Abstract

The paper introduces a multiple signal classification technique based method for fringe analysis. In the proposed method, the phase of a fringe pattern is locally approximated as a polynomial. The polynomial phase signal is then transformed to obtain signals comprising of only even- or odd-order polynomial coefficients. Subsequently, covariance matrix formulation is applied, and the two sets of coefficients are jointly estimated from the noise subspace of the covariance matrix using the multiple signal classification technique. The method allows simultaneous estimation of multiple coefficients and provides phase without the requirement of complex unwrapping algorithms. The effectiveness of the proposed method is validated through numerical simulation.

© 2012 Optical Society of America

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References

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  1. T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).
  2. P. K. Rastogi, ed., Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2000).
  3. G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50, 3–10 (2012).
  4. K. Creath, “Phase-measurement interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1988), pp. 349–393.
  5. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  6. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
    [CrossRef]
  7. L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24, 905–907 (1999).
    [CrossRef]
  8. F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223 (2002).
    [CrossRef]
  9. J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-d shape measurement,” Opt. Eng. 43, 895–899 (2004).
    [CrossRef]
  10. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
    [CrossRef]
  11. S. S. Gorthi and P. Rastogi, “Numerical analysis of fringe patterns recorded in holographic interferometry using high-order ambiguity function,” J. Mod. Opt. 56, 949–954 (2009).
    [CrossRef]
  12. S. S. Gorthi and P. Rastogi, “Windowed high-order ambiguity function method for fringe analysis,” Rev. Sci. Instrum. 80, 073109 (2009).
    [CrossRef]
  13. U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
    [CrossRef]
  14. Y. Wu, H. C. So, and H. Liu, “Subspace-based algorithm for parameter estimation of polynomial phase signals,” IEEE Trans. Signal Process. 56, 4977–4983 (2008).
    [CrossRef]
  15. R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag. 34, 276–280 (1986).
    [CrossRef]
  16. P. Stoica and R. Moses, Introduction to Spectral Analysis(Prentice Hall, 1997).
  17. E. Aboutanios and B. Mulgrew, “Iterative frequency estimation by interpolation on Fourier coefficients,” IEEE Trans. Signal Process. 53, 1237–1242 (2005).
    [CrossRef]

2012

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

2009

S. S. Gorthi and P. Rastogi, “Numerical analysis of fringe patterns recorded in holographic interferometry using high-order ambiguity function,” J. Mod. Opt. 56, 949–954 (2009).
[CrossRef]

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

2008

Y. Wu, H. C. So, and H. Liu, “Subspace-based algorithm for parameter estimation of polynomial phase signals,” IEEE Trans. Signal Process. 56, 4977–4983 (2008).
[CrossRef]

2007

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

2005

E. Aboutanios and B. Mulgrew, “Iterative frequency estimation by interpolation on Fourier coefficients,” IEEE Trans. Signal Process. 53, 1237–1242 (2005).
[CrossRef]

2004

J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-d shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

2002

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223 (2002).
[CrossRef]

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

1999

1997

1986

R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag. 34, 276–280 (1986).
[CrossRef]

1982

Aboutanios, E.

E. Aboutanios and B. Mulgrew, “Iterative frequency estimation by interpolation on Fourier coefficients,” IEEE Trans. Signal Process. 53, 1237–1242 (2005).
[CrossRef]

Barnes, T. H.

Creath, K.

K. Creath, “Phase-measurement interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1988), pp. 349–393.

Cuevas, F. J.

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223 (2002).
[CrossRef]

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef]

Gorthi, S. S.

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

S. S. Gorthi and P. Rastogi, “Numerical analysis of fringe patterns recorded in holographic interferometry using high-order ambiguity function,” J. Mod. Opt. 56, 949–954 (2009).
[CrossRef]

Ina, H.

Juptner, W. P. O.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Kemao, Q.

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

Kobayashi, S.

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

Liu, H.

Y. Wu, H. C. So, and H. Liu, “Subspace-based algorithm for parameter estimation of polynomial phase signals,” IEEE Trans. Signal Process. 56, 4977–4983 (2008).
[CrossRef]

Marroquin, J. L.

