GeometryTransformation 几何变换
对图像的几何变换本质上是一种线性变换,其数学本质为
Inew=TIold
即通过变换矩阵 T 将原图上的点的位置 Iold 变换到新的位置,从而得到新的图像 Inew
2D平面变换示意图(”Computer Vision: Algorithms and Applications”, RichardSzeliski)
- Translation 平移
- Euclidean(rigid, rotation) 旋转
- Scale 缩放;图中没有画出
- Similarity 相似变换;结合旋转,平移和缩放
- Affine 仿射变换;想象在similarity的基础上用两只手对图像进行按压拉伸
- Projective 投影变换;想象投影仪做的事情,将一个面投影到另外一个面的情况
仿射变换（Affine Transformation）和齐次坐标系（Homogeneous Coordinate)是计算机图形学中经常碰到的基本概念。这篇文章主要讲述什么是仿射变换和齐次坐标系，以及在图形系统中为什么要是用它们。不求全面，只为自己学习理解。
仿射变换其实是另外两种简单变换的叠加：一个是线性变换，一个是平移变换。统一平移变换和线性变换的一种变换我们起了个名字叫“仿射变换”。这个新的变换就不再单纯的是两个线性空间的映射了，而是变成了两个仿射空间的映射关系。为了更好地理解仿射变换，首先就要知道线性变换以及它的不足。在未说明的情况下，下面使用的是卡迪尔坐标系。
所谓线性变换是指两个线性空间的映射，一个变换
举个例子说明一下。建设
线性变换可以用矩阵来表示。假设
![\left[ {\begin{array}{c} x'\\y'\\ \end{array}} \right]=R(p)= \left[ {\begin{array}{cc} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \\ \end{array} } \right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bc%7D+x%27%5C%5Cy%27%5C%5C+%5Cend%7Barray%7D%7D+%5Cright%5D%3DR%28p%29%3D+%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bcc%7D+cos%28%5Ctheta%29+%26+-sin%28%5Ctheta%29+%5C%5C+sin%28%5Ctheta%29+%26+cos%28%5Ctheta%29+%5C%5C+%5Cend%7Barray%7D+%7D+%5Cright%5D&bg=ffffff&fg=555555&s=0 "\left[ {\begin{array}{c} x'\\y'\\ \end{array}} \right]=R(p)= \left[ {\begin{array}{cc} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \\ \end{array} } \right]")
![\left[ {\begin{array}{cc} x \\ y \\ \end{array} } \right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bcc%7D+x+%5C%5C+y+%5C%5C+%5Cend%7Barray%7D+%7D+%5Cright%5D&bg=ffffff&fg=555555&s=0 "\left[ {\begin{array}{cc} x \\ y \\ \end{array} } \right]")
![\left[ {\begin{array}{c} x'\\y'\\ \end{array}} \right]=S(p)= \left[ {\begin{array}{cc} S_x & 0 \\ 0 & S_y \\ \end{array} } \right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bc%7D+x%27%5C%5Cy%27%5C%5C+%5Cend%7Barray%7D%7D+%5Cright%5D%3DS%28p%29%3D+%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bcc%7D+S_x+%26+0+%5C%5C+0+%26+S_y+%5C%5C+%5Cend%7Barray%7D+%7D+%5Cright%5D&bg=ffffff&fg=555555&s=0 "\left[ {\begin{array}{c} x'\\y'\\ \end{array}} \right]=S(p)= \left[ {\begin{array}{cc} S_x & 0 \\ 0 & S_y \\ \end{array} } \right]")
![\left[ {\begin{array}{c} x \\ y \\ \end{array} } \right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bc%7D+x+%5C%5C+y+%5C%5C+%5Cend%7Barray%7D+%7D+%5Cright%5D&bg=ffffff&fg=555555&s=0 "\left[ {\begin{array}{c} x \\ y \\ \end{array} } \right]")
但是在卡迪尔坐标系中，平移变换却不能用矩阵来表示。一个平移变换T具有如下的形式
