Algorithm to match pairs from two sets

Question

I have two sets, each set is a listing of a pair of numbers

Set1 =[(x1, y1), (x2, y2), ..., (xN, yN)]
Set2 =[(a1, b1), (a2, b2), ..., (aN, bN)]

If plotted on an XY plane, Set1 and Set2 have the same basic shape, however the data points of set2 are a rotated/translated/scaled/noisy/skewed version of set1. The ordering of the pairs within each set is random. Is there an efficient way to determine which points in set1 correspond to their counterparts in set2?

Answer 1

You are looking for a family of algorithms that try to minimize the difference between two point clouds. This is a fairly hard problem to solve and there can be multiple solutions (for example, if you were given two cubes, there are many possible rotations that work).

One particularly popular approach is the ICP (iterative closest point) algorithm, which starts with a candidate guess and continuously refines it until some correctness criterion is reached or time expires. It might be a good starting point to check out.

Hope this helps!

Answer 2

Yes, assuming only Rotations, Scalings and Translations this can be done (except for the "noise" and "skew" part, that I'm not sure about).

One approach:

1. Determine the Centroid (2D mean) of each set.
2. Determine the Least-Squares (2D) slope of each set.
3. Determine the 2D Variance of each set.
4. Remap the first set, using the Centroid difference to Translate, the Slope differences to Rotate(*) and the Variance differences to Scale, so that both sets now have the same Centroid, Slope and Variance.
5. Sort both sets and then compare the points for equality/similarity. (Alternately, you can do an RSM (root-mean-square) difference between them as a measure of "noise"/difference).

(*-Note that Reflections and/or symmetries can cause problems with the Rotation part.)