Abstract

A computationally efficient technique for fringe analysis in digital holographic interferometry using a difference equation representation of the interference field is presented. The spatially varying coefficient of the difference equation is estimated accurately by constraining it in the subspace spanned by the linearly independent basis functions. The coefficient estimated provides an accurate estimation of the interference phase derivative and enables the linear estimation of the interference field. Thereupon, the interference phase is estimated using a simple unwrapping algorithm. The performance of the proposed method is validated with the help of simulation and experimental results.

© 2014 Optical Society of America

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References

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  1. G. K. Bhat, “A hybrid fringe analysis technique for the elimination of random noise in interferometric wrapped phase maps,” Opt. Commun. 111, 214–218 (1994).
    [CrossRef]
  2. F. Palacios, E. Gonçalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
    [CrossRef]
  3. R. Cusack, J. M. Huntley, and H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781–789 (1995).
    [CrossRef]
  4. K. A. Stetson, J. Wahid, and P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 36, 4830–4838 (1997).
    [CrossRef]
  5. S. S. Gorthi and P. Rastogi, “Improved high-order ambiguity-function method for the estimation of phase from interferometric fringes,” Opt. Lett. 34, 2575–2577 (2009).
    [CrossRef]
  6. 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 11, 065405 (2009).
    [CrossRef]
  7. Y. Zou, G. Pedrini, and H. Tiziani, “Derivatives obtained directly from displacement data,” Opt. Commun. 111, 427–432 (1994).
    [CrossRef]
  8. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Polynomial Wigner–Ville distribution-based method for direct phase derivative estimation from optical fringes,” J. Opt. A 11, 125402 (2009).
    [CrossRef]
  9. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Adaptive window Wigner–Ville-distribution-based method to estimate phase derivative from optical fringes,” Opt. Lett. 34, 3151–3153 (2009).
    [CrossRef]
  10. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
    [CrossRef]
  11. G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50, 3–10 (2012).
  12. A. S. Kayhan, “Representation and analysis of complex chirp signals,” Signal Process. 66, 111–116 (1998).
    [CrossRef]
  13. L. A. Liporace, “Linear estimation of nonstationary signals,” J. Acoust. Soc. Am. 58, 1288–1295 (1975).
    [CrossRef]

2012 (1)

G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50, 3–10 (2012).

2009 (4)

S. S. Gorthi and P. Rastogi, “Improved high-order ambiguity-function method for the estimation of phase from interferometric fringes,” Opt. Lett. 34, 2575–2577 (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 11, 065405 (2009).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Polynomial Wigner–Ville distribution-based method for direct phase derivative estimation from optical fringes,” J. Opt. A 11, 125402 (2009).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Adaptive window Wigner–Ville-distribution-based method to estimate phase derivative from optical fringes,” Opt. Lett. 34, 3151–3153 (2009).
[CrossRef]

2004 (2)

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
[CrossRef]

F. Palacios, E. Gonçalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

1998 (1)

A. S. Kayhan, “Representation and analysis of complex chirp signals,” Signal Process. 66, 111–116 (1998).
[CrossRef]

1997 (1)

1995 (1)

1994 (2)

G. K. Bhat, “A hybrid fringe analysis technique for the elimination of random noise in interferometric wrapped phase maps,” Opt. Commun. 111, 214–218 (1994).
[CrossRef]

Y. Zou, G. Pedrini, and H. Tiziani, “Derivatives obtained directly from displacement data,” Opt. Commun. 111, 427–432 (1994).
[CrossRef]

1975 (1)

L. A. Liporace, “Linear estimation of nonstationary signals,” J. Acoust. Soc. Am. 58, 1288–1295 (1975).
[CrossRef]

Bhat, G. K.

G. K. Bhat, “A hybrid fringe analysis technique for the elimination of random noise in interferometric wrapped phase maps,” Opt. Commun. 111, 214–218 (1994).
[CrossRef]

Cusack, R.

Gauthier, P.

Goldrein, H. T.

Gonçalves, E.

F. Palacios, E. Gonçalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

Gorthi, S. S.

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 11, 065405 (2009).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Polynomial Wigner–Ville distribution-based method for direct phase derivative estimation from optical fringes,” J. Opt. A 11, 125402 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Improved high-order ambiguity-function method for the estimation of phase from interferometric fringes,” Opt. Lett. 34, 2575–2577 (2009).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Adaptive window Wigner–Ville-distribution-based method to estimate phase derivative from optical fringes,” Opt. Lett. 34, 3151–3153 (2009).
[CrossRef]

Huntley, J. M.

Kayhan, A. S.

A. S. Kayhan, “Representation and analysis of complex chirp signals,” Signal Process. 66, 111–116 (1998).
[CrossRef]

Kemao, Q.

Liporace, L. A.

L. A. Liporace, “Linear estimation of nonstationary signals,” J. Acoust. Soc. Am. 58, 1288–1295 (1975).
[CrossRef]

Palacios, F.

F. Palacios, E. Gonçalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

Pedrini, G.

