This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Discrete Sibson Interpolation
March/April 2006 (vol. 12 no. 2)
pp. 243-253

Abstract—Natural-neighbor interpolation methods, such as Sibson's method, are well-known schemes for multivariate data fitting and reconstruction. Despite its many desirable properties, Sibson's method is computationally expensive and difficult to implement, especially when applied to higher-dimensional data. The main reason for both problems is the method's implementation based on a Voronoi diagram of all data points. We describe a discrete approach to evaluating Sibson's interpolant on a regular grid, based solely on finding nearest neighbors and rendering and blending d{\hbox{-}}\rm dimensional spheres. Our approach does not require us to construct an explicit Voronoi diagram, is easily implemented using commodity three-dimensional graphics hardware, leads to a significant speed increase compared to traditional approaches, and generalizes easily to higher dimensions. For large scattered data sets, we achieve two-dimensional (2D) interpolation at interactive rates and 3D interpolation (3D) with computation times of a few seconds.

[1] P. Alfeld, Mathematical Methods in Computer Aided Geometric Design, pp. 1-33, 1989.
[2] G. Farin, “Surfaces over Dirichlet Tessellations,” Computer Aided Geometric Design, vol. 7, nos. 1-4, pp. 281-292, 1990.
[3] K.E. HoffIII, J. Keyser, M. Lin, D. Manocha, and T. Culver, “Fast Computation of Generalized Voronoi Diagrams Using Graphics Hardware,” Proc. SIGGRAPH '99 Conf., pp. 277-286, Aug. 1999.
[4] W. Heidrich and H.-P. Seidel, “Realistic, Hardware-Accelerated Shading and Lighting,” Proc. SIGGRAPH '99 Conf., pp. 171-178, Aug. 1999.
[5] M.J. Harris, G. Coombe, T. Scheuermann, and A. Lastra, “Physically-Based Visual Simulation on Graphics Hardware,” Proc. Graphics Hardware 2002 Conf., pp. 109-118, Sept. 2002.
[6] J. Krüger and R. Westermann, “Linear Algebra Operators for GPU Implementation of Numerical Algorithms,” ACM Trans. Graphics, vol. 22, pp. 908-916, July 2003.
[7] C.J. Thompson, S. Hahn, and M. Oskin, “Using Modern Graphics Architectures for General-Purpose Computing: A Framework and Analysis,” Proc. 35th Ann. Int'l Symp. Microarchitecture, pp. 306-317, Nov. 2002.
[8] I. Amidror, “Scattered Data Interpolation Methods for Electronic Imaging Systems: A Survey,” J. Electronic Imaging, vol. 2, no. 11, pp. 157-176, 2002.
[9] T.A. Foley, D.A. Lane, G.M. Nielson, R. Franke, and H. Hagen, “Interpolation of Scattered Data on Closed Surfaces,” Computer Aided Geometric Design, vol. 7, nos. 1-4, pp. 303-312, 1990.
[10] R. Franke and G. Nielson, Geometric Modeling: Methods and Applications, pp. 131-160, Springer-Verlag, 1991.
[11] S.K. Lodha and R. Franke, “Scattered Data Techniques for Surfaces,” Proc. Dagstuhl Conf. Scientific Visualization, pp. 182-222, 1999.
[12] G. Nielson, “Scattered Data Modeling,” IEEE Computer Graphics and Applications, vol. 13, no. 1, pp. 60-70, Jan. 1993.
[13] G.M. Nielson, T.A. Foley, B. Hamann, and D. Lane, “Visualizing and Modeling Scattered Multivariate Data,” IEEE Computer Graphics and Applications, vol. 11, no. 3, pp. 47-55, May 1991.
[14] D. Shepard, “A Two-Dimensional Interpolation Function for Irregularly Spaced Data,” Proc. 23rd Nat'l Conf., pp. 517-524, Aug. 1968.
[15] R. Sibson, “A Vector Identity for the Dirichlet Tessellation,” Math. Proc. Cambridge Philosophical Soc., vol. 87, no. 1, pp. 151-155, 1980.
[16] S.J. Owen, “An Implementation of Natural Neighbor Interpolation in Three Dimensions,” master's thesis, Brigham Young Univ., 1992.
[17] N. Sukumar, B. Morann, and T. Belyschko, “The Natural Element Method in Solid Mechanics,” Int'l J. Numerical Methods in Eng., vol. 43, pp. 839-887, Nov. 1998.
[18] C. Sigg, R. Peikert, and M. Gross, “Signed Distance Transform Using Graphics Hardware,” Proc. IEEE Visualization '03 Conf., 2003.
[19] Y. Jang, M. Weiler, M. Hopf, J. Huang, D.S. Ebert, K.P. Gaither, and T. Ertl, “Interactively Visualizing Procedurally Encoded Scalar Fields,” Proc. EG/IEEE TCVG Symp. Visualization VisSym '04, O. Deussen et al. eds., 2004.
[20] Q. Fan, A. Efrat, V. Koltun, S. Krishnan, and S. Venkatasubramanian, “Hardware Assisted Natural Neighbour Interpolation,” Proc. Seventh Workshop Algorithm Eng. and Experiments (ALENEX), 2005.
[21] Y.L.H. Yee, “An Implementation of Natural Neighbor Interpolation in Three Dimensions,” master's thesis, Brigham Young Univ., 1992.
[22] J.-D. Boissonnat and F. Cazals, “Smooth Surface Reconstruction Via Natural Neighbour Interpolation Of Distance Functions,” Proc. 16th Ann. Symp. Computational Geometry, pp. 223-232, 2000.
[23] O. Kreylos and B. Hamann, “On Simulated Annealing and the Construction of Linear Spline Approximations for Scattered Data,” Proc. Joint EUROGRAPHICS and IEEE TVCG Symp. Visualization (VisSym '99), E. Groeller et al., eds., pp. 189-198, 1999.
[24] C.S. Co, B. Heckel, H. Hagen, B. Hamann, and K.I. Joy, “Hierarchical Clustering for Unstructured Volumetric Scalar Fields,” Proc. IEEE Visualization 2003 Conf., G. Turk et al., eds., pp. 325-332, Oct. 2003.

Index Terms:
Scattered data interpolation, natural-neighbor interpolation, graphics hardware.
Citation:
Sung W. Park, Lars Linsen, Oliver Kreylos, John D. Owens, Bernd Hamann, "Discrete Sibson Interpolation," IEEE Transactions on Visualization and Computer Graphics, vol. 12, no. 2, pp. 243-253, March-April 2006, doi:10.1109/TVCG.2006.27
Usage of this product signifies your acceptance of the Terms of Use.