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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] |
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| 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]].
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| ==Definition==
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| [[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.]]
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| 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
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| :<math>\sum_{i \in I} (p(x)-f_i)^2\theta(\|x-x_i\|)</math>
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| 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>.
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| In the example <math>\theta(s) = e^{-s^2}</math>.
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| ==See also==
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| *[[Local regression]]
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| *[[Diffuse element method]]
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| ==References==
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| {{Reflist}}
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| *[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 ]
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| *[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.
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| * [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
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| ==External links==
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| * [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]
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| [[Category:Mathematical optimization]]
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| [[Category:Regression analysis]]
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| [[Category:Least squares]]
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| {{mathapplied-stub}}
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