Abstract

Based on the windowed Fourier transform, the windowed Fourier ridges (WFR) algorithm and the windowed Fourier filtering algorithm (WFF) have been developed and proven effective for fringe pattern analysis. The WFR algorithm is able to estimate local frequency and phase by assuming the phase distribution in a local area to be a quadratic polynomial. In this paper, a general and detailed statistical analysis is carried out for the WFR algorithm when an exponential phase field is disturbed by additive white Gaussian noise. Because of the bias introduced by the WFR algorithm for phase estimation, a phase compensation method is proposed for the WFR algorithm followed by statistical analysis. The mean squared errors are derived for both local frequency and phase estimates using a first-order perturbation technique. These mean square errors are compared with Cramer–Rao bounds, which shows that the WFR algorithm with phase compensation is a suboptimal estimator. The above theoretical analysis and comparison are verified by Monte Carlo simulations. Furthermore, the WFR algorithm is shown to be slightly better than the WFF algorithm for quadratic phase.

© 2012 Optical Society of America

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References

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  1. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, application and implementation,” Opt. Laser. Eng. 45, 304–317 (2007).
    [CrossRef]
  2. Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47, 5408–5419 (2008).
    [CrossRef]
  3. S. Peleg and B. Porat, “Linear FM signal parameter estimation from discrete-time observations,” IEEE Trans. Aerosp. Electron. Syst. 27, 607–616 (1991).
    [CrossRef]
  4. B. Friedlander and J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
    [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. P. O’Shea, “A fast algorithm for estimating the parameters of a quadratic FM signal,” IEEE Trans. Signal Process. 52, 385–393 (2004).
    [CrossRef]
  7. 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]
  8. H. Cramer, Mathematical Methods of Statistics (Princeton University, 1999).
  9. C. R. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).
  10. A. C. Tamhane and D. D. Dunlop, Statistics and Data Analysis: From Elementary to Intermediate (Prentice-Hall, 2000).
  11. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics (Elsevier, 1988), Vol. XXVI, pp. 349–393.
  12. M. Takeda, H. Ina, and S. Kobayashi, “Fourier transform methods of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  13. U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2005).
  14. S. Peleg and B. Porat, “The achievable accuracy in estimating the instantaneous phase and frequency of a constant amplitude signal,” IEEE Trans. Signal Process. 41, 2216–2224 (1993).
    [CrossRef]
  15. Q. Kemao, W. Gao, and H. Wang, “Windowed Fourier filtered and quality guided phase unwrapping algorithm: On locally high-order polynomial phase,” Appl. Opt. 49, 1075–1079 (2010).
    [CrossRef]

2010 (2)

2009 (1)

2008 (1)

2007 (1)

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

2004 (1)

P. O’Shea, “A fast algorithm for estimating the parameters of a quadratic FM signal,” IEEE Trans. Signal Process. 52, 385–393 (2004).
[CrossRef]

1996 (1)

B. Friedlander and J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[CrossRef]

1993 (1)

S. Peleg and B. Porat, “The achievable accuracy in estimating the instantaneous phase and frequency of a constant amplitude signal,” IEEE Trans. Signal Process. 41, 2216–2224 (1993).
[CrossRef]

1991 (1)

S. Peleg and B. Porat, “Linear FM signal parameter estimation from discrete-time observations,” IEEE Trans. Aerosp. Electron. Syst. 27, 607–616 (1991).
[CrossRef]

1982 (1)

1945 (1)

C. R. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).

Cramer, H.

H. Cramer, Mathematical Methods of Statistics (Princeton University, 1999).

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics (Elsevier, 1988), Vol. XXVI, pp. 349–393.

Dunlop, D. D.

A. C. Tamhane and D. D. Dunlop, Statistics and Data Analysis: From Elementary to Intermediate (Prentice-Hall, 2000).

Francos, J. M.

B. Friedlander and J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[CrossRef]

Friedlander, B.

B. Friedlander and J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[CrossRef]

Gao, W.

Gorthi, S. S.

Ina, H.

Jueptner, W.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2005).

Kemao, Q.

Kobayashi, S.

O’Shea, P.

P. O’Shea, “A fast algorithm for estimating the parameters of a quadratic FM signal,” IEEE Trans. Signal Process. 52, 385–393 (2004).
[CrossRef]

Peleg, S.

S. Peleg and B. Porat, “The achievable accuracy in estimating the instantaneous phase and frequency of a constant amplitude signal,” IEEE Trans. Signal Process. 41, 2216–2224 (1993).
[CrossRef]

S. Peleg and B. Porat, “Linear FM signal parameter estimation from discrete-time observations,” IEEE Trans. Aerosp. Electron. Syst. 27, 607–616 (1991).
[CrossRef]

Porat, B.

S. Peleg and B. Porat, “The achievable accuracy in estimating the instantaneous phase and frequency of a constant amplitude signal,” IEEE Trans. Signal Process. 41, 2216–2224 (1993).
[CrossRef]

S. Peleg and B. Porat, “Linear FM signal parameter estimation from discrete-time observations,” IEEE Trans. Aerosp. Electron. Syst. 27, 607–616 (1991).
[CrossRef]

Rajshekhar, G.

