 Moving least squares

Moving least squares is a method of reconstructing continuous functions from a set of unorganized point samples via the calculation of a weighted least squares 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.
Contents
Definition
Consider a function and a set of sample points S = {(x_{i},f_{i})  f(x_{i}) = f_{i}} where and the f_{i}'s are real numbers. Then, the moving least square approximation of degree m at the point x is where minimizes the weighted leastsquare error
over all polynomials p of degree m in . θ(s) is the weight and it tends to zero as .
In the example .
See also
References
 Moving least squares response surface approximation: Formulation and metal forming applications Piotr Breitkopf; Hakim Naceur; Alain Rassineux; Pierre Villon, Computers and Structures, Volume 83, 1718, 2005.
 Generalizing the finite element method: diffuse approximation and diffuse elements, B Nayroles, G Touzot. Pierre Villon, P, Computational Mechanics Volume 10, pp 307318, 1992
External links
 An AsShortAsPossible Introduction to the Least Squares, Weighted Least Squares and Moving Least Squares Methods for Scattered Data Approximation and Interpolation
 THE APPROXIMATION POWER OF MOVING LEASTSQUARES
This applied mathematicsrelated article is a stub. You can help Wikipedia by expanding it.