变换

This paper illustrates the process where projective transformation and affine transformation are employed to solve advanced geometrical problems.

引言变换是我们在几何解题中常用的方法之一。初等几何中的变换方法有对称变换、平移变换、旋转变换和相似变换，高等几何中的变换有射影变换、仿射变换等。

This paper analyzes the property of FNT, the characteristics of Kernel Matrixes and the relationship between Kernel Matrixes of two-dimensional Fermat number transform whose modulus are odd prime and image data, and presents and proves the deduction of the theorem which demonstrates the relationship among different transform matrixes corresponding to different roots of unity under the same odd prime modulus of FNT, and reveals the essence of FNT whose modulus are odd prime.

分析二维Fermat数变换性质、正反变换核的特点、二维模为奇素数的二维Fermat数变换正反变换核以及与图像数据的关系，提出了不同单位根对应变换矩阵间的关系定理的推论，证明了该推论中模为奇素数的二维Fermat数变换的不同单位根对应的变换矩阵之间的关系，揭示了模为奇素数的二维Fermat数变换的本质，在构造离散图象数据Fermat数变换时为单位根的选择奠定了理论基础。

It is shown that many of the existing integral transforms (including their logically equivalents) such as chirplet transform, dispersion transform, wavelet transform, chirp-Fourier transform, short-time Fourier transform, Gabor transform, Fourier transform, cosine transform, sine transform, Hartley transform, Laplace transform, z transform, Mellin transform, Hilbert transform, autocorrelation function, cross-correlation function, and the energy and mean of a signal, can each be considered as a special case of the FMmlet transform with specific parameters. In fact, an inventory of subspaces of FMmlet transform runs into countless numbers. The subspaces mentioned above are merely a few among a zoo of subspaces. They are essentially obtained by cutting the transform space of FMmlet transform at different positions, and can be likened to the computed tomography in medical diagnosis. Through these subspaces we actually see different slices or profiles of the FMmlet transform.

将现有诸多变换置于统一的 FMmlet 变换中加以审视，发现 chirplet 变换、频散变换、小波变换、 chirp-Fourier 变换、短时 Fourier 变换、 Gabor 变换、 Fourier 变换、余弦变换、正弦变换、 Hartley 变换、 Laplace 变换、变换、 Mellin 变换、 Hilbert 变换、自相关函数、互相关函数、能量和均值等，均为 FMmlet 变换在其参数取特定值时的特例；上述诸变换之间的差别，主要在于变换空间的维数有别，以及在不同空间维上取值的不同；这些变换有如医学诊断中的 CT，均由压缩 FMmlet 变换域空间所致，可以说我们通过这些变换看到的，是 FMmlet 变换的不同剖面。

Contents : Chapter 4 Image Enhancement (contrast enhancement, gray transform, histogram dressing, image smoothing, sharpening, Pseudo-color and false color enhancement); Chapter 5 of Image Restoration (inverse filter rehabilitation, Wiener filtering method); Chapter 6 of image processing orthogonal transformation Fourier transform, discrete cosine transform (DCT, the Walsh transform, Based on the eigenvectors of transformation, Hotelling transform, SVD transform, wavelet transform, Mallat algorithm; Chapter 7 Image Coding Hoffman (Huffman coding, arithmetic coding, RIc, the bit-plane coding, predictive coding, DPCM.

内容：第4章图像增强（对比度增强、灰度变换法、直方图修整法、图像平滑、图像锐化、伪彩色和假彩色增强）；第5章图像复原（逆滤波复原、维纳滤波方法）；第6章图像处理中的正交变换（傅立叶变换、离散余弦变换、沃尔什变换、基于特征向量的变换、霍特林变换、SVD变换、小波变换、Mallat算法）；第7章图像压缩编码（霍夫曼编码、算术编码、游程编码、位平面编码、预测编码、DPCM、JPEG 2000编码

Mastery: The Z-Transform, The Region of Convergence for the Z-Transform, The Inverse Z-Transform, Properties of the Z-Transform, System Function Algebra and Block Diagram Representations.

教学内容： Z 变换； Z 变换收敛域；逆 Z 变换；由零极点图对傅立叶变换进行几何求值； Z 变换性质；几个常用 Z 变换对；利用 Z 变换分析和表征线性时不变系统；系统函数的代数属性与方框图表示；单边 Z 变换。

The Laplace Transform; The Region of Convergence for Laplace Transforms; The Inverse Laplace Transform; Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot; Properties of the Laplace Transform; Analysis and Characterization of LTI Systems Using the Laplace Transform; System Function Algebra and Block Diagram Representations; The Unilateral Laplace Transform..

教学内容：拉普拉斯变换；拉普拉斯变换收敛域；拉普拉斯反变换；由零极点图对傅立叶变换进行几何求值；拉普拉斯变换性质；常用拉普拉斯变换对；用拉普拉斯变换分析和表征线性时不变系统；系统函数的代数属性与方框图表示；单边拉普拉斯变换。

In the ***, all types and fundamental principles of Radon transform are summarized systematacially. This *** starts from the general linear Radon transform, and then researches how to use the damped least square method to find the solution of the forward transform. However, the least squares smoothing effect will lead to a certain degree of trails in the data of Radon domain, which led to the application\'s limitations. After that the implementation in Radon domain and relative merits of linear Radon transform, parabolic Radon transform and polynomial Radon transform which have time invariance are introduced, the conjugate gradient method that can solve numerical solution is also given. And then the high resolution Randon transform is discussed, it introduces the regularization theory and Bayes principle into inversion, which successfully remove the negative affect caused by discrete Radon transform. Finally, we introduce the optimal weighting similar coefficient Radon transform, which adapt to the time variant quality of hyperbolic Radon transform.

