Abstract

A method for processing fringe patterns containing additively superimposed multiple fringe sets is presented. It enables to analyze different fringe families present in a single image separately. The proposed method is based on a two-dimensional continuous wavelet transform. A robust ridge extraction algorithm for a single fringe set extraction is presented. The method is fully automatic and requires no user interference. Spectral separation of fringe families is not required. Simulations are presented to verify performance and advantage of the proposed method over the Fourier transform based technique. Method validity has been confirmed using experimental images.

© 2012 Optical Society of America

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References

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  1. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1990).
  2. D. W. Robinson and G. Reid, Interferogram Analysis: Digital Fringe Pattern Measurement (Institute of Physics Publishing, 1993).
  3. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis For Optical Testing (Marcel Dekker, 1998).
  4. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18, 1862–1870 (2001).
    [CrossRef]
  5. K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns. ii. stationary phase analysis of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18, 1871–1881 (2001).
    [CrossRef]
  6. M. Wielgus and K. Patorski, “Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations,” Appl. Opt. 50, 5513–5523 (2011).
    [CrossRef]
  7. Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006).
    [CrossRef]
  8. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722–8732 (2006).
    [CrossRef]
  9. A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt. 46, 6120–6126 (2007).
    [CrossRef]
  10. K. Pokorski and K. Patorski, “Visualization of additive-type Moiré and time-average fringe patterns using the continuous wavelet transform,” Appl. Opt. 49, 3640–3651 (2010).
    [CrossRef]
  11. K. Patorski and K. Pokorski, “Examination of singular scalar light fields using wavelet processing of fork fringes,” Appl. Opt. 50, 773–781 (2011).
    [CrossRef]
  12. K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-Moiré phenomena,” Opt. Express 19, 26065–26078 (2011).
    [CrossRef]
  13. P. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, 1969).
  14. D. Post, “The Moiré grid-analyzer method for strain analysis,” Exp. Mech. 5, 368–377 (1965).
    [CrossRef]
  15. J. M. Burch and C. Forno, “High resolution Moire photography,” Opt. Eng. 21, 214602 (1982).
    [CrossRef]
  16. J. M. Huntley and J. E. Field, “High resolution Moire photography: application to dynamic stress analysis,” Opt. Eng. 28, 288926 (1989).
    [CrossRef]
  17. T. Bertin-Mourot, C. Denoual, G. Deshors, P. F. Louvigné, and T. Thomas, “High speed photography of Moire fringes. Application to ceramics under impact,” J. Phys. IV 07, C3-311–C3-316 (1997).
    [CrossRef]
  18. M. Dadkhah, F. Wang, and A. Kobayashi, “Simultaneous on-line measurement of orthogonal displacement fields by Moiré interferometry,” Exp. Tech. 12, 28–30 (1988).
    [CrossRef]
  19. A. S. Kobayashi and Z. K. Guo, “Simultaneous measurement of u- and v-displacement fields by Moiré interferometry,” Exp. Tech. 17, 21–24 (1993).
    [CrossRef]
  20. L. Salbut, “Multichannel system for automatic analysis of u, v, w displacements in grating interferometry,” in Physical Research, W. Juptner and W. Osten, eds. 19 (Akademie Verlag, 1993), pp. 282–287.
  21. J. Schmit, K. Patorski, and K. Creath, “Simultaneous registration of in- and out-of-plane displacements in modified grating interferometry,” Opt. Eng. 36, 2240–2248 (1997).
    [CrossRef]
  22. N. Demoli and D. Vukicevic, “Detection of hidden stationary deformations of vibrating surfaces by use of time-averaged digital holographic interferometry,” Opt. Lett. 29, 2423–2425 (2004).
    [CrossRef]
  23. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Simultaneous multidimensional deformation measurements using digital holographic Moiré,” Appl. Opt. 50, 4189–4197 (2011).
    [CrossRef]
  24. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Estimation of multiple phases from a single fringe pattern in digital holographic interferometry,” Opt. Express 20, 1281–1291 (2012).
    [CrossRef]
  25. J. Vargas, J. A. Quiroga, and T. Belenguer, “Direct demodulation of closed-fringe interferograms based on active contours,” Opt. Lett. 35, 3550–3552 (2010).
    [CrossRef]
  26. K. Pokorski and K. Patorski, “Continuous wavelet transform processing of fringe patterns containing multiple fringe sets,” Proc. SPIE, 8697-29.
  27. J.-P. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, 2008).
  28. J. Kirby, “Which wavelet best reproduces the Fourier power spectrum,” Comput. Geosci. 31, 846–864 (2005).
    [CrossRef]
  29. S. Li, X. Wang, X. Su, and F. Tang, “Two-dimensional wavelet transform for reliability-guided phase unwrapping in optical fringe pattern analysis,” Appl. Opt. 51, 2026–2034 (2012).
    [CrossRef]
  30. J. Ma, Z. Wang, M. Vo, and L. Luu, “Parameter discretization in two-dimensional continuous wavelet transform for fast fringe pattern analysis,” Appl. Opt. 50, 6399–6408 (2011).
    [CrossRef]
  31. J. Ma, Z. Wang, B. Pan, T. Hoang, M. Vo, and L. Luu, “Two-dimensional continuous wavelet transform for phase determination of complex interferograms,” Appl. Opt. 50, 2425–2430 (2011).
    [CrossRef]
  32. S. Li, W. Chen, and X. Su, “Reliability-guided phase unwrapping in wavelet-transform profilometry,” Appl. Opt. 47, 3369–3377 (2008).
    [CrossRef]
  33. J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
    [CrossRef]
  34. K. Patorski, Handbook of the Moiré Fringe Technique(Elsevier, 1993).
  35. L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
    [CrossRef]
  36. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  37. R. Czarnek, “Three-mirror, four-beam Moiré interferometer and its capabilities,” Opt. Lasers Eng. 15, 93–101 (1991).
    [CrossRef]
  38. P. Post, B. Han, and P. Ifju, High Sensitivity Moiré (Springer Verlag, 1994).

