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Moving least squares

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 Title: Moving least squares Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

Moving least squares

Moving least squares is a method of reconstructing 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)-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}.