Abstract

Simultaneous measurement of multidimensional displacements using digital holographic interferometry involves multi-directional illumination of the deformed object and requires the reliable estimation of the resulting multiple interference phase distributions. The paper introduces an elegant method to simultaneously estimate the desired multiple phases from a single fringe pattern. The proposed method relies on modeling the reconstructed interference field as a piecewise multicomponent polynomial phase signal. Effectively, in a given region or segment, the reconstructed interference field is represented as the sum of different components i.e. complex signals with polynomial phases. The corresponding polynomial coefficients are estimated using the product high-order ambiguity function. To ensure proper matching of the estimated coefficients with the corresponding components, an amplitude based discrimination criterion is used. The main advantage of the proposed method is direct retrieval of multiple phases without the application of spatial carrier based filtering operations.

© 2011 OSA

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References

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  1. G. Pedrini, Y. L. Zou, and H. J. Tiziani, “Simultaneous quantitative evaluation of in-plane and out-of-plane deformations by use of a multidirectional spatial carrier,” Appl. Opt. 36, 786–792 (1997).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. P. Picart, E. Moisson, and D. Mounier, “Twin-sensitivity measurement by spatial multiplexing of digitally recorded holograms,” Appl. Opt. 42, 1947–1957 (2003).
    [CrossRef] [PubMed]
  7. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Simultaneous multidimensional deformation measurements using digital holographic moiré,” Appl. Opt. 50, 4189–4197 (2011).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. S. S. Gorthi and P. Rastogi, “Piecewise polynomial phase approximation approach for the analysis of reconstructed interference fields in digital holographic interferometry,” J. Opt. A: Pure Appl. Opt. 11 (2009).
    [CrossRef]
  12. S. S. Gorthi and P. Rastogi, “Windowed high-order ambiguity function method for fringe analysis,” Rev. Sci. Inst. 80, 073109 (2009).
    [CrossRef]
  13. S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express 18, 560–565 (2010).
    [CrossRef] [PubMed]
  14. S. Barbarossa, A. Scaglione, and G. B. Giannakis, “Product high-order ambiguity function for multicomponent polynomial-phase signal modeling,” IEEE Trans. Sig. Proc. 46, 691–708 (1998).
    [CrossRef]
  15. D. S. Pham and A. M. Zoubir, “Analysis of multicomponent polynomial phase signals,” IEEE Tran. Sig. Proc. 55, 56–65 (2007).
    [CrossRef]

2011 (1)

2010 (1)

2009 (2)

S. S. Gorthi and P. Rastogi, “Piecewise polynomial phase approximation approach for the analysis of reconstructed interference fields in digital holographic interferometry,” J. Opt. A: Pure Appl. Opt. 11 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Windowed high-order ambiguity function method for fringe analysis,” Rev. Sci. Inst. 80, 073109 (2009).
[CrossRef]

2008 (1)

Y. Morimoto, T. Matui, M. Fujigaki, and A. Matsui, “Three-dimensional displacement analysis by windowed phase-shifting digital holographic interferometry,” Strain 44, 49–56 (2008).
[CrossRef]

2007 (2)

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Laser. Eng. 45, 304–317 (2007).
[CrossRef]

D. S. Pham and A. M. Zoubir, “Analysis of multicomponent polynomial phase signals,” IEEE Tran. Sig. Proc. 55, 56–65 (2007).
[CrossRef]

2005 (1)

S. Okazawa, M. Fujigaki, Y. Morimoto, and T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Materials 3–4, 223–228 (2005).
[CrossRef]

2003 (2)

1999 (1)

1998 (1)

S. Barbarossa, A. Scaglione, and G. B. Giannakis, “Product high-order ambiguity function for multicomponent polynomial-phase signal modeling,” IEEE Trans. Sig. Proc. 46, 691–708 (1998).
[CrossRef]

1997 (2)

G. Pedrini, Y. L. Zou, and H. J. Tiziani, “Simultaneous quantitative evaluation of in-plane and out-of-plane deformations by use of a multidirectional spatial carrier,” Appl. Opt. 36, 786–792 (1997).
[CrossRef] [PubMed]

G. Pedrini and H. J. Tiziani, “Quantitative evaluation of two-dimensional dynamic deformations using digital holography,” Opt. Laser Tech. 29, 249–256 (1997).
[CrossRef]

1982 (1)

Barbarossa, S.

