Abstract

The recently developed technique of high-speed phase-shifting speckle interferometry combined with temporal phase unwrapping allows dynamic displacement fields to be measured, even for objects containing global discontinuities such as cracks or boundaries. However, when local speckle averaging is included, small phase errors introduced at each time step are accumulated along the time axis, yielding total phase values that depend strongly on the speckle rereference rate. We present an analysis of the errors introduced in the phase evaluation by three sources: intensity errors, velocity errors, and speckle decorrelation. These errors are analyzed when they act both independently and together, for the most commonly used phase-shifting algorithms, with computer-generated speckle patterns. It is shown that, in a controlled out-of-plane geometry, errors in the unwrapped phase map that are due to speckle decorrelation rise as the time between rereferencing events is increased, whereas those due to intensity and velocity errors are reduced. It is also shown that speckle decorrelation errors are typically more important than the intensity and velocity errors. These results provide guidance as to the optimal speckle rereferencing rate in practical applications of the technique.

© 2005 Optical Society of America

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References

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  1. P. K. Rastogi, “Techniques of displacement and deformation measurements in speckle metrology,” in Speckle Metrology, R. S. Sirohi, ed. (Dekker, 1993), pp. 41–98.
  2. J. M. Huntley, G. H. Kaufmann, D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999).
    [CrossRef]
  3. C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle in-terferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
    [CrossRef]
  4. J. M. Kilpatrick, A. J. Moore, J. S. Barton, J. D. C. Jones, M. Reeves, C. Buckberry, “Measurement of complex surface deformation by high-speed dynamic phase-stepped digital speckle pattern interferometry,” Opt. Lett. 25, 1068–1070 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  7. J. M. Huntley, “Automated analysis of speckle interferograms,” in Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastogi, ed. (Wiley, 2001), pp. 59–139.
  8. P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Vibration-induced phase errors in high-speed phase-shifting speckle-pattern interferometry,” Appl. Opt. 40, 2117–2125 (2001).
    [CrossRef]
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    [CrossRef]
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  13. E. Kolenovic, W. Osten, W. Juptner, “Non-linear speckle phase changes in the image plane caused by out of plane displacement,” Opt. Commun. 171, 333–344 (1999).
    [CrossRef]
  14. U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms,” J. Opt. Soc. Am. 11, 2011–2015 (1994).
    [CrossRef]
  15. S. Donati, G. Martini, “Speckle-pattern intensity and phase: second-order conditional statistics,” J. Opt. Soc. Am. 69, 1690–1694 (1979).
    [CrossRef]
  16. K. Creath, “Phase-shifting speckle interferometry,” in International Conference on Speckle, H. H. Arsenault, ed., SPIE556, 337–346 (1985).
    [CrossRef]
  17. I. Yamaguchi, “Fringe formation in speckle photography,” J. Opt. Soc. Am. A 1, 81–86 (1984).
    [CrossRef]

2001 (2)

2000 (1)

1999 (3)

1998 (1)

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle in-terferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

1997 (1)

J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131–150 (1997).
[CrossRef]

1994 (1)

U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms,” J. Opt. Soc. Am. 11, 2011–2015 (1994).
[CrossRef]

1993 (1)

1989 (1)

1984 (1)

1979 (1)

Barton, J. S.

Benckert, L. R.

Buckberry, C.

Coggrave, C. R.

Creath, K.

K. Creath, “Phase-shifting speckle interferometry,” in International Conference on Speckle, H. H. Arsenault, ed., SPIE556, 337–346 (1985).
[CrossRef]

Davila, A.

A. Davila, G. H. Kaufmann, D. Kerr, “Digital processing of ESPI addition fringes,” in Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Juptner, W. Osten, eds.(Akademie Verlag, 1993), pp. 339–346.

Donati, S.

Franze, B.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle in-terferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

Haible, P.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle in-terferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

Hand, D. P.

Huntley, J. M.

Joenathan, C.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle in-terferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

Jones, J. D. C.

Juptner, W.

E. Kolenovic, W. Osten, W. Juptner, “Non-linear speckle phase changes in the image plane caused by out of plane displacement,” Opt. Commun. 171, 333–344 (1999).
[CrossRef]

Kaufmann, G. H.

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Vibration-induced phase errors in high-speed phase-shifting speckle-pattern interferometry,” Appl. Opt. 40, 2117–2125 (2001).
[CrossRef]

J. M. Huntley, G. H. Kaufmann, D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999).
[CrossRef]

A. Davila, G. H. Kaufmann, D. Kerr, “Digital processing of ESPI addition fringes,” in Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Juptner, W. Osten, eds.(Akademie Verlag, 1993), pp. 339–346.

Kerr, D.

J. M. Huntley, G. H. Kaufmann, D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999).
[CrossRef]

A. Davila, G. H. Kaufmann, D. Kerr, “Digital processing of ESPI addition fringes,” in Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Juptner, W. Osten, eds.(Akademie Verlag, 1993), pp. 339–346.

