Shape modeling based on specifying the initial B-spline curve and scaled BFGS optimization method

Multimedia Tools and Applications volume 77, pages30331–30351(2018)Cite this article

Abstract

In this paper, we consider the problem of fitting the B-spline curves to a set of ordered points, by finding the control points and the location parameters. The presented method takes two main steps: specifying initial B-spline curve and optimization. The method determines the number and the position of control points such that the initial B-spline curve is very close to the target curve. The proposed method introduces a length parameter in which this allows us to adjust the number of the control points and increases the precision of the initial B-spline curve. Afterwards, the scaled BFGS algorithm is used to optimize the control points and the foot points simultaneously and generates the final curve. Furthermore, we present a new procedure to insert a new control point and repeat the optimization method, if it is necessary to modify the fitting accuracy of the generated B-spline fitting curve. Associated examples are also offered to show that the proposed approach performs accurately for complex shapes with a large number of data points and is able to generate a precise fitting curve with a high degree of approximation.

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References

  1. 1.

    Andrei N (2017) An adaptive scaled BFGS method for unconstrained optimization. Numerical Algorithms 77:1–20

    MathSciNet  Google Scholar 

  2. 2.

    Bergström P, Söderkvist I (2012) Fitting NURBS using separable least squares techniques. International Journal of Mathematical Modelling and Numerical Optimisation 3(4):319–334

    MATH  Google Scholar 

  3. 3.

    Bergström P, Edlund O, Söderkvist I (2012) Efficient computation of the gauss-newton direction when fitting NURBS using ODR. BIT Numer Math 52 (3):571–588

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bhimani J, Mi N, Leeser M, Yang Z (2017) Fim: performance prediction for parallel computation in iterative data processing applications. In: 2017 IEEE 10th international conference on cloud computing (CLOUD). IEEE, pp 359–366

  5. 5.

    Bhimani J, Yang Z, Leeser M, Mi N (2017) Accelerating big data applications using lightweight virtualization framework on enterprise cloud. In: High performance extreme computing conference (HPEC), 2017 IEEE. IEEE, pp 1–7

  6. 6.

    Borges CF, Pastva T (2002) Total least squares fitting of Bézier and B-spline curves to ordered data. Comput Aided Geom Des 19(4):275–289

    MATH  Google Scholar 

  7. 7.

    Broyden CG (1970) The convergence of a class of double-rank minimization algorithms 1. General considerations. IMA J Appl Math 6(1):76–90

    MATH  Google Scholar 

  8. 8.

    Carlier A, Leonard K, Hahmann S, Morin G, Collins M (2016) The 2D shape structure dataset: a user annotated open access database. Comput Graph 58:23–30

    MathSciNet  Google Scholar 

  9. 9.

    Carlson N, Gulliksson M, Kartalopoulos S, Buikis A, Mastorakis N, Vladareanu L (2008) Surface fitting with NURBS- a gauss newton with trust region approach. In: WSEAS international conference. Proceedings. Mathematics and computers in science and engineering. WSEAS, p 13

  10. 10.

    Cheng W, Li D (2010) Spectral scaling BFGS method. J Optim Theory Appl 146(2):305–319

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Deng C, Lin H (2014) Progressive and iterative approximation for least squares B-spline curve and surface fitting. Comput Aided Des 47:32–44

    MathSciNet  Google Scholar 

  12. 12.

    Di Fiore C, Zellini P (2001) Matrix algebras in optimal preconditioning. Linear Algebra Appl 335(1–3):1–54

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Di Fiore C, Fanelli S, Lepore F, Zellini P (2003) Matrix algebras in quasi-newton methods for unconstrained minimization. Numer Math 94(3):479–500

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Ebrahimi A, Loghmani G (2017) B-spline curve fitting by diagonal approximation BFGS methods. Iranian Journal of Science and Technology, Transactions A: Science:1–12. https://doi.org/10.1007/s40995-017-0347-1

  15. 15.

    Farin GE (2002) Curves and surfaces for CAGD: a practical guide. Morgan Kaufmann, San Mateo

    Google Scholar 

  16. 16.

    Fletcher R (1970) A new approach to variable metric algorithms. Comput J 13 (3):317–322

    MATH  Google Scholar 

  17. 17.

    Gálvez A, Iglesias A (2011) Efficient particle swarm optimization approach for data fitting with free knot B-splines. Comput Aided Des 43(12):1683–1692

    Google Scholar 

  18. 18.

    Gálvez A, Iglesias A (2013) Firefly algorithm for explicit B-spline curve fitting to data points. Math Probl Eng 2013 (Article ID 528215)

  19. 19.

    Gálvez A, Iglesias A (2013) From nonlinear optimization to convex optimization through firefly algorithm and indirect approach with applications to cad/cam. Sci World J 2013 (Article ID 283919)

  20. 20.

    Gálvez A, Iglesias A (2013) A new iterative mutually coupled hybrid ga–pso approach for curve fitting in manufacturing. Appl Soft Comput 13(3):1491–1504

    Google Scholar 

  21. 21.

