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59
Directional Statistics and Shape Analysis
, 1995
"... There have been various developments in shape analysis in the last decade. We describe here some relationships of shape analysis with directional statistics. For shape, rotations are to be integrated out or to be optimized over whilst they are the basis for directional statistics. However, various c ..."
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Cited by 794 (33 self)
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There have been various developments in shape analysis in the last decade. We describe here some relationships of shape analysis with directional statistics. For shape, rotations are to be integrated out or to be optimized over whilst they are the basis for directional statistics. However, various concepts are connected. In particular, certain distributions of directional statistics have emerged in shape analysis, such a distribution is Complex Bingham Distribution. This paper first gives some background to shape analysis and then it goes on to directional distributions and their applications to shape analysis. Note that the idea of using tangent space for analysis is common to both manifold as well. 1 Introduction Consider shapes of configurations of points in Euclidean space. There are various contexts in which k labelled points (or "landmarks") x 1 ; :::; x k in IR m are given and interest is in the shape of (x 1 ; :::; x k ). Example 1 The microscopic fossil Globorotalia truncat...
Geometric morphometrics: ten years of progress following the 'revolution'
 ITALIAN JOURNAL OF ZOOLOGY
, 2004
"... The analysis of shape is a fundamental part of much biological research. As the field of statistics developed, so have the sophistication of the analysis of these types of data. This lead to multivariate morphometrics in which suites of measurements were analyzed together using canonical variates an ..."
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Cited by 104 (5 self)
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The analysis of shape is a fundamental part of much biological research. As the field of statistics developed, so have the sophistication of the analysis of these types of data. This lead to multivariate morphometrics in which suites of measurements were analyzed together using canonical variates analysis, principal components analysis, and related methods. In the 1980s, a fundamental change began in the nature of the data gathered and analyzed. This change focused on the coordinates of landmarks and the geometric information about their relative positions. As a byproduct of such an approach, results of multivariate analyses could be visualized as configurations of landmarks back in the original space of the organism rather than only as statistical scatter plots. This new approach, called “geometric morphometrics”, had benefits that lead Rohlf and Marcus (1993) to proclaim a “revolution” in morphometrics. In this paper, we briefly update the discussion in that paper and summarize the advances in the ten years since the paper by Rohlf and Marcus. We also speculate on future directions in morphometric analysis.
Segmentation and Interpretation of MR Brain Images: An Improved Active Shape Model
, 1997
"... This paper reports a novel method for fully automated segmentation that is based on description of shape and its variation using Point Distribution Models (PDM). An improvement of the Active Shape procedure introduced by Cootes and Taylor to find new examples of previously learned shapes using PDMs ..."
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Cited by 76 (7 self)
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This paper reports a novel method for fully automated segmentation that is based on description of shape and its variation using Point Distribution Models (PDM). An improvement of the Active Shape procedure introduced by Cootes and Taylor to find new examples of previously learned shapes using PDMs is presented. The new method for segmentation and interpretation of deep neuroanatomic structures such as thalamus, putamen, ventricular system, etc. incorporates a priori knowledge about shapes of the neuroanatomic structures to provide their robust segmentation and labeling in MR brain images. The method was trained in 8 MR brain images and tested in 19 brain images by comparison to observerdefined independent standards. Neuroanatomic structures in all testing images were successfully identified. Computeridentified and observerdefined neuroanatomic structures agreed well. The average labeling error was 7 \Sigma 3%. Border positioning errors were quite small, with the average border posi...
A Unifying Theorem for Spectral Embedding and Clustering
, 2003
"... Spectral methods use selected eigenvectors of a data affinity matrix to obtain a data representation that can be trivially clustered or embedded in a lowdimensional space. We present a theorem that explains, for broad classes of affinity matrices and eigenbases, why this works: For successive ..."
