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Continuous Procrustes distance between two surfaces. (English) Zbl 1276.68157

The Procrustes distance is a measure for the similarity of two surfaces used in geometric morphometrics. In its original version, it is based on a sampling of “landmark points” on both surfaces which, for several reasons, is undesirable. In this article, the authors propose a continuous Procrustes distance for two-dimensional surfaces \(\mathcal{S}, \mathcal{S}' \subset \mathbb{R}^3\) as the infimum of the integral \(\int_{\mathcal{S}} \| Rx - \mathcal{C}x \|^2 d{\text{vol}_{\mathcal{S}}(x)}\) over all Euclidean transformations \(R\) of \(\mathbb{R}^3\) and all area-preserving maps \(\mathcal{C}: \mathcal{S} \to \mathcal{S}'\). The underlying assumption is that area-preserving maps will automatically provide a good choice for homologous point pairs. The existence of a minimizer can be guaranteed if \(\mathcal{C}\) is subject to a reasonably restricted class of bi-Lipschitz maps. The thus defined continuous Procrustes distance is indeed a metric in the class of surfaces under scrutiny modulo congruence.
The computation of the continuous Procrustes distance is unfeasible. In order to make it usable in practice, the authors propose to approximate \(\mathcal{C}\) by a conformal or anticonformal map and, subsequently, deform it to a nearby area-preserving map. This leads to a dramatic reduction of the search space. The theoretical justification is a statement that area-preserving diffeomorphisms have small conformal distortion provided the associated Procrustes distance is small. The authors describe strategies for actually computing the approximating (anti)conformal map and its deformation to an area-preserving map. Their viability is demonstrated by an actual implementation and illustrated with real world examples.

MSC:

68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
65K10 Numerical optimization and variational techniques
30C30 Schwarz-Christoffel-type mappings

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