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

Question

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

Answer 1

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.

Answer 2

The difference is the type of transform:

1. Procrustes finds the optimal (with respect to least squares distances of corresponding points) transform consisting of translation, scaling, reflection and rotation. So rigid transform (translation and rotation) plus global scaling and reflection.

2. Linear is different as it is missing translation. Also it allows for scaling each axis differently, and also allows shearing. Affine is linear plus translation so it is the most flexible of the mentioned transformations.

Therefore 1) Procrustes is basically shape preserving as shape is what remains after removing rigid, reflection and global scaling. 2) is not shape preserving as linear and affine can distort the shape quite a bit by their more flexible scaling and shearing capabilities.