Abstract

The present paper describes a fitting procedure capable of providing a smooth approximation of experimental data distributed on a bi-dimensional domain, e.g. the typical output of an interferometric technique. The procedure is based on the optimization of an analytical model defined on the whole domain by the B-spline formulation. In the paper rectangular, circular and polygonal convex domains are considered in details, but, according to the need of the operating conditions, the procedure can be extended to domains of different shapes. The proposed procedure was initially calibrated by an analytical case study: a thin square plate simply supported along the edges and loaded by a uniform pressure. Subsequently, by the operative parameters defined by the analyses carried out on the analytic data, the fitting procedure was applied on experimental data obtained by phase shifting speckle interferometry.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. M. Frocht, Photoelasticity, (John Wiley and Sons, 1941) Vol. I.
  2. D. Post, B. Han, and P. Ifju, High sensitivity moiré (Springer Verlag 1997).
  3. R. K. Erf, Speckle metrology (Academic Press 1978).
  4. U. Schnars and W. Jueptner, Digital holography (Springer 2005).
  5. K. Creath, "Temporal phase measurement methods," in Interferogram analysis, D. W. Robinson and G. T. Reid, eds., (Institute of Physics Publishing, 1993).
  6. M. Kujawinska, "Spatial phase measurement methods," in Interferogram analysis, D. W. Robinson and G. T. Reid, eds., (Institute of Physics Publishing, 1993).
  7. D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping (John Wiley & Sons, 1998).
  8. H. A. Aebischer and S. Waldner, "A simple and effective method for filtering speckle-interferometric phase fringe patterns," Opt. Commun. 162, 205-210 (1999).
    [CrossRef]
  9. M. J. Huang, Z. N. He and F. Z. Lee, "A novel methodology for enhancing the contrast of correlation fringes obtained by ESPI," Measurement 36, 93-100 (2004).
    [CrossRef]
  10. Q. Kemao, S. Hock Soon, and A. Asundi, "Smoothing filters in phase-shifting interferometry," Opt. Laser Technol. 35, 649-654 (2003).
    [CrossRef]
  11. M. I. Younus and M. S. Alam, "Enhanced phase stepped interferometry via appropriate filtering," J. Opt. Eng. 38, 1918-1923 (1999).
    [CrossRef]
  12. V. I. Vlad and D. Malacara, "Direct spatial reconstruction of optical phase from phase-modulated images," Prog. Opt. 33, 261-317 (1994).
    [CrossRef]
  13. J. Novak and A. Miks, "Least-squares fitting of wavefront using rational function," Opt. Lasers Eng. 44, 40-51 (2005).
  14. C. L. Bajaj and G. Xu, "Spline approximations of real algebraic surfaces," J. Symb. Comput. 23, 315-333 (1997).
    [CrossRef]
  15. B. Juttler and A. Felis, "Least-squares fitting of algebraic spline surfaces," Adv. Comput. Math. 17, 135-152 (2002).
    [CrossRef]
  16. F. Furgiuele, M. Muzzupappa and L. Pagnotta, "A full-field procedure for evaluating the elastic properties of advanced ceramics," Exp. Mech. 37, 285-291 (1997).
    [CrossRef]
  17. D. M. Bates and D. G. Watts, Nonlinear regression analysis and its applications (John Wiley & Sons, 1988).
    [CrossRef]
  18. V. B. Anand, Computer graphics and geometric modeling for engineers (John Wiley & Sons, 1993).
  19. A. Knight, Basics of Matlab and beyond (Chapman & Hall/CRC, 2000).
  20. S. Wolfram, Mathematica Book, 5th edition (Wolfram Media Inc, 2003).
  21. R. Szilard, Theory and analysis of plates: classical and numerical methods (Prentice-Hall, 1974).

2005 (1)

J. Novak and A. Miks, "Least-squares fitting of wavefront using rational function," Opt. Lasers Eng. 44, 40-51 (2005).

2004 (1)

M. J. Huang, Z. N. He and F. Z. Lee, "A novel methodology for enhancing the contrast of correlation fringes obtained by ESPI," Measurement 36, 93-100 (2004).
[CrossRef]

2003 (1)

Q. Kemao, S. Hock Soon, and A. Asundi, "Smoothing filters in phase-shifting interferometry," Opt. Laser Technol. 35, 649-654 (2003).
[CrossRef]

2002 (1)

B. Juttler and A. Felis, "Least-squares fitting of algebraic spline surfaces," Adv. Comput. Math. 17, 135-152 (2002).
[CrossRef]

1999 (2)