2012 (2)

2011 (6)

2010 (3)

K. Pokorski and K. Patorski, “Visualization of additive-type Moiré and time-average fringe patterns using the continuous wavelet transform,” Appl. Opt. 49, 3640–3651 (2010).
[CrossRef]

J. Vargas, J. A. Quiroga, and T. Belenguer, “Direct demodulation of closed-fringe interferograms based on active contours,” Opt. Lett. 35, 3550–3552 (2010).
[CrossRef]

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

2008 (1)

2007 (2)

2006 (2)

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006).
[CrossRef]

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722–8732 (2006).
[CrossRef]

2005 (1)

J. Kirby, “Which wavelet best reproduces the Fourier power spectrum,” Comput. Geosci. 31, 846–864 (2005).
[CrossRef]

2004 (1)

2001 (2)

1997 (2)

J. Schmit, K. Patorski, and K. Creath, “Simultaneous registration of in- and out-of-plane displacements in modified grating interferometry,” Opt. Eng. 36, 2240–2248 (1997).
[CrossRef]

T. Bertin-Mourot, C. Denoual, G. Deshors, P. F. Louvigné, and T. Thomas, “High speed photography of Moire fringes. Application to ceramics under impact,” J. Phys. IV 07, C3-311–C3-316 (1997).
[CrossRef]

1993 (1)

A. S. Kobayashi and Z. K. Guo, “Simultaneous measurement of u- and v-displacement fields by Moiré interferometry,” Exp. Tech. 17, 21–24 (1993).
[CrossRef]

1991 (1)

R. Czarnek, “Three-mirror, four-beam Moiré interferometer and its capabilities,” Opt. Lasers Eng. 15, 93–101 (1991).
[CrossRef]

1989 (1)

J. M. Huntley and J. E. Field, “High resolution Moire photography: application to dynamic stress analysis,” Opt. Eng. 28, 288926 (1989).
[CrossRef]

1988 (1)

M. Dadkhah, F. Wang, and A. Kobayashi, “Simultaneous on-line measurement of orthogonal displacement fields by Moiré interferometry,” Exp. Tech. 12, 28–30 (1988).
[CrossRef]

1982 (2)

1965 (1)

D. Post, “The Moiré grid-analyzer method for strain analysis,” Exp. Mech. 5, 368–377 (1965).
[CrossRef]

Abid, A. Z.

Ali, S. T.

J.-P. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, 2008).

Antoine, J.-P.

J.-P. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, 2008).

