Abstract

The method presented extracts the demodulated phase from only one fringe pattern. Locally, this method approaches the fringe pattern morphology with the help of a mathematical model. The degree of similarity between the mathematical model and the real fringe is estimated by minimizing a correlation function. To use an optimization process, we have chosen a polynomial form such as a mathematical model. However, the use of a polynomial form induces an identification procedure with the purpose of retrieving the demodulated phase. This method, polynomial modulated phase correlation, is tested on several examples. Its performance, in terms of speed and precision, is presented on very noised fringe patterns.

© 2005 Optical Society of America

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References

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  1. J. C. Dupre, A. Lagarde, “Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing,” Exp. Mech. 37, 393–397 (1997).
    [CrossRef]
  2. J. Villa, J. A. Quiroga, J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Lasers Eng. 41, 127–137 (2004).
    [CrossRef]
  3. H. Aben, L. Ainola, “Isochromatic fringes in photoelasticity,” J. Opt. Soc. Am. A 13, 750–755 (2000).
    [CrossRef]
  4. R. A. Tomlinson, E. A. Patterson, “The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
    [CrossRef]
  5. C. Breque, J. C. Dupre, F. Bremand, “Calibration of a system of projection moiré for relief measuring: biomechanical applications,” Opt. Lasers Eng. 41, 241–260 (2004).
    [CrossRef]
  6. L. Humbert, V. Valle, M. Cottron, “Experimental determination and empirical representation of out-of-plane displacements in a cracked elastic plate loaded in mode I,” Int. J. Solids Struct. 37, 5493–5504 (2000).
    [CrossRef]
  7. G. Mauvoisin, F. Bremand, A. Lagarde, “Three-dimensional shape reconstruction by phase-shifting shadow moiré,” Appl. Opt. 33, 2163–2169 (1994).
    [CrossRef] [PubMed]
  8. J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow Moiré to micro deformation measurements of curved surfaces,” Opt. Lasers Eng. 36, 29–40 (2001).
    [CrossRef]
  9. Y. Morimoto, M. Fujisaa, “Fringe pattern analysis by a phase shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
    [CrossRef]
  10. M. A. Gdeisat, D. R. Burton, M. J. Lalor, “Fringe pattern demodulation with a two-dimensional digital phase-locked loop algorithm,” Appl. Opt. 41, 5479–5487 (2002).
    [CrossRef] [PubMed]
  11. M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
    [CrossRef]
  12. M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
    [CrossRef] [PubMed]
  13. M. Servin, J. L. Marroquin, J. A. Quiroga, “Regularized quadrature and phase tracking from a single closed-fringe interferogram,” J Opt. Soc. Am A Opt. Image Sci. Vis 21, 411–419 (2004).
    [CrossRef] [PubMed]
  14. E. Robin, V. Valle, “Phase demodulation from a single fringe pattern based on a correlation technique,” Appl. Opt. 43, 4355–4361 (2004).
    [CrossRef] [PubMed]
  15. H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newtown-Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
    [CrossRef]
  16. M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, S. R. Mcneill, “Full-field speckle pattern image correlation with B-spline deformation function,” Exp. Mech. 42, 344–352 (2002).
    [CrossRef]
  17. H. W. Schreier, M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape function,” Exp. Mech. 42, 303–310 (2000).
    [CrossRef]
  18. G. Arfken, “The method of steepest descent,” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), Sec. 7.4, pp. 428–436.
  19. F. Bremand, “A phase unwrapping technique for object relief determination,” Opt. Lasers Eng. 21, 49–60 (1994).
    [CrossRef]
  20. B. Gutmann, H. Weber, “Phase unwrapping with the branch cut method: role of phase field direction,” App. Opt. 39, 4802–4816 (2000).
    [CrossRef]
  21. C. Wykes, C. Buckberry, M. Dale, M. Reeves, D. Towers, “Functional testing using rapid prototyped components and optical measurment,” Opt. Lasers Eng. 31, 411–424 (1999).
    [CrossRef]

2004 (4)

J. Villa, J. A. Quiroga, J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Lasers Eng. 41, 127–137 (2004).
[CrossRef]

C. Breque, J. C. Dupre, F. Bremand, “Calibration of a system of projection moiré for relief measuring: biomechanical applications,” Opt. Lasers Eng. 41, 241–260 (2004).
[CrossRef]

M. Servin, J. L. Marroquin, J. A. Quiroga, “Regularized quadrature and phase tracking from a single closed-fringe interferogram,” J Opt. Soc. Am A Opt. Image Sci. Vis 21, 411–419 (2004).
[CrossRef] [PubMed]

