相机标定--A Flexible New Technique for Camera Calibration

A Flexible New Technique for Camera Calibration
Technical Report
MSR-TR-98-71

大家可以参考一下这个文章： http://www.cnblogs.com/wangguchangqing/p/8335131.html

张正友的标定算法：提出简单实用的相机标定算法，使得普通研究者可以自己打印棋盘格来快速准确的完成相机标定任务。

2 Basic Equations
一个 2D point m = [u,v]T， 一个 3D point is denoted by M = [X,Y,Z]T, 它们对应的增广坐标形式分别为 m~ = [u,v,1]T M~ = [X,Y,Z,1]T
我们使用普通的 pinhole 对相机进行建模，a 3D point M and its image projection m的关系如下所示

其中 s is an arbitrary scale factor， (R,t), 称之为相机外参数,分别表示旋转矩阵和平移向量，表示世界坐标系和相机坐标系直接的联系， is the rotation and translation which relates the world coordinate system to the camera coordinate system

A, called the camera intrinsic matrix 相机内参数矩阵

(u 0 ,v 0 ) the coordinates of the principal point, α and β the scale factors in image u and v axes, and γ the parameter describing the skewness of the two image axes

2.2 Homography between the model plane and its image
we assume the model plane is on Z = 0 of the world coordinate system，我们假定标定板平面位于世界坐标系的 Z=0

张氏标定法中，将世界坐标系构建在棋盘格平面上，令棋盘格平面为Z=0的平面。则可得

因为 Z=0，所以 M = [X,Y ]T ，, M~ = [X,Y,1] T，a model point M and its image m is related by a homography H:

homography 平面上的点的对应关系， Z=0 作为标定板平面

2.3 Constraints on the intrinsic parameters 相机内参的约束
给定一个平面标定板的图像，其对应的 homography 可以被估计出来。我们使用 H =[h1 h2 h3] 来表示，则对于每张图像有如下关系

r1 和 r2 是标准正交向量， 所以 r1T*r2=0 以及 ||r1||=||r2||=1
一个 homography 有8个自由度，相机外参数有 6个变量（3 for rotation and 3 for translation），we can only obtain 2 constraints on the intrinsic parameters

2.4 Geometric Interpretation 几何解释
这里主要介绍的上面公式（3）和（4）的几何推导过程，与 absolute conic 联系起来

3 Solving Camera Calibration 相机标定的求解
3.1 Closed-form solution
假定 A 为相机内参数矩阵

其中

所以 the two fundamental constraints (3) and (4), from a given homography, can be rewritten as 2 homogeneous equations in b:

Once b is estimated, we can compute all camera intrinsic matrix A. See Appendix B for the details.

Once A is known, the extrinsic parameters for each image is readily computed

3.2 Maximum likelihood estimation

nonlinear minimization problem ： Levenberg-Marquardt Algorithm

3.3 Dealing with radial distortion
相机的径向畸变

Complete Maximum Likelihood Estimation

3.4 Summary
The recommended calibration procedure is as follows: