Procrustes analysis

The name Procrustes (Greek: Προκρούστης) refers to a bandit from Greek mythology who made his victims fit his bed either by stretching their limbs or cutting them off.

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.

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.

Optimal translation and scaling are determined with much simpler operations (see below).

Here we just consider objects made up from a finite number k of points in n dimensions.

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.

Removing the rotational component is more complex, as a standard reference orientation is not always available.

Consider two objects composed of the same number of points with scale and translation removed.

when the derivative is zero gives When the object is three-dimensional, the optimum rotation is represented by a 3-by-3 rotation matrix R, rather than a simple angle, and in this case singular value decomposition can be used to find the optimum value for R (see the solution for the constrained orthogonal Procrustes problem, subject to det(R) = 1).

Notice that other more complex definitions of Procrustes distance, and other measures of "shape difference" are sometimes used in the literature.

The same method can be applied to superimpose a set of three or more shapes, as far as the above mentioned reference orientation is used for all of them.

However, Generalized Procrustes analysis provides a better method to achieve this goal.

Generalized and ordinary Procrustes analysis differ only in their determination of a reference orientation for the objects, which in the former technique is optimally determined, and in the latter one is arbitrarily selected.

When only two shapes are compared, GPA is equivalent to ordinary Procrustes analysis.

The algorithm outline is the following: There are many ways of representing the shape of an object.

Bookstein obtains a representation of shape by fixing the position of two points called the bases line.

It is also common to consider shape and scale that is with translational and rotational components removed.

Shape analysis is used in biological data to identify the variations of anatomical features characterised by landmark data, for example in considering the shape of jaw bones.

[1] One study by David George Kendall examined the triangles formed by standing stones to deduce if these were often arranged in straight lines.