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.

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 $\mathbf {P} =\{\mathbf {c} _{i}=(\mathbf {x} _{i},\mathbf {y} _{i},\mathbf {z} _{i})\vert _{i=1}^{N}\subset \mathbb {R} ^{3}\}$
Let there exist a set of values of some function in scattered points $\mathbf {H} =\{\mathbf {h} _{i}\vert _{i=1}^{N}\subset \mathbb {R} \}$
Find a function $\mathbf {f} (\mathbf {x} )$ which will meet the condition $\mathbf {f} (\mathbf {x} )=1$ for points lying on the shape and $\mathbf {f} (\mathbf {x} )\neq 1$ for points not lying on the shape.
As J. C. Carr et al. showed [1] this function looks like $\mathbf {f} (\mathbf {x} )=\sum _{i=1}^{N}\lambda _{i}\varphi (\mathbf {x} ,\mathbf {c} _{i})$ where:

$\varphi$ — it is RBF; $\lambda$ — 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 $\mathbf {f} (\mathbf {x} )$ in interesting points x. A lack of such method is considerable complication [2] $\mathbf {O} (\mathbf {n} ^{2})$ for calculate RBF, solve system and determine surface.

Other similar methods

Reduce interpolation centres ( $\mathbf {O} (\mathbf {n} ^{2})$ for calculate RBF and solve system, $\mathbf {O} (\mathbf {m} \mathbf {n} )$ for determine surface)
Compactly supported RBF ( $\mathbf {O} (\mathbf {n} \log {\mathbf {n} })$ for calculate RBF, $\mathbf {O} (\mathbf {n} ^{1.2..1.5})$ for solve system, $\mathbf {O} (\mathbf {m} \log {\mathbf {n} })$ for determine surface)
FMM ( $\mathbf {O} (\mathbf {n} ^{2})$ for calculate RBF, $\mathbf {O} (\mathbf {n} \log {\mathbf {n} })$ for solve system, $\mathbf {O} (\mathbf {m} +\mathbf {n} \log {\mathbf {n} })$ 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.[3] For a 3D case a method is used in the tasks of 3D graphics by W. Qiang et al.[4] and modified by Babkov V.[5]

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References

Carr, J.C.; Beatson, R.K.; Cherrie, J.B.; Mitchell, T.J.; Fright, W.R.; McCallum B.C.; Evans, T.R. (2001), “Reconstruction and Representation of 3D Objects with Radial Basis Functions” ACM SIGGRAPH 2001, Los Angeles, CA, P. 67–76.
Bashkov, E.A.; Babkov, V.S. (2008) “Research of RBF-algorithm and his modifications apply possibilities for the construction of shape computer models in medical practice”. Proc Int. Conference "Simulation-2008", Pukhov Institute for Modelling in Energy Engineering, Archived 2011-07-22 at the Wayback Machine (in Russian)
Pouderoux, J. et al. (2004), “Adaptive hierarchical RBF interpolation for creating smooth digital elevathion models”, Proc. 12-th ACM Int. Symp. Advances in Geographical information Systems 2004, ACP Press, P. 232–240
Qiang, W.; Pan, Z.; Chun, C.; Jiajun, B. (2007), “Surface rendering for parallel slice of contours from medical imaging”, Computing in science & engineering, 9(1), January–February 2007, P 32–37
Babkov, V.S. (2008) “Modification of hierarchical RBF method for 3D-modelling based on laser scan result”. Proc. Int. Conference “Modern problems and achievement of radio, communication and informatics”, Zaporizhzhya National Technical University, Archived 2011-07-22 at the Wayback Machine (in Ukrainian)

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[1] Carr, J.C.; Beatson, R.K.; Cherrie, J.B.; Mitchell, T.J.; Fright, W.R.; McCallum B.C.; Evans, T.R. (2001), “Reconstruction and Representation of 3D Objects with Radial Basis Functions” ACM SIGGRAPH 2001, Los Angeles, CA, P. 67–76.

[2] Bashkov, E.A.; Babkov, V.S. (2008) “Research of RBF-algorithm and his modifications apply possibilities for the construction of shape computer models in medical practice”. Proc Int. Conference "Simulation-2008", Pukhov Institute for Modelling in Energy Engineering, Archived 2011-07-22 at the Wayback Machine (in Russian)

[3] Pouderoux, J. et al. (2004), “Adaptive hierarchical RBF interpolation for creating smooth digital elevathion models”, Proc. 12-th ACM Int. Symp. Advances in Geographical information Systems 2004, ACP Press, P. 232–240

[4] Qiang, W.; Pan, Z.; Chun, C.; Jiajun, B. (2007), “Surface rendering for parallel slice of contours from medical imaging”, Computing in science & engineering, 9(1), January–February 2007, P 32–37

[5] Babkov, V.S. (2008) “Modification of hierarchical RBF method for 3D-modelling based on laser scan result”. Proc. Int. Conference “Modern problems and achievement of radio, communication and informatics”, Zaporizhzhya National Technical University, Archived 2011-07-22 at the Wayback Machine (in Ukrainian)