CODEWITHSUNDEEP
c++opencvmatrixrotationtranslation
Teris
Teris

Reputation: 610

Extract Translation and Rotation from Fundamental Matrix

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;

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]

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

EDIT: 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]

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]

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). image http://imageshack.us/a/img802/5292/bildschirmfoto20130110u.png

and the other one, with the epilines of the points in the first image. image http://imageshack.us/a/img546/189/bildschirmfoto20130110uy.png

Upvotes: 18

Views: 28739

Answers (2)

user1993497
user1993497

Reputation: 116

Please take a look at this link:

DIKU-3DCV2/Handouts/Lecture16.pdf.

alternative link

Refer to Page 2. There are two possibilities for R. The first is UWVT and the second is UWTVT. You used the second. Try the first.

Upvotes: 10

MichalSzczep
MichalSzczep

Reputation: 365

The 8-point algorithm is the simplest method of computing fundamental matrix, but if care is taken you can perform it well. The key to obtain the good results is proper careful normalization of the input data before constructing the equations to solve. Many of algorithms can do it. Pixels point coordinate must be changed to camera coordinates, you do it in this line:

Mat E = A.t() * F * A;

However this assumption is not accurate. If camera calibration matrix K is known, then you may apply inverse to the point x to obtain the point expressed in normalized coordinates.

X_{norm}= K.inv()*X_{pix} where X_{pix}(2), z is equal 1.

In the case of the 8PA, a simple transformation of points improve and hence in the stability of the results. The suggested normalization is a translation and scaling of each image so that the centroid of the reference points is at origin of the coordinates and the RMS distance of the points from the origin is equal to \sqrt{2}. Note that it is recommended that the singularity condition should be enforced before denormalization.

Reference: check it if : you are still interested

Upvotes: 0

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