Kilpatrick, J. M.

Kolenovic, E.

E. Kolenovic, W. Osten, W. Juptner, “Non-linear speckle phase changes in the image plane caused by out of plane displacement,” Opt. Commun. 171, 333–344 (1999).
[CrossRef]

Martini, G.

Moore, A. J.

Osten, W.

E. Kolenovic, W. Osten, W. Juptner, “Non-linear speckle phase changes in the image plane caused by out of plane displacement,” Opt. Commun. 171, 333–344 (1999).
[CrossRef]

Rastogi, P. K.

P. K. Rastogi, “Techniques of displacement and deformation measurements in speckle metrology,” in Speckle Metrology, R. S. Sirohi, ed. (Dekker, 1993), pp. 41–98.

Reeves, M.

Ruiz, P. D.

Schnars, U.

U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms,” J. Opt. Soc. Am. 11, 2011–2015 (1994).
[CrossRef]

Shen, Y.

Sjodahl, M. S.

Tiziani, H. J.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle in-terferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

Yamaguchi, I.

Appl. Opt. (6)

J. Mod. Opt. (1)

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle in-terferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

J. Opt. Soc. Am. (2)

U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms,” J. Opt. Soc. Am. 11, 2011–2015 (1994).
[CrossRef]

S. Donati, G. Martini, “Speckle-pattern intensity and phase: second-order conditional statistics,” J. Opt. Soc. Am. 69, 1690–1694 (1979).
[CrossRef]

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

Opt. Commun. (1)

E. Kolenovic, W. Osten, W. Juptner, “Non-linear speckle phase changes in the image plane caused by out of plane displacement,” Opt. Commun. 171, 333–344 (1999).
[CrossRef]

Opt. Lasers Eng. (1)

J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131–150 (1997).
[CrossRef]

Opt. Lett. (1)

Other (4)

J. M. Huntley, “Automated analysis of speckle interferograms,” in Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastogi, ed. (Wiley, 2001), pp. 59–139.

A. Davila, G. H. Kaufmann, D. Kerr, “Digital processing of ESPI addition fringes,” in Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Juptner, W. Osten, eds.(Akademie Verlag, 1993), pp. 339–346.

P. K. Rastogi, “Techniques of displacement and deformation measurements in speckle metrology,” in Speckle Metrology, R. S. Sirohi, ed. (Dekker, 1993), pp. 41–98.

K. Creath, “Phase-shifting speckle interferometry,” in International Conference on Speckle, H. H. Arsenault, ed., SPIE556, 337–346 (1985).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

4f optical system used for speckle pattern simulation.

Fig. 2
Fig. 2

Standard deviation σϕ of wrapped phase errors due to speckle decorrelation plotted against the normalized speckle displacement ρ for the four-frame phase-shifting method. Continuous curve, theoretical value for σϕ. From top to bottom: ○, convolution of 1 × 1 pixels; *, 1 × 3 pixels; ◊, 1 × 9 pixels.

Fig. 3
Fig. 3

Standard deviation σϕ of unwrapped phase errors due to speckle decorrelation plotted against the normalized speckle displacement ρ for the four-frame method with 1 × 3 pixel averaging along a column of zero velocity. From top to bottom, rereferencing with τr = 4096, 500, 200, 50, 1 corresponding to Δρ = 1.0, 0.12, 0.048, 0.012, 0.00024, respectively.

Fig. 4
Fig. 4

Standard deviation σϕ of unwrapped phase errors due to velocity ( ϕ ˙ = 0.3927 rad frame−1) for the four-frame phase-shifting algorithm for a range of rereferencing event times τr. The plot for τr = ∞ has a uniform error offset of 0.1.

Fig. 5
Fig. 5

Standard deviation σϕ of unwrapped phase errors due to velocity ( ϕ ˙ = 0.3927 rad frame−1 ) for the five-frame phase-shifting algorithm for a range of rereferencing event times τr.

Fig. 6
Fig. 6

Standard deviation σϕ of unwrapped phase errors due to velocity ( ϕ ˙ = 0.3927 rad frame−1) for the Carré phase-shifting algorithm for a range of rereferencing event times τr.

Fig. 7
Fig. 7

Standard deviation σϕ of unwrapped phase errors due to intensity noise for the four-frame phase-shifting algorithm: effect of the rereferencing rate on intensity errors for noise levels with standard deviations s of 0.01 and 0.05 with respect to the mean gray level. The plot for s = 0.01 and τr = ∞ has a uniform error offset of 0.009.

Fig. 8
Fig. 8

Standard deviation σϕ of unwrapped phase errors due to intensity noise for the five-step phase-shifting algorithm: effect of the rereferencing rate on intensity errors for noise levels with standard deviations s of 0.01 and 0.05 with respect to the mean gray level. The plot for s = 0.01 and τr = ∞ has a uniform error offset of 0.0085.

