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123
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...
Principal Geodesic Analysis for the Study of Nonlinear Statistics of Shape
 TO APPEAR IEEE TRANSACTIONS ON MEDICAL IMAGING
, 2004
"... 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 ..."
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Cited by 181 (33 self)
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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 representations of shape. We have been developing representations of geometry based on the medial axis description or mrep. While the medial representation provides a rich language for variability in terms of bending, twisting, and widening, the medial parameters are not elements of a Euclidean vector space. They are in fact elements of a nonlinear Riemannian symmetric space. In this paper we develop the method of principal geodesic analysis, a generalization of principal component analysis to the manifold setting. We demonstrate its use in describing the variability of mediallydefined anatomical objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented.
Analysis of Planar Shapes Using Geodesic Paths on Shape Spaces
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2004
"... For analyzing shapes of planar, closed curves, we propose di#erential geometric representations of curves using their direction functions and curvature functions. Shapes are represented as elements of infinitedimensional spaces and their pairwise di#erences are quantified using the lengths of ge ..."
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Cited by 170 (37 self)
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For analyzing shapes of planar, closed curves, we propose di#erential geometric representations of curves using their direction functions and curvature functions. Shapes are represented as elements of infinitedimensional spaces and their pairwise di#erences are quantified using the lengths of geodesics connecting them on these spaces. We use a Fourier basis to represent tangents to the shape spaces and then use a gradientbased shooting method to solve for the tangent that connects any two shapes via a geodesic.
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.
Matching shape sequences in video with applications in human movement analysis
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2005
"... Abstract—We present an approach for comparing two sequences of deforming shapes using both parametric models and nonparametric methods. In our approach, Kendall’s definition of shape is used for feature extraction. Since the shape feature rests on a nonEuclidean manifold, we propose parametric mode ..."
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Cited by 99 (26 self)
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Abstract—We present an approach for comparing two sequences of deforming shapes using both parametric models and nonparametric methods. In our approach, Kendall’s definition of shape is used for feature extraction. Since the shape feature rests on a nonEuclidean manifold, we propose parametric models like the autoregressive model and autoregressive moving average model on the tangent space and demonstrate the ability of these models to capture the nature of shape deformations using experiments on gaitbased human recognition. The nonparametric model is based on Dynamic TimeWarping. We suggest a modification of the Dynamic timewarping algorithm to include the nature of the nonEuclidean space in which the shape deformations take place. We also show the efficacy of this algorithm by its application to gaitbased human recognition. We exploit the shape deformations of a person’s silhouette as a discriminating feature and provide recognition results using the nonparametric model. Our analysis leads to some interesting observations on the role of shape and kinematics in automated gaitbased person authentication. Index Terms—Shape, shape sequences, shape dynamics, comparison of shape sequences, gait recognition. 1
Representation and Detection of Deformable Shapes
 PAMI
, 2004
"... We describe some techniques that can be used to represent and detect deformable shapes in images. The main di#culty with deformable template models is the very large or infinite number of possible nonrigid transformations of the templates. This makes the problem of finding an optimal match of a ..."
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Cited by 95 (3 self)
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We describe some techniques that can be used to represent and detect deformable shapes in images. The main di#culty with deformable template models is the very large or infinite number of possible nonrigid transformations of the templates. This makes the problem of finding an optimal match of a deformable template to an image incredibly hard. Using a new representation for deformable shapes we show how to e#ciently find a global optimal solution to the nonrigid matching problem. The representation is based on the description of objects using triangulated polygons. Our matching algorithm can minimize a large class of energy functions, making it applicable to a wide range of problems. We present experimental results of detecting shapes in medical images and images of natural scenes. Our method does not depend on initialization and is very robust, yielding good matches even in images with high clutter.
Approximations of shape metrics and application to shape warping and empirical shape statistics
 FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
, 2004
"... This paper proposes a framework for dealing with several problems related to the analysis of shapes. Two related such problems are the definition of the relevant set of shapes and that of defining a metric on it. Following a recent research monograph by Delfour and Zolésio [11], we consider the cha ..."
