays an important role in the field of image processing based on partial differential equation.However,it has the problems of ill-posedness and sensitivity to noise.To address both problems,some authors have proposed the regularization(spatial,temporal and spatio-temporal regularization)of PM model.In these models,the regularizations are achieved by regularizing the gradient of the evolving function in the diffusion coefficient;in some literature,the well-posedness of the models have been discussed in the sense of weak solutions or in discrete situations.In our study,the temporal regularization is considered as the specific case of exponential smoothing;thus,the temporal regularization could be studied within the exponential smoothing framework.For that,we propose a new spatio-temporal regularization version of PM model;it is a system of two partial differential equations.Different from the existing regularized PM models,the regularization for our model is achieved by smoothing the evolving function in the diffusion coefficient instead of the evolving function gradient.This model has the advantages of both well-posedness and robustness to strong noise.A semi-implicit parallel splitting-up method is designed to solve the model efficiently,and the stability of the numerical scheme is also discussed.(3)Directional diffusion-based model for segmentation of two-phase imagesA two-phase image(such as document image)refers to an image with two different gray scale ranges.The area corresponding to one gray scale range is called the object(s),and the area corresponding to another gray scale range is called the background(s).For this type of image,the edges are the important feature;the edge-preservation for the objects is particularly important during the segmentation process.By directional diffusion,we propose an edge-preserving segmentation model with source term,and design a semi-implicit parallel splitting-up method to solved our model numerically.Experiments have confirmed the effectiveness of the model for the segmentation of two-phase image.(4)Binarization of degraded document images with low contrast or contrast variationThe low contrast or contrast variation of the text is one of the typical degraded features for document images.We propose a PDE-based model for the binarization of degraded document images with low contrast or contrast variation,which couples a selective shock filter and a time-dependent binary classifier to a fourth-order indirect diffusion process.The fourth-order indirect diffusion not only has the capability of noise immunity,but also preferably preserves the low-frequency components(such as weak edges of text).The role of the shock filter is to enhance the weak edges,and avoid the misalignment of the weak edges caused by diffusion.The time-dependent binary classifier plays an increasingly predominant role in the process of binarization.We design a semi-implicit parallel splitting-up method for solving our model.The proposed model is tested on DIBCO(Document Image Binarization COmpetition)datasets.The experimental results indicate that our model is very effective,compared with four related PDE models and four benchmark and recent non-PDE methods.