Rao, C. R.

C. R. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).

Rastogi, P.

Schnars, U.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2005).

Takeda, M.

Tamhane, A. C.

A. C. Tamhane and D. D. Dunlop, Statistics and Data Analysis: From Elementary to Intermediate (Prentice-Hall, 2000).

Wang, H.

Appl. Opt. (2)

Bull. Calcutta Math. Soc. (1)

C. R. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).

IEEE Trans. Aerosp. Electron. Syst. (1)

S. Peleg and B. Porat, “Linear FM signal parameter estimation from discrete-time observations,” IEEE Trans. Aerosp. Electron. Syst. 27, 607–616 (1991).
[CrossRef]

IEEE Trans. Signal Process. (3)

B. Friedlander and J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[CrossRef]

P. O’Shea, “A fast algorithm for estimating the parameters of a quadratic FM signal,” IEEE Trans. Signal Process. 52, 385–393 (2004).
[CrossRef]

S. Peleg and B. Porat, “The achievable accuracy in estimating the instantaneous phase and frequency of a constant amplitude signal,” IEEE Trans. Signal Process. 41, 2216–2224 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (1)

Opt. Laser. Eng. (1)

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

Opt. Lett. (1)

Other (4)

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2005).

H. Cramer, Mathematical Methods of Statistics (Princeton University, 1999).

A. C. Tamhane and D. D. Dunlop, Statistics and Data Analysis: From Elementary to Intermediate (Prentice-Hall, 2000).

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics (Elsevier, 1988), Vol. XXVI, pp. 349–393.

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Figures (6)

Fig. 1.
Fig. 1.

Simulated wrapped phase maps. (a) c2=0.001, (b) c2=0.02.

Fig. 2.
Fig. 2.

Theoretical and experimental MSEs by the WFR algorithm with phase compensation. (a) Local frequency estimate, (b) phase estimate.

Fig. 3.
Fig. 3.

The influence of c2 in the WFR algorithm with phase compensation. (a) Local frequency estimate, (b) phase estimate.

Fig. 4.
Fig. 4.

The influence of window truncation, 6σ+1=61 versus 10σ+1=101.

Fig. 5.
Fig. 5.

Phase estimate by the WFR algorithm with phase compensation and WFF algorithm.

Fig. 6.
Fig. 6.

Comparison of phase estimate by the WFR algorithm with phase compensation, HAF, and CPF algorithms.

Tables (1)

Tables Icon

Table 1. Theoretical MSEs of Local Frequency and Phase Estimates by the WFR Algorithm with Phase Compensation and Their CRBs

Equations (66)

