Abstract

Interferometers with low-coherence illumination allow noncontact measurement of rough-surface relief with a wide range of measurement definition by locating the visibility maxima of interference fringes. The problem is light scattering by the surface to be measured, which can cause distortion of low-coherence interferometric signals. We propose to use a stochastic fringe model and a Kalman filtering method for processing noisy low-coherence fringes dynamically. Prediction of the fringe’s signal value at each discretization step is based on all the information available before this step; the prediction error is used for dynamic correction of the estimates of the fringe envelope and phase. The advantages of the Kalman filtering method consist in its immunity to noise, optimal fringe evaluation, and data-processing speed.

© 2004 Optical Society of America

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References

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  1. T. Dresel, G. Häusler, H. Ventzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
    [CrossRef] [PubMed]
  2. T. Yoshimura, K. Kida, N. Masazumi, “Development of an image-processing system for a low coherence interferometer,” Opt. Commun. 117, 207–212 (1995).
    [CrossRef]
  3. L. Deck, P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33, 7334–7338 (1994).
    [CrossRef] [PubMed]
  4. J. G. Fujimoto, C. Pitris, S. A. Boppart, M. E. Bresinski, “Optical coherence tomography: an emerging technology for biomedical imaging and optical biopsy,” Neoplasia 2, 9–25 (2000).
    [CrossRef] [PubMed]
  5. A. F. Fercher, “Optical coherence tomography,” J. Biomed. Opt. 1, 157–173 (1996).
    [CrossRef] [PubMed]
  6. Q. Wang, Y. N. Ning, A. W. Palmer, K. T. V. Grattan, “Central fringe identification in a white light interferometer using a multi-stage-squaring signal processing scheme,” Opt. Commun. 117, 241–244 (1995).
    [CrossRef]
  7. I. Gurov, D. Sheynihovich, “Interferometric data analysis based on Markov nonlinear filtering methodology,” J. Opt. Soc. Am. A 17, 21–27 (2000).
    [CrossRef]
  8. I. Gurov, A. Djabiev, “3D surface reconstruction by vector nonlinear Markov filtering of white-light interference fringe visibility,” in Optical Biopsies and Microscopic Techniques III, I. J. Bigio, H. Schneckenburger, J. Slavik, K. Svanberg, P. M. Viallet, eds., Proc. SPIE3568, 178–184 (1999).
    [CrossRef]
  9. A. Zakharov, M. Volkov, I. Gurov, V. Temnov, K. Sokolovski-Tinten, D. von der Linde, “Interferometric diagnostics of ablation craters created by the femtosecond laser pulses,” J. Opt. Technol. (English-language translation of Opticheskii Zhurnal) 69, 478–482 (2002).
    [CrossRef]
  10. E. Alarousu, I. Gurov, J. Hast, R. Myllylä, T. Prykäri, A. Zakharov, “Diagnostics of internal random structure of wood fiber tissue by the dynamic stochastic fringe processing method,” in Proceedings of the International Workshop on Low-Coherence Interferometry, Spectroscopy and Optical Coherence Tomography [Saint Petersburg Institute of Fine Mechanics and Optics (Technical University), Saint Petersburg, Russia, 2002], pp. 16–25.
  11. I. Gurov, P. Hlubina, “Analysis of spectral modulated interferograms by recurrence nonlinear filtering method,” in Laser Optics 2000: Control of Laser Beam Characteristics and Nonlinear Methods for Wavefront Control, L. N. Soms, V. Sherstobitov, eds., Proc. SPIE4353, 281–286 (2001).
    [CrossRef]
  12. R. E. Kalman, “A new approach to linear filtering and prediction problems,” Trans. ASME 82, 35–45 (1960).
    [CrossRef]
  13. A. Lloyd, W. Ledermann, eds. Handbook of Applicable Mathematics, Vol. 6 (Statistics), (Wiley-Interscience, Chichester, UK, 1984), Chap. 20.
  14. I. Gurov, J. Vozniuk, “Rough surface shape retrieval in a fringe projection technique by the image enhancement and fringe tracing method,” in Proceedings of QCAV’2001, International Conference on Quality Control by Artificial Vision (Cépaduès-Éditions, Toulouse, France, 2001), Vol. 1, pp. 79–84.
  15. J. W. Goodman, “Statistical properties of laser speckle patterns” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–77.
  16. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968).