S. Barbarossa, A. Scaglione, and G. B. Giannakis, “Product high-order ambiguity function for multicomponent polynomial-phase signal modeling,” IEEE Trans. Sig. Proc. 46, 691–708 (1998).
[CrossRef]

Barnes, T. H.

Fujigaki, M.

Y. Morimoto, T. Matui, M. Fujigaki, and A. Matsui, “Three-dimensional displacement analysis by windowed phase-shifting digital holographic interferometry,” Strain 44, 49–56 (2008).
[CrossRef]

S. Okazawa, M. Fujigaki, Y. Morimoto, and T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Materials 3–4, 223–228 (2005).
[CrossRef]

Giannakis, G. B.

S. Barbarossa, A. Scaglione, and G. B. Giannakis, “Product high-order ambiguity function for multicomponent polynomial-phase signal modeling,” IEEE Trans. Sig. Proc. 46, 691–708 (1998).
[CrossRef]

Gorthi, S. S.

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

S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express 18, 560–565 (2010).
[CrossRef] [PubMed]

S. S. Gorthi and P. Rastogi, “Piecewise polynomial phase approximation approach for the analysis of reconstructed interference fields in digital holographic interferometry,” J. Opt. A: Pure Appl. Opt. 11 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Windowed high-order ambiguity function method for fringe analysis,” Rev. Sci. Inst. 80, 073109 (2009).
[CrossRef]

Ina, H.

Juptner, W.

Kemao, Q.

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Laser. Eng. 45, 304–317 (2007).
[CrossRef]

Klattenhoff, R.

Kobayashi, S.

Kolenovic, E.

Lai, S.

Matsui, A.

Y. Morimoto, T. Matui, M. Fujigaki, and A. Matsui, “Three-dimensional displacement analysis by windowed phase-shifting digital holographic interferometry,” Strain 44, 49–56 (2008).
[CrossRef]

Matui, T.

Y. Morimoto, T. Matui, M. Fujigaki, and A. Matsui, “Three-dimensional displacement analysis by windowed phase-shifting digital holographic interferometry,” Strain 44, 49–56 (2008).
[CrossRef]

S. Okazawa, M. Fujigaki, Y. Morimoto, and T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Materials 3–4, 223–228 (2005).
[CrossRef]

Moisson, E.

Morimoto, Y.

Y. Morimoto, T. Matui, M. Fujigaki, and A. Matsui, “Three-dimensional displacement analysis by windowed phase-shifting digital holographic interferometry,” Strain 44, 49–56 (2008).
[CrossRef]

S. Okazawa, M. Fujigaki, Y. Morimoto, and T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Materials 3–4, 223–228 (2005).
[CrossRef]

Mounier, D.

Okazawa, S.

S. Okazawa, M. Fujigaki, Y. Morimoto, and T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Materials 3–4, 223–228 (2005).
[CrossRef]

Osten, W.

Pedrini, G.

G. Pedrini, Y. L. Zou, and H. J. Tiziani, “Simultaneous quantitative evaluation of in-plane and out-of-plane deformations by use of a multidirectional spatial carrier,” Appl. Opt. 36, 786–792 (1997).
[CrossRef] [PubMed]

G. Pedrini and H. J. Tiziani, “Quantitative evaluation of two-dimensional dynamic deformations using digital holography,” Opt. Laser Tech. 29, 249–256 (1997).
[CrossRef]

Pham, D. S.

D. S. Pham and A. M. Zoubir, “Analysis of multicomponent polynomial phase signals,” IEEE Tran. Sig. Proc. 55, 56–65 (2007).
[CrossRef]

Picart, P.

Rajshekhar, G.

