Abstract

The reconstruction of surfaces from speckle interferometry data is a demanding data-analysis task that involves edge detection, edge completion, and image reconstruction from noisy data. We present an approach that makes optimal use of the experimental information to minimize the hampering influence of the noise. The experimental data are then analyzed with a combination of wavelet transform and Bayesian probability theory. Nontrivial examples are presented to illustrate the proposed technique.

© 1999 Optical Society of America

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References

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  1. W. Osten, Digitale Verarbeitung und Auswertung von Interferenzbildern (Akademie Verlag, Berlin, 1991).
  2. D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), p. 194.
  3. K. A. Stetson, J. Wahid, P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 36, 4830–4838 (1997).
    [CrossRef] [PubMed]
  4. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1989).
  5. R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, UK, 1989).
    [CrossRef]
  6. A. W. Koch, M. Ruprecht, O. Toedter, G. Häusler, Optische Messtechnik an Technischen Oberflächen (Expert Verlag, Renningen, Germany, 1998).
  7. K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), p. 94.
  8. E. Berger, W. von der Linden, V. Dose, M. Ruprecht, A. W. Koch, “Approach for the evaluation of speckle deformation measurements by application of the wavelet transformation,” Appl. Opt. 36, 7455–7460 (1997).
    [CrossRef]
  9. M. Zeller, “Flinkes Wellenspiel,” C’T 11, 258–264 (1994).
  10. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
    [CrossRef]
  11. G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, Boston, Mass., 1994).
  12. J. Froment, S. Mallat, “Second generation compact image coding with wavelets,” in Wavelets—A Tutorial in Theory and Applications, C. K. Chui, ed. (Academic, New York, 1992), p. 655.
  13. R. D. Rosenkrantz, ed., E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics (Reidel, Dordrecht, The Netherlands, 1983).
  14. E. T. Jaynes, “Prior probabilities,” in E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics, R. Rosenkrantz, ed. (Reidel, Dordrecht, The Netherlands, 1983), p. 114.
  15. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  16. J. Meinguet, “Multivariate interpolation at arbitrary points made simple,” ZAMP 30, 292–304 (1979).
    [CrossRef]

1997 (2)

1994 (1)

M. Zeller, “Flinkes Wellenspiel,” C’T 11, 258–264 (1994).

1979 (1)

J. Meinguet, “Multivariate interpolation at arbitrary points made simple,” ZAMP 30, 292–304 (1979).
[CrossRef]

Berger, E.

Creath, K.

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), p. 94.

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1989).

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
[CrossRef]

Dose, V.

Froment, J.

J. Froment, S. Mallat, “Second generation compact image coding with wavelets,” in Wavelets—A Tutorial in Theory and Applications, C. K. Chui, ed. (Academic, New York, 1992), p. 655.

Gauthier, P.

Häusler, G.

A. W. Koch, M. Ruprecht, O. Toedter, G. Häusler, Optische Messtechnik an Technischen Oberflächen (Expert Verlag, Renningen, Germany, 1998).

Jaynes, E. T.

E. T. Jaynes, “Prior probabilities,” in E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics, R. Rosenkrantz, ed. (Reidel, Dordrecht, The Netherlands, 1983), p. 114.

Jones, R.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, UK, 1989).
[CrossRef]

Kaiser, G.

G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, Boston, Mass., 1994).

Koch, A. W.

Mallat, S.

J. Froment, S. Mallat, “Second generation compact image coding with wavelets,” in Wavelets—A Tutorial in Theory and Applications, C. K. Chui, ed. (Academic, New York, 1992), p. 655.

Meinguet, J.

J. Meinguet, “Multivariate interpolation at arbitrary points made simple,” ZAMP 30, 292–304 (1979).
[CrossRef]

Osten, W.

W. Osten, Digitale Verarbeitung und Auswertung von Interferenzbildern (Akademie Verlag, Berlin, 1991).

Robinson, D. W.

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), p. 194.

Ruprecht, M.

Stetson, K. A.

Toedter, O.

A. W. Koch, M. Ruprecht, O. Toedter, G. Häusler, Optische Messtechnik an Technischen Oberflächen (Expert Verlag, Renningen, Germany, 1998).

von der Linden, W.

