Appropriate similarity metrics for multiple sets of 2d co-ordinates

Question

Pardon me if this has been asked/answered before...my searches did not bring it up.

I have a collection of 2D co-ordinate sets (on the scale of a 100K-500K points in each set) and I am looking for the most efficient way to measure the similarity of 1 set to the other. I know of the usuals: Cosine, Jaccard/Tanimoto etc. However hoping for some suggestions on any fast/efficient ones to measure similarity, especially ones that can cluster by similarity.

Edit 1: The image shows what I need to do, I need to cluster all the reds, blues and greens by their shape/orientatoin etc.

Answer 1

Try K-means algorithm. It dynamically calculated the centroid of each cluster and calculates distance to all the pointers and associates them to the nearest cluster.

Answer 2

It seems that the first step of any solution is going to be to find the centroid, or other reference point, of each shape, so that they can be compared regardless of absolute position.

One algorithm that comes to mind would be to start at the point nearest the centroid and walk to its nearest neighbors. Compare the offsets of those neighbors (from the centroid) between the sets being compared. Keep walking to the next-nearest neighbors of the centroid, or the nearest not-already-compared neighbors of the ones previously compared, and keep track of the aggregate difference (perhaps RMS?) between the two shapes. Also, at each step of this process calculate the rotational offset that would bring the two shapes into closest alignment [and whether mirroring affects it as well?]. When you are finished you will have three values for every pair of sets, including their direct similarity, their relative rotational offset (mostly only useful if they are close matches after rotation), and their similarity after rotation.

Answer 3

Since your clustering is based on a nearness-to-shape metric, perhaps you need some form of connected component labeling. UNION-FIND can give you a fast basic set primitive.

For union-only, start every point in a different set, and merge them if they meet some criterion of nearness, influenced by local colinearity since that seems important to you. Then keep merging until you pass some over-threshold condition for how difficult your merge is. If you treat it like line-growing (only join things at their ends) then some data structures become simpler. Are all your clusters open lines and curves? No closed curves, like circles?

The crossing lines are trickier to get right, you either have to find some way merge then split, or you set your merge criteria to extremely favor colinearity and you luck out on the crossing lines.