A robust multigrid approach for variational image registration models

Variational registration models are non-rigid and deformable imaging techniques for accurate registration of two images. As with other models for inverse problems using the Tikhonov regularization, they must have a suitably chosen regularization term as well as a data fitting term. One distinct feature of registration models is that their fitting term is always highly nonlinear and this nonlinearity restricts the class of numerical methods that are applicable. This paper first reviews the current state-of-the-art numerical methods for such models and observes that the nonlinear fitting term is mostly ‘avoided’ in developing fast multigrid methods. It then proposes a unified approach for designing fixed point type smoothers for multigrid methods. The diffusion registration model (second-order equations) and a curvature model (fourth-order equations) are used to illustrate our robust methodology. Analysis of the proposed smoothers and comparisons to other methods are given. As expected of a multigrid method, being many orders of magnitude faster than the unilevel gradient descent approach, the proposed numerical approach delivers fast and accurate results for a range of synthetic and real test images.     

A robust multigrid approach for variational image registration models

Variational registration models are non-rigid and deformable imaging techniques for accurate registration of two images. As with other models for inverse problems using the Tikhonov regularization, they must have a suitably chosen regularization term as well as a data fitting term. One distinct feature of registration models is that their fitting term is always highly nonlinear and this nonlinearity restricts the class of numerical methods that are applicable. This paper first reviews the current state-of-the-art numerical methods for such models and observes that the nonlinear fitting term is mostly ‘avoided’ in developing fast multigrid methods. It then proposes a unified approach for designing fixed point type smoothers for multigrid methods. The diffusion registration model (second-order equations) and a curvature model (fourth-order equations) are used to illustrate our robust methodology. Analysis of the proposed smoothers and comparisons to other methods are given. As expected of a multigrid method, being many orders of magnitude faster than the unilevel gradient descent approach, the proposed numerical approach delivers fast and accurate results for a range of synthetic and real test images.     

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.     

基于Huber正则化的红外与可见光图像融合

图像融合通常是指从多源信道采集同一目标图像,将互补的多焦点、多模态、多时相和/或多视点图像集成在一起,形成新图像的过程.在本文中,我们采用基于Huber正则化的红外与可见光图像的融合模型.该模型通过约束融合图像与红外图像相似的像素强度保持热辐射信息,以及约束融合图像与可见光图像相似的灰度梯度和像素强度保持图像的边缘和纹理等外观信息,同时能够改善图像灰度梯度相对较小区域的阶梯效应.为了最小化这种变分模型,我们结合增广拉格朗日方法(ALM)和量身定做有限点方法(TFPM)的思想设计数值算法,并给出了算法的收敛性分析.最后,我们将所提模型和算法与其他七种图像融合方法进行定性和定量的比较,分析了本文所提模型的特点和所提数值算法的有效性.     

High-order total variation-based Poissonian image deconvolution with spatially adapted regularization parameter

Images captured by image acquisition systems using photon-counting devices such as astronomical imaging, positron emission tomography and confocal microscopy imaging, are often contaminated by Poisson noise. Total variation (TV) regularization, which is a classic regularization technique in image restoration, is well-known for recovering sharp edges of an image. Since the regularization parameter is important for a good recovery, Chen and Cheng (2012) proposed an effective TV-based Poissonian image deblurring model with a spatially adapted regularization parameter. However, it has drawbacks since the TV regularization produces staircase artifacts. In this paper, in order to remedy the shortcoming of TV of their model, we introduce an extra high-order total variation (HTV) regularization term. Furthermore, to balance the trade-off between edges and the smooth regions in the images, we also incorporate a weighting parameter to discriminate the TV and the HTV penalty. The proposed model is solved by an iterative algorithm under the framework of the well-known alternating direction method of multipliers. Our numerical results demonstrate the effectiveness and efficiency of the proposed method, in terms of signal-to-noise ratio (SNR) and relative error (RelRrr).     

Galerkin finite element methods for two-dimensional RLW and SRLW equations

In this paper, Galerkin finite methods for two-dimensional regularized long wave and symmetric regularized long wave equation are studied. The discretization in space is by Galerkin finite element method and in time is based on linearized backward Euler formula and extrapolated Crank–Nicolson scheme. Existence and uniqueness of the numerical solutions have been shown by Brouwer fixed point theorem. The error estimates of linearlized Crank–Nicolson method for RLW and SRLW equations are also presented. Numerical experiments, including the error norms and conservation variables, verify the efficiency and accuracy of the proposed numerical schemes.     

