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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
Population shape regression from random design data
 IN: PROC. OF ICCV 2007
, 2007
"... Regression analysis is a powerful tool for the study of changes in a dependent variable as a function of an independent regressor variable, and in particular it is applicable to the study of anatomical growth and shape change. When the underlying process can be modeled by parameters in a Euclidean s ..."
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Cited by 79 (15 self)
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Regression analysis is a powerful tool for the study of changes in a dependent variable as a function of an independent regressor variable, and in particular it is applicable to the study of anatomical growth and shape change. When the underlying process can be modeled by parameters in a Euclidean space, classical regression techniques [13, 34] are applicable and have been studied extensively. However, recent work suggests that attempts to describe anatomical shapes using flat Euclidean spaces undermines our ability to represent natural biological variability [9, 11]. In this paper we develop a method for regression analysis of general, manifoldvalued data. Specifically, we extend NadarayaWatson kernel regression by recasting the regression problem in terms of Fréchet expectation. Although this method is quite general, our driving problem is the study anatomical shape change as a function of age from random design image data. We demonstrate our method by analyzing shape change in the brain from a random design dataset of MR images of 89 healthy adults ranging in age from 22 to 79 years. To study the small scale changes in anatomy, we use the infinite dimensional manifold of diffeomorphic transformations, with an associated metric. We regress a representative anatomical shape, as a function of age, from this population.
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.
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
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
T.: Segmentation of High Angular Resolution Diffusion MRI Modeled as a Field of von MisesFisher Mixtures
 In: European Conference on Computer Vision (ECCV). Volume 3953
, 2006
"... Abstract. High angular resolution diffusion imaging (HARDI) permits the computation of water molecule displacement probabilities over a sphere of possible displacement directions. This probability is often referred to as the orientation distribution function (ODF). In this paper we present a novel m ..."
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Cited by 20 (0 self)
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Abstract. High angular resolution diffusion imaging (HARDI) permits the computation of water molecule displacement probabilities over a sphere of possible displacement directions. This probability is often referred to as the orientation distribution function (ODF). In this paper we present a novel model for the diffusion ODF namely, a mixture of von MisesFisher (vMF) distributions. Our model is compact in that it requires very few variables to model complicated ODF geometries which occur specifically in the presence of heterogeneous nerve fiber orientation. We also present a Riemannian geometric framework for computing intrinsic distances, in closedform, and performing interpolation between ODFs represented by vMF mixtures. As an example, we apply the intrinsic distance within a hidden Markov measure field segmentation scheme. We present results of this segmentation for HARDI images of rat spinal cords – which show distinct regions within both the white and gray matter. It should be noted that such a fine level of parcellation of the gray and white matter cannot be obtained either from contrast MRI scans or Diffusion Tensor MRI scans. We validate the segmentation algorithm by applying it to synthetic data sets where the ground truth is known. 2 1
Statistical Shape Analysis of MultiObject Complexes
"... An important goal of statistical shape analysis is the discrimination between populations of objects, exploring group differences in morphology not explained by standard volumetric analysis. Certain applications additionally require analysis of objects in their embedding context by joint statistical ..."
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Cited by 19 (4 self)
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An important goal of statistical shape analysis is the discrimination between populations of objects, exploring group differences in morphology not explained by standard volumetric analysis. Certain applications additionally require analysis of objects in their embedding context by joint statistical analysis of sets of interrelated objects. In this paper, we present a framework for discriminant analysis of populations of 3D multiobject sets. In view of the driving medical applications, a skeletal object parametrization of shape is chosen since it naturally encodes thickening, bending and twisting. In a multiobject setting, we not only consider a joint analysis of sets of shapes but also must take into account differences in pose. Statistics on features of medial descriptions and pose parameters, which include rotational frames and distances, uses a Riemannian symmetric space instead of the standard Euclidean metric. Our choice of discriminant method is the distance weighted discriminant (DWD) because of its generalization ability in high dimensional, low sample size settings. Joint analysis of 10 subcortical brain structures in a pediatric autism study demonstrates that multiobject analysis of shape results in a better group discrimination than pose, and that the combination of pose and shape performs better than shape alone. Finally, given a discriminating axis of shape and pose, we can visualize the differences between the populations. 1.
GMAT: The Groupwise Medial Axis Transform for Fuzzy Skeletonization and Intelligent Pruning
"... Abstract. There is a frequent need to compute medial shape representations of each of a group of structures, e.g. for use in a medical study of anatomical shapes. We present a novel approach to skeletonization that leverages information provided from such a group. We augment the traditional medial a ..."
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Cited by 17 (2 self)
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Abstract. There is a frequent need to compute medial shape representations of each of a group of structures, e.g. for use in a medical study of anatomical shapes. We present a novel approach to skeletonization that leverages information provided from such a group. We augment the traditional medial axis transform with an additional coordinate stored at each medial locus, indicating the confidence that the branch on which that locus lies represents signal and not noise. This confidence is calculated based on the support given to that branch by corresponding branches in other skeletons in the group. We establish the aforementioned correspondence by a set of bipartite graph matchings using the Hungarian algorithm, and compute branch support based on similarity of computed geometric and topological features at each branch. This groupwise skeletonization approach supports an intelligent pruning algorithm, which we show to operate quickly and provide pruning in an intuitive manner. We show that the method is amenable to automatic detection of skeletal configurations with one, or more than one, topological class of skeletons. This is useful to medical studies which often involve patient groups whose structures may differ topologically. 1