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On Simulated Annealing and the Construction of Linear Spline Approximations for Scattered Data
January-March 2001 (vol. 7 no. 1)
pp. 17-31

Abstract—We describe a method to create optimal linear spline approximations to arbitrary functions of one or two variables, given as scattered data without known connectivity. We start with an initial approximation consisting of a fixed number of vertices and improve this approximation by choosing different vertices, governed by a simulated annealing algorithm. In the case of one variable, the approximation is defined by line segments; in the case of two variables, the vertices are connected to define a Delaunay triangulation of the selected subset of sites in the plane. In a second version of this algorithm, specifically designed for the bivariate case, we choose vertex sets and also change the triangulation to achieve both optimal vertex placement and optimal triangulation. We then create a hierarchy of linear spline approximations, each one being a superset of all lower-resolution ones.

[1] O. Kreylos and B. Hamann, “On Simulated Annealing and the Construction of Linear Spline Approximations for Scattered Data,” Proc. EUROGRAPHICS-IEEE TCCG Symp. Visualization Data Visualization '99, E. Gröller, H. Löffelmann, and W. Ribarsky, eds., pp. 189-198, 1999.
[2] G.-P. Bonneau, S. Hahmann, and G.M. Nielson, “BLaC-Wavelets: A Multi-Resolution Analysis with Non-Nested Spaces,” Proc. Visualization '96, R. Yagel and G.M. Nielson, eds., pp. 43-48, 1996.
[3] M. Ech, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuetzle, "Multiresolution Analysis of Arbitrary Meshes," Computer Graphics Proc. Ann. Conf. Series (Proc. Siggraph '95), pp. 173-182, 1995.
[4] T.S. Gieng, B. Hamann, K.I. Joy, G.L. Schussman, and I.J. Trotts, Constructing Hierarchies for Triangle Meshes IEEE Trans. Visualization and Computer Graphics, vol. 4, no. 2, pp. 145-161, Apr.-June 1998.
[5] B. Hamann, "A Data Reduction Scheme for Triangulated Surfaces," Computer Aided Geometric Design, vol. 11, no. 2, pp. 197-214 1994.
[6] I.J. Trotts, B. Hamann, K.I. Joy, and D.F. Wiley, “Simplification of Tetrahedral Meshes,” Proc. Visualization '98, D. Ebert, H. Hagen, and H.E. Rushmeier, eds., pp. 287-295, 1998.
[7] B. Hamann, B.W. Jordan, and D.F. Wiley, “On a Construction of a Hierarchy of Best Linear Spline Approximations Using Repeated Bisection,” IEEE Trans. Visualization and Computer Graphics, vol. 5, no. 1, pp. 30-46, Jan.-Mar. 1999.
[8] B. Hamann, O. Kreylos, G. Monno, and A.E. Uva, “Optimal Linear Spline Approximation of Digitized Models,” Proc. Int'l Conf. Information Visualization '99 (IV '99)-Computer Aided Geometric Design Symp., pp. 244-249, 1999.
[9] B. Heckel, A.E. Uva, and B. Hamann, “Clustering-Based Generation of Hierarchical Surface Models,” Proc. IEEE Visualization '98-Late Breaking Hot Topics, C.M. Wittenbrink and A. Varshney, eds., pp. 41-44, 1998.
[10] G.M. Nielson, “Scattered Data Modeling,” IEEE CG&A, Vol. 13, No. 1, Jan. 1993, pp. 60-70.
[11] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C, second ed. Cambridge Univ. Press, 1992.
[12] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equations of State Calculations by Fast Computing Machines,” J. Chemical Physics, vol. 21, pp. 1087-1092, 1953.
[13] L.L. Schumaker, "Computing Optimal Triangulations Using Simulated Annealing," Computer Aided Geometric Design, vol. 10, pp. 329-345, 1993.
[14] B. Delaunay, “Sur la sphere vide,” Otdelenie Matematicheskii i Estestvennyka Nauk, vol. 7, Izv. Akad. Nauk SSSR, pp. 793-800, 1934.
[15] M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf, Computational Geometry. New York: Springer-Verlag, 1990.
[16] H. Edelsbrunner, Algorithms in Combinatorical Geometry. EATCS Monographs in Computer Science, Berlin: Springer, 1987.
[17] H. Edelsbrunner and R. Seidel, “VoronoïDiagrams and Arrangements,” Discrete Computational Geometry, vol. 1, pp. 25-44, 1986.
[18] F.P. Preparata and M.I. Shamos, Computational Geometry. Springer-Verlag, 1985.
[19] L.J. Guibas, D.E. Knuth, and M. Sharir, “Randomized Incremental Construction of Delaunay and VoronoïDiagrams,” Proc. 17th Int'l Colloquium—Automata, Languages, and Programming, pp. 414-431, 1990.

Index Terms:
Function approximation, linear splines, simulated annealing, multiresolution approximation, data-dependent triangulation.
Oliver Kreylos, Bernd Hamann, "On Simulated Annealing and the Construction of Linear Spline Approximations for Scattered Data," IEEE Transactions on Visualization and Computer Graphics, vol. 7, no. 1, pp. 17-31, Jan.-March 2001, doi:10.1109/2945.910818
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