E. Robin, V. Valle, “Phase demodulation from a single fringe pattern based on a correlation technique,” Appl. Opt. 43, 4355–4361 (2004).
[CrossRef] [PubMed]

2002 (3)

M. A. Gdeisat, D. R. Burton, M. J. Lalor, “Fringe pattern demodulation with a two-dimensional digital phase-locked loop algorithm,” Appl. Opt. 41, 5479–5487 (2002).
[CrossRef] [PubMed]

M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, S. R. Mcneill, “Full-field speckle pattern image correlation with B-spline deformation function,” Exp. Mech. 42, 344–352 (2002).
[CrossRef]

R. A. Tomlinson, E. A. Patterson, “The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
[CrossRef]

2001 (1)

J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow Moiré to micro deformation measurements of curved surfaces,” Opt. Lasers Eng. 36, 29–40 (2001).
[CrossRef]

2000 (4)

L. Humbert, V. Valle, M. Cottron, “Experimental determination and empirical representation of out-of-plane displacements in a cracked elastic plate loaded in mode I,” Int. J. Solids Struct. 37, 5493–5504 (2000).
[CrossRef]

H. Aben, L. Ainola, “Isochromatic fringes in photoelasticity,” J. Opt. Soc. Am. A 13, 750–755 (2000).
[CrossRef]

H. W. Schreier, M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape function,” Exp. Mech. 42, 303–310 (2000).
[CrossRef]

B. Gutmann, H. Weber, “Phase unwrapping with the branch cut method: role of phase field direction,” App. Opt. 39, 4802–4816 (2000).
[CrossRef]

1999 (1)

C. Wykes, C. Buckberry, M. Dale, M. Reeves, D. Towers, “Functional testing using rapid prototyped components and optical measurment,” Opt. Lasers Eng. 31, 411–424 (1999).
[CrossRef]

1997 (2)

J. C. Dupre, A. Lagarde, “Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing,” Exp. Mech. 37, 393–397 (1997).
[CrossRef]

M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef] [PubMed]

1995 (1)

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

1994 (3)

F. Bremand, “A phase unwrapping technique for object relief determination,” Opt. Lasers Eng. 21, 49–60 (1994).
[CrossRef]

G. Mauvoisin, F. Bremand, A. Lagarde, “Three-dimensional shape reconstruction by phase-shifting shadow moiré,” Appl. Opt. 33, 2163–2169 (1994).
[CrossRef] [PubMed]

Y. Morimoto, M. Fujisaa, “Fringe pattern analysis by a phase shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

1989 (1)

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newtown-Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
[CrossRef]

Aben, H.

H. Aben, L. Ainola, “Isochromatic fringes in photoelasticity,” J. Opt. Soc. Am. A 13, 750–755 (2000).
[CrossRef]

Ainola, L.

H. Aben, L. Ainola, “Isochromatic fringes in photoelasticity,” J. Opt. Soc. Am. A 13, 750–755 (2000).
[CrossRef]

Arfken, G.

G. Arfken, “The method of steepest descent,” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), Sec. 7.4, pp. 428–436.

Boone, P.

J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow Moiré to micro deformation measurements of curved surfaces,” Opt. Lasers Eng. 36, 29–40 (2001).
[CrossRef]

Bremand, F.

C. Breque, J. C. Dupre, F. Bremand, “Calibration of a system of projection moiré for relief measuring: biomechanical applications,” Opt. Lasers Eng. 41, 241–260 (2004).
[CrossRef]

G. Mauvoisin, F. Bremand, A. Lagarde, “Three-dimensional shape reconstruction by phase-shifting shadow moiré,” Appl. Opt. 33, 2163–2169 (1994).
[CrossRef] [PubMed]

F. Bremand, “A phase unwrapping technique for object relief determination,” Opt. Lasers Eng. 21, 49–60 (1994).
[CrossRef]

Breque, C.

C. Breque, J. C. Dupre, F. Bremand, “Calibration of a system of projection moiré for relief measuring: biomechanical applications,” Opt. Lasers Eng. 41, 241–260 (2004).
[CrossRef]

Bruck, H. A.

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newtown-Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
[CrossRef]

Buckberry, C.

C. Wykes, C. Buckberry, M. Dale, M. Reeves, D. Towers, “Functional testing using rapid prototyped components and optical measurment,” Opt. Lasers Eng. 31, 411–424 (1999).
[CrossRef]

Burton, D. R.

Cottron, M.

