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276
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 198 (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.
Human Detection via Classification on Riemannian Manifolds, 2007
 In IEEE Conf. Comp. Vision and Pattern Recognition (CVPR
"... We present a new algorithm to detect humans in still images utilizing covariance matrices as object descriptors. Since these descriptors do not lie on a vector space, well known machine learning techniques are not adequate to learn the classifiers. The space of ddimensional nonsingular covariance ..."
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Cited by 169 (8 self)
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We present a new algorithm to detect humans in still images utilizing covariance matrices as object descriptors. Since these descriptors do not lie on a vector space, well known machine learning techniques are not adequate to learn the classifiers. The space of ddimensional nonsingular covariance matrices can be represented as a connected Riemannian manifold. We present a novel approach for classifying points lying on a Riemannian manifold by incorporating the a priori information about the geometry of the space. The algorithm is tested on INRIA human database where superior detection rates are observed over the previous approaches. IEEE CVPR 2007
A review of statistical approaches to level set segmentation: Integrating color, texture, motion and shape
 International Journal of Computer Vision
, 2007
"... Abstract. Since their introduction as a means of front propagation and their first application to edgebased segmentation in the early 90’s, level set methods have become increasingly popular as a general framework for image segmentation. In this paper, we present a survey of a specific class of reg ..."
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Cited by 163 (4 self)
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Abstract. Since their introduction as a means of front propagation and their first application to edgebased segmentation in the early 90’s, level set methods have become increasingly popular as a general framework for image segmentation. In this paper, we present a survey of a specific class of regionbased level set segmentation methods and clarify how they can all be derived from a common statistical framework. Regionbased segmentation schemes aim at partitioning the image domain by progressively fitting statistical models to the intensity, color, texture or motion in each of a set of regions. In contrast to edgebased schemes such as the classical Snakes, regionbased methods tend to be less sensitive to noise. For typical images, the respective cost functionals tend to have less local minima which makes them particularly wellsuited for local optimization methods such as the level set method. We detail a general statistical formulation for level set segmentation. Subsequently, we clarify how the integration of various low level criteria leads to a set of cost functionals and point out relations between the different segmentation schemes. In experimental results, we demonstrate how the level set function is driven to partition the image plane into domains of coherent color, texture, dynamic texture or motion. Moreover, the Bayesian formulation allows to introduce prior shape knowledge into the level set method. We briefly review a number of advances in this domain.
Pedestrian Detection Via Classification on Riemannian Manifolds
, 2008
"... Detecting different categories of objects in image and video content is one of the fundamental tasks in computer vision research. The success of many applications such as visual surveillance, image retrieval, robotics, autonomous vehicles, and smart cameras are conditioned on the accuracy of the det ..."
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Cited by 129 (3 self)
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Detecting different categories of objects in image and video content is one of the fundamental tasks in computer vision research. The success of many applications such as visual surveillance, image retrieval, robotics, autonomous vehicles, and smart cameras are conditioned on the accuracy of the detection process. Two main processing steps can be distinguished in a typical object detection algorithm. The first task is feature extraction, in which the most informative object descriptors regarding the detection process are obtained from the visual content. The second task is detection, in which the obtained object descriptors are utilized in a classification framework to detect the objects of interest. The feature extraction methods can be further categorized into two groups based on the representation. The first group of methods is the sparse representations, where a set of representative local regions is obtained as the result of an interest point detection algorithm. Reliable interest points should encapsulate valuable information about the local image content and remain stable under changes, such as in viewpoint and/or illumination. There exists an extensive literature on interest point detectors, and [14],[18],[21],[25], and [27] are only a few of the most commonly used methods that satisfy consistency over a large range of operating conditions.
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
Clinical DTMRI estimation, smoothing and fiber tracking with logEuclidean metrics
 in Proc. of ISBI’06
, 2006
"... Diffusion tensor MRI is an imaging modality that is gaining importance in clinical applications. However, in a clinical environment, data have to be acquired rapidly, often at the detriment of the image quality. We propose a new variational framework that specifically targets low quality DTMRI. The ..."
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Cited by 76 (13 self)
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Diffusion tensor MRI is an imaging modality that is gaining importance in clinical applications. However, in a clinical environment, data have to be acquired rapidly, often at the detriment of the image quality. We propose a new variational framework that specifically targets low quality DTMRI. The Rician nature of the noise on the images leads us to a maximum likelihood strategy to estimate the tensor field. To further reduce the noise, we optimally exploit the spatial correlation by adding to the estimation an anisotropic regularization term. This criterion is easily optimized thanks to the use of recently introduced LogEuclidean metrics. Results on real clinical data show promising improvements of fiber tracking in the brain and the spinal cord. 1.
