Maybe you could do some kind of PCA of the data and derive the affine transformation that maps one’s principal components into the others. If that’s not good enough, it could be a starting point for some kind of refinement. I’m not sure what ICP is.

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Yes, I found it: A Flexible New Technique for Camera Calibration, Appendix C. Now that I re-read it, I’m even less sure that it will be useful, anyway:

The problem considered in this section is to solve the best rotation matrix R to approximate a given

3 x 3 matrix Q. Here, “best” is in the sense of the smallest Frobenius norm of the difference R-Q.

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So this is my problem… I have two unstructured point clouds, containing different numbers of points, which have been sampled in a noisy manner. I want to find the “best” affine transformation between these sets of points. I’ve tried ICP, but the points are too noisy and I often fail to find enough pairs of points for the algorithm to succeed. I have a procedure to combine the sets once they are registered as closely as possible, but solving the registration problem in the presence of noise is very difficult. Does anyone have any suggestions, ideas, code? Thanks!