A stabilized multigrid solver for hyperelastic image registration

Image registration is a central problem in a variety of areas involving imaging techniques and is known to be challenging and ill‐posed. Regularization functionals based on hyperelasticity provide a powerful mechanism for limiting the ill‐posedness. A key feature of hyperelastic image registration approaches is their ability to model large deformations while guaranteeing their invertibility, which is crucial in many applications. To ensure that numerical solutions satisfy this requirement, we discretize the variational problem using piecewise linear finite elements, and then solve the discrete optimization problem using the Gauss–Newton method. In this work, we focus on computational challenges arising in approximately solving the Hessian system. We show that the Hessian is a discretization of a strongly coupled system of partial differential equations whose coefficients can be severely inhomogeneous. Motivated by a local Fourier analysis, we stabilize the system by thresholding the coefficients. We propose a Galerkin‐multigrid scheme with a collective pointwise smoother. We demonstrate the accuracy and effectiveness of the proposed scheme, first on a two‐dimensional problem of a moderate size and then on a large‐scale real‐world application with almost 9 million degrees of freedom.     

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.     

A variational formulation for physical noised image segmentation

Image segmentation is a hot topic in image science. In this paper we present a new variational segmentation model based on the theory of Mumford-Shah model. The aim of our model is to divide noised image, according to a certain criterion, into homogeneous and smooth regions that should correspond to structural units in the scene or objects of interest. The proposed region-based model uses total variation as a regularization term, and different fidelity term can be used for image segmentation in the cases of physical noise, such as Gaussian, Poisson and multiplicative speckle noise. Our model consists of five weighted terms, two of them are responsible for image denoising based on fidelity term and total variation term, the others assure that the three conditions of adherence to the data, smoothing, and discontinuity detection are met at once. We also develop a primal-dual hybrid gradient algorithm for our model. Numerical results on various synthetic and real images are provided to compare our method with others,these results show that our proposed model and algorithms are effective.     

A strong variational algorithm for delay systems

In this paper, we present a convergent extension of the first-order strong-variational algorithm by Mayne and Polak (Ref. 1) for solving optimal control problems with control constraints to delay systems. Although the algorithm is similar to the one presented in Ref. 1, the proof of convergence is different, since the differential dynamic techniques used by Mayne and Polak are not applicable.This work forms part of the author's PhD Dissertation and was conducted at the Imperial College of Science and Technology under a studentship awarded by the UK Science and Engineering Research Council. This assistance is gratefully acknowledged. The author also wishes to thank Dr. R. B. Vinter for his encouragement and help.     

On some new variational problems for image denoising

This paper is devoted to image denoising problems using multiresolution schemes related to variational problems. We start with the linear approach of Donoho and Johnstone, that is related to a well known diffusion‐type variational problem. In order to improve the behavior of this approach, we propose some new nonlinear variational problems more adapted to the problem of denoising. Moreover, the discretization is performed using nonlinear multiresolution schemes. In particular, we obtain some fast and well adapted schemes for the considered problem of denoising.     

A fast algorithm for a total variation based phase demodulation model

In this paper we introduce fast numerical algorithms for the solution of the model. For each variable, background illumination, amplitude modulation and phase map, we develop a fixed point method. Then, we write all three algorithms in the same framework and analyze their convergence rates, local smoothing factors by means of local Fourier analysis and present experimental evidence of their performance on synthetic and real world problems.     

A numerical technique for solving a class of fractional variational problems

This paper presents a numerical method for solving a class of fractional variational problems (FVPs) with multiple dependent variables, multi order fractional derivatives and a group of boundary conditions. The fractional derivative in the problem is in the Caputo sense. In the presented method, the given optimization problem reduces to a system of algebraic equations using polynomial basis functions. An approximate solution for the FVP is achieved by solving the system. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. We extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique.     

A nonlinear descent method for a variational inequality on a nonconvex set

In this paper, we consider a generalized variational inequality problem which involves the integrable cost mapping and a nonsmooth mapping with convex components. We propose a new gradient-type method which determines a stepsize by using the smooth part of the cost function. Thus, the method does not utilize analogs of derivatives of nonsmooth functions. We show that its convergence does not require additional assumptions.     

An FEM approximation for a fourth-order variational inequality of second kind

A fourth-order variational inequality of the second kind arising in a plate frictional bending problem is considered. By using regularization method, the original problem can be formulated as a differentiable variational equation, and the corresponding discrete FEM variational equation is presented afterwards. Abstract error estimates and error estimates of the approximation are derived in terms of energy norm and L^2-norm.     

Finite element analysis and application for a nonlinear diffusion model in image denoising

The stability analysis and error estimates are presented for a nonlinear diffusion model, which appears in image denoising and solved by a fully discrete time Galerkin method with kth (k ≥ 1) order conforming finite element spaces. Numerical experiments are provided with denoising several grayscale noisy images by our Galerkin method on bilinear finite elements. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 649–662, 2002; DOI 10.1002/num.10017     

Finite convergence of algorithms for nonlinear programs and variational inequalities

Algorithms for nonlinear programming and variational inequality problems are, in general, only guaranteed to converge in the limit to a Karush-Kuhn-Tucker point, in the case of nonlinear programs, or to a solution in the case of variational inequalities. In this paper, we derive sufficient conditions for nonlinear programs with convex feasible sets such that any convergent algorithm can be modified, by adding a convex subproblem with a linear objective function, to guarantee finite convergence in a generalized sense. When the feasible set is polyhedral, the subproblem is a linear program and finite convergence is obtained. Similar results are also developed for variational inequalities.The research of the first author was supported in part by the Office of Naval Research under Contract No. N00014-86-K-0173.The authors are indebted to Professors Olvi Mangasarian, Garth McCormick, Jong-Shi Pang, Hanif Sherali, and Hoang Tuy for helpful comments and suggestions and to two anonymous referees for constructive remarks and for bringing to their attention the results in Refs. 13 and 14.     