Fiber tractoriented statistics for quantitative diffusion tensor MRI analysis
 Medical Image Analysis
, 2006
"... Quantitative diffusion tensor imaging (DTI) has become the major imaging modality to study properties of white matter and the geometry of fiber tracts of the human brain. Clinical studies mostly focus on regional statistics of fractional anisotropy (FA) and mean diffusivity derived from tensors. Exi ..."
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Cited by 63 (11 self)
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Quantitative diffusion tensor imaging (DTI) has become the major imaging modality to study properties of white matter and the geometry of fiber tracts of the human brain. Clinical studies mostly focus on regional statistics of fractional anisotropy (FA) and mean diffusivity derived from tensors. Existing analysis techniques do not sufficiently take into account that the measurements are tensors, and thus require proper interpolation and statistics of tensors, and that regions of interest are fiber tracts with complex spatial geometry. We propose a new framework for quantitative tractoriented DTI analysis that systematically includes tensor interpolation and averaging, using nonlinear Riemannian symmetric space. A new measure of tensor anisotropy, called geodesic anisotropy (GA) is applied and compared with FA. As a result, tracts of interest are represented by the geometry of the medial spine attributed with tensor statistics (average and variance) calculated within crosssections. Feasibility of our approach is demonstrated on various fiber tracts of a single data set. A validation study, based on six repeated scans of the same subject, assesses the reproducibility of this new DTI data analysis framework. Preprint submitted to Medical Image Analysis Key words: Diffusion tensor interpolation, diffusion tensor statistics, DTI analysis, fiber tract modeling. ∗ Corresponding author. Email addresses:
Fast and simple calculus on tensors in the logEuclidean framework
 Proceedings of the 8th Int. Conf. on Medical Image Computing and ComputerAssisted Intervention  MICCAI 2005, Part I, volume 3749 of LNCS
"... Abstract. Computations on tensors have become common with the use of DTMRI. But the classical Euclidean framework has many defects, and affineinvariant Riemannian metrics have been proposed to correct them. These metrics have excellent theoretical properties but lead to complex and slow algorithms ..."
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Cited by 56 (8 self)
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Abstract. Computations on tensors have become common with the use of DTMRI. But the classical Euclidean framework has many defects, and affineinvariant Riemannian metrics have been proposed to correct them. These metrics have excellent theoretical properties but lead to complex and slow algorithms. To remedy this limitation, we propose new metrics called LogEuclidean. They also have excellent theoretical properties and yield similar results in practice, but with much simpler and faster computations. Indeed, LogEuclidean computations are Euclidean computations in the domain of matrix logarithms. Theoretical aspects are presented and experimental results for multilinear interpolation and regularization of tensor fields are shown on synthetic and real DTI data. 1 Introduction: Calculus
Riemannian elasticity: A statistical regularization framework for nonlinear registration
 in Proceedings of the 8th Int. Conf. on Medical Image Computing and ComputerAssisted Intervention  MICCAI 2005, Part II
"... Abstract. In intersubject registration, one often lacks a good model of the transformation variability to choose the optimal regularization. Some works attempt to model the variability in a statistical way, but the reintroduction in a registration algorithm is not easy. In this paper, we interpret ..."
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Cited by 44 (15 self)
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Abstract. In intersubject registration, one often lacks a good model of the transformation variability to choose the optimal regularization. Some works attempt to model the variability in a statistical way, but the reintroduction in a registration algorithm is not easy. In this paper, we interpret the elastic energy as the distance of the GreenSt Venant strain tensor to the identity, which reflects the deviation of the local deformation from a rigid transformation. By changing the Euclidean metric for a more suitable Riemannian one, we define a consistent statistical framework to quantify the amount of deformation. In particular, the mean and the covariance matrix of the strain tensor can be consistently and efficiently computed from a population of nonlinear transformations. These statistics are then used as parameters in a Mahalanobis distance to measure the statistical deviation from the observed variability, giving a new regularization criterion that we called the statistical Riemannian elasticity. This new criterion is able to handle anisotropic deformations and is inverseconsistent. Preliminary results show that it can be quite easily implemented in a nonrigid registration algorithms. 1