L. Humbert, V. Valle, M. Cottron, “Experimental determination and empirical representation of out-of-plane displacements in a cracked elastic plate loaded in mode I,” Int. J. Solids Struct. 37, 5493–5504 (2000).
[CrossRef]

Cuevas, F. J.

Dale, M.

C. Wykes, C. Buckberry, M. Dale, M. Reeves, D. Towers, “Functional testing using rapid prototyped components and optical measurment,” Opt. Lasers Eng. 31, 411–424 (1999).
[CrossRef]

Degrieck, J.

J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow Moiré to micro deformation measurements of curved surfaces,” Opt. Lasers Eng. 36, 29–40 (2001).
[CrossRef]

Dupre, J. C.

C. Breque, J. C. Dupre, F. Bremand, “Calibration of a system of projection moiré for relief measuring: biomechanical applications,” Opt. Lasers Eng. 41, 241–260 (2004).
[CrossRef]

J. C. Dupre, A. Lagarde, “Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing,” Exp. Mech. 37, 393–397 (1997).
[CrossRef]

Fujisaa, M.

Y. Morimoto, M. Fujisaa, “Fringe pattern analysis by a phase shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

Gdeisat, M. A.

Gómez-Pedrero, J. A.

J. Villa, J. A. Quiroga, J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Lasers Eng. 41, 127–137 (2004).
[CrossRef]

Gutmann, B.

B. Gutmann, H. Weber, “Phase unwrapping with the branch cut method: role of phase field direction,” App. Opt. 39, 4802–4816 (2000).
[CrossRef]

Humbert, L.

L. Humbert, V. Valle, M. Cottron, “Experimental determination and empirical representation of out-of-plane displacements in a cracked elastic plate loaded in mode I,” Int. J. Solids Struct. 37, 5493–5504 (2000).
[CrossRef]

Lagarde, A.

J. C. Dupre, A. Lagarde, “Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing,” Exp. Mech. 37, 393–397 (1997).
[CrossRef]

G. Mauvoisin, F. Bremand, A. Lagarde, “Three-dimensional shape reconstruction by phase-shifting shadow moiré,” Appl. Opt. 33, 2163–2169 (1994).
[CrossRef] [PubMed]

Lalor, M. J.

Malacara, D.

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

Marroquin, J. L.

M. Servin, J. L. Marroquin, J. A. Quiroga, “Regularized quadrature and phase tracking from a single closed-fringe interferogram,” J Opt. Soc. Am A Opt. Image Sci. Vis 21, 411–419 (2004).
[CrossRef] [PubMed]

M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef] [PubMed]

Mauvoisin, G.

Mcneill, S. R.

M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, S. R. Mcneill, “Full-field speckle pattern image correlation with B-spline deformation function,” Exp. Mech. 42, 344–352 (2002).
[CrossRef]

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newtown-Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
[CrossRef]

Morimoto, Y.

Y. Morimoto, M. Fujisaa, “Fringe pattern analysis by a phase shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

Patterson, E. A.

R. A. Tomlinson, E. A. Patterson, “The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
[CrossRef]

Peng Cheng, M. A.

M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, S. R. Mcneill, “Full-field speckle pattern image correlation with B-spline deformation function,” Exp. Mech. 42, 344–352 (2002).
[CrossRef]

Peters, W. H.

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newtown-Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
[CrossRef]

Quiroga, J. A.

M. Servin, J. L. Marroquin, J. A. Quiroga, “Regularized quadrature and phase tracking from a single closed-fringe interferogram,” J Opt. Soc. Am A Opt. Image Sci. Vis 21, 411–419 (2004).
[CrossRef] [PubMed]

J. Villa, J. A. Quiroga, J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Lasers Eng. 41, 127–137 (2004).
[CrossRef]

Reeves, M.

C. Wykes, C. Buckberry, M. Dale, M. Reeves, D. Towers, “Functional testing using rapid prototyped components and optical measurment,” Opt. Lasers Eng. 31, 411–424 (1999).
[CrossRef]

Robin, E.

Rodriguez-Vera, R.

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

Schreier, H. W.

M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, S. R. Mcneill, “Full-field speckle pattern image correlation with B-spline deformation function,” Exp. Mech. 42, 344–352 (2002).
[CrossRef]

H. W. Schreier, M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape function,” Exp. Mech. 42, 303–310 (2000).
[CrossRef]

Servin, M.

M. Servin, J. L. Marroquin, J. A. Quiroga, “Regularized quadrature and phase tracking from a single closed-fringe interferogram,” J Opt. Soc. Am A Opt. Image Sci. Vis 21, 411–419 (2004).
[CrossRef] [PubMed]

M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef] [PubMed]

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

Sutton, M. A.

