Abstract

A method for demodulating fringe patterns containing contrast reversals is proposed. It consists of two steps. First, the absolute value of the fringe intensity distribution with its background removed is calculated. Then, two dimensional continuous wavelet transform with enhanced ridge extraction algorithm is applied to extract the fringe phase map. Proposed approach allows to dispose of phase jumps along the contrast reversal bands. The method requires only one image and has no special demands concerning the fringe pattern design. Method validity and robustness is confirmed using experimentally acquired time-averaged interferograms of vibrating silicon micromembranes.

© 2013 OSA

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2013 (1)

2012 (4)

H. M. Subhash, “Full-field and single-shot full-field optical coherence tomography: A novel technique for biomedical imaging applications,” Adv. Opt. Technol.2012, 435408 (2012).

Z. Sunderland, K. Patorski, and K. Pokorski, “Evaluation of optical parameters of quasi-parallel plates with single-frame interferogram analysis methods,” Photonics Lett. Poland2, 63–65 (2012).

J. Ma, Z. Wang, M. Vo, and B. Pan, “Wavelet selection in two-dimensional continuous wavelet transform technique for optical fringe pattern analysis,” J. Opt.14, 065403 (2012).
[CrossRef]

X. Wang, J. Gao, and W. Chen, “A new tiling scheme for 2-d continuous wavelet transform with different rotation parameters at different scales resulting in a tighter frame,” IEEE Signal Process. Lett.19, 407–410 (2012).
[CrossRef]

2011 (3)

2010 (2)

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] [PubMed]

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]

2009 (1)

M. Ragulskis and Z. Navickas, “Interpretation of fringes produced by time-averaged projection moiré,” Strain45, doi: (2009), and references therein.
[CrossRef]

2008 (2)

2007 (3)

2006 (3)

K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng.45, 085602 (2006).
[CrossRef]

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] [PubMed]

2004 (2)

2003 (2)

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry in the MEMS field,” Proc. SPIE5145, 1–16 (2003).
[CrossRef]

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE5145, 23–32 (2003).
[CrossRef]

2001 (1)

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, “Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization,” Proc. SPIE4400, 51–60 (2001).
[CrossRef]

2000 (2)

P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt.39, 2658–2663 (2000).
[CrossRef]

J. G. Fujimoto, W. Drexler, U. Morgner, F. Kärtner, and E. Ippen, “Optical coherence tomography: High resolution imaging using echoes of light,” Opt. Photon. News11, 24–31 (2000).
[CrossRef]

1997 (1)

1996 (1)

1994 (3)

1993 (1)

G. Xian-Yu Su, von Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun.98, 141–150 (1993).
[CrossRef]

1990 (1)

R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibration patterns using electro-optic holography,” Proc. SPIE1162, 456–467 (1990).
[CrossRef]

1988 (1)

1984 (1)

1971 (1)

1932 (1)

Abid, A. Z.

Ai, C.

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]

Bioucas-Dias, J.

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

Bosseboeuf, A.

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry in the MEMS field,” Proc. SPIE5145, 1–16 (2003).
[CrossRef]

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, “Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization,” Proc. SPIE4400, 51–60 (2001).
[CrossRef]

Burke, J.

Burton, D. R.

Chavel, P.

Chen, W.

X. Wang, J. Gao, and W. Chen, “A new tiling scheme for 2-d continuous wavelet transform with different rotation parameters at different scales resulting in a tighter frame,” IEEE Signal Process. Lett.19, 407–410 (2012).
[CrossRef]

Choi, E. S.

Choi, W. J.

Danaie, K.

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, “Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization,” Proc. SPIE4400, 51–60 (2001).
[CrossRef]

de Groot, P.

Dean, T.

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE5145, 23–32 (2003).
[CrossRef]

Descour, M.

Doval, A. F.

J. D. R. Valera, J. D. C. Jones, and A. F. Doval, “Whole-field vibration phase measurement with electronic speckle pattern interferometry (espi),” Proc. SPIE2248, 241–248 (1994).
[CrossRef]

Drexler, W.

J. G. Fujimoto, W. Drexler, U. Morgner, F. Kärtner, and E. Ippen, “Optical coherence tomography: High resolution imaging using echoes of light,” Opt. Photon. News11, 24–31 (2000).
[CrossRef]

Fairman, P. S.

Fujimoto, J. G.

