Principal geodesic analysis for the study of nonlinear statistics of shape

  title={Principal geodesic analysis for the study of nonlinear statistics of shape},
  author={P. Thomas Fletcher and Conglin Lu and Stephen M. Pizer and Sarang C. Joshi},
  journal={IEEE Transactions on Medical Imaging},
A primary goal of statistical shape analysis is to describe the variability of a population of geometric objects. A standard technique for computing such descriptions is principal component analysis. However, principal component analysis is limited in that it only works for data lying in a Euclidean vector space. While this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points, it does not handle more complex… 
Shape Variation of Medial Axis Representations via Principal Geodesic Analysis on Symmetric Spaces
This chapter presents a general method called principal geodesic analysis (PGA) for computing the variability of manifold-valued data and demonstrates the use of PGA to describe the shape variability of medial representations, and results are shown on a hippocampus data set.
Statistics of Pose and Shape in Multi-object Complexes Using Principal Geodesic Analysis
This paper discusses the decoupling of pose and shape in multi-object sets using different normalization settings and introduces new methods of describing the statistics of object pose using a novel extension of PGA, which previously has been used for global shape statistics.
Principal component geodesics for planar shape spaces
The numerical findings back the notion that the Euclidean approximation is good for highly concentrated data and for low concentration, however, the error can be strongly notable.
Multi-Object Statistics using Principal Geodesic Analysis in a Longitudinal Pediatric Study Anonymous CVPR submission Paper ID 245
  • 2006
A main focus of statistical shape analysis is the description of variability of a population of geometric objects. In this paper, we present work in progress towards modeling the shape and pose
A Surface-Theoretic Approach for Statistical Shape Modeling
A novel approach for nonlinear statistical shape modeling that is invariant under Euclidean motion and thus alignment-free is presented, facilitating Riemannian analysis of large shape populations accessible through longitudinal and multi-site imaging studies providing increased statistical power.
Discrimination analysis using multi-object statistics of shape and pose
This paper discusses the decoupling of pose and shape in multi-object sets using different normalization settings, and introduces methods of describing the statistics of object pose and object shape, both separately and simultaneously using a novel extension of PGA.
The differential geometry of landmark shape manifolds: metrics, geodesics, and curvature
The study of shapes and their similarities is central in computer vision, in that it allows to recognize and classify objects from their representation. One has the interest of defining a distance
A Riemannian Statistical Shape Model using Differential Coordinates
We propose a novel Riemannian framework for statistical analysis of shapes that is able to account for the nonlinearity in shape variation. By adopting a physical perspective, we introduce a
Statistical Analysis of Organs’ Shapes and Deformations: The Riemannian and the Affine Settings in Computational Anatomy
This chapter details the extension of this framework to Lie groups endowed with the affine symmetric connection, a more invariant but non-metric structure on transformation groups, and provides strong theoretical bases for the use of one-parameter subgroups and diffeomorphisms parametrized by stationary velocity fields (SVF).
Discrete Geodesic Regression in Shape Space
The proposed method is applied to the analysis of root growth in botany and the morphological changes of brain structures due to aging and is based on a variational time discretization of geodesics.


Statistics of shape via principal geodesic analysis on Lie groups
  • P. Fletcher, Conglin Lu, S. Joshi
  • Mathematics, Computer Science
    2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings.
  • 2003
This paper shows that medial descriptions are in fact elements of a Lie group, and develops methodology based on Lie groups for the statistical analysis of medially-defined anatomical objects.
Gaussian Distributions on Lie Groups and Their Application to Statistical Shape Analysis
Analogous to principal component analysis of covariance in Euclidean spaces, principal geodesic analysis on Lie groups is defined for the study of anatomical variability in medially-defined objects and results of applying this framework on a population of hippocampi in a schizophrenia study are presented.
The shape-space l. k m whose points a represent the shapes of not totally degenerate /c-ads in IR m is introduced as a quotient space carrying the quotient metric. When m = 1, we find that Y\ = S k ~
Size and Shape Spaces for Landmark Data in Two Dimensions
Biometric studies of the forms of organisms usually consider size and shape variations in the geometric configuration of landmarks, points that correspond biologically from form to form. The size
A Survey of the Statistical Theory of Shape
This is a review of the current state of the "theory of shape" introduced by the author in 1977. It starts with a definition of "shape" for a set of k points in m dimensions. The first task is to
A Statistical Theory of Shape
A general approach for defining shape and finding its density, expressed in the densities for the individual points, is developed and some examples, that can be computed analytically, are given.
Medial Models Incorporating Object Variability for 3D Shape Analysis
A novel approach that incorporates variability of an object population into the generation of a characteristic 3D shape model based on a fine-scale spherical harmonics boundary description and a coarse-scale sampled medial description is described.
Elastic model-based segmentation of 3-D neuroradiological data sets
A new technique for the automatic model-based segmentation of three-dimensional (3-D) objects from volumetric image data based on a hierarchical parametric object description rather than a point distribution model, which shows that invariant object surface parametrization provides a good approximation to automatically determine object homology.
Multiscale deformable model segmentation and statistical shape analysis using medial descriptions
This paper presents a multiscale framework based on a medial representation for the segmentation and shape characterization of anatomical objects in medical imagery and presents a novel statistical shape analysis approach based on the medial descriptions that examines shape via separate intuitive categories, such as global variability at the coarse scale and localization at the fine scale.
Probabilities and statistics on Riemannian manifolds: Basic tools for geometric measurements
  • X. Pennec
  • Computer Science, Mathematics
  • 1999
Measurements of geometric primitives, such as rotations or rigid transformations, are often noisy and we need to use statisti cs either to reduce the uncertainty or to compare measurements. Unfor