数字图像处理中的一些数学概念:矩阵,函数,导数,微分,积分

      数字图像处理对于数学还是有一定要求的,说它的要求高也高,往深里面去了,确实很难。但是作为初学者的我,如果一开始就被这些数学知识给难住了,那接下来的学习也不用谈了。学习是一个讲究循序渐进的过程。需要从理解开始,再逐步的深入。先从最最简单的概念入手。扫盲各种概念。要知道,对于数学而言,它的各种概念和定理可是非常多的。所以一定要耐下心一点点的夯实基础。如下是一些自己的资料收集和学习心得。比较简单,没有难的,因为难的我也不会啊。有不对的地方,欢迎拍砖并修正。


矩阵的运算

加减法:两个矩阵相加减,即他们相同位置的元素相加减。

注:只有对于同型矩阵(行列数相等),加减法才有意义。

矩阵的加减法符合交换律和结合律:

     即  A  + B = B + A     交换律

         (A + B) + C = A + (B + C)  结合律

数乘:数λ乘以A,即将矩阵A中的每个元素都乘以λ。记λA或Aλ。

数乘满足结合律,分配率

   (λμ)A = λ(μA);(λ + μ)A = λA +μA;结合律

  λ (A + B) = λA +λB;  分配率

乘法:AB = C,即用A的第i行与B的第j列相乘,取乘积之和,得到C的第i行j列元素。

数字图像处理中的一些数学概念:矩阵,函数,导数,微分,积分_第1张图片

矩阵相乘必须满足:A的列数 = B的行数。

矩阵的乘法不满足交换律,比如AB未必等于BA。


 

函数反函数

函数有三种常用表示法:图形表示法;表格表示法;解析表示法.见到最多的就是解析式表示法。

即用公式来表示函数的方法。

反函数

定义:设函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image004.gif的定义域是http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image005.gif,值域是http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image006.gif.若对任何http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image007.gif,在http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image005.gif内有唯一确定的http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image002.gif使http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image004.gif,则称这样形成的函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image002.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image004.gif的反函数,记为http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image008.gif,相应地,也称函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image057.gif是直接函数.

 函数是用自变量x来表示因变量y,反函数,就是用y来表示x。

求反函数的步骤:先从函数 http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image004.gif中解出 http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image008.gif,再置换http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image002.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image003.gif,就得到 http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image004.gif的 反函数 http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0104/image009.gif


 

导数和几种求导方法

导数概念引入的目的是为了描述连续变量变化率

定义:设函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image001.gif,当自变量在点http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image002.gif处有一个增量http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image036.gif,即自变量从http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image002.gif变化到http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image037.gif时,函数相应的增量为http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image009.gif如果极限http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image038.gif存在,则称函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image039.gif在点http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image002.gif可导,且这个极限值为函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image040.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image041.gif处的导数,记为http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image042.gif,即

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image043.gif  

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image039.gif在点 http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image002.gif处的导数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image042.gif也可用下列记号来表示:

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image049.gif, http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image050.gif, http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image051.gif

由导数的定义,求导数的方法可概括为以下三个步骤:
  (1) 求增量 求出当自变量从http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image003.gif变化到http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image071.gif时,函数相应的改变量

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image072.gif

  (2) 算比值 求出函数的平均变化率

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image073.gif

  (3) 取极限 求自变量的增量http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image074.gif时函数的瞬时变化率,即导数

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0301/image075.gif

高阶导数:

高阶导数就是从一阶导数开始逐级求导。

对于函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0304/image001.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0304/image002.gif称为它的一阶导数;
一阶导数的导数称为它的二阶导数,记为 http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0304/image007.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0304/image008.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0304/image009.gif,即http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0304/image010.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0304/image011.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0304/image012.gif


微分

微分的引入是为了描述连续变量变化量

定义:设函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image020.gif在某区间上有定义,http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image005.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image021.gif在这区间内,若函数的增量http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image022.gif可表示为

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image023.gif

其中http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image024.gif为不依赖于http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image014.gif的常数,而http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image025.gif是比http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image014.gif高阶的无穷小,则称函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image020.gif在点http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image026.gif可微,http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image027.gif称为http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image020.gif在点http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image028.gif处相应于自变量增量http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image014.gif的微分,记为http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image029.gif, 即

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0305/image030.gif

由定义可知,微分是描述增量,即△y。

这里引出一个问题:导数和微分的关系是什么呢?

函数的微分除以自变量的微分等于函数的导数,因此导数也称为“微商”,意即“微分之商”.

强调一下,导数是描述变化微分是描述变化

其实这些概念也没有那么可怕。只是其中的变化千万种,各种变化又精妙无穷,实在是要感叹数学家们的贡献。我等若能窥得其中一二,便受用无穷。

 

微分中值定理(费马定理,罗尔定理,拉格朗日中值定理)

中值定理揭示了函数在某区间上的整体性质,与函数在该区间内部某一点处的导数之间的关系,因而称为中值定理.中值定理有着重大的理论价值,因而也称为微分基本定理.

费马定理
  若函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image050.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image042.gif处可导,并且http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image042.gif是极值点,则http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image055.gif

  根据导数的几何意义,某点导数为零,即曲线在该点的切线斜率为零,亦即曲线在该点的切线平行于http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image011.gif轴.

  费马定理的几何意义为:若http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image042.gif是函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image058.gif的极值点,且函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image058.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image042.gif处可导,则曲线http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image009.gif在点http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image061.gif处的切线一定是水平的.

