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Preconditioned Iterative Methods for Algebraic Systems from Multiplicative Half-Quadratic Regularization Image Restorations

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Numerical Mathematics(Theory,Methods and Applications) 2010年第4期3卷 461-474页

作者： michael k.ng Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong Kong

Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored *** this paper,we consider a class of convex and edge-preserving regularization functions,i.e.,multiplicative half-quadratic regularizations,and we use the Newton method to solve the correspondingly reduced systems of nonlinear *** each Newton iterate,the preconditioned conjugate gradient method,incorporated with a constraint preconditioner,is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived,which can be used to estimate the convergence speed of the preconditioned conjugate gradient *** use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably well.

关键词： Edge-preserving image restoration multiplicative half-quadratic regularization Newton method preconditioned conjugate gradient method constraint preconditioner eigenvalue bounds

ON ALGORITHMS FOR AUTOMATIC DEBLURRIng FROM A SIngLE IMAGE

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Journal of Computational Mathematics 2012年第1期30卷 80-100页

作者： Wei Wang michael k.ng Department of Mathematics Tongji UniversityShanghai 200092China Centre for Mathematical Imaging and Vision and Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong KongChina

In this paper, we study two variational blind deblurring models for a single linage,The first model is to use the total variation prior in both image and blur, while the second model is to use the flame based prior in both image and blur. The main contribution of this paper is to show how to employ the generalized cross validation （GCV） method efficiently and automatically to estimate the two regularization parameters associated with the priors in these two blind motion deblurring models. Our experimental results show that the visual quality of restored images by the proposed method is very good, and they are competitive with the tested existing methods. We will also demonstrate the proposed method is also very efficient.

关键词： Blind deconvolution Iterative methods Total variation Framelet Generalizedcross validation.

A Nonlocal Total Variation Model for Image Decomposition: Illumination and Reflectance

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Numerical Mathematics(Theory,Methods and Applications) 2014年第3期7卷 334-355页

作者： Wei Wang michael k.ng Department of Mathematics Tongji UniversityShanghaiChina Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong Kong

In this paper,we study to use nonlocal bounded variation(NLBV)techniques to decompose an image intensity into the illumination and reflectance *** considering spatial smoothness of the illumination component and nonlocal total variation(NLTV)of the reflectance component in the decomposition framework,an energy functional is *** establish the theoretical results of the space of NLBV functions such as lower semicontinuity,approximation and *** essential properties of NLBV functions are important tools to show the existence of solution of the proposed energy *** results on both grey-level and color images are shown to illustrate the usefulness of the nonlocal total variation image decomposition model,and demonstrate the performance of the proposed method is better than the other testing methods.

关键词： Image decomposition illumination reflectance nonlocal total variation iterative method

kernel Density Estimation Based Multiphase Fuzzy Region Competition Method for Texture Image Segmentation

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Communications in Computational Physics 2010年第8期8卷 623-641页

作者： Fang Li michael k.ng Department of Mathematics East China Normal UniversityShanghai 200241China Centre for Mathematical Imaging and Vision and Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong Kong

In this paper,we propose a multiphase fuzzy region competition model for texture image *** the functional,each region is represented by a fuzzy membership function and a probability density function that is estimated by a nonparametric kernel density *** overall algorithm is very efficient as both the fuzzy membership function and the probability density function can be implemented *** apply the proposed method to synthetic and natural texture images,and synthetic aperture radar *** experimental results have shown that the proposed method is competitive with the other state-of-the-art segmentation methods.

关键词： Texture multiphase region competition kernel density estimation fuzzy membership function total variation

Parallel Active Subspace Decomposition for Tensor Robust Principal Component Analysis

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Communications on Applied Mathematics and Computation 2021年第2期3卷 221-241页

作者： michael k.ng Xue-Zhong Wang Department of Mathematics The University of Hong KongHong KongChina School of Mathematics and Statistics Hexi University734000 ZhangyeGansu ProvinceChina

Tensor robust principal component analysis has received a substantial amount of attention in various *** existing methods,normally relying on tensor nuclear norm minimization,need to pay an expensive computational cost due to multiple singular value decompositions at each *** overcome the drawback,we propose a scalable and efficient method,named parallel active subspace decomposition,which divides the unfolding along each mode of the tensor into a columnwise orthonormal matrix(active subspace)and another small-size matrix in *** a transformation leads to a nonconvex optimization problem in which the scale of nuclear norm minimization is generally much smaller than that in the original *** solve the optimization problem by an alternating direction method of multipliers and show that the iterates can be convergent within the given stopping criterion and the convergent solution is close to the global optimum solution within the prescribed *** results are given to demonstrate that the performance of the proposed model is better than the state-of-the-art methods.

