从基本矩阵中提取翻译和旋转 [英] Extract Translation and Rotation from Fundamental Matrix

问题描述

我正在尝试从计算的基本矩阵中检索翻译和旋转向量。我使用OpenCV和一般的方法是从维基百科。我的代码是这样的：

I am trying to retrieve translation and rotation vectors from a computed fundamental Matrix. I do use OpenCV and the general approach is from wikipedia. My Code is like this:

//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棋盘计算的旋转每个图像（我计划获得外部参数没有棋盘）。例如我得到这个：

At the end I try to compare the estimated rotation to the one I computed using the chessboard which is in every Image (I plan to get the extrinsic parameters without the chessboard). For example I get this:

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]

很清楚似乎有一个问题，

So clearly there seems to be a problem, I just can't figure out what it could be.

编辑：
下面是使用未转换的vt（显然是从另一个场景）的结果：

Here are the results I got with the untransposed vt(obviously from another scene):

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]

这是我计算的相机矩阵，错误（约0.17 ...）。

Here is my computed camera matrix, the error was pretty low(about 0.17...).

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

这里是我尝试重新投影cube ...
相机0，立方体是轴对齐的，旋转和平移是（0，0，0）。

Here are the results I get when trying to reproject a cube... Camera 0, the cube is axis-aligned, rotation and translation are (0, 0, 0).

和另一个，与第一个图像中的点的epilines。

and the other one, with the epilines of the points in the first image.

推荐答案

请查看此链接：

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

参考第2页.R有两种可能性。第一种是U * W * VT，第二种是U * WT * VT。你使用第二个。尝试第一个。

Refer to Page 2. There are two possibilities for R. The first is U*W*VT and the second is U*WT*VT. You used the second. Try the first.

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