Cylinder set measure - Revision history

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	'~~''Moving least squares''' is a method of reconstructing [[continuous function]]~~s from a [[set (mathematics)\|set]] of unorganized point samples via the calculation of a weighted [[least squares]] [[measure (mathematics)\|measure]] biased towards the region around the point at which the reconstructed value is ~~requested~~.		The writer's title is Christy Brookins. He is an purchase clerk and it's some thing he truly appreciate. I am really fond of to go to karaoke but I've been using on new issues lately. Ohio is where my home is but my spouse desires us to move.<br><br>My webpage [http://www.weddingwall.com.au/groups/easy-advice-for-successful-personal-development-today/ best psychic readings]

	~~In [[computer graphics]], the moving least squares method~~ is ~~useful for reconstructing a surface from a set of points. Often~~ it ~~is used to create a 3D surface from a [[point cloud]] through either [[downsampling]] or [[upsampling]]~~.

	~~==Definition==~~
	[[Image:Moving_Least_Squares2.png\|thumb\|200px\|width=250\|length=150\|Here is a 2D example. The circles are the samples and the polygon is a linear interpolation. The blue curve is a smooth interpolation of ~~order 3.]]~~
	~~Consider a function <math>f: \mathbb{R}^n \~~to ~~\mathbb{R}</math> and a set of sample points <math>S = \{ (x_i,f_i) \| f(x_i) = f_i \} </math> where <math>x_i \in \mathbb{R}^n</math> and the <math>f_i</math>~~'~~s are real numbers~~. ~~Then, the moving least square approximation of degree <math>m</math> at the point <math>x</math>~~ is ~~<math>\tilde{p}(x)</math>~~ where ~~<math>\tilde{p}</math> minimizes the weighted least-square error~~
	~~:<math>\sum_{i \in I} (p(x)-f_i)^2\theta(\\|x-x_i\\|)</math>~~
	~~over all polynomials <math>p</math> of degree <math>m</math> in <math>\mathbb{R}^n</math>. <math>\theta(s)</math>~~ is ~~the weight and it tends~~ to ~~zero as <math>s\to \infty</math>~~.

	~~In the example~~ <~~math~~>~~\theta(s) = e^{-s^2}~~<~~/math~~>.

	~~==See also==~~
	*[[Local regression]]
	*[[Diffuse element method]]

	~~==References==~~
	~~{{Reflist}}~~
	*[http://dl.~~acm~~.~~org/citation~~.~~cfm?id=301704 The approximation power of moving least squares] David Levin, Mathematics of Computation, Volume 67, 1517-1531, 1998 [http:~~//~~www.ams.org/mcom/1998~~-67-~~224/S0025~~-~~5718~~-98-~~00974~~-0/~~S0025-5718-98-00974-0.pdf~~ ]
	*[http://www.sciencedirect.com/science/article/pii/S0045794905000726/ Moving least squares response surface approximation: Formulation and metal forming applications] Piotr Breitkopf; Hakim Naceur; Alain Rassineux; Pierre Villon, Computers and Structures, Volume 83, 17-18, 2005.
	* [http://www.springerlink.com/content/v7164702238848p1/ Generalizing the finite element method: diffuse approximation and diffuse elements], B Nayroles, G Touzot. Pierre Villon, P, Computational Mechanics Volume 10, pp 307-318, 1992

	~~==External links==~~
	* [http://www.nealen.net/projects/mls/asapmls.pdf An As-Short-As-Possible Introduction to the Least Squares, Weighted Least Squares and Moving Least Squares Methods for Scattered Data Approximation and Interpolation]

	~~[[Category:Mathematical optimization]]~~
	~~[[Category:Regression analysis]]~~
	~~[[Category:Least squares]]~~

