Fundamental matrix from transformation

Question

I have two images with known corresponding 2D points, the intrinsic parameters of the cameras and the 3D transformation between the cameras. I want to calculate the 2D reprojection error from one image to the other.

To do so, I thought about getting a fundamental matrix from the transformation, so I can compute the point-to-line distance between the points and the corresponding epipolar lines. How can I get the fundamental matrix?

I know that E = R * [t] and F = K^(-t) * E * K^(-1), where E is the essential matrix and [t] is the skew-symmetric matrix of the translation vector. However, this returns a null matrix if the motion is pure rotation (t = [0 0 0]). I know that in this case a homography explains the motion better than the fundamental matrix, so that I can compare the norm of the translation vector with a small threshold to choose a fundamental matrix or a homogaphy. Is there a better way of doing this?

Answer 1

"I want to calculate the 2D reprojection error from one image to the other."

Then go and calculate it. Your setup is calibrated, so you don't need anything other than a known piece of 3D geometry. Forget about the epipolar error, which may as well be undefined if your camera motion is (close to) a pure rotation.

Take an object of known size and shape (for example, a checkerboard), work out its location in 3D space from one camera view (for a checkerboard you can fit a homography between its physical model and its projection, then decompose it into [R|t]). Then project the now-located 3D shape into the other camera given that camera's calibrated parameters, and compare the projection with the object's actual image.