M. I. Younus and M. S. Alam, "Enhanced phase stepped interferometry via appropriate filtering," J. Opt. Eng. 38, 1918-1923 (1999).
[CrossRef]

H. A. Aebischer and S. Waldner, "A simple and effective method for filtering speckle-interferometric phase fringe patterns," Opt. Commun. 162, 205-210 (1999).
[CrossRef]

1997 (2)

F. Furgiuele, M. Muzzupappa and L. Pagnotta, "A full-field procedure for evaluating the elastic properties of advanced ceramics," Exp. Mech. 37, 285-291 (1997).
[CrossRef]

C. L. Bajaj and G. Xu, "Spline approximations of real algebraic surfaces," J. Symb. Comput. 23, 315-333 (1997).
[CrossRef]

1994 (1)

V. I. Vlad and D. Malacara, "Direct spatial reconstruction of optical phase from phase-modulated images," Prog. Opt. 33, 261-317 (1994).
[CrossRef]

Aebischer, H. A.

H. A. Aebischer and S. Waldner, "A simple and effective method for filtering speckle-interferometric phase fringe patterns," Opt. Commun. 162, 205-210 (1999).
[CrossRef]

Alam, M. S.

M. I. Younus and M. S. Alam, "Enhanced phase stepped interferometry via appropriate filtering," J. Opt. Eng. 38, 1918-1923 (1999).
[CrossRef]

Asundi, A.

Q. Kemao, S. Hock Soon, and A. Asundi, "Smoothing filters in phase-shifting interferometry," Opt. Laser Technol. 35, 649-654 (2003).
[CrossRef]

Bajaj, C. L.

C. L. Bajaj and G. Xu, "Spline approximations of real algebraic surfaces," J. Symb. Comput. 23, 315-333 (1997).
[CrossRef]

Felis, A.

B. Juttler and A. Felis, "Least-squares fitting of algebraic spline surfaces," Adv. Comput. Math. 17, 135-152 (2002).
[CrossRef]

Furgiuele, F.

F. Furgiuele, M. Muzzupappa and L. Pagnotta, "A full-field procedure for evaluating the elastic properties of advanced ceramics," Exp. Mech. 37, 285-291 (1997).
[CrossRef]

He, Z. N.

M. J. Huang, Z. N. He and F. Z. Lee, "A novel methodology for enhancing the contrast of correlation fringes obtained by ESPI," Measurement 36, 93-100 (2004).
[CrossRef]

Hock Soon, S.

Q. Kemao, S. Hock Soon, and A. Asundi, "Smoothing filters in phase-shifting interferometry," Opt. Laser Technol. 35, 649-654 (2003).
[CrossRef]

Huang, M. J.

M. J. Huang, Z. N. He and F. Z. Lee, "A novel methodology for enhancing the contrast of correlation fringes obtained by ESPI," Measurement 36, 93-100 (2004).
[CrossRef]

Juttler, B.

B. Juttler and A. Felis, "Least-squares fitting of algebraic spline surfaces," Adv. Comput. Math. 17, 135-152 (2002).
[CrossRef]

Kemao, Q.

Q. Kemao, S. Hock Soon, and A. Asundi, "Smoothing filters in phase-shifting interferometry," Opt. Laser Technol. 35, 649-654 (2003).
[CrossRef]

Lee, F. Z.

M. J. Huang, Z. N. He and F. Z. Lee, "A novel methodology for enhancing the contrast of correlation fringes obtained by ESPI," Measurement 36, 93-100 (2004).
[CrossRef]

Malacara, D.

V. I. Vlad and D. Malacara, "Direct spatial reconstruction of optical phase from phase-modulated images," Prog. Opt. 33, 261-317 (1994).
[CrossRef]

Miks, A.

J. Novak and A. Miks, "Least-squares fitting of wavefront using rational function," Opt. Lasers Eng. 44, 40-51 (2005).

Muzzupappa, M.

F. Furgiuele, M. Muzzupappa and L. Pagnotta, "A full-field procedure for evaluating the elastic properties of advanced ceramics," Exp. Mech. 37, 285-291 (1997).
[CrossRef]

Novak, J.

J. Novak and A. Miks, "Least-squares fitting of wavefront using rational function," Opt. Lasers Eng. 44, 40-51 (2005).

Pagnotta, L.

F. Furgiuele, M. Muzzupappa and L. Pagnotta, "A full-field procedure for evaluating the elastic properties of advanced ceramics," Exp. Mech. 37, 285-291 (1997).
[CrossRef]

Vlad, V. I.

V. I. Vlad and D. Malacara, "Direct spatial reconstruction of optical phase from phase-modulated images," Prog. Opt. 33, 261-317 (1994).
[CrossRef]

Waldner, S.

