Hierarchical RBF
Encyclopedia
A hierarchical RBF is an interpolation
method based on RBF
.
Interpolation
based on RBF
used for the construction of shape models in 3D computer graphics
(see Stanford Bunny
on picture), treatment of results from a 3D scanner
, terrain reconstruction and others.
This problem often named "large scattered data point set interpolation".
The idea of method (for example in 3D) consists of the following:
— it is RBF
;
— it is coefficients which are the solution of the system show on picture:
for determination of surface it is necessary to estimate the value of function
in interesting points x.
A lack of such method is considerable complication
for calculate RBF
, solve system and determine surface.
of intricate problem on the great number of simple (see picture).
In this case hierarchical division of space containing points on elementary parts, the system of small dimension solves in each of which. The calculation of surface in this case is taken to the hierarchical (on the basis of tree-structure
) calculation of interpolant. A method for a 2D
case is offered Pouderoux J. et al. For a 3D
case a method is used in the tasks of 3D graphics
by W. Qiang et al. and modified by Babkov V.
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
method based on RBF
Radial basis function
A radial basis function is a real-valued function whose value depends only on the distance from the origin, so that \phi = \phi; or alternatively on the distance from some other point c, called a center, so that \phi = \phi...
.
Interpolation
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
based on RBF
Radial basis function
A radial basis function is a real-valued function whose value depends only on the distance from the origin, so that \phi = \phi; or alternatively on the distance from some other point c, called a center, so that \phi = \phi...
used for the construction of shape models in 3D computer graphics
3D computer graphics
3D computer graphics are graphics that use a three-dimensional representation of geometric data that is stored in the computer for the purposes of performing calculations and rendering 2D images...
(see Stanford Bunny
Stanford Bunny
The Stanford Bunny is a computer graphics 3D test model developed by Greg Turk and Marc Levoy in 1994 at Stanford University.The Bunny consists of data describing 69,451 triangles determined by 3D scanning a ceramic figurine of a rabbit. The data can be used to test various graphics algorithms;...
on picture), treatment of results from a 3D scanner
3D scanner
A 3D scanner is a device that analyzes a real-world object or environment to collect data on its shape and possibly its appearance . The collected data can then be used to construct digital, three dimensional models....
, terrain reconstruction and others.
This problem often named "large scattered data point set interpolation".
The idea of method (for example in 3D) consists of the following:
- let the scattered points be presented a set
- let the exist a set of values of some function in scattered points
- find a function
which will meet next condition: 
for points lies on shape and 
for points not lies on shape - as J. C. Carr et al. showed this function looks like
where:
— it is RBFRadial basis function
A radial basis function is a real-valued function whose value depends only on the distance from the origin, so that \phi = \phi; or alternatively on the distance from some other point c, called a center, so that \phi = \phi...
;
— it is coefficients which are the solution of the system show on picture:for determination of surface it is necessary to estimate the value of function
in interesting points x.A lack of such method is considerable complication
for calculate RBFRadial basis function
A radial basis function is a real-valued function whose value depends only on the distance from the origin, so that \phi = \phi; or alternatively on the distance from some other point c, called a center, so that \phi = \phi...
, solve system and determine surface.
Other similar methods
- Reduce interpolation centres (
for calculate RBFRadial basis functionA radial basis function is a real-valued function whose value depends only on the distance from the origin, so that \phi = \phi; or alternatively on the distance from some other point c, called a center, so that \phi = \phi...
and solve system,
for determine surface) - Compactly supported RBFRadial basis functionA radial basis function is a real-valued function whose value depends only on the distance from the origin, so that \phi = \phi; or alternatively on the distance from some other point c, called a center, so that \phi = \phi...
(
for calculate RBFRadial basis functionA radial basis function is a real-valued function whose value depends only on the distance from the origin, so that \phi = \phi; or alternatively on the distance from some other point c, called a center, so that \phi = \phi...
,
for solve system, 
for determine surface) - FMMFast Multipole MethodThe fast multipole method is a mathematical technique that was developed to speed up the calculation of long-ranged forces in the n-body problem...
(
for calculate RBFRadial basis functionA radial basis function is a real-valued function whose value depends only on the distance from the origin, so that \phi = \phi; or alternatively on the distance from some other point c, called a center, so that \phi = \phi...
,
for solve system, 
for determine surface)
Hierarchical algorithm
An idea of hierarchical algorithm is an acceleration of calculations due to decompositionDecomposition (computer science)
Decomposition in computer science, also known as factoring, refers to the process by which a complex problem or system is broken down into parts that are easier to conceive, understand, program, and maintain.- Overview :...
of intricate problem on the great number of simple (see picture).
In this case hierarchical division of space containing points on elementary parts, the system of small dimension solves in each of which. The calculation of surface in this case is taken to the hierarchical (on the basis of tree-structure
Tree (data structure)
In computer science, a tree is a widely-used data structure that emulates a hierarchical tree structure with a set of linked nodes.Mathematically, it is an ordered directed tree, more specifically an arborescence: an acyclic connected graph where each node has zero or more children nodes and at...
) calculation of interpolant. A method for a 2D
2D computer graphics
2D computer graphics is the computer-based generation of digital images—mostly from two-dimensional models and by techniques specific to them...
case is offered Pouderoux J. et al. For a 3D
3D computer graphics
3D computer graphics are graphics that use a three-dimensional representation of geometric data that is stored in the computer for the purposes of performing calculations and rendering 2D images...
case a method is used in the tasks of 3D graphics
3D computer graphics
3D computer graphics are graphics that use a three-dimensional representation of geometric data that is stored in the computer for the purposes of performing calculations and rendering 2D images...
by W. Qiang et al. and modified by Babkov V.