A nonlinear diffusion model for image restoration

In this paper, we study the ill posed Perona-Malik equation of image processing^[14] and the regularized P-M model i.e. C-model proposed by Catte et al.^[4]. The authors present the convex compound of these two models in the form of the system of partial differential equations. The weak solution for the equations is proved in detail. The additive operator splitting (AOS) algorithm for the proposed model is also given. Finally, we show some numeric experimental results on images.     

A METHOD TO APPROACH OPTIMAL RESTORATION IN IMAGE RESTORATION PROBLEMS WITHOUT NOISE ENERGY INFORMATION

This paper proposes a new image restoration technique, in which the resulting regularized image approximates the optimal solution steadily. The affect of the regular-ization operator and parameter on the lower band and upper band energy of the residue of the regularized image is theoretically analyzed by employing wavelet transform. This paper shows that regularization operator should generally be lowstop and highpass. So this paper chooses a lowstop and highpass operator as regularization operator, and construct an optimization model which minimizes the mean squares residue of regularized solution to determine regularization parameter. Although the model is random, on the condition of this paper, it can be solved and yields regularization parameter and regularized solu-tion. Otherwise, the technique has a mechanism to predict noise energy. So, without noisei nformation, it can also work and yield good restoration results.     

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.     

Nonlinear regression modeling using regularized local likelihood method

We introduce a nonlinear regression modeling strategy, using a regularized local likelihood method. The local likelihood method is effective for analyzing data with complex structure. It might be, however, pointed out that the stability of the local likelihood estimator is not necessarily guaranteed in the case that the structure of system is quite complex. In order to overcome this difficulty, we propose a regularized local likelihood method with a polynomial function which unites local likelihood and regularization. A crucial issue in constructing nonlinear regression models is the choice of a smoothing parameter, the degree of polynomial and a regularization parameter. In order to evaluate models estimated by the regularized local likelihood method, we derive a model selection criterion from an information-theoretic point of view. Real data analysis and Monte Carlo experiments are conducted to examine the performance of our modeling strategy.     

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

<正>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 image.In 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 equations.At 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 method.We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably well.     

A hybrid convex variational model for image restoration

We propose a new hybrid model for variational image restoration using an alternative diffusion switching non-quadratic function with a parameter. The parameter is chosen adaptively so as to minimize the smoothing near the edges and allow the diffusion to smooth away from the edges. This model belongs to a class of edge-preserving regularization methods proposed in the past, the ?-function formulation. This involves a minimizer to the associated energy functional. We study the existence and uniqueness of the energy functional of the model. Using real and synthetic images we show that the model is effective in image restoration.     

An edge driven wavelet frame model for image restoration

Wavelet frame systems are known to be effective in capturing singularities from noisy and degraded images. In this paper, we introduce a new edge driven wavelet frame model for image restoration by approximating images as piecewise smooth functions. With an implicit representation of image singularities sets, the proposed model inflicts different strength of regularization on smooth and singular image regions and edges. The proposed edge driven model is robust to both image approximation and singularity estimation. The implicit formulation also enables an asymptotic analysis of the proposed models and a rigorous connection between the discrete model and a general continuous variational model. Finally, numerical results on image inpainting and deblurring show that the proposed model is compared favorably against several popular image restoration models.     

Modeling and analysis of the unilateral contact of a piezoelectric body with a conductive support

We consider a mathematical model which describes the quasistatic process of contact between a piezoelectric body and an electrically conductive support, the so-called foundation. We model the material's behavior with a nonlinear electro-viscoelastic constitutive law; the contact is frictionless and is described with the Signorini condition and a regularized electrical conductivity condition. We derive a variational formulation for the problem and then we prove the existence of a unique weak solution to the model. The proof is based on arguments of nonlinear equations with multivalued maximal monotone operators and fixed point. Then we introduce a fully discrete scheme, based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. We treat the unilateral contact conditions by using an augmented Lagrangian approach. We implement this scheme in a numerical code then we present numerical simulations in the study of two-dimensional test problems, together with various comments and interpretations.     

Well-posedness for a class of fourth order diffusions for image processing

A number of image denoising models based on higher order parabolic partial differential equations (PDEs) have been proposed in an effort to overcome some of the problems attendant to second order methods such as the famous Perona–Malik model. However, there is little analysis of these equations to be found in the literature. In this paper, methods of maximal regularity are used to prove the existence of unique local solutions to a class of fourth order PDEs for noise removal. The proof is laid out explicitly for two newly proposed fourth order models, and an outline is given for how to apply the techniques to other proposed models.     

