Abstract
The Gaussian distribution is the basis for many methods used in the statistical analysis of shape. One such method is principal component analysis, which has proven to be a powerful technique for describing the geometric variability of a population of objects. The Gaussian framework is well understood when the data being studied are elements of a Euclidean vector space. This is the case for geometric objects that are described by landmarks or dense collections of boundary points. We have been using medial representations, or m-reps, for modelling the geometry of anatomical objects. The medial parameters are not elements of a Euclidean space, and thus standard PCA is not applicable. In our previous work we have shown that the m-rep model parameters are instead elements of a Lie group. In this paper we develop the notion of a Gaussian distribution on this Lie group. We then derive the maximum likelihood estimates of the mean and the covariance of this distribution. Analogous to principal component analysis of covariance in Euclidean spaces, we define principal geodesic analysis on Lie groups for the study of anatomical variability in medially-defined objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Cootes, T.F., Taylor, C.J., Cooper, D.H., Graham, J.: Active shape models - their trainin gand application. Computer Vision and Image Understanding 61, 38–59 (1995)
Cootes, T.F., Edwards, G.J., Taylor, C.J.: Active appearance models. In: Fifth European Conference on Computer Vision, pp. 484–498 (1998)
Kelemen, A., Székely, G., Gerig, G.: Three-dimensional model-based segmentation. Transactions Medical Imaging 18, 828–839 (1999)
Jolliffe, I.T.: Principal Component Analysis. Springer, Heidelberg (1986)
Csernansky, J., Joshi, S., Wang, L., Haller, J., Gado, M., Miller, J., Grenander, U., Miller, M.: Hippocampal morphometry in schizophrenia via high dimensional brain mapping. Proceedings National Academy of Sciences, 11406–11411 (1998)
Joshi, S., Pizer, S., Fletcher, P.T., Yushkevich, P., Thall, A., Marron, J.S.: Multiscale deformable model segmentation and statistical shape analysis using medial descriptions. Transactions on Medical Imaging 21 (2002)
Blum, H., Nagel, R.: Shape description using weighted symmetric axis features. Pattern Recognition 10, 167–180 (1978)
Thall, A.: Fast C2 interpolating subdivision surfaces using iterative inversion of stationary subdivision rules (2002), http://midag.cs.unc.edu/pub/papers/Thall_TR02-001.pdf
Duistermaat, J.J., Kolk, J.A.C.: Lie Groups. Springer, Heidelberg (2000)
Curtis, M.L.: Matrix Groups. Springer, Heidelberg (1984)
Mardia, K.V.: Directional Statistics. John Wiley and Sons, Chichester (1999)
Kendall, D.G.: Shape manifolds, Procrustean metrics, and complex projective spaces. Bulletin of the London Mathematical Society 16, 18–121 (1984)
Srivastava, A., Klassen, E.: Monte-Carlo extrinsic estimators of manifold-valued parameters. IEEE Transactions on Signal Processing 50, 299–308 (2001)
Moakher, M.: Means and averaging in the group of rotations. SIAM Journal on Matrix Analysis and Applications 24, 1–16 (2002)
Grenander, U.: Probabilities on Algebraic Structures. John Wiley and Sons, Chichester (1963)
Lee, J.M.: Riemannian Manifolds: An Introduction to Curvature. Springer, Heidelberg (1997)
Buss, S.R., Fillmore, J.P.: Spherical averages and applications to spherical splines and interpolation. ACM Transactions on Graphics 20, 95–126 (2001)
Goodall, C.: Procrustes methods in the statistical analysis of shape. Journal of the Royal Statistical Society 53, 285–339 (1991)
Styner, M., Gerig, G.: Medial models incorporating object variability for 3D shape analysis. In: Information Processing in Medical Imaging, pp. 502–516 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fletcher, P.T., Joshi, S., Lu, C., Pizer, S.M. (2003). Gaussian Distributions on Lie Groups and Their Application to Statistical Shape Analysis. In: Taylor, C., Noble, J.A. (eds) Information Processing in Medical Imaging. IPMI 2003. Lecture Notes in Computer Science, vol 2732. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45087-0_38
Download citation
DOI: https://doi.org/10.1007/978-3-540-45087-0_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40560-3
Online ISBN: 978-3-540-45087-0
eBook Packages: Springer Book Archive