ProcrustesAbstands

ProcrustesAbstands, commonly referred to as the Procrustes distance, is a measure used in shape analysis to quantify how dissimilar two shapes are after removing the effects of translation, rotation, and, in some variants, scaling. It is defined for configurations of corresponding landmarks in a common space and is invariant to position, orientation, and size (depending on the variant chosen).

In practice, each shape is represented as a configuration of corresponding points (landmarks). The typical procedure is to center each configuration by subtracting its centroid, and optionally scale to a common size. The remaining problem is to find the optimal rotation (and, if included, scaling) that minimizes the sum of squared distances between corresponding landmarks. The Procrustes distance is the minimized residual error, often expressed as the Frobenius norm of the difference between the aligned configurations. The rotation is obtained from an orthogonal Procrustes problem, usually solved via a singular value decomposition of the cross-covariance matrix. If scaling is included, the optimal scale is determined jointly with the rotation; if scaling is excluded, only rotation (and translation) is optimized.

Variants of the Procrustes distance exist. The standard Procrustes distance typically includes translation removal and optional scaling, yielding a measure of pure shape difference. The orthogonal Procrustes distance fixes scale and focuses on rotational alignment. Generalized and partial Procrustes approaches extend the idea to align multiple shapes or to use only subsets of landmarks.

Applications of ProcrustesAbstands span morphometrics, biology, computer vision, and pattern recognition, where shapes must be compared independently of position, orientation, and size. Limitations include sensitivity to landmark choice and potential impact from outliers or missing data.