  我们将导数为零的点称为驻点.则费马定理又可表述为: 可导的极值点一定是
驻点

罗尔定理
  设函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image058.gif满足
  (1)在闭区间http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image078.gif上连续,
  (2)在开区间http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image080.gif内可导,
  (3)http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image082.gif
  则至少存在一点http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image084.gif,使得http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image086.gif,即http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image058.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image080.gif内至少有一个驻点

  罗尔定理的几何意义是:如果函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image058.gif满足所给的三个条件时,则在区间内部至少能找到一个点,该点处切线是水平的.

拉格朗日中值定理

设函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image058.gif在闭区间http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image078.gif上连续,在开区间http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image080.gif内可导,则在http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image080.gif内至少存在一点http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image103.gif,使得

即:                                                                                                                       

注意到结论中,等式http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image112.gif的右端正好是连接A点和B点的直线的斜率(两点的纵坐标之差与两点的横坐标之差的比值),所以拉格朗日中值定理的几何意义是:在http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image080.gif内至少能找到一点,该点处的切线平行于弦http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0401/image115.gif

数字图像处理中的一些数学概念:矩阵,函数,导数,微分,积分_第2张图片

拉格朗日中值定理是罗尔中值定理的推广,它反映了可导函数在闭区间上的整体平均变化率与某点的局部变化率的关系。


 

积分

不定积分

函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image008.gif在区间http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image004.gif上的全体原函数称为http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image008.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image004.gif上的不定积分,记为http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image025.gif.其中称记号http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image026.gif称为积分号http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image008.gif为被积函数,http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image027.gif为被积表达式,http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image028.gif为积分变量.

记号http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image025.gif表示http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image029.gif的全体原函数,只需找出http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image029.gif的一个原函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image030.gif,则http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image030.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image031.gif就是http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image029.gif的全体原函数,即http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image025.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image032.gif

例:

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image010.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image012.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image011.gif的一个原函数,所以http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image011.gif的全体原函数为http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0501/image033.gif

由定理可知,不定积分是一个函数族。是不确定的。求不定积分的过程,是找原函数的过程。同时,不定积分又称原函数反导数

 

定积分

 定义:定积分就是上述类型的和式的极限,记为http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image031.gif,即

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image032.gif

  其中http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image033.gif叫做被积函数,http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image034.gif叫做被积表达式,http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image009.gif叫做积分变量,http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image035.gif叫做积分下限,http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image036.gif叫做积分上限,http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image013.gif叫做积分区间.

定积分是对固定的区间,由自变量得因变量,并对因变量进行累计求和的过程。也称为对变量进行积分。因为自变量是连续的,所以需要用到极限。积分的过程也是在求和。

定积分的几何意义
  (1) 当http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image038.gif时,曲边梯形在http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image009.gif轴的上方(如图所示),定积分http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image039.gif表示曲边梯形的面积,即 http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image040.gif

数字图像处理中的一些数学概念:矩阵,函数,导数,微分,积分_第3张图片

  (2) 当http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image042.gif时,曲边梯形在http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image009.gif轴的下方(如图所示),定积分http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image039.gif表示曲边梯形的面积的负值,即http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image043.gif

数字图像处理中的一些数学概念:矩阵,函数,导数,微分,积分_第4张图片


  (3) 当http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image011.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image005.gif上有正、有负时,则由http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image011.gif,直线http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image006.gif以及http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image009.gif轴所围的图形,部分位于http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image009.gif轴上方,部分位于http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image009.gif轴下方(如图所示).http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image045.gif表示的是上、下图形面积的代数和:http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image009.gif轴上方图形的面积之和-http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image009.gif轴下方图形的面积之和.例如出现下图所示的情况时,则有http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image045.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0505/image046.gif

数字图像处理中的一些数学概念:矩阵,函数,导数,微分,积分_第5张图片

定积分是为了求函数f(X)在区间[a,b]中的图像包围的面积

定积分和不定积分的区别

定积分是一个

不定积分是一个函数表达式

定积分可以用牛顿-莱布尼兹公式来计算。除此之外,定积分和不定积分并没有任何关系!只是名字比较接近,容易混淆而已。

 

牛顿-莱布尼兹公式

定理:设在区间http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0507/image001.gif上,http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0507/image002.gif是连续函数http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0507/image003.gif的一个原函数,则

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0507/image004.gif

  此公式称为牛顿-莱布尼兹公式.为了便于使用上述公式,记

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0507/image005.gif,或http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0507/image006.gif

  则 http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0507/image007.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0507/image008.gif,或http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0507/image007.gifhttp://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/images/0507/image009.gif

牛顿-莱布尼茨公式,给定积分提供了一个有效而简便的计算方法,大大简化了定积分的计算过程。

 

如上只是一些公式的简单的罗列,目的也是为了,看到这些概念的时候,能联想到概念的背后是什么东西。不至于太陌生。具体公式的背后,则各有千秋。不得不感叹数学的博大精深。然而,深的,我也不懂啊,读书时候,数学不好,工作了,就全忘记了。只能在用到的时候,再翻出来细细研究。

引用:

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/ch0605.html

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/ch0102.html

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/ch0104.html

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/ch0301.html

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/ch0304.html

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/ch0305.html

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/ch0401.html

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/ch0511.html

http://www2.edu-edu.com.cn/lesson_crs78/self/j_0022/soft/ch0507.html

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