Abstract

A subspace-based method is applied to phase shifting interferometry for obtaining in real time values of phase shifts between data frames at each pixel point. A generalized phase extraction algorithm then allows for computing the phase distribution. The method is applicable to spherical beams and is capable of handling nonsinusoidal waveforms in an effective manner. Numerical simulations demonstrate phase measurement with high accuracy even in the presence of noise.

© 2005 Optical Society of America

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References

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Appl. Opt. (2)

ICASSP 1979 (1)

G. Bienvenu, “Influence of the spatial coherence of the background noise on high resolution passive methods,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Washington, DC, 306-309 (1979)

ICASSP 1983 (1)

A. J. Barabell, “Improving the resolution performance of eigenstructure-based direction-finding algorithms,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Boston, MA, 336-339 (1983).

IEEE Proceedings-F (1993) (1)

T. Söderström and P. Stoica, “Accuracy of higher-order Yule-Walker methods for frequency estimation of complex sine waves,” IEEE Proceedings-F 140, 71-80 (1993).

IEEE Trans. Acoustics, Speech, Signal Pr (1)

J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing 36, 1846-1853 (1988)
[CrossRef]

IEEE Transactions on Signal Processing (1)

B. D. Rao and K. V. S. Hari, “Weighted subspace methods and spatial smoothing: analysis and comparison,” IEEE Transactions on Signal Processing 41, 788-803 (1993).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Express (1)

Proceedings RADC 1979 (1)

R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” in Proceedings RADC, Spectral Estimation Workshop, Rome, NY, (243-258) 1979.

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Figures (4)

Fig. 1.
Fig. 1.

Plot of the diagonal of S versus frequency for noiseless signal and signal with SNR=10 dB.

Fig. 2.
Fig. 2.

Plots of phase step values α (in degrees) obtained using forward-backward approach at an arbitrary pixel point with different values of N and m.

Fig. 3.
Fig. 3.

Plot shows typical wrapped phase φ (in radians) for phase step values obtained from Fig. 2(d) for 30dB noise.

Fig. 4.
Fig. 4.

Plot shows typical absolute error obtained in computation of phase φ (in radians) for phase step values determined from Fig. 2(d) for 30dB noise.

Equations (21)

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I ( t ) = I d c + k = 1 κ a k e i k φ k u k t + k = 1 κ a k * e i k φ k * ( u k * ) t + η ( t )
for   t = 0 , 1 , m , , N 1
r ( p ) = E [ I ( t ) I * ( t p ) ] = n = 0 2 κ A n 2 e i ω n p + σ 2 δ p , 0
R I = E [ I c ( t ) I ( t ) ] = [ r ( 0 ) r * ( 1 ) . . r * ( m 1 ) r ( 1 ) . . . . . . . . . . . . . r * ( 1 ) r ( m 1 ) . . . r ( 0 ) ]
R I = APA c R s + σ 2 I R ε
P = [ A 0 2 0 . 0 0 A 1 2 . . . . . . 0 . . A 2 κ 2 ]
R I G = G [ λ n + 1 0 . . 0 0 λ n + 2 . . . . . . . . . . . . 0 0 0 . . λ m ] = σ 2 G = APA c G + σ 2 G
a T ( z 1 ) G ̂ G ̂ c a ( z ) = 0
φ ( x , y ) = 2 π λ ( x x ) 2 + ( x y ) 2
R ̂ I = 1 2 N t = m N [ [ I * ( t 1 ) I * ( t 2 ) . . I * ( t m ) ] [ I ( t 1 ) I ( t 2 ) . . I ( t m ) ] + [ I * ( t m ) . . I * ( t 2 ) I * ( t 1 ) ] [ I ( t m ) . . I ( t 2 ) I ( t 1 ) ] ]
[ e i κ α 0 e i κ α 0 i ( κ 1 ) α 0 1 e i κ α 1 e i κ α 1 . 1 . . . . e i κ α ( N 1 ) . . 1 ] [ κ κ * . I dc ] = [ I 0 I 1 . I N 1 ]
I ( t ) = I dc + k = 1 κ a k e ik φ e i α kt + k = 1 κ a k e ik φ e i α kt + η ( t ) ;
r ( p ) = E { I ( t ) I * ( t p ) }
I ( t ) = I dc + a 1 e i φ e i α t + a 1 e i φ e i α t + η ( t )
I * ( t p ) = I dc + a 1 e i φ e i α ( t p ) + a 1 e i φ e i α ( t p ) + η * ( t p )
r ( p ) = E { I ( t ) I * ( t p ) } = E { I dc 2 + I dc a 1 e i φ e i α t + I dc a 1 e i φ e i α t + e i α p ( a 1 2 + I dc a 1 e i φ e i α t + a 1 2 e 2 i φ e 2 i α t ) + e i α p ( a 1 2 + I dc a 1 e i φ e i α t + a 1 2 e 2 i φ e 2 i α t ) + η ( t ) η * ( t p ) }
r ( p ) = E { I dc 2 + c 1 + e i α p ( a 1 2 + c 2 ) + e i α p ( a 1 2 + c 3 ) + η ( t ) η * ( t p ) }
r ( p ) = A 0 2 + A 1 2 e i α p + A 2 2 e i α p + σ 2 δ p , 0
E { η ( k ) η * ( j ) } = σ 2 δ k , j E { η ( k ) η ( k ) } = 0 }
0 2 π e i ψ d ψ = 0
r ( p ) = E [ I ( t ) I * ( t p ) ] = n = 0 2 κ A n 2 e i ω n p + σ 2 δ p , 0

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