# 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 $f:\mathbb {R} ^{n}\to \mathbb {R}$ 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

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