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Let's say you have two objects, each described by some 2D corresponding points. In order to compare these two shapes, you can multiple algorithms:

  1. Procrustes analysis
  2. Search the Linear / Affine transformation (with least-squares)
  3. Other ...

My question is; what's the difference between the first two? For me, it seems like Procrustes is the same as finding the Linear/Affine transformation, only divided in 3 steps (translation, rotation and than scaling). And they both try to minimize square distance

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    $\begingroup$ Procrustes is rotation and scaling. Affine also allows translation. $\endgroup$ – mathreadler Dec 9 '17 at 14:23
  • $\begingroup$ And in relation to linear transformation (which doesn't includes translation)? $\endgroup$ – tuuttuut Dec 9 '17 at 14:26
  • $\begingroup$ A linear transformation can not have translation unless you blow up the space slightly by adding a dimension or two. $\endgroup$ – mathreadler Dec 9 '17 at 14:27
  • $\begingroup$ Linear transformation can also mirror which is not a rotation or scaling. Well technlically you can say it is a scaling with scale factor negative I guess. $\endgroup$ – mathreadler Dec 9 '17 at 14:29
  • $\begingroup$ So, is there any essential difference between a linear transformation and Procrustes analysis? I'm asking because I'm working on a image processing application where I need to conclude if two objects have the same shape. And to me, they seem exactly the same.. The only thing I can think of is that the iterative character of the Procrustes way could deliver better results is some cases $\endgroup$ – tuuttuut Dec 9 '17 at 14:33
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Based on my own experience with computer vision, Procrustes analysis is just the process of solving the problem of fitting one set of points to another using rotations, translations and scaling (and in some contexts, reflection.)

Finding such a transformation with least squares is simply one solution to the Procrustes problem. Part of the Procrustes approach is choosing what quantity you are minimizing, and the sum of squares of the point differences is one particular choice. We could have made a different choice like minimizing the maximum difference between corresponding points.

So as you can see, I didn’t not think the two are distinguished by the transformations they use, I think that the latter is just one technique for solving the former problem-type.

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