Abstract

For data processing in conventional phase shifting interferometry, Fourier transform, and least-squares-fitting techniques, a whole interferometric data series is required. We propose a new interferometric data processing methodology based on a recurrent nonlinear procedure. The signal value is predicted from the previous step to the next step, and the prediction error is used for nonlinear correction of an a priori estimate of the parameters phase, visibility, or frequency of interference fringes. Such a recurrent procedure is correct on the condition that the noise component be a Markov stochastic process realization. The accuracy and stability of the recurrent Markov nonlinear filtering algorithm were verified by computer simulations. It was discovered that the main advantages of the proposed methodology are dynamic data processing, phase error minimization, and high noise immunity against the influence of non-Gaussian noise correlated with the signal and the automatic solution of the phase unwrapping problem.

© 2000 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  27. I. P. Gurov, D. V. Sheynihovich, “Non-linear phase estimation by computer-aided Markov filtering method: accuracy investigations,” in Proceedings of the IEEE TENCON ’96 Conference, Perth, Australia (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 758–762.
  28. I. P. Gurov, D. V. Sheynikhovich, “Determination of phase characteristics of an interference pattern by the method of nonlinear Markov filtering,” Opt. Spectrosc. (USSR) 83, 137–142 (1997).

1997 (1)

I. P. Gurov, D. V. Sheynikhovich, “Determination of phase characteristics of an interference pattern by the method of nonlinear Markov filtering,” Opt. Spectrosc. (USSR) 83, 137–142 (1997).

1995 (2)

1994 (2)

1993 (1)

1992 (2)

C. T. Farrell, M. A. Player, “Phase step measurement and variable step algorithms in phase shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

R. Jozwicki, M. Kujawinska, L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

1991 (2)

I. P. Gurov, I. M. Nagibina, “The structure of multichannel interference measuring systems for precision control of the objects geometric characterization,” Izv. Vyssh. Uchebn. Zaved. Priborostr. 34, 59–66 (1991) (in Russian).

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

1990 (1)

D. K. Bowen, D. G. Chetwynd, D. R. Schwarzenberger, “Sub-nanometre displacements calibration using x-ray interferometry,” Meas. Sci. Technol. 1, 107–119 (1990).
[CrossRef]

1988 (1)

K. Creath, “Phase measurement interferometry technique,” Prog. Opt. 26, 349–383 (1988).
[CrossRef]

1987 (2)

1983 (2)

1982 (2)

1974 (1)

1960 (1)

R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction theory,” Basic Eng. 82, 35–40 (1960).
[CrossRef]

Bowen, D. K.

D. K. Bowen, D. G. Chetwynd, D. R. Schwarzenberger, “Sub-nanometre displacements calibration using x-ray interferometry,” Meas. Sci. Technol. 1, 107–119 (1990).
[CrossRef]

Brangaccio, D. J.

Bruning, J. H.

Bucy, R. S.

R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction theory,” Basic Eng. 82, 35–40 (1960).
[CrossRef]

Burow, R.

Chetwynd, D. G.

D. K. Bowen, D. G. Chetwynd, D. R. Schwarzenberger, “Sub-nanometre displacements calibration using x-ray interferometry,” Meas. Sci. Technol. 1, 107–119 (1990).
[CrossRef]

Creath, K.

K. Creath, “Phase measurement interferometry technique,” Prog. Opt. 26, 349–383 (1988).
[CrossRef]

de Groot, P. J.

Eiju, T.

Ellsner, K. E.

Farrant, D. I.

Farrell, C. T.

C. T. Farrell, M. A. Player, “Phase step measurement and variable step algorithms in phase shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

Gallagher, J. E.

Grevenkamp, J. E.

J. E. Grevenkamp, J. H. Bruning, “Phase-shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992).

Grzanna, J.

Gurov, I. P.

I. P. Gurov, D. V. Sheynikhovich, “Determination of phase characteristics of an interference pattern by the method of nonlinear Markov filtering,” Opt. Spectrosc. (USSR) 83, 137–142 (1997).

I. P. Gurov, I. M. Nagibina, “The structure of multichannel interference measuring systems for precision control of the objects geometric characterization,” Izv. Vyssh. Uchebn. Zaved. Priborostr. 34, 59–66 (1991) (in Russian).

I. P. Gurov, D. V. Sheynikhovich, “Noise-immune phase-shifting interferometric system based on Markov non-linear filtering method,” in Statistical and Stochastic Methods for Image Processing, E. R. Dougherty, F. Preteux, J. L. Davidson, eds., Proc. SPIE2823, 121–125 (1996).
[CrossRef]

I. P. Gurov, D. V. Sheynihovich, “Non-linear phase estimation by computer-aided Markov filtering method: accuracy investigations,” in Proceedings of the IEEE TENCON ’96 Conference, Perth, Australia (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 758–762.

Han, G. S.

Hariharan, P.

Herriott, D. R.

Hibino, K.

