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

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References

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  1. M. M. Frocht, Photoelasticity, Vol. I (John Wiley and Sons, 1941).
  2. D. Post, B. Han, and P. Ifju, High sensitivity moiré (Springer Verlag1997).
  3. R. K. Erf, Speckle metrology (Academic Press1978).
  4. U. Schnars and W. Jueptner, Digital holography (Springer2005).
  5. K. Creath, “Temporal phase measurement methods,” in Interferogram analysis, D.W. Robinson and G.T. Reid, ed. (Institute of Physics Publishing, 1993).
  6. M. Kujawinska, “Spatial phase measurement methods,” in Interferogram analysis, D.W. Robinson and G.T. Reid, ed. (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,” Progress in Optics 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. Symbolic 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, The 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. Symbolic Comput. 23, 315–333, (1997).
[Crossref]

1994 (1)

V. I. Vlad and D. Malacara, “Direct spatial reconstruction of optical phase from phase-modulated images,” Progress in Optics 33, 261–317, (1994).
[Crossref]

1974 (1)

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

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]

Anand, V. B.

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

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. Symbolic Comput. 23, 315–333, (1997).
[Crossref]

Bates, D. M.

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

Creath, K.

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

Erf, R. K.

R. K. Erf, Speckle metrology (Academic Press1978).

Felis, A.

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

Frocht, M. M.

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

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]

Ghiglia, D. C.

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

Han, B.

D. Post, B. Han, and P. Ifju, High sensitivity moiré (Springer Verlag1997).

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]

Ifju, P.

D. Post, B. Han, and P. Ifju, High sensitivity moiré (Springer Verlag1997).

Jueptner, W.

U. Schnars and W. Jueptner, Digital holography (Springer2005).

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]

Knight, A.

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

Kujawinska, M.

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

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,” Progress in Optics 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]

Post, D.

D. Post, B. Han, and P. Ifju, High sensitivity moiré (Springer Verlag1997).

Pritt, M. D.

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

Schnars, U.

U. Schnars and W. Jueptner, Digital holography (Springer2005).

Szilard, R.

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

Vlad, V. I.

V. I. Vlad and D. Malacara, “Direct spatial reconstruction of optical phase from phase-modulated images,” Progress in Optics 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]

Watts, D. G.

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

Wolfram, S.

S. Wolfram, The Mathematica book, 5th edition (Wolfram Media inc, 2003).

Xu, G.

C. L. Bajaj and G. Xu, “Spline approximations of real algebraic surfaces,” J. Symbolic 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. Symbolic Comput. (1)

C. L. Bajaj and G. Xu, “Spline approximations of real algebraic surfaces,” J. Symbolic 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).

Progress in Optics (1)

V. I. Vlad and D. Malacara, “Direct spatial reconstruction of optical phase from phase-modulated images,” Progress in Optics 33, 261–317, (1994).
[Crossref]

Other (12)

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

D. Post, B. Han, and P. Ifju, High sensitivity moiré (Springer Verlag1997).

R. K. Erf, Speckle metrology (Academic Press1978).

U. Schnars and W. Jueptner, Digital holography (Springer2005).

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

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram analysis, D.W. Robinson and G.T. Reid, ed. (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, The Mathematica book, 5th edition (Wolfram Media inc, 2003).

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

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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.

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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.

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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.

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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.

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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.

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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).

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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.

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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.

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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.

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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).

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

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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 ,

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