2002 (1)

2000 (2)

I. Gurov, D. Sheynihovich, “Interferometric data analysis based on Markov nonlinear filtering methodology,” J. Opt. Soc. Am. A 17, 21–27 (2000).
[CrossRef]

J. G. Fujimoto, C. Pitris, S. A. Boppart, M. E. Bresinski, “Optical coherence tomography: an emerging technology for biomedical imaging and optical biopsy,” Neoplasia 2, 9–25 (2000).
[CrossRef] [PubMed]

1996 (1)

A. F. Fercher, “Optical coherence tomography,” J. Biomed. Opt. 1, 157–173 (1996).
[CrossRef] [PubMed]

1995 (2)

Q. Wang, Y. N. Ning, A. W. Palmer, K. T. V. Grattan, “Central fringe identification in a white light interferometer using a multi-stage-squaring signal processing scheme,” Opt. Commun. 117, 241–244 (1995).
[CrossRef]

T. Yoshimura, K. Kida, N. Masazumi, “Development of an image-processing system for a low coherence interferometer,” Opt. Commun. 117, 207–212 (1995).
[CrossRef]

1994 (1)

1992 (1)

1960 (1)

R. E. Kalman, “A new approach to linear filtering and prediction problems,” Trans. ASME 82, 35–45 (1960).
[CrossRef]

Alarousu, E.

E. Alarousu, I. Gurov, J. Hast, R. Myllylä, T. Prykäri, A. Zakharov, “Diagnostics of internal random structure of wood fiber tissue by the dynamic stochastic fringe processing method,” in Proceedings of the International Workshop on Low-Coherence Interferometry, Spectroscopy and Optical Coherence Tomography [Saint Petersburg Institute of Fine Mechanics and Optics (Technical University), Saint Petersburg, Russia, 2002], pp. 16–25.

Boppart, S. A.

J. G. Fujimoto, C. Pitris, S. A. Boppart, M. E. Bresinski, “Optical coherence tomography: an emerging technology for biomedical imaging and optical biopsy,” Neoplasia 2, 9–25 (2000).
[CrossRef] [PubMed]

Bresinski, M. E.

J. G. Fujimoto, C. Pitris, S. A. Boppart, M. E. Bresinski, “Optical coherence tomography: an emerging technology for biomedical imaging and optical biopsy,” Neoplasia 2, 9–25 (2000).
[CrossRef] [PubMed]

de Groot, P.

Deck, L.

Djabiev, A.

I. Gurov, A. Djabiev, “3D surface reconstruction by vector nonlinear Markov filtering of white-light interference fringe visibility,” in Optical Biopsies and Microscopic Techniques III, I. J. Bigio, H. Schneckenburger, J. Slavik, K. Svanberg, P. M. Viallet, eds., Proc. SPIE3568, 178–184 (1999).
[CrossRef]

Dresel, T.

Fercher, A. F.

A. F. Fercher, “Optical coherence tomography,” J. Biomed. Opt. 1, 157–173 (1996).
[CrossRef] [PubMed]

Fujimoto, J. G.

J. G. Fujimoto, C. Pitris, S. A. Boppart, M. E. Bresinski, “Optical coherence tomography: an emerging technology for biomedical imaging and optical biopsy,” Neoplasia 2, 9–25 (2000).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–77.

Grattan, K. T. V.

Q. Wang, Y. N. Ning, A. W. Palmer, K. T. V. Grattan, “Central fringe identification in a white light interferometer using a multi-stage-squaring signal processing scheme,” Opt. Commun. 117, 241–244 (1995).
[CrossRef]

Gurov, I.

