Distributions on the Special Manifolds

Chikuse, Yasuko

doi:10.1007/978-0-387-21540-2_2

Cited by 24 publications

(39 citation statements)

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“…This is because the structural tensor (18) is positive definite, and, therefore, the pixels of J(x) defined as above are no more elements of an Euclidean space, but rather of a nonlinear manifold. Fortunately, the availability of kernel-based methods for estimating probability densities defined over nonlinear manifolds makes it possible to apply our approach in this situation, as well [41].…”

Section: B Examples Of Discriminative Featuresmentioning

confidence: 99%

“…It is interesting to note that, in this case, the velocity V(x) is given by the log-likelihood ratio (40) Thus, for example, in the case when the feature image J(x) is formed by image intensities (i.e., J(x) = I(x)), and the densities p − (z) and p + (z) are Gaussian with their mean values equal to μ − and μ + , and their variances equal to , respectively, the velocity (40) becomes (41) where the estimates (42) are supposed to be recomputed at each iteration.…”

Section: Comparative Studymentioning

confidence: 99%

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Image Segmentation Using Active Contours Driven by the Bhattacharyya Gradient Flow

Michailovich

Rathi

Tannenbaum

2007

IEEE Trans. on Image Process.

298

242

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This paper addresses the problem of image segmentation by means of active contours, whose evolution is driven by the gradient flow derived from an energy functional that is based on the Bhattacharyya distance. In particular, given the values of a photometric variable (or of a set thereof), which is to be used for classifying the image pixels, the active contours are designed to converge to the shape that results in maximal discrepancy between the empirical distributions of the photometric variable inside and outside of the contours. The above discrepancy is measured by means of the Bhattacharyya distance that proves to be an extremely useful tool for solving the problem at hand. The proposed methodology can be viewed as a generalization of the segmentation methods, in which active contours maximize the difference between a finite number of empirical moments of the "inside" and "outside" distributions. Furthermore, it is shown that the proposed methodology is very versatile and flexible in the sense that it allows one to easily accommodate a diversity of the image features based on which the segmentation should be performed. As an additional contribution, a method for automatically adjusting the smoothness properties of the empirical distributions is proposed. Such a procedure is crucial in situations when the number of data samples (supporting a certain segmentation class) varies considerably in the course of the evolution of the active contour. In this case, the smoothness properties of the empirical distributions have to be properly adjusted to avoid either over-or underestimation artifacts. Finally, a number of relevant segmentation results are demonstrated and some further research directions are discussed.

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Section: B Examples Of Discriminative Featuresmentioning

confidence: 99%

Section: Comparative Studymentioning

confidence: 99%

Image Segmentation Using Active Contours Driven by the Bhattacharyya Gradient Flow

Michailovich

Rathi

Tannenbaum

2007

IEEE Trans. on Image Process.

298

242

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“…The covariance in this case is defined in terms of the rows and columns of the matrix and is not necessarily orthogonally invariant. A connection between this distribution and James' is made by Chikuse [7]. In modern random matrix theory, the most common Gaussian distribution for symmetric matrices is the Gaussian Orthogonal Ensemble (GOE), which was developed independently in the physics literature (Mehta [24]).…”

Section: Introduction Consider the Signal-plus-noise Modelmentioning

confidence: 99%

Inference for eigenvalues and eigenvectors of Gaussian symmetric matrices

Schwartzman¹,

Mascarenhas²,

Taylor³

2008

Ann. Statist.

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This article presents maximum likelihood estimators (MLEs) and log-likelihood ratio (LLR) tests for the eigenvalues and eigenvectors of Gaussian random symmetric matrices of arbitrary dimension, where the observations are independent repeated samples from one or two populations. These inference problems are relevant in the analysis of diffusion tensor imaging data and polarized cosmic background radiation data, where the observations are, respectively, 3 × 3 and 2 × 2 symmetric positive definite matrices. The parameter sets involved in the inference problems for eigenvalues and eigenvectors are subsets of Euclidean space that are either affine subspaces, embedded submanifolds that are invariant under orthogonal transformations or polyhedral convex cones. We show that for a class of sets that includes the ones considered in this paper, the MLEs of the mean parameter do not depend on the covariance parameters if and only if the covariance structure is orthogonally invariant. Closed-form expressions for the MLEs and the associated LLRs are derived for this covariance structure.

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“…There exist only a few distributions on the Stiefel or Grassmann manifolds [7], [8], the most popular being the Bingham or von Mises Fisher (vMF) distributions which have proven to be relevant in a number of applications including meteorology, biology, medicine, image analysis (see [7] and references therein), modeling of multipath communications channels [9] and shape analysis [10]. These distributions depend on a matrix whose range space is "close" to that of U , along with a concentration parameter that rules the distance between the subspaces.…”

Section: B Prior Distributionsmentioning

confidence: 99%

“…These distributions depend on a matrix whose range space is "close" to that of U , along with a concentration parameter that rules the distance between the subspaces. More specifically, we assume that R (U ) is close to a given subspace spanned by the columns of an orthonormal matrixŪ and we consider the following prior distributions for U [7], [8]:…”

Section: B Prior Distributionsmentioning

confidence: 99%

Bayesian estimation of a subspace

Besson

Dobigeon

Tourneret

2011

2011 Conference Record of the Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR)

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Abstract-We consider the problem of subspace estimation in a Bayesian setting. First, we revisit the conventional minimum mean square error (MSE) estimator and explain why the MSE criterion may not be fully suitable when operating in the Grassmann manifold. As an alternative, we propose to carry out subspace estimation by minimizing the mean square distance between the true subspace U and its estimate, where the considered distance is a natural metric on the Grassmann manifold. We show that the resulting estimator is no longer the posterior mean of U but entails computing the principal eigenvectors of the posterior mean of U U T . Illustrative examples involving a linear Gaussian model for the data and a Bingham or von Mises Fisher prior distribution for U are presented. In the former case the minimum mean square distance (MMSD) estimator is obtained in closed-form while, in the latter case, a Markov chain Monte Carlo method is used to approximate the MMSD estimator. The method is shown to provide accurate estimates even when the number of samples is lower than the dimension of U . Finally, an application to hyperspectral imagery is presented.Index Terms-Subspace estimation, Bayesian inference, minimum distance error.

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Distributions on the Special Manifolds

Cited by 24 publications

References 0 publications

Image Segmentation Using Active Contours Driven by the Bhattacharyya Gradient Flow

Image Segmentation Using Active Contours Driven by the Bhattacharyya Gradient Flow

Inference for eigenvalues and eigenvectors of Gaussian symmetric matrices

Bayesian estimation of a subspace

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