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Asked: 2026-05-18T01:12:26+00:00

Problem Given a set of known cartesian points (set A), and a 2d transformation

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Problem

Given a set of known cartesian points (set A), and a 2d transformation (rotation, translation, scale) of some subset of those points (set B), find the orientation of the subset (rotation, translation, scale) relative to the original set of points.

I.E. Suppose I take a "picture" of a known set of 2d points on a wall. I want to know what position the camera was in relative to "upright and centered" when the picture was taken. Some of the points may not be visible in the picture (they may be occluded). (in this analogy, assume the camera is orthoganal and always pointed directly at the plane of the wall, so you don’t need to take distortion or perspective into account)

Proposed approach:

Step 1: Scale B to the same "range" as A

Don’t know how; open to suggestions. Maybe take the area of a convex hull around all the points in B, and scale it to nearly that of the convex hull around A. This is tricky, because points may be missing from B.

Step 2: Match some arbitrary point in "B" to its twin in "A"

Pick some random point in set B. Call this point K. Somehow take a "fingerprint" of K relative to all the other points in B (using distance only). Find its match in A by fingerprinting all points in A and taking the point with the most similar fingerprint of K.

Step 3: Rotate B (around K) until all points in B are aligned with a point in A

Multiple solutions are possible, so keep rotating though 360d looking for solutions.

That’s just shooting from the hip, I may be way off base. Anyone have any ideas?

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1 Answer

  1. Editorial Team

    Assuming you don’t actually know the correspondence between the points in the two clouds, you could try a statistical approach.

    First, compute the mean x0 of the original cloud, then compute the mean x1 of the subset cloud. The difference of the mean vectors, x1-x0, is a good estimate of the required translation.

    Now, subtract the relevant mean vector from each set to give two clouds centered at the origin. Compute the covariance matrix for each cloud and find its eigenvalues and eigenvectors. The required rotation can be found from the eigenvectors, while the scaling corresponds to the eigenvalues.

    Compose all of this and you should have a good statistical estimate of the desired transform. Obviously, its quality will be a function of how well the subset spans the original set.

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