原理
下图表示了小孔成像模型（图片及公式参考opencv官方资料）
这个图里涉及4个坐标系：
- 世界坐标系：其坐标原点可视情况而定，可以表示空间的物体，单位为长度单位，比如mm，用矩阵表示；
- 相机坐标系：以摄像机光心为原点（在针孔模型中也就是针孔为中心），z轴与光轴重合，也就是z轴指向相机的前方（与成像平面垂直），x轴与y轴的正方向与世界坐标系平行，单位为长度单位，比如mm，用矩阵表示；
- 图像物理坐标系（也叫成像平面坐标系）：用物理长度单位表示像素的位置，坐标原点为摄像机光轴与图像物理坐标系的交点位置。坐标系为图上o-xy，单位为长度单位，比如mm，用矩阵表示。
- 像素坐标系：坐标原点在左上角，以像素为单位，有明显的范围限制，即用于表示全画面的像素长和像素长宽，矩阵表示。
以下公式描述了、、和之间的转换关系。
以上公式中，和表示1个像素有多少长度，即用传感器的尺寸除以像素数量，比如2928.384umx2205.216um的传感的分辨率为2592x1944，每个像素的大小即约1.12um。
表示焦距，在上图中根据相似三角形，P点和p点具有以下关系：
即 ，可见：越大，和越大，越大，和越小。
和表示中心点在像素坐标系中的位置。
要求像素坐标系中某像素点对应在世界坐标系中的位置，需要知道相机的内参、外参，相机的内参可以通过标定获得，外参可以人为设定。
第一步，将像素坐标变换到相机坐标系：
两边乘以K的逆后推导出：
第二步，从相机坐标系变换到世界坐标系：
将方程乘以，可以推导出：
代码
通过输入相机的内参，旋转向量，平移向量和像素坐标，可以通过以下函数求出对应的世界坐标点。
以下代码中需求注意要对平移向量取转置，将1x3矩阵变为3x1矩阵后，才能实现3x3矩阵和3x1矩阵的乘法运算。
void cameraToWorld(InputArray cameraMatrix, InputArray rV, InputArray tV, vector imgPoints, vector &worldPoints)
{
Mat invK64, invK;
invK64 = cameraMatrix.getMat().inv();
invK64.convertTo(invK, CV_32F);
Mat r, t, rMat;
rV.getMat().convertTo(r, CV_32F);
tV.getMat().convertTo(t, CV_32F);
Rodrigues(r, rMat);
//计算 invR * T
Mat invR = rMat.inv();
//cout << "invR\n" << invR << endl;
//cout << "t\n" << t << t.t() << endl;
Mat transPlaneToCam;
if(t.size() == Size(1, 3)){
transPlaneToCam = invR * t;//t.t();
}
else if(t.size() == Size(3, 1)){
transPlaneToCam = invR * t.t();
}
else{
return;
}
//cout << "transPlaneToCam\n" << transPlaneToCam << endl;
int npoints = (int)imgPoints.size();
//cout << "npoints\n" << npoints << endl;
for (int j = 0; j < npoints; ++j){
Mat coords(3, 1, CV_32F);
Point3f pt;
coords.at(0, 0) = imgPoints[j].x;
coords.at(1, 0) = imgPoints[j].y;
coords.at(2, 0) = 1.0f;
//[x,y,z] = invK * [u,v,1]
Mat worldPtCam = invK * coords;
//cout << "worldPtCam:" << worldPtCam << endl;
//[x,y,1] * invR
Mat worldPtPlane = invR * worldPtCam;
//cout << "worldPtPlane:" << worldPtPlane << endl;
//zc
float scale = transPlaneToCam.at(2) / worldPtPlane.at(2);
//cout << "scale:" << scale << endl;
Mat scale_worldPtPlane(3, 1, CV_32F);
//scale_worldPtPlane.at(0, 0) = worldPtPlane.at(0, 0) * scale;
//zc * [x,y,1] * invR
scale_worldPtPlane = scale * worldPtPlane;
//cout << "scale_worldPtPlane:" << scale_worldPtPlane << endl;
//[X,Y,Z]=zc*[x,y,1]*invR - invR*T
Mat worldPtPlaneReproject = scale_worldPtPlane - transPlaneToCam;
//cout << "worldPtPlaneReproject:" << worldPtPlaneReproject << endl;
pt.x = worldPtPlaneReproject.at(0);
pt.y = worldPtPlaneReproject.at(1);
//pt.z = worldPtPlaneReproject.at(2);
pt.z = 1.0f;
worldPoints.push_back(pt);
}
}
def cameraToWorld(self, cameraMatrix, r, t, imgPoints):
invK = np.asmatrix(cameraMatrix).I
rMat = np.zeros((3, 3), dtype=np.float64)
cv2.Rodrigues(r, rMat)
#print('rMat=', rMat)
#计算 invR * T
invR = np.asmatrix(rMat).I #3*3
#print('invR=', invR)
transPlaneToCam = np.dot(invR , np.asmatrix(t)) #3*3 dot 3*1 = 3*1
#print('transPlaneToCam=', transPlaneToCam)
worldpt = []
coords = np.zeros((3, 1), dtype=np.float64)
