Statistics of shape via principal geodesic analysis on Lie groups

Principal component analysis has proven to be useful for understanding geometric variability in populations of parameterized objects. The statistical framework is well understood when the parameters of the objects are elements of a Euclidean vector space. This is certainly the case when the objects are described via landmarks or as a dense collection of boundary points. We have been developing representations of geometry based on the medial axis description or m-rep. Although this description has proven to be effective, the medial parameters are not naturally elements of a Euclidean space. In this paper we show that medial descriptions are in fact elements of a Lie group. We develop methodology based on Lie groups for the statistical analysis of medially-defined anatomical objects.

[1]  K. Mardia.  Directional statistics and shape analysis , 1999 .

[2]  A. Kelemen,et al.  Three-dimensional model-based segmentation of brain MRI , 1998, Proceedings. Workshop on Biomedical Image Analysis (Cat. No.98EX162).

[3]  Maher Moakher,et al.  Means and Averaging in the Group of Rotations , 2002, SIAM J. Matrix Anal. Appl..

[4]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .

[5]  Michael I. Miller,et al.  Hilbert-Schmidt Lower Bounds for Estimators on Matrix Lie Groups for ATR , 1998, IEEE Trans. Pattern Anal. Mach. Intell..

[6]  HARRY BLUM,et al.  Shape description using weighted symmetric axis features , 1978, Pattern Recognit..

[7]  M. Turk,et al.  Eigenfaces for Recognition , 1991, Journal of Cognitive Neuroscience.

[8]  Andrew Thall,et al.  Fast C 2 Interpolating Subdivision Surfaces using Iterative Inversion of Stationary Subdivision Rules , 2003 .

[9]  R. Carter.  Lie Groups , 1970, Nature.

[10]  Adam Krzyzak,et al.  Learning and Design of Principal Curves , 2000, IEEE Trans. Pattern Anal. Mach. Intell..

[11]  Anuj Srivastava,et al.  Monte Carlo extrinsic estimators of manifold-valued parameters , 2002, IEEE Trans. Signal Process..

[12]  Guido Gerig,et al.  Elastic model-based segmentation of 3-D neuroradiological data sets , 1999, IEEE Transactions on Medical Imaging.

[13]  U. Grenander,et al.  Hippocampal morphometry in schizophrenia by high dimensional brain mapping. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

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

[15]  Timothy F. Cootes,et al.  Active Appearance Models , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

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

[17]  Samuel R. Buss,et al.  Spherical averages and applications to spherical splines and interpolation , 2001, TOGS.

[18]  Martin Styner,et al.  Medial Models Incorporating Object Variability for 3D Shape Analysis , 2001, IPMI.

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

[20]  Paul A. Yushkevich,et al.  Multiscale deformable model segmentation and statistical shape analysis using medial descriptions , 2002, IEEE Transactions on Medical Imaging.

[21]  Vijay Kumar,et al.  Metrics and Connections for Rigid-Body Kinematics , 1999, Int. J. Robotics Res..