A. Zakharov, M. Volkov, I. Gurov, V. Temnov, K. Sokolovski-Tinten, D. von der Linde, “Interferometric diagnostics of ablation craters created by the femtosecond laser pulses,” J. Opt. Technol. (English-language translation of Opticheskii Zhurnal) 69, 478–482 (2002).
[CrossRef]

I. Gurov, D. Sheynihovich, “Interferometric data analysis based on Markov nonlinear filtering methodology,” J. Opt. Soc. Am. A 17, 21–27 (2000).
[CrossRef]

I. Gurov, J. Vozniuk, “Rough surface shape retrieval in a fringe projection technique by the image enhancement and fringe tracing method,” in Proceedings of QCAV’2001, International Conference on Quality Control by Artificial Vision (Cépaduès-Éditions, Toulouse, France, 2001), Vol. 1, pp. 79–84.

I. Gurov, A. Djabiev, “3D surface reconstruction by vector nonlinear Markov filtering of white-light interference fringe visibility,” in Optical Biopsies and Microscopic Techniques III, I. J. Bigio, H. Schneckenburger, J. Slavik, K. Svanberg, P. M. Viallet, eds., Proc. SPIE3568, 178–184 (1999).
[CrossRef]

I. Gurov, P. Hlubina, “Analysis of spectral modulated interferograms by recurrence nonlinear filtering method,” in Laser Optics 2000: Control of Laser Beam Characteristics and Nonlinear Methods for Wavefront Control, L. N. Soms, V. Sherstobitov, eds., Proc. SPIE4353, 281–286 (2001).
[CrossRef]

E. Alarousu, I. Gurov, J. Hast, R. Myllylä, T. Prykäri, A. Zakharov, “Diagnostics of internal random structure of wood fiber tissue by the dynamic stochastic fringe processing method,” in Proceedings of the International Workshop on Low-Coherence Interferometry, Spectroscopy and Optical Coherence Tomography [Saint Petersburg Institute of Fine Mechanics and Optics (Technical University), Saint Petersburg, Russia, 2002], pp. 16–25.

Hast, J.

E. Alarousu, I. Gurov, J. Hast, R. Myllylä, T. Prykäri, A. Zakharov, “Diagnostics of internal random structure of wood fiber tissue by the dynamic stochastic fringe processing method,” in Proceedings of the International Workshop on Low-Coherence Interferometry, Spectroscopy and Optical Coherence Tomography [Saint Petersburg Institute of Fine Mechanics and Optics (Technical University), Saint Petersburg, Russia, 2002], pp. 16–25.

Häusler, G.

Hlubina, P.

I. Gurov, P. Hlubina, “Analysis of spectral modulated interferograms by recurrence nonlinear filtering method,” in Laser Optics 2000: Control of Laser Beam Characteristics and Nonlinear Methods for Wavefront Control, L. N. Soms, V. Sherstobitov, eds., Proc. SPIE4353, 281–286 (2001).
[CrossRef]

Kalman, R. E.

R. E. Kalman, “A new approach to linear filtering and prediction problems,” Trans. ASME 82, 35–45 (1960).
[CrossRef]

Kida, K.

T. Yoshimura, K. Kida, N. Masazumi, “Development of an image-processing system for a low coherence interferometer,” Opt. Commun. 117, 207–212 (1995).
[CrossRef]

Masazumi, N.

T. Yoshimura, K. Kida, N. Masazumi, “Development of an image-processing system for a low coherence interferometer,” Opt. Commun. 117, 207–212 (1995).
[CrossRef]

Myllylä, R.

