Principal Geodesic Analysis for the Study of Nonlinear Statistics of Shape

Fletcher, P. Thomas; Lu, C.; Pizer, Stephen M.; Joshi, Sarang

doi:10.1109/tmi.2004.831793

Cited by 681 publications

(637 citation statements)

References 27 publications

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“…Each structure is displayed in its mean orientation, position, and scale in the global coordinate frame. The average orientation for each structure was computed using methods for averaging in curved spaces [39]. I used the arithmetic mean of position and the geometric mean of scale.…”

Section: Resultsmentioning

confidence: 99%

“…Kent, for example, describes PCA in the Procrustes tangent space [66], and Cootes, et al discuss PCA for correspondence models [21]. A generalization of PCA to nonlinear distances on manifolds, called principal geodesic analysis (PGA), has been developed by Fletcher, et al [39]. While PGA o↵ers a much more accurate approach by accounting for the true nonlinear distances between shape samples, multivariate statistical methods on PGA loadings are not as well understood by researchers, and standard PCA remains the most common approach.…”

Section: Principal Component Analysismentioning

confidence: 99%

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Shape Modeling and Analysis with Entropy-Based Particle Systems

Cates

Fletcher

Styner

et al. 2007

Lecture Notes in Computer Science

139

224

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Many important fields of basic research in medicine and biology routinely employ tools for the statistical analysis of collections of similar shapes. Biologists, for example, have long relied on homologous, anatomical landmarks as shape models to characterize the growth and development of species. Increasingly, however, researchers are exploring the use of more detailed models that are derived computationally from three-dimensional images and surface descriptions. While computationally-derived models of shape are promising new tools for biomedical research, they also present some significant engineering challenges, which existing modeling methods have only begun to address.In this dissertation, I propose a new computational framework for statistical shape modeling that significantly advances the state-of-the-art by overcoming many of the limitations of existing methods. The framework uses a particle-system representation of shape, with a fast correspondence-point optimization based on information content. The optimization balances the simplicity of the model (compactness) with the accuracy of the shape representations by using two commensurate entropy metrics and no free parameters. The idea is to maximize both the geometric accuracy and the statistical simplicity of the shape model, in accordance with the principle of parsimony in model selection. The nonparametric representation allows the method to be applied to a larger class of problems than existing methods, including nonspherical surfaces, open surfaces, and sets of multiple surfaces.The relative simplicity of the surface representation and the low number of free parameters results in a framework that is easy to use and can operate directly on image segmentations. In collaboration with scientists from several important areas of biomedicine, I have demonstrated that the proposed method is indeed an e↵ective tool for scientific research.The specific research contributions of this dissertation are as follows. First, I describe a mathematical framework and a robust numerical algorithm for computing optimized correspondence-point shape models using an entropy-based optimization and particle-system technology. Second, I develop a series of extensions of the framework to more general classes of shape analysis problems, including the analysis of multiple-object complexes, the generalization to correspondence based on generic functions of position, an extension to handle surfaces with open boundaries, and shape modeling with simple regression. Third, I describe the application of statistical hypothesis testing, regression analysis, and multiple-analysis of covariance to the proposed shape models. I also introduce new techniques for visualization and interpretation of these statistics. Finally, in cooperation with biomedical researchers, I present validation of the above research contributions by their successful application to real-world research problems.

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Section: Resultsmentioning

confidence: 99%

Section: Principal Component Analysismentioning

confidence: 99%

Shape Modeling and Analysis with Entropy-Based Particle Systems

Cates

Fletcher

Styner

et al. 2007

Lecture Notes in Computer Science

139

224

View full text Add to dashboard Cite

show abstract

“…However, a linear PCA cannot describe object rotations and the modeling cannot be extended to model points and normals. Non-linear modeling is achieved by principle geodesic analysis (PGA), developed by Fletcher et al [14]. PGA extends linear PCA into nonlinear space using "curved statistics" and is a natural generalization of PCA for data that are parameterized as curved manifolds.…”

Section: Methodsmentioning

confidence: 99%

“…In the m-rep pose normalization, the sum-of-squared geodesic distances, instead of Euclidean distances, between corresponding medial atoms is minimized, as described in [14]. In this paper we discuss two types of pose normalization in the context of multiple object sets.…”

Section: Object Representation By a Mesh Of Medial Samplesmentioning

confidence: 99%

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Statistics of Pose and Shape in Multi-object Complexes Using Principal Geodesic Analysis

Styner

Gorczowski²,

Fletcher³

et al. 2006

Lecture Notes in Computer Science

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Abstract.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 variability of sets of multiple objects. Principal geodesic analysis (PGA) is the extension of the standard technique of principal component analysis (PCA) into the nonlinear Riemannian symmetric space of pose and our medial m-rep shape description, a space in which use of PCA would be incorrect. In this paper, we discuss the decoupling of pose and shape in multi-object sets using different normalization settings. Further, we introduce new methods of describing the statistics of object pose using a novel extension of PGA, which previously has been used for global shape statistics. These new pose statistics are then combined with shape statistics to form a more complete description of multi-object complexes. We demonstrate our methods in an application to a longitudinal pediatric autism study with object sets of 10 subcortical structures in a population of 20 subjects. The results show that global scale accounts for most of the major mode of variation across time. Furthermore, the PGA components and the corresponding distribution of different subject groups vary significantly depending on the choice of normalization, which illustrates the importance of global and local pose alignment in multi-object shape analysis.

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Evolutionary Deformable Models for Medical Image Segmentation: A Genetic Algorithm Approach to Optimizing Learned, Intuitive, and Localized Medial‐Based Shape Deformation

McIntosh¹,

Hamarneh²

2010

Genetic and Evolutionary Computation

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We present a novel evolutionary computing based approach to medical image segmentation. Our method complements the image-pixel integration power of deformable shape models with the high-level control mechanisms of genetic algorithms (GA). Specifically, GA alleviate typical deformable model weaknesses pertaining to model initialization, deformation parameter selection, and energy functional local minima through the simultaneous evolution of a large number of models. Furthermore, we constrain the evolution, and thus reduce the size of the search-space, by using statistically-based deformable models whose deformations are intuitive (stretch, bulge, bend) and driven in-terms of principal modes of variation of a learned mean shape. We demonstrate our work through its application to corpus callosum segmentation in mid-sagittal brain magnetic resonance images (MRI).

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Principal Geodesic Analysis for the Study of Nonlinear Statistics of Shape

Cited by 681 publications

References 27 publications

Shape Modeling and Analysis with Entropy-Based Particle Systems

Shape Modeling and Analysis with Entropy-Based Particle Systems

Statistics of Pose and Shape in Multi-object Complexes Using Principal Geodesic Analysis

Evolutionary Deformable Models for Medical Image Segmentation: A Genetic Algorithm Approach to Optimizing Learned, Intuitive, and Localized Medial‐Based Shape Deformation

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