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In [[computer graphics]], a '''hierarchical RBF''' is an [[interpolation]] method based on [[Radial basis function]]s (RBF). Hierarchical RBF interpolation has applications in the construction of shape models in [[3d computer graphics|3D computer graphics]] (see [[Stanford Bunny]] image below), treatment of results from a [[3D scanner]], [[terrain]] reconstruction and others.
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[[Image:MyBunny.gif]]
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This problem informally named "large scattered data point set interpolation".
 
The idea of method (for example in 3D) consists of the following:
* let the scattered points be presented a set <math>\mathbf{P}=\{\mathbf{c}_i=(\mathbf{x}_i,\mathbf{y}_i,\mathbf{z}_i)\vert^{N}_{i=0} \subset \mathbb{R}^3\}</math>
* let the exist a set of values of some function in scattered points <math>\mathbf{H}=\{\mathbf{h}_i \vert^{N}_{i=0}\subset \mathbb{R}\}</math>
* find a function <math>\mathbf{f}(\mathbf{x})</math> which will meet next condition: <math>\mathbf{f}(\mathbf{x})=1</math> for points lies on shape and <math>\mathbf{f}(\mathbf{x})\neq1</math> for points not lies on shape
* as J. C. Carr et al. showed <ref>Carr, J.C.; Beatson, R.K.; Cherrie, J.B.; Mitchell, T.J.; Fright, W.R.; McCallum B.C.; Evans, T.R. (2001), “Reconstruction and Representation of 3D Objects with Radial Basis Functions” ACM SIGGRAPH 2001, Los Angeles, CA, P. 67–76.</ref> this function looks like <math>\mathbf{f}(\mathbf{x})=\sum_{i=1}^N \lambda_i \varphi(\mathbf{x},\mathbf{c}_i)</math> where:
<math>\varphi</math> &mdash; it is [[Radial basis function|RBF]];
<math>\lambda</math> &mdash; it is coefficients which are the solution of the [[Linear system of equations|system]] show on picture:
 
[[Image:System.gif]]
 
for determination of surface it is necessary to estimate the value of function <math>\mathbf{f}(\mathbf{x})</math> in interesting  points ''x''.
A lack of such method is considerable complication <ref>Bashkov, E.A.; Babkov, V.S. (2008) “Research of RBF-algorithm and his modifications apply
possibilities for the construction of shape computer models in medical practic”. Proc Int.
Conference "Simulation-2008", Pukhov Institute for Modelling in Energy Engineering, [http://babkov.name/article/2008-09.pdf] (in Russian)</ref> <math>\mathbf{O}(\mathbf{n}^2)</math> for calculate [[Radial basis function|RBF]], solve [[Linear system of equations|system]] and determine surface.
 
==Other similar methods==
* Reduce interpolation centres (<math>\mathbf{O}(\mathbf{n}^2)</math> for calculate [[Radial basis function|RBF]] and solve [[Linear system of equations|system]], <math>\mathbf{O}(\mathbf{m}\mathbf{n})</math> for determine surface)
* Compactly supported [[Radial basis function|RBF]] (<math>\mathbf{O}(\mathbf{n}\log{\mathbf{n}})</math> for calculate [[Radial basis function|RBF]], <math>\mathbf{O}(\mathbf{n}^{1.2..1.5})</math> for solve [[Linear system of equations|system]], <math>\mathbf{O}(\mathbf{m}\log{\mathbf{n}})</math> for determine surface)
* [[Fast multipole method|FMM]]  (<math>\mathbf{O}(\mathbf{n}^2)</math> for calculate [[Radial basis function|RBF]], <math>\mathbf{O}(\mathbf{n}\log{\mathbf{n}})</math> for solve [[Linear system of equations|system]], <math>\mathbf{O}(\mathbf{m}+\mathbf{n}\log{\mathbf{n}})</math> for determine surface)
 
==Hierarchical algorithm==
An idea of [[hierarchical]] [[Algorythm|algorithm]] is an acceleration of calculations due to [[Decomposition (computer science)|decomposition]] of intricate problem on the great number of simple (see picture). [[File:Hierarchical algorithm flow chart.gif]]
 
In this case [[hierarchical]] division of space containing points on elementary parts, the [[Linear system of equations|system]] of small dimension solves in each of which. The calculation of surface in this case is taken to the [[hierarchical]] (on the basis of [[Tree (data structure)|tree-structure]]) calculation of interpolant. A method for a [[2D computer graphics|2D]] case is offered Pouderoux J. et al.<ref>Pouderox, J. et al. (2004), “Adaptive hierarchical RBF interpolation for creating smooth digital elevathion models”, Proc. 12-th ACM Int. Symp. Advances in Geographical information Systems 2004, ACP Press, P. 232&ndash;240</ref> For a [[3D computer graphics|3D]] case a method is used in the tasks of [[3D computer graphics|3D graphics]] by W. Qiang et al.<ref>Qiang, W.; Pan, Z.; Chun, C.; Jiajun, B. (2007), “Surface rendering for parallel slice of contours from medical imaging”, Computing in science & engineering, 9(1), January&ndash;February 2007, P 32&ndash;37</ref> and modified by Babkov V.<ref>Babkov, V.S. (2008) “Modification of hierarchical RBF method for 3D-modelling based on laser scan result”. Proc. Int. Conference “Modern problems and achievement of radio, communication
and informatics”, Zaporizhzhya National Technical University, [http://babkov.name/article/2008-08.pdf] (in Ukrainian)</ref>
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Hierarchical Rbf}}
[[Category:Geometric algorithms]]
[[Category:Computer graphics]]
[[Category:Interpolation]]

Latest revision as of 03:39, 6 November 2014

by Nas, is very fitting and the film agrees with it. Medical word press themes give you the latest medical designs. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. After activating, you will find their website link and get the activation code from their website.

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