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Object oriented data analysis: Sets of trees
 The Annals of Statistics
"... Object Oriented Data Analysis is the statistical analysis of populations of complex objects. In the special case of Functional Data Analysis, these data objects are curves, where standard Euclidean approaches, such as principal components analysis, have been very successful. Recent developments in m ..."
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Cited by 29 (8 self)
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Object Oriented Data Analysis is the statistical analysis of populations of complex objects. In the special case of Functional Data Analysis, these data objects are curves, where standard Euclidean approaches, such as principal components analysis, have been very successful. Recent developments in medical image analysis motivate the statistical analysis of populations of more complex data objects which are elements of mildly nonEuclidean spaces, such as Lie Groups and Symmetric Spaces, or of strongly nonEuclidean spaces, such as spaces of treestructured data objects. These new contexts for Object Oriented Data Analysis create several potentially large new interfaces between mathematics and statistics. This point is illustrated through the careful development of a novel mathematical framework for statistical analysis of populations of tree structured objects. 1. Introduction Object Oriented Data Analysis (OODA) is the statistical analysis of data sets of complex objects. The area is understood through consideration
Multiscale Medial ShapeBased Analysis of Image Objects
, 2003
"... Medial representation of a threedimensional (3D) object or an ensemble of 3D objects involves capturing the object interior as a locus of medial atoms, each atom being two vectors of equal length joined at the tail at the medial point. Medial representation has a variety of beneficial properties, ..."
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Cited by 27 (1 self)
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Medial representation of a threedimensional (3D) object or an ensemble of 3D objects involves capturing the object interior as a locus of medial atoms, each atom being two vectors of equal length joined at the tail at the medial point. Medial representation has a variety of beneficial properties, among the most important of which are 1) its inherent geometry, provides an objectintrinsic coordinate system and thus provides correspondence between instances of the object in and near the object(s); 2) it captures the object interior and is, thus, very suitable for deformation; and 3) it provides the basis for an intuitive objectbased multiscale sequence leading to efficiency of segmentation algorithms and trainability of statistical characterizations with limited training sets. As a result of these properties, medial representation is particularly suitable for the following image analysis tasks; how each operates will be described and will be illustrated by results:
Multiple Rotation Averaging
"... • Single Rotation Averaging: Several estimates are obtained of a single rotation, which are then averaged to give the best estimate. • Multiple Rotation Averaging: Relative rotations Rij are given, and absolute ..."
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Cited by 17 (4 self)
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• Single Rotation Averaging: Several estimates are obtained of a single rotation, which are then averaged to give the best estimate. • Multiple Rotation Averaging: Relative rotations Rij are given, and absolute
Motion Compression using Principal Geodesics Analysis
 EUROGRAPHICS
, 2009
"... Due to the growing need for large quantities of human animation data in the entertainment industry, it has become a necessity to compress motion capture sequences in order to ease their storage and transmission. We present a novel, lossy compression method for human motion data that exploits both te ..."
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Cited by 16 (2 self)
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Due to the growing need for large quantities of human animation data in the entertainment industry, it has become a necessity to compress motion capture sequences in order to ease their storage and transmission. We present a novel, lossy compression method for human motion data that exploits both temporal and spatial coherence. Given one motion, we first approximate the poses manifold using Principal Geodesics Analysis (PGA) in the configuration space of the skeleton. We then search this approximate manifold for poses matching endeffectors constraints using an iterative minimization algorithm that allows for realtime, datadriven inverse kinematics. The compression is achieved by only storing the approximate manifold parametrization along with the endeffectors and root joint trajectories, also compressed, in the output data. We recover poses using the IK algorithm given the endeffectors trajectories. Our experimental results show that considerable compression rates can be obtained using our method, with few reconstruction and perceptual errors.
Robust statistics on Riemannian manifolds via the geometric median
 In IEEE Conference on Computer Vision and Pattern Recognition (CVPR
, 2008
"... The geometric median is a classic robust estimator of centrality for data in Euclidean spaces. In this paper we formulate the geometric median of data on a Riemannian manifold as the minimizer of the sum of geodesic distances to the data points. We prove existence and uniqueness of the geometric med ..."
