Abstract

Time-averaged fringe patterns in vibration testing of MEMS (microelectromechanical systems) are unaffected by carrier displacements. They are additive superimposition type moirés. These features and Hilbert transform vulnerability to additive trend are utilized for visualization of centers of dark Bessel fringes. Two frames with shifted carrier are subtracted for background and noise correction. Two normalized images of this pattern are calculated with slightly different bias levels and subtracted. The method does not require precise phase shifting between two frames, cosinusoidal carrier and linear recording. It enables detecting light power variations and phase shifting nonuniformities. Synthetic and experimental results corroborate the robustness of the method.

©2013 Optical Society of America

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2012 (3)

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

Q. Yuan, Z. Gao, Y. Zhou, G. Chu, and C. Zhang, “Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical aperture spherical surfaces,” Opt. Lasers Eng. 50(11), 1568–1574 (2012).
[Crossref]

M. Trusiak, K. Patorski, and M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express 20(21), 23463–23479 (2012).
[Crossref] [PubMed]

2011 (2)

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(19), 3640–3651 (2010).
[Crossref] [PubMed]

M. Olfatnia, V. R. Singh, T. Xu, J. M. Miao, and L. S. Ong, “Analysis of the vibration modes of piezoelectric circular microdiaphragms,” J. Micromech. Microeng. 20(8), 085013 (2010).
[Crossref]

2009 (6)

M. Ragulskis and Z. Navickas, “Interpretation of fringes produced by time-averaged projection moiré,” Strain 45, (2009), doi: (and references therein).
[Crossref]

I. Amidror and R. D. Hersch, “The role of Fourier theory and of modulation in the prediction of visible moiré effects,” J. Mod. Opt. 56(9), 1103–1118 (2009).
[Crossref]

M. Ragulskis, L. Saunoriene, and R. Maskeliunas, “The structure of moiré grating lines and its influence to time-averaged fringes,” Exp. Tech. 33(2), 60–64 (2009).
[Crossref]

M. Ragulskis, A. Aleksa, and R. Maskeliunas, “Contrast enhancement of time-averaged fringes based on moving average mapping functions,” Opt. Lasers Eng. 47(7–8), 768–773 (2009).
[Crossref]

L. Xiong and S. Jia, “Phase-error analysis and elimination for nonsinusoidal waveforms in Hilbert transform digital-fringe projection profilometry,” Opt. Lett. 34(15), 2363–2365 (2009).
[Crossref] [PubMed]

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Normalization of fringe patterns using the bidimensional empirical mode decomposition and the Hilbert transform,” Appl. Opt. 48(36), 6862–6869 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (1)

2006 (4)

K. Patorski, A. Styk, L. Bruno, and P. Szwaykowski, “Tilt-shift error detection in phase-shifting interferometry,” Opt. Express 14(12), 5232–5249 (2006).
[Crossref] [PubMed]

L. Saunoriene and M. Ragulskis, “Visualization of fringes in time averaged moiré patterns,” Inf. Technol. Control 35(3), 249–254 (2006).

K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng. 45(8), 085602 (2006).
[Crossref]

E. Hong, S. Trolier-McKinstry, R. Smith, S. V. Krishnaswamy, and C. B. Freidhoff, “Vibration of micromachined circular piezoelectric diaphragms,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53(4), 697–706 (2006).
[PubMed]

2005 (1)

C. Gorecki, M. Jozwik, and L. Salbut, “Multifunctional interferometric platform for on-chip testing the micromechanical properties of MEMS/MOEMS,” J. Microlith., Microfab., Microsyst. 4(4), 041402 (2005).

2003 (3)

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry techniques in the MEMS field,” Proc. SPIE 5145, 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. SPIE 5145, 23–32 (2003).
[Crossref]

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4–6), 221–227 (2003).
[Crossref]

2002 (1)

2001 (3)

2000 (1)

1994 (2)

G. O. Rosvold, “Video-based vibration analysis using projected fringes,” Appl. Opt. 33(5), 775–786 (1994).
[Crossref] [PubMed]

R. J. Pryputniewicz, “A hybrid approach to deformation analysis,” Proc. SPIE 2342, 282–296 (1994).
[Crossref]

1992 (1)

1990 (3)

J. A. Lin, J. Hsu, and S. G. Shiue, “Quantitative three-beam Ronchi test,” Appl. Opt. 29(13), 1912–1918 (1990).
[Crossref] [PubMed]

