When a homography is a valid model
projective geometry assumptions.
homography holds for pure rotation or a planar scene; it fails with translation plus 3D parallax, where epipolar geometry applies.
thinking homography works for any two views.
What's really being asked
The question probes understanding of multi-view geometry, specifically the assumptions baked into using a single homography and what breaks them.
The full answer
Validity conditions: a homography, a 3x3 projective transform between image planes, correctly maps all correspondences in exactly two cases. First, pure rotation: the camera rotates about its optical center with zero translation, so depth cancels and any scene works. Second, planar scene: all observed points lie on a single plane, so even with camera translation the points map by one homography. Failure: when the camera translates and the scene has 3D depth, near and far points shift differently, producing parallax that no single homography can capture, causing ghosting in stitching. General model: in that case the two views are related by epipolar geometry, encoded by the fundamental matrix for uncalibrated cameras or the essential matrix for calibrated ones, which constrains corresponding points to epipolar lines rather than a one-to-one mapping.
The mistakes people make
Claiming a homography aligns any two photos, ignoring parallax. Or saying it only needs the same camera, missing the rotation-or-plane condition. Or confusing the fundamental matrix with a homography.
What usually comes next
Why does panorama capture instruct you to rotate, not walk. What artifacts does parallax create in stitching. What does the fundamental matrix encode. How would you detect that a homography model is inadequate.
A concrete example
Standing still and panning a phone to shoot a panorama satisfies the pure-rotation condition, so consecutive frames align with homographies cleanly and the seams blend without artifacts. This is exactly why panorama apps instruct you to pivot in place rather than walk. If you instead walk sideways while shooting a room with both foreground and background objects, the near and far objects shift by different amounts, so a single homography cannot align both and the stitch shows ghosting. To handle that translating case you would estimate the fundamental or essential matrix and reconstruct depth, since the relationship is now full epipolar geometry rather than a planar mapping.
Interview question
Under which condition is a single homography a valid model between two views of a general 3D scene?
- a.The camera translates sideways with large depth variation
- b.The scene contains many objects at varied depths
- c.The camera rotates about its optical center with no translationCorrect
- d.The two cameras have different focal lengths only
Why? this is the answer
Pure rotation about the optical center makes depth cancel, so one homography maps all points; a planar scene also works. Translation with depth variation causes parallax that a homography cannot model.
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Read the original → visionbook.mit.edu
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- #multi-view-geometry
- #image-stitching
- #parallax
- #fundamental-matrix
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