Using a level set approach for image segmentation under interpolation conditions

In this paper, we propose a new 2D segmentation model including geometric constraints, namely interpolation conditions, to detect objects in a given image. We propose to apply the deformable models to an explicit function using the level set approach (Osher and Sethian [24]); so, we avoid the classical problem of parameterization of both segmentation representation and interpolation conditions. Furthermore, we allow this representation to have topological changes. A problem of energy minimization on a closed subspace of a Hilbert space is defined and introducing Lagrange multipliers enables us to formulate the corresponding variational problem with interpolation conditions. Thus the explicit function evolves, while minimizing the energy and it stops evolving when the desired outlines of the object to detect are reached. The stopping term, as in the classical deformable models, is related to the gradient of the image. Numerical results are given. AMS subject classification 74G65, 46-xx, 92C55     

A class of spline functions for landmark‐based image registration

A class of spline functions, called Lobachevsky splines, is proposed for landmark‐based image registration. Analytic expressions of Lobachevsky splines and some of their properties are given, reasoning in the context of probability theory. Because these functions have simple analytic expressions and compact support, landmark‐based transformations can be advantageously defined using them. Numerical results point out accuracy and stability of Lobachevsky splines, comparing them with Gaussians and thin plate splines. Moreover, an application to a real‐life case (cervical X‐ray images) shows the effectiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Implementation of high‐order variational models made easy for image processing

High‐order variational models are powerful methods for image processing and analysis, but they can lead to complicated high‐order nonlinear partial differential equations that are difficult to discretise to solve computationally. In this paper, we present some representative high‐order variational models and provide detailed descretisation of these models and numerical implementation of the split Bregman algorithm for solving these models using the fast Fourier transform. We demonstrate the advantages and disadvantages of these high‐order models in the context of image denoising through extensive experiments. The methods and techniques can also be used for other applications, such as image decomposition, inpainting and segmentation. Copyright © 2016 John Wiley & Sons, Ltd.     

On a two-fluid model of two-phase compressible flows and its numerical approximation

We consider a two-fluid model of two-phase compressible flows. First, we derive several forms of the model and of the equations of state. The governing equations in all the forms contain source terms representing the exchanges of momentum and energy between the two phases. These source terms cause unstability for standard numerical schemes. Using the above forms of equations of state, we construct a stable numerical approximation for this two-fluid model. That only the source terms cause the oscillations suggests us to minimize the effects of source terms by reducing their amount. By an algebraic operator, we transform the system to a new one which contains only one source term. Then, we discretize the source term by making use of stationary solutions. We also present many numerical tests to show that while standard numerical schemes give oscillations, our scheme is stable and numerically convergent.     

Comparing numerical methods for Boussinesq equation model problem

This article applies three methods to solve a class of nonlinear differential equations. We obtain the exact solution and numerical solution of the Boussinesq equation for certain initial condition. Comparsion of the results with those of other methods have led us to significant consequences. The numerical solutions are compared with the known analytical solutions. The numerical results demonstrate that those methods are quite accurate and readily implemented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

求解图像分割CV模型的BB算法

给出图像分割的一种新算法——BB算法.该方法的优点在于利用迭代过程中当前点和前一点的信息确定搜索步长,从而更有效地搜索最优解.为此,首先通过变分水平集方法将CV模型转化为最优化问题;其次,将BB算法引入该优化问题进行求解;然后,对BB算法进行收敛性分析,为该算法应用在CV模型中提供了理论依据;最后将该方法与已有的最速下降法、共轭梯度法的分割结果进行比较.结果表明,跟其他两种方法相比,BB算法在保证较好分割效果的前提下,提高了算法的速度和性能.     

A class of one-dimensional geometric variational problems with energy involving the curvature and its derivatives: a geometric measure-theoretic approach

The notions of Legendrian and Gaussian towers are defined and investigated. Then applications in the context of one-dimensional geometric variational problems with the energy involving the curvature and its derivatives are provided. Particular attention is paid to the case when the functional is defined on smooth boundaries of plane sets.      

On a New Diffeomorphic Multi-Modality Image Registration Model and Its Convergent Gauss-Newton Solver

In this work, we propose a new variational model for multi-modal image registration and present an efficient numerical implementation. The model minimizes a new functional based on using reformulated normalized gradients of the images as the fidelity term and higher-order derivatives as the regularizer. A key feature of the model is its ability of guaranteeing a diffeomorphic transformation which is achieved by a control term motivated by the quasi-conformal map and Beltrami coefficient. The existence of the solution of this model is established. To solve the model numerically, we design a Gauss-Newton method to solve the resulting discrete optimization problem and prove its convergence; a multilevel technique is employed to speed up the initialization and avoid likely local minima of the underlying functional. Finally, numerical experiments demonstrate that this new model can deliver good performances for multi-modal image registration and simultaneously generate an accurate diffeomorphic transformation.     

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.