Rastogi, P.

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

S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express 18, 560–565 (2010).
[CrossRef] [PubMed]

S. S. Gorthi and P. Rastogi, “Windowed high-order ambiguity function method for fringe analysis,” Rev. Sci. Inst. 80, 073109 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Piecewise polynomial phase approximation approach for the analysis of reconstructed interference fields in digital holographic interferometry,” J. Opt. A: Pure Appl. Opt. 11 (2009).
[CrossRef]

Scaglione, A.

S. Barbarossa, A. Scaglione, and G. B. Giannakis, “Product high-order ambiguity function for multicomponent polynomial-phase signal modeling,” IEEE Trans. Sig. Proc. 46, 691–708 (1998).
[CrossRef]

Takeda, M.

Tan, S. M.

Tiziani, H. J.

G. Pedrini, Y. L. Zou, and H. J. Tiziani, “Simultaneous quantitative evaluation of in-plane and out-of-plane deformations by use of a multidirectional spatial carrier,” Appl. Opt. 36, 786–792 (1997).
[CrossRef] [PubMed]

G. Pedrini and H. J. Tiziani, “Quantitative evaluation of two-dimensional dynamic deformations using digital holography,” Opt. Laser Tech. 29, 249–256 (1997).
[CrossRef]

von Kopylow, C.

Watkins, L. R.

Zou, Y. L.

Zoubir, A. M.

D. S. Pham and A. M. Zoubir, “Analysis of multicomponent polynomial phase signals,” IEEE Tran. Sig. Proc. 55, 56–65 (2007).
[CrossRef]

Appl. Mech. Materials (1)

S. Okazawa, M. Fujigaki, Y. Morimoto, and T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Materials 3–4, 223–228 (2005).
[CrossRef]

Appl. Opt. (4)

IEEE Tran. Sig. Proc. (1)

D. S. Pham and A. M. Zoubir, “Analysis of multicomponent polynomial phase signals,” IEEE Tran. Sig. Proc. 55, 56–65 (2007).
[CrossRef]

IEEE Trans. Sig. Proc. (1)

S. Barbarossa, A. Scaglione, and G. B. Giannakis, “Product high-order ambiguity function for multicomponent polynomial-phase signal modeling,” IEEE Trans. Sig. Proc. 46, 691–708 (1998).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

S. S. Gorthi and P. Rastogi, “Piecewise polynomial phase approximation approach for the analysis of reconstructed interference fields in digital holographic interferometry,” J. Opt. A: Pure Appl. Opt. 11 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (1)

Opt. Laser Tech. (1)

G. Pedrini and H. J. Tiziani, “Quantitative evaluation of two-dimensional dynamic deformations using digital holography,” Opt. Laser Tech. 29, 249–256 (1997).
[CrossRef]

Opt. Laser. Eng. (1)

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Laser. Eng. 45, 304–317 (2007).
[CrossRef]

Opt. Lett. (1)

Rev. Sci. Inst. (1)

S. S. Gorthi and P. Rastogi, “Windowed high-order ambiguity function method for fringe analysis,” Rev. Sci. Inst. 80, 073109 (2009).
[CrossRef]

Strain (1)

Y. Morimoto, T. Matui, M. Fujigaki, and A. Matsui, “Three-dimensional displacement analysis by windowed phase-shifting digital holographic interferometry,” Strain 44, 49–56 (2008).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Dual-beam illumination of a diffuse object

Fig. 2
Fig. 2

(a) Third order (m = 3) PHAF spectrum for (a) A1 = A2, (b) A1 < A2 and (c) A1 > A2. (d) Second order (m = 2) PHAF spectrum. (e) First order (m = 1) PHAF spectrum. (f) Original vs estimated first phase in radians.

Fig. 3
Fig. 3

(a) Third order (m = 3) PHAF spectrum. (b) Second order (m = 2) PHAF spectrum. (c) First order (m = 1) PHAF spectrum. (d) Original vs estimated second phase in radians. (e) Estimation error in radians for first phase. (f) Estimation error in radians for second phase.

