Finding an optimal overlay of two point clouds

Question

Imagine I have two sets of points:

pointSetPerfect = Flatten[Table[{i, j}, {i, 10, 40, 10}, {j, 10, 40, 10}], 1];

pointSetNoise = Table[pointSetPerfect[[i]] + RandomReal[{-2, 2} - 100], {i, 1, Length[pointSetPerfect]}];

r = RotationTransform[47.5 Degree, {0, 2}];
pointSetNoise = r[pointSetNoise];

ListPlot[{pointSetPerfect, pointSetNoise}]

Without knowing what I did above to generate pointSetNoise, I would like to overlay, as best as I can, a set of points on pointSetNoise with the same interpoint spacings as in the point set pointSetPerfect, and return these coordinates. If necessary, I could also click on the image to specify by eye a (very approximate) 1-to-1 mapping between points in either set.

Is there a nice way of doing this in Mathematica v9?

Answer 1

try:

f = FindGeometricTransform[pointSetNoise, pointSetPerfect, 
                           "Transformation" -> "Rigid", Method -> "FindFit"]

I know it's too short for an answer, but that's it.

The result can be tested like this:

ListPlot[{f[[2]][pointSetPerfect], pointSetNoise}, Axes -> False, Frame -> True]