An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems

Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous solution. A well known mesh condition for the DMP satisfaction by the linear finite element solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the finite element solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem. Several variants of the new condition are obtained. Based on one of them, two metric tensors for use in anisotropic mesh generation are developed to account for DMP satisfaction and the combination of DMP satisfaction and mesh adaptivity. Numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors.     

A general strategy for anisotropic diffusion in MR image denoising and enhancement

Anisotropic diffusion (AD) has proven to be very effective in the denoising of magnetic resonance (MR) images. The result of AD filtering is highly dependent on several parameters, especially the conductance parameter. However, there is no automatic method to select the optimal parameter values. This paper presents a general strategy for AD filtering of MR images using an automatic parameter selection method. The basic idea is to estimate the parameters through an optimization step on a synthetic image model, which is different from traditional analytical methods. This approach can be easily applied to more sophisticated diffusion models for better denoising results. We conducted a systematic study of parameter selection for the AD filter, including the dynamic parameter decreasing rate, the parameter selection range for different noise levels and the influence of the image contrast on parameter selection. The proposed approach was validated using both simulated and real MR images. The model image generated using our approach was shown to be highly suitable for the purpose of parameter optimization. The results confirm that our method outperforms most state-of-the-art methods in both quantitative measurement and visual evaluation. By testing on real images with different noise levels, we demonstrated that our method is sufficiently general to be applied to a variety of MR images.     

The adaptive sinogram restoration algorithm based on anisotropic diffusion by energy minimization for low-dose X-ray CT

Apparent streak artifacts will present in reconstructed images due to excessive quantum noise in low-dose X-ray imaging process. Estimating a noise-free sinogram to satisfy the filtered back-projection (FBP) reconstruction is an effective way to solve this problem. In this paper, we propose a novel sinogram noise reduction method by energy minimization. An adaptive smoothness parameter based on a modified anisotropic diffusion coefficient is applied for an optimal estimation. The smoothness parameter can make the method effectively adjust the degree of smoothness according to the noise level and the region feature in the sinogram. Visual effect together with quantitative analysis of the experimental result shows the developed approach has the excellent performance in protection of the edge and removal of streak artifacts in the reconstructed image.     

A Fourier method for the fractional diffusion equation describing sub-diffusion

In this paper, a fractional partial differential equation (FPDE) describing sub-diffusion is considered. An implicit difference approximation scheme (IDAS) for solving a FPDE is presented. We propose a Fourier method for analyzing the stability and convergence of the IDAS, derive the global accuracy of the IDAS, and discuss the solvability. Finally, numerical examples are given to compare with the exact solution for the order of convergence, and simulate the fractional dynamical systems.     

Comparing anisotropic diffusion filters for the enhancement of sodium magnetic resonance images

The anisotropic diffusion (AND) filter, an image processing technique derived from physics, was applied to low-resolution sodium magnetic resonance imaging (MRI) to examine the possibilities of image enhancement by postprocessing. We compared six different variants of AND filters. Besides the qualitative good results on phantom measurements, quantitative analyses on MRI of human kidney yielded major improvements in noise reduction and other quality measures: the noise (i.e., the standard deviation in the image background) could be reduced to 1%-2% of its original value, while linear filters (Gaussian, Fermi, Hamming) achieved a reduction to 42%-64%. Besides that, less than 5% of structures and intensities are lost when using AND filters. Comparing the different variants, the two-dimensional and the three-dimensional AND filter outperformed the histogram-of-gradient and tensor-based AND filter. We envision that by using these AND filters, quantitative analysis of sodium MRI of kidney could be improved.     

Efficient anisotropic filtering of diffusion tensor images

To improve the accuracy of structural and architectural characterization of living tissue with diffusion tensor imaging, an efficient smoothing algorithm is presented for reducing noise in diffusion tensor images. The algorithm is based on anisotropic diffusion filtering, which allows both image detail preservation and noise reduction. However, traditional numerical schemes for anisotropic filtering have the drawback of inefficiency and inaccuracy due to their poor stability and first order time accuracy. To address this, an unconditionally stable and second order time accuracy semi-implicit Craig-Sneyd scheme is adapted in our anisotropic filtering. By using large step size, unconditional stability allows this scheme to take much fewer iterations and thus less computation time than the explicit scheme to achieve a certain degree of smoothing. Second-order time accuracy makes the algorithm reduce noise more effectively than a first order scheme with the same total iteration time. Both the efficiency and effectiveness are quantitatively evaluated based on synthetic and in vivo human brain diffusion tensor images, and these tests demonstrate that our algorithm is an efficient and effective tool for denoising diffusion tensor images.     

