# What is the difference between Procrustes analysis and the Linear Transformation in terms of Shape Analysis?

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

• Procrustes is rotation and scaling. Affine also allows translation. – mathreadler Dec 9 '17 at 14:23
• And in relation to linear transformation (which doesn't includes translation)? – tuuttuut Dec 9 '17 at 14:26
• A linear transformation can not have translation unless you blow up the space slightly by adding a dimension or two. – mathreadler Dec 9 '17 at 14:27
• 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. – mathreadler Dec 9 '17 at 14:29
• 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 – tuuttuut Dec 9 '17 at 14:33