Skip to main content
SpringerLink
Log in

Univariate cubic L 1 interpolating splines based on the first derivative and on 5-point windows: analysis, algorithm and shape-preserving properties

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

In this paper, univariate cubic L 1 interpolating splines based on the first derivative and on 5-point windows are introduced. Analytical results for minimizing the local spline functional on 5-point windows are presented and, based on these results, an efficient algorithm for calculating the spline coefficients is set up. It is shown that cubic L 1 splines based on the first derivative and on 5-point windows preserve linearity of the original data and avoid extraneous oscillation. Computational examples, including comparison with first-derivative-based cubic L 1 splines calculated by a primal affine algorithm and with second-derivative-based cubic L 1 splines, show the advantages of the first-derivative-based cubic L 1 splines calculated by the new algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Auquiert, P., Gibaru, O., Nyiri, E.: C 1 and C 2-continuous polynomial parametric L p splines (p≥1). Comput. Aided Geom. Des. 24, 373–394 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Auquiert, P., Gibaru, O., Nyiri, E.: On the cubic L 1 spline interpolant to the Heaviside function. Numer. Algorithms 46, 321–332 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bulatov, D., Lavery, J.E.: Reconstruction and texturing of 3D urban terrain from uncalibrated monocular images using L 1 splines. Photogramm. Eng. Remote Sens. 76, 439–449 (2010)

    Google Scholar 

  4. Cheng, H., Fang, S.-C., Lavery, J.E.: Univariate cubic L 1 splines: a geometric programming approach. Math. Methods Oper. Res. 56, 197–229 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheng, H., Fang, S.-C., Lavery, J.E.: An efficient algorithm for generating univariate cubic L 1 splines. Comput. Optim. Appl. 29, 219–253 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheng, H., Fang, S.-C., Lavery, J.E.: Shape-preserving properties of univariate cubic L 1 splines. J. Comput. Appl. Math. 174, 361–382 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng, H., Fang, S.-C., Lavery, J.E.: A geometric programming framework for univariate cubic L 1 smoothing splines. Ann. Oper. Res. 133, 229–248 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jin, Q., Lavery, J.E., Fang, S.-C.: Univariate cubic L 1 interpolating splines: analytical results for linearity, convexity and oscillation on 5-point windows. Algorithms 3, 276–293 (2010)

    Article  MathSciNet  Google Scholar 

  9. Lavery, J.E.: Univariate cubic L p splines and shape-preserving, multiscale interpolation by univariate cubic L 1 splines. Comput. Aided Geom. Des. 17, 319–336 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lavery, J.E.: Shape-preserving, multiscale fitting of univariate data by cubic L 1 smoothing splines. Comput. Aided Geom. Des. 17, 715–727 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lavery, J.E.: Shape-preserving, multiscale interpolation by bi- and multivariate cubic L 1 splines. Comput. Aided Geom. Des. 18, 321–343 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lavery, J.E.: Shape-preserving, multiscale interpolation by univariate curvature-based cubic L 1 splines in Cartesian and polar coordinates. Comput. Aided Geom. Des. 19, 257–273 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lavery, J.E.: Shape-preserving, first-derivative-based parametric and nonparametric cubic L 1 spline curves. Comput. Aided Geom. Des. 23, 276–296 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lavery, J.E.: Shape-preserving univariate cubic and higher-degree L 1 splines with function-value-based and multistep minimization principles. Comput. Aided Geom. Des. 26, 1–16 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lin, Y.-M., Zhang, W., Wang, Y., Fang, S.-C., Lavery, J.E.: Computationally efficient models of urban and natural terrain by non-iterative domain decomposition with L 1 smoothing splines. In: Proceedings of the 25th Army Science Conference, Department of the Army, Washington, DC, USA, Nov. 2006; 9 pages

    Google Scholar 

  16. Nyiri, E., Gibaru, O., Auquiert, P.: Fast \(L_{1}^{k}\)–C k polynomial spline interpolation algorithm with shape-preserving properties. Comput. Aided Geom. Des. 28, 65–74 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, Y., Fang, S.-C., Lavery, J.E.: A geometric programming approach for bivariate cubic L 1 splines. Comput. Math. Appl. 49, 481–514 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, Y., Fang, S.-C., Lavery, J.E.: A compressed primal-dual method for bivariate cubic L 1 splines. Comput. Math. Appl. 201, 69–87 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yu, L., Jin, Q., Lavery, J.E., Fang, S.-C.: Univariate cubic L 1 interpolating splines: spline functional, window size and analysis-based algorithm. Algorithms 3, 311–328 (2010)

    Article  MathSciNet  Google Scholar 

  20. Zhang, W., Wang, Y., Fang, S.-C., Lavery, J.E.: Cubic L 1 splines on triangulated irregular networks. Pac. J. Optim. 2, 289–317 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Zhao, Y., Fang, S.-C., Lavery, J.E.: Geometric dual formulation for first-derivative-based univariate cubic L 1 splines. J. Glob. Optim. 40, 589–621 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Management Science and Engineering, Zhejiang University, Hangzhou, China

    Qingwei Jin

  2. Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, 27695-7906, USA

    Qingwei Jin, Lu Yu, John E. Lavery & Shu-Cherng Fang

  3. Mathematical Sciences Division, Army Research Office, Army Research Laboratory, P.O. Box 12211, Research Triangle Park, NC, 27709-2211, USA

    John E. Lavery

Authors
  1. Qingwei Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lu Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. John E. Lavery
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shu-Cherng Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qingwei Jin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jin, Q., Yu, L., Lavery, J.E. et al. Univariate cubic L 1 interpolating splines based on the first derivative and on 5-point windows: analysis, algorithm and shape-preserving properties. Comput Optim Appl 51, 575–600 (2012). https://doi.org/10.1007/s10589-011-9426-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-011-9426-y

Keywords

Search

Navigation