从基本矩阵中提取翻译和轮换

Question

我试图从计算的基本矩阵中检索平移和旋转向量.我确实使用OpenCV,一般方法来自维基百科.我的代码是这样的:

//Compute Essential Matrix
Mat A = cameraMatrix(); //Computed using chessboard
Mat F = fundamentalMatrix(); //Computed using matching keypoints
Mat E = A.t() * F * A;

//Perfrom SVD on E
SVD decomp = SVD(E);

//U
Mat U = decomp.u;

//S
Mat S(3, 3, CV_64F, Scalar(0));
S.at<double>(0, 0) = decomp.w.at<double>(0, 0);
S.at<double>(1, 1) = decomp.w.at<double>(0, 1);
S.at<double>(2, 2) = decomp.w.at<double>(0, 2);

//V
Mat V = decomp.vt; //Needs to be decomp.vt.t(); (transpose once more)

//W
Mat W(3, 3, CV_64F, Scalar(0));
W.at<double>(0, 1) = -1;
W.at<double>(1, 0) = 1;
W.at<double>(2, 2) = 1;

cout << "computed rotation: " << endl;
cout << U * W.t() * V.t() << endl;
cout << "real rotation:" << endl;
Mat rot;
Rodrigues(images[1].rvec - images[0].rvec, rot); //Difference between known rotations
cout << rot << endl;

最后,我尝试将估计的旋转与我使用每个图像中的棋盘计算的旋转进行比较(我计划在没有棋盘的情况下获得外部参数).例如我得到这个:

computed rotation:
[0.8543027125286542, -0.382437675069228, 0.352006107978011;
  0.3969758209413922, 0.9172325022900715, 0.03308676972148356;
  0.3355250705298953, -0.1114717965690797, -0.9354127247453767]

real rotation:
[0.9998572365450219, 0.01122579241510944, 0.01262886032882241;
  -0.0114034800333517, 0.9998357441946927, 0.01408706050863871;
  -0.01246864754818991, -0.01422906234781374, 0.9998210172891051]

很明显,似乎有一个问题,我只是无法弄清楚它可能是什么.

编辑:这是我用未转置的vt得到的结果(显然来自另一个场景):

computed rotation: 
[0.8720599858028177, -0.1867080200550876, 0.4523842353671251;
 0.141182538980452, 0.9810442195058469, 0.1327393312518831;
-0.4685924368239661, -0.05188790438313154, 0.8818893204535954]
real rotation
[0.8670861432556456, -0.427294988334106, 0.2560871201732064;
 0.4024551137989086, 0.9038194629873437, 0.1453969040329854;
-0.2935838918455123, -0.02300806966752995, 0.9556563855167906]

这是我的计算相机矩阵,误差非常低(大约0.17 ......).

[1699.001342509651, 0, 834.2587265398068;
  0, 1696.645251354618, 607.1292618175946;
  0, 0, 1]

以下是我尝试重新投影立方体时得到的结果...相机0,立方体是轴对齐的,旋转和平移是(0,0,0). 图片http://imageshack.us/a/img802/5292/bildschirmfoto20130110u.png

和另一个,与第一个图像中的点的epilines. 图片http://imageshack.us/a/img546/189/bildschirmfoto20130110uy.png

Answer 1

请看一下这个链接:

http://isit.u-clermont1.fr/~ab/Classes/DIKU-3DCV2/Handouts/Lecture16.pdf.

参见第2页.R有两种可能性.第一种是U W VT,第二种是U WT VT.你使用了第二个.试试第一个.