Abstract

The objective of this paper is to describe an integral approach -based on the use of a super-resolution frequency estimation method - to phase shifting interferometry, starting from phase step estimation to phase evaluation at each point on the object surface. Denoising is also taken into consideration for the case of a signal contaminated with white Gaussian noise. The other significant features of the proposal are that it caters to the presence of multiple PZTs in an optical configuration, is capable of determining the harmonic content in the signal and effectively eliminating their influence on measurement, is insensitive to errors arising from PZT miscalibration, is applicable to spherical beams, and is a robust performer even in the presence of white Gaussian intensity noise.

© 2004 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  52. M. Dendrinos, S. Bakamidis, and G. Carayannis, �??Speech enhancement from noise: A regenerative approach,�?? Speech Communication 10, 45-57 (1991).
    [CrossRef]
  53. P. C. Hansen and S. H. Jensen, �??FIR filter representations of reduced-rank noise reduction,�?? IEEE Transactions on Signal Processing 46, 1737-1741 (1998).
    [CrossRef]
  54. P. K. Rastogi and E. Denarié, �??Visualization of in-plane displacement fields using phase shifting holographic moiré: application to crack detection and propagation,�?? Appl. Opt. 31, 2402-2404 (1992).
    [CrossRef] [PubMed]
  55. P. K. Rastogi, M. Barillot, and G. Kaufmann, �??Comparative phase shifting holographic interferometry,�?? Appl. Opt. 30, 722-728 (1991).
    [CrossRef] [PubMed]
  56. P. K. Rastogi, �??Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,�?? J. Mod. Opt. 38, 1251-1263 (1991).
    [CrossRef]
  57. R. Kumaresan and D. W. Tufts, �??Estimating the angles of arrival of multiple plane waves,�?? IEEE Transactions on Aerospace and Electronic Systems AES-19, 134-139 (1983).
    [CrossRef]
  58. K. B. Hill, S. A. Basinger, R. A. Stack, and D. J. Brady, �??Noise and information in interferometric cross correlators,�?? Appl. Opt. 36, 3948-3958 (1997).
    [CrossRef] [PubMed]

App. Mathematics & Computation (1)

B. Raphael and I. F. C. Smith, �??A direct stochastic algorithm for global search,�?? App. Mathematics & Computation 146/2-3, 729-758 (2003).
[CrossRef]

App. Opt. (7)

P. K. Rastogi, �??Phase shifting applied to four-wave holographic interferometers,�?? App. Opt. 31, 1680-1681 (1992).
[CrossRef]

P. K. Rastogi, �??Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,�?? App. Opt. 32, 3669-3675 (1993).
[CrossRef]

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, �??Digital wavefront measuring interferometer for testing optical surfaces and lenses, App. Opt. 13, 2693-2703 (1974).
[CrossRef]

Y. Zhu and T. Gemma, �??Method for designing error-compensating phase-calculation algorithms for phase-shifting interferometry,�?? App. Opt. 40, 4540-4546 (2001).
[CrossRef]

P. Hariharan, B. F. Oreb, and T. Eiju, �??Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,�?? App. Opt. 26, 2504-2506 (1987).
[CrossRef]

Y. �??Y. Cheng and J. C. Wyant, �??Phase-shifter calibration in phase-shifting interferometry,�?? App. Opt. 24, 3049-3052 (1985).
[CrossRef]

P. de Groot and L. L. deck, �??Numerical simulations of vibration in phase-shifting interferometry,�?? App. Opt. 35, 2172-2178 (1996).
[CrossRef]

Appl. Opt. (14)

P. L. Wizinowich, �??Phase-shifting interferometry in the presence of vibration: a new algorithm and system,�?? Appl. Opt. 29, 3271-3279 (1990).
[CrossRef] [PubMed]

K. Kinnnstaetter, A. W. Lohmann, J Schwider, and N. Streibl, �??Accuracy of phase shifting interferometry,�?? Appl. Opt. 27, 5082-5089 (1988).
[CrossRef]

Ch. Ai and J. C. Wyant, �??Effect of spurious reflection on phase shift interferometry,�?? Appl. Opt. 27, 3039-3045 (1988).
[CrossRef] [PubMed]

J. Schmit and K. Creath, �??Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,�?? Appl. Opt. 34, 3610-3619 (1995).
[CrossRef] [PubMed]

