Some Geometry of the Cone of Nonnegative Definite Matrices and Weights of Associated X<sup>2</sup>Distribution

This paper deals with the computation of the histogram of tensor images, that is, images where at each pixel is given a n × n positive definite symmetric matrix, SPD(n). An approach based on orthogonal series density estimation is introduced, which is particularly useful for the case of measures based on Riemannian metrics. By considering SPD(n) as the space of the covariance matrices of multivariate gaussian distributions, we obtain the corresponding density estimation for the measure of both the Fisher metric and the Wasserstein metric. Experimental results on the application of such histogram estimation to DTI image segmentation, texture segmentation and texture recognition are included.

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“…The modification of the density of the volume measure is expressed by the Jacobian of the change of coordinates [8]:…”

Section: Two Parametrisations Of Spd(n)mentioning

confidence: 99%

“…Let us write µ Lebes 1 the Lebesgue measure of the parametrization 1 of SPD(n) and µ Lebes 2 the Lebesgue measure of parametrization 2. The computation of the Jacobian of the coordinate change gives the following relation [8]:…”

Section: Two Parametrisations Of Spd(n)mentioning

confidence: 99%

Computing Histogram of Tensor Images Using Orthogonal Series Density Estimation and Riemannian Metrics

Chevallier

Angulo

2014

2014 22nd International Conference on Pattern Recognition

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“…These three cases are summarized as m+rank H 3. Now, it remains to check the boundary condition (12). Since c(x) is bounded, the boundary condition (12) holds obviously for m 3.…”

Section: Proposed Estimatorsmentioning

confidence: 99%

“…In order to determine the function c(X), we need the explicit form of the second fundamental form of the boundary of C*=&C, which has been derived by Kuriki and Takemura [12].…”

Section: Estimation Of Nonnegative Definite Mean Matrixmentioning

confidence: 99%

Shrinkage Estimation towards a Closed Convex Set with a Smooth Boundary

Kuriki

Takemura

2000

Journal of Multivariate Analysis

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We give James Stein type estimators of a multivariate normal mean vector by shrinkage towards a closed convex set K with a smooth or piecewise smooth boundary. The rate of shrinkage is determined by the curvature of the boundary of K at the projection point onto K. By considering a sequence of polytopes K j converging to K, we show that a particular estimator we propose is the limit of a sequence of shrinkage estimators towards K j given by M. E. Bock (1982). In fact our estimators reduce to the James Stein estimator and to the Bock estimator when K is a point and a convex polyhedron, respectively. Therefore they can be considered as natural extensions of these estimators. Furthermore we apply the same method to the problem of improving the restricted mle by shrinkage towards the origin in the multivariate normal mean model where the mean vector is restricted to a closed convex cone with a smooth or piecewise smooth boundary. We demonstrate our estimators in two settings, one shrinking to a ball and the other shrinking to the cone of nonnegative definite matrices. Academic PressAMS 1991 subject classifications: 62F10, 60D05, 62F30.

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“…Initially, we have been investigating the coefficients of the % w w 2 distribution appearing in order restricted inference [8][9][10] from geometric viewpoint. During this investigation, we have recognized that the tube method leads to approximation of the upper tail probability of maximum-type statistics in the form of linear combination of w 2 distributions, which can be regarded as a generalization of the % w w 2 distribution.…”

Section: Introductionmentioning

confidence: 99%

Application of tube formula to distributional problems in multiway layouts

Kuriki

Takemura

2002

Appl Stoch Models Bus & Ind

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SUMMARYRecently, an integral geometric method called the tube method has been actively developed. The tube method gives us a powerful tool to tackle problems where conventional matrix theory such as the singular value decomposition cannot be applied. The aim of this paper is to survey several recent applications of the tube method to distributional problems in multiway layouts which are of practical importance but hardly handled by conventional methods. Null distributions of test statistics of the following three testing problems are discussed: (i) A test for interaction in three-way layout based on the three-way analogue of the largest singular value. (ii) Testing independence in ordered categorical data by maximizing row and column scores under order restriction. (iii) Detecting a change point in two-way layout with ordinal factors.

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Some Geometry of the Cone of Nonnegative Definite Matrices and Weights of Associated X²Distribution

Cited by 10 publications

References 12 publications

Computing Histogram of Tensor Images Using Orthogonal Series Density Estimation and Riemannian Metrics

Computing Histogram of Tensor Images Using Orthogonal Series Density Estimation and Riemannian Metrics

Shrinkage Estimation towards a Closed Convex Set with a Smooth Boundary

Application of tube formula to distributional problems in multiway layouts

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Some Geometry of the Cone of Nonnegative Definite Matrices and Weights of Associated X2Distribution

Cited by 10 publications

References 12 publications

Computing Histogram of Tensor Images Using Orthogonal Series Density Estimation and Riemannian Metrics

Computing Histogram of Tensor Images Using Orthogonal Series Density Estimation and Riemannian Metrics

Shrinkage Estimation towards a Closed Convex Set with a Smooth Boundary

Application of tube formula to distributional problems in multiway layouts

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Product

Resources

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Some Geometry of the Cone of Nonnegative Definite Matrices and Weights of Associated X²Distribution