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Visualizing and modeling unstructured data

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Abstract

Scientific data are often sampled at unstructured spatial locations because of physical constraints, yet most visualization software applies only to gridded or regular data. We discuss several effective techniques for representing scalar and vector-valued functions that interpolate to irregularly located data. Special attention is given to the situations in which the sampling domain is a 2D plane, a 3D volume, or a closed 3D surface. The interpolants can be evaluated on a fine regular grid and they can then be visualized with conventional techniques. Instead of giving a comprehensive survey of this subject, we concentrate on several methods that were developed in the last couple of years.

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References

  1. Alfeld P (1989) Scattered data interpolation in three or more variables. In: Lyche T, Schumaker LL (eds) Mathematical methods in CAGD. Academic Press, New York, pp. 1–33

    Google Scholar 

  2. Bacchelli-Montefusco L, Casciola G (1989) Algorithm 677:C 1 surface interpolation. ACM Trans Math Software 15:365–374

    Google Scholar 

  3. Barnhill RE (1977) Representation and approximation of surfaces. In: Rice JR (ed) Mathematical software III. Academic Press, New York, pp 69–120

    Google Scholar 

  4. Barnhill RE (1985) Surfaces in computer aided geometric design: a survey with new results. Comput Aided Geometric Design 2:1–17

    Google Scholar 

  5. Barnhill RE (ed) (1991) Geometry processing for design and manufacturing. SIAM, Philadelphia

    Google Scholar 

  6. Barnhill RE, Birkhoff G, Gordon WJ (1973) Smooth interpolation in triangles. J Approx Theor 8:114–128

    Google Scholar 

  7. Barnhill RE, Foley TA (1991) Methods for constructing surfaces on surfaces. In: Hagen H, Roller D (eds) Geometric modeling: methods and their applications. Springer, Heidelberg, pp 1–15

    Google Scholar 

  8. Barnhill RE, Foley TA, Lane DA (1992) Interpolating scattered multivariate data as a function of time. Comput Aided Geometric Design 9:337–348

    Google Scholar 

  9. Barnhill RE, Piper BR, Rescorla KL (1987) Interpolation to arbitrary data on a surface. In: Farin G (ed) Geometric modeling. SIAM, Philadelphia, pp 281–289

    Google Scholar 

  10. Barnhill RE, Ou HS (1990) Surfaces defined on surfaces. Comput Aided Geometric Design 7:323–336

    Google Scholar 

  11. Buhmann MD (1988) Convergence of univariate quasi-interpolation using multiquadrics. IMA J Numer Anal 8:365–383

    Google Scholar 

  12. Buhmann MD, Powell MJD (1990) Radial basis function interpolation on an infinite regular grid. In: Mason JC, Cox MG (eds) Algorithms for approximation II. Chapman and Hall, London, pp 146–169

    Google Scholar 

  13. Carlson RE, Foley TA (1991) The parameterR 2 in multiquadric interpolation. Comput Math Appl 21:29–42

    Google Scholar 

  14. Chiyokura H, Kimura F (1983) Design of solids with free-form surfaces. Comput Graph 17:289–298

    Google Scholar 

  15. Clough RW, Tocher JL (1965) Finite element stiffness matrices for analysis of plates in bending. Proc Conference Matrix Methods in Structure Mechanics, Air Force Institute of Technology, Wright-Patterson AFB, Ohio

  16. Drebin RA, Carpenter L, Hanrahan P (1988) Volume rendering. Comput Graph 22:65–74

    Google Scholar 

  17. Duchon J (1975) Splines minimizing rotation invariant seminorms in Sobolev spaces. In: Schempp W, Zeller K (eds) Multivariate approximation theory. Birkhäuser, Basel, pp 85–100

    Google Scholar 

  18. Dyn N, Levin D (1983) Iterative solution of systems originating from integral equations and surface interpolation. SIAM J Numer Anal 20:377–390

    Google Scholar 

  19. Dyn N, Levin D, Rippa S (1990a) Algorithms for the construction of data dependent triangulations. In: Mason JC, Cox MG (eds) Algorithms for approximation II. Chapman and Hall, London