Asundi, A. K.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Belenguer, T.

Bertin-Mourot, T.

T. Bertin-Mourot, C. Denoual, G. Deshors, P. F. Louvigné, and T. Thomas, “High speed photography of Moire fringes. Application to ceramics under impact,” J. Phys. IV 07, C3-311–C3-316 (1997).
[CrossRef]

Bioucas-Dias, J.

J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[CrossRef]

Bone, D. J.

Burch, J. M.

J. M. Burch and C. Forno, “High resolution Moire photography,” Opt. Eng. 21, 214602 (1982).
[CrossRef]

Burton, D. R.

Chen, W.

Creath, K.

J. Schmit, K. Patorski, and K. Creath, “Simultaneous registration of in- and out-of-plane displacements in modified grating interferometry,” Opt. Eng. 36, 2240–2248 (1997).
[CrossRef]

Czarnek, R.

R. Czarnek, “Three-mirror, four-beam Moiré interferometer and its capabilities,” Opt. Lasers Eng. 15, 93–101 (1991).
[CrossRef]

Dadkhah, M.

M. Dadkhah, F. Wang, and A. Kobayashi, “Simultaneous on-line measurement of orthogonal displacement fields by Moiré interferometry,” Exp. Tech. 12, 28–30 (1988).
[CrossRef]

Demoli, N.

Denoual, C.

T. Bertin-Mourot, C. Denoual, G. Deshors, P. F. Louvigné, and T. Thomas, “High speed photography of Moire fringes. Application to ceramics under impact,” J. Phys. IV 07, C3-311–C3-316 (1997).
[CrossRef]

Deshors, G.

T. Bertin-Mourot, C. Denoual, G. Deshors, P. F. Louvigné, and T. Thomas, “High speed photography of Moire fringes. Application to ceramics under impact,” J. Phys. IV 07, C3-311–C3-316 (1997).
[CrossRef]

Field, J. E.

J. M. Huntley and J. E. Field, “High resolution Moire photography: application to dynamic stress analysis,” Opt. Eng. 28, 288926 (1989).
[CrossRef]

Forno, C.

J. M. Burch and C. Forno, “High resolution Moire photography,” Opt. Eng. 21, 214602 (1982).
[CrossRef]

Gdeisat, M. A.

Gorthi, S. S.

Guo, Z. K.

A. S. Kobayashi and Z. K. Guo, “Simultaneous measurement of u- and v-displacement fields by Moiré interferometry,” Exp. Tech. 17, 21–24 (1993).
[CrossRef]

Han, B.

P. Post, B. Han, and P. Ifju, High Sensitivity Moiré (Springer Verlag, 1994).

Hoang, T.

Huang, L.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Huntley, J. M.

J. M. Huntley and J. E. Field, “High resolution Moire photography: application to dynamic stress analysis,” Opt. Eng. 28, 288926 (1989).
[CrossRef]

Ifju, P.

P. Post, B. Han, and P. Ifju, High Sensitivity Moiré (Springer Verlag, 1994).

Ina, H.

Kemao, Q.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Kirby, J.

J. Kirby, “Which wavelet best reproduces the Fourier power spectrum,” Comput. Geosci. 31, 846–864 (2005).
[CrossRef]

Kobayashi, A.

M. Dadkhah, F. Wang, and A. Kobayashi, “Simultaneous on-line measurement of orthogonal displacement fields by Moiré interferometry,” Exp. Tech. 12, 28–30 (1988).
[CrossRef]

Kobayashi, A. S.

A. S. Kobayashi and Z. K. Guo, “Simultaneous measurement of u- and v-displacement fields by Moiré interferometry,” Exp. Tech. 17, 21–24 (1993).
[CrossRef]

Kobayashi, S.

Lalor, M. J.

Larkin, K. G.

Li, S.

Lilley, F.

Louvigné, P. F.

T. Bertin-Mourot, C. Denoual, G. Deshors, P. F. Louvigné, and T. Thomas, “High speed photography of Moire fringes. Application to ceramics under impact,” J. Phys. IV 07, C3-311–C3-316 (1997).
[CrossRef]

Luu, L.

Ma, H.

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006).
[CrossRef]

Ma, J.

Malacara, D.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis For Optical Testing (Marcel Dekker, 1998).

