Abstract

The standard tool to estimate the phase of a sequence of phase-shifted interferograms is the Phase Shifting Algorithm (PSA). The performance of PSAs to a sequence of interferograms corrupted by non-white additive noise has not been reported before. In this paper we use the Frequency Transfer Function (FTF) of a PSA to generalize previous white additive noise analysis to non-white additive noisy interferograms. That is, we find the ensemble average and the variance of the estimated phase in a general PSA when interferograms corrupted by non-white additive noise are available. Moreover, for the special case of additive white-noise, and using the Parseval’s theorem, we show (for the first time in the PSA literature) a useful relationship of the PSA’s noise robustness; in terms of its FTF spectrum, and in terms of its coefficients. In other words, we find the PSA’s estimated phase variance, in the spectral space as well as in the PSA’s coefficients space.

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References

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2009

1997

Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt. 36(1), 271–276 (1997).
[CrossRef] [PubMed]

K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shiftinginterferometry,” Appl. Opt. 36, 2064–2093 (1997).

1995

C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. 12(9), 1997–2008 (1995).
[CrossRef]

1990

1987

1983

1982

1974

Brangaccio, D. J.

Brophy, C. P.

Bruning, J. H.

Burow, R.

Cywiak, M.

Eiju, T.

Elssner, K. E.

Estrada, J. C.

Gallagher, J. E.

Grzanna, J.

Hariharan, P.

Herriott, D. R.

Hibino, K.

K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shiftinginterferometry,” Appl. Opt. 36, 2064–2093 (1997).

Merkel, K.

Morgan, C. J.

Mosiño, J. F.

Oreb, B. F.

Quiroga, J. A.

Rathjen, C.

C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. 12(9), 1997–2008 (1995).
[CrossRef]

Rosenfeld, D. P.

Schwider, J.

Servin, M.

Spolaczyk, R.

Surrel, Y.

White, A. D.

Appl. Opt.

J. Opt. Soc. Am.

C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. 12(9), 1997–2008 (1995).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Other

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, Taylor & Francis CRC Press, 2th edition (2005).

A. Papoulis, Probability Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill Series in Electrical Engineering, 2001).

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Figures (4)

Fig. 1
Fig. 1

Convolution product S(t) of a sequence of N interferograms I(t) and a digital filter (the PSA) h(t) having also N samples. The middle-point occurs at tm = N-1.

Fig. 2
Fig. 2

Phasor representation of the output signal S and its corrupting noise nH . The noiseless signal (b/2)H(ω 0)exp[] is inclined an angle φ. The output-noise nH has a variance of σS 2 = E{nH 2 } and its random angle is uniformly distributed within [0,2π], so its phasor may point anywhere within the circle shown. The noisy estimated phase is φ-hat.

Fig. 3
Fig. 3

Least-Squares PSA’s impulse response hLS (t), and its spectral response HLS (ω). The frequency axis is normalized to the carrier ω 0 = 2π/6.

Fig. 4
Fig. 4

Impulse response hSH (t) and its spectral plot |HSH (ω)| of the 5-steps SH-PSA.

Equations (20)

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I ( x , y , k ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) + ω 0 k ] + n ( x , y , k ) ; k = 0 , ... , N 1 .
tan [ ϕ ^ ( x , y ) ] = k = 0 N 1 c k I ( k ) sin ( ω 0 k ) k = 0 N 1 c k I ( k ) cos ( ω 0 k ) .
S = k = 0 N 1 c k I ( k ) cos ( ω 0 k ) + i [ k = 0 N 1 c k I ( k ) sin ( ω 0 k ) ] = k = 0 N 1 c k I ( k ) e i ω 0 k .
S ( t ) = [ I ( t ) h ( t ) ] = { k = 0 N 1 I ( k ) δ ( t k ) } { k = 0 N 1 c k δ ( t k ) e i ω 0 k } .
I ( ω ) H ( ω ) = { a δ ( ω ) + b 2 δ ( ω ω 0 ) e i ϕ + b 2 δ ( ω + ω 0 ) e i ϕ + η ( ω ) } { k = 0 N 1 c k e i k ( ω ω 0 ) } ,
S = [ I ( t ) * h ( t ) ] | t m = N 1 = b 2 H ( ω 0 ) e i [ ϕ + ω 0 t m ] + n H ( t m ) .
E { n H 2 ( t m ) } = E { S 2 } = σ S 2 = 1 2 π π π | η ( ω ) | 2 | H ( ω ) | 2 d ω .
E { S } = b 2 H ( ω 0 ) e i ϕ ,     E { S 2 } = 1 2 π π π | η ( ω ) | 2 | H ( ω ) | 2 d ω .
E { ϕ ^ 2 } = σ ϕ ^ = 2 tan 1 ( σ S / 2 ( b / 2 ) H ( ω 0 ) ) σ S H ( ω 0 ) ( b / 2 ) ,
E { ϕ ^ } = ϕ , a n d E { ϕ ^ 2 } = 1 ( b / 2 ) 2 1 | H ( ω 0 ) | 2 [ 1 2 π π π | η ( ω ) | 2 | H ( ω ) | 2 d ω ] .
σ ϕ ^ 2 = σ n 2 / ( b / 2 ) 2 | H ( ω 0 ) | 2 [ 1 2 π π π | H ( ω ) | 2 d ω ] = σ n 2 / ( b / 2 ) 2 | k = 0 N 1 c k | 2 [ k = 0 N 1 | c k | 2 ] .
S = b e i ϕ + n H .
E { ϕ ^ 2 } = E { n 2 } b 2 1 2 π π π | H ( ω ) | 2 d ω = ( ο n 2 b 2 ) k = 0 N 1 | c k | 2 .
tan [ ϕ ^ L S ( x , y ) ] = [ k = 0 N 1 I ( k ) sin ( ω 0 k ) ] / [ k = 0 N 1 I ( k ) cos ( ω 0 k ) ] , ω 0 = 2 π N .
h L S ( t ) = k = 0 N 1 e i ω 0 k δ ( t k ) , ω 0 = 2 π N ,
H L S ( ω ) = F [ h L S ( t ) ] = k = 0 N 1 e i k ( ω ω 0 ) , ω 0 = 2 π N .
E { ϕ ^ L S 2 } = σ n 2 ( b / 2 ) 2 [ k = 0 N 1 | c k | 2 ] / | k = 0 N 1 c k | 2 = σ n 2 ( b / 2 ) 2 N N 2 = 1 N σ n 2 ( b / 2 ) 2 .
tan [ ϕ ^ S H ( x , y ) ] = 2 [ I ( t 1 ) I ( t 3 ) ] I ( t ) 2 I ( t 2 ) + I ( t 4 ) .
h S H ( t ) = δ ( t ) + 2 e i ω 0 δ ( t 1 ) + 2 e 2 i ω 0 δ ( t 2 ) + 2 e 3 i ω 0 δ ( t 3 ) + e 4 i ω 0 δ ( t 4 ) .
E { ϕ ^ S H 2 } = σ n 2 ( b / 2 ) 2 [ k = 0 4 | c k | 2 ] / | k = 0 4 c k | 2 = 14 64 σ n 2 ( b / 2 ) 2 = 0.91 ( 1 5 σ n 2 ( b / 2 ) 2 ) .

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