Matrix differential equation: Difference between revisions

Revision as of 04:35, 1 January 2014

In computer graphics, a hierarchical RBF is an interpolation method based on Radial basis functions (RBF). Hierarchical RBF interpolation has applications in the construction of shape models in 3D computer graphics (see Stanford Bunny image below), treatment of results from a 3D scanner, terrain reconstruction and others.

File:MyBunny.gif

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 $\mathbf {P} =\{\mathbf {c} _{i}=(\mathbf {x} _{i},\mathbf {y} _{i},\mathbf {z} _{i})\vert _{i=0}^{N}\subset \mathbb {R} ^{3}\}$
let the exist a set of values of some function in scattered points $\mathbf {H} =\{\mathbf {h} _{i}\vert _{i=0}^{N}\subset \mathbb {R} \}$
find a function $\mathbf {f} (\mathbf {x} )$ which will meet next condition: $\mathbf {f} (\mathbf {x} )=1$ for points lies on shape and $\mathbf {f} (\mathbf {x} )\neq 1$ for points not lies on shape
as J. C. Carr et al. showed ^[1] this function looks like $\mathbf {f} (\mathbf {x} )=\sum _{i=1}^{N}\lambda _{i}\varphi (\mathbf {x} ,\mathbf {c} _{i})$ where:

$\varphi$ — it is RBF; $\lambda$ — it is coefficients which are the solution of the system show on picture:

File:System.gif

for determination of surface it is necessary to estimate the value of function $\mathbf {f} (\mathbf {x} )$ in interesting points x. A lack of such method is considerable complication ^[2] $\mathbf {O} (\mathbf {n} ^{2})$ for calculate RBF, solve system and determine surface.

Other similar methods

Reduce interpolation centres ( $\mathbf {O} (\mathbf {n} ^{2})$ for calculate RBF and solve system, $\mathbf {O} (\mathbf {m} \mathbf {n} )$ for determine surface)
Compactly supported RBF ( $\mathbf {O} (\mathbf {n} \log {\mathbf {n} })$ for calculate RBF, $\mathbf {O} (\mathbf {n} ^{1.2..1.5})$ for solve system, $\mathbf {O} (\mathbf {m} \log {\mathbf {n} })$ for determine surface)
FMM ( $\mathbf {O} (\mathbf {n} ^{2})$ for calculate RBF, $\mathbf {O} (\mathbf {n} \log {\mathbf {n} })$ for solve system, $\mathbf {O} (\mathbf {m} +\mathbf {n} \log {\mathbf {n} })$ for determine surface)

Hierarchical algorithm

An idea of hierarchical algorithm is an acceleration of calculations due to 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 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-structure) calculation of interpolant. A method for a 2D case is offered Pouderoux J. et al.^[3] For a 3D case a method is used in the tasks of 3D graphics by W. Qiang et al.^[4] and modified by Babkov V.^[5]

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

↑ 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.
↑ 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, [1] (in Russian)
↑ 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. 232–240
↑ 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–February 2007, P 32–37
↑ 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, [2] (in Ukrainian)

[1] 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.

[2] 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, [1] (in Russian)

[3] 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. 232–240

[4] 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–February 2007, P 32–37

[5] 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, [2] (in Ukrainian)

[1]

[2]

[3]

[4]

[5]

@@ Line 1: / Line 1: @@
+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.
+[[Image:MyBunny.gif]]
-They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. Medical word press themes give you the latest medical designs. The Word - Press Dashboard : an administrative management tool that supports FTP content upload  2. If you are using videos on your site then this is the plugin to use. After activating, you will find their website link and get the activation code from their website. <br><br>Most Word - Press web developers can provide quality CMS website solutions and they price their services at reasonable rates. Best of all, you can still have all the functionality that you desire when you use the Word - Press platform. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. Apart from these, you are also required to give some backlinks on other sites as well. Moreover, many Word - Press themes need to be purchased and designing your own WP site can be boring. <br><br>It is also popular because willing surrogates,as well as egg and sperm donors,are plentiful. To sum up, ensure that the tactics are aiming to increase the ranking and attracting the maximum intended traffic in the major search engines. For a much deeper understanding of simple wordpress themes", check out  Upon browsing such, you'll be able to know valuable facts. Thousands of plugins are available in Word - Press plugin's library which makes the task of selecting right set of plugins for your website a very tedious task. " Thus working with a Word - Press powered web application, making any changes in the website design or website content is really easy and self explanatory. <br><br>If you beloved this posting and you would like to acquire much more details concerning [http://ad4.fr/wordpress_backup_6346078 wordpress backup] kindly pay a visit to the page. Word - Press installation is very easy and hassle free. But the Joomla was created as the CMS over years of hard work. Next you'll go by way of to your simple Word - Press site. A whole lot worse, your site will likely be useless as well as your merchandise won't sell if no one has the endurance to wait for the web pages to load. Now all you have to do is log into your Word - Press site making use of the very same username and password that you initially had in your previous site. <br><br>Many developers design websites and give them to the clients, but still the client faces problems to handle the website. When you sign up with Wordpress, you gain access to several different templates and plug-in that allow you to customize your blog so that it fits in with your business website design seamlessly. The days of spending a lot of time and money to have a website built are long gone. You should stay away from plugins that are full of flaws and bugs. As for performing online business, websites and blogs are the only medium that are available to interact with customers and Word - Press perform this work with the help of cross-blog communication tools, comments and  full user registration plug-ins.
+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]]

Matrix differential equation: Difference between revisions

Revision as of 04:35, 1 January 2014

Other similar methods

Hierarchical algorithm

References

Navigation menu

Search