Compute homography given rotation and translation between two cameras

Question

I know that one can compute the homography matrix by using at least four correpondence points.

I was wondering if and how can I obtain a homography matrix if I already know the rotation and translation between two cameras, including the camera intrinsics?

I found something that looked like this

H= KRK^-1

but this assumes a pure rotation. What would be the case for a pure translation? And what if I want to warp an image with the homography matrix that is not from points purely on a plane?

I'm somewhat confused right now and would really really appreciate any explanations! Thank you in advance!

Answer 1

If there is a nonzero translation, and the image contains more than a plane (or you are not looking at very far away things) then the images are not related by a homography. You can convince yourself that this is the case by noticing that some points visible in one image may be occluded in the other one.

If the image shows a plane among other things, and you estimate a homography using only point correspondences on that plane, than the homography will correctly transform all points on that plane, but will map incorrectly all other points.

Answer 2

Given the rotation R and translation t and a plane with surface normal n and a (signed) distance d from the origin in the scene with calibration cameras K and K' for each camera the homography induced by the plane is

This is the subject of chapter 13 (Example 13.2) of the canon on the subject by Hartley and Zisserman.