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 approximation of order 3.

Consider a function $f:\mathbb {R} ^{n}\to \mathbb {R}$ and a set of sample points $S=\{(x_{i},f_{i})|f(x_{i})=f_{i}\}$. 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
• Moving average

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