基于PICCS的非局部TGV能谱CT重建算法

能谱CT将宽谱划分为窄谱,导致通道内光子数目明显减少,加大了噪声影响,故从噪声投影中重建出高质量图像是能谱CT的一个研究热点.传统全变分(total variational,TV)容易造成重建图像中出现块状伪影等问题,总广义全变分(total generalized variation,TGV)算法可以逼近任意阶函数,再结合非局部均值算法的思想,同时考虑到不同能谱通道下重建图像的相关性,将高质量全能谱重建图像作为先验图像指导能谱CT重建,提出了基于先验图像约束压缩感知(prior image constrained compressed sensing,PICCS)的非局部TGV重建算法.实验结果表明,所提算法在抑制噪声的同时能够有效复原图像细节及边缘信息,且收敛速度快.     

Variational image inpainting

Inpainting is an image interpolation problem with broad applications in image and vision analysis. Described in the current expository paper are our recent efforts in developing universal inpainting models based on the Bayesian and variational principles. Discussed in detail are several variational inpainting models built upon geometric image models, the associated Euler‐Lagrange PDEs and their geometric and dynamic interpretations, as well as effective computational approaches. Novel efforts are then made to further extend this systematic variational framework to the inpainting of oscillatory textures, interpolation of missing wavelet coefficients as in the wireless transmission of JPEG2000 images, as well as light‐adapted inpainting schemes motivated by Weber's law in visual perception. All these efforts lead to the conclusion that unlike many familiar image processors such as denoising, segmentation, and compression, the performance of a variational/Bayesian inpainting scheme much more crucially depends on whether the image prior model well resolves the spatial coupling (or geometric correlation) of image features. As a highlight, we show that the Besov image models appear to be less interesting for image inpainting in the wavelet domain, highly contrary to their significant roles in thresholding‐based denoising and compression. Thus geometry is the single most important keyword throughout this paper. © 2005 Wiley Periodicals, Inc.     

A fractional partial differential equation based multiscale denoising model for texture image

Traditional integer‐order partial differential equation based image denoising approach can easily lead edge and complex texture detail blur, thus its denoising effect for texture image is always not well. To solve the problem, we propose to implement a fractional partial differential equation (FPDE) based denoising model for texture image by applying a novel mathematical method—fractional calculus to image processing from the view of system evolution. Previous studies show that fractional calculus has some unique properties that it can nonlinearly enhance complex texture detail in digital image processing, which is obvious different with integer‐order differential calculus. The goal of the modeling is to overcome the problems of the existed denoising approaches by utilizing the aforementioned properties of fractional differential calculus. Using classic definition and property of fractional differential calculus, we extend integer‐order steepest descent approach to fractional field to implement fractional steepest descent approach. Then, based on the earlier fractional formulas, a FPDE based multiscale denoising model for texture image is proposed and further analyze optimal parameters value for FPDE based denoising model. The experimental results prove that the ability for preserving high‐frequency edge and complex texture information of the proposed fractional denoising model are obviously superior to traditional integral based algorithms, as for texture detail rich images. Copyright © 2013 John Wiley & Sons, Ltd.     

Modular solvers for image restoration problems using the discrepancy principle

Many problems in image restoration can be formulated as either an unconstrained non‐linear minimization problem, usually with a Tikhonov‐like regularization, where the regularization parameter has to be determined; or as a fully constrained problem, where an estimate of the noise level, either the variance or the signal‐to‐noise ratio, is available. The formulations are mathematically equivalent. However, in practice, it is much easier to develop algorithms for the unconstrained problem, and not always obvious how to adapt such methods to solve the corresponding constrained problem. In this paper, we present a new method which can make use of any existing convergent method for the unconstrained problem to solve the constrained one. The new method is based on a Newton iteration applied to an extended system of non‐linear equations, which couples the constraint and the regularized problem, but it does not require knowledge of the Jacobian of the irregularity functional. The existing solver is only used as a black box solver, which for a fixed regularization parameter returns an improved solution to the unconstrained minimization problem given an initial guess. The new modular solver enables us to easily solve the constrained image restoration problem; the solver automatically identifies the regularization parameter, during the iterative solution process. We present some numerical results. The results indicate that even in the worst case the constrained solver requires only about twice as much work as the unconstrained one, and in some instances the constrained solver can be even faster. Copyright © 2002 John Wiley & Sons, Ltd.     

