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.

Definition

Consider a function $f: \mathbb^n \to \mathbb$ and a set of sample points $S = \$. Then, the moving least square approximation of degree $m$ at the point $x$ is $\tilde(x)$ where $\tilde$ minimizes the weighted least-square error
:$\sum_ (p(x_i)-f_i)^2\theta(\, x-x_i\, )$
over all polynomials $p$ of degree $m$ in $\mathbb^n$. $\theta(s)$ is the weight and it tends to zero as $s\to \infty$.

In the example $\theta(s) = e^$. The smooth interpolator of "order 3" is a quadratic interpolator.

See also

*Local regression
* Diffuse element method
*Moving average

References

The approximation power of moving least squares
David Levin, Mathematics of Computation, Volume 67, 1517-1531, 199
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

Least squares

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