J. G. Fujimoto, W. Drexler, U. Morgner, F. Kärtner, and E. Ippen, “Optical coherence tomography: High resolution imaging using echoes of light,” Opt. Photon. News11, 24–31 (2000).
[CrossRef]

Gao, J.

X. Wang, J. Gao, and W. Chen, “A new tiling scheme for 2-d continuous wavelet transform with different rotation parameters at different scales resulting in a tighter frame,” IEEE Signal Process. Lett.19, 407–410 (2012).
[CrossRef]

Gdeisat, M. A.

Gorecki, C.

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE5145, 23–32 (2003).
[CrossRef]

Greivenkamp, J. E.

Hibino, K.

Hoang, T.

Hovanesian, J. D.

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]

Hung, Y.

Ippen, E.

J. G. Fujimoto, W. Drexler, U. Morgner, F. Kärtner, and E. Ippen, “Optical coherence tomography: High resolution imaging using echoes of light,” Opt. Photon. News11, 24–31 (2000).
[CrossRef]

Jacobelli, A.

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE5145, 23–32 (2003).
[CrossRef]

Jones, J. D. C.

J. D. R. Valera, J. D. C. Jones, and A. F. Doval, “Whole-field vibration phase measurement with electronic speckle pattern interferometry (espi),” Proc. SPIE2248, 241–248 (1994).
[CrossRef]

Jozwik, M.

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE5145, 23–32 (2003).
[CrossRef]

Juskaitis, R.

Kacperski, J.

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE5145, 23–32 (2003).
[CrossRef]

Kärtner, F.

J. G. Fujimoto, W. Drexler, U. Morgner, F. Kärtner, and E. Ippen, “Optical coherence tomography: High resolution imaging using echoes of light,” Opt. Photon. News11, 24–31 (2000).
[CrossRef]

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]

Lalor, M. J.

Larkin, K. G.

Lee, B. H.

Lilley, F.

Lowman, A. E.

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.

Mack, V.

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).

Morgner, U.

J. G. Fujimoto, W. Drexler, U. Morgner, F. Kärtner, and E. Ippen, “Optical coherence tomography: High resolution imaging using echoes of light,” Opt. Photon. News11, 24–31 (2000).
[CrossRef]

Murenzi, R.

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

Na, J.

Navickas, Z.

M. Ragulskis and Z. Navickas, “Interpretation of fringes produced by time-averaged projection moiré,” Strain45, doi: (2009), and references therein.
[CrossRef]

Neil, M. A. A.

Oreb, B. F.

Osterberg, H.

Pan, B.

J. Ma, Z. Wang, M. Vo, and B. Pan, “Wavelet selection in two-dimensional continuous wavelet transform technique for optical fringe pattern analysis,” J. Opt.14, 065403 (2012).
[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] [PubMed]

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.

K. Patorski and M. Trusiak, “Highly contrasted Bessel fringe minima visualization for time-averaged vibration profilometry using Hilbert transform two-frame processing,” Opt. Express21, 16863–16881 (2013).
[CrossRef] [PubMed]

Z. Sunderland, K. Patorski, and K. Pokorski, “Evaluation of optical parameters of quasi-parallel plates with single-frame interferogram analysis methods,” Photonics Lett. Poland2, 63–65 (2012).

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] [PubMed]

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] [PubMed]

A. Styk and K. Patorski, “Fizeau interferometer for quasi parallel optical plate testing,” Proc. SPIE7063, 70630P–70630P–9 (2008).
[CrossRef]

A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity modulation determination,” Appl. Opt.46, 4613–4624 (2007).
[CrossRef] [PubMed]

K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng.45, 085602 (2006).
[CrossRef]

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE5145, 23–32 (2003).
[CrossRef]

K. Pokorski and K. Patorski, “Comprehensive fringe pattern processing using continuous wavelet transform,” Fringe 2013, doi:33 pp. 225–228 (Springer-VerlagBerlin Heidelberg, 2014).
[CrossRef]

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

Petitgrand, S.

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry in the MEMS field,” Proc. SPIE5145, 1–16 (2003).
[CrossRef]

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, “Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization,” Proc. SPIE4400, 51–60 (2001).
[CrossRef]

Pokorski, K.

Z. Sunderland, K. Patorski, and K. Pokorski, “Evaluation of optical parameters of quasi-parallel plates with single-frame interferogram analysis methods,” Photonics Lett. Poland2, 63–65 (2012).

