Abstract

The paper presents approaches based on traditional phase shifting, flexible least-squares, and signal processing methods in dual phase shifting interferometry primarily applied to holographic moiré for retrieving multiple phases. The study reveals that these methods cannot be applied straightforward to retrieve phase information and discusses the constraints associated with these methods. Since the signal processing method is the most efficient among these approaches, the paper discusses significant issues involved in the successful implementation of the concept. In this approach the knowledge of the pair of phase steps is of paramount interest. Thus the paper discusses the choice of the pair of phase steps that can be applied to the phase shifting devices (PZTs) in the presence of noise. In this context, we present a theoretical study using Cramér-Rao bound with respect to the selection of the pair of phase step values in the presence of noise.

© 2006 Optical Society of America

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References

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    [CrossRef]
  4. J. E. Grievenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).
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  7. A. Patil and P. Rastogi, "Estimation of multiple phases in holographic moiré in presence of harmonics and noise using minimum-norm algorithm," Opt. Express 13, 4070-4084 (2005), <a href= http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4070>http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4070</a>.
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  9. A. Patil and P. Rastogi, "Maximum-likelihood estimator for dual phase extraction in holographic moiré," Opt. Lett. 17, 2227-2229 (2005).
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    [CrossRef]
  15. Abhijit Patil, Benny Raphael, and Pramod Rastogi, "Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search," Opt. Lett. 12, 1381-1383 (2004).
    [CrossRef]

A Survey of Matrix Theory and Matrix Ine

M. Marcus and H. Minc, "Vandermonde Matrix," in A Survey of Matrix Theory and Matrix Inequalities (Dover, New York, 1992), pp. 15-16.

Appl. Opt.

J. Mod. Opt.

P. K. Rastogi, "Phase shifting four wave holographic interferometry," J. Mod. Opt. 39, 677-680 (1992).
[CrossRef]

J. Opt. Soc. Am. A

A. Patil and P. Rastogi, "Rotational invariance approach for the evaluation of multiple phases in interferometry in presence of nonsinusoidal waveforms and noise," J. Opt. Soc. Am. A 9, 1918-1928 (2005).
[CrossRef]

G. Lai and T. Yatagai, "Generalized phase-shifting interferometry," J. Opt. Soc. Am. A 8, 822-827 (1991).
[CrossRef]

Opt. Eng

J. E. Grievenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).

Opt. Express

Opt. Lett.

Abhijit Patil, Benny Raphael, and Pramod Rastogi, "Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search," Opt. Lett. 12, 1381-1383 (2004).
[CrossRef]

A. Patil and P. Rastogi, "Maximum-likelihood estimator for dual phase extraction in holographic moiré," Opt. Lett. 17, 2227-2229 (2005).
[CrossRef]

Other

<a href= http://www.mathworks.com/>http://www.mathworks.com/</a>.

<a href= http://mathworld.wolfram.com/VandermondeMatrix.html>http://mathworld.wolfram.com/VandermondeMatrix.html</a>.

K. M. Hoffman and R. Kunze, "Linear Algebra, 2nd ed.," (Prentice Hall, 1971).

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Figures (10)

Fig. 1.
Fig. 1.

Schematic of holographic moiré.

Fig. 2.
Fig. 2.

Moiré fringes (512 × 512 pixels) corresponding to Eq. (1). In (a) the random phases Φ ran1 = Φ ran2 = 0, while in (b)Φ ran1 = Φ ran2 ≠ 0.

Fig. 3.
Fig. 3.

The map corresponding to the wrapped sum of phases φ1 + φ2 obtained using Eq (3) for the fringe map in (a) Fig. 2(a) and (b) Fig. 2(b).

Fig. 4.
Fig. 4.

Typical plot for wrapped sum of phases φ12 (in radians) shown in Fig. 3(a). In plot the pixels in the central row from pixel (256, 0) till pixel (256, 127) is shown. In the plot, R1 shows the discontinuity in phase since φ1 - φ2 = ±2π.

