Moving least squares

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

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

$(y_{i},{\mathbf {x}}_{i}),i=1,2,\dots ,n.$

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:

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

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,

$S({\mathbf {a}}({\mathbf {x}}))=\sum _{i=1}^{n}w_{i}({\mathbf {x}})(y_{i}-f({\mathbf {x}}_{i};{\mathbf {a}}({\mathbf {x}})))^{2},$

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

• Function approximation
• Weighted least squares
• Optimization