Y. Zou, G. Pedrini, and H. Tiziani, “Derivatives obtained directly from displacement data,” Opt. Commun. 111, 427–432 (1994).
[CrossRef]

Rajshekhar, G.

G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50, 3–10 (2012).

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Polynomial Wigner–Ville distribution-based method for direct phase derivative estimation from optical fringes,” J. Opt. A 11, 125402 (2009).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Adaptive window Wigner–Ville-distribution-based method to estimate phase derivative from optical fringes,” Opt. Lett. 34, 3151–3153 (2009).
[CrossRef]

Rastogi, P.

G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50, 3–10 (2012).

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Polynomial Wigner–Ville distribution-based method for direct phase derivative estimation from optical fringes,” J. Opt. A 11, 125402 (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 11, 065405 (2009).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Adaptive window Wigner–Ville-distribution-based method to estimate phase derivative from optical fringes,” Opt. Lett. 34, 3151–3153 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Improved high-order ambiguity-function method for the estimation of phase from interferometric fringes,” Opt. Lett. 34, 2575–2577 (2009).
[CrossRef]

Ricardo, J.

F. Palacios, E. Gonçalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

Stetson, K. A.

Tiziani, H.

Y. Zou, G. Pedrini, and H. Tiziani, “Derivatives obtained directly from displacement data,” Opt. Commun. 111, 427–432 (1994).
[CrossRef]

Valin, J. L.

F. Palacios, E. Gonçalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

Wahid, J.

Zou, Y.

Y. Zou, G. Pedrini, and H. Tiziani, “Derivatives obtained directly from displacement data,” Opt. Commun. 111, 427–432 (1994).
[CrossRef]

Appl. Opt. (3)

J. Acoust. Soc. Am. (1)

L. A. Liporace, “Linear estimation of nonstationary signals,” J. Acoust. Soc. Am. 58, 1288–1295 (1975).
[CrossRef]

J. Opt. A (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 11, 065405 (2009).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Polynomial Wigner–Ville distribution-based method for direct phase derivative estimation from optical fringes,” J. Opt. A 11, 125402 (2009).
[CrossRef]

Opt. Commun. (3)

Y. Zou, G. Pedrini, and H. Tiziani, “Derivatives obtained directly from displacement data,” Opt. Commun. 111, 427–432 (1994).
[CrossRef]

G. K. Bhat, “A hybrid fringe analysis technique for the elimination of random noise in interferometric wrapped phase maps,” Opt. Commun. 111, 214–218 (1994).
[CrossRef]

F. Palacios, E. Gonçalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

Opt. Lasers Eng. (1)

G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50, 3–10 (2012).

Opt. Lett. (2)

Signal Process. (1)

A. S. Kayhan, “Representation and analysis of complex chirp signals,” Signal Process. 66, 111–116 (1998).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Estimation of (a) phase (in radians) and (b) phase derivative (in radians/pixel) computed using different BFs. Error in the estimation of (c) phase (in radians) and (d) phase derivative (in radians/pixel) as a function of basis dimension.

Fig. 2.
Fig. 2.

Simulated (a) interference phase and (b) fringe pattern.

Fig. 3.
Fig. 3.

Estimated (a) phase (in radians) and (b) phase derivative (in radians/pixel). Error in the estimation of (c) the phase (in radians) and (d) the phase derivative (in radians/pixel).

Fig. 4.
Fig. 4.

(a) Experimental fringe pattern, (b) estimated phase (in radians), and (c) estimated phase derivative (in radians/pixel).

Tables (1)

Tables Icon

Table 1. Error in the Estimation of the Phase (in Radians) and Phase Derivative (in Radians/Pixel) Calculated for Different BFs

Equations (22)

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Γ[n,m]=ejθ[n,m]+ϵ[n,m],
Γ[n]=ejθ[n]+ϵ[n].
θ˙[n]=θ[n]θ[n1].
Γ[n]=ej(θ˙[n]+θ[n1])+ϵ[n]=ejθ˙[n]ejθ[n1]+ϵ[n].
Γ[n]=c[n]Γ[n1]+ϵ[n],
c[n]=i=0Lciβi[n],
Γ[n]=(i=0Lciβi[n])Γ[n1]+ϵ[n].
Γ^[n]=(i=0Lciβi[n])Γ[n1].
ϵ[n]=Γ[n]Γ^[n].
ϵ2=|ϵ[1]|2+|ϵ[2]|2++|ϵ[N]|2.
(p,q)=j=1Npjqj*,
ϵ2=(ϵ,ϵ)
ϵ2=(ϵ,ϵ),=Γ22(Γ,Γ^)+Γ^2.
Sik={βi[n]Γ[nk]};
rij=(Si1,Sj1),r0j=(S00,Sj1),
ϵ2=Γ22CTr0+CTRC.
RC=r0.
θ˙[n]=arg(c[n]).
βi[n]=(nN)i.
βi[n]={cosiπn2Nievensin(i+1)πn2Niodd.
β0[n]=1,β1[n]=nN;βi+1=1(i+1)((2i+1)(nN)βi[n]iβi1[n]).
β0[n]=1,β1[n]=nN;βi+1=2(nN)βi[n]βi1[n].

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