R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibration patterns using electro-optic holography,” Proc. SPIE 1162, 456–467 (1990).
[Crossref]

J. Li, X.-Y. Su, and L.-R. Guo, “Improved Fourier transform profilometry for the automatic measurement of the three-dimensional object shapes,” Opt. Eng. 29(12), 1439–1444 (1990).
[Crossref]

1989 (1)

1988 (3)

1985 (1)

K. Patorski and L. Salbut, “Optical differentiation of distorted gratings using Talbot and double diffraction interferometry,” Opt. Acta (Lond.) 32(11), 1323–1331 (1985).
[Crossref]

1976 (1)

1971 (2)

1969 (1)

N. E. Molin and K. A. Stetson, “Measuring combination mode vibration patterns by hologram interferometry,”J. Phys. E. 2(7), 609–612 (1969).
[Crossref]

Aleksa, A.

M. Ragulskis, A. Aleksa, and R. Maskeliunas, “Contrast enhancement of time-averaged fringes based on moving average mapping functions,” Opt. Lasers Eng. 47(7–8), 768–773 (2009).
[Crossref]

Amidror, I.

I. Amidror and R. D. Hersch, “The role of Fourier theory and of modulation in the prediction of visible moiré effects,” J. Mod. Opt. 56(9), 1103–1118 (2009).
[Crossref]

Apostol, D.

Bernini, M. B.

Bone, D. J.

Bosseboeuf, A.

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry techniques in the MEMS field,” Proc. SPIE 5145, 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. SPIE 4400, 51–60 (2001).
[Crossref]

Bruno, L.

Bryngdahl, O.

Chen, M.

Choi, E. S.

Choi, W. J.

Chu, G.

Q. Yuan, Z. Gao, Y. Zhou, G. Chu, and C. Zhang, “Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical aperture spherical surfaces,” Opt. Lasers Eng. 50(11), 1568–1574 (2012).
[Crossref]

Chung, M. S.

Damian, V.

Danaie, K.

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, “Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization,” Proc. SPIE 4400, 51–60 (2001).
[Crossref]

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. SPIE 5145, 23–32 (2003).
[Crossref]

Dobroiu, A.

Ellingsrud, S.

Eschbach, R.

Federico, A.

Freidhoff, C. B.

E. Hong, S. Trolier-McKinstry, R. Smith, S. V. Krishnaswamy, and C. B. Freidhoff, “Vibration of micromachined circular piezoelectric diaphragms,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53(4), 697–706 (2006).
[PubMed]

Gao, Z.

Q. Yuan, Z. Gao, Y. Zhou, G. Chu, and C. Zhang, “Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical aperture spherical surfaces,” Opt. Lasers Eng. 50(11), 1568–1574 (2012).
[Crossref]

Gorecki, C.

C. Gorecki, M. Jozwik, and L. Salbut, “Multifunctional interferometric platform for on-chip testing the micromechanical properties of MEMS/MOEMS,” J. Microlith., Microfab., Microsyst. 4(4), 041402 (2005).

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. SPIE 5145, 23–32 (2003).
[Crossref]

Guo, H.

Guo, L.-R.

J. Li, X.-Y. Su, and L.-R. Guo, “Improved Fourier transform profilometry for the automatic measurement of the three-dimensional object shapes,” Opt. Eng. 29(12), 1439–1444 (1990).
[Crossref]

Hersch, R. D.

I. Amidror and R. D. Hersch, “The role of Fourier theory and of modulation in the prediction of visible moiré effects,” J. Mod. Opt. 56(9), 1103–1118 (2009).
[Crossref]

Hong, E.

E. Hong, S. Trolier-McKinstry, R. Smith, S. V. Krishnaswamy, and C. B. Freidhoff, “Vibration of micromachined circular piezoelectric diaphragms,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53(4), 697–706 (2006).
[PubMed]

Hovanesian, J. D.

Hrebesh, M. S.

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

Hsu, J.

Hung, Y. Y.

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. SPIE 5145, 23–32 (2003).
[Crossref]

Jia, S.

Jozwik, M.

C. Gorecki, M. Jozwik, and L. Salbut, “Multifunctional interferometric platform for on-chip testing the micromechanical properties of MEMS/MOEMS,” J. Microlith., Microfab., Microsyst. 4(4), 041402 (2005).