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis For Optical Testing (Marcel Dekker, 1998).

Murenzi, R.

J.-P. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, 2008).

Oldfield, M. A.

Pan, B.

J. Ma, Z. Wang, B. Pan, T. Hoang, M. Vo, and L. Luu, “Two-dimensional continuous wavelet transform for phase determination of complex interferograms,” Appl. Opt. 50, 2425–2430 (2011).
[CrossRef]

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Patorski, K.

Pokorski, K.

Post, D.

D. Post, “The Moiré grid-analyzer method for strain analysis,” Exp. Mech. 5, 368–377 (1965).
[CrossRef]

Post, P.

P. Post, B. Han, and P. Ifju, High Sensitivity Moiré (Springer Verlag, 1994).

Quiroga, J. A.

Rajshekhar, G.

Rastogi, P.

Reid, G.

D. W. Robinson and G. Reid, Interferogram Analysis: Digital Fringe Pattern Measurement (Institute of Physics Publishing, 1993).

Robinson, D. W.

D. W. Robinson and G. Reid, Interferogram Analysis: Digital Fringe Pattern Measurement (Institute of Physics Publishing, 1993).

Salbut, L.

L. Salbut, “Multichannel system for automatic analysis of u, v, w displacements in grating interferometry,” in Physical Research, W. Juptner and W. Osten, eds. 19 (Akademie Verlag, 1993), pp. 282–287.

Schmit, J.

J. Schmit, K. Patorski, and K. Creath, “Simultaneous registration of in- and out-of-plane displacements in modified grating interferometry,” Opt. Eng. 36, 2240–2248 (1997).
[CrossRef]

Schwider, J.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1990).

Servin, M.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis For Optical Testing (Marcel Dekker, 1998).

Su, X.

Takeda, M.

Tang, F.

Theocaris, P.

P. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, 1969).

Thomas, T.

T. Bertin-Mourot, C. Denoual, G. Deshors, P. F. Louvigné, and T. Thomas, “High speed photography of Moire fringes. Application to ceramics under impact,” J. Phys. IV 07, C3-311–C3-316 (1997).
[CrossRef]

Trusiak, M.

Valadao, G.

J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[CrossRef]

Vandergheynst, P.

J.-P. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, 2008).

Vargas, J.

Vo, M.

Vukicevic, D.

Wang, F.

M. Dadkhah, F. Wang, and A. Kobayashi, “Simultaneous on-line measurement of orthogonal displacement fields by Moiré interferometry,” Exp. Tech. 12, 28–30 (1988).
[CrossRef]

Wang, X.

Wang, Z.

Wielgus, M.

Appl. Opt. (10)

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722–8732 (2006).
[CrossRef]

A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt. 46, 6120–6126 (2007).
[CrossRef]

K. Pokorski and K. Patorski, “Visualization of additive-type Moiré and time-average fringe patterns using the continuous wavelet transform,” Appl. Opt. 49, 3640–3651 (2010).
[CrossRef]

K. Patorski and K. Pokorski, “Examination of singular scalar light fields using wavelet processing of fork fringes,” Appl. Opt. 50, 773–781 (2011).
[CrossRef]

M. Wielgus and K. Patorski, “Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations,” Appl. Opt. 50, 5513–5523 (2011).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Simultaneous multidimensional deformation measurements using digital holographic Moiré,” Appl. Opt. 50, 4189–4197 (2011).
[CrossRef]

S. Li, X. Wang, X. Su, and F. Tang, “Two-dimensional wavelet transform for reliability-guided phase unwrapping in optical fringe pattern analysis,” Appl. Opt. 51, 2026–2034 (2012).
[CrossRef]

J. Ma, Z. Wang, M. Vo, and L. Luu, “Parameter discretization in two-dimensional continuous wavelet transform for fast fringe pattern analysis,” Appl. Opt. 50, 6399–6408 (2011).
[CrossRef]

J. Ma, Z. Wang, B. Pan, T. Hoang, M. Vo, and L. Luu, “Two-dimensional continuous wavelet transform for phase determination of complex interferograms,” Appl. Opt. 50, 2425–2430 (2011).
[CrossRef]

S. Li, W. Chen, and X. Su, “Reliability-guided phase unwrapping in wavelet-transform profilometry,” Appl. Opt. 47, 3369–3377 (2008).
[CrossRef]