Robust Error Estimates for Adaptive Phase Field Simulations

Phase field equations are commonly used as a regularized model, where bulk phases are separated by interface regions that have a thickness of the order γ. Their numerical analysis is well established for a fixed parameter size γ, but conventional error estimates depend exponentially on γ⁻¹ and thus become useless in the relevant case if γ→0. Technical applications include e.g. the simulation of Sn–Cu alloys for the production of lead free solder or Ni–Al alloys used for rotor blade surfaces. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Clusters, outliers, and regression: fixed point clusters

Fixed point clustering is a new stochastic approach to cluster analysis. The definition of a single fixed point cluster (FPC) is based on a simple parametric model, but there is no parametric assumption for the whole dataset as opposed to mixture modeling and other approaches. An FPC is defined as a data subset that is exactly the set of non-outliers with respect to its own parameter estimators. This paper concentrates upon the theoretical foundation of FPC analysis as a method for clusterwise linear regression, i.e., the single clusters are modeled as linear regressions with normal errors. In this setup, fixed point clustering is based on an iteratively reweighted estimation with zero weight for all outliers. FPCs are non-hierarchical, but they may overlap and include each other. A specification of the number of clusters is not needed. Consistency results are given for certain mixture models of interest in cluster analysis. Convergence of a fixed point algorithm is shown. Application to a real dataset shows that fixed point clustering can highlight some other interesting features of datasets compared to maximum likelihood methods in the presence of deviations from the usual assumptions of model based cluster analysis.     

Model order reduction of finite element models: improved component mode synthesis

The finite element (FE) approach constitutes an essential methodology when modelling the elastic properties of structures in various research disciplines such as structural mechanics, engine dynamics and so on. Because of increased accuracy requirements, the FE method results in discretized models, which are described by higher order ordinary differential equations, or, in FE terms, by a large number of degrees of freedom (DoF). In this regard, the application of an additional methodology, referred to as the model order reduction (MOR) or DoF condensation, is rather compulsory. Herein, a reduced dimension set of ordinary differential equations is generated, i.e. the initially large number of DoF is condensed, while aiming to keep the dynamics of the original model as intact as possible. In the commercially available FE software tools, the static and the component mode syntheses (CMS) are the only available integrated condensation methods. The latter represents the state of the art generating well-correlated reduced order models (ROMs), which can be further utilized for FE or multi-body systems simulations. Taking into consideration the information loss of the CMS, which is introduced by its part-static nature, the improved CMS (ICMS) method is proposed. Here the algorithmic scheme of the standard CMS is adopted, which is qualitatively improved by adequately considering the advantageous characteristics of another MOR approach, the so-called improved reduction system method. The ICMS results in better correlated reduced order models in comparison to all the aforementioned methods, while preserving the required structural properties of the original FE model.     

基于奇异值分解建立的一种新的正则化方法

根据紧算子的奇异系统理论，引入一种正则化滤子函数，从而建立一种新的正则化方法来求解右端近似给定的第一类算子方程，并给出了正则解的误差分析。通过正则参数的先验选取，证明了正则解的误差具有渐进最优阶。      

An Algorithm for image removals and decompositions without inverse matrices

Partial Differential Equation (PDE) based methods in image processing have been actively studied in the past few years. One of the effective methods is the method based on a total variation introduced by Rudin, Oshera and Fatemi (ROF) [L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992) 259–268]. This method is a well known edge preserving model and an useful tool for image removals and decompositions. Unfortunately, this method has a nonlinear term in the equation which may yield an inaccurate numerical solution. To overcome the nonlinearity, a fixed point iteration method has been widely used. The nonlinear system based on the total variation is induced from the ROF model and the fixed point iteration method to solve the ROF model is introduced by Dobson and Vogel [D.C. Dobson, C.R. Vogel, Convergence of an iterative method for total variation denoising, SIAM J. Numer. Anal. 34 (5) (1997) 1779–1791]. However, some methods had to compute inverse matrices which led to roundoff error. To address this problem, we developed an efficient method for solving the ROF model. We make a sequence like Richardson’s method by using a fixed point iteration to evade the nonlinear equation. This approach does not require the computation of inverse matrices. The main idea is to make a direction vector for reducing the error at each iteration step. In other words, we make the next iteration to reduce the error from the computed error and the direction vector.