Moses, R.

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

Mulgrew, B.

E. Aboutanios and B. Mulgrew, “Iterative frequency estimation by interpolation on Fourier coefficients,” IEEE Trans. Signal Process. 53, 1237–1242 (2005).
[CrossRef]

Rajshekhar, G.

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

Rastogi, P.

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

S. S. Gorthi and P. Rastogi, “Numerical analysis of fringe patterns recorded in holographic interferometry using high-order ambiguity function,” J. Mod. Opt. 56, 949–954 (2009).
[CrossRef]

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

Rastogi, P. K.

P. K. Rastogi, ed., Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2000).

Schmidt, R. O.

R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag. 34, 276–280 (1986).
[CrossRef]

Schnars, U.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Servin, M.

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223 (2002).
[CrossRef]

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef]

So, H. C.

Y. Wu, H. C. So, and H. Liu, “Subspace-based algorithm for parameter estimation of polynomial phase signals,” IEEE Trans. Signal Process. 56, 4977–4983 (2008).
[CrossRef]

Sossa-Azuela, J. H.

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223 (2002).
[CrossRef]

Stoica, P.

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

Takeda, M.

Tan, S. M.

Watkins, L. R.

Weng, J.

J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-d shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

Wu, Y.

Y. Wu, H. C. So, and H. Liu, “Subspace-based algorithm for parameter estimation of polynomial phase signals,” IEEE Trans. Signal Process. 56, 4977–4983 (2008).
[CrossRef]

Zhong, J.

J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-d shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

Appl. Opt.

IEEE Trans. Antennas Propag.

R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag. 34, 276–280 (1986).
[CrossRef]

IEEE Trans. Signal Process.

Y. Wu, H. C. So, and H. Liu, “Subspace-based algorithm for parameter estimation of polynomial phase signals,” IEEE Trans. Signal Process. 56, 4977–4983 (2008).
[CrossRef]

E. Aboutanios and B. Mulgrew, “Iterative frequency estimation by interpolation on Fourier coefficients,” IEEE Trans. Signal Process. 53, 1237–1242 (2005).
[CrossRef]

J. Mod. Opt.

S. S. Gorthi and P. Rastogi, “Numerical analysis of fringe patterns recorded in holographic interferometry using high-order ambiguity function,” J. Mod. Opt. 56, 949–954 (2009).
[CrossRef]

J. Opt. Soc. Am.

Meas. Sci. Technol.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Opt. Commun.

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223 (2002).
[CrossRef]

Opt. Eng.

J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-d shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

Opt. Lasers Eng.

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

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

Opt. Lett.

Rev. Sci. Instrum.

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

Other

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

K. Creath, “Phase-measurement interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1988), pp. 349–393.

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

P. K. Rastogi, ed., Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2000).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Function f(κ) for cubic phase signal. (b) Original versus estimated phase in radians. (c) Estimation error in radians.

Fig. 2.
Fig. 2.

Function f(κ), original versus estimated phase in radians, and estimation error in radians at SNR of 25 dB in (a)–(c), SNR of 20 dB in (d)–(f), and SNR of 15 dB in (g)–(i).

Fig. 3.
Fig. 3.

(a) Function f(κ) for quadratic phase signal. (b) Original versus estimated phase in radians. (c) Estimation error in radians.

Fig. 4.
Fig. 4.

(a) Fringe pattern; (b) estimated phase in radians; (c) estimation error in radians using the proposed method; (d) estimation error in radians using the windowed Fourier transform (WFT) method.

