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60
Unbiased diffeomorphic atlas construction for computational anatomy
 NEUROIMAGE
, 2004
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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.
Covariance tracking using model update based on lie algebra
 in IEEE Conference on Computer Vision and Pattern Recognition
, 2006
"... We propose a simple and elegant algorithm to track nonrigid objects using a covariance based object description and a Lie algebra based update mechanism. We represent an object window as the covariance matrix of features, therefore we manage to capture the spatial and statistical properties as well ..."
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Cited by 127 (8 self)
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We propose a simple and elegant algorithm to track nonrigid objects using a covariance based object description and a Lie algebra based update mechanism. We represent an object window as the covariance matrix of features, therefore we manage to capture the spatial and statistical properties as well as their correlation within the same representation. The covariance matrix enables efficient fusion of different types of features and modalities, and its dimensionality is small. We incorporated a model update algorithm using the Lie group structure of the positive definite matrices. The update mechanism effectively adapts to the undergoing object deformations and appearance changes. The covariance tracking method does not make any assumption on the measurement noise and the motion of the tracked objects, and provides the global optimal solution. We show that it is capable of accurately detecting the nonrigid, moving objects in nonstationary camera sequences while achieving a promising detection rate of 97.4 percent.
Principal geodesic analysis on symmetric spaces: Statistics of diffusion tensors
 IN PROC. COMPUTER VISION APPROACHES TO MEDICAL IMAGE ANALYSIS
, 2004
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Statistics on diffeomorphisms via tangent space representations
 NeuroImage
, 2004
"... In this paper, we present a linear setting for statistical analysis of shape and an optimization approach based on a recent derivation of a conservation of momentum law for the geodesics of diffeomorphic flow. Once a template is fixed, the space of initial momentum becomes an appropriate space for s ..."
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Cited by 87 (11 self)
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In this paper, we present a linear setting for statistical analysis of shape and an optimization approach based on a recent derivation of a conservation of momentum law for the geodesics of diffeomorphic flow. Once a template is fixed, the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point along the geodesic is completely determined by the momentum at the origin through geodesic shooting equations. The space of initial momentum provides a linear representation of the nonlinear diffeomorphic shape space in which linear statistical analysis can be applied. Specializing to the landmark matching problem of Computational Anatomy, we derive an algorithm for solving the variational problem with respect to the initial momentum and demonstrate principal component analysis (PCA) in this setting with threedimensional face and hippocampus databases.
Riemannian Geometry for the Statistical Analysis of Diffusion Tensor Data
 Signal Processing
, 2007
"... The tensors produced by diffusion tensor magnetic resonance imaging (DTMRI) represent the covariance in a Brownian motion model of water diffusion. Under this physical interpretation, diffusion tensors are required to be symmetric, positivedefinite. However, current approaches to statistical analy ..."
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Cited by 81 (1 self)
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The tensors produced by diffusion tensor magnetic resonance imaging (DTMRI) represent the covariance in a Brownian motion model of water diffusion. Under this physical interpretation, diffusion tensors are required to be symmetric, positivedefinite. However, current approaches to statistical analysis of diffusion tensor data, which treat the tensors as linear entities, do not take this positivedefinite constraint into account. This difficulty is due to the fact that the space of diffusion tensors does not form a vector space. In this paper we show that the space of diffusion tensors is a type of curved manifold known as a Riemannian symmetric space. We then develop methods for producing statistics, namely averages and modes of variation, in this space. We show that these statistics preserve natural geometric properties of the tensors, including the constraint that their eigenvalues be positive. The symmetric space formulation also leads to a natural definition for interpolation of diffusion tensors and a new measure of anisotropy. We expect that these methods will be useful in the registration of diffusion tensor images, the production of statistical atlases from diffusion tensor data, and the quantification of the anatomical variability caused by disease. The framework presented in this paper should also be useful in other applications where symmetric, positivedefinite tensors arise, such as mechanics and computer vision. 1
2dshape analysis using conformal mapping
 Proc. IEEE Conf. Computer Vision and Pattern Recognition
, 2004
"... The study of 2D shapes and their similarities is a central problem in the field of vision. It arises in particular from the task of classifying and recognizing objects from their observed silhouette. Defining natural distances between 2D shapes creates a metric space of shapes, whose mathematical st ..."
