In the first stage of Wiener filtering, the noisy image is decomposed by SWT. Then, the signal variances of wavelet coefficients are estimated in the elliptic windows. In the second stage of Wiener filtering, the image restored by Inverse SWT is decomposed by DTCWT. Then, the adaptive windows determined by the autocorrelation function of wavelet coefficients` energy distribution are used to estimate the signal variance of wavelet coefficients. Finally, the image is restored by Inverse DTCWT.
在第一次局部维纳滤波中，用静态小波变换对含噪图像进行分解，然后利用椭圆方向窗来估计不同方向子带的各点信号的方差；在第二次的局部维纳滤波中，第一次局部维纳滤波恢复后的图像被对偶数复小波变换分解后，利用由子带能量自相关函数确定的自适应窗来估计不同方向子带的各点信号的方差，然后利用逆对偶数复小波变换对图像进行恢复。
In Chapter 2, a steady permeability identification problem in elliptic equa-tions for two-dimensional orthotropic medium is considered. We construct afunctional, obtain its first and second Gateaux differential, and prove that thefunctional is strictly convex on a convex admissible function set. We seek to iden-tify the permeability by minimization of the above functional.
第二章研究了二维稳态正交异性介质的渗透率反演问题，我们构造了一个泛函，求得该泛函的一、二阶Gateaux微分，证明了它在一个凸的容许集上的严格凸性，并使用极小化泛函即变分方法来求解此问题。
Under this background, we developed an equation to solve the phase problem from the amplitude of exit wave function, which derived from Schr? dinger equation as follows. whereφis amplitude of the exit wave function, I is the intensity of exit wave function, which is measurable, αis the phase which is unknown. The above expression is a two dimension elliptic partial differential equation.
在此背景下，本文从薛定谔方程出发，推导出如下的直接从波函数振幅模拟出射面波函数位相的表达式，在(2)式中φ为出射面波函数得振幅，I为出射波函数的强度，为实验可测的数据，α为位相，是要求的未知数，上式为二元二次椭圆偏微分方程，可以通过数值计算方法求解。
In long wave limit, these elliptic function wave patterns may degenerate into single valued localized excitations dromion and multi-lumps.
椭圆函数传播波之间的相互作用是非弹性的，而在长波极限下相应相干孤子激发Dromion，Lumps之间的相互作用可以是弹性的，也可以是非弹性的。
In long wave limit, i. e, the modulus of the Jacobi elliptic function m→1, some elliptic function waves may degenerate into solitary wave solutions such as bell-like solitary wave, kink solitary wave, saddle-like solitary wave and line soliton as well.
长波极限下即椭圆函数模量m→1时，一些椭圆函数波退化得到相应的非线性孤立波解，如钟形孤波，单扭结孤波，马鞍型双扭结孤波，线孤子等。
When A is an even integer, the operator obtained by us include the example given by Boca. When A is an odd integer, we give a new closed form of projection operator in terms of elliptic function.
这些算子在A为偶数时包括了Boca的著名例子，而在A为奇数时给出了新的由椭圆函数表达的有限形式。
It can measure various elliptic and circular pores, calculate parameters, locate center point and give out-of-roundness, which meet demands of many areas. Users can adopt elliptic measuring method to circles for higher precision if need. In this system, many algorithms have been studied on image preprocessing, manipulation and elliptic parameter calculation according to pore image characteristics, which improve the whole system performance and can be used for reference by similar machines. With the system, a function library is offered which can be used for further development.
本文的贡献主要是：1、提供了对多种椭圆和圆类型孔径进行参数检测、中心点标定和不圆度的检测，涵盖了多个领域的孔径检测要求；2、用户可以根据需要采用椭圆测量的方法对圆进行参数测量和中心点标定，提高了测量精度；3、在图像预处理、图像分割、椭圆和圆参数计算的软件编程中，比较研究了多种算法，根据椭圆孔图像的实际情况采用了一些实用有效的算法，提高了系统的整体性能并对同类产品具有借鉴作用；4、提供了整个系统软件的函数库，用户如有需要可在本系统软件取得相关数据的基础上进行二次开发。
Finally, a new function transform was proposed to solve nonlinear evolution equations with the help of the solutions of elliptic equation, and many new results are obtained
五、对椭圆方程进行讨论，提出新的函数变换并利用椭圆方程的解来求解非线性发展方程，从而得到大量新的周期波解。
A new technique is presented for maximizing the band-edge selectivity of elliptic filters, and then the equal ripple parameter of the quasi-elliptic function can be calculated when the band-edge selectivities of the elliptic function and quasi-elliptic function are the same.