E. Alarousu, I. Gurov, J. Hast, R. Myllylä, T. Prykäri, A. Zakharov, “Diagnostics of internal random structure of wood fiber tissue by the dynamic stochastic fringe processing method,” in Proceedings of the International Workshop on Low-Coherence Interferometry, Spectroscopy and Optical Coherence Tomography [Saint Petersburg Institute of Fine Mechanics and Optics (Technical University), Saint Petersburg, Russia, 2002], pp. 16–25.

Ning, Y. N.

Q. Wang, Y. N. Ning, A. W. Palmer, K. T. V. Grattan, “Central fringe identification in a white light interferometer using a multi-stage-squaring signal processing scheme,” Opt. Commun. 117, 241–244 (1995).
[CrossRef]

Palmer, A. W.

Q. Wang, Y. N. Ning, A. W. Palmer, K. T. V. Grattan, “Central fringe identification in a white light interferometer using a multi-stage-squaring signal processing scheme,” Opt. Commun. 117, 241–244 (1995).
[CrossRef]

Pitris, C.

J. G. Fujimoto, C. Pitris, S. A. Boppart, M. E. Bresinski, “Optical coherence tomography: an emerging technology for biomedical imaging and optical biopsy,” Neoplasia 2, 9–25 (2000).
[CrossRef] [PubMed]

Prykäri, T.

E. Alarousu, I. Gurov, J. Hast, R. Myllylä, T. Prykäri, A. Zakharov, “Diagnostics of internal random structure of wood fiber tissue by the dynamic stochastic fringe processing method,” in Proceedings of the International Workshop on Low-Coherence Interferometry, Spectroscopy and Optical Coherence Tomography [Saint Petersburg Institute of Fine Mechanics and Optics (Technical University), Saint Petersburg, Russia, 2002], pp. 16–25.

Sheynihovich, D.

Sokolovski-Tinten, K.

Temnov, V.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968).

Ventzke, H.

Volkov, M.

von der Linde, D.

Vozniuk, J.

I. Gurov, J. Vozniuk, “Rough surface shape retrieval in a fringe projection technique by the image enhancement and fringe tracing method,” in Proceedings of QCAV’2001, International Conference on Quality Control by Artificial Vision (Cépaduès-Éditions, Toulouse, France, 2001), Vol. 1, pp. 79–84.

Wang, Q.

Q. Wang, Y. N. Ning, A. W. Palmer, K. T. V. Grattan, “Central fringe identification in a white light interferometer using a multi-stage-squaring signal processing scheme,” Opt. Commun. 117, 241–244 (1995).
[CrossRef]

Yoshimura, T.

T. Yoshimura, K. Kida, N. Masazumi, “Development of an image-processing system for a low coherence interferometer,” Opt. Commun. 117, 207–212 (1995).
[CrossRef]

Zakharov, A.

A. Zakharov, M. Volkov, I. Gurov, V. Temnov, K. Sokolovski-Tinten, D. von der Linde, “Interferometric diagnostics of ablation craters created by the femtosecond laser pulses,” J. Opt. Technol. (English-language translation of Opticheskii Zhurnal) 69, 478–482 (2002).
[CrossRef]

E. Alarousu, I. Gurov, J. Hast, R. Myllylä, T. Prykäri, A. Zakharov, “Diagnostics of internal random structure of wood fiber tissue by the dynamic stochastic fringe processing method,” in Proceedings of the International Workshop on Low-Coherence Interferometry, Spectroscopy and Optical Coherence Tomography [Saint Petersburg Institute of Fine Mechanics and Optics (Technical University), Saint Petersburg, Russia, 2002], pp. 16–25.