![\left[ {\begin{array}{c} x'\\y'\\ \end{array}} \right]=T(p)= I \left[ {\begin{array}{cc} x \\ y \\ \end{array} } \right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bc%7D+x%27%5C%5Cy%27%5C%5C+%5Cend%7Barray%7D%7D+%5Cright%5D%3DT%28p%29%3D+I+%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bcc%7D+x+%5C%5C+y+%5C%5C+%5Cend%7Barray%7D+%7D+%5Cright%5D&bg=ffffff&fg=555555&s=0 "\left[ {\begin{array}{c} x'\\y'\\ \end{array}} \right]=T(p)= I \left[ {\begin{array}{cc} x \\ y \\ \end{array} } \right]")
![\left[ {\begin{array}{c} t_x \\ t_y \\ \end{array} } \right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bc%7D+t_x+%5C%5C+t_y+%5C%5C+%5Cend%7Barray%7D+%7D+%5Cright%5D&bg=ffffff&fg=555555&s=0 "\left[ {\begin{array}{c} t_x \\ t_y \\ \end{array} } \right]")
我们可以很容易地验证，平移变换T是不能写成两个矩阵乘积形式的。使用齐次坐标系很好的解决了这个问题(可能还有其它的原因)。齐次坐标系统其实是用高维坐标来表示一个低维的点，就好比我们用（x,1)来表示一个长度值一样，其实用一个x就可以了，但是用高一维的表示，在有的时候会带来便利。一个N维的卡迪尔坐标系中的一个点
现在回来让我们看看使用齐次坐标时，对应的线性变换是什么形式。假设
![\left[ {\begin{array}{c} x'\\y'\\1\\ \end{array}} \right]=R(p)= \left[ {\begin{array}{ccc} cos(\theta) & -sin(\theta) & 0 \\ sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 1\\\end{array} } \right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bc%7D+x%27%5C%5Cy%27%5C%5C1%5C%5C+%5Cend%7Barray%7D%7D+%5Cright%5D%3DR%28p%29%3D+%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bccc%7D+cos%28%5Ctheta%29+%26+-sin%28%5Ctheta%29+%26+0+%5C%5C+sin%28%5Ctheta%29+%26+cos%28%5Ctheta%29+%26+0+%5C%5C+0+%26+0+%26+1%5C%5C%5Cend%7Barray%7D+%7D+%5Cright%5D&bg=ffffff&fg=555555&s=0 "\left[ {\begin{array}{c} x'\\y'\\1\\ \end{array}} \right]=R(p)= \left[ {\begin{array}{ccc} cos(\theta) & -sin(\theta) & 0 \\ sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 1\\\end{array} } \right]")
![\left[ {\begin{array}{c} x \\ y \\ 1 \\ \end{array} } \right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bc%7D+x+%5C%5C+y+%5C%5C+1+%5C%5C+%5Cend%7Barray%7D+%7D+%5Cright%5D&bg=ffffff&fg=555555&s=0 "\left[ {\begin{array}{c} x \\ y \\ 1 \\ \end{array} } \right]")
![\left[ {\begin{array}{c} x'\\y'\\1\\ \end{array}} \right]=S(p)= \left[ {\begin{array}{ccc} S_x & 0 & 0 \\ 0 & S_y & 0 \\ 0 & 0 & 1\\ \end{array} } \right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bc%7D+x%27%5C%5Cy%27%5C%5C1%5C%5C+%5Cend%7Barray%7D%7D+%5Cright%5D%3DS%28p%29%3D+%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bccc%7D+S_x+%26+0+%26+0+%5C%5C+0+%26+S_y+%26+0+%5C%5C+0+%26+0+%26+1%5C%5C+%5Cend%7Barray%7D+%7D+%5Cright%5D&bg=ffffff&fg=555555&s=0 "\left[ {\begin{array}{c} x'\\y'\\1\\ \end{array}} \right]=S(p)= \left[ {\begin{array}{ccc} S_x & 0 & 0 \\ 0 & S_y & 0 \\ 0 & 0 & 1\\ \end{array} } \right]")