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. SPIE 5145, 23–32 (2003).
[Crossref]

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. SPIE 5145, 23–32 (2003).
[Crossref]

Kaufmann, G. H.

Kim, J.

Kozak, S.

Krishnaswamy, S. V.

E. Hong, S. Trolier-McKinstry, R. Smith, S. V. Krishnaswamy, and C. B. Freidhoff, “Vibration of micromachined circular piezoelectric diaphragms,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53(4), 697–706 (2006).
[PubMed]

Larkin, K. G.

Lee, B. H.

Li, J.

J. Li, X.-Y. Su, and L.-R. Guo, “Improved Fourier transform profilometry for the automatic measurement of the three-dimensional object shapes,” Opt. Eng. 29(12), 1439–1444 (1990).
[Crossref]

Lim, J. S.

Lin, J. A.

Maskeliunas, R.

M. Ragulskis, L. Saunoriene, and R. Maskeliunas, “The structure of moiré grating lines and its influence to time-averaged fringes,” Exp. Tech. 33(2), 60–64 (2009).
[Crossref]

M. Ragulskis, A. Aleksa, and R. Maskeliunas, “Contrast enhancement of time-averaged fringes based on moving average mapping functions,” Opt. Lasers Eng. 47(7–8), 768–773 (2009).
[Crossref]

Miao, J. M.

M. Olfatnia, V. R. Singh, T. Xu, J. M. Miao, and L. S. Ong, “Analysis of the vibration modes of piezoelectric circular microdiaphragms,” J. Micromech. Microeng. 20(8), 085013 (2010).
[Crossref]

Molin, N. E.

N. E. Molin and K. A. Stetson, “Measuring combination mode vibration patterns by hologram interferometry,”J. Phys. E. 2(7), 609–612 (1969).
[Crossref]

Na, J.

Nascov, V.

Navickas, Z.

M. Ragulskis and Z. Navickas, “Interpretation of fringes produced by time-averaged projection moiré,” Strain 45, (2009), doi: (and references therein).
[Crossref]

Oldfield, M. A.

Olfatnia, M.

M. Olfatnia, V. R. Singh, T. Xu, J. M. Miao, and L. S. Ong, “Analysis of the vibration modes of piezoelectric circular microdiaphragms,” J. Micromech. Microeng. 20(8), 085013 (2010).
[Crossref]

Ong, L. S.

M. Olfatnia, V. R. Singh, T. Xu, J. M. Miao, and L. S. Ong, “Analysis of the vibration modes of piezoelectric circular microdiaphragms,” J. Micromech. Microeng. 20(8), 085013 (2010).
[Crossref]

Patorski, K.

M. Trusiak, K. Patorski, and M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express 20(21), 23463–23479 (2012).
[Crossref] [PubMed]

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(28), 5513–5523 (2011).
[Crossref] [PubMed]

K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express 19(27), 26065–26078 (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(19), 3640–3651 (2010).
[Crossref] [PubMed]

A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity modulation determination,” Appl. Opt. 46(21), 4613–4624 (2007).
[Crossref] [PubMed]

K. Patorski, A. Styk, L. Bruno, and P. Szwaykowski, “Tilt-shift error detection in phase-shifting interferometry,” Opt. Express 14(12), 5232–5249 (2006).
[Crossref] [PubMed]

K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng. 45(8), 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. SPIE 5145, 23–32 (2003).
[Crossref]

K. Patorski, “Moire methods in interferometry,” Opt. Lasers Eng. 8(3-4), 147–170 (1988).
[Crossref]

K. Patorski and S. Kozak, “Self-imaging with nonparabolic approximation of spherical wave fronts,” J. Opt. Soc. Am. A 5(8), 1322–1327 (1988).
[Crossref]

K. Patorski and L. Salbut, “Optical differentiation of distorted gratings using Talbot and double diffraction interferometry,” Opt. Acta (Lond.) 32(11), 1323–1331 (1985).
[Crossref]

M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. (to be published).

Petitgrand, S.

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry techniques in the MEMS field,” Proc. SPIE 5145, 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. SPIE 4400, 51–60 (2001).
[Crossref]

Pokorski, K.

Pryputniewicz, R. J.