    Google Scholar 

  20. Dyn N, Levin D, Rippa S (1990b) Data dependent triangulations for piecewise linear interpolation. IMA J Numer Anal

  21. Farin G (1985) A modified Clough-Tocher interpolant. Comput Aided Geometric Design 2:19–27

    Google Scholar 

  22. Farin G (1990a) Curves and surfaces for computer aided geometric design. Academic Press, San Diego

    Google Scholar 

  23. Farin G (1990b) Surfaces over Dirichlet tessellations. Comput Aided Geometric Design 7:281–292

    Google Scholar 

  24. Foley TA (1986) Scattered data interpolation and approximation with error bounds. Comput Aided Geometric Design 3:163–177

    Google Scholar 

  25. Foley TA (1987) Interpolation and approximation of 3D and 4D scattered data. Comput Math Appl 13:711–740

    Google Scholar 

  26. Foley TA (1990) Interpolation of scattered data on a spherical domain. In: Mason JC, Cox MG (eds) Algorithms for approximation II. Chapman and Hall, London, pp 303–310

    Google Scholar 

  27. Foley TA, Lane DA (1990) Visualization of irregular multivariate data. Proc Visualization'90 IEEE Computer Society, Los Alamitos, Calif., pp 247–254

  28. Foley TA, Lane DA, Nielson GM (1990) Towards animating ray-traced volume visualization. Visualization Comput Animat J 1:2–8

    Google Scholar 

  29. Foley TA, Lane DA, Nielson GM, Franke R, Hagen H (1990) Interpolation of scattered data on closed surfaces. Comput Aided Geometric Design 7:303–312

    Google Scholar 

  30. Foley TA, Lane DA, Nielson GM, Ramaraj R (1990) Visualizing functions over a sphere. IEEE Comput Graph Appl 10:32–40

    Google Scholar 

  31. Foley TA, Opitz K (1992) Hybrid cubic Bézier triangle patches. In: Lyche T, Schumaker LL (eds) Mathematical methods in computer aided geometric design II. Academic Press, New York pp 275–286

    Google Scholar 

  32. Franke R (1979) A critical comparison of some methods for interpolation of scattered data. Technical Report NPS-53-79-003 Naval Postgraduate School

  33. Franke R (1982a) Scattered data interpolation: tests of some methods. Math Comp 38:181–200

    Google Scholar 

  34. Franke R (1982b) Smooth interpolation of scattered data by local thin plate splines. Comput Math Appl 8:273–281

    Google Scholar 

  35. Franke R (1987) Recent advances in the approximation of surfaces from scattered data. In: Schumaker LL, Chui CC, Utreras F (eds) Topics in multivariate approximations. Academic Press, New York, pp 175–184

    Google Scholar 

  36. Franke R, Nielson GM (1980) Smooth interpolation to large sets of scattered data. Int J Numer Methods Eng 15:1691–1704

    Google Scholar 

  37. Franke R, Nielson GM (1991) Scattered data interpolation: a tutorial and survey. In: Hagen H, Roller D (eds) Geometric modeling: methods and their applications. Springer, Berlin Heidelberg New York pp 131–160

    Google Scholar 

  38. Franke R, Schumaker LL (1987) A bibliography of multivariate approximation. In: Schumaker LL, Chui CC, Utreras F (eds) Topics in multivariate approximations. Academic Press, New York, pp 275–335

    Google Scholar 

  39. Freeden W (1984) Spherical spline interpolation — basic theory and computational aspects. J Comput Appl Math 11:367–375

    Google Scholar 

  40. Grosse E (1990) A catalog of algorithms for approximation. In: Mason JC, Cox MG (eds) Algorithms for approximation II. Chapman and Hall, London

    Google Scholar 

  41. Hagen H, Pottmann H (1989) Curvature continuous triangular interpolants. In: Lyche T, Schumaker LL (eds) Mathematical methods in computer aided geometric design. Academic Press. New York, pp 373–384

