分数阶微积分在图像处理中的研究综述*

综述了关于分数阶微积分理论在数字图像底层处理中的应用研究,具体包括:分数阶微积分和分数阶偏微分方程的基本理论及分数阶傅里叶变换的基本性质;分数阶微分滤波器的构造及在图像增强中的应用研究;分数阶积分滤波器的构造及在图像去噪中的应用研究;分数阶偏微分方程在图像去噪中的应用研究。最后,总结了分数阶微积分理论在图像处理中已取得的研究成果,并结合已有的基于分数阶微积分理论的图像底层处理模型,展望了分数阶微积分理论在图像处理中的应用前景。     

分数阶偏微分方程在图像处理中的应用

分数阶偏微分方程在图像处理中的应用已受到了广泛的关注,尤其在图像去噪和图像超分辨率（SR）重建方面,目前的研究成果已显示了分数阶应用的优势与效果。对分数阶微积分在图像处理中的作用进行了分析;介绍并讨论了分数阶偏微分方程在图像去噪和图像超分辨率重建中的相关理论与模型;通过仿真实验表明,基于分数阶偏微分方程的方法在去噪和减少阶梯效应等方面比整数阶偏微分方程更具有优势;最后指出了未来的相关研究问题。     

几种分数阶微分数值解法的比较研究

研究分数阶算法特性,基于抛物线插值方法,对分数阶微分计算设计了一种数值计算方法.明确了几种算法的特点,并进行了数值仿真,对于函数分数阶微分及分数阶微分方程的求解,在计算精度与计算时间开销方面,与迭代方法、线性插值方法进行了比较,结果表明迭代方法的计算精度和时间开销两方面都比较好,抛物线插值方法的计算精度高于线性插值,但线性插值在时间开销方面占优势,研究结果可为实际选择分数阶微分数值计算方法提供科学依据.     

时间分数阶四阶扩散方程的显-隐和隐-显差分格式

时间分数阶四阶扩散方程是一类重要的发展型偏微分方程,其数值解的研究有重要的科学意义和工程实际价值.本文针对时间分数阶四阶扩散方程,研究一类显-隐(E-I)差分格式和隐-显(I-E)差分格式解法,该方法基于经典隐式和经典显式格式相结合构造而成,分析E-I和I-E两种差分格式解的存在唯一性、稳定性和收敛性.理论分析和数值试验结果证实本文E-I差分格式和I-E差分格式无条件稳定,具有空间2阶精度,时间2-α阶精度.在计算精度一致的要求下,E-I和I-E差分格式较经典隐式差分格式具有省时性,其计算时间相比古典隐格式减少约70%,研究表明本文格式求解时间分数阶四阶扩散方程是有效的.     

基于分数阶偏微分方程的图像去噪新模型

将分数阶微分理论和全变分方法相结合应用于图像去噪,提出了一种基于分数阶偏微分方程的图像去噪新模型。该模型很好地继承了现有的全变分(TV)模型去噪效果与保持图像边缘细节特征的优点,同时利用分数阶微分运算特有的幅频特性优势,较好地保留了图像平滑区域中灰度变化不大的纹理细节。实验结果表明：一方面,与现有去噪方法相比,新模型不仅具有较强的抑制噪声能力,而且能较好地保持图像边缘特征,还能保留更多的图像纹理细节信息,优于常用的整数阶偏微分图像去噪方法;另一方面,从峰值信噪比的对比实验可以看出该模型去噪效果优于其他方法,较好地达到了去噪目的,是一种有效、实用的图像去噪模型。     

一类四阶各向异性扩散方程在图像放大中的应用

在文中我们首先分析了进行图像放大时各向异性偏微分方程优于各向同性偏微分方程,随后我们分析了在本文中不同四阶模型的扩散方向.为了消除低阶偏微分方程在处理图像中出现的块状效应的影响,同时保证方程为各向异性扩散,我们构造了两个各向异性的四阶偏微分方程,并且分别从数据和放大图像效果两方面来说明我们给出的模型优于文中提到的其它四个模型.     

基于反步法的耦合分数阶反应扩散系统边界输出反馈控制

针对具有空间依赖耦合系数的分数阶反应扩散系统, 利用反步法设计了基于观测器的边界输出反馈控制器, 证明了观测增益和控制增益核函数矩阵方程的适定性. 针对误差系统和输出反馈的闭环系统, 利用分数阶Lyapunov方法分析了系统的Mittag-Leffler稳定性, 且利用Wirtinger不等式改进了耦合系统稳定的条件. 当系统具有空间依赖的耦合系数时, 难以求得控制增益和观测增益核函数的解析解, 为此, 给出了核函数偏微分方程的数值解方法. 数值仿真验证了理论结果.     

基于分数阶微分的图像增强

通过理论分析得出分数阶微分可以大幅提升信号高频成分,增强信号的中频成分、非线性保留信号的甚低频,据此得出分数阶微分应用于图像增强将使图像边缘明显突出、纹理更加清晰和图像平滑区域信息得以保留的增强图像;然后由经典的分数阶微分定义出发,推导出了分数阶差分方程,构建了近似的Tiansi微分算子.通过图像增强的实验表明:采用基于分数阶微分算子的图像增强方法,其增强图像的视觉效果明显优于传统的微分锐化(整数微分)方法.文中方法为拓展分数阶微分的应用领域进行了有意义的探索.     

