Efficient Piecewise-Cubic Polynomial Curve Approximation Using Uniform Metric

  • Kim, Jae-Hoon (Division of Computer Engineering, Pusan University of Foreign Studies)
  • Published : 2008.09.30

Abstract

We present efficient algorithms for solving the piecewise-cubic approximation problems in the plane. Given a set D of n points in the plane, we find a piecewise-cubic polynomial curve passing through only the points of a subset S of D and approximating the other points using the uniform metric. The goal is to minimize the size of S for a given error tolerance $\varepsilon$, called the min-# problem, or to minimize the error tolerance $\varepsilon$ for a given size of S, called the min-$\varepsilon$ problem. We give algorithms with running times O($n^2$ logn) and O($n^3$) for both problems, respectively.

Keywords

piecewise-cubic;approximation;uniform metric

References

  1. S.L. Hakimi and E.F. Schmeichel, 'Fitting polygonal functions to a set of points in the plane', Computer Vision, Graphics, and Image Processing, vol. 53, pp. 132-136, 1991
  2. H. Imai and M. Iri, 'Computational-geometric methods for polygonal approximations of a curve', Computer Vision, Graphics, and Image Processing, vol. 36, pp. 31-41, 1986 https://doi.org/10.1016/S0734-189X(86)80027-5
  3. G. Barequet, D.Z. Chen, O. Daescu, M.T. Goodrich, and J. Snoeyink, 'Efficiently approximating polygonal paths in three and higher dimensions', Pro. 14th Annual ACM Symp. on Computational Geometry, pp. 317-326, 1998
  4. F.P. Preparata and M.I. Shamos, 'Computational geometry: An introduction', Springer-Verlag, 1985
  5. W.S. Chan and F. Chin, 'Approximation of polygonal curves with minimum number of line segments or minimum error', Int. J. of Computational Geometry and Applications, vol. 6(1), pp. 59-77, 1996 https://doi.org/10.1142/S0218195996000058
  6. A. Melkman and J. O'Rourks, 'On polygonal chain approximation', Computational Morphology, pp. 87-95, 1988
  7. D.Z. Chen and O. Daescu, 'Space-efficient algorithms for approximating polygonal curves in two-dimensional space', The 4th Annual Int. Computing and Combinatorics Conference, pp. 45-54, 1998 https://doi.org/10.1007/3-540-68535-9_8
  8. K.R. Varadarajan, 'Approximating monotone polygonal curves using the uniform metric', Pro. 12th Annual ACM Symp. on Computational Geometry, pp. 311-318, 1996