R. J. Pryputniewicz, “A hybrid approach to deformation analysis,” Proc. SPIE 2342, 282–296 (1994).
[Crossref]

R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibration patterns using electro-optic holography,” Proc. SPIE 1162, 456–467 (1990).
[Crossref]

Quiroga, J. A.

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4–6), 221–227 (2003).
[Crossref]

Ragulskis, M.

M. Ragulskis, A. Aleksa, and R. Maskeliunas, “Contrast enhancement of time-averaged fringes based on moving average mapping functions,” Opt. Lasers Eng. 47(7–8), 768–773 (2009).
[Crossref]

M. Ragulskis, L. Saunoriene, and R. Maskeliunas, “The structure of moiré grating lines and its influence to time-averaged fringes,” Exp. Tech. 33(2), 60–64 (2009).
[Crossref]

M. Ragulskis and Z. Navickas, “Interpretation of fringes produced by time-averaged projection moiré,” Strain 45, (2009), doi: (and references therein).
[Crossref]

L. Saunoriene and M. Ragulskis, “Visualization of fringes in time averaged moiré patterns,” Inf. Technol. Control 35(3), 249–254 (2006).

Rosvold, G. O.

Ryu, S. Y.

Salbut, L.

C. Gorecki, M. Jozwik, and L. Salbut, “Multifunctional interferometric platform for on-chip testing the micromechanical properties of MEMS/MOEMS,” J. Microlith., Microfab., Microsyst. 4(4), 041402 (2005).

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. SPIE 5145, 23–32 (2003).
[Crossref]

K. Patorski and L. Salbut, “Optical differentiation of distorted gratings using Talbot and double diffraction interferometry,” Opt. Acta (Lond.) 32(11), 1323–1331 (1985).
[Crossref]

Saunoriene, L.

M. Ragulskis, L. Saunoriene, and R. Maskeliunas, “The structure of moiré grating lines and its influence to time-averaged fringes,” Exp. Tech. 33(2), 60–64 (2009).
[Crossref]

L. Saunoriene and M. Ragulskis, “Visualization of fringes in time averaged moiré patterns,” Inf. Technol. Control 35(3), 249–254 (2006).

Servin, M.

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4–6), 221–227 (2003).
[Crossref]

Shiue, S. G.

Singh, V. R.

M. Olfatnia, V. R. Singh, T. Xu, J. M. Miao, and L. S. Ong, “Analysis of the vibration modes of piezoelectric circular microdiaphragms,” J. Micromech. Microeng. 20(8), 085013 (2010).
[Crossref]

Smith, R.

E. Hong, S. Trolier-McKinstry, R. Smith, S. V. Krishnaswamy, and C. B. Freidhoff, “Vibration of micromachined circular piezoelectric diaphragms,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53(4), 697–706 (2006).
[PubMed]

Stetson, K. A.

R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibration patterns using electro-optic holography,” Proc. SPIE 1162, 456–467 (1990).
[Crossref]

N. E. Molin and K. A. Stetson, “Measuring combination mode vibration patterns by hologram interferometry,”J. Phys. E. 2(7), 609–612 (1969).
[Crossref]

Styk, A.

Su, X.-Y.

J. Li, X.-Y. Su, and L.-R. Guo, “Improved Fourier transform profilometry for the automatic measurement of the three-dimensional object shapes,” Opt. Eng. 29(12), 1439–1444 (1990).
[Crossref]

Suzuki, T.

Szwaykowski, P.

Trolier-McKinstry, S.

E. Hong, S. Trolier-McKinstry, R. Smith, S. V. Krishnaswamy, and C. B. Freidhoff, “Vibration of micromachined circular piezoelectric diaphragms,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53(4), 697–706 (2006).
[PubMed]

Trusiak, M.

Wei, C.

Wielgus, M.

Xiong, L.

Xu, T.

M. Olfatnia, V. R. Singh, T. Xu, J. M. Miao, and L. S. Ong, “Analysis of the vibration modes of piezoelectric circular microdiaphragms,” J. Micromech. Microeng. 20(8), 085013 (2010).
[Crossref]

Yahiaoui, R.

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, “Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization,” Proc. SPIE 4400, 51–60 (2001).
[Crossref]

Yokozeki, S.

Yuan, Q.

Q. Yuan, Z. Gao, Y. Zhou, G. Chu, and C. Zhang, “Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical aperture spherical surfaces,” Opt. Lasers Eng. 50(11), 1568–1574 (2012).
[Crossref]

Zhang, C.