Spin diffusion in anisotropic Heisenberg chains: S≥1/2

In this paper, we investigate spin diffusion in Heisenberg chains with uniaxial nearest-neighbor interactions. The approach followed is based on an analysis of the infinite-temperature longitudinal spin density and spin current correlation functions. For S=1/2, exact results are presented for the time-dependent correlation functions in the XY limit. Away from this limit, the second and fourth moments of the Fourier transform of the spin density correlation function provide information about spin dynamics for arbitrary values of the spin. The moments are used in an assessment of the accuracy of the Gaussian approximation for the spin diffusion constant for S=1/2. The general behavior of the Gaussian approximation when S>1/2 is discussed, and numerical results for the spin diffusion constant are presented for S=1/2, 1, 3/2, 2 and in the classical limit. A moment-based criterion for the boundary in reciprocal space between diffusive and non-diffusive dynamics that applies to arbitrary values of the spin is presented.     

各向异性扩散滤波远探测声波测井图像降噪方法*

声波远探测测井技术近年来在复杂油气藏的构造识别和储层评价中发挥着重要作用。该技术利用来自井外的反射波对井旁地质构造进行准确成像,但由于反射波具有幅度低、受幅度强的井孔直达波干扰等特点,实际数据提取到的反射波信噪比往往比较低,需要对反射波进行降噪处理。非线性各向异性扩散滤波能够在滤除图像噪声的同时保留图像边缘及细节等信息,在地震数据处理和医学图像去噪中都有广泛应用。该文从各向异性扩散滤波的基本原理入手,将提取到的井外反射波信号当作图像,采用不同扩散张量进行处理,通过含噪声的模拟数据处理验证了该方法的处理效果并建立起适合于远探测测井的数据处理流程,实际远探测数据处理结果进一步表明其具有较好的应用前景。     

A semi-implicit lattice method for simulating flow

We propose a new semi-implicit lattice numerical method for modeling fluid flow that depends only on local primitive variable information (density, pressure, velocity) and not on relaxed upstream distribution function values. This method has the potential for reducing parallel processor communication and permitting larger time steps than the lattice-Boltzmann method. Several benchmark problems are solved to demonstrate the accuracy of the method.     

Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular media

Effect of scattering on radiative heat transfer in two-dimensional rectangular media by the finite-volume method has been studied. Compared with the existing solutions, it shows that the result obtained by the finite-volume method is reliable. Furthermore, relative errors caused by the approximation that linear and nonlinear anisotropic scattering media is simplified to isotropic scattering media have been studied.     

基于各向异性扩散的单幅图像去雾算法

针对单幅图像去雾算法容易产生光晕现象且去雾后图像细节不突出的问题,提出了一种基于各向异性扩散的去雾算法.首先在基于像素的暗通道先验假设的基础上计算出初始大气传输函数,使用Perona-Malik偏微分方程模型求解出精细化的大气传输函数,再经过最小值校正,最终得到准确的大气传输函数。为了估计大气光,对基于像素的亮通道图像进行像素排序,从中选取出可靠的大气光向量。实验结果表明,提出的算法能够恢复更多的图像细节,同时有效地抑制了光晕现象。     

An anisotropic images segmentation and bias correction method

Intensity inhomogeneities cause considerable difficulty in the quantitative analysis of magnetic resonance (MR) images. Thus, bias field correction is a necessary step before quantitative analysis of MR data can be undertaken. This paper presents an anisotropic approach to bias correction and segmentation for images with intensity inhomogeneities and noise. Intensity-based methods are usually applied to estimate the bias field; however, most of them only concern the intensity information. When the images have noise or slender topological objects, these methods cannot obtain accurate results or bias fields. We use structure information to construct an anisotropic Gibbs field and combine the anisotropic Gibbs field with the Bayesian framework to segment images while estimating the bias fields. Our method is able to capture bias of quite general profiles. Moreover, it is robust to noise and slender topological objects. The proposed method has been used for images of various modalities with promising results.     

Finite difference approximations for a fractional advection diffusion problem

The use of the conventional advection diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative diffusion models have been proposed. Fractional space derivatives are used to model anomalous diffusion or dispersion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion model. When a fractional derivative replaces the second derivative in a diffusion or dispersion model, it leads to enhanced diffusion, also called superdiffusion. We consider a one-dimensional advection–diffusion model, where the usual second-order derivative gives place to a fractional derivative of order α$\alpha$, with 1<α?2$1<\alpha ?2$. We derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection–diffusion equation. We present the order of accuracy of the schemes and in order to show its convergence we prove they are stable under certain conditions. In the end we present a test problem.     