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Cited by 66 (0 self)
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Spectral methods use selected eigenvectors of a data affinity matrix to obtain a data representation that can be trivially clustered or embedded in a lowdimensional space. We present a theorem that explains, for broad classes of affinity matrices and eigenbases, why this works: For successively smaller eigenbases (i.e., using fewer and fewer of the affinity matrix's dominant eigenvalues and eigenvectors), the angles between "similar" vectors in the new representation shrink while the angles between "dissimilar" vectors grow. Specifically, the sum of the squared cosines of the angles is strictly increasing as the dimensionality of the representation decreases. Thus spectral methods work because the truncated eigenbasis amplifies structure in the data so that any heuristic postprocessing is more likely to succeed. We use this result to construct a nonlinear dimensionality reduction (NLDR) algorithm for data sampled from manifolds whose intrinsic coordinate system has linear and cyclic axes, and a novel clusteringbyprojections algorithm that requires no postprocessing and gives superior performance on "challenge problems" from the recent literature.
Shape modeling and analysis with entropybased particle systems
 In Proceedings of the 20th International Conference on Information Processing in Medical Imaging
, 2007
"... 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. I ..."
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Cited by 27 (14 self)
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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 threedimensional images and surface descriptions. While computationallyderived 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 stateoftheart by overcoming many of the limitations of existing methods. The framework uses a particlesystem representation of shape, with a fast correspondencepoint 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
Intrinsic shape analysis: Geodesic PCA for Riemannian manifolds modulo isometric lie group actions
 Statistica Sinica
, 2010
"... Abstract: A general framework is laid out for principal component analysis (PCA) on quotient spaces that result from an isometric Lie group action on a complete Riemannian manifold. If the quotient is a manifold, geodesics on the quotient can be lifted to horizontal geodesics on the original manifol ..."
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Cited by 27 (2 self)
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Abstract: A general framework is laid out for principal component analysis (PCA) on quotient spaces that result from an isometric Lie group action on a complete Riemannian manifold. If the quotient is a manifold, geodesics on the quotient can be lifted to horizontal geodesics on the original manifold. Thus, PCA on a manifold quotient can be pulled back to the original manifold. In general, however, the quotient space may no longer carry a manifold structure. Still, horizontal geodesics can be welldefined in the general case. This allows for the concept of generalized geodesics and orthogonal projection on the quotient space as the key ingredients for PCA. Generalizing a result of Bhattacharya and Patrangenaru (2003), geodesic scores can be defined outside a null set. Building on that, an algorithmic method to perform PCA on quotient spaces based on generalized geodesics is developed. As a typical example where nonmanifold quotients appear, this framework is applied to Kendall’s shape spaces. In fact, this work has been motivated by an application occurring in forest biometry where the current method of Euclidean linear approximation is unsuitable for performing PCA. This is illustrated by a data example of
"Shape Activity": A Continuous State HMM for Moving/Deforming Shapes with Application to Abnormal Activity Detection
"... The aim is to model "activity" performed by a group of moving and interacting objects (which can be people or cars or different rigid components of the human body) and use the models for abnormal activity detection. Previous approaches to modeling group activity include cooccurrence stati ..."
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Cited by 25 (11 self)
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The aim is to model "activity" performed by a group of moving and interacting objects (which can be people or cars or different rigid components of the human body) and use the models for abnormal activity detection. Previous approaches to modeling group activity include cooccurrence statistics (individual and joint histograms) and Dynamic Bayesian Networks, neither of which is applicable when the number of interacting objects is large. We treat the objects as point objects (referred to as "landmarks") and propose to model their changing configuration as a moving and deforming "shape" (using Kendall's shape theory for discrete landmarks). A continuous state Hidden Markov Model (HMM) is defined for landmark shape dynamics in an activity. The configuration of landmarks at a given time forms the observation vector and the corresponding shape and the scaled Euclidean motion parameters form the hidden state vector. An abnormal activity is then defined as a change in the shape activity model, which could be slow or drastic and whose parameters are unknown. Results are shown on a real abnormal activity detection problem involving multiple moving objects.