考虑分数阶梯度的雾天图像增强偏微分方程模型

为了在去雾的同时增强图像中的纹理细节信息,提高图像亮度,改善图像质量,提出一个雾天图像增强的分数阶偏微分方程模型.将分数阶微分与大气散射物理模型结合,建立了去雾图像的分数阶梯度场;为了突出图像的纹理细节信息,避免出现边缘过度增强或细节纹理增强不够的现象,构造了分数阶梯度场的增强函数,使分数阶梯度场随着梯度模的变化达到非线性增强的效果;在梯度域建立能量泛函,使雾天图像梯度场逼近增强梯度场,通过变分法得到分数阶偏微分方程图像增强模型;最后用有限差分法对模型进行数值求解.实验结果表明,文中模型在去雾的同时,能够有效地提高图像的对比度和清晰度,是一种有效的雾天图像增强模型.     

基于1~2阶分数阶微分的图像增强算法

在利用分数阶微分进行图像增强时,现有方法大多是基于0~1阶分数阶微分,而基于1~2阶分数阶微分的方法较少。为此,分析1~2阶分数阶微分对图像增强的作用,基于1~2阶分数阶微分构造一种用于图像增强的掩模算子。实验结果表明,该算子优于常用的频域法和空域法,比现有的一些0~1阶分数阶微分算子具有更好的图像增强效果。     

一种自适应分数阶偏微分图像增强模型

针对传统的自适应分数阶偏微分方程图像增强算法对图像暗区纹理区域的增强不足的缺点,考虑到人眼对光感的敏感程度不同,将亮度对视觉的影响因素考虑进传统的自适应分数阶偏微分方程图像增强算法。以梯度和灰度值为参数,建立了一种新的自适应分数阶偏微分图像增强模型。该模型改善了传统算法对暗区图像增强不足的缺点,图像增强后的平均梯度提升明显,很好地改善了图像的视觉效果。实验结果说明本算法具有一定的有效性。     

The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics

A non-standard finite difference scheme is developed to solve the linear partial differential equations with time- and space-fractional derivatives. The Grunwald–Letnikov method is used to approximate the fractional derivatives. Numerical illustrations that include the linear inhomogeneous time-fractional equation, linear space-fractional telegraph equation, linear inhomogeneous fractional Burgers equation and the fractional wave equation are investigated to show the pertinent features of the technique. Numerical results are presented graphically and reveal that the non-standard finite difference scheme is very effective and convenient for solving linear partial differential equations of fractional order.     

Quasi-wavelet method for time-dependent fractional partial differential equation

In this decade, many new applications in engineering and science are governed by a series of fractional partial differential equations. In this paper, we propose a novel numerical method for a class of time-dependent fractional partial differential equations. The time variable is discretized by using the second order backward differentiation formula scheme, and the quasi-wavelet method is used for spatial discretization. The stability and convergence properties related to the time discretization are discussed and theoretically proven. Numerical examples are obtained to investigate the accuracy and efficiency of the proposed method. The comparisons of the present numerical results with the exact analytical solutions show that the quasi-wavelet method has distinctive local property and can achieve accurate results.     

The Exp-function Method for Some Time-fractional Differential Equations

In this article, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the Exp-function method are employed for constructing the exact solutions of nonlinear time fractional partial differential equations in mathematical physics. As a result, some new exact solutions for them are successfully established. It is indicated that the solutions obtained by the Exp-function method are reliable, straightforward and effective method for strongly nonlinear fractional partial equations with modified Riemann-Liouville derivative by Jumarie's. This approach can also be applied to other nonlinear time and space fractional differential equations.      

A numerical method for solving a fractional partial differential equation through converting it into an NLP problem

In this paper, we propose a new approach for solving fractional partial differential equations, which is very easy to use and can also be applied to equations of other types. The main advantage of the method lies in its flexibility for obtaining the approximate solutions of time fractional and space fractional equations. Using this approach, we convert a fractional partial differential equation into a nonlinear programming problem. Several numerical examples are used to demonstrate the effectiveness and accuracy of the method.     

The BEM for numerical solution of partial fractional differential equations

A numerical method is presented for the solution of partial fractional differential equations (FDEs) arising in engineering applications and in general in mathematical physics. The solution procedure applies to both linear and nonlinear problems described by evolution type equations involving fractional time derivatives in bounded domains of arbitrary shape. The method is based on the concept of the analog equation, which in conjunction with the boundary element method (BEM) enables the spatial discretization and converts a partial FDE into a system of coupled ordinary multi-term FDEs. Then this system is solved using the numerical method for the solution of such equations developed recently by Katsikadelis. The method is illustrated by solving second order partial FDEs and its efficiency and accuracy is validated.     

Fractional partial differential equation denoising models for texture image

In this paper,a set of fractional partial differential equations based on fractional total variation and fractional steepest descent approach are proposed to address the problem of traditional drawbacks of PM and ROF multi-scale denoising for texture image.By extending Green,Gauss,Stokes and Euler-Lagrange formulas to fractional field,we can find that the integer formulas are just their special case of fractional ones.In order to improve the denoising capability,we proposed 4 fractional partial differential equation based multiscale denoising models,and then discussed their stabilities and convergence rate.Theoretic deduction and experimental evaluation demonstrate the stability and astringency of fractional steepest descent approach,and fractional nonlinearly multi-scale denoising capability and best value of parameters are discussed also.The experiments results prove that the ability for preserving high-frequency edge and complex texture information of the proposed denoising models are obviously superior to traditional integral based algorithms,especially for texture detail rich images.     

Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science

In this paper, we suggest a fractional functional for the variational iteration method to solve the linear and nonlinear fractional order partial differential equations with fractional order initial and boundary conditions by using the modified Riemann-Liouville fractional derivative proposed by G. Jumarie. Fractional order Lagrange multiplier has been considered. Solution has been plotted for different values of α.