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Harmonic deformation of Delaunay triangulations [article]

Pablo A. Ferrari, Rafael M. Grisi, Pablo Groisman
2012 arXiv   pre-print
We construct harmonic functions on random graphs given by Delaunay triangulations of ergodic point processes as the limit of the zero-temperature harness process.  ...  The Delaunay triangulation of this points does not necessarily coincide with the harmonic deformation of the original Delaunay triangulation.  ...  If such an H exists, the resulting graph is the harmonic deformation of the Delaunay triangulation of S • .  ... 
arXiv:1012.1677v2 fatcat:f53weuswhfbztarrymazf65kla

Harmonic deformation of Delaunay triangulations

Pablo A. Ferrari, Rafael M. Grisi, Pablo Groisman
2012 Stochastic Processes and their Applications  
Application to random walk in Delaunay triangulation.  ...  Y S t : random walk in the Delaunay triangulation with generator L S f (s) = s ′ ∈S a(s, s ′ )[f (s ′ ) − f (s)] Since the graph H is harmonic, H(Y S t ) is a martingale and so satisfies the invariance  ...  Proof of (a) and (b) of the Theorem (a) Existence of a harmonic surface with inclination 1 will be a consequence of (b) Convergence of the harness process to a harmonic surface.  ... 
doi:10.1016/j.spa.2012.02.003 fatcat:jp5nuvwrfvbjhcognv4gbdlduq

Direct Spherical Harmonic Transform of a Triangulated Mesh

M. Mousa, R. Chaine, S. Akkouche
2006 Journal of Graphics Tools  
Spherical harmonic transform plays an important role in research in shape description.  ...  This paper describes a fast and accurate technique for computing spherical harmonic coecients directly from the description of the mesh.  ...  transform of the triangulated mesh M .  ... 
doi:10.1080/2151237x.2006.10129219 fatcat:dgg7nj4icndgxgqhwuvef5eiuy

Multiscale approximation of piecewise smooth two-dimensional functions using normal triangulated meshes

Maarten Jansen, Richard Baraniuk, Sridhar Lavu
2005 Applied and Computational Harmonic Analysis  
Multiresolution triangulation meshes are widely used in computer graphics for representing three-dimensional (3-d) shapes.  ...  We show that normal mesh subdivision is an efficient triangulation, thanks to its local adaptivity to the discontinuities.  ...  Multiscale Image Triangulations Quadtree triangulations Consider the construction of a multiscale triangulated function representation using the principle of subdivision (vertex refinement).  ... 
doi:10.1016/j.acha.2005.02.006 fatcat:hrgodyqyzjdpxj3lnaueoenayq

Face Transformation With Harmonic Models by the Finite-Volume Method With Delaunay Triangulation

Zi-Cai Li, John Y Chiang, Ching Y Suen
2010 IEEE Transactions on Systems Man and Cybernetics Part B (Cybernetics)  
Index Terms-Delaunay triangulation, face transformation, finite-volume method (FVM), harmonic models.  ...  The main method in this paper is a combination of the finite-volume method (FVM) with Delaunay triangulation to solve the Laplace equations in the harmonic transformation of face images.  ...  We solicit the harmonic transformation.  ... 
doi:10.1109/tsmcb.2010.2042955 pmid:20363682 fatcat:ves6drrad5ffnmrg74fldg2cfa

Properties of the Delaunay triangulation

Oleg R. Musin
1997 Proceedings of the thirteenth annual symposium on Computational geometry - SCG '97  
HARMONIC INDEX OF TRIANGULATION This index haa its origin in the theory of the so called " harmonic maps".  ...  For planar triangulation tof a set S let denote by hrm(t) (harmonic index of triangulation t)the sum of h-m of its triangles: hrm(t) =~hmn(A~) A,ct Our main result for harmonic index of planar triangulation  ... 
doi:10.1145/262839.263061 dblp:conf/compgeom/Musin97 fatcat:oclbi53trzcp3apc3hlsmcgeau

Hodge's harmonic p-sets and Pontrjagin classes [article]

Jonathan Fine
1998 arXiv   pre-print
This paper shows how Hodge's theory of harmonic p-sets (a discrete version of his theory of harmonic forms) allows a new approach to be taken to the problem of providing a combinatorial definition of the  ...  Now say that η is harmonic (for the given triangulation) if both η and η * are closed.  ...  Hodge realised that a triangulation is a discrete form of a metric, and that given a triangulation a discrete version of the harmonic theory can be developed.  ... 
arXiv:math/9802077v1 fatcat:zole2sxunrezlnubvjadfwyvre

Development of a shape specification based on the waviness parameter of tapered roller bearing

D. Carrera, I. Miguel, E. Padilla, V. Zaera, P. Dubón, S. Santo Domingo, J.J. Aguilar
2019 Procedia Manufacturing  
Ovality and triangulation specification. Harmonic relations These two values are derived from the harmonic amplitude specification.  ...  harmonic amplitude Table 1 . 1 Harmonic amplitude measurements ranges Table 2 . 2 Ovality and triangulation specification Bore diameter / Outer diameter [mm] Ovality (µm) Triangulation (µm) Over To  ... 
doi:10.1016/j.promfg.2019.10.002 fatcat:aq37ua2dffbctkyvx43wdobzk4