Appl. Opt. (2)

J. Biomed. Opt. (1)

A. F. Fercher, “Optical coherence tomography,” J. Biomed. Opt. 1, 157–173 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

J. Opt. Technol. (1)

Neoplasia (1)

J. G. Fujimoto, C. Pitris, S. A. Boppart, M. E. Bresinski, “Optical coherence tomography: an emerging technology for biomedical imaging and optical biopsy,” Neoplasia 2, 9–25 (2000).
[CrossRef] [PubMed]

Opt. Commun. (2)

T. Yoshimura, K. Kida, N. Masazumi, “Development of an image-processing system for a low coherence interferometer,” Opt. Commun. 117, 207–212 (1995).
[CrossRef]

Q. Wang, Y. N. Ning, A. W. Palmer, K. T. V. Grattan, “Central fringe identification in a white light interferometer using a multi-stage-squaring signal processing scheme,” Opt. Commun. 117, 241–244 (1995).
[CrossRef]

Trans. ASME (1)

R. E. Kalman, “A new approach to linear filtering and prediction problems,” Trans. ASME 82, 35–45 (1960).
[CrossRef]

Other (7)

A. Lloyd, W. Ledermann, eds. Handbook of Applicable Mathematics, Vol. 6 (Statistics), (Wiley-Interscience, Chichester, UK, 1984), Chap. 20.

I. Gurov, J. Vozniuk, “Rough surface shape retrieval in a fringe projection technique by the image enhancement and fringe tracing method,” in Proceedings of QCAV’2001, International Conference on Quality Control by Artificial Vision (Cépaduès-Éditions, Toulouse, France, 2001), Vol. 1, pp. 79–84.

J. W. Goodman, “Statistical properties of laser speckle patterns” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–77.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968).

E. Alarousu, I. Gurov, J. Hast, R. Myllylä, T. Prykäri, A. Zakharov, “Diagnostics of internal random structure of wood fiber tissue by the dynamic stochastic fringe processing method,” in Proceedings of the International Workshop on Low-Coherence Interferometry, Spectroscopy and Optical Coherence Tomography [Saint Petersburg Institute of Fine Mechanics and Optics (Technical University), Saint Petersburg, Russia, 2002], pp. 16–25.

I. Gurov, P. Hlubina, “Analysis of spectral modulated interferograms by recurrence nonlinear filtering method,” in Laser Optics 2000: Control of Laser Beam Characteristics and Nonlinear Methods for Wavefront Control, L. N. Soms, V. Sherstobitov, eds., Proc. SPIE4353, 281–286 (2001).
[CrossRef]

I. Gurov, A. Djabiev, “3D surface reconstruction by vector nonlinear Markov filtering of white-light interference fringe visibility,” in Optical Biopsies and Microscopic Techniques III, I. J. Bigio, H. Schneckenburger, J. Slavik, K. Svanberg, P. M. Viallet, eds., Proc. SPIE3568, 178–184 (1999).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic of a low-coherence interferometer.

Fig. 2
Fig. 2

Normalized low-coherence fringe signal.

Fig. 3
Fig. 3

(a) Fringe envelope realizations and (b) unwrapped phase realizations modeled by Eq. (7).

Fig. 4
Fig. 4

Schematic of the interferometric setup.

Fig. 5
Fig. 5

Initial 3-D rough-surface relief presented as (a) a gray-level map and (b) a 3-D relief reconstruction shown as a convex form.

Fig. 6
Fig. 6

Low-coherence images obtained for different reference-mirror positions in the schematic of Fig. 4.

Fig. 7
Fig. 7

Dynamic fringe-envelope estimates with (a) positive and (b) negative envelope sign.

Fig. 8
Fig. 8

(a) Surface-relief reconstruction and (b) relief reconstruction error presented by gray-level maps.

Fig. 9
Fig. 9

Experimental estimate of the probability density function of the surface-relief reconstruction error.

Fig. 10
Fig. 10

(a) Low-coherence fringes with variable frequency and (b) unwrapped fringe phase recovered dynamically by the discrete nonlinear Kalman filtering method.

Equations (44)

Equations on this page are rendered with MathJax. Learn more.

