Compute fundamental matrix from camera calibration

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Hello,

I try to compute the fundamental matrix given the following camera calibration parameters:

• camera Matrix 1/2 (camMat1, camMat2)
• rotation Vector 1/2 (rotVec1, rotVec2)
• translation Vector 1/2 (transVec1, transVec2)

According to the following formula the fundamental matrix F is computed by:

F = inverse(transpose(camMat1)) * R * S * inverse(camMat2)

Anyway, i am quite a bit lost how to compute R and S. I know that R is the rotation matrix which brings image 1 into image 2. Also i know that S is the translation vector to transform image 1 into image 2. My plan would be:

1) Rodrigues both rotation Vectors and substract rotation Matrix 1 from rotation Matrix 2

cv::Mat rotMat1, rotMat2;
cv::Rodrigues(rotVec1[0], rotMat1);
cv::Rodrigues(rotVec2[0], rotMat2);
cv::Mat R = rotMat2 - rotMat1;

2) Substract translation Vector 1 from translation Vector 2

T = transVec2 - transVec1

3) Compose S

S = [0,-T[3], T[2]; T[3], 0, -T[1]; -T[2], T[1], 0];

Would this be correct? Any help on this topic would be appreciated. I hope this is not offtopic since it is not directly related to OpenCV.

Edit: I worked in the solution you provided. I cross checked it with cv::stereoCalib(). Unfortunatly the matrizes does not match. Any suggestions what i did wrong?

cv::Mat computeFundMat(cv::Mat camMat1, cv::Mat camMat2, vector<cv::Mat> rotVec1, 
    vector<cv::Mat> rotVec2, vector<cv::Mat> transVec1, vector<cv::Mat> transVec2)
{
    cv::Mat rotMat1(3, 3, CV_64F), rotMat2(3, 3, CV_64F);
    cv::Mat transVec2toWorld(3, 1, CV_64F);
    cv::Mat R(3, 3, CV_64F), T(3, 1, CV_64F), S(cv::Mat::zeros(3, 3, CV_64F));
    cv::Mat F(3, 3, CV_64F);
    //Convert rotation vector into rotation matrix 
    cv::Rodrigues(rotVec1.at(0), rotMat1);
    cv::Rodrigues(rotVec2.at(0), rotMat2);
    //Transform prameters of camera 2 into world
    rotMat2 = rotMat2.t();
    transVec2toWorld = (-rotMat2 * transVec2.at(0));
    //Compute R, T to rotate, translate image b into image a
    R = rotMat2 * rotMat1; //or --> rotMat2 * rotMat1 ???
    T = transVec1.at(0) + transVec2toWorld;
    //Compose skew matrix (Format: S = 
    // [0,-T[3], T[2];   -> 1
    S.at<double>(0, 1) = -T.at<double>(2); S.at<double>(0, 2) = T.at<double>(1); //-> 1
    //  T[3], 0, -T[1];  -> 2
    S.at<double>(1, 0) = T.at<double>(2); S.at<double>(1, 2) = -T.at<double>(0); //-> 2
    // -T[2], T[1], 0];) -> 3
    S.at<double>(2, 0) = -T.at<double>(1); S.at<double>(2, 1) = T.at<double>(0); //-> 3
    //Compute fundamental matrix (F = inverse(transpose(camMat2)) * R * S * inverse(camMat1))
    return cv::Mat(camMat2.t().inv() * R * S * camMat1.inv());
}