本文系统总结了Radon变换的各种类型及其基本原理，从τ- p变换(即一般线性Radon变换)出发，研究利用阻尼最小二乘平方法求解其正变换，但是最小平方法的平滑作用会导致Radon域数据存在一定程度的拖尾效应，给这种方法的应用带来了很大的局限性；然后先容了具有时不变性的线性、抛物和多项式Radon变换在频率域的实现及其优缺点，给出了求解其数值解的共轭梯度算法；接着讨论了高分辨率Radon变换，它把正则化思想和贝叶斯原理引进到了反演中，很好的克服了离散Radon变换产生的截断效应对结果的不利影响；最后先容了适应双曲Radon变换时变性质的高分辨率最优相似系数加权Radon变换。

Including: basic coordinate changing: MOVE (move the entire or a part of figure to other location of coordinate); ZOOM (used for export map with different size); CIRCUMROTATION;仿射变换 integration move, zoom and circumrotation

坐标变换包括：基本坐标变换：平移（将图形的一部分或全部移动到迪卡尔坐标系中另外的位置）、缩放（用于输出大小不同的图形）、旋转；仿射变换（综合考虑图形的平移、旋转和缩放的变换）。

Including: basic coordinate changing: MOVE (move the entire or a part of figure to other location of coordinate); ZOOM (used for export map with different size); CIRCUMROTATION;仿射变换 integration move, zoom and circumrotation

包括：基本坐标变换：平移（将图形的一部分或全部移动到迪卡尔坐标系中另外的位置）、缩放（用于输出大小不同的图形）、旋转；仿射变换（综合考虑图形的平移、旋转和缩放的变换）。

Only with such characteristics, the movement equations can be expressed as matrices, and the idea of transforming the movement equations to the simplest form through a nonlinear transformation can be realized;(2) The form of Zi =Yi + YTH2i Y + Y7H3i Y(2)+ Y(2)T H4i Y(2)+ YTH5i Y(3) is adhibited in the nonlinear transformation, so that the multivalued problem caused by the nonlinear transformation is avoided, and the higher order transformation can be taken next;(3) The fourth order nonlinear transformation matrices H21,H31,H41 and H51 are derived, by which the original movement equations of electric power system is transformed to Jodan form in Z space;(4) By use of the fourth order nonlinear transformation, the approximate expression of the stability boundary is obtained, in Z space it is Z1= 0,in Y space it is Y1 + YTH21 Y + YTH31 Y(2)-i- Y(2) TH41 Y(2)+YTH51 Y(3)= 0;(5) The criterion used in this paper to judge whether the system critical unstable is simple and quick;(6) The method used in this paper is a direct method, and no need to construct an energy function.

正是由 于电力系统的运动方程具有这样的特性，才能写成矩阵的形式，通过非线性变换将电力系统的运动方程变换为最简单的线性形式的思想才能得以实现；（2）将通常运用于电力系统暂态稳定性分析的Normal Form变换的形式由 Yi＝ Zi＋ ZTh2riZ变形为 Zi＝ Yi＋YTH2iY＋YTH3iY（2）＋Y（2）TH4iY（2）＋YTH5iY（3），从而使得在对持续故障轨线实施同样的非线性变换以确定临界切除时间时，避免了非线性变换带来的多值性的问题，而只有在没有多值性问题的困扰下，才能采用较高阶的变换：（3）推导出了将原始电力系统系统的运动方程变换到Z空间的约当形式的非线性变换矩阵H21、H31、H41、HS1：（4）在运用四阶了「线性变换的情况下，给出了受扰动后系统的稳定边界的近似的解析表达，在Z空间为Z1=0，在y空间为： Y1+YTH21Y+YTH31Y(2)+Y(2)TH41Y(2)+YTH51Y(3)=0 （5）确定临界失稳的判据简单、快捷：对于一个复杂的电力系统，其稳定边界是相当复杂的一个高维曲面，即便是已知系统稳定边界的解析表达，要求出系统持续故障轨线何时与这一高维曲面相交，在数学上几乎是不可能实现的。

The absorption and distribution of chromium were studied in ryeusing nutrient culture technique and pot experiment.

采用不同浓度K2CrO4(0，0.4，0.8和1.2 mmol/L)的Hoagland营养液处理黑麦幼苗，测定铬在黑麦体内的亚细胞分布、铬化学形态及不同部位的积累。

By analyzing theory foundation of mathematical morphology in the digital image processing, researching morphology arithmetic of the binary Image, discussing two basic forms for the least structure element: dilation and erosion.

通过分析数学形态学在图像中的理论基础，研究二值图像的形态分析算法，探讨最小结构元素的两种基本形态：膨胀和腐蚀；分析了数学形态学复杂算法的基本原理，把数学形态学的部分并行处理理念引入到家实际应用中。