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Distance regression by Gauss–Newton-type methods and iteratively re-weighted least-squares

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Abstract

We discuss the problem of fitting a curve or surface to given measurement data. In many situations, the usual least-squares approach (minimization of the sum of squared norms of residual vectors) is not suitable, as it implicitly assumes a Gaussian distribution of the measurement errors. In those cases, it is more appropriate to minimize other functions (which we will call norm-like functions) of the residual vectors. This is well understood in the case of scalar residuals, where the technique of iteratively re-weighted least-squares, which originated in statistics (Huber in Robust statistics, 1981) is known to be a Gauss–Newton-type method for minimizing a sum of norm-like functions of the residuals. We extend this result to the case of vector-valued residuals. It is shown that simply treating the norms of the vector-valued residuals as scalar ones does not work. In order to illustrate the difference we provide a geometric interpretation of the iterative minimization procedures as evolution processes.

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References

  1. Aigner M, Jüttler B (2007) Approximation flows in shape manifolds. In: Chenin P, Lyche T, Schumaker L (eds) Curve and surface design: Avignon 2006, Nashboro Press, pp 1–10

  2. Aigner M, Šír Z, Jüttler B (2007) Evolution-based least-squares fitting using Pythagorean hodograph spline curves. Comput Aided Geom Des 24: 310–322

    Article  Google Scholar 

  3. Al-Subaihi I, Watson GA (2004) The use of the l 1 and l ∞ norms in fitting parametric curves and surfaces to data. Appl Numer Anal Comput Math 1: 363–376

    Article  MATH  MathSciNet  Google Scholar 

  4. Alhanaty M, Bercovier M (2001) Curve and surface fitting and design by optimal control methods. Comput Aided Des 33: 167–182

    Article  Google Scholar 

  5. Atieg A, Watson GA (2003) A class of methods for fitting a curve or surface to data by minimizing the sum of squares of orthogonal distances. J Comput Appl Math 158: 277–296

    Article  MATH  MathSciNet  Google Scholar 

  6. Blake A, Isard M (1998) Active contours. Springer, New York

    Google Scholar 

  7. Boggs PT, Byrd RH, Schnabel RB (1987) A stable and efficient algorithm for nonlinear orthogonal distance regression. J Sci Stat Comput 8(6): 1052–1078

    Article  MATH  MathSciNet  Google Scholar 

  8. Bube KP, Langan RT (1997) Hybrid ℓ 1/ℓ 2 minimization with application to tomography. Geophysics 62: 1183–1195

    Article  Google Scholar 

  9. Holland P, Welsch R (1977) Robust regression using iteratively re-weighted least-squares. Commun Stat Theory Methods 9(A6): 813–827

    Google Scholar 

  10. Hoschek J, Lasser D (1993) Fundamentals of computer aided geometric design. A K Peters, Wellesley

    MATH  Google Scholar 

  11. Huber PJ (1981) Robust statistics. Wiley, New York

    Book  MATH  Google Scholar 

  12. Jüttler B (1998) Computational methods for parametric discrete ℓ 1 and ℓ ∞ curve fitting. Int J Shape Model 4: 21–34

    Article  MATH  Google Scholar 

  13. Kelley CT (1999) Iterative methods for optimization. Society for Industrial and Applied Mathematics, Philadelphia

  14. Liu Y, Wang W (2008) A revisit to least squares orthogonal distance fitting of parametric curves and surfaces. In: Chen F, Jüttler B (eds) Advances in geometric modeling and processing, GMP 2008, Springer, pp 384–397

  15. Mahadevan V, Narasimha-Iyer H, Roysam B, Tanenbaum HL (2004) Robust model-based vasculature detection in noisy biomedical images. IEEE Trans Inf Technol Biomed 8(3): 360–376

    Article  Google Scholar 

  16. McCullagh P, Nelder J (1998) Generalized linear models. Chapman & Hall, London

    Google Scholar 

  17. Osborne MR (1985) Finite algorithms in optimization and data analysis. Wiley, New York

    MATH  Google Scholar 

  18. Pottmann H, Leopoldseder S, Hofer M, Steiner T, Wang W (2005) Industrial geometry: recent advances and applications in CAD. Comput Aided Des 37: 751–766

    Article  Google Scholar 

  19. Rogers D, Fog N (1989) Constrained B-spline curve and surface fitting. Comput Aided Des 21: 641–648

    Article  MATH  Google Scholar 

  20. Sarkar B, Menq CH (1991) Parameter optimization in approximating curves and surfaces to measurement data. Comput Aided Geom Des 8: 267–280

    Article  MATH  MathSciNet  Google Scholar 

  21. Speer T, Kuppe M, Hoschek J (1998) Global reparametrization for curve approximation. Comput Aided Geom Des 15: 869–877

    Article  MATH  MathSciNet  Google Scholar 

  22. Wang W, Pottmann H, Liu Y (2006) Fitting B-spline curves to point clouds by curvature-based squared distance minimization. ACM Trans Graph 25(2): 214–238

    Article  Google Scholar 

  23. Watson GA (2000) Approximation in normed linear spaces. J Comput Appl Math 121: 1–36

    Article  MATH  MathSciNet  Google Scholar 

  24. Watson GA (2001) On the Gauss–Newton method for ℓ 1 orthogonal distance regression. IMA J Num Anal 22: 345–357

    Article  Google Scholar 

  25. Wedderburn RWM (1974) Quasi-likelihood functions, generalized linear models, and the Gauss– Newton method. Biometrika 61(3): 439–447

    MATH  MathSciNet  Google Scholar 

  26. Yamamoto T (2000) Historical developments in convergence analysis for Newton’s and Newton-like methods. J Comput Appl Math 124(1/2): 1–23

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Geometry, Johannes Kepler University, Linz, Austria

    Martin Aigner & Bert Jüttler

Authors
  1. Martin Aigner
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  2. Bert Jüttler
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Correspondence to Bert Jüttler.

Additional information

Communicated by C.H. Cap.

Supported by the Austrian Science Fund (FWF) through project S9202.

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Aigner, M., Jüttler, B. Distance regression by Gauss–Newton-type methods and iteratively re-weighted least-squares. Computing 86, 73–87 (2009). https://doi.org/10.1007/s00607-009-0055-6

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  • DOI: https://doi.org/10.1007/s00607-009-0055-6

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