Fig. 5.
Fig. 5.

The map corresponding to the wrapped sum of phases φ12 obtained using Eq. (3) for the fringe map in (a) Fig. 2(a) and (b) Fig. 2(b).

Fig. 6.
Fig. 6.

The map corresponding to the wrapped difference of phases φ12 obtained using Eq. (4) for the fringe map in (a) Fig. 2(a) and (b) Fig. 2(b). The phase map shows the recurrence of the carrier fringes while extracting the moiré.

Fig. 7.
Fig. 7.

Typical plot for wrapped difference of phases φ12 (in radians) shown in Fig. 6(a). In plot the pixels in the central row from pixel (256, 0) till pixel (256, 511) is shown. In the plot, R2 shows the discontinuity in phase since φ12 = ±2π.

Fig. 8.
Fig. 8.

The figure shows the final wrapped difference of phases obtained after removing the φ12 = ±2π discontinuity. Figure (a) shows the wrapped difference of phases for the fringe map in (a) Fig. 2(a) while (b) shows the wrapped phase for fringe in Fig. 2(b).

Fig. 9.
Fig. 9.

The plots show the MSE for β–α with respect to β–α as percentage of α, and SNR. The plots are shown for data frames N = 11, 15, and 20. The plots a, c, and e, are without the denoising procedure, while plots in b, d, and f are with the denoising step. In the plot the red shade represents the Cramér-Rao lower bound while the yellow shade represents the MSE obtained using the annihilation filter method. MSE is represented in log scale.

Fig. 10.
Fig. 10.

The plots show the bounds for retrieving the phase steps α and β for 0 to 70 dB SNR. The values of β in (b), (d), and (e) are selected as 1.2, 1.5, and 1.8 times of α = π/6, respectively. The line with circle dots shows the error bounds obtained using the annihilation method while the simple line represents the bounds given by the CRB.

Equations (31)

