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I have a process that produces samples on a sphere. I don't know the center or radius of the sphere, and the samples are highly non-uniform -- as a somewhat example, I have 90% of them in one quadrant of the sphere, and another octant might be completely empty. (Let's assume that the convex hull of the points includes the center of the sphere, but my point is that their mean could be very far away etc.) The samples are also somewhat noisy.

I have series A of these observations on this sphere, and then another series B: B is the same points A, but they have been rotated by some matrix (and have their own noise applied).

I would like to find the rotation matrix. Of course I could do a general curve fit, but to give that a good starting point / do something that will run faster, I'm considering using the correlation matrix. Since these are vectors in 3D, I get a 3x3 correlation matrix.

If the samples were perfectly uniformly distributed and there was no noise, then the correlation matrix would literally give me the rotation matrix. With noise, one could imagine that the it should be a very good approximation of the rotation matrix (optimal?). But given that my points are not very uniformly distributed, I don't think this would work. Any ideas for faster approaches then a whole raw curve fit?

Alex Meiburg
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  • (1) Do you mean sphere (a set of points at a constant distance from their center) or ball (a set of points within a given distance from the center)? (2) Are the points in series B explicitly paired with their counterparts in series A or not? (3) Could you confirm that the two spheres have a common center, as suggested by the assertion that all you need is the correlation matrix? (4) Exactly what "correlation matrix" do you refer to? The full matrix of correlations among A and B is $6\times 6.$ (5) What can you say or assume about the distribution of this "noise"? – whuber Jul 04 '18 at 00:31
  • (1) I mean a sphere. The points are only on a shell. (2) Yes, there is a pairing between A and B, that I know; this is why I can easily compute things like correlations between them. (3) Regarding centers: they do not share the same center, no. If the distribution was uniform then the center could be estimated fairly well by the mean of course, but as it is... (4) When I said correlation matrix, I mean the correlation between component of the A series each with component of the B series. (5) Let's say noise is additive normal, standard deviation significantly smaller than the sphere's radius. – Alex Meiburg Jul 06 '18 at 23:50

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