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Abstract

I want to discuss the problem from differential geometry of describing those plane curves C which minimize the integral
$$\int\limits_C {(\alpha k^2 + \beta )ds.}$$
Here α and β are constants, kis the curvature of C, ds the arc length and, to make the fewest boundary conditions, we mean minimizing for infinitesimal variations of C on a compact set not containing the endpoints of C. Alternately, one may minimize
$$\int\limits_C {k^2 ds}$$
over variations of C which preserve the total length.

Keywords

Computer Vision Theta Function Infinitesimal Variation Independent Normal Random Variable Piecewise Polynomial Interpolation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    G. Birkhoff, H. Burchard & D. Thomas, Non-linear interpolation by splines, pseudosplines and elastica, General Motors Research Lab. report 468, 1965.Google Scholar
  2. [2]
    G. Birkhoff & C.R. De Boor, Piecewise polynomial interpolation and approximation, in Approximation of Functions, ed. by H. Garabedian, Elsevier, 1965.Google Scholar
  3. [3]
    R. Bryant & P. Griffiths, Reduction for constrained variational problems and ∫/k 2/2 ds, Am. J. Math., vol. 108, pp. 525–570, 1986.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Lausanne, 1744.Google Scholar
  5. [5]
    M. Golumb & J. Jerome, Equilibria of the curvature functional and manifolds of non-linear interpolating spline curves, Siam J. Math. Anal., vol. 13, pp. 421–458, 1982.MathSciNetCrossRefGoogle Scholar
  6. [6]Donald Geman, Random fields and inverse problems in imaging, Math. Dept., U. Mass., preprint. Google Scholar
  7. [7]
    Ulf Grenander, Lectures in Pattern Theory, vol. 1–3, Springer-Verlag, 1981.Google Scholar
  8. [8]
    B.K.P. Horn, The curve of least energy, ACM Trans, on Math. Software, vol. 9, pp. 441–460, 1983.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    M. Kaas, A. Witkin & D. Terzopoulos, Snakes: Active contour models, Proc. 1st Int. Conf. Comp. Vision, pp.259–268, IEEE, 1987.Google Scholar
  10. [10]
    Gaetano Kanizsa, Organization in Vision: Essays on Gestalt Perception, Praeger Scientific, 1979.Google Scholar
  11. [11]
    A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, Camb. Univ. Press, 4th ed., 1927.MATHGoogle Scholar
  12. [12]
    H. McKean, Stochastic integrals, Academic Press, 1969.MATHGoogle Scholar
  13. [13]
    D. Mumford, Tata Lectures on Theta, vol. 1, Birkhauser-Boston, 1983.MATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • David Mumford

There are no affiliations available

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