H. A. Aebischer and S. Waldner, "A simple and effective method for filtering speckle-interferometric phase fringe patterns," Opt. Commun. 162, 205-210 (1999).
[CrossRef]

Xu, G.

C. L. Bajaj and G. Xu, "Spline approximations of real algebraic surfaces," J. Symb. Comput. 23, 315-333 (1997).
[CrossRef]

Younus, M. I.

M. I. Younus and M. S. Alam, "Enhanced phase stepped interferometry via appropriate filtering," J. Opt. Eng. 38, 1918-1923 (1999).
[CrossRef]

Adv. Comput. Math. (1)

B. Juttler and A. Felis, "Least-squares fitting of algebraic spline surfaces," Adv. Comput. Math. 17, 135-152 (2002).
[CrossRef]

Exp. Mech. (1)

F. Furgiuele, M. Muzzupappa and L. Pagnotta, "A full-field procedure for evaluating the elastic properties of advanced ceramics," Exp. Mech. 37, 285-291 (1997).
[CrossRef]

J. Opt. Eng. (1)

M. I. Younus and M. S. Alam, "Enhanced phase stepped interferometry via appropriate filtering," J. Opt. Eng. 38, 1918-1923 (1999).
[CrossRef]

J. Symb. Comput. (1)

C. L. Bajaj and G. Xu, "Spline approximations of real algebraic surfaces," J. Symb. Comput. 23, 315-333 (1997).
[CrossRef]

Measurement (1)

M. J. Huang, Z. N. He and F. Z. Lee, "A novel methodology for enhancing the contrast of correlation fringes obtained by ESPI," Measurement 36, 93-100 (2004).
[CrossRef]

Opt. Commun. (1)

H. A. Aebischer and S. Waldner, "A simple and effective method for filtering speckle-interferometric phase fringe patterns," Opt. Commun. 162, 205-210 (1999).
[CrossRef]

Opt. Laser Technol. (1)

Q. Kemao, S. Hock Soon, and A. Asundi, "Smoothing filters in phase-shifting interferometry," Opt. Laser Technol. 35, 649-654 (2003).
[CrossRef]

Opt. Lasers Eng. (1)

J. Novak and A. Miks, "Least-squares fitting of wavefront using rational function," Opt. Lasers Eng. 44, 40-51 (2005).

Prog. Opt. (1)

V. I. Vlad and D. Malacara, "Direct spatial reconstruction of optical phase from phase-modulated images," Prog. Opt. 33, 261-317 (1994).
[CrossRef]

Other (12)

M. M. Frocht, Photoelasticity, (John Wiley and Sons, 1941) Vol. I.

D. Post, B. Han, and P. Ifju, High sensitivity moiré (Springer Verlag 1997).

R. K. Erf, Speckle metrology (Academic Press 1978).

U. Schnars and W. Jueptner, Digital holography (Springer 2005).

K. Creath, "Temporal phase measurement methods," in Interferogram analysis, D. W. Robinson and G. T. Reid, eds., (Institute of Physics Publishing, 1993).

M. Kujawinska, "Spatial phase measurement methods," in Interferogram analysis, D. W. Robinson and G. T. Reid, eds., (Institute of Physics Publishing, 1993).

D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping (John Wiley & Sons, 1998).

D. M. Bates and D. G. Watts, Nonlinear regression analysis and its applications (John Wiley & Sons, 1988).
[CrossRef]

V. B. Anand, Computer graphics and geometric modeling for engineers (John Wiley & Sons, 1993).

A. Knight, Basics of Matlab and beyond (Chapman & Hall/CRC, 2000).

S. Wolfram, Mathematica Book, 5th edition (Wolfram Media Inc, 2003).

R. Szilard, Theory and analysis of plates: classical and numerical methods (Prentice-Hall, 1974).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1.

Blending functions of a B-spline non-periodic uniform surface with h=k=3 (parabolic approximation), n+1=4 and m+1=4 (16 control points in all).The z-values of the functions are represented as phase maps just for a better visualization.

Fig. 2.
Fig. 2.

Cubic blending functions (h=k=4) of a B-spline surface defined on a circular domain. Non-periodic formulation and 6 control points were used in the radial direction, while periodic formulation and 16 control points were used in the circumferential direction, n+1=6 and m+1=16 (96 control points in all). Again the z-values of the functions are represented as phase maps just for a better visualization.

Fig. 3.
Fig. 3.

A generic quadrangular domain: a) the geometric transformation of the coordinate system; b) the geometric constructions for the calculation of the transformation equations.

Fig. 4.
Fig. 4.