B. Zhao and Y. Surrel, �??Effect of quantization error on the computed phase of phase-shifting measurements,�?? Appl. Opt. 36, 2070-2075 (1997).
[CrossRef] [PubMed]

Y. Surrel, �??Phase stepping: a new self-calibrating algorithm,�?? Appl. Opt. 32, 3598-3600 (1993).
[CrossRef] [PubMed]

Y. Surrel, �??Design of algorithms for phase measurements by the use of phase stepping,�?? Appl. Opt. 35, 51-60 (1996).
[CrossRef] [PubMed]

J. van Wingerden, H. J. Frankena, and C. Smorenburg, �??Linear approximation for measurement errors in phase shifting interferometry,�?? Appl. Opt. 30, 2718-2729 (1991).
[CrossRef] [PubMed]

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, �??Digital wave-front measuring interferometry: some systematic error sources,�?? Appl. Opt. 22, 3421-3432 (1983).
[CrossRef] [PubMed]

P. K. Rastogi and E. Denarié, �??Visualization of in-plane displacement fields using phase shifting holographic moiré: application to crack detection and propagation,�?? Appl. Opt. 31, 2402-2404 (1992).
[CrossRef] [PubMed]

P. K. Rastogi, M. Barillot, and G. Kaufmann, �??Comparative phase shifting holographic interferometry,�?? Appl. Opt. 30, 722-728 (1991).
[CrossRef] [PubMed]

X Chen, M. Gramaglia, and J. A. Yeazell, �??Phase-shifting interferometry with uncalibrated phase shifts,�?? Appl. Opt. 39, 585-591 (2000).
[CrossRef]

G. �??S. Han and S. �??W. Kim, �??Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,�?? Appl. Opt. 33, 7321-7325 (1994).
[CrossRef] [PubMed]

K. B. Hill, S. A. Basinger, R. A. Stack, and D. J. Brady, �??Noise and information in interferometric cross correlators,�?? Appl. Opt. 36, 3948-3958 (1997).
[CrossRef] [PubMed]

Com. Graphics and Image Process. (1)

F. L. Bookstein, �??Fitting conic sections to scattered data,�?? Com. Graphics and Image Process. 9, 56-71 (1979).
[CrossRef]

IEEE Trans. on Acoustics, Speech, & Sign (1)

J. J. Fuchs, �??Estimating the number of sinusoids in additive white noise,�?? IEEE Transactions on Acoustics, Speech, and Signal Processing 36, 1846-1853 (1988).
[CrossRef]

IEEE Trans. on Aerospace and Elec. Syst. (1)

R. Kumaresan and D. W. Tufts, �??Estimating the angles of arrival of multiple plane waves,�?? IEEE Transactions on Aerospace and Electronic Systems AES-19, 134-139 (1983).
[CrossRef]

IEEE Transactions on Signal Processing (1)

P. C. Hansen and S. H. Jensen, �??FIR filter representations of reduced-rank noise reduction,�?? IEEE Transactions on Signal Processing 46, 1737-1741 (1998).
[CrossRef]

J. Mod. Opt. (2)

P. K. Rastogi, �??Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,�?? J. Mod. Opt. 38, 1251-1263 (1991).
[CrossRef]

C. Joenathan and B. M. Khorana, �??Phase measurement by differentiating interferometric fringes, �??J. Mod. Opt. 39, 2075-2087 (1992).
[CrossRef]

J. Opt. Soc. Am. A (7)

Meas. Sci. Technol. (5)

B. Zhao, �??A statistical method for fringe intensity-correlated error in phase-shifting measurement: the effect of quantization error on the N-bucket algorithm,�?? Meas. Sci. Technol. 8, 147-153 (1997).
[CrossRef]

Q. Kemao, S. Fangjun, and W. Xiaoping, �??Determination of the best phase step of the Carré algorithm in phase shifting interferometry,�?? Meas. Sci. Technol. 11, 1220-1223 (2000).
[CrossRef]

B. V. Dorrío and J. L. Fernández, �??Phase-evaluation methods in whole-field optical measurement techniques,�?? Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

C. T. Farrell and M. A. Player, �??Phase step measurement and variable step algorithms in phase-shifting interferometry,�?? Meas. Sci. Technol. 3, 953-958 (1992).
[CrossRef]