容易验证, 
![\left[ {\begin{array}{c} x'\\y'\\1\\ \end{array}} \right]=T(p)= \left[ {\begin{array}{ccc} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1\\ \end{array} } \right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bc%7D+x%27%5C%5Cy%27%5C%5C1%5C%5C+%5Cend%7Barray%7D%7D+%5Cright%5D%3DT%28p%29%3D+%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bccc%7D+1+%26+0+%26+t_x+%5C%5C+0+%26+1+%26+t_y+%5C%5C+0+%26+0+%26+1%5C%5C+%5Cend%7Barray%7D+%7D+%5Cright%5D&bg=ffffff&fg=555555&s=0 "\left[ {\begin{array}{c} x'\\y'\\1\\ \end{array}} \right]=T(p)= \left[ {\begin{array}{ccc} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1\\ \end{array} } \right]")
最后，我们给出仿射变换稍微正式点的定义。一个仿射变换T，可以表示成一个线性变换A后平移t：T(p)=Ap+t，其中p是待变换的点齐次坐标表示。T可以表示成如下的形式：
![\bf{T}=\left[ {\begin{array}{cccc} a_11&a_12&a_13&t_1\\ a_21&a_22&a_23&t_2\\ a_31&a32&a33&t_3\\ 0&0&0&1\\ \end{array}} \right]](https://s0.wp.com/latex.php?latex=%5Cbf%7BT%7D%3D%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bcccc%7D+a_11%26a_12%26a_13%26t_1%5C%5C+a_21%26a_22%26a_23%26t_2%5C%5C+a_31%26a32%26a33%26t_3%5C%5C+0%260%260%261%5C%5C+%5Cend%7Barray%7D%7D+%5Cright%5D&bg=ffffff&fg=555555&s=0 "\bf{T}=\left[ {\begin{array}{cccc} a_11&a_12&a_13&t_1\\ a_21&a_22&a_23&t_2\\ a_31&a32&a33&t_3\\ 0&0&0&1\\ \end{array}} \right]")
其中，
![\bf{A}=\left[ {\begin{array}{ccc} a_11&a_12&a_13\\ a_21&a_22&a_23\\ a_31&a32&a33\\ \end{array}} \right]](https://s0.wp.com/latex.php?latex=%5Cbf%7BA%7D%3D%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bccc%7D+a_11%26a_12%26a_13%5C%5C+a_21%26a_22%26a_23%5C%5C+a_31%26a32%26a33%5C%5C+%5Cend%7Barray%7D%7D+%5Cright%5D&bg=ffffff&fg=555555&s=0 "\bf{A}=\left[ {\begin{array}{ccc} a_11&a_12&a_13\\ a_21&a_22&a_23\\ a_31&a32&a33\\ \end{array}} \right]")
![\bf{t}=\left[ {\begin{array}{c} t_1\\ t_2\\ t_3\\ \end{array}} \right]](https://s0.wp.com/latex.php?latex=%5Cbf%7Bt%7D%3D%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7Bc%7D+t_1%5C%5C+t_2%5C%5C+t_3%5C%5C+%5Cend%7Barray%7D%7D+%5Cright%5D&bg=ffffff&fg=555555&s=0 "\bf{t}=\left[ {\begin{array}{c} t_1\\ t_2\\ t_3\\ \end{array}} \right]")
仿射变换之所以重要，另一个重要的原因是仿射变换后不改变点的共线/共面性，而且还保持比例，这对图形系统尤其重要。例如，根据这个性质，如果我们要变换一个三角形，只需要对三个定点v1,v2,v3进行变换T就可以了，对于原先边v1v2上的点，变换后一定还在边后T(v1)T(v2)上。
总结一下，仿射变换是线性变换后进行平移变换（其实也是齐次空间的线性变换），使用齐次坐标使得仿射变换可以以统一的矩阵形式进行表示。