关键词： Principal component analysis Low-rank tensors Nuclear norm minimization Active subspace decomposition Matrix factorization

Lipschitz and Total-Variational Regularization for Blind Deconvolution

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Communications in Computational Physics 2008年第6期4卷 195-206页

作者： Yu-Mei Huang michael k.ng Centre for Mathematical Imaging and Vision and Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong Kong. Department of Mathematics Lanzhou UniversityGansuChina.

In[3],Chan and Wong proposed to use total variational regularization for both images and point spread functions in blind *** experimental results show that the detail of the restored images cannot be *** this paper,we consider images in Lipschitz spaces,and propose to use Lipschitz regularization for images and total variational regularization for point spread functions in blind *** experimental results show that such combination of Lipschitz and total variational regularization methods can recover both images and point spread functions quite well.

关键词： Lipschitz regularization total variational regularization blind deconvolution texture Poisson singular integral alternating iterative algorithm.

Spectral Analysis for Preconditioning of Multi-Dimensional Riesz Fractional Diffusion Equations

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Numerical Mathematics(Theory,Methods and Applications) 2022年第3期15卷 565-591页

作者： Xin Huang Xue-Lei Lin michael k.ng Hai-Wei Sun Department of Mathematics University of MacaoMacao SARChina Shenzhen JL Computational Science and Applied Research Institute Shenzhen 518131China Beijing Computational Science Research Center Beijing 100193China Department of Mathematics The University of HongkongHong Kong SARChina

In this paper,we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion *** finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives,which generates symmetric positive definite ill-conditioned multi-level Toeplitz *** preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear ***,we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval(12,32)and thus the preconditioned conjugate gradient method converges linearly within an iteration number independent of the discretization ***,the proposed method can be extended to handle ill-conditioned multi-level Toeplitz matrices whose blocks are generated by functions with zeros of fractional *** theoretical results fill in a vacancy in the *** examples are presented to show the convergence performance of the proposed preconditioner that is better than other preconditioners.

关键词： Multi-dimensional Riesz fractional derivative multi-level Toeplitz matrix sine transform based preconditioner preconditioned conjugate gradient method

On Multivariate Markov Chains for Common and Non-Common Objects in Multiple Networks

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Numerical Mathematics(Theory,Methods and Applications) 2012年第3期5卷 384-402页

作者： Xutao Li Wen Li michael k.ng Yunming Ye Department of Computer Science Shenzhen Graduate SchoolHarbin Institute of TechnologyShenzhen518055China School of Mathematical Sciences South China Normal UniversityGuangzhou510631China Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong Kong

Node importance or centrality evaluation is an important methodology for network *** this paper,we are interested in the study of objects appearing in several *** common objects are important in network-network interactions via object-object *** main contribution of this paper is to model multiple networks where there are some common objects in a multivariate Markov chain framework,and to develop a method for solving common and non-common objects’stationary probability distributions in the *** stationary probability distributions can be used to evaluate the importance of common and non-common objects via network-network *** experimental results based on examples of co-authorship of researchers in different conferences and paper citations in different categories have shown that the proposed model can provide useful information for researcher-researcher interactions in networks of different conferences and for paperpaper interactions in networks of different categories.

关键词： transition probability multivariate Markov chains stationary probability distribution irreducible multiple networks

Nonnegative Low Rank Matrix Completion by Riemannian Optimalization Methods

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Annals of Applied Mathematics 2023年第2期39卷 181-205页

作者： Guang-Jing Song michael k.ng School of Mathematics and Information Sciences Weifang UniversityWeifangShandong 261061China Department of Mathematics The University of Hong KongHong KongChina

In this paper,we study Riemannian optimization methods for the problem of nonnegative matrix completion that is to recover a nonnegative low rank matrix from its partial observed *** the underlying matrix incohence conditions,we show that when the number m of observed entries are sampled independently and uniformly without replacement,the inexact Riemannian gradient descent method can recover the underlying n_(1)-by-n_(2)nonnegative matrix of rank r provided that m is of O(r^(2)slog^(2)s),where s=max{n_(1),n_(2)}.Numerical examples are given to illustrate that the nonnegativity property would be useful in the matrix *** particular,we demonstrate the number of samples required to recover the underlying low rank matrix with using the nonnegativity property is smaller than that without using the property.

关键词： Manifolds tangent spaces nonnegative matrices low rank