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Moving least squares

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Title: Moving least squares  
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Subject: Diffuse element method, Least squares, Meshfree methods, Regression analysis, Growth curve (statistics)
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Moving least squares

Moving least squares is a method of reconstructing 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.

Definition

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 f: \mathbb{R}^n \to \mathbb{R} and a set of sample points S = \{ (x_i,f_i) | f(x_i) = f_i \} where x_i \in \mathbb{R}^n and the f_i's are real numbers. Then, the moving least square approximation of degree m at the point x is \tilde{p}(x) where \tilde{p} minimizes the weighted least-square error

\sum_{i \in I} (p(x)-f_i)^2\theta(\|x-x_i\|)

over all polynomials p of degree m in \mathbb{R}^n. \theta(s) is the weight and it tends to zero as s\to \infty.

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

See also

References

  • The approximation power of moving least squares David Levin, Mathematics of Computation, Volume 67, 1517-1531, 1998 [1]
  • 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.
  • 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

  • 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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