Abstract

We have developed an accurate and robust phase-estimation method in phase-shifting electronic speckle pattern interferometry. Unlike other methods that assume a constant phase within a fitting window, our method treats the phase variation with a gradient. A cost function that can utilize the information of pixel positions is formulated on the basis of a least-squares criterion. Powell’s iteration method is applied to it to derive the phase and its gradient. An automatic consistency-checking routine and an algorithm that improves the initial guess of the iteration are developed for severe situations with large noise and steep phase variations.

© 2003 Optical Society of America

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References

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  1. J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131–150 (1997).
    [CrossRef]
  2. C. K. Hong, H. S. Ryu, H. C. Lim, “Least-squares fitting of the phase map obtained in phase-shifting electronic speckle pattern interferometry,” Opt. Lett. 20, 931–933 (1995).
    [CrossRef] [PubMed]
  3. H. A. Aebischer, S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
    [CrossRef]
  4. H. S. Ryu, C. K. Hong, “Maximum-likelihood phase estimation in phase-shifting electronic speckle pattern interferometry,” J. Opt. Soc. Am. A 14, 1051–1057 (1997).
    [CrossRef]
  5. J. L. Marroquin, M. Servin, R. Rodriguez Vera, “Adaptive quadrature filters for multiple phase-stepping images,” Opt. Lett. 23, 238–240 (1998).
    [CrossRef]
  6. P. Picart, J. C. Pascal, J. M. Breteau, “Systematic errors of phase-shifting speckle interferometry,” Appl. Opt. 40, 2107–2116 (2001).
    [CrossRef]
  7. H. van Brug, P. A. A. M. Somers, “Temporal phase unwrapping with two or four images per time frame: a comparison,” in Interferometry ’99: Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 358–365 (1999).
    [CrossRef]
  8. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroquin, R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
    [CrossRef]
  9. J. M. Huntley, H. Saldner, “Temporal phase unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [CrossRef] [PubMed]
  10. M. Owner-Peterson, P. Damgaard Jenson, “Computer-aided speckle pattern interferometry deformation analysis by fringe manipulation,” NonDestr. Test. Inst. 21, 422–426 (1988).
  11. M. Lehmann, “Decorrelation-induced phase errors in phase-shifting speckle interferometry,” Appl. Opt. 36, 3657–3667 (1997).
    [CrossRef] [PubMed]
  12. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK1988), pp. 408–420.
  13. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 51–54.

2001

1999

M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroquin, R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
[CrossRef]

H. A. Aebischer, S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

1998

1997

1995

1993

1988

M. Owner-Peterson, P. Damgaard Jenson, “Computer-aided speckle pattern interferometry deformation analysis by fringe manipulation,” NonDestr. Test. Inst. 21, 422–426 (1988).

Aebischer, H. A.

H. A. Aebischer, S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

Breteau, J. M.

Cuevas, F. J.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK1988), pp. 408–420.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 51–54.

Hong, C. K.

Huntley, J. M.

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

J. M. Huntley, H. Saldner, “Temporal phase unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[CrossRef] [PubMed]

Jenson, P. Damgaard

M. Owner-Peterson, P. Damgaard Jenson, “Computer-aided speckle pattern interferometry deformation analysis by fringe manipulation,” NonDestr. Test. Inst. 21, 422–426 (1988).

Lehmann, M.

Lim, H. C.

Malacara, D.

Marroquin, J. L.

Owner-Peterson, M.

M. Owner-Peterson, P. Damgaard Jenson, “Computer-aided speckle pattern interferometry deformation analysis by fringe manipulation,” NonDestr. Test. Inst. 21, 422–426 (1988).

Pascal, J. C.

Picart, P.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK1988), pp. 408–420.

Rodriguez-Vera, R.

Ryu, H. S.

Saldner, H.

Servin, M.

Somers, P. A. A. M.

H. van Brug, P. A. A. M. Somers, “Temporal phase unwrapping with two or four images per time frame: a comparison,” in Interferometry ’99: Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 358–365 (1999).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK1988), pp. 408–420.

van Brug, H.

H. van Brug, P. A. A. M. Somers, “Temporal phase unwrapping with two or four images per time frame: a comparison,” in Interferometry ’99: Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 358–365 (1999).
[CrossRef]

Vera, R. Rodriguez

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK1988), pp. 408–420.

