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

{

for (int i = 0; i

{

/* 假设定标板放在世界坐标系中z=0的平面上 */

Point3f tempPoint;

tempPoint.x = i*square_size.height;

tempPoint.y = j*square_size.width;

tempPoint.z = 0;

tempPointSet.push_back(tempPoint);

}

}

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。