Scalable Representation - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 356

30.1.1 Radial Basis Functions
Previously, most radial basis function (RBF) encoding work has concentrated on
surface fitting. In the early 1970s, RBFs were used to interpolate geography surface
data using multiquadric RBFs [ 13 ] and results showed that RBFs are a good inter-
polation basis function for smooth surface data sets. Hardy [ 14 ] presented 20 years
of discovery in the theory and applications of multiquadric RBFs and surveys RBF
work from 1968 to 1988. Franke [ 10 ] showed scattered data interpolation and tests
using several methods, such as the inverse distance weighted method, the rectangle
based blending method, and the triangle based blending method. He compared these
methods and showed that Hardy's multiquadric approach is best. Since Franke's
work, multiquadrics have been considered the best basis function in most surface
fitting research. After Franke's survey, Franke and Nielson [ 12 ] collected more work
on surface fitting and presented their research by surveying and comparing several
techniques. For better interpolation, the least squares approach were used by Franke
and Hagen [ 11 ]. For the approximation of surface fitting, knot (center) selection [ 23 ]
was introduced using thin plate splines by Dirichlet tessellation. Through knot selec-
tion, encoded data can be reduced and a small number of basis function can represent
the whole data set.
Although RBFs have been used to reconstruct surfaces by approximating scattered
data sets, they were primarily used for mesh reduction of surface representations [ 4 ,
25 , 32 , 39 ]. In more recent work on surface fitting, Carr et al. [ 4 ] showed surface
fitting as an approximation using multiquadric RBFs. They iteratively added basis
functions using a greedy algorithm by computing fitting errors, where basis functions
were added at larger error points. In their work, the zero level set implicit surface
of the distance function was fit and energy (error) was minimized for the smoothest
interpolant. Ohtake et al. [ 31 ] also showed the fitting of implicit surfaces. They
selected centers based on the density of data points. More basis functions were
added in higher density areas. By linking the RBF approximation and the partition
of unity method [ 30 ], Ohtake el al. presented a robust approximation for noisy data.
Volume fitting using RBFs was introduced by Nielson et al. [ 28 , 29 ], where they
extended surface fitting methods to volume fitting. Their approaches showed good
approximation of volume data. In more recent work on volume fitting, Co et al. [ 5 ]
showed a hierarchical representation of volumetric data sets based on clusters com-
puted by Principal Component Analysis (PCA). A level of detail representation was
extracted by either the hierarchical level or the error. Jang et al. [ 16 , 17 ] and Weiler
et al. [ 41 ] proposed a functional representation approach for interactive volume ren-
dering. Their approaches were designed for any scattered datasets and directly vol-
ume rendered the basis functions without resampling. Moreover, using ellipsoidal
basis functions, they improved the functional representation statistically and visu-
ally [ 16 ]. Recently, Ledergerber et al. [ 19 ] applied amoving least square to interpolate
the volumetric data and Vuçini et al. [ 40 ] reconstructed non-uniform volumetric data
by B-splines.