基于图像熵的各向异性扩散相干斑噪声抑制

本文提出了一种基于图像熵的各向异性扩散滤波方法. 该方法使用图像熵作为边缘检测算子, 避免了由于均值和方差等统计量的估计带来的误差, 提高了边缘检测能力. 试验结果表明该方法能够获得比传统方法更好的相干斑噪声抑制效果. 关键词： 相干斑抑制 各向异性扩散 图像熵 合成孔径雷达     

页岩气滑脱、扩散传输机理耦合新方法

针对页岩气流动计算中所用耦合机理不同的现状,且为了厘清滑脱和各种扩散之间的关系,首先采用理论分析和数学模型的方法,根据定义和微观运动机制对滑脱和各种扩散进行了分析,然后在考虑吸附层页岩气分子所占空间对气体流动影响的情况下,提出了"壁联扩散"的概念来表征克努森扩散和表面扩散的总效应,并指出壁联扩散和滑脱效应等同,由此提出了壁联扩散和滑脱效应在流动计算中可互换而不重复叠加的耦合新方法.实例验证表明,当毛细管半径从5 nm增大到2000 nm,壁联扩散和滑脱效应的质量通量相对误差较小,在绝大部分范围内都小于10%,且在整个孔径范围内两者平均值相差1.4×10~(-6)kg·m~(-2)·s~(-1),即平均值的相对误差仅为5.8%,该方法可以满足工程计算的需要.考虑到参数选取、机理数学模型有待完善等方面的影响,新方法的论证存在进一步提升的空间.壁联扩散的提出具有实际开发意义和多重研究意义,耦合新方法的提出阐明了滑脱和各种扩散之间的关系,防止了页岩纳米级孔隙中流动机理的重复叠加,能较好改变页岩气流动计算耦合方法不一致的现状,为页岩气开发定量计算指明新方向.     

A second-order accurate numerical method for the two-dimensional fractional diffusion equation

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow, finance and others. In this paper, we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation. This numerical method combines the alternating directions implicit (ADI) approach with a Crank–Nicolson discretization and a Richardson extrapolation to obtain an unconditionally stable second-order accurate finite difference method. The stability and the consistency of the method are established. Numerical solutions for an example super-diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method.     

A fast method for the solution of the Helmholtz equation

In this paper, we consider the numerical solution of the Helmholtz equation, arising from the study of the wave equation in the frequency domain. The approach proposed here differs from those recently considered in the literature, in that it is based on a decomposition that is exact when considered analytically, so the only degradation in computational performance is due to discretization and roundoff errors. In particular, we make use of a multiplicative decomposition of the solution of the Helmholtz equation into an analytical plane wave and a multiplier, which is the solution of a complex-valued advection–diffusion–reaction equation. The use of fast multigrid methods for the solution of this equation is investigated. Numerical results show that this is an efficient solution algorithm for a reasonable range of frequencies.     

A fast chaos-based symmetric image cryptosystem with an improved diffusion scheme

In recent years, a number of chaos-based image cryptosystems have been proposed to meet the increasing demand for real-time secure image transmission. In this paper, an improved diffusion scheme named continuous diffusion strategy is proposed to promote the efficiency of the conventional permutation–diffusion type image cipher. The new scheme contains a supplementary diffusion procedure after the conventional diffusion process and the control parameters are altered by the cipher image after the first diffusion procedure. As a result, the difference can be introduced at the beginning and spread out to the whole image, and hence the same level of security can be achieved with fewer overall rounds. Moreover, to further enhance the confusion effect of the diffusion operation, an intensive diffusion approach is proposed, using stretched key stream elements to perform a cyclic shift to the cipher pixels. Extensive cryptanalysis has been performed using differential analysis, key space analysis, key sensitivity analysis and various statistical analyses. Experiment results demonstrate that the new scheme has a high level of security and fast encryption speed for practical image encryption.     

A Gaussian distribution model of anisotropic magnetorheological elastomers

A Gaussian distribution model was developed to examine the field-induced performance of anisotropic magnetorheological elastomers. The developed model was based on the assumption that the iron particles in magnetorheological elastomers aggregate into a large number of parallel body-centered tetragonal structure columns whose length obeys the Gaussian distribution. By using multi-pole approximation with local field effect and taking into account the nonlinearity and saturation of particle magnetization, the field-induced shear modulus was calculated as a function of distribution and dimension of the particle structures, the external magnetic field and the dynamic shear strain. Compared with other modes as well as the published experimental results, this model shows a remarkable improvement in accurately predicting the behavior of the magnetorheological elastomers.     

Equation of state for anisotropic spheres

We present a new class of exact interior solutions for anisotropic spheres to the Einstein field equations with a prescribed energy density. This category of solutions has similar energy density profiles to the models of Chaisi and Maharaj (Gen. Rel. Grav. 37, 1177–1189, 2005) whose approach we follow in the integration process. A distinguishing feature of the solutions presented is that they satisfy a barotropic equation of state linearly relating the radial pressure to the energy density.