for imgpt in imgPoints:
coords[0][0] = imgpt[0][0]
coords[1][0] = imgpt[0][1]
coords[2][0] = 1.0
worldPtCam = np.dot(invK , coords) #3*3 dot 3*1 = 3*1
#print('worldPtCam=', worldPtCam)
#[x,y,1] * invR
worldPtPlane = np.dot(invR , worldPtCam) #3*3 dot 3*1 = 3*1
#print('worldPtPlane=', worldPtPlane)
#zc
scale = transPlaneToCam[2][0] / worldPtPlane[2][0]
#print("scale: ", scale)
#zc * [x,y,1] * invR
scale_worldPtPlane = np.multiply(scale , worldPtPlane)
#print("scale_worldPtPlane: ", scale_worldPtPlane)
#[X,Y,Z]=zc*[x,y,1]*invR - invR*T
worldPtPlaneReproject = np.asmatrix(scale_worldPtPlane) - np.asmatrix(transPlaneToCam) #3*1 dot 1*3 = 3*3
#print("worldPtPlaneReproject: ", worldPtPlaneReproject)
pt = np.zeros((3, 1), dtype=np.float64)
pt[0][0] = worldPtPlaneReproject[0][0]
pt[1][0] = worldPtPlaneReproject[1][0]
pt[2][0] = 0
worldpt.append(pt.T.tolist())
#print('worldpt:',worldpt)
return worldpt
验证
先使用projectPoints生成像素点：
Vec3f eulerAngles;//欧拉角
vector translation_vectors;/* 每幅图像的平移向量 */
Mat rotationMatrix = eulerAnglesToRotationMatrix(eulerAngles);
*pR_matrix = rotationMatrix;
cvRodrigues2(pR_matrix, pnew_vec, 0); //从旋转矩阵求旋转向量
Mat mat_tmp(pnew_vec->rows, pnew_vec->cols, pnew_vec->type, pnew_vec->data.fl);
cv::Mat distortion_coeffs1 = cv::Mat(1, 5, CV_32FC1, cv::Scalar::all(0)); /* 摄像机的5个畸变系数：k1,k2,p1,p2,k3 */
projectPoints(tempPointSet, mat_tmp, translation_vectors[i], intrinsic_matrix, distortion_coeffs1, image_points2);
使用以下欧拉角：
[0, 0, 0]//欧拉角度，表示平面和相机的角度
旋转向量：[0, 0, 0]
对应的平移向量，表示空间坐标原点相对在相平面原点偏移x=134mm，y=132mm，z=200mm。
原始:[134.0870803179094, 132.7580766544178, 200.3789038923399]
生成空间坐标点：
Size board_size = Size(11,8);
Size square_size = Size(30, 30);
vector tempPointSet;
for (int j = 0; j 经projectPoints计算后对应的像素空间点是：
projectPoints(tempPointSet, mat_tmp, translation_vectors[i], intrinsic_matrix, distortion_coeffs1, image_points2);
cout << "原始空间点:\n" << image_points2 << endl;
[1194.8174, 1074.1355;
1285.1735, 1074.1355;
1375.5295, 1074.1355;
1465.8856, 1074.1355;
1556.2417, 1074.1355;
1646.5978, 1074.1355;
1736.9539, 1074.1355;
1827.3099, 1074.1355;
1917.666, 1074.1355;
2008.0221, 1074.1355;
2098.3782, 1074.1355;
1194.8174, 1164.5713;
1285.1735, 1164.5713;
1375.5295, 1164.5713;
1465.8856, 1164.5713;
1556.2417, 1164.5713;
1646.5978, 1164.5713;
1736.9539, 1164.5713;
1827.3099, 1164.5713;
1917.666, 1164.5713;
2008.0221, 1164.5713;
2098.3782, 1164.5713;
1194.8174, 1255.0072;
1285.1735, 1255.0072;
1375.5295, 1255.0072;
1465.8856, 1255.0072;
1556.2417, 1255.0072;
1646.5978, 1255.0072;
1736.9539, 1255.0072;
1827.3099, 1255.0072;
1917.666, 1255.0072;
2008.0221, 1255.0072;
2098.3782, 1255.0072;
1194.8174, 1345.443;
1285.1735, 1345.443;
1375.5295, 1345.443;
1465.8856, 1345.443;
1556.2417, 1345.443;
1646.5978, 1345.443;
1736.9539, 1345.443;
1827.3099, 1345.443;
1917.666, 1345.443;
2008.0221, 1345.443;
2098.3782, 1345.443;
1194.8174, 1435.8789;
1285.1735, 1435.8789;
1375.5295, 1435.8789;