C. T. Farrell and M. A. Player, �??Phase-step insensitive algorithms for phase-shifting interferometry,�?? Meas. Sci. Technol. 5, 648-652 (1994).
[CrossRef]

Metrologia (1)

P. Carré, �??Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,�?? Metrologia 2, 13-23 (1966).
[CrossRef]

Opt. and Lasers Eng. (1)

G. T. Reid, �??Automatic fringe pattern analysis: a review,�?? Opt. and Lasers Eng. 7, 37-68 (1986).
[CrossRef]

Opt. and Lasers in Eng. (1)

G. Stoilov and T. Dragostinov, �??Phase-stepping interferometry: five-frame algorithm with an arbitrary step,�?? Opt. and Lasers in Eng. 28, 61-69 (1997).
[CrossRef]

Opt. Eng. (3)

J. E. Grievenkamp, �??Generalized data reduction for heterodyne interferometry,�?? Opt. Eng. 23, 350-352 (1984).

J. Schwider, O. Falkenstorfer, H. Schreiber, and A. Zoller, �??New compensating four-phase algorithm for phase-shift interferometry,�?? Opt. Eng. 32, 1883-1885 (1993).
[CrossRef]

R. Józwicki, M. Kujawinska, and M. Salbut, �??New contra old wavefront measurement concepts for interferometric optical testing, Opt. Eng. 31, 422-433 (1992).
[CrossRef]

Opt. Exp. (1)

K. G. Larkin, �??A self-calibrating phase-shifting algorithm based on the natural demodulation of twodimensional fringe patterns,�?? Opt. Exp. 9, 236-253 (2001).
[CrossRef]

Opt. Let. (1)

L. Z. Cai, Q. Liu, and X. L. Yang, �??Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,�?? Opt. Let. 28, 1808-1810 (2003).
[CrossRef]

Opt. Lett. (3)

Pattern Anal. and Machine Intelligence. (1)

A. Fitzgibbon, M. Pilu, and R. B. Fisher, �??Direct least square fitting of ellipses,�?? Pattern Anal. and Machine Intelligence. 21, 474-480 (1999).

Proc. SPIE (1)

W. Jüptner, T. Kreis, and H. Kreitlow, �??Automatic evaluation of holographic interferograms by reference beam phase shifting,�?? In W. F. Fagan, ed., Industrial Applications of Laser Technology, Proc. Of Soc. Photo- Opt. Instr. Eng., 1553, 569-582 (1991).

Speech Communication (1)

M. Dendrinos, S. Bakamidis, and G. Carayannis, �??Speech enhancement from noise: A regenerative approach,�?? Speech Communication 10, 45-57 (1991).
[CrossRef]

Springer Series in Optical Sciences (1)

K. Creath, �??Phase-shifting holographic interferometry,�?? Holographic Interferometry, P. K. Rastogi, ed. (Springer Series in Optical Sciences, Berlin 1994), Vol.68, pp. 109-150.

Other (3)

T. Kreis, �??Holographic interferometry Principles and Methods,�?? Akademie Verlag, 1996, pp. 101-170.

J. E. Greivenkamp and J. H. Bruning, Phase shifting interferometry Optical Shop Testing ed D. Malacara (New York: Wiley ) 501-598 (1992).

P. Stoica and R. Moses, Introduction to Spectral Analysis (Prentice Hall, New Jersey, 1997).

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Figures (7)

Fig. 1.
Fig. 1.

Plot showing the magnitude of diagonal values in matrix S versus N data points. From the plot the number of harmonics κ in the signal can be computed using M = 2 Hκ+1. In figure (o) and (+) represent the diagonal values for noiseless and noisy (SNR=10 dB) signals, respectively.

Fig. 2.
Fig. 2.

Holographic moiré for κ = 1 and for noise levels (a) 0 dB (b) 10 dB , and (c) 60 dB additive white Gaussian noise.

Fig. 3.
Fig. 3.

Holographic moiré for κ=2 and for noise levels (a) 0 dB (b) 10 dB , and (c) 60 dB additive white Gaussian noise.

Fig. 4.
Fig. 4.

Plot of phase steps α and β versus noise when computation is done over (a) eighteen (b) twenty seven, and (c) thirty six frames. Plots in. (b) and (c) are obtained after applying the denoising procedure. Large number of frames are due to the presence of two PZTs (H = 2) and two harmonics κ = 2, which in turn impose lower limit on data samples (4Hκ+ 2) as eighteen for phase step estimation.

