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