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Cited by 91 (19 self)
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This paper proposes a framework for dealing with several problems related to the analysis of shapes. Two related such problems are the definition of the relevant set of shapes and that of defining a metric on it. Following a recent research monograph by Delfour and Zolésio [11], we consider the characteristic functions of the subsets of R 2 and their distance functions. The L² norm of the difference of characteristic functions, the L∞ and the W 1,2 norms of the difference of distance functions define interesting topologies, in particular the wellknown Hausdorff distance. Because of practical considerations arising from the fact that we deal with
Face Localization via Shape Statistics
, 1995
"... In this paper, a face localization system is proposed in which local detectors are coupled with a statistical model of the spatial arrangement of facial features to yield robust performance. The outputs from the local detectors are treated as candidate locations and constellations are formed from th ..."
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Cited by 88 (8 self)
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In this paper, a face localization system is proposed in which local detectors are coupled with a statistical model of the spatial arrangement of facial features to yield robust performance. The outputs from the local detectors are treated as candidate locations and constellations are formed from these. The effects of translation, rotation, and scale are eliminated by mapping to a set of shape variables. The constellations are then ranked according to the likelihood that the shape variables correspond to a face versus an alternative model. Incomplete constellations, which occur when some of the true features are missed, are handled in a principled way. 1 Introduction The problem of face recognition has received considerable attention in the literature [11, 24, 21, 4, 19, 17, 22, 10]; however, in most of these studies, the faces were either embedded in a benign background or were assumed to have been presegmented. For any of these recognition algorithms to work in realworld applicati...
Recognition of Planar Object Classes
 In Proc. IEEE Comput. Soc. Conf. Comput. Vision and Pattern Recogn
, 1996
"... We present a new framework for recognizing planar object classes, which is based on local feature detectors and a probabilistic model of the spatial arrangement of the features. The allowed object deformations are represented through shape statistics, which are learned from examples. Instances of an ..."
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Cited by 80 (12 self)
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We present a new framework for recognizing planar object classes, which is based on local feature detectors and a probabilistic model of the spatial arrangement of the features. The allowed object deformations are represented through shape statistics, which are learned from examples. Instances of an object in an image are detected by finding the appropriate features in the correct spatial configuration. The algorithm is robust with respect to partial occlusion, detector false alarms, and missed features. A 94% success rate was achieved for the problem of locating quasifrontal views of faces in cluttered scenes. 1 Introduction Many early pattern recognition algorithms were based on template matching [13], which is optimal for detecting a known signal in white noise. However, since the underlying assumption that "the signal is known exactly" rarely holds true, considerable effort has been devoted to extending this method to handle variability in the target signal. For example, approach...
Embedding Gestalt Laws in Markov Random Fields  a theory for shape modeling and perceptual organization
, 1999
"... The goal of this paper is to study a mathematical framework of 2D object shape modeling and learning for middle level vision problems, such as image segmentation and perceptual organization. For this purpose, we pursue generic shape models which characterize the most common features of 2D object sha ..."
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Cited by 77 (10 self)
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The goal of this paper is to study a mathematical framework of 2D object shape modeling and learning for middle level vision problems, such as image segmentation and perceptual organization. For this purpose, we pursue generic shape models which characterize the most common features of 2D object shapes. In this paper, shape models are learned from observed natural shapes based on a minimax entropy learning theory (Zhu and Mumford 1997, Zhu, Wu and Mumford 1997)[31, 32]. The learned shape models are Gibbs distributions dened on Markov random elds (MRFs). The neighborhood structures of these MRFs correspond to Gestalt laws {colinearity, cocircularity, proximity, parallelism, and symmetry. Thus both contourbased and regionbased features are accounted for. Stochastic Markov chain Monte Carlo (MCMC) algorithms are proposed for learning and model verication. Furthermore, this paper provides a quantitative measure for the socalled nonaccidental statistics, and thus justies some empi...