Fig. 9
Fig. 9

Standard deviation σϕ of unwrapped phase errors due to intensity noise for the Carré phase-shifting algorithm: effect of the rereferencing rate on intensity errors for noise levels with standard deviations s of 0.01 and 0.05 with respect to the mean gray level. The plot for s = 0.01 and τr = ∞ has a uniform error offset of 0.023.

Fig. 10
Fig. 10

Combined errors of decorrelation and velocity for a computer-generated tilting plane for the four-frame phase-shifting algorithm, where each tilt corresponds to a number of tilt fringes across the field of view of F = 1/20 along the horizontal direction x0 and with no rereferencing (τr = ∞).

Fig. 11
Fig. 11

Rereferencing effects of three phase-shifting algorithms on the normalized integral of the error χ for F = 1, with circles representing the Carré phase-shifting algorithm, diamonds the four-frame phase-shifting algorithm, and asterisks the five-frame phase-shifting algorithm.

Fig. 12
Fig. 12

Rereferencing effects of three phase-shifting algorithms on the normalized integral of the error χ for F = 2, with circles representing the Carré phase-shifting algorithm, diamonds the four-frame phase-shifting algorithm, and asterisks the five-frame phase-shifting algorithm.

Fig. 13
Fig. 13

Rereferencing effects of three phase shifting algorithms on the normalized integral of the error χ for F = 1 and addition of intensity noise with a standard deviation of 0.05. Circles represent the Carré phase shifting algorithm, diamonds the four-frame phase-shifting algorithm, and asterisks the five-frame phase-shifting algorithm.

Fig. 14
Fig. 14

Rms phase error for the five-frame phase-step formula after ρ = 0.390, as a function of local nondimensional surface velocity ξ, for two different tilt rates (δF = 1 and δF = 2) and for three nondimensional pupil-plane speckle displacements between rereferencing events (Δρ = 0.01, ∞, and the optimal value of Δρ = 0.078). Dashed and dotted lines, Δρ = 0.01 for tilt rates δF = 1 and δF = 2, respectively; circles and continuous lines, Δρ = ∞ for tilt rates δF = 1 and δF = 2, respectively; diamonds and asterisks, Δρ = 0.078 for tilt rates δF = 1 and δF = 2, respectively. Spatial smoothing with a 3 × 3 kernel was used.

Equations (20)

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Z ( t ) = { t = 0 M 1 I ( t + t ) [ a ( t ) + i b ( t ) ] } α * ( t ) , t = 0 , 1 , 2 , , N t M ,
N ( t ) = Im [ Z ( t ) ] , D ( t ) = Re [ Z ( t ) ] .
N ( t ) = sign [ I ( t + 2 ) I ( t + 3 ) ] × ( [ I ( t ) I ( t + 3 ) + I ( t + 1 ) I ( t + 2 ) ] { 3 [ I ( t + 1 ) I ( t + 2 ) ] I ( t ) + I ( t + 3 ) } ) 1 / 2 ,
D ( t ) = I ( t + 1 ) + I ( t + 2 ) I ( t ) I ( t + 3 ) ,
Δ Φ w ( t 2 , t 1 ) = tan 1 [ N ( t 2 ) D ( t 1 ) D ( t 2 ) N ( t 1 ) D ( t 2 ) D ( t 1 ) + N ( t 2 ) N ( t 1 ) ] .
d ( t ) = NINT { [ Δ Φ w ( t , 0 ) Δ Φ w ( t 1 , 0 ) ] / 2 π } , t = 2 , 3 , , N t M ,
υ ( t ) = t = 2 t d ( t ) , t = 2 , 3 , , N t M , υ ( 1 ) = 0 ,
Δ Φ u ( t , 0 ) = Δ Φ w ( t , 0 ) 2 π υ ( t ) , t = 1 , 2 , , N t M .
Δ Φ u ( t , 0 ) = Δ Φ u ( t , t κ ) + k = 2 κ Δ Φ u ( t k , t k 1 ) + Δ Φ u ( t 1 , 0 ) .
I ( x 4 , y 4 , t ) = | T 1 { W T [ A o ( x 0 , y 0 , t ) ] } + A r ( x 4 , y 4 , t ) | 2 , x j , y j = 0 , 1 , 2 , , N , j = 0 , 1 , 2 , , 4 ,
A t ( x o , y o , t ) = exp ( i 2 π F x 0 / N Δ 0 ) ,
A t ( x o , y o , t ) = exp ( i 4 π Ω x 0 / λ ) ,
Ω = λ F / 2 N Δ 0 .
ρ = d / D a .
ρ = f λ F / N L N Δ 2 Δ 0 .
N = λ f / Δ 0 Δ 2 ,
ρ = F / N L .
σ ϕ 2 = π 2 3 π arcsin | μ | + arcsin 2 | μ | 1 2 n = 1 μ 2 n n 2 ,
U 0 = exp ( i ϕ ˙ p t ) ,
ξ = V V N = 4 δ F x 0 N Δ 0 ,

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