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] [PubMed]

K. Pokorski and K. Patorski, “Comprehensive fringe pattern processing using continuous wavelet transform,” Fringe 2013, doi:33 pp. 225–228 (Springer-VerlagBerlin Heidelberg, 2014).
[CrossRef]

Pryputniewicz, R. J.

R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibration patterns using electro-optic holography,” Proc. SPIE1162, 456–467 (1990).
[CrossRef]

Ragulskis, M.

M. Ragulskis and Z. Navickas, “Interpretation of fringes produced by time-averaged projection moiré,” Strain45, doi: (2009), and references therein.
[CrossRef]

Rahman, M.

Reid, G.

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

Richards-Kortum, R.

Robinson, D.W.

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

Rogers, J.

Rosvold, G.

Ryu, S. Y.

Salbut, L.

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE5145, 23–32 (2003).
[CrossRef]

Servin, M.

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

Sokolov, K.

Stetson, K. A.

R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibration patterns using electro-optic holography,” Proc. SPIE1162, 456–467 (1990).
[CrossRef]

Strand, T. C.

Styk, A.

A. Styk and K. Patorski, “Fizeau interferometer for quasi parallel optical plate testing,” Proc. SPIE7063, 70630P–70630P–9 (2008).
[CrossRef]

A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity modulation determination,” Appl. Opt.46, 4613–4624 (2007).
[CrossRef] [PubMed]

K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng.45, 085602 (2006).
[CrossRef]

Subhash, H. M.

H. M. Subhash, “Full-field and single-shot full-field optical coherence tomography: A novel technique for biomedical imaging applications,” Adv. Opt. Technol.2012, 435408 (2012).

Sunderland, Z.

Z. Sunderland, K. Patorski, and K. Pokorski, “Evaluation of optical parameters of quasi-parallel plates with single-frame interferogram analysis methods,” Photonics Lett. Poland2, 63–65 (2012).

Theocaris, P. S.

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

Timoshenko, S.

S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1959).

Tkaczyk, T.

Trusiak, M.

Valadao, G.

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Figures (8)

Fig. 1
Fig. 1

Proposed algorithm diagram.

Fig. 2
Fig. 2

Time-averaged interferogram of a vibrating circular silicon micromembrane, frez = 833 kHz [5].

Fig. 3
Fig. 3

Preprocessing steps: (a) original interferogram, (b) its background distribution, and (c) absolute value of the original image with its background removed.

Fig. 4
Fig. 4

Scale and angle maps preparation for the second ridge extraction procedure: (a) and (e), scales and angles extracted using the direct maximum algorithm; (b) and (f), modulation map and angle mask; (c) and (g), masked scale and angle maps; (d) and (h), scale and angle maps extrapolated, filtered and rounded.

Fig. 5
Fig. 5

Proposed method carrier demodulation results: (a) original image, (b) extracted phase, (c) reconstructed image.

Fig. 6
Fig. 6

Proposed method accuracy evaluation results: (a) error distribution relative to the TPS method, (b) error distribution relative to the standard 2D CWT processing. In the experimental setup described in [5] the error of 1 rad corresponds to OPD of approx. 53 nm (laser diode wavelength λ = 670 nm).

Fig. 7
Fig. 7

Results of multiplicative superimposition of: (a) two identical interferograms of static micromembrane, (b) static micromembrane interferogram and the interferogram obtained by our method, (c) two indentical interferograms generated by our algorithm as the original interferogram reconstruction. In all cases the two overlapped images were lateraly displaced by 30 pixels, the image dimensions were 350 × 350 pixels.

Fig. 8
Fig. 8

Results of the rectified subtractive superimposition of co-phasial (a) and out-of-phase (b) interferograms of static and vibrating micromembranes. Double frequency carrier is modulated by functions 1 ∓ J0, respectively. No moiré fringes are generated over the membrane area, which proves the same mean shape of the vibrating and stationary membranes.

Equations (5)

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

I vibr = K ( x , y ) { 1 + C stat ( x , y ) J 0 [ 4 π λ a 0 ] cos φ vibr } ,
S 2 D ( s , b , θ ) = s η R 2 ψ * ( s 1 r θ ( x b ) ) f ( x ) d 2 x ,
ψ ( x ) = 2 | x / σ i σ k | 2 σ 2 e i k x e | x | 2 2 σ ,
φ c ( x , y ) = c arctan ( s 2 2 φ x 2 ) c arctan ( s 2 2 φ y 2 ) ,
I subtr = | ( 1 + cos α ) ( 1 + J 0 cos α ) | = | ( 1 J 0 ) cos α | ;

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