Dimension Reduction on Polyspheres with Application to Skeletal Representations

We present a novel method that adaptively deforms a polysphere (a product of spheres) into a single high dimensional sphere which then allows for principal nested spheres (PNS) analysis. Applying our method to skeletal representations of simulated bodies as well as of data from real human hippocampi yields promising results in view of dimension reduction. Specifically in comparison to composite PNS (CPNS), our method of principal nested deformed spheres (PNDS) captures essential modes of variation by lower dimensional representations.

[1]  J. Gower.  Generalized procrustes analysis , 1975 .

[2]  Nicholas Ayache,et al.  Principal Spine Shape Deformation Modes Using Riemannian Geometry and Articulated Models , 2006, AMDO.

[3]  J. Damon.  Smoothness and geometry of boundaries associated to skeletal structures, II: Geometry in the Blum case , 2004, Compositio Mathematica.

[4]  J. Damon.  Global geometry of regions and boundaries via skeletal and medial integrals , 2007 .

[5]  Stefan Sommer,et al.  Horizontal Dimensionality Reduction and Iterated Frame Bundle Development , 2013, GSI.

[6]  Nicholas Ayache,et al.  A Log-Euclidean Framework for Statistics on Diffeomorphisms , 2006, MICCAI.

[7]  A. Munk,et al.  Intrinsic shape analysis: Geodesic principal component analysis for Riemannian manifolds modulo Lie group actions. Discussion paper with rejoinder. , 2010 .

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

[9]  Stephan Huckemann,et al.  Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces , 2006, Advances in Applied Probability.

[10]  J. S. Marron,et al.  Nested Sphere Statistics of Skeletal Models , 2013, Innovations for Shape Analysis, Models and Algorithms.

[11]  J. Marron,et al.  Analysis of principal nested spheres. , 2012, Biometrika.

[12]  Kaleem Siddiqi,et al.  Medial Representations: Mathematics, Algorithms and Applications , 2008 .

[13]  B. Afsari.  Riemannian Lp center of mass: existence, uniqueness, and convexity , 2011 .

[14]  James Stephen Marron,et al.  Analysis of Rotational Deformations From Directional Data , 2015 .