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f(x)=f0(x)+n(x),
f0(x)=bexp[jφ(x)],
φ(x)=c0+c1x+0.5c2x2,
ci=φ(i)(0),i=0,1,2.
ω(x)=dφdx=c1+c2x.
f(x)=f0(x),
Sf(u;ξ)=Sf0(u;ξ)=f(x)g(ux)exp(jξx)dx,
g(xu)=1(πσ2)1/4exp[(xu)22σ2],
Sf(u;ξ)=A(u;ξ)exp[jΦ(u;ξ)],
A(u;ξ)=b(4πσ21+σ4c22)1/4exp{σ2[ξω(u)]22(1+σ4c22)},
Φ(u;ξ)=φ(u)ξuσ4c2[ξω(u)]22(1+σ4c22)+0.5arctan(σ2c2).
ω^(u)=argmaxξ|Sf(u;ξ)|=argmaxξ|Sf(u;ξ)exp(jξu)|,
φ^(u)=angle{Sf[u;ω^(u)]exp[jω^(u)u]}0.5arctan(σ2c^2),
c^2=[ω^(u2)ω^(u1)]/(u2u1),
n(x)=nr(x)+jni(x),
Sf(u;ξ)=Sf0(u;ξ)+Sn(u;ξ),
Sn(u;ξ)=n(x)g(xu)exp(jξx)dx.
E[(δω)2]=Var(δω)=(1+σ4c22)5/28πσ3SNR,
δφ(δφ)1+(δφ)2+(δφ)3,
(δφ)1=Im{Sn[u;ω(u)]exp[jω(u)u]/V0},
(δφ)2=σ4c22(1+σ4c22)[δω(u)]2,
(δφ)3=σ2δc22(1+σ4c22),
V0=b(4πσ21+σ4c22)1/4exp{j[φ(u)+0.5arctan(σ2c2)]},
E(δφ)σc2(1+σ4c22)3/216πSNR,
E[(δφ)2](1+σ4c22)1/24πσSNR[1+k1(c2,SNR,σ)+k2(c2,σ,s)+k3(c2,σ,s,l)],
k1(c2,SNR,σ)=3σ3c22(1+σ4c22)5/264πSNR,
k2(c2,s,σ)=(1+σ4c22)2s2σ2{1[1s22σ2(1+σ4c22)]exp[s2(1+σ4c22)4σ2]},
k3(c2,σ,s,l)=14s{lexp[l2(1+σ4c22)4σ2](ls)exp[(ls)2(1+σ4c22)4σ2]},
E[(δφ)2](1+σ4c22)1/24πσSNR,
fN(Ω)=|gN(Ω)|2=gN(Ω)gN*(Ω),
δΩBA,
A=2Re[gN(Ω0)2gN*(Ω0)Ω2+gN(Ω0)ΩgN*(Ω0)Ω],
B=2Re[gN(Ω0)δgN*(v0)Ω+gN(Ω0)ΩδgN*(Ω0)].
E[(δΩ)2]E(B2)A2.
δω(u)BA=(1+σ4c22)5/4bσ2(4πσ2)1/4Im(exp{j[c00.5c2u2+0.5arctan(σ2c2)]}n*(x)g(xu)(xu)exp(jωx)dx),
Sf0(u;ω)=b(4πσ21+σ4c22)1/4exp{j[c00.5c2u2+0.5arctan(σ2c2)]},
Sf0(u;ω)ξ=juSf0(u;ω),
2Sf0*(u;ω)ξ2=Sf0*(u;ω)[u2+σ2(1+jσ2c2)],
Sn*(u;ω)=n*(x)g(ux)exp(jωx)dx,
Sn*(u;ω)ξ=jn*(x)g(ux)xexp(jωx)dx.
E[δω(u)]=0.
E[δω(u1)·δω(u2)](1+σ4c22)5/28πσ3SNR[1(1+σ4c22)s22σ2]exp[(1+σ4c22)s24σ2],
E{[δω(u1)]2}=E{[δω(u2)]2}=Vω(1+σ4c22)5/28πσ3SNR,
δc2=[δω(u2)δω(u1)]/(u2u1),
E[(δc2)2]2s2Vω{1[1s22σ2(σ4c22+1)]exp[s2(σ4c22+1)4σ2]}.
ξ=ω^(u)=c1+c2u+δω(u).
V^=Sf[u;ω^(u)]exp[jω^(u)u]=Sf0[u;ω^(u)]exp[jω^(u)u]+Sn[u;ω^(u)]exp[jω^(u)u].
V^Sf0[u;ω(u)]exp[jω(u)u]+Sn[u;ω(u)]exp[jω(u)u]0.5f0(x)(δω)2(ux)2exp[jω(ux)]g(ux)dx,
V^V0(1+τη),
V0=Sf0[u;ω(u)]exp[jω(u)u]=b(4πσ21+σ4c22)1/4exp{j[φ(u)+0.5arctan(σ2c2)]},
τ=Sn[u;ω(u)]exp[jω(u)u]/V0,
η=σ2(1+jσ2c2)2(1+σ4c22)[δω(u)]2.
log[C(1+D)]logC+D.
logV^log[b(4πσ21+σ4c22)1/4]+j[φ(u)+0.5arctan(σ2c2)]+τη.
Φ^=angle(V^)=Im[log(V^)]φ(u)+Im(τ)Im(η)+0.5arctan(σ2c2).
δφIm(τ)Im(η)+0.5arctan(σ2c2)0.5arctan(σ2c^2)=Im(τ)σ4c22(1+σ4c22)[δω(u)]20.5σ2δc21+σ4c22=(δφ)1+(δφ)2+(δφ)3,
E(δφ)E[(δφ)2]=E{σ4c22(1+σ4c22)[δω(u)]2}=σc2(1+σ4c22)3/216πSNR.
E[(δφ)2]=E[(δφ)12]+E[(δφ)22]+E[(δφ)32]+2E[(δφ)1(δφ)2]+2E[(δφ)2(δφ)3]+2E[(δφ)1(δφ)3].
E[(δφ)12]=Var[Im(τ)]=(1+σ4c22)1/24πσSNR.
E[(δφ)22]=σ8c224(1+σ4c22)2E{[δω(u)]4}σ8c224(1+σ4c22)23(E{[δω(u)]2})2=3σ2c22(1+σ4c22)3256π(SNR)2.
E[(δφ)32]=σ(1+σ4c22)1/216πs2SNR{1[1s22σ2(σ4c22+1)]exp[s2(σ4c22+1)4σ2]}.
E[(δφ)1(δφ)3]=(1+σ4c22)1/216πσsSNR{(uu1)exp[(uu1)2(1+σ4c22)4σ2](uu2)exp[(uu2)2(1+σ4c22)4σ2]}.
E[(δφ)2](1+σ4c22)1/24πσSNR[1+k1(c2,SNR,σ)+k2(c2,σ,s)+k3(c2,σ,s,l)],
k1(c2,SNR,σ)=3σ3c22(1+σ4c22)5/264πSNR,
k2(c2,σ,s)=(1+σ4c22)2s2σ2{1[1s22σ2(1+σ4c22)]exp[s2(1+σ4c22)4σ2]},
k3(c2,σ,s,l)=14s{lexp[l2(1+σ4c22)4σ2](ls)exp[(ls)2(1+σ4c22)4σ2]},

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