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Cited by 16 (0 self)
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The geometric median is a classic robust estimator of centrality for data in Euclidean spaces. In this paper we formulate the geometric median of data on a Riemannian manifold as the minimizer of the sum of geodesic distances to the data points. We prove existence and uniqueness of the geometric median on manifolds with nonpositive sectional curvature and give sufficient conditions for uniqueness on positively curved manifolds. Generalizing the Weiszfeld procedure for finding the geometric median of Euclidean data, we present an algorithm for computing the geometric median on an arbitrary manifold. We show that this algorithm converges to the unique solution when it exists. This method produces a robust central point for data lying on a manifold, and should have use in a variety of vision applications involving manifolds. We give examples of the geometric median computation and demonstrate its robustness for three types of manifold data: the 3D rotation group, tensor manifolds, and shape spaces. 1.
Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations
 European Conference on Computer Vision
, 2010
"... Abstract. Manifolds are widely used to model nonlinearity arising in a range of computer vision applications. This paper treats statistics on manifolds and the loss of accuracy occurring when linearizing the manifold prior to performing statistical operations. Using recent advances in manifold comp ..."
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Cited by 15 (5 self)
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Abstract. Manifolds are widely used to model nonlinearity arising in a range of computer vision applications. This paper treats statistics on manifolds and the loss of accuracy occurring when linearizing the manifold prior to performing statistical operations. Using recent advances in manifold computations, we present a comparison between the nonlinear analog of Principal Component Analysis, Principal Geodesic Analysis, in its linearized form and its exact counterpart that uses true intrinsic distances. We give examples of datasets for which the linearized version provides good approximations and for which it does not. Indicators for the differences between the two versions are then developed and applied to two examples of manifold valued data: outlines of vertebrae from a study of vertebral fractures and spacial coordinates of human skeleton endeffectors acquired using a stereo camera and tracking software.
How to put probabilities on homographies
 IEEE Trans. Patt. Anal. Mach. Intell
, 2005
"... Abstract—We present a family of “normal ” distributions over a matrix group together with a simple method for estimating its parameters. In particular, the mean of a set of elements can be calculated. The approach is applied to planar projective homographies, showing that using priors defined in thi ..."
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Cited by 12 (0 self)
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Abstract—We present a family of “normal ” distributions over a matrix group together with a simple method for estimating its parameters. In particular, the mean of a set of elements can be calculated. The approach is applied to planar projective homographies, showing that using priors defined in this way improves object recognition. Index Terms—Homography, lie groups, normal distribution, Bayesian statistics, geodesics.
Lie bodies: A manifold representation of 3D human shape
 in ECCV
, 2012
"... Abstract. Threedimensional object shape is commonly represented in terms of deformations of a triangular mesh from an exemplar shape. Existing models, however, are based on a Euclidean representation of shape deformations. In contrast, we argue that shape has a manifold structure: For example, sum ..."
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Cited by 11 (4 self)
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Abstract. Threedimensional object shape is commonly represented in terms of deformations of a triangular mesh from an exemplar shape. Existing models, however, are based on a Euclidean representation of shape deformations. In contrast, we argue that shape has a manifold structure: For example, summing the shape deformations for two people does not necessarily yield a deformation corresponding to a valid human shape, nor does the Euclidean difference of these two deformations provide a meaningful measure of shape dissimilarity. Consequently, we define a novel manifold for shape representation, with emphasis on body shapes, using a new Lie group of deformations. This has several advantages. First we define triangle deformations exactly, removing nonphysical deformations and redundant degrees of freedom common to previous methods. Second, the Riemannian structure of Lie Bodies enables a more meaningful definition of body shape similarity by measuring distance between bodies on the manifold of body shape deformations. Third, the group structure allows the valid composition of deformations. This is important for models that factor body shape deformations into multiple causes or represent shape as a linear combination of basis shapes. Finally, body shape variation is modeled using statistics on manifolds. Instead of modeling Euclidean shape variation with Principal Component Analysis we capture shape variation on the manifold using Principal Geodesic Analysis. Our experiments show consistent visual and quantitative advantages of Lie Bodies over traditional Euclidean models of shape deformation and our representation can be easily incorporated into existing methods.