共查询到20条相似文献,搜索用时 15 毫秒
1.
Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements 总被引:2,自引:0,他引:2
Xavier Pennec 《Journal of Mathematical Imaging and Vision》2006,25(1):127-154
In medical image analysis and high level computer vision, there is an intensive use of geometric features like orientations,
lines, and geometric transformations ranging from simple ones (orientations, lines, rigid body or affine transformations,
etc.) to very complex ones like curves, surfaces, or general diffeomorphic transformations. The measurement of such geometric
primitives is generally noisy in real applications and we need to use statistics either to reduce the uncertainty (estimation),
to compare observations, or to test hypotheses. Unfortunately, even simple geometric primitives often belong to manifolds
that are not vector spaces. In previous works [1, 2], we investigated invariance requirements to build some statistical tools
on transformation groups and homogeneous manifolds that avoids paradoxes. In this paper, we consider finite dimensional manifolds
with a Riemannian metric as the basic structure. Based on this metric, we develop the notions of mean value and covariance
matrix of a random element, normal law, Mahalanobis distance and χ2 law. We provide a new proof of the characterization of Riemannian centers of mass and an original gradient descent algorithm
to efficiently compute them. The notion of Normal law we propose is based on the maximization of the entropy knowing the mean
and covariance of the distribution. The resulting family of pdfs spans the whole range from uniform (on compact manifolds)
to the point mass distribution. Moreover, we were able to provide tractable approximations (with their limits) for small variances
which show that we can effectively implement and work with these definitions. 相似文献
2.
P. Thomas Fletcher 《International Journal of Computer Vision》2013,105(2):171-185
This paper develops the theory of geodesic regression and least-squares estimation on Riemannian manifolds. Geodesic regression is a method for finding the relationship between a real-valued independent variable and a manifold-valued dependent random variable, where this relationship is modeled as a geodesic curve on the manifold. Least-squares estimation is formulated intrinsically as a minimization of the sum-of-squared geodesic distances of the data to the estimated model. Geodesic regression is a direct generalization of linear regression to the manifold setting, and it provides a simple parameterization of the estimated relationship as an initial point and velocity, analogous to the intercept and slope. A nonparametric permutation test for determining the significance of the trend is also given. For the case of symmetric spaces, two main theoretical results are established. First, conditions for existence and uniqueness of the least-squares problem are provided. Second, a maximum likelihood criteria is developed for a suitable definition of Gaussian errors on the manifold. While the method can be generally applied to data on any manifold, specific examples are given for a set of synthetically generated rotation data and an application to analyzing shape changes in the corpus callosum due to age. 相似文献
3.
Semi-Supervised Learning on Riemannian Manifolds 总被引:1,自引:0,他引:1
We consider the general problem of utilizing both labeled and unlabeled data to improve classification accuracy. Under the assumption that the data lie on a submanifold in a high dimensional space, we develop an algorithmic framework to classify a partially labeled data set in a principled manner. The central idea of our approach is that classification functions are naturally defined only on the submanifold in question rather than the total ambient space. Using the Laplace-Beltrami operator one produces a basis (the Laplacian Eigenmaps) for a Hilbert space of square integrable functions on the submanifold. To recover such a basis, only unlabeled examples are required. Once such a basis is obtained, training can be performed using the labeled data set. Our algorithm models the manifold using the adjacency graph for the data and approximates the Laplace-Beltrami operator by the graph Laplacian. We provide details of the algorithm, its theoretical justification, and several practical applications for image, speech, and text classification. 相似文献
4.
Søren Hauberg François Lauze Kim Steenstrup Pedersen 《Journal of Mathematical Imaging and Vision》2013,46(1):103-120
In recent years there has been a growing interest in problems, where either the observed data or hidden state variables are confined to a known Riemannian manifold. In sequential data analysis this interest has also been growing, but rather crude algorithms have been applied: either Monte Carlo filters or brute-force discretisations. These approaches scale poorly and clearly show a missing gap: no generic analogues to Kalman filters are currently available in non-Euclidean domains. In this paper, we remedy this issue by first generalising the unscented transform and then the unscented Kalman filter to Riemannian manifolds. As the Kalman filter can be viewed as an optimisation algorithm akin to the Gauss-Newton method, our algorithm also provides a general-purpose optimisation framework on manifolds. We illustrate the suggested method on synthetic data to study robustness and convergence, on a region tracking problem using covariance features, an articulated tracking problem, a mean value optimisation and a pose optimisation problem. 相似文献
5.
Neural Processing Letters - Covariance matrices have attracted increasing attention for data representation in many computer vision tasks. The nonsingular covariance matrices are regarded as points... 相似文献
6.
Roberto Giambò Fabio Giannoni Paolo Piccione 《Mathematics of Control, Signals, and Systems (MCSS)》2004,16(4):278-296
We introduce a framework for the study of one-dimensional variational problems of arbitrary order of regularity on manifolds. As an application, we discuss the problem of higher-order interpolation in a Riemannian manifold.
Date received: January 10, 2003. Date revised: May 22, 2003. 相似文献
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Nonlinear Mean Shift over Riemannian Manifolds 总被引:1,自引:0,他引:1
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-vector 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. 相似文献
9.
The paper concerns the strong uniform consistency and the asymptotic distribution of the kernel density estimator of random
objects on a Riemannian manifolds, proposed by Pelletier (Stat. Probab. Lett., 73(3):297–304, 2005). The estimator is illustrated via one example based on a real data.
