Geodesic Paths for Time-Dependent Covariance Matrices in a Riemannian Manifold

Time-dependent covariance matrices are important in remote sensing and hyperspectral detection theory. The difficulty is that C(t) is usually available only at two endpoints C(t 0 ) = A and C(t 1 ) = B where is needed. We present the Riemannian manifold of positive definite symmetric matrices as a f...

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Bibliographic Details
Published in:IEEE geoscience and remote sensing letters 2014-09, Vol.11 (9), p.1499-1503
Main Authors: Ben-David, Avishai, Marks, Justin
Format: Article
Language:English
Subjects:
Background characterization
Covariance matrices
detection algorithms
Eigenvalues and eigenfunctions
geodesic path
Hyperspectral imaging
hyperspectral remote sensing
Manifolds
matched filters
Remote sensing
Riemannian manifold
signal processing algorithms
Signal to noise ratio
statistical modeling
time-dependent covariance matrices
Vectors
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Summary:Time-dependent covariance matrices are important in remote sensing and hyperspectral detection theory. The difficulty is that C(t) is usually available only at two endpoints C(t 0 ) = A and C(t 1 ) = B where is needed. We present the Riemannian manifold of positive definite symmetric matrices as a framework for predicting a geodesic time-dependent covariance matrix. The geodesic path A→B is the shortest and most efficient path (minimum energy). Although there is no guarantee that data will necessarily follow a geodesic path, the predicted geodesic C(t) is of value as a concept. The path for the inverse covariance is also geodesic and is easily computed. We present an interpretation of C(t) with coloring and whitening operators to be a sum of scaled, stretched, contracted, and rotated ellipses.
ISSN:1545-598X
1558-0571
DOI:10.1109/LGRS.2013.2296833