Fig. 5.
Fig. 5.

Plot of error in the computation of phase φ 1 when SNR = 30 dB obtained a) without and b) with the application of the denoising procedure.

Fig. 6.
Fig. 6.

Plot of error in the computation of phase φ 2 when SNR = 30 dB obtained a) without and b) with the application of the denoising procedure.

Fig. 7.
Fig. 7.

Wrapped phase distributions φ 1 (solid line) and φ 2 (broken line) as functions of pixel position when SNR = 30 dB. Denoising procedure has been applied for obtaining these results.

Equations (38)

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

I ( x , y ; m ) = I dc + k = 1 κ a k exp [ ik ( φ 1 + ) ] + k = 1 κ a k exp [ ik ( φ 1 + ) ] +
k = 1 κ b k exp [ ik ( φ 2 + ) ] + k = 1 κ b k exp [ ik ( φ 2 + ) ] ; for m = 1,2 , …… , N
Φ = φ 1 φ 2
Ψ = φ 1 + φ 2
I n ( x , y ; m ) = I dc + k = 1 κ k u k m + k = 1 κ k * ( u k * ) m + k = 1 κ k v k m + k = 1 κ k * ( v k * ) m ,
for m = n + 1 = 1,2 , …… , N
I ( z ) = n = 0 N 1 I n z n = n = 0 N 1 I dc z n + n = 0 N 1 k = 1 κ k u k ( n + 1 ) z n + n = 0 N 1 k = 1 κ k * ( u k * ) ( n + 1 ) z n +
n = 0 N 1 k = 1 κ k v k ( n + 1 ) z n + n = 0 N 1 k = 1 κ k * ( v k * ) ( n + 1 ) z n
I ( z ) = I dc ( 1 z N 1 z 1 ) + k = 1 κ k e iαk ( 1 e iαkN z N ) [ D 1 ( z ) P 1 ( z ) ] +
k = 1 κ k * e iαk ( 1 e −iαkN z N ) [ D 2 ( z ) P 2 ( z ) ] +
k = 1 κ k e iβk ( 1 e iβkN z N ) [ D 3 ( z ) P 3 ( z ) ] +
k = 1 κ k * e iβk ( 1 e iβkN z N ) [ D 4 ( z ) P 4 ( z ) ]
D 1 ( z ) = j = 1 , j k κ ( 1 e iαj z 1 ) D 2 ( z ) = j = 1 , j k κ ( 1 e iαj z 1 ) , , = j = 0 κ 1 d 1 j z j = j = 0 κ 1 d 2 j z j D 3 ( z ) = j = 1 , j k κ ( 1 e iβj z 1 ) D 4 ( z ) = j = 1 , j k κ ( 1 e iβj z 1 ) , , = j = 0 κ 1 d 3 j z j = j = 0 κ 1 d 4 j z j
P 1 ( z ) = k = 1 κ ( 1 e iαk z 1 ) , P 2 ( z ) = k = 1 κ ( 1 e iαk z 1 ) , P 3 ( z ) = k = 1 κ ( 1 e iβk z 1 )
and P 4 ( z ) = k = 1 κ ( 1 e iβk z 1 ) .
I ( z ) = I dc ( 1 z N 1 z 1 ) + { k = 1 κ k e iαk [ j = 0 κ 1 d 1 j z j j = 0 κ 1 d 1 j e iαkN z ( j + N ) ] k = 1 κ ( 1 e iαk z 1 ) } +
{ k = 1 κ k * e iαk [ j = 0 κ 1 d 2 j z j j = 0 κ 1 d 2 j e iαkN z ( j + N ) ] k = 1 κ ( 1 e iαk z 1 ) } +
{ k = 1 κ k e iβk [ j = 0 κ 1 d 3 j z j j = 0 κ 1 d 3 j e iβkN z ( j + N ) ] k = 1 κ ( 1 e iβk z 1 ) } +
{ k = 1 κ k * e iβk [ j = 0 κ 1 d 4 j z j j = 0 κ 1 d 4 j e iβkN z ( j + N ) ] k = 1 κ ( 1 e iβk z 1 ) }