提出了标准椭圆函数的边带优化方法，在此基础上使标准椭圆函数与修正后的准椭圆函数在边带上的衰减相等，得到准椭圆函数的等波纹系数。
By using the elliptic function and conformal transformation theory, a close form solution to this problem is obtained.
运用椭圆函数和保角变换理论，获得了该问题严格的闭合解。
If f is a Morse function then Sf may meet So non-elliptic Z. c.
如果f是莫尔斯作用那么Sf也许遇见那么非椭圆Z.c。
The contents are the following:In chapter two, the existence and multiplicity results for the following equation of p-Laplacian type are obtained. For the elliptic quasilinear hemivariational inequality involving the p-Laplacian operator, in order to use the mountain pass theorem proving the existence result, the authors usually need to use the uniform convexity of the Sobolev space to prove the energy function satisfies the PS condition. But for the p-Laplacian type equation mentioned above, this method is no use. To overcome this difficulty, the potential function is assumed to be convex, then I prove the existence result and by using the extension of the Ricceri theorem, the multiplicity result for the problem is obtained.
在第二章我们首先考虑关于以下p-Laplacian型(p-Laplacian type)方程非平凡解及多解的存在性对于带有p-Laplacian算子的椭圆拟线性半边分不等式问题，为应用非光滑的山路引理证明解的存在性，在证明方程所对应的能量泛函满足非光滑的PS条件时，需利用Sobolev空间的一致凸性，但是对于具有更一般形式的算子的p-Laplacian型方程，不具备上述性质，在文中为克服这一困难，本人对位势泛函做了一致凸的假设，从而证明了解的存在性，并应用推广的Ricceri定理，证明了方程三个解的存在性。
The approach of realization of the elliptic function complementary filter has been given.
提出了椭圆函数互补滤波器的实现条件。
Each of these three methods uses a different mathematic background, including one-way hash function, Chinese remainder theorem, and elliptic curve cryptosystem, to ensure the security of the secret key, and to meet the requirements of dynamic management such as solution to problems like adding and deleting groups, modifying relationships among groups, and changing of secret key, etc.
这三种方法分别使用不同的数学背景，包括单向杂凑函数、中国余数定理、以及椭圆曲线密码系统，用以确保秘密金钥的安全性，并且达到动态管理问题的需求，诸如：群组的新增与删除、群组关系的异动、以及更改秘密金钥等问题的解决方案。
We present and extend some methods such as the tanh-function method, the Jacobi elliptic function method, the ansatz method and so on.
本文对一些求解方法进行了研究，特别是Lie对称方法，分析并改进了前人的理论和算法，并在计算机符号系统Maple上给出了相应的实现。
The threshold condition for fracture is also expressed by another quarter-elliptic function.
疲劳裂纹扩展停止的门槛值条件也可用椭圆方程表达。
The NODE can be solved by F-expansion, which can be thought of as a generalization or concentration of the Jacobi elliptic function expansion method.
在求非线性数学物理方程的精确解方面，最近提出的F-展开法[3~13]是很有效的工具。
The existence of solutions is primary to check the reasonability. In Chapter 2, the 4-order elliptic equation can describe the structures of airplanes, ships and buildings. By using classical methods, such as, upper and lower solutions, an auxiliary function and Shauder fixed point theory, we achieve the existence under the presence of a couple supper and lower solutions.
解的存在性是鉴定模型合理性的首要前提。4-阶椭圆方程的边值问题可用于描述飞机、轮船及建筑物的结构模型，在第二章中，利用经典的分析方法，如上、下解方法，辅助的截断函数，Schauder不动点定理，在一对上、下解存在的假设下，我们首先论证了解的存在性。
This is a kind of evolutionary elliptic free boundary problems, in which the free boundary changes with time and at each fixed time the unknown function satisfies a boundary value problem for an equation of elliptic type.
详细本文分两部分。第一部分讨论拟定常电加工问题。这是一类发展型的椭圆自由边界问题，其自由边界依赖于时间的变化而不断演化，在每一固定时间，未知函数满足椭圆型方程的边值问题。
Namely, we discrete some function surfaces including sphere, elliptic paraboloid and hyperboloid paraboloid as samples, and give the error estimation in order to compare, analyze and explain results.
我们以离散型函数曲面（包括球面，椭球面，双曲抛物面）作为样本，对各种方法的实验结果进行误差统计和分析比较。