Ina, H.

Jordache, N.

Joshi, G. A.

Jozwicki, R.

R. Jozwicki, M. Kujawinska, L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

Kalman, R. E.

R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction theory,” Basic Eng. 82, 35–40 (1960).
[CrossRef]

Kim, S. W.

Kobayashi, S.

Kujawinska, M.

R. Jozwicki, M. Kujawinska, L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

Lai, G.

Larkin, K. G.

Merkel, K.

Mironov, M. A.

V. I. Tikhonov, M. A. Mironov, Markov Processes (Soviet Radio, Moscow, 1977) (in Russian).

M. S. Yarlykov, M. A. Mironov, Markov Theory of Stochastic Processes Estimation (Radio I svyaz, Moscow, 1993) (in Russian).

Nagibina, I. M.

I. P. Gurov, I. M. Nagibina, “The structure of multichannel interference measuring systems for precision control of the objects geometric characterization,” Izv. Vyssh. Uchebn. Zaved. Priborostr. 34, 59–66 (1991) (in Russian).

Nussbaumer, H. J.

H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithms (Springer-Verlag, Berlin, 1982).

Oreb, B. F.

Pandit, S. M.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Player, M. A.

C. T. Farrell, M. A. Player, “Phase step measurement and variable step algorithms in phase shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

Robinson, D. W.

Roddier, C.

Roddier, F.

Rosenfeld, D. P.

Salbut, L.

R. Jozwicki, M. Kujawinska, L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

Sarrel, Y.

Schwarzenberger, D. R.

D. K. Bowen, D. G. Chetwynd, D. R. Schwarzenberger, “Sub-nanometre displacements calibration using x-ray interferometry,” Meas. Sci. Technol. 1, 107–119 (1990).
[CrossRef]

Schwider, J.

Sheynihovich, D. V.

I. P. Gurov, D. V. Sheynihovich, “Non-linear phase estimation by computer-aided Markov filtering method: accuracy investigations,” in Proceedings of the IEEE TENCON ’96 Conference, Perth, Australia (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 758–762.

Sheynikhovich, D. V.

I. P. Gurov, D. V. Sheynikhovich, “Determination of phase characteristics of an interference pattern by the method of nonlinear Markov filtering,” Opt. Spectrosc. (USSR) 83, 137–142 (1997).

I. P. Gurov, D. V. Sheynikhovich, “Noise-immune phase-shifting interferometric system based on Markov non-linear filtering method,” in Statistical and Stochastic Methods for Image Processing, E. R. Dougherty, F. Preteux, J. L. Davidson, eds., Proc. SPIE2823, 121–125 (1996).
[CrossRef]

Spolaczyk, R.

Takeda, M.

Tikhonov, V. I.

V. I. Tikhonov, M. A. Mironov, Markov Processes (Soviet Radio, Moscow, 1977) (in Russian).

White, A. D.

Wyant, J. C.

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18, 65–71 (1982).

Yarlykov, M. S.

M. S. Yarlykov, M. A. Mironov, Markov Theory of Stochastic Processes Estimation (Radio I svyaz, Moscow, 1993) (in Russian).

Yatagai, T.

Appl. Opt. (7)

Basic Eng. (1)

R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction theory,” Basic Eng. 82, 35–40 (1960).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Priborostr. (1)

I. P. Gurov, I. M. Nagibina, “The structure of multichannel interference measuring systems for precision control of the objects geometric characterization,” Izv. Vyssh. Uchebn. Zaved. Priborostr. 34, 59–66 (1991) (in Russian).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Laser Focus (1)

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18, 65–71 (1982).

Meas. Sci. Technol. (2)

D. K. Bowen, D. G. Chetwynd, D. R. Schwarzenberger, “Sub-nanometre displacements calibration using x-ray interferometry,” Meas. Sci. Technol. 1, 107–119 (1990).
[CrossRef]

C. T. Farrell, M. A. Player, “Phase step measurement and variable step algorithms in phase shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

Opt. Eng. (1)

R. Jozwicki, M. Kujawinska, L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

Opt. Spectrosc. (USSR) (1)

I. P. Gurov, D. V. Sheynikhovich, “Determination of phase characteristics of an interference pattern by the method of nonlinear Markov filtering,” Opt. Spectrosc. (USSR) 83, 137–142 (1997).

Prog. Opt. (1)

K. Creath, “Phase measurement interferometry technique,” Prog. Opt. 26, 349–383 (1988).
[CrossRef]

Other (8)

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithms (Springer-Verlag, Berlin, 1982).

J. E. Grevenkamp, J. H. Bruning, “Phase-shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992).

E. Lloyd, ed. Statistics, Vol.6 of Handbook of Applicable Mathematics, Walter Ledermann, ed. (Wiley, Chichester, UK, 1984).

M. S. Yarlykov, M. A. Mironov, Markov Theory of Stochastic Processes Estimation (Radio I svyaz, Moscow, 1993) (in Russian).

