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.


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


Full Text: DOI arXiv


[1] Ahlfors, Lectures on quasiconformal mappings 38 (2006) · doi:10.1090/ulect/038
[2] Besl, A method for registration of 3-D shapes, IEEE Trans. Pattern Anal. Mach. Intell. 14 (2) pp 239– (1992) · Zbl 05110721 · doi:10.1109/34.121791
[3] Boyer , D. M. Lipman , Y. Clair , E. S. Puente , J. Patel , B. A. Funkhouser , T. A. Jernvall , J. Daubechies , I. New algorithms to automatically quantify the geometric similarity of anatomical surfaces
[4] Bronstein, Generalized multidimensional scaling: A framework for isometry-invariant partial surface matching, Proc. Natl. Acad. Sci. USA 103 (5) pp 1168– (2006) · Zbl 1160.65306 · doi:10.1073/pnas.0508601103
[5] Dacorogna, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1) pp 1– (1990) · Zbl 0707.35041 · doi:10.1016/S0294-1449(16)30307-9
[6] Dominitz, Texture mapping via optimal mass transport, IEEE Transactions on Visualization and Computer Graphics 16 (3) pp 419– (2010) · doi:10.1109/TVCG.2009.64
[7] Eggert, Estimating 3-D rigid body transformations: a comparison of four major algorithms, Mach. Vision Appl. 9 (5-6) pp 272– (1997) · doi:10.1007/s001380050048
[8] Eldar, The farthest point strategy for progressive image sampling, IEEE Transactions on Image Processing 6 (9) pp 1305– (1997) · doi:10.1109/83.623193
[9] Ghosh, Feature-driven deformation for dense correspondence, Proc. SPIE 7261 (2009) · doi:10.1117/12.811463
[10] Grenander, Computational anatomy: an emerging discipline, Quart. Appl. Math. 56 (4) pp 617– (1998) · Zbl 0952.92016 · doi:10.1090/qam/1668732
[11] Gu, Computational conformal geometry (2008)
[12] Imayoshi, An introduction to Teichmüller spaces (1992) · Zbl 0754.30001 · doi:10.1007/978-4-431-68174-8
[13] Lipman, On surface comparison and symmetry, Third Workshop on Non-Rigid Shape Analysis and Deformable Image Alignment (in conjunction with CVPR’10)
[14] Lipman, Mobius voting for surface correspondence, ACM Transactions on Graphics (Proc. SIGGRAPH) 28 (3) (2009) · doi:10.1145/1531326.1531378
[15] Lipman, Conformal Wasserstein distance: II. Computational aspects and extensions, Math. Comp. · Zbl 1281.65034
[16] Mémoli, Gromov-Hausdorff distances in Euclidean spaces, IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops (Anchorage, 2008) pp 1–
[17] Mémoli, On the use of Gromov-Hausdorff distances for shape comparison, Symposium on Point Based Graphics 2007 (Eurographics/IEEE Computer Society VGTC Symposium Proceedings) pp 81–
[18] Mémoli, A theoretical and computational framework for isometry invariant recognition of point cloud data, Found. Comput. Math 5 (3) pp 313– (2005) · Zbl 1101.53022 · doi:10.1007/s10208-004-0145-y
[19] Micheli , M. Michor , P. W. Mumford , D. Sectional curvature in terms of the cometric, with applications to the Riemannian manifolds of landmarks · Zbl 1276.37047
[20] Miller, On the metrics and Euler-Lagrange equations of computational anatomy, Annual Review of Biomedical Engineering 4 (1) pp 375– (2002) · doi:10.1146/annurev.bioeng.4.092101.125733
[21] Mitteroecker, Advances in geometric morphometrics, Evolutionary Biology 36 (2) pp 235– (2009) · doi:10.1007/s11692-009-9055-x
[22] Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (2) pp 286– (1965) · Zbl 0141.19407 · doi:10.1090/S0002-9947-1965-0182927-5
[23] Pottmann, Registration without ICP, Comput. Vis. Image Underst. 95 (1) pp 54– (2004) · Zbl 02179938 · doi:10.1016/j.cviu.2004.04.002
[24] Rangarajan, Information Processing in Medical Imaging pp 29– (1997) · doi:10.1007/3-540-63046-5_3
[25] Rusinkiewicz, Efficient variants of the ICP algorithm, Proceedings of the Third International Conference on 3-D Digital Imaging and Modeling (Quebec City, 2001) pp 145–
[26] Smolyanov, Brownian motion on a manifold as limit of stepwise conditioned standard Brownian motions, Stochastic processes, physics and geometry: new interplays, II (Leipzig, 1999) pp 589– (2000)
[27] Spivak, A comprehensive introduction to differential geometry 2 (1999) · Zbl 1213.53001
[28] Trouvé, Local geometry of deformable templates, SIAM J. Math. Anal. 37 (1) pp 17– (2005) · Zbl 1090.58008 · doi:10.1137/S0036141002404838
[29] Wang, Conformal geometry and its applications on 3D shape matching, recognition, and stitching, IEEE Trans. Pattern Anal. Mach. Intell. 29 (7) pp 1209– (2007) · Zbl 05340891 · doi:10.1109/TPAMI.2007.1050
[30] Wendland, Scattered data approximation (2005)
[31] Wirth, A continuum mechanical approach to geodesics in shape space, Int. J. Comput. Vis. 93 (3) pp 293– (2011) · Zbl 1235.68309 · doi:10.1007/s11263-010-0416-9
[32] Younes, Shapes and diffeomorphisms (2010) · Zbl 1205.68355 · doi:10.1007/978-3-642-12055-8
[33] Younes, A metric on shape spaces with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (1) pp 25– (2008) · Zbl 1142.58013 · doi:10.4171/RLM/506
[34] Zeng, 3D face matching and registration based on hyperbolic Ricci flow, IEEE Conference on Computer Vision and Pattern Recognition Workshops (Anchorage, 2008) pp 1–
[35] Zeng, 3D non-rigid surface matching and registration based on holomorphic differentials, Proceedings of the 10th European Conference on Computer Vision: Part III (Marseilles, 2008) pp 1– (2008)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.