Adaptive total variation regularization based scheme for Poisson noise removal

To better preserve the edge features, this paper investigates an adaptive total variation regularization based variational model for removing Poisson noise. This edge‐preserving scheme comprises a spatially adaptive diffusivity coefficient, which adjusts the diffusion strength automatically. Compared with the classical total variation based one, numerical simulations distinctly indicate the superiority of our proposed strategy in maintaining the small details while denoising Poissonian image. Copyright © 2012 John Wiley & Sons, Ltd.     

Image denoising using the Gaussian curvature of the image surface

A number of high‐order variational models for image denoising have been proposed within the last few years. The main motivation behind these models is to fix problems such as the staircase effect and the loss of image contrast that the classical Rudin–Osher–Fatemi model [Leonid I. Rudin, Stanley Osher and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), pp. 259–268] and others also based on the gradient of the image do have. In this work, we propose a new variational model for image denoising based on the Gaussian curvature of the image surface of a given image. We analytically study the proposed model to show why it preserves image contrast, recovers sharp edges, does not transform piecewise smooth functions into piecewise constant functions and is also able to preserve corners. In addition, we also provide two fast solvers for its numerical realization. Numerical experiments are shown to illustrate the good performance of the algorithms and test results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1066–1089, 2016     

A primal‐dual method for the Meyer model of cartoon and texture decomposition

In this paper, we study the original Meyer model of cartoon and texture decomposition in image processing. The model, which is a minimization problem, contains an l₁‐based TV‐norm and an l_∞‐based G‐norm. The main idea of this paper is to use the dual formulation to represent both TV‐norm and G‐norm. The resulting minimization problem of the Meyer model can be given as a minimax problem. A first‐order primal‐dual algorithm can be developed to compute the saddle point of the minimax problem. The convergence of the proposed algorithm is theoretically shown. Numerical results are presented to show that the original Meyer model can decompose better cartoon and texture components than the other testing methods.     

On a nonlinear multigrid algorithm with primal relaxation for the image total variation minimisation

Digital image restoration has drawn much attention in the recent years and a lot of research has been done on effective variational partial differential equation models and their theoretical studies. However there remains an urgent need to develop fast and robust iterative solvers, as the underlying problem sizes are large. This paper proposes a fast multigrid method using primal relaxations. The basic primal relaxation is known to get stuck at a ‘local’ non-stationary minimum of the solution, which is usually believed to be ‘non-smooth’. Our idea is to utilize coarse level corrections, overcoming the deadlock of a basic primal relaxation scheme. A further refinement is to allow non-regular coarse levels to correct the solution, which helps to improve the multilevel method. Numerical experiments on both 1D and 2D images are presented.     

A Compound Algorithm of Denoising Using Second-Order and Fourth-Order Partial Differential Equations

In this paper, we propose a compound algorithm for the image restoration. The algorithm is a convex combination of the ROF model and the LLT model with a parameter function 6. The numerical experiments demonstrate that our compound algorithm is efficient and preserves the main advantages of the two models. In particular, the errors of the compound algorithm in L2 norm between the exact images and corresponding restored images are the smallest among the three models. For images with strong noises, the restored images of the compound algorithm are the best in the corresponding restored images. The proposed algorithm combines the fixed point method, an improved AMG method and the Krylov acceleration. It is found that the combination of these methods is efficient and robust in the image restoration.     

