Moving least squares

| Main Article
| Discussion
| Related Articles  [?]
| Bibliography  [?]
| External Links  [?]
| Citable Version  [?]

|
|
|
|
|
|
|
|

This editable Main Article is under development and subject to a disclaimer.

[edit intro]

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

(yi,𝐱i),i=1,2,…,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(𝐱;𝐚(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(𝐚(𝐱))=∑i=1nwi(𝐱)(yi−f(𝐱i;𝐚(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 xi.

## See also[edit]

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