Q. Yuan, Z. Gao, Y. Zhou, G. Chu, and C. Zhang, “Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical aperture spherical surfaces,” Opt. Lasers Eng. 50(11), 1568–1574 (2012).
[Crossref]

Zhou, Y.

Q. Yuan, Z. Gao, Y. Zhou, G. Chu, and C. Zhang, “Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical aperture spherical surfaces,” Opt. Lasers Eng. 50(11), 1568–1574 (2012).
[Crossref]

Adv. Opt. Technol. (1)

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

Appl. Opt. (12)

J. S. Lim, J. Kim, and M. S. Chung, “Additive type moire with computer image processing,” Appl. Opt. 28(13), 2677–2680 (1989).
[Crossref] [PubMed]

J. A. Lin, J. Hsu, and S. G. Shiue, “Quantitative three-beam Ronchi test,” Appl. Opt. 29(13), 1912–1918 (1990).
[Crossref] [PubMed]

G. O. Rosvold, “Video-based vibration analysis using projected fringes,” Appl. Opt. 33(5), 775–786 (1994).
[Crossref] [PubMed]

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
[Crossref] [PubMed]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
[Crossref] [PubMed]

S. Yokozeki and T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt. 10(7), 1575–1580 (1971).
[Crossref] [PubMed]

J. D. Hovanesian and Y. Y. Hung, “Moiré contour-sum, contour-difference, and vibration analysis of arbitrary objects,” Appl. Opt. 10(12), 2734–2738 (1971).
[Crossref] [PubMed]

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Normalization of fringe patterns using the bidimensional empirical mode decomposition and the Hilbert transform,” Appl. Opt. 48(36), 6862–6869 (2009).
[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(19), 3640–3651 (2010).
[Crossref] [PubMed]

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(28), 5513–5523 (2011).
[Crossref] [PubMed]

A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity modulation determination,” Appl. Opt. 46(21), 4613–4624 (2007).
[Crossref] [PubMed]

J. Na, W. J. Choi, E. S. Choi, S. Y. Ryu, and B. H. Lee, “Image restoration method based on Hilbert transform for full-field optical coherence tomography,” Appl. Opt. 47(3), 459–466 (2008).
[Crossref] [PubMed]

Exp. Tech. (1)

M. Ragulskis, L. Saunoriene, and R. Maskeliunas, “The structure of moiré grating lines and its influence to time-averaged fringes,” Exp. Tech. 33(2), 60–64 (2009).
[Crossref]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

E. Hong, S. Trolier-McKinstry, R. Smith, S. V. Krishnaswamy, and C. B. Freidhoff, “Vibration of micromachined circular piezoelectric diaphragms,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53(4), 697–706 (2006).
[PubMed]

Inf. Technol. Control (1)

L. Saunoriene and M. Ragulskis, “Visualization of fringes in time averaged moiré patterns,” Inf. Technol. Control 35(3), 249–254 (2006).

J. Microlith., Microfab., Microsyst. (1)

C. Gorecki, M. Jozwik, and L. Salbut, “Multifunctional interferometric platform for on-chip testing the micromechanical properties of MEMS/MOEMS,” J. Microlith., Microfab., Microsyst. 4(4), 041402 (2005).

J. Micromech. Microeng. (1)

M. Olfatnia, V. R. Singh, T. Xu, J. M. Miao, and L. S. Ong, “Analysis of the vibration modes of piezoelectric circular microdiaphragms,” J. Micromech. Microeng. 20(8), 085013 (2010).
[Crossref]

J. Mod. Opt. (1)

I. Amidror and R. D. Hersch, “The role of Fourier theory and of modulation in the prediction of visible moiré effects,” J. Mod. Opt. 56(9), 1103–1118 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

J. Phys. E. (1)

N. E. Molin and K. A. Stetson, “Measuring combination mode vibration patterns by hologram interferometry,”J. Phys. E. 2(7), 609–612 (1969).
[Crossref]

Opt. Acta (Lond.) (1)

K. Patorski and L. Salbut, “Optical differentiation of distorted gratings using Talbot and double diffraction interferometry,” Opt. Acta (Lond.) 32(11), 1323–1331 (1985).
[Crossref]

Opt. Commun. (1)

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4–6), 221–227 (2003).
[Crossref]