Equations (42)

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Γ(x,y)=a(x,y)exp(jϕ(x,y))+η(x,y),
Γ(x)=a(x)exp(jϕ(x))+η(x).
ϕi(x)=m=0Kαmxmx[Ns/2,Ns/2],
Γi(x)=a(x)exp(jm=0Kαmxm)+ηi(x),
ϒ1(x)=Γi(x)Γi(x)x[1:Ns/2]=a2exp(2jm=0Mα2mx2m)+η1(x)
ϒ2(x)=Γi(x)Γi*(x)x[1:Ns/2]=a2exp(2jm=0Mα2m+1x2m+1)+η2(x),
ϒ3(x)=ϒ2(x+1)ϒ2*(x1)x[2:Ns/21]=a4exp(2jm=0Mβ2mx2m)+η3(x),
β2m=2p=mMα2p+1(2p+12m)m[0:M]
(nr)=n!r!(nr)!.
y(x)=ϒ1(x)+ϒ3(x)x[2:Ns/21].
y(x)=a2exp(2jm=0Mα2mx2m)+a4exp(2jm=0Mβ2mx2m)+ϵ(x)=a2exp(2jα0)exp(2jm=1Mα2mx2m)+a4exp(2jβ0)exp(2jm=1Mβ2mx2m)+ϵ(x)=[exp(2jm=1Mα2mx2m)exp(2jm=1Mβ2mx2m)]×[a2exp(2jα0)a4exp(2jβ0)]+ϵ(x),
v(κ1)=[exp(2jm=1Mα2m22m)exp(2jm=1Mα2m(Ns/21)2m)],
v(κ2)=[exp(2jm=1Mβ2m22m)exp(2jm=1Mβ2m(Ns/21)2m)].
y=As+ϵ,
y=[y(2)y(Ns/21)],s=[a2exp(2jα0)a4exp(2jβ0)],ϵ=[ϵ(2)ϵ(Ns/21)]
A=[v(κ1)v(κ2)]=[exp(2jm=1Mα2m22m)exp(2jm=1Mβ2m22m)exp(2jm=1Mα2m(Ns/21)2m)exp(2jm=1Mβ2m(Ns/21)2m)].
R=E{yyH}APAH+Q,
vH(κ)GGHv(κ)=0ifκ=κ1orκ2.
{κ^1,κ^2}=argmaxκ1vH(κ)G^G^Hv(κ).
[α^3α^2M+1]=[(31)(2M+12M1)0(2M+11)][β^2β^2M],
Γi1(x)=Γi(x)exp(j(α^2x2++α^2M+1x2M+1))a(x)exp(j(α0+α1x)),
α^1=argmaxω|x=Ns/2Ns/2Γi1(x)exp(jωx)|.
Γi0(x)=Γi1(x)exp(jα^1x)a(x)exp(jα0)
α^0=angle{1Ns+1x=Ns/2Ns/2Γi0(x)}.
ϒ1(x)=Γi(x)Γi(x)x[1:Ns/2]=a2exp(2jm=0Mα2mx2m)+η1(x)
ϒ2(x)=Γi(x)Γi*(x)x[1:Ns/2]=a2exp(2jm=0M1α2m+1x2m+1)+η2(x).
ϒ3(x)=ϒ1(x+1)ϒ1*(x1)x[2:Ns/21]=a4exp(2jm=0M1β2mx2m+1)+η3(x),
y(x)=ϒ2(x)+ϒ3(x)x[2:Ns/21].
Γ(x)=exp(jϕ(x))+η(x)x[128,128],
ϕ(x)=α0+α1x+α2x2+α3x3
ϒ1(x)=exp(2j(α0+α2x2))+η1(x),
ϒ2(x)=exp(2j(α1x+α3x3))+η2(x),
ϒ3(x)=exp(2j(2(α1+α3)+6α3x2))+η3(x).
y(x)=exp(2j(α0+α2x2))+exp(2j(2(α1+α3)+6α3x2))+ϵ(x).
f(κ)=1vH(κ)G^G^Hv(κ),
ϒ1(x)=ϒ1(x)exp(jκmaxx2){exp(2jα0)ifκmax=κ1exp(2j(α0+α2x26α3x3))ifκmax=κ2.
Γ(x)=exp(jϕ(x))+η(x)x[128,128],
ϕ(x)=α0+α1x+α2x2
ϒ1(x)=exp(2j(α0+α2x2))+η1(x),
ϒ2(x)=exp(2jα1x)+η2(x),
ϒ3(x)=exp(2j(4α2x))+η3(x).
y(x)=exp(2jα1x)+exp(2j(4α2x))+ϵ(x).

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