OpenCV鱼眼相机标定模型

本文对OpenCV鱼眼相机标定模型进行了整理

前情提要：关于相机成像畸变模型，可戳相机成像畸变模型与鱼眼相机模型

鱼眼相机模型

$X$是世界坐标系中一点$P$的坐标，$R$是旋转矩阵，$T$是平移向量，点$P$在相机坐标系下的坐标可表示为

$$X_c=RX+T$$

$x，y，z$是$X_c$的三个坐标，可表示为

$$x=Xc_1$$ $$y=Xc_2$$ $$z=Xc_3$$

P点小孔投影的坐标为$[a;b]$，其中

$$a = \frac{x}{z}$$ $$b = \frac{y}{z}$$ $$r^2=a^2+b^2$$ $$\theta=atan(r)$$

鱼眼畸变模型

$$\theta_d=\theta(1+k_1\theta^2+k_2\theta^4+k_3\theta^6+k_4\theta^8)$$

畸变点的坐标$[x’;y’]$

$$x'=(\frac{\theta_d}{r})a$$ $$y'=(\frac{\theta_d}{r})b$$

最终的像素坐标系坐标$[u;v]$

$$u=f_x(x'+\alpha y')+c_x$$ $$v=f_xy'+c_y$$

鱼眼图像标定、校正代码

1.cv::fisheye::calibrate()

首先使用标定板标定相机。该函数输入角点，返回鱼眼相机内参K、D和外参R、T。
其中，K是33的相机内参矩阵，D是畸变系数，R是旋转向量，T是平移向量

2.cv::getOptimalNewCameraMatrix

Returns the new camera intrinsic matrix based on the free scaling parameter.

根据比例因子返回新的相机内参矩阵

3.cv::fishheye::initUndistortRectifyMap

Computes undistortion and rectification maps for image transform by cv::remap(). If D is empty zero distortion is used, if R or P is empty identity matrixes are used.

该函数计算原始图像和矫正图像之间的转换关系，输入是鱼眼相机内参，输出是两者间的映射关系map1和map2。

4.cv::remap

The function remap transforms the source image using the specified map:

$$dst(x,y)=src(map_x(x,y),map_y(x,y))$$

where values of pixels with non-integer coordinates are computed using one of available interpolation methods. mapx and mapy can be encoded as separate floating-point maps in map1 and map2 respectively, or interleaved floating-point maps of (x,y) in map1, or fixed-point maps created by using convertMaps. The reason you might want to convert from floating to fixed-point representations of a map is that they can yield much faster (2x) remapping operations. In the converted case, map1 contains pairs (cvFloor(x), cvFloor(y)) and map2 contains indices in a table of interpolation coefficients.

使用map1和map2进行矫正