    Gálvez A, Iglesias A (2016) Particle-based meta-model for continuous breakpoint optimization in smooth local-support curve fitting. Appl Math Comput 275:195–212

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Gálvez A, Iglesias A, Puig-Pey J (2012) Iterative two-step genetic-algorithm-based method for efficient polynomial B-spline surface reconstruction. Inf Sci 182(1):56–76

    MathSciNet  Google Scholar 

  23. 23.

    Gill PE, Leonard MW (2001) Reduced-hessian quasi-newton methods for unconstrained optimization. SIAM J Optim 12(1):209–237

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Goldfarb D (1970) A family of variable-metric methods derived by variational means. Math Comput 24(109):23–26

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Goshtasby AA (2000) Grouping and parameterizing irregularly spaced points for curve fitting. ACM Trans Graph (TOG) 19(3):185–203

    MathSciNet  Google Scholar 

  26. 26.

    Hasegawa AY, Tormena C, Parpinelli RS (2014) Bézier curve parametrization using a multiobjective evolutionary algorithm. IJCSA 11(2):1–18

    Google Scholar 

  27. 27.

    Hoschek J (1988) Intrinsic parametrization for approximation. Comput Aided Geom Des 5(1):27–31

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Hoschek J (1988) Spline approximation of offset curves. Comput Aided Geom Des 5(1):33–40

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Hoschek J, Lasser D, Schumaker LL (1993) Fundamentals of computer aided geometric design. AK Peters, Ltd, Natick

    Google Scholar 

  30. 30.

    Iglesias A, Gálvez A, Collantes M (2016) Four adaptive memetic bat algorithm schemes for Bézier curve parameterization. In: Transactions on computational science XXVIII. Springer, pp 127–145

  31. 31.

    Irshad M, Khalid S, Hussain MZ, Sarfraz M (2016) Outline capturing using rational functions with the help of genetic algorithm. Appl Math Comput 274:661–678

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Isard M, Blake A (1998) Active contours. Springer-Verlag

  33. 33.

    Javidrad F (2012) An accelerated simulated annealing method for B-spline curve fitting to strip-shaped scattered points. International Journal of CAD/CAM 12(1):9–19

    Google Scholar 

  34. 34.

    Khan MA (2012) A new method for video data compression by quadratic Bézier curve fitting. SIViP 6(1):19–24

    Google Scholar 

  35. 35.

    Laurent-Gengoux P, Mekhilef M (1993) Optimization of a nurbs representation. Comput Aided Des 25(11):699–710

    MATH  Google Scholar 

  36. 36.

    Leu M, Peng X, Zhang W (2005) Surface reconstruction for interactive modeling of freeform solids by virtual sculpting. CIRP Ann Manuf Technol 54(1):131–134

    Google Scholar 

  37. 37.

    Liu Y, Wang W (2008) A revisit to least squares orthogonal distance fitting of parametric curves and surfaces. In: International conference on geometric modeling and processing. Springer, pp 384–397

  38. 38.

    Liu H, Shao J, Wang H, Chang B (2015) An adaptive sizing bfgs method for unconstrained optimization. Calcolo 52(2):233–244

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Lu F, Milios EE (1994) Optimal spline fitting to planar shape. Signal Process 37(1):129–140

    MATH  Google Scholar 

  40. 40.

    Masood A, Sarfraz M (2009) Capturing outlines of 2D objects with bézier cubic approximation. Image Vis Comput 27(6):704–712

    Google Scholar 

  41. 41.

    Nocedal J, Wright S (2006) Numerical optimization. Springer Science & Business Media, Berlin

    Google Scholar 

  42. 42.

    Park H (2001) Choosing nodes and knots in closed B-spline curve interpolation to point data. Comput Aided Des 33(13):967–974

    Google Scholar 

  43. 43.

    Park H, Lee JH (2007) B-spline curve fitting based on adaptive curve refinement using dominant points. Comput Aided Des 39(6):439–451

    Google Scholar 

  44. 44.

    Piegl LA, Tiller W (2000) Surface approximation to scanned data. The visual computer 16(7):386–395

    MATH  Google Scholar 

  45. 45.

    Piegl L, Tiller W (2012) The NURBS book. Springer Science & Business Media, Berlin

    Google Scholar 

  46. 46.

    Plass M, Stone M (1983) Curve-fitting with piecewise parametric cubics. In: ACM SIGGRAPH computer graphics, vol 17. ACM, pp 229–239

  47. 47.

    Pottmann H, Hofer M (2003) Geometry of the squared distance function to curves and surfaces. In: Visualization and mathematics III. Springer, pp 221–242

  48. 48.

    Pottmann H, Leopoldseder S, Hofer M (2002) Approximation with active B-spline curves and surfaces. In: 10th pacific conference on computer graphics and applications, 2002. Proceedings. IEEE, pp 8–25

  49. 49.

    Prasad M, Fitzgibbon A (2006) Single view reconstruction of curved surfaces. In: 2006 IEEE computer society conference on computer vision and pattern recognition (CVPR’06), vol 2. IEEE, pp 1345–1354

  50. 50.

    Rogers DF, Fog N (1989) Constrained B-spline curve and surface fitting. Comput Aided Des 21(10):641–648

    MATH  Google Scholar 

  51. 51.