Opt. Eng. (2)

J. Li, X.-Y. Su, and L.-R. Guo, “Improved Fourier transform profilometry for the automatic measurement of the three-dimensional object shapes,” Opt. Eng. 29(12), 1439–1444 (1990).
[Crossref]

K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng. 45(8), 085602 (2006).
[Crossref]

Opt. Express (3)

Opt. Lasers Eng. (4)

M. Ragulskis, A. Aleksa, and R. Maskeliunas, “Contrast enhancement of time-averaged fringes based on moving average mapping functions,” Opt. Lasers Eng. 47(7–8), 768–773 (2009).
[Crossref]

M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. (to be published).

K. Patorski, “Moire methods in interferometry,” Opt. Lasers Eng. 8(3-4), 147–170 (1988).
[Crossref]

Q. Yuan, Z. Gao, Y. Zhou, G. Chu, and C. Zhang, “Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical aperture spherical surfaces,” Opt. Lasers Eng. 50(11), 1568–1574 (2012).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (5)

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, “Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization,” Proc. SPIE 4400, 51–60 (2001).
[Crossref]

R. J. Pryputniewicz and K. A. Stetson, “Measurement of vibration patterns using electro-optic holography,” Proc. SPIE 1162, 456–467 (1990).
[Crossref]

R. J. Pryputniewicz, “A hybrid approach to deformation analysis,” Proc. SPIE 2342, 282–296 (1994).
[Crossref]

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry techniques in the MEMS field,” Proc. SPIE 5145, 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. SPIE 5145, 23–32 (2003).
[Crossref]

Strain (1)

M. Ragulskis and Z. Navickas, “Interpretation of fringes produced by time-averaged projection moiré,” Strain 45, (2009), doi: (and references therein).
[Crossref]

Other (7)

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

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

W. Osten, Digital Processing and Evaluation of Interference Images (Akademie Verlag, 1991).

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “A novel approach of fast and adaptive bidimensional empirical mode decomposition,” in Proceedings of IEEE International Conference on Acoustic, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, 2008), pp. 1313–1316.
[Crossref]

M. Trusiak and K. Patorski, “Space domain interpretation of incoherent moiré superimpositions using FABEMD,” Proc. SPIE 8697, 18th Czech-Polish-Slovak Optical Conference on Wave and Quantum Aspects of Contemporary Optics, 869704 (December 18, 2012).
[Crossref]

K. A. Stetson, “Holographic vibration analysis,” in Holographic Nondestructive Testing, R.E. Erf ed. (Academic Press, 1984).

K. Patorski, “The self-imaging phenomenon and applications,” in Progress in Optics, E. Wolf, ed., vol. 27, 1–103 (North Holland, 1989).

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Figures (18)

Fig. 1
Fig. 1 Exemplary time-averaged interferogram for a circular active silicon membrane [5,9,10] vibrating at resonance frequency 659 kHz (a), its magnified local region (b), and the result of subtracting two time-averaged frames with mutual carrier phase shift by π (c) for the same local region as shown in (b). Note significant improvement of the pattern quality.

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Fig. 2
Fig. 2 Flow chart of the proposed algorithm.

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Fig. 3
Fig. 3 Results of processing simulated cosinusoidal fringe carrier cos(x) with amplitude modulation by the Bessel function J0(x,y): (a) frame1, (b) dif, (c) modu1, (d) norm1, (e) modu2, (f) norm2, (g) dif_m, (h) dif_n; please see text for symbol explanations. Noise free, DC = 0, bias = 0.025 (i.e., in the double normalization process the DC difference between the frames has been increased by 2.5% of the amplitude value).

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Fig. 4
Fig. 4 Horizontal cross section (256th pixel row) of the contour pattern of centers of dark Bessel fringes, Fig. 3(h).

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Fig. 5
Fig. 5 (a) dif_m and (b) dif_n in the case of Fig. 3 but with noise SNR = 2. Root mean square (RMS) value computed for dif_n with Fig. 3(h) as a reference is equal to 0.21.

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Fig. 6
Fig. 6 Calculated algorithm output results: (a) dif_m and (b) dif_n for nonuniform DC level change between frame1 and frame2 described by (c) DC = sin(x); (d) background intensity distribution extracted from dif using the FABEMD algorithm, (e) dif_m and (f) dif_n computed after background (residue) removal from dif. Note that extracted DC term is of opposite sign to the one simulated and added to frame2, Fig. 6(c), as it was subtracted generating the dif pattern.