Fig. 4
Fig. 4

(a) Original Δϕ1(x, y) in radians. (b) Original Δϕ2(x, y) in radians. (c) Fringe pattern. (d) Fourier spectrum |γ(ωxy)|

Fig. 5
Fig. 5

(a) Estimated Δϕ1(x, y) in radians. (b) Estimation error for Δϕ1(x, y) in radians. (c) Estimated Δϕ2(x, y) in radians. (d) Estimation error for Δϕ2(x, y) in radians.

Equations (18)

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Γ ( x , y ) = A 1 ( x , y ) exp [ j Δ ϕ 1 ( x , y ) ] + A 2 ( x , y ) exp [ j Δ ϕ 2 ( x , y ) ] + η ( x , y )
Δ ϕ 1 i ( y ) = m = 0 M a i m y m
Δ ϕ 2 i ( y ) = m = 0 M b i m y m
Γ i ( y ) = A 1 i ( y ) exp [ j ( m = 0 M a i m y m ) ] + A 2 i ( y ) exp [ j ( m = 0 M b i m y m ) ] + η i ( y )
x 1 ( y ) = Γ i ( y ) , x 2 ( y ; τ k , 1 ) = x 1 ( y ) x 1 * ( y τ k , 1 ) , x m ( y ; τ k , m 1 ) = x m 1 ( y ; τ k , m 2 ) x m 1 * ( y τ k , m 1 ; τ k , m 2 )
X m ( ω ; τ k , m 1 ) = DFT [ x m ( y ; τ k , m 1 ) ] = y = 0 N 1 x m ( y ; τ k , m 1 ) exp [ j ω y ]
X K , m ( ω ) = k = 1 K X m ( P m ( τ k , m 1 , τ 1 , m 1 ) ω , τ k , m 1 )
P m ( τ k , m 1 , τ 1 , m 1 ) = [ τ k , m 1 τ 1 , m 1 ] m 1
a i m , b i m = ω max m ! ( τ 1 , m 1 ) m 1
Γ ( y ) = A 1 exp [ j ( a 0 + a 1 y + a 2 y 2 + a 3 y 3 ) ] + A 2 exp [ j ( b 0 + b 1 y + b 2 y 2 + b 3 y 3 ) ] y [ 1 , N ] , N = 256
Γ 2 ( y ) = Γ ( y ) exp [ j a ^ 3 y 3 ] A 1 exp [ j ( a 0 + a 1 y + a 2 y 2 ) ] + A 2 exp [ j ( b 0 + b 1 y + b 2 y 2 + ( b 3 a ^ 3 ) y 3 ) ]
Γ 1 ( y ) = Γ 2 ( y ) exp [ j a ^ 2 y 2 ] A 1 exp [ j ( a 0 + a 1 y ) ] + A 2 exp [ j ( b 0 + b 1 , y + ( b 2 a ^ 2 ) y 2 + ( b 3 a ^ 3 ) y 3 ) ]
Γ 0 ( y ) = Γ 1 ( y ) exp [ j a ^ 1 y ] A 1 exp [ j a 0 ] + A 2 exp [ j ( b 0 + ( b 1 a ^ 1 ) y + ( b 2 a ^ 2 ) y 2 + ( b 3 a ^ 3 ) y 3 ) ]
a ^ 0 = angle [ 1 N y = 1 N Γ 0 ( y ) ]
A ^ 1 = | 1 N y = 1 N Γ 0 ( y ) |
Γ ( y ) = Γ ( y ) A ^ 1 exp [ j ( a ^ 0 + a ^ 1 y + a ^ 2 y 2 + a ^ 3 y 3 ) ] A 2 exp [ j ( b 0 + b 1 y + b 2 y 2 + b 3 y 3 ) ]
Γ ( x , y ) = A 1 exp [ j Δ ϕ 1 ( x , y ) ] + A 2 exp [ j Δ ϕ 2 ( x , y ) ]
γ ( ω x , ω y ) = DFT [ Γ ( x , y ) ]

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