M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, S. R. Mcneill, “Full-field speckle pattern image correlation with B-spline deformation function,” Exp. Mech. 42, 344–352 (2002).
[CrossRef]

H. W. Schreier, M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape function,” Exp. Mech. 42, 303–310 (2000).
[CrossRef]

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newtown-Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
[CrossRef]

Tomlinson, R. A.

R. A. Tomlinson, E. A. Patterson, “The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
[CrossRef]

Towers, D.

C. Wykes, C. Buckberry, M. Dale, M. Reeves, D. Towers, “Functional testing using rapid prototyped components and optical measurment,” Opt. Lasers Eng. 31, 411–424 (1999).
[CrossRef]

Valle, V.

E. Robin, V. Valle, “Phase demodulation from a single fringe pattern based on a correlation technique,” Appl. Opt. 43, 4355–4361 (2004).
[CrossRef] [PubMed]

L. Humbert, V. Valle, M. Cottron, “Experimental determination and empirical representation of out-of-plane displacements in a cracked elastic plate loaded in mode I,” Int. J. Solids Struct. 37, 5493–5504 (2000).
[CrossRef]

Van Paepegem, W.

J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow Moiré to micro deformation measurements of curved surfaces,” Opt. Lasers Eng. 36, 29–40 (2001).
[CrossRef]

Villa, J.

J. Villa, J. A. Quiroga, J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Lasers Eng. 41, 127–137 (2004).
[CrossRef]

Weber, H.

B. Gutmann, H. Weber, “Phase unwrapping with the branch cut method: role of phase field direction,” App. Opt. 39, 4802–4816 (2000).
[CrossRef]

Wykes, C.

C. Wykes, C. Buckberry, M. Dale, M. Reeves, D. Towers, “Functional testing using rapid prototyped components and optical measurment,” Opt. Lasers Eng. 31, 411–424 (1999).
[CrossRef]

App. Opt. (1)

B. Gutmann, H. Weber, “Phase unwrapping with the branch cut method: role of phase field direction,” App. Opt. 39, 4802–4816 (2000).
[CrossRef]

Appl. Opt. (4)

Exp. Mech. (5)

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newtown-Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
[CrossRef]

M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, S. R. Mcneill, “Full-field speckle pattern image correlation with B-spline deformation function,” Exp. Mech. 42, 344–352 (2002).
[CrossRef]

H. W. Schreier, M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape function,” Exp. Mech. 42, 303–310 (2000).
[CrossRef]

J. C. Dupre, A. Lagarde, “Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing,” Exp. Mech. 37, 393–397 (1997).
[CrossRef]

R. A. Tomlinson, E. A. Patterson, “The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
[CrossRef]

Int. J. Solids Struct. (1)

L. Humbert, V. Valle, M. Cottron, “Experimental determination and empirical representation of out-of-plane displacements in a cracked elastic plate loaded in mode I,” Int. J. Solids Struct. 37, 5493–5504 (2000).
[CrossRef]

J Opt. Soc. Am A Opt. Image Sci. Vis (1)

M. Servin, J. L. Marroquin, J. A. Quiroga, “Regularized quadrature and phase tracking from a single closed-fringe interferogram,” J Opt. Soc. Am A Opt. Image Sci. Vis 21, 411–419 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

H. Aben, L. Ainola, “Isochromatic fringes in photoelasticity,” J. Opt. Soc. Am. A 13, 750–755 (2000).
[CrossRef]

Opt. Eng. (1)

Y. Morimoto, M. Fujisaa, “Fringe pattern analysis by a phase shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

Opt. Lasers Eng. (6)

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow Moiré to micro deformation measurements of curved surfaces,” Opt. Lasers Eng. 36, 29–40 (2001).
[CrossRef]

C. Breque, J. C. Dupre, F. Bremand, “Calibration of a system of projection moiré for relief measuring: biomechanical applications,” Opt. Lasers Eng. 41, 241–260 (2004).
[CrossRef]

J. Villa, J. A. Quiroga, J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Lasers Eng. 41, 127–137 (2004).
[CrossRef]

F. Bremand, “A phase unwrapping technique for object relief determination,” Opt. Lasers Eng. 21, 49–60 (1994).
[CrossRef]

C. Wykes, C. Buckberry, M. Dale, M. Reeves, D. Towers, “Functional testing using rapid prototyped components and optical measurment,” Opt. Lasers Eng. 31, 411–424 (1999).
[CrossRef]

Other (1)

G. Arfken, “The method of steepest descent,” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), Sec. 7.4, pp. 428–436.

