In statistics, Procrustes analysis is a form of statistical shape analysis used to analyse the distribution of a set of shapes. The name Procrustes () refers to a bandit from Greek mythology who made his victims fit his bed either by stretching their limbs or cutting them off.
In mathematics: an orthogonal Procrustes problem is a method which can be used to find out the optimal rotation and/or reflection (i.e., the optimal orthogonal linear transformation) for the Procrustes Superimposition (PS) of an object with respect to another. a constrained orthogonal Procrustes problem, subject to det(R) = 1 (where R is an orthogonal matrix), is a method which can be used to determine the optimal rotation for the PS of an object with respect to another (reflection is not allowed). In some contexts, this method is called the Kabsch algorithm.
When a shape is compared to another, or a set of shapes is compared to an arbitrarily selected reference shape, Procrustes analysis is sometimes further qualified as classical or ordinary, as opposed to generalized Procrustes analysis (GPA), which compares three or more shapes to an optimally determined "mean shape".
Introduction
To compare the shapes of two or more objects, the objects must be first optimally "superimposed". Procrustes superimposition (PS) is performed by optimally translating, rotating and uniformly scaling the objects. In other words, both the placement in space and the size of the objects are freely adjusted. The aim is to obtain a similar placement and size, by minimizing a measure of shape difference called the Procrustes distance between the objects. This is sometimes called full, as opposed to partial PS, in which scaling is not performed (i.e. the size of the objects is preserved). Notice that, after full PS, the objects will exactly coincide if their shape is identical. For instance, with full PS two spheres with different radii will always coincide, because they have exactly the same shape. Conversely, with partial PS they will never coincide. This implies that, by the strict definition of the term shape in geometry, shape analysis should be performed using full PS. A statistical analysis based on partial PS is not a pure shape analysis as it is not only sensitive to shape differences, but also to size differences. Both full and partial PS will never manage to perfectly match two objects with different shape, such as a cube and a sphere, or a right hand and a left hand.
In some cases, both full and partial PS may also include reflection. Reflection allows, for instance, a successful (possibly perfect) superimposition of a right hand to a left hand. Thus, partial PS with reflection enabled preserves size but allows translation, rotation and reflection, while full PS with reflection enabled allows translation, rotation, scaling and reflection.
Optimal translation and scaling are determined with much simpler operations (see below).
Ordinary Procrustes analysis
Here we just consider objects made up from a finite number k of points in n dimensions. Often, these points are selected on the continuous surface of complex objects, such as a human bone, and in this case they are called landmark points.
The shape of an object can be considered as a member of an equivalence class formed by removing the translational, rotational and uniform scaling components.
Translation
For example, translational components can be removed from an object by translating the object so that the mean of all the object's points (i.e. its centroid) lies at the origin.
Mathematically: take points in two dimensions, say
.
The mean of these points is where
Now translate these points so that their mean is translated to the origin , giving the point .
Uniform scaling
Likewise, the scale component can be removed by scaling the object so that the root mean square distance (RMSD) from the points to the translated origin is 1. This RMSD is a statistical measure of the object's scale or size:
The scale becomes 1 when the point coordinates are divided by the object's initial scale:
.
Notice that other methods for defining and removing the scale are sometimes used in the literature.