This research was partially supported by Grants X-094 from the Universidad de Buenos Aires, pid 5505 from conicet and pav 120 and pict 21407 from anpcyt, Argentina. 相似文献
10.
In this article we extend the computational geometric curve reconstruction approach to the curves embedded in the Riemannian manifold. We prove that the minimal spanning tree, given a sufficiently dense sample, correctly reconstructs the smooth arcs which can be used to reconstruct closed and simple curves in Riemannian manifolds. The proof is based on the behavior of the curve segment inside the tubular neighborhood of the curve. To take care of the local topological changes of the manifold, the tubular neighborhood is constructed in consideration with the injectivity radius of the underlying Riemannian manifold. We also present examples of successfully reconstructed curves and show applications of curve reconstruction to ordering motion frames. 相似文献
11.
We prove consistency results for two types of density estimators on a closed, connected Riemannian manifold under suitable regularity conditions. The convergence rates are consistent with those in Euclidean space as well as those obtained for a previously proposed class of kernel density estimators on closed Riemannian manifolds. The first estimator is the uniform mixture of heat kernels centered at each observation, a natural extension of the usual Gaussian estimator to Riemannian manifolds. The second is an approximate heat kernel (AHK) estimator that is motivated by more practical considerations, where observations occur on a manifold isometrically embedded in Euclidean space whose structure or heat kernel may not be completely known. We also provide some numerical evidence that the predicted convergence rate is attained for the AHK estimator. 相似文献
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A Riemannian Framework for Tensor Computing 总被引:22,自引:0,他引:22
Xavier Pennec Pierre Fillard Nicholas Ayache 《International Journal of Computer Vision》2006,66(1):41-66
Tensors are nowadays a common source of geometric information. In this paper, we propose to endow the tensor space with an
affine-invariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of positive definite
symmetric matrices is replaced by a regular and complete manifold 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 previously shown that the Riemannian metric provides a powerful framework for generalizing statistics to manifolds.
In this paper, we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms
such as interpolation, filtering, diffusion and restoration of missing data. For instance, most interpolation and Gaussian
filtering schemes can be tackled efficiently through a weighted mean computation. Linear and anisotropic diffusion schemes
can be adapted to our Riemannian framework, through partial differential evolution equations, provided that the metric of
the tensor space is taken into account. For that purpose, we provide intrinsic numerical schemes to compute the gradient and
Laplace-Beltrami operators. Finally, to enforce the fidelity to the data (either sparsely distributed tensors or complete
tensors fields) we propose least-squares criteria based on our invariant Riemannian distance which are particularly simple
and efficient to solve. 相似文献
16.
Contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another. In this work, we present the fundamental results of contraction theory in an intrinsic, coordinate-free setting, with the presentation highlighting the underlying geometric foundation of contraction theory and the resulting stability properties. We provide coordinate-free proofs of the main results for autonomous vector fields, and clarify the assumptions under which the results hold. We state and prove several interesting corollaries to the main result, study cascade and feedback interconnections of contracting systems, study some simple examples, and highlight how contraction theory has arisen independently in other scientific disciplines. We conclude by illustrating the developed theory for the case of gradient dynamics. 相似文献
17.
Many classification algorithms see a reduction in performance when tested on data with properties different from that used for training. This problem arises very naturally in face recognition where images corresponding to the source domain (gallery, training data) and the target domain (probe, testing data) are acquired under varying degree of factors such as illumination, expression, blur and alignment. In this paper, we account for the domain shift by deriving a latent subspace or domain, which jointly characterizes the multifactor variations using appropriate image formation models for each factor. We formulate the latent domain as a product of Grassmann manifolds based on the underlying geometry of the tensor space, and perform recognition across domain shift using statistics consistent with the tensor geometry. More specifically, given a face image from the source or target domain, we first synthesize multiple images of that subject under different illuminations, blur conditions and 2D perturbations to form a tensor representation of the face. The orthogonal matrices obtained from the decomposition of this tensor, where each matrix corresponds to a factor variation, are used to characterize the subject as a point on a product of Grassmann manifolds. For cases with only one image per subject in the source domain, the identity of target domain faces is estimated using the geodesic distance on product manifolds. When multiple images per subject are available, an extension of kernel discriminant analysis is developed using a novel kernel based on the projection metric on product spaces. Furthermore, a probabilistic approach to the problem of classifying image sets on product manifolds is introduced. We demonstrate the effectiveness of our approach through comprehensive evaluations on constrained and unconstrained face datasets, including still images and videos. 相似文献
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Xu Song-Gang Zhang Yun-Xiang Yong Jun-Hai 《IEEE transactions on pattern analysis and machine intelligence》2010,32(2):231-241
A wide range of applications in computer intelligence and computer graphics require computing geodesics accurately and efficiently. The fast marching method (FMM) is widely used to solve this problem, of which the complexity is O(Nlog N), where N is the total number of nodes on the manifold. A fast sweeping method (FSM) is proposed and applied on arbitrary triangular manifolds of which the complexity is reduced to O(N). By traversing the undigraph, four orderings are built to produce two groups of interfering waves, which cover all directions of characteristics. The correctness of this method is proved by analyzing the coverage of characteristics. The convergence and error estimation are also presented. 相似文献