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Cited by 80 (8 self)
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The study of 2D shapes and their similarities is a central problem in the field of vision. It arises in particular from the task of classifying and recognizing objects from their observed silhouette. Defining natural distances between 2D shapes creates a metric space of shapes, whose mathematical structure is inherently relevant to the classification task. One intriguing metric space comes from using conformal mappings of 2D shapes into each other, via the theory of Teichmüller spaces. In this space every simple closed curve in the plane (a “shape”) is represented by a ‘fingerprint ’ which is a diffeomorphism of the unit circle to itself (a differentiable and invertible, periodic function). More precisely, every shape defines to a unique equivalence class of such diffeomorphisms up to right multiplication by aMöbius map. The fingerprint does not change if the shape is varied by translations and scaling and any such equivalence class comes from some shape. This coset space, equipped with the infinitesimal WeilPetersson (WP) Riemannian norm is a metric space. In this space, it appears very likely to be true that the shortest path between each two shapes is unique, and is given by a geodesic connecting them. Their distance from each other is given by integrating the WPnorm along that geodesic. In this paper we concentrate on solving the “welding ” problem of “sewing” together conformally the interior and exterior of the unit circle, glued on the unit circle by a given diffeomorphism, to obtain the unique 2D shape associated with this diffeomorphism. This will allow us to go back and forth between 2D shapes and their representing diffeomorphisms in this “space of shapes”. 1
Simultaneous multiple 3D motion estimation via mode finding on Lie groups
 In Proc. 10th intl. conf. on computer vision
, 2005
"... We propose a new method to estimate multiple rigid motions from noisy 3D point correspondences in the presence of outliers. The method does not require prior specification of number of motion groups and estimates all the motion parameters simultaneously. We start with generating samples from the rig ..."
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Cited by 39 (11 self)
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We propose a new method to estimate multiple rigid motions from noisy 3D point correspondences in the presence of outliers. The method does not require prior specification of number of motion groups and estimates all the motion parameters simultaneously. We start with generating samples from the rigid motion distribution. The motion parameters are then estimated via mode finding operations on the sampled distribution. Since rigid motions do not lie on a vector space, classical statistical methods can not be used for mode finding. We develop a mean shift algorithm which estimates modes of the sampled distribution using the Lie group structure of the rigid motions. We also show that proposed mean shift algorithm is general and can be applied to any distribution having a matrix Lie group structure. Experimental results on synthetic and real image data demonstrate the superior performance of the algorithm. 1.
Nonlinear Mean Shift over Riemannian Manifolds
, 2009
"... The original mean shift algorithm is widely applied for nonparametric clustering in vector spaces. In this paper we generalize it to data points lying on Riemannian manifolds. This allows us to extend mean shift based clustering and filtering techniques to a large class of frequently occurring non ..."
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Cited by 37 (1 self)
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The original mean shift algorithm is widely applied for nonparametric clustering in vector spaces. In this paper we generalize it to data points lying on Riemannian manifolds. This allows us to extend mean shift based clustering and filtering techniques to a large class of frequently occurring nonvector spaces in vision. We present an exact algorithm and prove its convergence properties as opposed to previous work which approximates the mean shift vector. The computational details of our algorithm are presented for frequently occurring classes of manifolds such as matrix Lie groups, Grassmann manifolds, essential matrices and symmetric positive definite matrices. Applications of the mean shift over these manifolds are shown.
The geometric median on Riemannian manifolds with application to robust atlas estimation
 NEUROIMAGE 45 (2009) S143–S152
, 2009
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