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

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

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

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 $\theta(s) = e^{-s^2}$. The smooth interpolator of "order 3" is a quadratic interpolator.

See also

• Local regression
• Diffuse element method

References

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