Find Homography matrix from Fundamental matrix

Question

I'm trying to compute Homography matrix H given a set of correspondences and the Fundamental matrix F.

From the principle of Epipolar geometry I know this can be done by cross product of epiline and F from Epipole Geometry

[e_ij] x F_ij = H_ij

I'm using OpenCV for finding Fundamental matrix F from set of matches between two views using cv::findFundamentalMat().

My question is that how can I find e_ij and how to use it in order to compute H. In OpenCV there is a function cv::computeCorrespondEpilines() that finds epilines corespond to each given point.

It's worth mentioning that I'm not interested in computing H directly from set of matches but only from computed Fundamental matrix.

Thanks

Answer 1

First, notice that the equation (C29) you mentioned from your link uses a line l with same coordinates as e_ij, which is the epipole in image j. Therefore, l=e_ij is not an epiline, since dot( e_ij , e_ij ) == norm(e_ij)² == 1 != 0.

If I stick to the notations given in your link, you can compute the epipole e_ij as the left null-vector of F_ij. You can obtain it by calling cv::SVD::solveZ on the transpose of F_ij. Then, as is mentioned in your link, the homography H_ij (which maps points from image i to image j) can be computed as H_ij = [e_ij]_x F_ij, where the notation [e_ij]_x refers to the 3x3 skew-symmetric operator. A definition of this notation can be found in the Wikipedia article on cross-product.

However, be aware that such an homography H_ij defines a mapping from image i to image j via the plane backprojected from image j using the line with same coordinates as e_ij. In general, this will give a result which is very different from the result returned by cv::findHomography, where the resulting homography is a mapping from image i to image j via the dominant plane in the observed scene. Hence, you will be able to approximately register the two images using the homography returned by cv::findHomography, but in general this will not be the case for the homography obtained using the method above.