Chapter 10 - Linear Methods for Nonlinear Maps: Procrustes Fits, Thin-Plate Splines, and the Biometric Analysis of Shape Variability

Abstract

A landmark-driven or landmark and-curve-driven unwarping is capable of substantially increasing the precision of clinically or scientifically interesting comparisons or regression slopes and decreasing the sample sizes required for reliable detection of patterns in statistical parametric maps. The statistical methods of Procrustes tangent space are the ordinary, familiar methods of multivariate statistical analysis. The theorems that control their application here—invariances against changes of basis, asymptotic efficiency of permutation tests, etc.—are no different from those in any other domain of application. The visualization that suits this context, the thin-plate spline, is a nearly linear formulation governed by a set of optimality relationships of its own. Procrustes distance is defined as an optimum (minimum), which can be proved to exist even though the domain of optimization is not compact. The usual theorems about the singular-value decomposition (SVD)—such as optimality of low-rank approximation—apply here along with the usual invariances (to orthogonal rotations) that give one the freedom that one needs to seek out interesting bases for feature space after a contemplation of the pattern of the findings.