V. I. Tikhonov, M. A. Mironov, Markov Processes (Soviet Radio, Moscow, 1977) (in Russian).

I. P. Gurov, D. V. Sheynikhovich, “Noise-immune phase-shifting interferometric system based on Markov non-linear filtering method,” in Statistical and Stochastic Methods for Image Processing, E. R. Dougherty, F. Preteux, J. L. Davidson, eds., Proc. SPIE2823, 121–125 (1996).
[CrossRef]

I. P. Gurov, D. V. Sheynihovich, “Non-linear phase estimation by computer-aided Markov filtering method: accuracy investigations,” in Proceedings of the IEEE TENCON ’96 Conference, Perth, Australia (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. 2, pp. 758–762.

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Figures (6)

Fig. 1
Fig. 1

Spectral density of the stochastic process described by Eq. (6).

Fig. 2
Fig. 2

Structure of the system for optimal nonlinear processing of phase shifting interferometric data.

Fig. 3
Fig. 3

Phase recovered by the MNLF algorithm [curve (a)], and the PSI method [curve (b)].

Fig. 4
Fig. 4

Typical phase error of the MNLF algorithm.

Fig. 5
Fig. 5

Histogram of the phase error (the estimated phase error is presented in Fig. 4).

Fig. 6
Fig. 6

Reconstruction of the phase function in the case of variable envelope and nonuniform background: (a) intensity distribution, (b) phase function, and (c) phase error.

Equations (32)

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S(x, y, Ө)=Sb(x, y)+Se(x, y)cos Φ(x, y, Өϕ)+Na(x, y),
Sk=S(xk, yk, Өk),
Skd=(Skd, Sk-1d ,, Sk-Nd)T
Smc=(S0mc, S1mc ,, Skmc ,, SKmc)T
Sk=Sbk+Sek cos(Φk+Nphk)+Nak,
dNph(x)dx=-αNph(x)+αw(x),
GNph(u)=α2N02(α2+u2).
Nphk=Nphk-1(1-α)Δxk+αwk,
Ξk=A[1+V exp(-Ck2)cos(Φk+Nphk)]+Nak.
Ө=[Φ, U, C, B, V, Nph]T
Ξk=Sk(xk, Өk)+Nak.
dӨdx=F(x, Ө)+G(x, Ө)N(x), M[N(x)]=0,
M[N(x)N(x+χ)]=N02δ(χ),
F(x, Ө)=[U, 0, B, 0, 0, -αNph],
G(x, Ө)=Gisa(6×6)matrix,
Gij=0,i, j=1, , 5,G66=α.
Өk=Өkk-1+Rkk-1Dk[Ξk-S(xk, Өkk-1)-B2/2]Lk-1,
Rk=Rkk-1(I-DkDkTRkk-1Lk-1),
Dk=Өkk-1TS(xk, Өkk-1)=-AVkk-1 sin[Φkk-1+(Nph)kk-1]00-AVkk-1 sin[Φkk-1+(Nph)kk-1].
p(x, Ө)t=T{p(x, Ө)}+2N0p(x, Ө)×{ξ(x)s(x, Ө)-s(x, Ө)Mps×[s(x, Ө)]-ξ(x)Mps[s(x, Ө)]+(Mps[s(x, Ө)])2},
T{p(x, Ө)}=-Ө[F(x, Ө)p(x, Ө)]+122Ө2×[N(x)p(x, Ө)].
Өk=Өkk-1+Rkk-1Өkk-1TST(xk, Өkk-1)×Lk-1[Өk-S(xk, Өkk-1)-12B2],
Rk=Rkk-1-Rkk-1Өkk-1TST(xk, Өkk-1)×Lk-1Өkk-1TST(xk, Өkk-1)TRkk-1,
B2=Rkk-1Өkk-1TTӨkk-1TS(xk, Өkk-1),
Lk=Na+Өkk-1TST(xk, Өkk-1)T×Rkk-1Өkk-1TST(xk, Өkk-1)+12B3,
[B3]ij=p,q,r,l=1n2si(xk, Ө)ӨpӨqrprrql2sj(xk,Ө)ӨrӨl.
Өkk-1=F(x, Өk-1),
Dk=Өkk-1TS(xk, Өkk-1)=(-A exp[-(Ckk-1)2]Vkk-1 sin[Φkk-1+(Nph)kk-1], 0, -ACkk-1 exp[-(Ckk-1)2]{1+Vkk-1×sin[Φkk-1+(Nph)kk-1]}, 0, 0, -A exp[-(Ckk-1)2]Vkk-1 sin[Φkk-1+(Nph)kk-1])T.
B2=i, j=16rij2S(xk, Өkk-1)ӨiӨj,
B2=2SΦ2(r11+2r16+r66)+2SΦC(2r13+2r63)+2SC2r33.
B2=2SΦΦ(r11+2r14+r44),
B3=2SΦΦ2(r11+2r14+r44)2.

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