High-quality surface remeshing using harmonic maps-Part II: Surfaces with high genus and of large aspect ratio

E. Marchandise, C. Carton de Wiart, W. G. Vos, C. Geuzaine, J.-F. Remacle
2011 International Journal for Numerical Methods in Engineering  
In [1] , it has been demonstrated that harmonic parametrizations can be used as input for surface meshers to produce high quality triangulations.  ...  the above mentioned limitations of harmonic maps.  ...  We compare the quality of the initial STL triangulation (obtained through a segmentation procedure) with the quality of the remeshed pelvis based on a laplacian harmonic map.  ... 
doi:10.1002/nme.3099 fatcat:rvrtuzo63ncvxddq6ouhwy6owq

High-quality surface remeshing using harmonic maps

J.-F. Remacle, C. Geuzaine, G. Compère, E. Marchandise
2010 International Journal for Numerical Methods in Engineering  
We show that, with the proposed approach, we are able to recover high quality meshes from both low input STL triangulations and complex surfaces defined by many CAD patches.  ...  In this paper, we present an efficient and robust technique for surface remeshing based on harmonic maps.  ...  Remeshing algorithm. 1) Initial triangulation, 2) Harmonic map u(x) and v(x) and 3) new mesh based on the harmonic map.  ... 
doi:10.1002/nme.2824 fatcat:4xkfnurwfvgdtaxbko6va4wemq

Non-Convex Planar Harmonic Maps [article]

Shahar Z. Kovalsky, Noam Aigerman, Ingrid Daubechies, Michael Kazhdan, Jianfeng Lu, Stefan Steinerberger
2020 arXiv   pre-print
In the discrete case, we use an analog of the cone condition to characterize invertible discrete-harmonic piecewise-linear maps of triangulations.  ...  graphs - RKC's discrete counterpart - which proves the invertibility of piecewise linear maps of triangulated domains satisfying a discrete-harmonic principle, into a convex planar polygon.  ...  Maps of triangulations have been further studied in [6, 7, 8] , which consider discrete-harmonic embeddings of triangulated surfaces with disk topology.  ... 
arXiv:2001.01322v1 fatcat:mdryt2p765aylekhcob4xykbwq

Harmonic Functions for Data Reconstruction on 3D Manifolds [article]

Li Chen, Feng Luo
2011 arXiv   pre-print
This paper employs harmonic functions, a generalization of triangulation because linearity is a form of harmonic. Therefore, local harmonic initialization is more sophisticated then triangulation.  ...  This paper has also a philosophical advantage over triangulation meshes. People usually use triangulation for data reconstruction.  ...  In the past, people usually use triangulation for data reconstruction. This paper employs harmonic functions, a generalization of triangulation, because linearity is a form of harmonic.  ... 
arXiv:1102.0200v3 fatcat:jr4zzsj3rjd6xmc5xzr4ge4nzu

Construction of 3D Shape Models of Femoral Articular Cartilage Using Harmonic Maps [chapter]

A. D. Brett, C. J. Taylor
2000 Lecture Notes in Computer Science  
In general these models will be close to being planar discs which makes the use of harmonic mapping particularly suitable for our application.  ...  In this paper we describe a method of generating the pairwise correspondences using piecewise-linear harmonic maps of the surfaces which is constrained to be diffeomorphic.  ...  Harmonic Maps A harmonic mapping h : D → P uniquely maps a (triangulated) topological disk D in R 3 to a polygonal region in R 2 whilst minimizing metric dispersion.  ... 
doi:10.1007/978-3-540-40899-4_129 fatcat:bzgkjd4j3rannbnmozgglzcv2y

A spectral characterization of the Delaunay triangulation

Renjie Chen, Yin Xu, Craig Gotsman, Ligang Liu
2010 Computer Aided Geometric Design  
A small harmonic index means the triangle is "fatter". Theorem. (See Musin, 1997 .) The harmonic index obtains its minimum on the Delaunay triangulation. Proof.  ...  Among others, he defined the harmonic index of a triangulation: H(T ) = t∈T a 2 t + b 2 t + c 2 t A t where t is a triangle in the triangulation T with edge lengths a t , b t , c t , and area A t .  ... 
doi:10.1016/j.cagd.2010.02.002 fatcat:v6i3m3qitvg2re5j4iceihvxei

Rigidity of Infinite Hexagonal Triangulation of the Plane [article]

Tianqi Wu and Xianfeng Gu and Jian Sun
2013 arXiv   pre-print
The proof uses the concept of quasi-harmonic functions to unfold the properties of the mesh.  ...  is the regular hexagonal triangulation.  ...  inner angles of the triangulation} = θ < π/2, then for any constant c ∈ V , the function ∆ c w is quasi-harmonic and its harmonic factor m(θ) depends only on θ.  ... 
arXiv:1306.3630v1 fatcat:43b5zuyve5fjrmgwe5soidhawq
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