    Google Scholar 

  42. Hagen H, Schreiber T, Gschwind E (1990) Methods of surface interrogation. Proc Visualization'90 IEEE Computer Society, Los Alamitos, Calif., pp 187–193

  43. Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76:1905–1915

    Google Scholar 

  44. Hardy RL (1990) Theory and applications of the multiquadric-biharmonic method. Comput Math Appl 19:163–208

    Google Scholar 

  45. Hardy RL, Goepfert WM (1975) Least squares prediction of gravity anomalies, geoidal undulations, and deflections of the vertical with multiquadric harmonic functions. Geophys Res Lett 2:423–426

    Google Scholar 

  46. Hardy RL, Nelson SA (1986) A multiquadric biharmonic representation and approximation of disturbing potential. Geophys Res Lett 13:18–21

    Google Scholar 

  47. Herron GF (1985) A characterization of certainC 1 discrete triangular interpolants. SIAM J Numer Anal 22:811–819

    Google Scholar 

  48. Hoschek J, Lasser D (1989) Grundlagen der geometrischen Datenverarbeitung. Teubner

  49. Jackson IRH (1988) Convergence properties of radial basis functions. Constructive Approx 4:243–264

    Google Scholar 

  50. Kansa EJ (1990) Multiquadrics — a scattered data approximation scheme with applications to fluid dynamics — I. Surface approximations and partial derivative estimates. Comput Math Appl 19:127–145

    Google Scholar 

  51. Kansa EJ (1990) Multiquadrics — a scattered data approximation scheme with applications to fluid dynamics — II. Solutions to parabolic, hyperbolic and elliptical partial differential equations. Comput Math Appl 19:147–161

    Google Scholar 

  52. Kaufman A (1990) Volume visualization. IEEE Computer Society Press, Los Alamitos, Calif.

    Google Scholar 

  53. Kelly AD, Malin AC, Nielson GM (1988) Terrain simulation using a model of stream erosion. Comput Graph 22:163–168

    Google Scholar 

  54. Lancaster P, Salkauskas K (1986) Curve and surface fitting: an introduction. Academic Press, New York

    Google Scholar 

  55. Lawson CL (1977) Software forC 1 surface interpolation. In: Rice JR (ed) Mathematical software III. Academic Press, New York, pp 161–194

    Google Scholar 

  56. Lawson CL (1984)C 1 surface interpolation for data on a sphere. Rocky Mountain J Math 14:177–184

    Google Scholar 

  57. Levoy M (1988) Display of surfaces from volume data. IEEE Comput Graph Appl 8:29–37

    Google Scholar 

  58. Lorenson WE, Cline HE (1987) Marching cubes: a high resolution 3D surface construction algorithm. Comput Graph 21:163–169

    Google Scholar 

  59. Madych WR, Nelson SA (1988) Multivariate interpolation and conditionally positive definite functions. J Approx Theor Appl 4:77–89

    Google Scholar 

  60. Miechelli CA (1986) Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr Approx 2:11–22

    Google Scholar 

  61. Nielson GM (1979) The side-vertex method for interpolation in triangles. J Approx Theor 318–336

  62. Nielson GM (1983) A method for interpolation of scattered data based upon a minimum norm network. Math Comp 40:253–271

    Google Scholar 

  63. Nielson GM (1987) Coordinate free scattered data interpolation. In: Schumaker LL, Chui CC, Utreras F (eds) Topics in multivariate approximations. Academic Press. New York, pp 175–184

    Google Scholar 

  64. Nielson GM, Foley TA (1989) A survey of applications of an affine invariant norm. In: Lyche T, Schumaker LL (eds) Mathematical methods in CAGD. Academic Press, New York, pp 445–467

    Google Scholar 

  65. Nielson GM, Foley TA, Hamann B, Lane D (1991) Visualizing and modeling scattered multivariate data. IEEE Comput Graph Appl 11:47–55