	~~{{mathapplied-stub}}~~

en>Mgkrupa: /* References */ Added {{Functional Analysis}} footer

2014-01-10T03:09:01Z

References: Added {{Functional Analysis}} footer

← Older revision		Revision as of 05:09, 10 January 2014
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	~~Claude~~ is ~~her title~~ and ~~she totally digs that title~~. ~~Her spouse~~ and ~~her selected to reside~~ in ~~Alabama~~. ~~The factor~~ I ~~adore most flower arranging and now I have time to consider on new issues~~. ~~Managing individuals~~ is ~~how she makes money~~ and ~~she will not alter~~ it ~~anytime soon~~.<br><br>~~my blog :~~: [http://~~Freebusinesscollege~~.~~com~~//~~read_blog~~/~~198199~~/~~great~~-~~ideas~~-~~about~~-~~auto~~-~~repair~~-~~that~~-~~anyone~~-~~can~~-~~use auto warranty~~]		'''Moving least squares''' is a method of reconstructing [[continuous function]]s from a [[set (mathematics)\|set]] of unorganized point samples via the calculation of a weighted [[least squares]] [[measure (mathematics)\|measure]] biased towards the region around the point at which the reconstructed value is requested.

			In [[computer graphics]], the moving least squares method is useful for reconstructing a surface from a set of points. Often it is used to create a 3D surface from a [[point cloud]] through either [[downsampling]] or [[upsampling]].

			==Definition==
			[[Image:Moving_Least_Squares2.png\|thumb\|200px\|width=250\|length=150\|Here is a 2D example. The circles are the samples and the polygon is a linear interpolation. The blue curve is a smooth interpolation of order 3.]]
			Consider a function <math>f: \mathbb{R}^n \to \mathbb{R}</math> and a set of sample points <math>S = \{ (x_i,f_i) \| f(x_i) = f_i \} </math> where <math>x_i \in \mathbb{R}^n</math> and the <math>f_i</math>'s are real numbers. Then, the moving least square approximation of degree <math>m</math> at the point <math>x</math> is <math>\tilde{p}(x)</math> where <math>\tilde{p}</math> minimizes the weighted least-square error
			:<math>\sum_{i \in I} (p(x)-f_i)^2\theta(\\|x-x_i\\|)</math>
			over all polynomials <math>p</math> of degree <math>m</math> in <math>\mathbb{R}^n</math>. <math>\theta(s)</math> is the weight and it tends to zero as <math>s\to \infty</math>.

			In the example <math>\theta(s) = e^{-s^2}</math>.

			==See also==
			*[[Local regression]]
			*[[Diffuse element method]]

			==References==
			{{Reflist}}
			*[http://dl.acm.org/citation.cfm?id=301704 The approximation power of moving least squares] David Levin, Mathematics of Computation, Volume 67, 1517-1531, 1998 [http://www.ams.org/mcom/1998-67-224/S0025-5718-98-00974-0/S0025-5718-98-00974-0.pdf ]
			*[http://www.sciencedirect.com/science/article/pii/S0045794905000726/ Moving least squares response surface approximation: Formulation and metal forming applications] Piotr Breitkopf; Hakim Naceur; Alain Rassineux; Pierre Villon, Computers and Structures, Volume 83, 17-18, 2005.
			* [http://www.springerlink.com/content/v7164702238848p1/ Generalizing the finite element method: diffuse approximation and diffuse elements], B Nayroles, G Touzot. Pierre Villon, P, Computational Mechanics Volume 10, pp 307-318, 1992

			==External links==
			* [http://www.nealen.net/projects/mls/asapmls.pdf An As-Short-As-Possible Introduction to the Least Squares, Weighted Least Squares and Moving Least Squares Methods for Scattered Data Approximation and Interpolation]

			[[Category:Mathematical optimization]]
			[[Category:Regression analysis]]
			[[Category:Least squares]]

			{{mathapplied-stub}}

en>ChrisGualtieri: /* References */Typo fixing, typos fixed: , → , (2) using AWB

2012-07-20T00:39:48Z

References: Typo fixing, typos fixed: , → , (2) using AWB

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