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I ( P ) = I dc { 1 + V [ cos φ 1 ( P ) + cos φ 2 ( P ) ] }
I ( P ) = I dc { 1 + V [ cos ( φ 1 ( P ) + α ) + cos ( φ 2 ( P ) + β ) ] }
Φ + = 2 tan 1 [ 2 ( I 2 I 4 ) 2 I 3 I 1 I 5 ] , for φ 1 ( P ) φ 2 ( P ) ( 2 χ + 1 ) π
Φ = 2 tan 1 [ 2 ( I 2 I 4 ) 2 I 3 I 1 I 5 ] , for φ 1 ( P ) φ 2 ( P ) ( 2 χ + 1 ) π
φ 1 n ʹ j ʹ = 2 π ( n ʹ n 0 ) 2 + ( j ʹ j 0 ) 2 λ 1 + Φ ran 1
φ 2 n ʹ j ʹ = 2 π ( n ʹ n 0 ) 2 + ( j ʹ p 0 ) 2 λ 2 + Φ ran 2
I n = I dc + γ cos α n + δ sin α n + μ cos β n + ν sin β n for n = 0,1,2 , N 1
E ( P ) = n = 0 N 1 [ I dc + γ cos α n + δ sin α n + μ cos β n + ν sin β n I n ] 2
A = [ N cos α n sin α n cos β n sin β n cos α n cos 2 α n cos α n sin α n cos α n cos β n cos α n sin β n sin α n cos α n sin α n sin 2 α n sin α n cos β n sin α n sin β n cos β n cos α n cos β n sin α n cos β n cos 2 β n sin β n cos β n sin β n cos α n sin β n sin α n sin β n sin β n cos β n sin 2 β n ]
Φ + = tan 1 δμ + ν γ δν μ γ
Φ = tan 1 γν + μδ γμ νδ
[ exp ( j α 0 ) exp ( j α 0 ) exp ( j β 0 ) exp ( j β ) 1 exp ( j α 1 ) exp ( j α 1 ) exp ( j β 1 ) exp ( j β 1 ) 1 exp ( j κ α N ) exp ( j κ α N ) exp ( j β N ) exp ( j β N ) 1 ] [ * * I dc ] = [ I 0 I 1 I N 1 ]
I n = I dc + k = 1 κ a k exp [ jk ( φ 1 + ) ] + k = 1 κ a k exp [ jk ( φ 1 + ) ]
k = 1 κ b k exp [ jk ( φ 2 + ) ] + k = 1 κ b k exp [ jk ( φ 2 + ) ] ;
I n = I dc + k = 1 κ ʹ k u k n + k = 1 κ ʹ k * ( u n * ) n + k = 1 κ k ϑ k n + k = 1 κ k * ( ϑ k * ) n
P ( z ) = ( z 1 ) k = 1 κ ( z u k ) ( z u k * ) ( z ϑ k ) ( z ϑ k * )
= k = 0 4 κ + 2 P k z k
k = 0 4 κ + 2 P k I n k = 0 m { 4 κ + 2 , 4 κ + 3 , , N 1 }
E { Ψ ̂ Ψ ̂ T } J 1
J = { [ Ψ log p ( I ʹ ) ] [ Ψ log p ( I ʹ ) ] T }
E { Ψ ̂ r 2 } J r , r 1 , for r = 1,2,3 , , 8
p I ʹ Ψ = ( 1 2 πσ ) N exp [ 1 2 σ 2 n = 0 N 1 ( I ʹ n I n ) 2 ]
log p = constant 1 2 σ 2 n = 0 N 1 ( I ʹ n I n ) 2
log p ψ r = 1 σ 2 n = 0 N 1 I n ψ r ( I ʹ n I n )
J r , s = E { 1 σ 4 n = 0 N 1 I n ψ r ( I ʹ n I n ) l = 0 N 1 I l ψ r ( I ʹ l I l ) }
J r , s = 1 σ 2 n = 0 N 1 I n ψ r I n ψ r
J = [ J I dc 1 , I dc 1 J I dc 1 , I dc 2 J I dc 1 , V 1 J I dc 1 , V 2 J I dc 1 , φ 1 J I dc 1 , φ 2 J I dc 1 , α J I dc 1 , β J I dc 1 , I dc 1 J I dc 2 , I dc 2 J I dc 2 , V 1 J I dc 2 , V 2 J I dc 2 , φ 1 J I dc 2 , φ 2 J I dc 2 , α J I dc , β J V 1 , I dc 1 J V 1 , I dc 2 J V 1 , V 1 J V 1 , V 2 J V 1 , φ 1 J V 1 , φ 2 J V 1 , α J V 1 , β J V 2 , I dc 1 J V 2 , I dc 2 J V 2 , V 1 J V 2 , V 2 J V 2 , φ 1 J V 2 , φ 2 J V 2 , α J V 2 β J V 1 , I dc 1 J φ 1 , I dc 2 J φ 1 , V 1 J φ 1 , V 2 J φ 1 , φ 1 J φ 1 , φ 2 J φ 1 , α J φ 1 , β J φ 2 , I dc 1 J φ 2 , I dc 2 J φ 2 , V 1 J φ 2 , V 2 J φ 2 , φ 1 J φ 2 , φ 2 J φ 2 , α J φ 2 , β J α , I dc 1 J α , I dc 2 J α , V 1 J α , V 2 J α , φ 1 J α , φ 2 J α , α J α , β J β , I dc 1 J β , I dc 2 J β , V 1 J β , V 2 J β , φ 1 J β , φ 2 J β , α J β , β ]
E [ q q ̂ 2 ] = E [ ( q q ̂ ) ( q q ̂ ) T ]
= E [ U T ( Ψ Ψ ̂ ) ( Ψ Ψ ̂ ) T U ]
= U T E [ ( Ψ Ψ ̂ ) ( Ψ Ψ ̂ ) T ] U
U T J 1 U

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