双目相机本质矩阵Essential和基础矩阵Fundament

作者: Kerwin_H | 来源:发表于2021-05-25 13:26 被阅读0次

To quote Thomas Jefferson: "He who receives an idea from me, receives instruction himself without lessening mine; as he who lights his taper at mine, receives light without darkening me. That ideas should freely spread from one to another over the globe, for the moral and mutual instruction of man, and improvement of his condition, seems to have been peculiarly and benevolently designed by nature, when she made them, like fire, expansible over all space, without lessening their density at any point, and like the air in which we breathe, move, and have our physical being, incapable of confinement or exclusive appropriation." Jefferson was one awesome dude.

准备知识

世界坐标系到图像坐标系

点在线上的向量表达方式：
$[u,v,1]*[a,b,c]=0$ --> (u,v)点在 $ax+by+c=0$ 直线上
其中[u v 1]为点（u,v）的齐次表达方式

在射影几何中，要通过两个不同的点求连接这两点的直线，两点作叉乘即可。
syms x1 y1 x2 y2
cross([x1 y1 1],[x2 y2 1]) = [ y1 - y2, x2 - x1, x1y2 - x2y1]
方程ax + by + c = 0:
$a = y1-y2; b = x2 - x1 ; c = x1*y2-x2*y1;$

基础矩阵Fundamental,F是任意两个相机相对关系的内参，换言之，给定了两个相机的相对位置关系和相机内参，F即被确定。
本质矩阵Essential,E是两个标准相机相对关系的内参，换言之，给定两个已标定相机的相对位置关系，E即被确定。

正文

极线约束

给定左图像中的一个点，我们不需要在整个右图像中寻找相应的点。
“极线约束”将搜索空间缩小为一维直线。

左右相机坐标点的对应关系

R为左右相机相对旋转矩阵
T为左右相机相对平移矩阵
Pr为空间点P在右相机的坐标
Pl为空间点P在左相机的坐标

image.png

T x Pl得到垂直于面OrPOl的向量，该向量乘以面内的向量结果为0；

叉乘转点乘
变换后得到本质矩阵
本质矩阵特点：
3*3矩阵，但秩为2 -- 左右各有一个零空间
只依赖相机的外参R和T