E1i(t)=ai(t)exp(j2πνit+ϕ1),
E2i(t)=rai(t+τ)exp[j2πνi(t+τ)],
I(τ)=|E1i(t)+E2i(t)|2i=I0[1+V(τ)cos(2πν0τ)],
γ(τ)=(1/I0)ai(t)ai(t+τ)i
s(z)=1+A(z)cos Φ(z),
s(z)=A(z)cos ϕ(z)+n(z),
dθ/dz=f(θ)+w(z),
dA/dz=-2σ(z-z0)A(z)+wA(z),
dϕ/dz=4πu0+wϕ(z),
dw/dz=-αw(z)+w0(z),
s(k)=H(k)θ(k)+n(k)
θ(k)=F(k-1)θ(k-1)+w(k).
H(k)=[cos ϕ(k)0],
F(k)=1-2σ(k-k0)Δz-αA001-αϕ,
θ(k)=F(k-1)θ(k-1)+P(k)[s(k)-H(k)F(k-1)θ(k-1)],
A(k)=A(k/k-1)+PA(k)[s(k)-A(k/k-1)cos ϕ(k-1)],
A(k/k-1)=[1-2σ(k-k0-1)Δz-αA]A(k-1),
ϕ(k)=(1-αϕ)ϕ(k-1)+Pϕ(k)[s(k)-A(k/k-1)cos ϕ(k-1)].
PA(k)=σ˜A2cos ϕ(k-1)[σ˜A2 cos2 ϕ(k-1)+σn2]-1=(σA2/σn2)cos ϕ(k-1),
A(k)=A(k/k-1)+P0 cos ϕ(k-1)×{s(k)-A(k/k-1)cos ϕ(k-1)}.
H(k)=[cos ϕ(k-1), -A(k)sin ϕ(k-1)],
θ=(A, δϕ)T,ϕ(k)=ϕ(k-1)+δϕ(k).
Pϕ(k)=-σ˜ϕ2A(k/k-1)sin ϕ(k-1)σ˜A2 cos2 ϕ(k-1)+σ˜ϕ2A2(k/k-1)sin2 ϕ(k-1)+σn2,
δz=(δΦ/2π)(λ/2),
ps(z, θ)z=T{ps(z, θ)}+[F(z, θ)-F(z, θ)]ps(z, θ),
F(z, θ)=(1/Gn)[sobs(z)-s(z, θ)]2
θ(k)=F(k-1)θ(k-1)+P(k)v(k),
R(k)=[I-P(k)H(k)]Rpr(k)[I-P(k)H(k)]T+P(k)RnP(k)T,
-Rpr(k)HT(k)+P(k)[H(k)Rpr(k)HT(k)+Rn]=0.
P(k)=Rpr(k)HT(k)[H(k)Rpr(k)HT(k)+Rn]-1,
Rpr(k)=F(k-1)R(k-1)FT(k-1)+Rw.
θ(k)=F(k-1)θ(k-1)+P(k)[s(k)-H(k)F(k-1)θ(k-1)],
R(k)=[I-P(k)H(k)]Rpr(k).
I(ξ, η, t)=|E(ξ, η, t)|2,
I(ξ, η)=i=1n|A(ξ, η, λi)|2,
A(ξ, η, λ)=Ar(λ)+(x0, y0)A(ξ, η, x0, y0, λ),
Ar(λ)=C(λ)exp[j(4π/λ)(f1+f2+Δh)],
A(ξ, η, x0, y0, λ)=A0(x0, y0, λ)r(x0, y0)Ω(ξ, η, x0, y0)×exp[j(2π/λ)d(ξ, η, x0, y0)],
A0(x0, y0, λ)=C(λ)exp[j(2π/λ)h(x0, y0)],
d(ξ, η, x0, y0)=d0+Δ1+d1+Δ2+d2,
x1=-x0f2f1+h+ξ(f1+h)f1f2h=-x0f1h-ξ(f1+h)f1f2h,
y1=-y0f2f1+h+η(f1+h)f1f2h=-y0f1h-η(f1+h)f1f2h,
x2=-x0f1h-ξf12-f2hf2h,
y2=-y0f1h-ηf12-f2hf2h.

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