Moving least squares - Alchetron, The Free Social Encyclopedia

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 : R n → R and a set of sample points S = { ( x i , f i ) | f ( x i ) = f i } where x i ∈ R n and the f i 's are real numbers. Then, the moving least square approximation of degree m at the point x is p ~ ( x ) where p ~ minimizes the weighted least-square error

∑ i ∈ I ( p ( x i ) − f i ) 2 θ ( ∥ x − x i ∥ )

over all polynomials p of degree m in R n . θ ( s ) is the weight and it tends to zero as s → ∞ .

In the example θ ( s ) = e − s 2 . The smooth interpolator of "order 3" is a quadratic interpolator.

References

Moving least squares Wikipedia

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