Comput. Geosci. (1)

J. Kirby, “Which wavelet best reproduces the Fourier power spectrum,” Comput. Geosci. 31, 846–864 (2005).
[CrossRef]

Exp. Mech. (1)

D. Post, “The Moiré grid-analyzer method for strain analysis,” Exp. Mech. 5, 368–377 (1965).
[CrossRef]

Exp. Tech. (2)

M. Dadkhah, F. Wang, and A. Kobayashi, “Simultaneous on-line measurement of orthogonal displacement fields by Moiré interferometry,” Exp. Tech. 12, 28–30 (1988).
[CrossRef]

A. S. Kobayashi and Z. K. Guo, “Simultaneous measurement of u- and v-displacement fields by Moiré interferometry,” Exp. Tech. 17, 21–24 (1993).
[CrossRef]

IEEE Trans. Image Process. (1)

J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Phys. IV (1)

T. Bertin-Mourot, C. Denoual, G. Deshors, P. F. Louvigné, and T. Thomas, “High speed photography of Moire fringes. Application to ceramics under impact,” J. Phys. IV 07, C3-311–C3-316 (1997).
[CrossRef]

Opt. Eng. (4)

J. Schmit, K. Patorski, and K. Creath, “Simultaneous registration of in- and out-of-plane displacements in modified grating interferometry,” Opt. Eng. 36, 2240–2248 (1997).
[CrossRef]

J. M. Burch and C. Forno, “High resolution Moire photography,” Opt. Eng. 21, 214602 (1982).
[CrossRef]

J. M. Huntley and J. E. Field, “High resolution Moire photography: application to dynamic stress analysis,” Opt. Eng. 28, 288926 (1989).
[CrossRef]

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lasers Eng. (2)

R. Czarnek, “Three-mirror, four-beam Moiré interferometer and its capabilities,” Opt. Lasers Eng. 15, 93–101 (1991).
[CrossRef]

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Opt. Lett. (2)

Other (9)

L. Salbut, “Multichannel system for automatic analysis of u, v, w displacements in grating interferometry,” in Physical Research, W. Juptner and W. Osten, eds. 19 (Akademie Verlag, 1993), pp. 282–287.

P. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, 1969).

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1990).

D. W. Robinson and G. Reid, Interferogram Analysis: Digital Fringe Pattern Measurement (Institute of Physics Publishing, 1993).

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis For Optical Testing (Marcel Dekker, 1998).

K. Pokorski and K. Patorski, “Continuous wavelet transform processing of fringe patterns containing multiple fringe sets,” Proc. SPIE, 8697-29.

J.-P. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, 2008).

K. Patorski, Handbook of the Moiré Fringe Technique(Elsevier, 1993).

P. Post, B. Han, and P. Ifju, High Sensitivity Moiré (Springer Verlag, 1994).

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Figures (4)

Fig. 1.
Fig. 1.

Reliability map construction: (a) simulated image containing two fringe sets, (b) one of the scalograms corresponding to pixel A, with clearly visible two maxima corresponding to two fringe sets present in the image (a); high value corresponding to higher scale is divided by π on the angle axis due to the fact that the wavelet rotation of π does not affect the wavelet coefficient modulus; the distance (in pixels) between maxima is calculated as min(d1,d2), and (c) gray scale reliability map.

Fig. 2.
Fig. 2.

Method accuracy evaluation: (a) simulated fringe pattern generated by additive superposition of two fringe families, (b) one family wrapped phase extracted using proposed method, and (c) phase determination error distribution (rad).

Fig. 3.
Fig. 3.

(a) Simulated fringe pattern with additive Gaussian noise and nonuniform modulation and background intensity distributions, (b), (c) fringe families extracted using the 2D CWT based method; note the presence of errors in the area of overlapping fringe spatial frequencies, and (d) fast Fourier transform (FFT) of (a).

Fig. 4.
Fig. 4.

(a) Additively superimposed experimental Moiré interferograms and (b), (c) both fringe sets separated using proposed algorithm.

Equations (3)

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S2D(s,b,θ)=sηR2ψ*(s1rθ(xb))f(x)d2x,
ψ(x)=eikxem|x|2,
ϵ=I0πmexpπ2m,

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