Moving least squares

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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[edit]

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 and a set of sample points where and the 's are real numbers. 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.

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