Wahid, J.

Wykes, C.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, UK, 1989).
[CrossRef]

Zeller, M.

M. Zeller, “Flinkes Wellenspiel,” C’T 11, 258–264 (1994).

Appl. Opt. (2)

C’T (1)

M. Zeller, “Flinkes Wellenspiel,” C’T 11, 258–264 (1994).

ZAMP (1)

J. Meinguet, “Multivariate interpolation at arbitrary points made simple,” ZAMP 30, 292–304 (1979).
[CrossRef]

Other (12)

W. Osten, Digitale Verarbeitung und Auswertung von Interferenzbildern (Akademie Verlag, Berlin, 1991).

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), p. 194.

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1989).

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, UK, 1989).
[CrossRef]

A. W. Koch, M. Ruprecht, O. Toedter, G. Häusler, Optische Messtechnik an Technischen Oberflächen (Expert Verlag, Renningen, Germany, 1998).

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), p. 94.

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
[CrossRef]

G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, Boston, Mass., 1994).

J. Froment, S. Mallat, “Second generation compact image coding with wavelets,” in Wavelets—A Tutorial in Theory and Applications, C. K. Chui, ed. (Academic, New York, 1992), p. 655.

R. D. Rosenkrantz, ed., E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics (Reidel, Dordrecht, The Netherlands, 1983).

E. T. Jaynes, “Prior probabilities,” in E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics, R. Rosenkrantz, ed. (Reidel, Dordrecht, The Netherlands, 1983), p. 114.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

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Figures (8)

Fig. 1
Fig. 1

Twyman–Green phase-shifting speckle interferometer: laser, argon-ion laser with beam expander; BS, beam splitter; MO, measurement object; RO, reference object with piezoelectric transducer; CA, computer-controlled high-voltage amplifier; CCD, CCD camera with objective lens; PC, personal computer with framegrabber.

Fig. 2
Fig. 2

Phase-shifting images of a surface slope measurement with a bias phase step of π/2: (a) Φ0, (b) Φ90, (c) Φ180, (d) Φ270.

Fig. 3
Fig. 3

Edge detection (a) in a binary image and (b) the result.

Fig. 4
Fig. 4

Unwrapping (a) without and (b) with disturbances, (c) selection of good data.

Fig. 5
Fig. 5

Wrapped image (a) by selection of good data and (b) the bias offset.

Fig. 6
Fig. 6

Unwrapped image of Fig. 5(a).

Fig. 7
Fig. 7

Spline reconstruction of the most probable surface of Fig. 6.

Fig. 8
Fig. 8

Spline reconstruction of the surface in Fig. 6 with a simulated step.

Equations (22)

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ΔΦ=arctand1n2-n1d2n1n2+d1d2,
Φ0=arctann/d,
Φ90=arctand/-n,
Φ180=arctan-n/-d,
Φ270=arctan-d/n.
f˜s, x=1s- ψ*x-xsfxdx.
ψsx, y=gsx, y,
gsx, y=12πx exp-12x2+y2s2
f˜sx, y=sf*gsx, y,
|f˜sx, y|=f˜s1x, y2+f˜s2x, y21/2
αsx, y=arctanf˜s2x, yf˜s1x, y.
Δx2=x2¯-x¯2,
x¯=1nxline x,
pρ|d, I=pd|ρ, Ipρ|Ipd|I
pρ|I  exp-N2 ln Φp,
Φρ=i,jpxx2xi, yj+ρyy2xi, yj+2ρxy2xi, yj,
ρxxxi, yj=ρi,jx2=ρi+1,j-2ρi,j+ρi-1,j,
ρyyxi, yj=ρi,jy2=ρi,j+1-2ρi,j+ρi,j-1,
ρxyxi, yj=ρi,jxy=14ρi+1,j-ρi+1,j-1-ρi-1,j+1+ρi-1,j-1,
pd|ρ, I  exp-N2 ln χ˜2p,
χ˜2=i,jdxi, yj-ρxi, yj2
pρ|d, I  exp-N2 ln Φpexp-N2 ln χ˜2ρ,

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