Recent advances of variational model in medical imaging and applications to computer aided surgery

In this paper, we review some mathematical models in medical image processing. Due to the superiority in modeling and computation, variational methods have been proven to be powerful techniques, which have been extremely popular and dramatically improved in the past two decades. On one hand, many models have been proposed for nearly all kinds of applications. On the other hand, a lot of models can be globally optimized and also many computation tools have been introduced. Under the variational framework, we focus on two basic problems in medical imaging: image restoration and segmentation, which are core components for kinds of specific tasks. For image restoration, we discuss some models on both additive and multiplicative noises. For image segmentation, we review some models on both whole image segmentation and specific target delineation, with the later being a key step in computer aided surgery. Additionally, we present some models on liver delineation and give their applications to living donor liver transplantation.     

修正的TV-Stokes模型的新快速算法

基于Tai等人的前期工作,本文研究修正的TV-Stokes图像去噪模型,提出一些新的求解该两步模型的快速算法.我们利用对偶形式和多重网格方法得到一个求解第1步的快速算法.给出另外一种新的求解光滑的切向量场的保不可压性质的算法.在第2步中,我们提出一类有效的全新算法:首先通过计算Poisson方程得到具有光滑法向量场的函数g,然后利用Jia和Zhao的方法得到恢复的图像.新算法的运算速度非常快,用于图像恢复的CPU时间少于0.1 s.数值结果显示新的快速算法是有效的和稳定的,恢复图像的质量也超过了一般去噪方法.     

Macro‐hybrid mixed variational two‐phase flow through fractured porous media

Macro‐hybrid mixed variational models of two‐phase flow, through fractured porous media, are analyzed at the mesoscopic and macroscopic levels. The mesoscopic models are treated in terms of nonoverlapping domain decompositions, in such a manner that the porous rock matrix system and the fracture network interact across rock–rock, rock–fracture, and fracture–fracture interfaces, with flux transmission conditions dualized. Alternatively, the models are scaled to a macroscopic level via an asymptotic process, where the width of the fractures tends to zero, and the fracture network turns out to be an interface system of one less spatial dimension, with variable high permeability. The two‐phase flow is characterized by a fractional flow dual mixed variational model. Augmented two‐field and three‐field variational reformulations are presented for regularization, internal approximations, and macro‐hybrid mixed finite element implementation. Also abstract proximal‐point penalty‐duality algorithms are derived and analyzed for parallel computing. Copyright © 2016 John Wiley & Sons, Ltd.     

Lie Group Spectral Variational Integrators

We present a new class of high-order variational integrators on Lie groups. We show that these integrators are symplectic and momentum-preserving, can be constructed to be of arbitrarily high order, or can be made to converge geometrically. Furthermore, these methods are capable of taking very large time-steps. We demonstrate the construction of one such variational integrator for the rigid body and discuss how this construction could be generalized to other related Lie group problems. We close with several numerical examples which demonstrate our claims and discuss further extensions of our work.     

A variational approach for restoring images corrupted by noisy blur kernels and additive noise

In this paper, we study a deblurring algorithm for distorted images by random impulse response. We propose and develop a convex optimization model to recover the underlying image and the blurring function simultaneously. The objective function is composed of 3 terms: the data‐fitting term between the observed image and the product of the estimated blurring function and the estimated image, the squared difference between the estimated blurring function and its mean, and the total variation regularization term for the estimated image. We theoretically show that under some mild conditions, the resulting objective function can be convex in which the global minimum value is unique. The numerical results confirm that the peak‐to‐signal‐noise‐ratio and structural similarity of the restored images by the proposed algorithm are the best when the proposed objective function is convex. We also present a proximal alternating minimization scheme to solve the resulting minimization problem. Numerical examples are presented to demonstrate the effectiveness of the proposed model and the efficiency of the numerical scheme.     

Homotopy method for a mean curvature-based denoising model

Variational image denoising models based on regularization of gradients have been extensively studied. The total variation model by Rudin, Osher, and Fatemi (1992) [38] can preserve edges well but for images without edges (jumps), the solution to this model has the undesirable staircasing effect. To overcome this, mean curvature-based energy minimization models offer one approach for restoring both smooth (no edges) and nonsmooth (with edges) images. As such models lead to fourth order (instead of the usual second order) nonlinear partial differential equations, development of fast solvers is a challenging task. Previously stabilized fixed point methods and their associated multigrid methods were developed but the underlying operators must be regularized by a relatively large parameter. In this paper, we first present a fixed point curvature method for solving such equations and then propose a homotopy approach for varying the regularized parameter so that the Newton type method becomes applicable in a predictor-corrector framework. Numerical experiments show that both of our methods are able to maintain all important information in the image, and at the same time to filter out noise.     