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Fig. 7
Fig. 7 Images dif_n in the case of various data recording nonlinearities: (a) linear detection, (b) b = 0.5, c = d = e = 0, (c) b = 1.5, c = d = e = 0, (d) b = c = 1, c = d = e = 0, (e) b = c = d = 1, e = 0 (f) case with 5th order nonlinearities with all coefficients equal to 1. Corresponding RMS values are equal to 0.02, 0.08, 0.15, 0.2 and 0.27, respectively.

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Fig. 8
Fig. 8 The results of processing two time-averaged frames with a binary amplitude carrier (mutual phase shift of π) leading to Bessel fringe contouring. (a) frame1, (b) dif, (c) modu1, (d) norm1, (e) dif_m, (f) dif_n. Note considerable departures of modulation distribution modu1 from previously presented one, Fig. 3(c).

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Fig. 9
Fig. 9 Results of subsequent computation stages and final contours in the case of the tilt-shift error producing 1% change of the spatial period of second frame. Figures 9(a)-9(f) show, correspondingly: frame1, dif, modu2, norm2, dif_m and dif_n (RMS = 0.19).

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Fig. 10
Fig. 10 Results of subsequent computation stages and final contours in the case of single-frame processing of experimental data. Figures 10(a)-10(d) show, correspondingly: frame1 with nonuniform background intensity, modu1 calculated for frame1 with bias removed using the FABEMD algorithm, dif_m and dif_n.

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Fig. 11
Fig. 11 Set of four π/2 phase shifted experimental time-averaged frames (frame2-frame5). First frame (frame1, reference pattern with phase shift equal to 0) is shown in Fig. 10(a).

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Fig. 12
Fig. 12 (a) Result dif(1,3) of subtracting π phase shifted frame1 and frame3 (Fig. 10(a) and Fig. 11(b)), (b) modulation distribution modu(1,3) of dif(1,3), (c) difference between modulation distributions dif_m(1,3) calculated with slightly different DC (bias = 0.025), and (d) difference between two normalized fringe patterns, denoted as dif_n(1,3).

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Fig. 13
Fig. 13 (a) Result dif(2,4) of subtracting π phase shifted frame2 and frame4 (Figs. 11(a) and 11(c)), and its subsequent computation stages (b) modu(2,4), (c) dif_m(2,4) and (d) final contours dif_n(2,4).

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Fig. 14
Fig. 14 (a) Result dif(3,5) of subtracting π phase shifted frame3 and frame5 (Figs. 11(b) and 11(d)), and its subsequent computation stages (b) modu(3,5), (c) dif_m(3,5) and (d) final contours dif_n(3,5).

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Fig. 15
Fig. 15 (a) High-pass filtered pattern dif(1,3) using the FABEMD algorithm with residue excluded for the background illumination error correction, (b) modulation distribution of corrected fringe pattern, (c) dif_m and (d) final contours dif_n.

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Fig. 16
Fig. 16 (a) Time-averaged frame for a circular silicon micro-membrane vibrating at 930 kHz and the results of two frame Hilbert transform processing: (b) modulation distribution and (c) dark fringe center contour lines.

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Fig. 17
Fig. 17 (a) Time-averaged frame for a circular silicon micro-membrane vibrating at 365 kHz and the results of two frame Hilbert transform processing: (b) modulation distribution and (c) dark fringe center contour lines.

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Fig. 18
Fig. 18 (a) Time-averaged frame of a square silicon micro-membrane vibrating at 172 kHz and the results of two frame Hilbert transform processing: (b) modulation distribution and (c) dark fringe center contour lines.

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Equations (3)

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

I vib (x,y)=K(x,y){1+ C stat (x,y) J 0 [(4π/λ) a 0 (x,y)]cos φ vib (x,y)},
I 2frame (x,y)=2K(x,y) C stat (x,y) J 0 [(4π/λ) a 0 (x,y)]cos φ vib (x,y),
frame1_nl_rec=frame1+b (frame1) 2 +c (frame1) 3 +d (frame1) 4 +e (frame1) 5 ; frame2_nl_rec=frame2+b (frame2) 2 +c (frame2) 3 +d (frame2) 4 +e (frame2) 5 ;

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