Moving least squares
Encyclopedia
Moving least squares is a method of reconstructing continuous function
s from a set of unorganized point samples via the calculation of a weighted least squares
measure
biased towards the region around the point at which the reconstructed 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 point cloud
through either downsampling
or upsampling
.
and a set of sample points 
where 
and the 
's are real numbers. Then, the moving least square approximation of degree 
at the point 
is 
where 
minimizes the weighted least-square error
over all polynomials
of degree 
in 
. 
is the weight and it tends to zero as 
.
In the example
.
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s from a set of unorganized point samples via the calculation of a weighted least squares
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
biased towards the region around the point at which the reconstructed value is requested.
In computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
, 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
Point cloud
A point cloud is a set of vertices in a three-dimensional coordinate system. These vertices are usually defined by X, Y, and Z coordinates, and typically are intended to be representative of the external surface of an object....
through either downsampling
Downsampling
In signal processing, downsampling is the process of reducing the sampling rate of a signal. This is usually done to reduce the data rate or the size of the data....
or upsampling
Upsampling
Upsampling is the process of increasing the sampling rate of a signal. For instance, upsampling raster images such as photographs means increasing the resolution of the image....
.
Definition
Consider a function and a set of sample points where and the 's are real numbers. Then, the moving least square approximation of degree at the point is where minimizes the weighted least-square errorover all polynomials
of degree in . is the weight and it tends to zero as .In the example
.