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Visualizing bregman voronoi diagrams

Frank Nielsen, Jean-Daniel Boissonnat, Richard Nock
2007 Proceedings of the twenty-third annual symposium on Computational geometry - SCG '07  
Generalizing Voronoi Diagrams Voronoi diagrams widely studied in comp. geometry [AK'00]: Abstract Voronoi diagrams [Klein'89], etc.  ...  diagram equivalence classes Since Vor(S; d 2 ) = Vor(S; d 2 2 ), the ordinary Voronoi diagram is interpreted as a Bregman Voronoi diagram.  ...  Nock On Bregman Voronoi Diagrams , straight and geodesic triangulations are the same ordinary Delaunay triangulations.) F. Nielsen, J.-D. Boissonnat and R. Nock On Bregman Voronoi Diagrams  ... 
doi:10.1145/1247069.1247089 dblp:conf/compgeom/NielsenBN07 fatcat:do7vrdsztrdctpaxbnjbf7pfji

Quantum Voronoi diagrams and Holevo channel capacity for 1-qubit quantum states

Frank Nielsen, Richard Nock
2008 2008 IEEE International Symposium on Information Theory  
We show that these diagrams can be computed as Bregman Voronoi diagrams for the corresponding Bregman generator acting on Hermitian density matrices.  ...  This implies that these dual diagrams can be derived from power diagrams of balls in the Laguerre geometry, and allows one to prove by equivalence that the von Neumann quantum Voronoi diagram on the degenerated  ...  Theorem 3.1: The von Neumann quantum Voronoi diagrams are Bregman Voronoi diagrams that can be computed from equivalent power diagrams.  ... 
doi:10.1109/isit.2008.4594955 dblp:conf/isit/NielsenN08 fatcat:5pe5nltwhfhdraktmfkqoen4t4

Bregman Voronoi Diagrams

Jean-Daniel Boissonnat, Frank Nielsen, Richard Nock
2010 Discrete & Computational Geometry  
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 connection with Bregman Voronoi diagrams  ...  Accordingly, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions.  ...  Fig. 2 2 Visualizing the Bregman divergence.  ... 
doi:10.1007/s00454-010-9256-1 fatcat:lik3travqzdfhbm2md3p2t5mfi

Skew Jensen-Bregman Voronoi Diagrams [chapter]

Frank Nielsen, Richard Nock
2011 Lecture Notes in Computer Science  
This paper includes novel extensions to matrix-based Jensen-Bregman divergences, and present the general framework of skew Jensen-Bregman Voronoi diagrams that include Bregman Voronoi diagrams as particular  ...  We study the geometric properties and combinatorial complexities of both the Voronoi diagrams and the centroidal Voronoi diagrams induced by such as class of divergences.  ...  Namely, the (normalized) skew Jensen-Bregman Voronoi diagrams become a regular Bregman Voronoi diagram [6] . Figure 9 depicts several skew left-sided/right-sided Jensen-Bregman Voronoi diagrams.  ... 
doi:10.1007/978-3-642-25249-5_4 fatcat:7eclqfkzjbexbdosy3vc6kgtzm

Bregman Voronoi Diagrams: Properties, Algorithms and Applications [article]

Frank Nielsen, Jean-Daniel Boissonnat, Richard Nock
2007 arXiv   pre-print
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  ...  Accordingly, Bregman Voronoi diagrams allow to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions.  ...  First-type affine Bregman Voronoi diagram (red), second-type Bregman Voronoi diagram (blue) and symmetrized Bregman Voronoi diagram (green). 1. computing the first-type Bregman Voronoi diagram vorF (S)  ... 
arXiv:0709.2196v1 fatcat:7juz25kny5fi3g45qkag3c4xze

Geometry and clustering with metrics derived from separable Bregman divergences [article]

Erika Gomes-Gonçalves, Henryk Gzyl, Frank Nielsen
2018 arXiv   pre-print
We investigate fixed rate quantization and its codebook Voronoi diagrams, and report on experimental performances of partition-based, hierarchical, and soft clustering algorithms with respect to these  ...  Riemann-Bregman distances.  ...  Figure 3 : 3 Four types of Voronoi diagrams induced by a Bregman generator: The leftsided Bregman Voronoi diagram (blue), the right-sided Bregman Voronoi diagram (red), the symmetrized Bregman Voronoi  ... 
arXiv:1810.10770v1 fatcat:2esjghsk5rc2zlctppwjdkneza

A Generic Statistics-Based Tessellation Method of Voronoi Diagram

Shun Kang
2015 Journal of Systems Science and Information  
AbstractIn terms of distance function and spatial continuity in Voronoi diagram, a generic generating method of Voronoi diagram, named statistical Voronoi diagram, is proposed in this paper based upon  ...  In the light of Mahalanobis distance, not only ordinary Voronoi and weighted Voronoi are implemented, but also the theory of Voronoi diagram is improved further.  ...  weighted Voronoi diagrams in raster [3, 4] ; centroidal Voronoi tessellations on applications and algorithms [5, 6] ; Bregman Voronoi diagrams with Bregman divergence [7] and the fast dynamic Voronoi  ... 
doi:10.1515/jssi-2015-0568 fatcat:bq34azz36vdfzc5aktbn2jn4vu

