Image Restoration Model With Multi-regularizations And Numerical Implementations

Digital image processing is one of the most popular interdisciplinary research fields to-day,covering the research topics of computational mathematics,information science,computer technology and other applied sciences.Image processing constitutes of several important di-rections,including image restoration(image reconstruction from complete or incomplete data),image enhancement(image denoising,image deblurring),image compression,image encoding and decoding,image segmentation,image patching,color image processing and others.These different image processing technologies have been widely used in different applied areas such as biomedical imaging,geographic remote sensing,aerospace imaging,industrial production,in-formation transmission.The mathematical models describing these image processing problems are essentially the problems to find the stable numerical solutions by the ill-posed problems,which are covering different mathematical disciplines such as partial differential equations,op-timization theory,regularization methods,and numerical calculations.Since the different image recovering models and input data,there are lots of problems about both theoretical analysis and numerical calculations in image processing to be studied furthermore.In the generalized senses,image restoration includes the process of image denoising and deblurring,which is the process to recover the image based on complete or incomplete data.It is a kind of image processing problem which is more extensive than the standard image enhance-ment.The main tasks and difficulties in image restoration are to catch the picture characteristics such as interfaces and textures from incomplete frequency data.This topic is mainly to study the image recovery from incomplete frequency data,with important applications in medical image recovery,that is,to reconstruct the identifiable and observable medical images from incomplete noisy data,which should be the diagnosed medical images for doctors.In the image restoring process,it is necessary to keep the interface characteristics of the objective images while the noises are removed as much as possible.In this thesis,three different image restoration models with multiple regularization penalty terms are studied to restore the images,that is to recover the images with additive noise and multiplicative noise from incomplete sampling data,and then to find the main applications for recovering the medical images.The theoretical analysis of the reconstruction schemes,the choice strategies for multi-regularization parameters,as well as the stability and convergence of the reconstruction algorithms are established.More precisely,the thesis has the following five chapters.The first chapter summarizes the background and existing work related to image restora-tion,and introduces the mathematical representations of the image restoration models with mul-tiple regularization penalty terms.The theory of compressed sensing(CS)is also introduced,including the sparse representation,the constructions of the models,algorithms and the mea-surement matrix(or sampling matrix).Based on the CS theory,the random band sampling(RBS)method based on incomplete frequency data applied in the image reconstruction algo-rithms are discussed.Finally,the main research topics and innovation points in the thesis are stated.In chapter two,the image restoration based on incomplete frequency data with additive noise are studied for piecewise smooth images.The multi-regularization image restoration model is established,which is consisted of the data matching term in frequency domain,total variation regularization penalty term and Frobenius norm penalty term in spatial domain.The properties of the minimizer of the cost functional and the error estimates on the regularizing so-lution are theoretically established.Then a double recursion scheme is proposed to restore the piecewise smooth images.The Bregman iteration scheme with the fixed point method is used to solve the corresponding nonlinear Euler-Lagrange equation.A new algorithm is proposed by the block symmetry coefficient matrix in the linearized Euler-Lagrange equation at each iteration step,i.e.,the properties of the block matrices in numerical implementation are applied to accel-erate the double recursion scheme.By implementing the proposed double recursion scheme,the satisfactory reconstructions are obtained by random sampling data.Numerical implementa-tions demonstrate the feasibility and validity of the proposed algorithm with satisfactory edges preservations.The third chapter considers the image restoration using random sampling frequency data with multiplicative noise.In terms of the image sparsity properties under wavelet base function expansion,an optimization model is proposed with a data fitting term in frequency domain and two regularizing terms(l~1sparse penalty term and total variation penalty term)specifying image sparsity and edge preservation on the restored image.For this model,the choice strategies for the regularizing parameters are rigorously set up together with corresponding error estimate on the restored image.A double recursion algorithm is constructed creatively,in which the cost functional with data-fitting term in frequency domain is minimized using the Bregman iteration scheme as the exterior loop.By deriving the gradient of the cost functional explicitly,the mini-mizer of the cost functional at each Bregman iteration step is also obtained by an inner iteration process,including directly iteration and adjoint conjugate gradient method with Tikhonov reg-ularization(ACGM).The double recursion scheme is implemented stably and efficiently due to the special structures of the regularizing iterative matrices.Numerical implementations are given to show the validity of the proposed scheme.In the forth chapter,the image restoration models with simplified l~1sparse penalty and total variation(TV)penalty term are studied.Firstly,an unconstrained optimization model with two non-smooth regularization penalty terms is proposed.With different smoothing approximation functions,such as Charbonnier function and Huber function,the regularization penalty terms are approximately smoothed.Based on the alternating iteration and augmented Lagrangian multiplier method(ALM),we introduce the gradient operator as a new variable,and a con-strained optimization model is then proposed with an alternative iteration scheme containing a nested loop(AIS).In this scheme,the first sub-optimization problem could get the explicit solution through shrinkage thresholding in the inner loop.The second sub-optimization prob-lem is to minimize the modified Lagrangian function with regularization penalty in the exterior loop,which is to calculate the approximative linearized solution by solving the nonlinear Euler-Lagrange equation with adaptive regularizing parameter.Due to the special structure of the coefficient matrix in the Euler equation and the smooth approximations on the penalty terms,the proposed scheme AIS can be implemented stably and efficiently.Theoretical analysis on the convergence of the iterative scheme is proposed creatively,and it is pointed out that the it-erative sequence generated by the AIS is almost convergent.Numerical experiments show that the simplified image restoration models with multi-regularizations and the proposed scheme can effectively recover the piecewise smooth images,medical images and their rotated images in a short time.Finally,the last chapter of the thesis summarizes the main results of the work,and it is also pointed some future overlooks to the future research directions.