Abstract
Scattered data is, by definition, irregularly spaced. Uniform surface schemes are generally not well adapted to the locally varying nature of such data. Conversely, triangular B-spline surfaces are more flexible in that they can be built over arbitrary triangulations and thus can be adapted to the scattered data.
This paper discusses the use of DMS spline surfaces for approximation of scattered data. A method is provided for automatically triangulating the domain containing the points and generating basis functions over this triangulation. A surface approximating the data is then found by a combination of least squares and bending energy minimization This combination serves both to generate a smooth surface and to accommodate for gaps in the data. Examples are presented which demonstrate the effectiveness of the technique for mathematical, geographical and other data sets.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
S. Auerbach, R. H. J. Gmelig Meyling, M. Neamtu, and H. Schaeben. Approximation and geometric modeling with simplex B-splines associated with irregular triangles. Computer Aided Geometric Design, 8: 67–87, 1991.
G. Celniker and D. Gossard. Deformable curve and surface finite-elements for free-form shape design. In Proceedings of SIGGRAPH ‘81, pages 257–265. ACM SIGGRAPH, acm Press, 1991.
W. Dahmen and C. A. Micchelli. On the linear independence of multivariate B-splines, I. triangulations of simploids. SIAM Journal on Numerical Analysis, 19 (5): 993–1012, October 1982.
W. Dahmen, C. A. Micchelli, and H.-P. Seidel. Blossoming begets B-spline bases built better by B-patches. Mathematics of Computation, 59 (199): 97–115, July 1992.
P. Dierckx. Curve and Surface Fitting with Splines. Monographs on Numerical Analysis. Clarendon Press, 1993.
A. A. Elassal and V. M. Caruso. Digital Elevation Models. USGS Digital Cartographic Data Standards. U.S. Geological Survey, 1985.
P. Fong and H.-P. Seidel. An implementation of triangular B-spline surfaces over arbitrary triangulations. Computer Aided Geometric Design, 10: 267–275, 1993.
S. Fortune. Voronoi diagrams and Delaunay triangulations. In D.-Z. Du and F.K. Hwang, editors, Computing in Euclidean Geometry, pages 193–233. World Scientific Publishing, 1994.
G. Greiner. Surface construction based on variational principles. In P. J. Laurent, A. Le Méhauté, and L L Schumaker, editors, Wavelets, Images and Surface Fitting. A K Peters, 1994.
G. Greiner. Variational design and fairing of spline surfaces. In Proceedings of EUROGRAPHICS ‘84, pages C.143-C. 154. Eurographics Association, Blackwell, 1994.
M. Halstead, M. Kass, and T. DeRose. Efficient, fair interpolation using Catmull-Clark surfaces. In Proceedings of SIGGRAPH ‘83, pages 35–44. ACM SIGGRAPH, acm Press, 1993.
K. Höllig. Multivariate splines. SIAM Journal on Numerical Analysis, 19 (5): 1013–1031, October 1982.
J. Hoschek and D. Lasser. Grundlagen der geometrischen Datenverarbeitung. Teubner, Stuttgart, 1989.
C. A. Micchelli. On a numerically efficient method for computing multivariate B-splines. In W. Schempp and K. Zeller, editors, Multivariate Approximation Theory. Birkhäuser, Basel, 1979.
G. M. Nielson. Scattered data modeling. IEEE Computer Graphics and Applications, 13 (1): 60–70, January 1993.
R. Pfeifle. Approximation and Interpolation using Quadratic Triangular B-Splines. PhD thesis, Universität Erlangen, Computer Graphics Group, 1996.
H. Samet. Design and Analysis of Spatial Data Structures. Addison-Wesley, 1990.
L. L. Schumaker. Fitting surfaces to scattered data. In G. G. Lorentz, C. K. Chui, and L. L. Schumaker, editors, Approximation Theory. Academic Press, New York, 1976.
H.-P. Seidel. Polar forms and triangular B-Spline surfaces. In Blossoming: The New Polar-Form Approach to Spline Curves and Surfaces SIGGRAPH ‘81 Course Notes # 26. ACM SIGGRAPH, 1991.
C. R. Traas. Practice of bivariate quadratic simplicial splines. In W. Dahmen, M. Gasca, and C. A. Micchelli, editors, Computation of Curves and Surfaces, NATO ASI Series, pages 383–422. Kluwer Academic Publishers, Dordrecht, 1990.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 B. G. Teubner Stuttgart
About this chapter
Cite this chapter
Pfeifle, R., Seidel, HP. (1996). Scattered Data Approximation with Triangular B-Splines. In: Hoschek, J., Kaklis, P.D. (eds) Advanced Course on FAIRSHAPE. Vieweg+Teubner Verlag. https://doi.org/10.1007/978-3-322-82969-6_20
Download citation
DOI: https://doi.org/10.1007/978-3-322-82969-6_20
Publisher Name: Vieweg+Teubner Verlag
Print ISBN: 978-3-519-02634-1
Online ISBN: 978-3-322-82969-6
eBook Packages: Springer Book Archive