Procrustes analysis

https://doi.org/10.1351/goldbook.10108
Comparison of shapes of multi-dimensional objects by a series of geometrical transformations to minimise the sum of squared distances between the transformed and target structures while maintaining the internal structure of the objects.
Notes:
  1. For two objects defined by \(\boldsymbol{X}\) and \(\boldsymbol{Y}\) the manipulation orthogonal rotation/reflection matrix \(\boldsymbol{R}\) Procrustes analysis minimises: \(\parallel\! \boldsymbol{Y} - \boldsymbol{X}\boldsymbol{R}\parallel ^{2}\) subject to \(\boldsymbol{R}^{\rm{T}}\boldsymbol{R} = \boldsymbol{R}\boldsymbol{R}^{\rm{T}} = 1\).
  2. Ordinary, or classical, Procrustes analysis is when an object is compared to one other object, which may be a reference shape. Generalized Procrustes analysis compares three or more shapes to an optimally-determined mean shape.
Source:
PAC, 2016, 88, 407. (Vocabulary of concepts and terms in chemometrics (IUPAC Recommendations 2016)) on page 423 [Terms] [Paper]