Moving Least Squares

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Moving least squares is a method of approximating a continuous functions from a set of eventually unorganized point samples via the calculation of a weighted least squares measure biased towards the region around the point at which the approximation 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 cloud of points through either downsampling or upsampling.

Problem statement[edit]

Consider the problem of adjusting an approximation of some function to best fit a given data set. The data set consist of n points

${\displaystyle (y_i,{\mathbf{𝐱}}_i),i=1,2,\dots ,n.}$

We define an approximation in a similar way as in the weighted least squares, but in such a way that its adjustable coefficients depend on the independent variables:

${\displaystyle y=f(\mathbf{𝐱};\mathbf{𝐚}(\mathbf{𝐱})),}$

where y is the dependent variable, x are the independent variables, and a(x) are the non-constant adjustable parameters of the model. In each point x where the approximation should be evaluated, we calculate the local values of these parameters such that the model best fits the data according to a defined error criterion. The parameters are obtained by minimization of the weighted sum of squares of errors,

${\displaystyle S(\mathbf{𝐚}(\mathbf{𝐱}))={\displaystyle \sum _{i=1}^n}{w}_i(\mathbf{𝐱})(y_i-f({\mathbf{𝐱}}_i;\mathbf{𝐚}(\mathbf{𝐱})){)}^2,}$

with respect to the adjustable parameters of the model a(x) in the point of evaluation of the approximation. Note that weights are replaced by weighting functions, which are usually bell-like functions centered around x_i.

See also[edit]

• Function approximation
• Weighted least squares
• Optimization (mathematics)