Procrustesanalysen

Procrustes analysis is a statistical method used in shape analysis to compare configurations of landmarks by removing differences due to translation, rotation, and scaling. This allows shapes to be studied within a common coordinate frame and to quantify residual variation that reflects true shape differences.

In ordinary Procrustes analysis, two configurations X and Y of n landmarks in d dimensions are aligned by translating to the origin, then optimally rotating and scaling X to match Y. The objective is to minimize the squared distance between corresponding points after transformation: ||sXR + t − Y||_F^2, where R is an orthogonal matrix and s > 0. If the coordinates are centered, translation t is zero and the problem reduces to finding the best rotation and scale. The optimal rotation R can be obtained from the singular value decomposition of X^T Y, and the scale s is derived from the traces of X^T X and X^T Y.

Generalized Procrustes analysis extends the method to multiple configurations. Each shape is aligned to a common mean shape, and the mean is updated iteratively as all configurations are re-aligned, until convergence. GPA is widely used in geometric morphometrics to produce a consensus shape and to study variation among specimens.

The method is named after Procrustes, a figure from Greek mythology, and its statistical formulation was developed in the late 20th century by researchers such as Gower and later Bookstein and colleagues. Applications span biology, anthropology, medicine, computer vision, and pattern recognition. Limitations include the need for a consistent set of homologous landmarks, sensitivity to outliers, and the assumption that corresponding points are meaningful across shapes. Robust or partial variants exist that address some of these issues.

Software implementations are available in R (for example, geomorph) and Python (SciPy and related libraries), enabling both simple Procrustes alignment and generalized analyses.