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Figures (10)

Fig. 1
Fig. 1

Example of demodulation by pMPC with n = 4: (a) studied fringe pattern, (b) wrapped phase without orientation, (c) field of inclination, (d) field of orientation, i.e., field of unwrapped inclination, (e) wrapped phase, (f) unwrapped phase, (g) 3D view of the unwrapped phase.

Fig. 2
Fig. 2

Examples of noised fringe patterns: (a) noise = 20%, (b) noise = 60%, (c) noise = 80%, (d) noise = 20% and analyzed by MPC, (e) noise = 60% and analyzed by MPC, (f) noise = 80% and analyzed by MPC, (g) noise = 20% and analyzed by pMPC with n = 4, (h) noise = 60% and analyzed by pMPC with n = 4, (i) noise = 80% and analyzed by pMPC with n = 4.

Fig. 3
Fig. 3

Standard deviation evolution of MPC and pMPC: Noised fringe patterns.

Fig. 4
Fig. 4

Examples of degraded fringe patterns: (a) signal = 10% and disturbance = 160%, (b) signal = 30% and disturbance = 160%, (c) signal = 10% and disturbance = 40%, (d) signal = 30% and disturbance = 40%.

Fig. 5
Fig. 5

Three-dimensional map of algorithms in versus to signal and disturbance: (a) MPC map, (b) pMPC map with n = 3, (c) pMPC map with n = 4, (d) pMPC map with n = 5.

Fig. 6
Fig. 6

Standard deviation evolution of MPC and pMPC: degraded fringe patterns by contrast and background illumination variations.

Fig. 7
Fig. 7

Examples of fringe patterns with different pitch values: (a) p = 30 pixels, (b) p = 50 pixels, (c) p = 100 pixels, (d) fringe pattern with p = 30 analyzed by MPC, (e) fringe pattern with p = 50 analyzed by MPC, (f) fringe pattern with p = 100 analyzed by MPC, (g) fringe pattern with p = 30 analyzed by pMPC with n = 4, (h) fringe pattern with p = 50 analyzed by pMPC with n = 4, (i) fringe pattern with p = 100 analyzed by pMPC with n = 4.

Fig. 8
Fig. 8

Standard deviation evolution of MPC and pMPC: Influence of pitch.

Fig. 9
Fig. 9

Demodulation of interferogram: (a) fringe pattern, (b) wrapped phase, (c) unwrapped phase.

Fig. 10
Fig. 10

Demodulation of fringe pattern obtained by electronic speckle pattern interferometry: (a) fringe pattern, (b) wrapped phase, (c) unwrapped phase.

Equations (12)

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

f ( ξ , γ ) = A cos [ θ ( ξ , γ ) ] + B ,
θ ( ξ , γ ) = 2 π p ( x + ξ ) cos ( α ) + 2 π p ( y + γ ) sin ( α ) + φ .
ψ N ξ , γ ( A , B , φ , p , α ) = N ξ , γ [ I ( x + ξ , y + γ ) f ( ξ , γ ) ] 2 d ξ d γ .
P n ( X ) = i = 0 n C i X i ,
X = ( x + ξ ) cos ( α ) + ( y + γ ) sin ( α ) .
ψ N ξ , γ ( C 0 , C 1 , , C n , α ) = N ξ , γ ( I ( x + ξ , y + γ ) P n ( ξ , γ ) ) 2 d ξ d γ .
C ( n ) = n ( n + 1 ) 2 1 with C ( 0 ) = 0 .
A cos ( 2 π p X + φ ) + B = B + A cos ( φ ) 2 A sin ( φ ) π p X + + A ( 2 π X ) cos ( φ + n π 2 ) p n n ! .
A cos ( 2 π p X + φ ) + B = A cos ( φ ) + B 2 A sin ( φ ) π p X 2 A cos ( φ ) π 2 p 2 X 2 + 4 A sin ( φ ) π 3 3 p 3 X 3
C 0 = A cos ( φ ) + B , C 1 = 2 A sin ( φ ) π p , C 2 = 2 A cos ( φ ) π 2 p 2 , C 3 = 4 A sin ( φ ) π 3 3 p 3 .
p = 2 π 2 C 1 3 C 3 , φ = arctan ( π C 1 p C 2 ) ,
A = 2 C 1 2 ( C 2 2 C 1 ) 6 C 3 , B = 3 C 0 C 3 C 2 C 1 3 C 3 .

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