Further results on the hyperbolic Voronoi diagrams [article]

Frank Nielsen, Richard Nock
2014 arXiv   pre-print
We investigate the hyperbolic Voronoi diagram in the hyperboloid model and show how it reduces to a Klein-type model using central projections.  ...  In Euclidean geometry, it is well-known that the k-order Voronoi diagram in R^d can be computed from the vertical projection of the k-level of an arrangement of hyperplanes tangent to a convex potential  ...  HVDs from lower envelopes The Voronoi diagram of a set P = {p 1 , ..., p n } of n points in R d w.r.t.  ... 
arXiv:1410.1036v1 fatcat:zcovdun6pnd6fbijdkffogpney

Optimal voronoi tessellations with hessian-based anisotropy

Max Budninskiy, Beibei Liu, Fernando de Goes, Yiying Tong, Pierre Alliez, Mathieu Desbrun
2016 ACM Transactions on Graphics  
Our approach sharply contrasts with previous anisotropic versions of Voronoi diagrams as it employs first-type Bregman diagrams, a generalization of power diagrams where sites are augmented with not only  ...  We thus refer to the resulting anisotropic polytopal meshes as Optimal Voronoi Tessellations.  ...  Bregman diagrams.  ... 
doi:10.1145/2980179.2980245 fatcat:yygraqrx4rabxhfg3a4sycy5fi

Power to the Points: Validating Data Memberships in Clusterings [article]

Parasaran Raman, Suresh Venkatasubramanian
2013 arXiv   pre-print
We also demonstrate useful visualizations of these scores; these might prove useful within an interactive analytics framework.  ...  We hope to develop better visualizations to accompany this method. And more generally we plan on studying other unsupervised learning tasks where local validation is important.  ...  It has been shown [6] that the Bregman Voronoi diagram on a collection of sites s 1 , . . . , s k defined as V i = {p|D φ (p, s i ) ≤ D φ (p, s j )∀j = i} is a convex polyhedron just like in the case  ... 
arXiv:1305.4757v1 fatcat:4hhjc43qjfcgpotdrdzt2d7bya

Voronoi Diagrams for Quantum States and Its Application to a Numerical Estimation of a Quantum Channel Capacity [article]

Kimikazu Kato
2008 arXiv   pre-print
Especially we introduce the concept of the Voronoi diagram and the smallest enclosing ball problem to the space of quantum states.  ...  [80] showed some properties of Voronoi diagram with respect to Bregman divergence, which is generalization of Kullback-Leibler divergence. Using the Voronoi diagram, Nielsen et al.  ...  Voronoi diagrams are not only useful for such applications as numerical calculation and visualization, but also useful for theoretical interpre-tation of a geometric object.  ... 
arXiv:0803.3109v2 fatcat:k3mtauzkt5gyde3hbim5scscwi

Power to the Points: Validating Data Memberships in Clusterings

Parasaran Raman, Suresh Venkatasubramanian
2013 2013 IEEE 13th International Conference on Data Mining  
It lends itself to easy visualization and can prove useful as part of an interactive visual analytics framework.  ...  We can also extend our Voronoi-based definition of affinity to clusterings in Bregman spaces [21] and kernel spaces [22] .  ...  Instead of constructing the Voronoi diagram, we will construct the power diagram defined in Section III.  ... 
doi:10.1109/icdm.2013.147 dblp:conf/icdm/RamanV13 fatcat:5xge7b74k5botg5h4576mgawd4

An Obstruction to Delaunay Triangulations in Riemannian Manifolds

Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh, Nikolay Martynchuk
2017 Discrete & Computational Geometry  
Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds.  ...  The Delaunay complex of P is the nerve of the Voronoi diagram.  ...  Thus every Voronoi face corresponds to a Delaunay simplex. The Voronoi cells give a decomposition of M, denoted Vor M (P ), called the Voronoi diagram.  ... 
doi:10.1007/s00454-017-9908-5 fatcat:xbcayt3btjg43py6kg264b67dq

An obstruction to Delaunay triangulations in Riemannian manifolds [article]

Jean-Daniel Boissonnat and Ramsay Dyer and Arijit Ghosh and Nikolay Martynchuk
2016 arXiv   pre-print
Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds.  ...  The Delaunay complex of P is the nerve of the Voronoi diagram.  ...  A schematic view of the Voronoi diagram associated with σ is depicted in Figure 3 . The circumumcentres of σ are the Voronoi vertices shown in the diagram.  ... 
arXiv:1612.02905v1 fatcat:ta7p4jsk2bahvouo2ajrov3jp4

Wavelet frame based surface reconstruction from unorganized points

Bin Dong, Zuowei Shen
2011 Journal of Computational Physics  
Another popular technique is to construct a polyhedral surface from the input set of points utilizing the Voronoi diagram [2, 4, 3, 41, 23] , which is closely related to Voronoi-based algorithms for medial  ...  This leads to the alternative split Bregman algorithm for (3.2) . The derivation of splitting Bregman algorithm in [34, 8] is based on Bregman distance. It was recently shown (see e.g.  ... 
doi:10.1016/j.jcp.2011.07.022 fatcat:se7iswshx5azjl2pkoalyt7ruu
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