Hierarchical RBF


In computer graphics, hierarchical RBF is an interpolation method based on radial basis functions. Hierarchical RBF interpolation has applications in treatment of results from a 3D scanner, terrain reconstruction, and the construction of shape models in 3D computer graphics.
This problem is informally named as "large scattered data point set interpolation."

Method

The steps of the interpolation method are as follows:
  1. Let the scattered points be presented as set
  2. Let there exist a set of values of some function in scattered points
  3. Find a function that will meet the condition for points lying on the shape and for points not lying on the shape
As J. C. Carr et al. showed, this function takes the form where is a radial basis function and are the coefficients that are the solution of the following system of equations">Linear system of equations">system of equations:
For determination of surface, it is necessary to estimate the value of function in specific points x. A lack of such method is a considerable complication on the order of to calculate RBF, solve system, and determine surface.

Other methods

Hierarchical algorithm

A hierarchical algorithm allows for an acceleration of calculations due to decomposition of intricate problems on the great number of simple.
In this case, hierarchical division of space contains points on elementary parts, and the system of small dimension solves for each. The calculation of surface in this case is taken to the hierarchical calculation of interpolant. A method for a 2D case is offered by 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.