1465.8856, 1435.8789;
1556.2417, 1435.8789;
1646.5978, 1435.8789;
1736.9539, 1435.8789;
1827.3099, 1435.8789;
1917.666, 1435.8789;
2008.0221, 1435.8789;
2098.3782, 1435.8789;
1194.8174, 1526.3147;
1285.1735, 1526.3147;
1375.5295, 1526.3147;
1465.8856, 1526.3147;
1556.2417, 1526.3147;
1646.5978, 1526.3147;
1736.9539, 1526.3147;
1827.3099, 1526.3147;
1917.666, 1526.3147;
2008.0221, 1526.3147;
2098.3782, 1526.3147;
1194.8174, 1616.7506;
1285.1735, 1616.7506;
1375.5295, 1616.7506;
1465.8856, 1616.7506;
1556.2417, 1616.7506;
1646.5978, 1616.7506;
1736.9539, 1616.7506;
1827.3099, 1616.7506;
1917.666, 1616.7506;
2008.0221, 1616.7506;
2098.3782, 1616.7506;
1194.8174, 1707.1864;
1285.1735, 1707.1864;
1375.5295, 1707.1864;
1465.8856, 1707.1864;
1556.2417, 1707.1864;
1646.5978, 1707.1864;
1736.9539, 1707.1864;
1827.3099, 1707.1864;
1917.666, 1707.1864;
2008.0221, 1707.1864;
2098.3782, 1707.1864]
经函数求出的空间坐标点是：
vector worldPoint;
cameraToWorld(intrinsic_matrix, mat_tmp, translation_vec_tmp, image_points2, worldPoint);
cout << "计算空间点:\n" << worldPoint << endl;
[0, 0, 1;
30, 0, 1;
60, 0, 1;
90.000015, 0, 1;
120.00002, 0, 1;
149.99995, 0, 1;
179.99998, 0, 1;
209.99998, 0, 1;
239.99998, 0, 1;
270, 0, 1;
300, 0, 1;
0, 29.999985, 1;
30, 29.999985, 1;
60, 29.999985, 1;
90.000015, 29.999985, 1;
120.00002, 29.999985, 1;
149.99995, 29.999985, 1;
179.99998, 29.999985, 1;
209.99998, 29.999985, 1;
239.99998, 29.999985, 1;
270, 29.999985, 1;
300, 29.999985, 1;
0, 60.000015, 1;
30, 60.000015, 1;
60, 60.000015, 1;
90.000015, 60.000015, 1;
120.00002, 60.000015, 1;
149.99995, 60.000015, 1;
179.99998, 60.000015, 1;
209.99998, 60.000015, 1;
239.99998, 60.000015, 1;
270, 60.000015, 1;
300, 60.000015, 1;
0, 89.999969, 1;
30, 89.999969, 1;
60, 89.999969, 1;
90.000015, 89.999969, 1;
120.00002, 89.999969, 1;
149.99995, 89.999969, 1;
179.99998, 89.999969, 1;
209.99998, 89.999969, 1;
239.99998, 89.999969, 1;
270, 89.999969, 1;
300, 89.999969, 1;
0, 120.00002, 1;
30, 120.00002, 1;
60, 120.00002, 1;
90.000015, 120.00002, 1;
120.00002, 120.00002, 1;
149.99995, 120.00002, 1;
179.99998, 120.00002, 1;
209.99998, 120.00002, 1;
239.99998, 120.00002, 1;
270, 120.00002, 1;
300, 120.00002, 1;
0, 149.99998, 1;
30, 149.99998, 1;
60, 149.99998, 1;
90.000015, 149.99998, 1;
120.00002, 149.99998, 1;
149.99995, 149.99998, 1;
179.99998, 149.99998, 1;
209.99998, 149.99998, 1;
239.99998, 149.99998, 1;
270, 149.99998, 1;
300, 149.99998, 1;
0, 179.99998, 1;
30, 179.99998, 1;
60, 179.99998, 1;
90.000015, 179.99998, 1;
120.00002, 179.99998, 1;
149.99995, 179.99998, 1;
179.99998, 179.99998, 1;
209.99998, 179.99998, 1;
239.99998, 179.99998, 1;
270, 179.99998, 1;
300, 179.99998, 1;
0, 209.99998, 1;
30, 209.99998, 1;
60, 209.99998, 1;
90.000015, 209.99998, 1;
120.00002, 209.99998, 1;
149.99995, 209.99998, 1;
179.99998, 209.99998, 1;
209.99998, 209.99998, 1;
239.99998, 209.99998, 1;
270, 209.99998, 1;
300, 209.99998, 1]
可以对比按11*8格和30mm/格所生成空间坐标点结果，基本一致。