变分不等式与互补问题、双层规划与平衡约束数学规划问题的若干进展

考虑有限维变分不等式与互补问题、双层规划以及均衡约束的数学规划问题. 在简单介绍这些问题之后,重点介绍近年来这些领域中发展迅速的几个研究方向,包括对称锥互补问题的理论与算法、变分不等式的投影收缩算法、随机变分不等式与随机互补问题的模型与方法、双层规划以及均衡约束数学规划问题的新方法. 最后提出几个进一步研究的方向.     

Applications of variational methods to Sturm–Liouville boundary‐value problem for fourth‐order impulsive differential equations

In this paper, we study Sturm–Liouville boundary‐value problem for fourth‐order impulsive differential equations. Applying variational methods, several new existence results are obtained. Copyright © 2013 John Wiley & Sons, Ltd.     

A Retinex-based total variation approach for image segmentation and bias correction

Image segmentation methods usually suffer from intensity inhomogeneity problem caused by many factors such as spatial variations in illumination (or bias fields of imaging devices). In order to address this problem, this paper proposes a Retinex-based variational model for image segmentation and bias correction. According to Retinex theory, the input inhomogeneous image can be decoupled into illumination bias and reflectance parts. The main contribution of this paper is to consider piecewise constant of the reflectance, and thereby introduce the total variation term in the proposed model for correcting and segmenting the input image. This is different from the existing model in which the spatial smoothness of the illumination bias is employed only. The existence of the minimizers to the variational model is established. Furthermore, we develop an efficient algorithm to solve the model numerically by using the alternating minimization method. Our experimental results are reported to demonstrate the effectiveness of the proposed method, and its performance is competitive with that of the other testing methods.     

Multi-Label Markov Random Fields as an Efficient and Effective Tool for Image Segmentation,Total Variations and Regularization

One of the classical optimization models for image segmentation is the well known Markov Random Fields (MRF) model. This model is a discrete optimization problem, which is shown here to formulate many continuous models used in image segmentation. In spite of the presence of MRF in the literature, the dominant perception has been that the model is not effective for image segmentation. We show here that the reason for the non-effectiveness is due to the lack of access to the optimal solution. Instead of solving optimally, heuristics have been engaged. Those heuristic methods cannot guarantee the quality of the solution nor the running time of the algorithm. Worse still, heuristics do not link directly the input functions and parameters to the output thus obscuring what would be ideal choices of parameters and functions which are to be selected by users in each particular application context.We describe here how MRF can model and solve efficiently several known continuous models for image segmentation and describe briefly a very efficient polynomial time algorithm, which is provably fastest possible, to solve optimally the MRF problem. The MRF algorithm is enhanced here compared to the algorithm in Hochbaum (2001) by allowing the set of assigned labels to be any discrete set. Other enhancements include dynamic features that permit adjustments to the input parameters and solves optimally for these changes with minimal computation time. Several new theoretical results on the properties of the algorithm are proved here and are demonstrated for images in the context of medical and biological imaging. An interactive implementation tool for MRF is described, and its performance and flexibility in practice are demonstrated via computational experiments.We conclude that many continuous models common in image segmentation have discrete analogs to various special cases of MRF and as such are solved optimally and efficiently, rather than with the use of continuous techniques, such as PDE methods, which restrict the type of functions used and furthermore, can only guarantee convergence to a local minimum.     

Application of He's variational iteration method and Adomian's decomposition method to Sawada–Kotera–Ito seventh‐order equation

In this article, the Sawada–Kotera–Ito seventh‐order equation is studied. He's variational iteration method and Adomian's decomposition method (ADM) are applied to obtain solution of this equation. We compare these methods together. The study highlights the significant features of the employed methods and its capability of handling completely integrable equations. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 887–897, 2011