Homography vs fundamental matrix degeneracy
epipolar geometry degeneracies.
homography fits planar scenes or pure rotation; fitting a fundamental matrix there is degenerate because points lack depth variation.
assuming the fundamental matrix always works.
What's really being asked
The interviewer checks deeper multi-view geometry: not just which model to use, but why one model becomes numerically degenerate under certain configurations.
The full answer
Homography scenario: when all observed 3D points lie on a single plane, such as photographing a building facade, a poster, or the ground, or when the camera undergoes pure rotation about its center, the two views are related exactly by a homography. Degeneracy of the fundamental matrix: the fundamental matrix encodes epipolar geometry that depends on the cameras having a baseline and the scene having depth variation so points constrain the epipoles. If all points are coplanar, the homography already explains every correspondence, and the epipolar constraint no longer uniquely fixes the fundamental matrix; an entire family of fundamental matrices is consistent with the data, so the estimate is rank-deficient and unstable, swinging wildly with noise. Likewise, pure rotation gives zero baseline, so there is no parallax and no meaningful epipolar geometry. Practically, robust pipelines detect this by checking whether a homography explains the matches well and switching models accordingly, as in GRIC-based model selection.
The mistakes people make
Saying the fundamental matrix always works for any two images. Or claiming the issue is just too few points rather than a geometric degeneracy. Or treating homography and fundamental matrix as interchangeable.
What usually comes next
Why does coplanarity make the fundamental matrix underdetermined. How does model selection like GRIC decide between H and F. What happens to RANSAC in a degenerate case. How does adding off-plane points restore stability.
A concrete example
You photograph a flat painting from two positions. Every correspondence lies on the painting's plane, so a homography aligns them perfectly. Trying to estimate the fundamental matrix yields an unstable, noise-driven result because the coplanar points cannot pin down a unique epipolar geometry.
Interview question
Why is estimating the fundamental matrix unstable when all correspondences lie on one plane?
- a.Because coplanar points are too noisy to detect reliably
- b.Because a family of fundamental matrices fits coplanar points, leaving it underdeterminedCorrect
- c.Because the fundamental matrix requires grayscale images
- d.Because planes always cause zero camera translation
Why? this is the answer
Coplanar points are fully explained by a homography, so the epipolar constraint cannot uniquely fix the fundamental matrix, making it rank-deficient. The degeneracy is geometric, not about noise, color, or translation.
Just read this? Test yourself on what you have been reading.
- #fundamental-matrix
- #homography
- #epipolar-geometry
- #degeneracy
- #model-selection
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