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36
A Riemannian Framework for Tensor Computing
 INTERNATIONAL JOURNAL OF COMPUTER VISION
, 2006
"... Positive definite symmetric matrices (socalled tensors in this article) are nowadays a common source of geometric information. In this paper, we propose to provide the tensor space with an affineinvariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of ..."
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Cited by 282 (27 self)
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Positive definite symmetric matrices (socalled tensors in this article) are nowadays a common source of geometric information. In this paper, we propose to provide the tensor space with an affineinvariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular manifold of constant curvature without boundaries (null eigenvalues are at the infinity), the geodesic between two tensors and the mean of a set of tensors are uniquely defined, etc. We have
Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements
, 1999
"... Measurements of geometric primitives, such as rotations or rigid transformations, are often noisy and we need to use statistics either to reduce the uncertainty or to compare measurements. Unfortunately, geometric primitives often belong to manifolds and not vector spaces. We have already shown [9] ..."
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Cited by 203 (24 self)
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Measurements of geometric primitives, such as rotations or rigid transformations, are often noisy and we need to use statistics either to reduce the uncertainty or to compare measurements. Unfortunately, geometric primitives often belong to manifolds and not vector spaces. We have already shown [9] that generalizing too quickly even simple statistical notions could lead to paradoxes. In this article, we develop some basic probabilistic tools to work on Riemannian manifolds: the notion of mean value, covariance matrix, normal law, Mahalanobis distance and χ² test. We also present an efficient algorithm to compute the mean value and tractable approximations of the normal and χ² laws for small variances.
Group Actions, Homeomorphisms, and Matching: A General Framework
, 2001
"... This paper constructs metrics on the space of images I defined as orbits under group actions G. The groups studied include the finite dimensional matrix groups and their products, as well as the infinite dimensional diffeomorphisms examined in Trouvé (1999, Quaterly of Applied Math.) and Dupuis et a ..."
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Cited by 133 (7 self)
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This paper constructs metrics on the space of images I defined as orbits under group actions G. The groups studied include the finite dimensional matrix groups and their products, as well as the infinite dimensional diffeomorphisms examined in Trouvé (1999, Quaterly of Applied Math.) and Dupuis et al. (1998). Quaterly of Applied Math.). Leftinvariant metrics are defined on the product G × I thus allowing the generation of transformations of the background geometry as well as the image values. Examples of the application of such metrics are presented for rigid object matching with and without signature variation, curves and volume matching, and structural generation in which image values are changed supporting notions such as tissue creation in carrying one image to another.
Web Data Mining
, 2006
"... Vol. 53: Soft Computing Approach to Pattern Recognition and Image Processing ..."
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Cited by 49 (3 self)
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Vol. 53: Soft Computing Approach to Pattern Recognition and Image Processing
LandmarkBased Registration Using Features Identified Through Differential Geometry
 HANDBOOK OF MEDICAL IMAGING PROCESSING AND ANALYSIS. I. BANKMAN EDITOR. ACADEMIC PRESS. 2000.
, 2000
"... Registration of 3D medical images consists in computing the “best” transformation between two acquisitions, or equivalently, determines the point to point correspondence between the images. Registration algorithms are usually based either on features extracted from the image (featurebased approache ..."
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Cited by 47 (8 self)
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Registration of 3D medical images consists in computing the “best” transformation between two acquisitions, or equivalently, determines the point to point correspondence between the images. Registration algorithms are usually based either on features extracted from the image (featurebased approaches) or on the optimization of a similarity measure of the images intensities (intensitybased or iconic approaches). Another classification criterion is the type of transformation sought (e.g. rigid or nonrigid). In this chapter, we concentrate on featurebased approaches for rigid registration, similar approaches for nonrigid registration being reported in another set of publication [35, 36]. We show how to reduce the dimension of the registration problem by first extracting a surface from the 3D image, then landmark curves on this surface and possibly landmark points on these curves. This concept proved its efficiency through many applications in medical image analysis as we will see in the sequel. This work has been for a long time a central investigation topic of the Epidaure team [2] and we can only reflect here on a small part of the research done in this area. We present in the first section the notions of crest lines and extremal points and how these differential geometry features can be extracted from 3D images. In Section 2, we focus on the different rigid registration algorithms that we used to register such features. The last section analyzes the possible errors in this registration scheme and demonstrates that a very accurate registration could be achieved.
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 38 (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.
Spatial Normalization and Averaging of Diffusion Tensor MRI Data Sets
, 2002
"... Diffusion tensor magnetic resonance imaging (DTMRI) is unique in providing information about both the structural integrity and the orientation of white matter fibers in vivo and, through “tractography”, revealing the trajectories of white matter tracts. DTMRI is therefore a promising technique for ..."
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Cited by 36 (1 self)
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Diffusion tensor magnetic resonance imaging (DTMRI) is unique in providing information about both the structural integrity and the orientation of white matter fibers in vivo and, through “tractography”, revealing the trajectories of white matter tracts. DTMRI is therefore a promising technique for detecting differences in white matter architecture between different subject populations. However, while studies involving analyses of group averages of scalar quantities derived from DTMRI data have been performed, as yet there have been no similar studies involving the whole tensor. Here we present the first step towards realizing such a study, i.e., the spatial normalization of whole tensor data sets. The approach is illustrated by spatial normalization of 10 DTMRI data sets to a standard
Channel smoothing: Efficient robust smoothing of lowlevel signal features
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2006
"... In this paper, we present a new and efficient method to implement robust smoothing of lowlevel signal features: Bspline channel smoothing. This method consists of three steps: encoding of the signal features into channels, averaging of the channels, and decoding of the channels. We show that line ..."
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Cited by 36 (22 self)
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In this paper, we present a new and efficient method to implement robust smoothing of lowlevel signal features: Bspline channel smoothing. This method consists of three steps: encoding of the signal features into channels, averaging of the channels, and decoding of the channels. We show that linear smoothing of channels is equivalent to robust smoothing of the signal features if we make use of quadratic Bsplines to generate the channels. The linear decoding from Bspline channels allows the derivation of a robust error norm, which is very similar to Tukey’s biweight error norm. We compare channel smoothing with three other robust smoothing techniques: nonlinear diffusion, bilateral filtering, and meanshift filtering, both theoretically and on a 2D orientationdata smoothing task. Channel smoothing is found to be superior in four respects: It has a lower computational complexity, it is easy to implement, it chooses the global minimum error instead of the nearest local minimum, and it can also be used on nonlinear spaces, such as orientation space.
FeatureBased Registration of Medical Images: Estimation and Validation of the Pose Accuracy
, 1998
"... We provide in this article a generic framework for pose estimation from geometric features. We propose more particularly two algorithms: a gradient descent on the Riemannian least squares distance and on the Mahalanobis distance. For each method, we provide a way to compute the uncertainty of the re ..."
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Cited by 23 (19 self)
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We provide in this article a generic framework for pose estimation from geometric features. We propose more particularly two algorithms: a gradient descent on the Riemannian least squares distance and on the Mahalanobis distance. For each method, we provide a way to compute the uncertainty of the resulting transformation. The analysis and comparison of the algorithms show their advantages and drawbacks and point out the very good prediction on the transformation accuracy. An application in medical image analysis validates the uncertainty estimation on real data and demonstrates that, using adapted and rigorous tools, we can detect very small modiøcations in medical images. We believe that these algorithms could be easily embedded in many applications and provide a thorough basis for computing many image statistics.
Toward a Generic Framework for Recognition Based on Uncertain Geometric Features
 Journal of Computer Vision Research
, 1998
"... this paper is to ..."
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