1,796 research outputs found
Bregman Voronoi Diagrams: Properties, Algorithms and Applications
The Voronoi diagram of a finite set of objects is a fundamental geometric
structure that subdivides the embedding space into regions, each region
consisting of the points that are closer to a given object than to the others.
We may define many variants of Voronoi diagrams depending on the class of
objects, the distance functions and the embedding space. In this paper, we
investigate a framework for defining and building Voronoi diagrams for a broad
class of distance functions called Bregman divergences. Bregman divergences
include not only the traditional (squared) Euclidean distance but also various
divergence measures based on entropic functions. Accordingly, Bregman Voronoi
diagrams allow to define information-theoretic Voronoi diagrams in statistical
parametric spaces based on the relative entropy of distributions. We define
several types of Bregman diagrams, establish correspondences between those
diagrams (using the Legendre transformation), and show how to compute them
efficiently. We also introduce extensions of these diagrams, e.g. k-order and
k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set
of points and their connexion with Bregman Voronoi diagrams. We show that these
triangulations capture many of the properties of the celebrated Delaunay
triangulation. Finally, we give some applications of Bregman Voronoi diagrams
which are of interest in the context of computational geometry and machine
learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures
Analytic Expressions for Stochastic Distances Between Relaxed Complex Wishart Distributions
The scaled complex Wishart distribution is a widely used model for multilook
full polarimetric SAR data whose adequacy has been attested in the literature.
Classification, segmentation, and image analysis techniques which depend on
this model have been devised, and many of them employ some type of
dissimilarity measure. In this paper we derive analytic expressions for four
stochastic distances between relaxed scaled complex Wishart distributions in
their most general form and in important particular cases. Using these
distances, inequalities are obtained which lead to new ways of deriving the
Bartlett and revised Wishart distances. The expressiveness of the four analytic
distances is assessed with respect to the variation of parameters. Such
distances are then used for deriving new tests statistics, which are proved to
have asymptotic chi-square distribution. Adopting the test size as a comparison
criterion, a sensitivity study is performed by means of Monte Carlo experiments
suggesting that the Bhattacharyya statistic outperforms all the others. The
power of the tests is also assessed. Applications to actual data illustrate the
discrimination and homogeneity identification capabilities of these distances.Comment: Accepted for publication in the IEEE Transactions on Geoscience and
Remote Sensing journa
A Graph-based approach to derive the geodesic distance on Statistical manifolds: Application to Multimedia Information Retrieval
In this paper, we leverage the properties of non-Euclidean Geometry to define
the Geodesic distance (GD) on the space of statistical manifolds. The Geodesic
distance is a real and intuitive similarity measure that is a good alternative
to the purely statistical and extensively used Kullback-Leibler divergence
(KLD). Despite the effectiveness of the GD, a closed-form does not exist for
many manifolds, since the geodesic equations are hard to solve. This explains
that the major studies have been content to use numerical approximations.
Nevertheless, most of those do not take account of the manifold properties,
which leads to a loss of information and thus to low performances. We propose
an approximation of the Geodesic distance through a graph-based method. This
latter permits to well represent the structure of the statistical manifold, and
respects its geometrical properties. Our main aim is to compare the graph-based
approximation to the state of the art approximations. Thus, the proposed
approach is evaluated for two statistical manifolds, namely the Weibull
manifold and the Gamma manifold, considering the Content-Based Texture
Retrieval application on different databases
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