Journal ArticleDOI
Geodesic Regression and the Theory of Least Squares on Riemannian Manifolds
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TLDR
Specific examples are given for a set of synthetically generated rotation data and an application to analyzing shape changes in the corpus callosum due to age, which can be generally applied to data on any manifold.Abstract:
This paper develops the theory of geodesic regression and least-squares estimation on Riemannian manifolds. Geodesic regression is a method for finding the relationship between a real-valued independent variable and a manifold-valued dependent random variable, where this relationship is modeled as a geodesic curve on the manifold. Least-squares estimation is formulated intrinsically as a minimization of the sum-of-squared geodesic distances of the data to the estimated model. Geodesic regression is a direct generalization of linear regression to the manifold setting, and it provides a simple parameterization of the estimated relationship as an initial point and velocity, analogous to the intercept and slope. A nonparametric permutation test for determining the significance of the trend is also given. For the case of symmetric spaces, two main theoretical results are established. First, conditions for existence and uniqueness of the least-squares problem are provided. Second, a maximum likelihood criteria is developed for a suitable definition of Gaussian errors on the manifold. While the method can be generally applied to data on any manifold, specific examples are given for a set of synthetically generated rotation data and an application to analyzing shape changes in the corpus callosum due to age.read more
Citations
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References
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Book
Differential Geometry, Lie Groups, and Symmetric Spaces
TL;DR: In this article, the structure of semisimplepleasure Lie groups and Lie algebras is studied. But the classification of simple Lie algesbras and of symmetric spaces is left open.
Journal ArticleDOI
Statistical shape analysis: clustering, learning, and testing
TL;DR: This work presents tools for hierarchical clustering of imaged objects according to the shapes of their boundaries, learning of probability models for clusters of shapes, and testing of newly observed shapes under competing probability models.
Book
Statistical Shape Analysis: With Applications in R
Ian L. Dryden,Kanti V. Mardia +1 more
TL;DR: In this article, the authors proposed a planar procrustes analysis for two-dimensional data and showed that it is possible to estimate the size and shape of a shape in images.
Book
An introduction to differentiable manifolds and Riemannian geometry
TL;DR: In this article, the authors present a revised edition of one of the classic mathematics texts published in the last 25 years, which includes updated references and indexes and error corrections and will continue to serve as the standard text for students and professionals in the field.
Journal ArticleDOI
Shape manifolds, procrustean metrics, and complex projective spaces
TL;DR: In this article, the shape-space l. k m whose points represent the shapes of not totally degenerate /c-ads in IR m is introduced as a quotient space carrying the quotient metric.