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.