Bicycle chain shape models

In this paper we introduce landmark-based pre-shapes which allow mixing of anatomical landmarks and pseudo-landmarks, constraining consecutive pseudo-landmarks to satisfy planar equidistance relations. This defines naturally a structure of Riemannian manifold on these preshapes, with a natural action of the group of planar rotations. Orbits define the shapes. We develop a geodesic generalized procrustes analysis procedure for a sample set on such a preshape spaces and use it to compute principal geodesic analysis. We demonstrate it on an elementary synthetic example as well on a dataset of manually annotated vertebra shapes from x-ray. We re-landmark them consistently and show that PGA captures the variability of the dataset better than its linear counterpart, PCA.

[1]  D. Mumford,et al.  An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach , 2006, math/0605009.

[2]  Timothy F. Cootes,et al.  Active Shape Models-Their Training and Application , 1995, Comput. Vis. Image Underst..

[3]  Alex M. Andrew,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition) , 2000 .

[4]  Anuj Srivastava,et al.  Analysis of planar shapes using geodesic paths on shape spaces , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[5]  R. Fildes.  Journal of the Royal Statistical Society (B): Gary K. Grunwald, Adrian E. Raftery and Peter Guttorp, 1993, “Time series of continuous proportions”, 55, 103–116.☆ , 1993 .

[6]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid , 2012 .

[7]  T. K. Carne,et al.  Shape and Shape Theory , 1999 .

[8]  L. Noakes.  A Global algorithm for geodesics , 1998, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[9]  D. Kendall.  SHAPE MANIFOLDS, PROCRUSTEAN METRICS, AND COMPLEX PROJECTIVE SPACES , 1984 .

[10]  Nico Karssemeijer,et al.  Proceedings of the 20th international conference on Information processing in medical imaging , 2007 .

[11]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[12]  Anuj Srivastava,et al.  Geodesics Between 3D Closed Curves Using Path-Straightening , 2006, ECCV.

[13]  P. Thomas Fletcher,et al.  Principal geodesic analysis for the study of nonlinear statistics of shape , 2004, IEEE Transactions on Medical Imaging.

[14]  C. Goodall.  Procrustes methods in the statistical analysis of shape , 1991 .

[15]  Timothy F. Cootes,et al.  A minimum description length approach to statistical shape modeling , 2002, IEEE Transactions on Medical Imaging.

[16]  Anuj Srivastava,et al.  Elastic-string models for representation and analysis of planar shapes , 2004, Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004..

[17]  Jean-Pierre Dedieu,et al.  Symplectic methods for the approximation of the exponential map and the Newton iteration on Riemannian submanifolds , 2005, J. Complex..

[18]  F. Bookstein,et al.  Morphometric Tools for Landmark Data: Geometry and Biology , 1999 .

[19]  Martin Styner,et al.  Shape Modeling and Analysis with Entropy-Based Particle Systems , 2007, IPMI.

[20]  David G. Kendall,et al.  Shape & Shape Theory , 1999 .