    Sarfraz M, Masood A (2007) Capturing outlines of planar images using Bézier cubics. Comput Graph 31(5):719–729

    Google Scholar 

  52. 52.

    Sarfraz M, Riyazuddin M, Baig M (2006) Capturing planar shapes by approximating their outlines. J Comput Appl Math 189(1):494–512

    MathSciNet  MATH  Google Scholar 

  53. 53.

    Sarkar B, Menq CH (1991) Parameter optimization in approximating curves and surfaces to measurement data. Comput Aided Geom Des 8(4):267–290

    MathSciNet  MATH  Google Scholar 

  54. 54.

    Sato H (2001) Moving average filter. US Patent 6,304,133

  55. 55.

    Saux E, Daniel M (2003) An improved hoschek intrinsic parametrization. Comput Aided Geom Des 20(8):513–521

    MathSciNet  MATH  Google Scholar 

  56. 56.

    Sevaux M, Mineur Y (2007) A curve-fitting genetic algorithm for a styling application. Eur J Oper Res 179(3):895–905

    MATH  Google Scholar 

  57. 57.

    Shanno DF (1970) Conditioning of quasi-newton methods for function minimization. Math Comput 24(111):647–656

    MathSciNet  MATH  Google Scholar 

  58. 58.

    Speer T, Kuppe M, Hoschek J (1998) Global reparametrization for curve approximation. Comput Aided Geom Des 15(9):869–877

    MathSciNet  MATH  Google Scholar 

  59. 59.

    Vassilev TI (1996) Fair interpolation and approximation of B-splines by energy minimization and points insertion. Comput Aided Des 28(9):753–760

    Google Scholar 

  60. 60.

    Wang X, Cheng FF, Barsky BA (1997) Energy and B-spline interproximation. Comput Aided Des 29(7):485–496

    Google Scholar 

  61. 61.

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

    Google Scholar 

  62. 62.

    Yang H, Wang W, Sun J (2004) Control point adjustment for B-spline curve approximation. Comput Aided Des 36(7):639–652

    Google Scholar 

  63. 63.

    Yang YJ, Cao S, Yong JH, Zhang H, Paul JC, Sun JG, Gu H (2008) Approximate computation of curves on b-spline surfaces. Comput Aided Des 40(2):223–234

    Google Scholar 

  64. 64.

    Yang Z, Wang J, Evans D, Mi N (2016) Autoreplica: automatic data replica manager in distributed caching and data processing systems. In: 2016 IEEE 35th international performance computing and communications conference (IPCCC). IEEE, pp 1–6

  65. 65.

    Yang Z, Hoseinzadeh M, Andrews A, Mayers C, Evans DT, Bolt RT, Bhimani J, Mi N, Swanson S (2017) Autotiering: automatic data placement manager in multi-tier all-flash datacenter. In: 36th IEEE international performance computing and communications conference. IEEE

  66. 66.

    Yang Z, Hoseinzadeh M, Wong P, Artoux J, Mayers C, Evans DT, Bolt RT, Bhimani J, Mi N, Swanson S (2017) H-nvme: a hybrid framework of nvme-based storage system in cloud computing environment. In: 36th IEEE international performance computing and communications conference (IPCCC). IEEE

  67. 67.

    Yoshimoto F, Moriyama M, Harada T (1999) Automatic knot placement by a genetic algorithm for data fitting with a spline. In: International conference on shape modeling and applications, 1999. Proceedings. Shape modeling international’99. IEEE, pp 162–169

  68. 68.

    Yoshimoto F, Harada T, Yoshimoto Y (2003) Data fitting with a spline using a real-coded genetic algorithm. Comput Aided Des 35(8):751–760

    Google Scholar 

  69. 69.

    Yuan Y (1991) A modified BFGS algorithm for unconstrained optimization. IMA J Numer Anal 11(3):325–332

    MathSciNet  MATH  Google Scholar 

  70. 70.

    Zhao X, Zhang C, Yang B, Li P (2011) Adaptive knot placement using a gmm-based continuous optimization algorithm in B-spline curve approximation. Comput Aided Des 43(6):598–604

    Google Scholar 

  71. 71.

    Zheng W, Bo P, Liu Y, Wang W (2012) Fast B-spline curve fitting by l-BFGS. Comput Aided Geom Des 29(7):448–462

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank anonymous referees for their helpful comments and useful suggestions which improved our paper considerably.

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Affiliations

  1. Computer Geometry and Dynamical Systems Laboratory, Faculty of Mathematical Sciences, Yazd University, Yazd, Iran

    A. Ebrahimi & G. Barid Loghmani

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  1. A. Ebrahimi
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  2. G. Barid Loghmani
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Correspondence to G. Barid Loghmani.

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Ebrahimi, A., Barid Loghmani, G. Shape modeling based on specifying the initial B-spline curve and scaled BFGS optimization method. Multimed Tools Appl 77, 30331–30351 (2018). https://doi.org/10.1007/s11042-018-6109-z

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Keywords

  • Geometric modeling
  • Curve fitting
  • Initial B-spline curve
  • Optimization method
  • Scaled BFGS method