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Moving least squares is a method of reconstructing
continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s from a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of unorganized point samples via the calculation of a
weighted least squares Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (''heteroscedasticity'') is incorporated into ...
measure biased towards the region around the point at which the reconstructed value is requested. In
computer graphics Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...
, 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 A point cloud is a discrete set of data Point (geometry), points in space. The points may represent a 3D shape or object. Each point Position (geometry), position has its set of Cartesian coordinates (X, Y, Z). Points may contain data other than ...
through either
downsampling In digital signal processing, downsampling, compression, and decimation are terms associated with the process of ''resampling'' in a multi-rate digital signal processing system. Both ''downsampling'' and ''decimation'' can be synonymous with ''co ...
or
upsampling In digital signal processing, upsampling, expansion, and interpolation are terms associated with the process of sample rate conversion, resampling in a multi-rate digital signal processing system. ''Upsampling'' can be synonymous with ''expansion'' ...
. In numerical analysis to handle contributions of geometry where it is difficult to obtain discretizations, the moving least squares methods have also been used and generalized to solve
PDEs In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how ...
on curved surfaces and other geometries. This includes numerical methods developed for curved surfaces for solving scalar parabolic PDEs and vector-valued hydrodynamic PDEs. In machine learning, moving least squares methods have also been used to develop model classes and learning methods. This includes function regression methods and neural network function and operator regression approaches, such as GMLS-Nets.


Definition

Consider a function f: \mathbb^n \to \mathbb and a set of sample points S = \ . Then, the moving least square approximation of degree m at the point x is \tilde(x) where \tilde minimizes the weighted least-square error :\sum_ (p(x_i)-f_i)^2\theta(\, x-x_i\, ) over all polynomials p of degree m in \mathbb^n. \theta(s) is the weight and it tends to zero as s\to \infty. In the example \theta(s) = e^. The smooth interpolator of "order 3" is a quadratic interpolator.


See also

*
Local regression Local regression or local polynomial regression, also known as moving regression, is a generalization of the moving average and polynomial regression. Its most common methods, initially developed for scatterplot smoothing, are LOESS (locally ...
*
Diffuse element method In numerical analysis the diffuse element method (DEM) or simply diffuse approximation is a meshfree method. The diffuse element method was developed by B. Nayroles, G. Touzot and Pierre Villon at the Universite de Technologie de Compiegne, in 199 ...
*
Moving average In statistics, a moving average (rolling average or running average or moving mean or rolling mean) is a calculation to analyze data points by creating a series of averages of different selections of the full data set. Variations include: #Simpl ...


References


The approximation power of moving least squares
David Levin, Mathematics of Computation, Volume 67, 1517-1531, 199
Moving least squares response surface approximation: Formulation and metal forming applications
Piotr Breitkopf; Hakim Naceur; Alain Rassineux; Pierre Villon, Computers and Structures, Volume 83, 17-18, 2005.
Generalizing the finite element method: diffuse approximation and diffuse elements
B Nayroles, G Touzot. Pierre Villon, P, Computational Mechanics Volume 10, pp 307-318, 1992


External links


An As-Short-As-Possible Introduction to the Least Squares, Weighted Least Squares and Moving Least Squares Methods for Scattered Data Approximation and Interpolation
{{Least squares and regression analysis Least squares