I ( z ) P ( z ) = D ( z )
P ( z ) = ( 1 z 1 ) k = 1 4 κ ( 1 e iαk z 1 ) ( 1 e iαk z 1 ) ( 1 e iβk z 1 ) ( 1 e iβk z 1 )
= k = 0 4 κ + 1 p k z k
C 1 ( z ) = k = 1 κ I dc ( 1 z N ) P 1 ( z ) P 2 ( z ) P 3 ( z ) P 4 ( z )
= j = 0 4 κ C 1 j z j j = 0 4 κ C 1 j + N z ( j + N )
I n p n = D n
p n = p 0 δ ( n ) + p 1 δ ( n 1 ) + p 2 δ ( n 2 ) + ……. + p 4 κ + 1 δ ( n 4 κ 1 )
k = 0 4 κ + 1 I n k p k = D n ; for n = 0,1 , …… , N , ….. , N + 4 κ
I n ( n + 1 ) = { 0 I n ( n + 1 ) 0 for n < 0 0 n N 1 , n N
and δ ( n ) = { 0 1 for n = 0 else where
[ I 0 ( 1 ) 0 0 . . 0 I 1 ( 2 ) I 0 ( 1 ) 0 . . 0 . . . . . . . . . . . . -------------- -------------- -------------- -------------- -------------- -------------- I 4 κ + 1 ( 4 κ + 2 ) I 4 κ ( 4 κ + 1 ) . . . I 0 ( 1 ) I 4 κ + 2 ( 4 κ + 3 ) I 4 κ + 1 ( 4 κ + 2 ) . . . I 1 ( 2 ) . . . . . . . . . . . . I N 2 ( N 1 ) I N 3 ( N 2 ) . . . I N 4 κ 3 ( N 4 κ 2 ) -------------- -------------- -------------- -------------- -------------- -------------- I N 1 ( N ) I N 2 ( N 1 ) . . . I N 4 κ 2 ( N 4 κ 1 ) 0 I N 1 ( N ) I N 2 ( N 1 ) . . . 0 0 . . . . 0 . . . . . 0 . . . I N 1 ( N ) I N 2 ( N 1 ) 0 0 0 0 0 I N 1 ( N ) ] [ p 0 p 1 p 2 . . . p 4 κ + 1 ] = [ D 0 D 1 . D 4 κ ------ 0 0 . . 0 ------ D N . . . . D N + 4 κ ]
[ I 4 κ + 1 ( 4 κ + 2 ) I 4 κ ( 4 κ + 1 ) . . . I 0 ( 1 ) I 4 κ + 2 ( 4 κ + 3 ) I 4 κ + 1 ( 4 κ + 2 ) . . . I 1 ( 2 ) . . . . . . I N 3 ( N 2 ) I N 4 ( N 3 ) . . . I N 4 κ 4 ( N 4 κ 3 ) I N 2 ( N 1 ) I N 3 ( N 2 ) . . . I N 4 κ 3 ( N 4 κ 2 ) ] R [ p 0 p 1 . . p 4 κ + 1 ] P = [ 0 0 0 . 0 ]
I ( x , y ; m ) = I dc + a 1 exp [ i ( φ 1 + ) ] + a 1 exp [ i ( φ 1 + ) ] T 1 +
b 1 exp [ i ( φ 2 + ) ] + b 1 exp [ i ( φ 2 + ) ] ; for m T 2 = 1,2 , …… , N .
Ī n = 1 r q + 1 j = 0 N 2 Z M ( n j + 1 , j + 1 ) ; for n = 0 , 1,2 , ….. N 2 ;
I n ( x , y ; m ) = k = 2 2 a k exp ( i { 2 π [ ( x x 0 ) 2 + ( y y 0 ) 2 ] λ 1 ξ + } ) +
k = 2 2 b k exp ( i { 2 π [ ( x p 0 ) 2 + ( y y 0 ) 2 ] λ 2 ξ + } )
; for m = n + 1 = 1,2 , …… N
[ e α 1 e α 1 e β 1 e β 1 e i ( κ 1 ) α 1 . . 1 e α 2 e α 2 e β 2 e β 2 e i ( κ 1 ) α 2 . . 1 . . . . . . . . . . . . . . . . e α N e α N e β N e β N e i ( κ 1 ) α N . . 1 ] [ κ κ * κ . I dc ] = [ I 0 I 1 I 2 . I N 1 ]

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