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Hierarchical RBF

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Hierarchical RBF

In computer graphics, a hierarchical RBF is an interpolation method based on Radial basis functions (RBF). Hierarchical RBF interpolation has applications in the construction of shape models in 3D computer graphics (see Stanford Bunny image below), treatment of results from a 3D scanner, terrain reconstruction and others.

Contents

This problem is informally 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 as set P = { c i = ( x i , y i , z i ) | i = 0 N ⊂ R 3 }
  • Let there exist a set of values of some function in scattered points H = { h i | i = 0 N ⊂ R }
  • Find a function f ( x ) which will meet the condition f ( x ) = 1 for points lying on the shape and f ( x ) ≠ 1 for points not lying on the shape.
  • As J. C. Carr et al. showed this function looks like f ( x ) = ∑ i = 1 N λ i φ ( x , c i ) where:

    • φ — it is RBF; λ — 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 f ( x ) in interesting points x. A lack of such method is considerable complication O ( n 2 ) for calculate RBF, solve system and determine surface.

    Other similar methods

  • Reduce interpolation centres ( O ( n 2 ) for calculate RBF and solve system, O ( m n ) for determine surface)
  • Compactly supported RBF ( O ( n log ⁡ n ) for calculate RBF, O ( n 1.2..1.5 ) for solve system, O ( m log ⁡ n ) for determine surface)
  • FMM ( O ( n 2 ) for calculate RBF, O ( n log ⁡ n ) for solve system, O ( m + n log ⁡ n ) for determine surface)

    • Hierarchical algorithm

    An idea of hierarchical algorithm is an acceleration of calculations due to decomposition 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.

    References

    Hierarchical RBF Wikipedia
    (Text) CC BY-SA


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