# 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 and a set of sample points . Then, the moving least square approximation of degree at the point is where minimizes the weighted least-square error

over all polynomials of degree in . is the weight and it tends to zero as .

In the example . The smooth interpolator of "order 3" is a quadratic interpolator.

## 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