Waldner, S.

H. A. Aebischer, S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

NonDestr. Test. Inst.

M. Owner-Peterson, P. Damgaard Jenson, “Computer-aided speckle pattern interferometry deformation analysis by fringe manipulation,” NonDestr. Test. Inst. 21, 422–426 (1988).

Opt. Commun.

H. A. Aebischer, S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

Opt. Lasers Eng.

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

Opt. Lett.

Other

H. van Brug, P. A. A. M. Somers, “Temporal phase unwrapping with two or four images per time frame: a comparison,” in Interferometry ’99: Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 358–365 (1999).
[CrossRef]

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK1988), pp. 408–420.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 51–54.

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Figures (7)

Fig. 1
Fig. 1

Sample distributions of the phasors. Each point represents a pixel in a 5×5 fitting window. (a) For the 2×4 phase-shifting scheme. z is the phasor to be estimated, and zk,l is the phasor of pixel (k, l). (b) For the 2×2 phase-shifting scheme. The inclination angle of the line is the half of the phase to be estimated.

Fig. 2
Fig. 2

Sample results of an out-of-plane deformation measurement by the SA and MPLS methods. (a) One of the four 640×480 images of the 2×2 phase-shifting scheme. (b) Phase map obtained by pixel-by-pixel calculation. (c) Phase map obtained by the SA method with m=5. (d) Phase map obtained by the MPLS method with m=5. (e), (f) Three-dimensional plots of the phase maps corresponding to the rectangular subregions marked in (c) and (d), respectively.

Fig. 3
Fig. 3

Phase maps obtained from the undersampled data. (a) By the SA method. (b) By the MPLS method. m=5.

Fig. 4
Fig. 4

One-dimensional illustration of the phase consistency. The solid circles and the arrows represent the phase and its gradient, respectively, obtained by Powell’s method. The open circles represent the phase value estimated from the phase and the gradient at the neighboring pixel.

Fig. 5
Fig. 5

Sample optimization process with the undersampled data. (a) Phase map obtained with the SA method. (b) Phase map obtained by Powell’s iteration with the phase map of (a) as the initial guess. (c) Consistency map of (b). The black area represents the inconsistent pixels. (d) Phase map obtained with the minima of the grid calculation at the inconsistent pixels.

Fig. 6
Fig. 6

(a) Noise dependence of the phase errors of the SA and MPLS methods. Noise magnitude of 1 is equal to the reference beam intensity. (b) Gradient dependence of the phase errors of the two methods.

Fig. 7
Fig. 7

Performance test of the MPLS method. (a) Synthesized model phase map composed of Gaussian functions. (b) Cosine fringes. (c) Pixel-by-pixel phase map. (d) Phase map obtained by the SA method. (e) Phase map obtained by the MPLS method. (f) Map of the number of iterations for (e).

Equations (19)

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An=Ir+Io+2IrIocos(ϕ+δn),
Bn=Ir+Io+2IrIocos(ϕ+ψ+δn),
ψ=arg(z)=tan-1(y/x),
x=(B1-B3)(A1-A3)+(B4-B2)(A4-A2),
y=(B4-B2)(A1-A3)-(B1-B3)(A4-A2),
z=16IrIo exp(iψ),
X=(A1-B2),
Y=(A2-B1),
Z=4IrIocos(ϕ+ψ/2)exp(iψ/2).
D(z)=k,l|zk,l-z|2=k,l(xk,l2+yk,l2+z2)-2zk,l(xk,l cos ψ+yk,l sin ψ),
S(ψ)=-k,l(xk,l cos ψ+yk,l sin ψ).
ψ=argk,lzk,l.
x=X2-Y2,
y=2XY,
z=16IrIo cos2(ϕ+ψ/2)exp(iψ),
ψ(k, l)=ψ+dk,lψ=ψ+kα+lβ
S(ψ, α, β)=-k,l[xk,l cos(ψ+kα+lβ)+yk,l sin(ψ+kα+lβ)].
Δpq12 [|W(ψp+dqpψp-ψq)|+|W(ψq+dpqψq-ψp)|]
pIo(Io)=(M/Io)MIoM-1 exp(-MIo/Io)Γ(M),

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