Optimization Letters

, Volume 6, Issue 7, pp 1265–1280 | Cite as

Adaptive image interpolation by cardinal splines in piecewise constant tension

  • Satoshi Matsumoto
  • Masaru Kamada
  • Renchin-Ochir Mijiddorj
Original Paper

Abstract

The cardinal spline in tension is modified to allow for different tensions in different sampling intervals. Varying the tension in proportion to an index of sharp change in image brightness, we obtain image interpolation results with less ringing artifacts compared to those by the cubic spline interpolation.

Keywords

Image interpolation Splines Dynamical systems 

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References

  1. 1.
    Ahlberg J.H., Nilson E.N., Walsh J.L.: The Theory of Splines and Their Applications. Academic Press, London (1967)MATHGoogle Scholar
  2. 2.
    Asuni, N., Giachetti, A.: Accuracy improvements and artifacts removal in edge based image interpolation. In: Proceedings of the 3rd International Conference Computer Vision Theory and Applications, pp. 58–65 (2008)Google Scholar
  3. 3.
    Barendt, S., Fischer, B., Modersitzki, J.: A kernel representation for exponential splines with global tension. Proc. SPIE, 7245, 0I-1–0I-10 (2009)Google Scholar
  4. 4.
    Berg K., Ehtamo H.: Interpretation of Lagrange multipliers in nonlinear pricing problem. Optim. Lett. 4, 275–285 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Canny J.: A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 8, 679–698 (1986)CrossRefGoogle Scholar
  6. 6.
    Chuah C.-S., Leou J.J.: An adaptive image interpolation algorithm for image/video processing. Pattern Recognit. 34, 2383–2393 (2001)MATHCrossRefGoogle Scholar
  7. 7.
    de Boor C.: Best approximation properties of spline functions of odd degree. J. Math. Mech. 12, 747–750 (1963)MathSciNetMATHGoogle Scholar
  8. 8.
    Hao, J., Moloney, C.: A new direction adaptive scheme for image interpolation. In: Proceedings of the 2002 International Conference on Image Processing, vol. 3, pp. 369–372 (2002)Google Scholar
  9. 9.
    Hashemi S.M., Mokarami S., Nasrabadi E.: Dynamic shortest path problems with time-varying costs. Optim. Lett. 4, 147–156 (2010)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ho Y.C., Kalman R.E., Narendra K.S.: Controllability of linear dynamical systems. Contrib. Diff. Eqs. 1, 189–213 (1963)MathSciNetGoogle Scholar
  11. 11.
    Holladay J.C.: Smoothest curve approximation Math. Tables Aids Comput. 11, 223–243 (1957)MathSciNetGoogle Scholar
  12. 12.
    Hou H.S., Andrews H.C.: Cubic splines for image interpolation and digital filtering. IEEE Trans. Acoustic Speech Signal Process. 26, 508–517 (1978)MATHCrossRefGoogle Scholar
  13. 13.
    Hu M., Tan J.: Adaptive osculatory rational interpolation for image processing. J. Comput. Appl. Math. 195, 46–53 (2006)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Kalman R.E.: A new approach to linear filtering and prediction problems. Trans. ASME 82(Series D), 35–45 (1960)Google Scholar
  15. 15.
    Kamada M., Toraichi K., Mori R.: Periodic spline orthonormal bases. J. Approx. Theory 55, 27–34 (1988)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Kim, J., Jin, S., Lee, H., Jeong, J.: New edge-guided interpolation algorithm based on weighted edge pattern analysis. In: Proceedings of the IEEE Picture Coding Symposium 2009. available online at the IEEE Xplore (2009)Google Scholar
  17. 17.
    Li X., Orchard M.T.: New edge-directed interpolation. IEEE Trans. Image Process. 10, 1521–1527 (2001)CrossRefGoogle Scholar
  18. 18.
    Liang L.: Image interpolation by blending kernels. IEEE Signal Process. Lett. 15, 805–808 (2008)CrossRefGoogle Scholar
  19. 19.
    Schoenberg I.J.: Contributions to the problem of approximation of equidistant data by analytic functions, Part A: On the problem of smoothing or graduation, a first class of analytic approximation formulas. Quart. Appl. Math. 4, 45–99 (1946)MathSciNetGoogle Scholar
  20. 20.
    Schoenberg I.J.: On interpolation by spline functions and its minimal properties. In: Butzer, P.L., Korevaar, J. (eds) On Approximation Theory, pp. 109–129. Birkhauser, Basel (1964)Google Scholar
  21. 21.
    Schoenberg I.J.: Cardinal Spline Interpolation. SIAM, Philadelphia (1973)MATHCrossRefGoogle Scholar
  22. 22.
    Schweikert D.G.: An interpolation curve using a spline in tension. J. Math. Phys. 45, 312–317 (1966)MathSciNetMATHGoogle Scholar
  23. 23.
    Sontag E.D.: Mathematical Control Theory. Springer, New York (1990)MATHCrossRefGoogle Scholar
  24. 24.
    Thévenaz P., Blu T., Unser M.: Interpolation revisited. IEEE Trans. Med. Imaging 19, 739–758 (2000)CrossRefGoogle Scholar
  25. 25.
    Thévenaz P., Blu T., Unser M.: Image interpolation and resampling. In: Bankman, I. (eds) Handbook of Medical Imaging: Processing and Analysis, pp. 393–420. Academic Press, San Diego (2000)Google Scholar
  26. 26.
    Toraichi K., Yang S., Kamada M., Mori R.: Two-dimensional spline interpolation for image reconstruction. Pattern Recognit 21, 275–284 (1988)CrossRefGoogle Scholar
  27. 27.
    Unser M., Aldroubi A.: Fast B-spline transforms for continuous image representation and interpolation. IEEE Trans. Patt. Anal. Mach. Intell. 13, 277–285 (1991)CrossRefGoogle Scholar
  28. 28.
    Unser M., Aldroubi A., Eden M.: Polynomial spline signal approximations: filter design and asymptotic equivalence with Shannon’s sampling theorem. IEEE Trans. Info. Th. 38, 95–103 (1992)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Zhang L., Wu X.: An edge-guided image interpolation algorithm via directional filtering and data fusion. IEEE Trans. Image Process. 15, 2226–2238 (2006)CrossRefGoogle Scholar
  30. 30.
    Zhao Y.B., Fang S.-C., Lavery J.E.: Geometric dual formulation for first-derivative-based univariate cubic L 1 splines. J. Global Optim. 40, 589–621 (2008)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Satoshi Matsumoto
    • 1
  • Masaru Kamada
    • 1
  • Renchin-Ochir Mijiddorj
    • 2
  1. 1.Department of Computer and Information Sciences, Faculty of EngineeringIbaraki UniversityHitachi, IbarakiJapan
  2. 2.Department of Programming and Didactics, School of Computer Science and Information TechnologyMongolian State University of EducationUlaanbaatarMongolia

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