    Google Scholar 

  66. Nielson GM, Franke R (1984) A method for construction of surfaces under tension. Rocky Mountain J Math 14:203–221

    Google Scholar 

  67. Nielson GM, Hamann B (1990) Techniques for the interactive visualization of volumetric data. Proc Visualization'90, IEEE Computer Society, Los Alamitos, Calif., pp 45–50

  68. Nielson GM, Hamann B (1991) The asymptotic decider: resolving the ambiguity in marching cubes. Proc Visualization'91, IEEE Computer Society, Los Alamitos, Calif., pp 83–91

  69. Nielson GM, Ramaraj R (1987) Interpolation over a sphere based upon a minimum norm network. Comput Aided Geometric Design 4:41–57

    Google Scholar 

  70. Pottmann H (1991) Scattered data interpolation of scattered data based upon generalized minimum norm networks. Constr Approx 7:247–256

    Google Scholar 

  71. Pottmann H (1992) Interpolation on surfaces using minimum norm networks. Comput Aided Geometric Design 9:51–67

    Google Scholar 

  72. Pottmann H, Eck M (1990) Modified multiquadric methods for scattered data interpolation over a sphere. Comput Aided Geometric Design 7:313–322

    Google Scholar 

  73. Pottmann H, Hagen H, Divivier A (1991) Visualizing functions on a surface. Visualization Comput Animat J 2:52–58

    Google Scholar 

  74. Powell MJD (1987) Radial basis functions for multivariate interpolation: a review. In: Mason JC, Cox MG (eds) Algorithms for approximation. Oxford University Press, Oxford, pp 143–167

    Google Scholar 

  75. Powell MJD (1991) The theory of radial basis function approximation in 1990. In: Light W (ed) Advances in numerical analysis II: wavelets, subdivision algorithms and radial functions. Oxford University Press, Oxford, pp 105–210

    Google Scholar 

  76. Quak E, Schumaker LL (1990) Cubic spline fitting using data dependent triangulations. Comput Aided Geometric Design 7:293–301

    Google Scholar 

  77. Ramaraj R (1986) Interpolation and display of scattered data over a sphere. MS thesis, Computer Science Department, Arizona State University, Tempe, Ariz.

    Google Scholar 

  78. Renka RJ (1984) Interpolation of data on the surface of a sphere. ACM Trans Math Software 10:417–436

    Google Scholar 

  79. Renka RL (1988) Algorithm 660: QSHEP2D: quadratic Shepard method for bivariate interpolation to scattered data. Trans Math Software 14:149–150

    Google Scholar 

  80. Schumaker LL (1976) Fitting surfaces to scattered data. In: Lorentz GG, Chui CK, Schumaker LL (eds) Approximation theory, Academic Press, New York

    Google Scholar 

  81. Schumaker LL (1987) Triangulation methods. In: Schumaker LL, Chui C, Utreras F (eds) Topics in multivariate approximations. Academic Press, New York, pp 219–232

    Google Scholar 

  82. Schweikert B (1966) An interpolation curve using a spline in tension. J Math Phys 45:312–317

    Google Scholar 

  83. Sibson R (1981) A brief description of the natural neighbor interpolant. In: Barnett V (ed) Interpolating multivariate data. Wiley, New York

    Google Scholar 

  84. Stead SE (1984) Estimation of gradients from scattered data. Rocky Mountain J Math 14:219–232

    Google Scholar 

  85. Whaba G (1981) Spline interpolation and smoothing on a sphere. SIAM J Sci Stat Comput 2:5–16

    Google Scholar 

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Authors and Affiliations

  1. Computer Science Department, Arizona State University, 85287-5406, Tempe, AZ, USA

    T. A. Foley & G. M. Nielson

  2. Informatik Department, Universität Kaiserslautern, P.O. Box 3049, D-67653, Kaiserslautern, Germany

    H. Hagen

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  1. T. A. Foley
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  3. G. M. Nielson
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Correspondence to T. A. Foley.

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Foley, T.A., Hagen, H. & Nielson, G.M. Visualizing and modeling unstructured data. The Visual Computer 9, 439–449 (1993). https://doi.org/10.1007/BF01888718

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