Matrix differential equation: Difference between revisions

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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 04:39, 6 November 2014

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