Shape Activity”: a continuousstate HMM for moving/deforming shapes with application to abnormal activity detection
 IEEE Transactions on Image Processing
, 2005
"... Abstract—The aim is to model “activity ” performed by a group of moving and interacting objects (which can be people, cars, or different rigid components of the human body) and use the models for abnormal activity detection. Previous approaches to modeling group activity include cooccurrence statis ..."
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Cited by 23 (2 self)
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Abstract—The aim is to model “activity ” performed by a group of moving and interacting objects (which can be people, cars, or different rigid components of the human body) and use the models for abnormal activity detection. Previous approaches to modeling group activity include cooccurrence statistics (individual and joint histograms) and dynamic Bayesian networks, neither of which is applicable when the number of interacting objects is large. We treat the objects as point objects (referred to as “landmarks”) and propose to model their changing configuration as a moving and deforming “shape ” (using Kendall’s shape theory for discrete landmarks). A continuousstate hidden Markov model is defined for landmark shape dynamics in an activity. The configuration of landmarks at a given time forms the observation vector, and the corresponding shape and the scaled Euclidean motion parameters form the hiddenstate vector. An abnormal activity is then defined as a change in the shape activity model, which could be slow or drastic and whose parameters are unknown. Results are shown on a real abnormal activitydetection problem involving multiple moving objects. Index Terms—Abnormal acitivity detection, activity recognition, hidden Markov model (HMM), landmark shape dynamics, particle filtering, shape activity. I.
H.: Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces
 Advances in Applied Probability 38
, 2006
"... Classical principal component analysis on manifolds, e.g. on Kendall’s shape spaces, is carried out in the tangent space of a Euclidean mean equipped with a Euclidean metric. We propose a method of principal component analysis for Riemannian manifolds based on geodesics of the intrinsic metric and p ..."
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Cited by 19 (7 self)
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Classical principal component analysis on manifolds, e.g. on Kendall’s shape spaces, is carried out in the tangent space of a Euclidean mean equipped with a Euclidean metric. We propose a method of principal component analysis for Riemannian manifolds based on geodesics of the intrinsic metric and provide for a numerical implementation in case of spheres. This method allows e.g. to compare principal component geodesics of different data samples. In order to determine principal component geodesics, we show that in general, due to curvature, the principal component geodesics do not pass through the intrinsic mean. As a consequence other means, different from the intrinsic mean, enter the setting allowing for several choices of a definition for geodesic variance. In conclusion we apply our method to the space of planar triangular shapes and compare our findings with standard Euclidean principal component analysis.
1 Rotation Invariant Kernels and Their Application to Shape Analysis
"... Shape analysis requires invariance under translation, scale and rotation. Translation and scale invariance can be realized by normalizing shape vectors with respect to their mean and norm. This maps the shape feature vectors onto the surface of a hypersphere. After normalization, the shape vectors c ..."
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Cited by 16 (4 self)
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Shape analysis requires invariance under translation, scale and rotation. Translation and scale invariance can be realized by normalizing shape vectors with respect to their mean and norm. This maps the shape feature vectors onto the surface of a hypersphere. After normalization, the shape vectors can be made rotational invariant by modelling the resulting data using complex scalar rotation invariant distributions defined on the complex hypersphere, e.g., using the complex Bingham distribution. However, the use of these distributions is hampered by the difficulty in estimating their parameters and the nonlinear nature of their formulation. In the present paper, we show how a set of kernel functions, that we refer to as rotation invariant kernels, can be used to convert the original nonlinear problem into a linear one. As their name implies, these kernels are defined to provide the much needed rotation invariance property allowing one to bypass the difficulty of working with complex spherical distributions. The resulting approach provides an easy, fast mechanism for 2D & 3D shape analysis. Extensive validation using a variety of shape modelling and classification problems demonstrates the accuracy of this proposed approach.