A generic convex domain whose boundary is a polyline: a) the geometric transformation of the coordinate system; b) the geometric constructions for the calculation of the transformation equations.

Fig. 5.
Fig. 5.

A generic convex domain whose boundary is a polyline with a hole: a) partition into sub-regions of the domain in the real coordinate system; b) transformation of a single sub-region into a regular rectangular region; c) transformation of a single sub-region into a regular circular sector.

Fig. 6.
Fig. 6.

Geometry of the loading configuration: a) square thin plate (L/t=100, ν=0.3) simply supported along the edges and loaded by a uniform pressure; b) simulated wrapped phase map of the out-of-plane displacements obtained by eq. (7).

Fig. 7.
Fig. 7.

a) Geometry of the loading configuration: S1, S2 and S3 are the support points, F is a punctual load applied in the middle of the specimen. The area observed by the interferometer is emphasized by projecting on it an experimental fringe pattern. b) Layout of Michelson interferometer for measuring out-of-plane displacements.

Fig. 8.
Fig. 8.

Low level of load: a) experimental data; b) fitted data; c) histogram of the error for known (continuous line) and unknown (dots) deformation field, experimental phase maps of the known deformation field in the upper-right corner, standard deviation of the known (σk) and unknown (σu) deformation field in the upper-left corner.

Fig. 9.
Fig. 9.

High level of load: a) experimental data; b) fitted data; c) histogram of the error for known (continuous line) and unknown (dots) deformation field, experimental phase maps of the known deformation field in the upper-right corner, standard deviation of the known (σk) and unknown (σu) deformation field in the upper-left corner.

Tables (2)

Tables Icon

Table 1. Maximum error and standard deviation (expressed as percentage of the maximum displacements) obtained by varying the order of the polynomial (h,k) and the number of control points (n+1,m+1).

Tables Icon

Table 2. Standard deviation evaluated on the data with various type and levels of noise generated numerically.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

P x y = i = 0 n j = 0 m W i , j N i , h ( x ) N j , k ( y ) ,
N i , 1 ( ξ ) = { 1 for ξ i ξ ξ i + 1 0 otherwise , N i , k ( ξ ) = ξ ξ i ξ i + k 1 ξ i N i , k 1 ( ξ ) + ξ i + k ξ ξ i + k ξ i + 1 N i + 1 , k 1 ( ξ )
[ N 1 , h ( x 1 ) N 1 , k ( y 1 ) N n , h ( x 1 ) N m , k ( y 1 ) N 1 , h ( x l ) N 1 , k ( y l ) N n , h ( x l ) N m , k ( y l ) N 1 , h ( x p ) N 1 , k ( y p ) N n , h ( x p ) N m , k ( y p ) ] { W 1,1 W n , m } = { v 1 v l v p } MW = V
{ x 1 = 2 y 1 = 4 { x 2 = 6 y 2 = 1 { x 3 = 5 y 3 = 6 { x 4 = 3 y 4 = 5 ,
{ ξ = 17 x 17 y + 34 18 x 24 y + 251 η = 3 x + 4 y 22 10 x 10 y + 27 . { x = 2 ( 1 ξ ) ( 1 η ) + 6 ξ ( 1 η ) + 5 ξ η + 3 ( 1 ξ ) η y = 4 ( 1 ξ ) ( 1 η ) + 1 ξ ( 1 η ) + 6 ξ η + 5 ( 1 ξ ) η
{ r = ( x x 1 ) ( y 2 y 3 ) ( y y 1 ) ( x 2 x 3 ) x 1 ( y 3 y 2 ) + x 2 ( y 1 y 3 ) + x 3 ( y 2 y 1 ) θ = θ max [ y 2 ( x x 1 ) x 2 ( y y 1 ) + x 1 y x y 1 ] θ min [ y 3 ( x x 1 ) x 3 ( y y 1 ) + x 1 y x y 1 ] ( x x 1 ) ( y 2 y 3 ) ( y y 1 ) ( x 2 x 3 ) { x = ( 1 r ) x 1 + r θ max θ θ max θ min x 2 + r θ θ min θ max θ min x 3 y = ( 1 r ) y 1 + r θ max θ θ max θ min y 2 + r θ θ min θ max θ min y 3 ,
w x y = 16 p π 6 E t 3 12 ( 1 v 2 ) n = 0 m = 0 sin ( 2 n 1 ) π x a sin ( 2 m 1 ) π y b ( 2 n 1 ) ( 2 m 1 ) ( ( 2 n 1 ) 2 a 2 + ( 2 m 1 ) 2 b 2 ) 2 ,

Metrics