Abstract

We introduce a new, to our knowledge, method using wavelets and probability theory for the evaluation of speckle interference patterns for quantitative out-of-plane deformation measurements of rough surfaces of nontransparent solids. The experiment uses a conventional Twyman–Green interferometer setup. The speckle interference patterns are obtained by the common method of subtraction of images taken before and after a surface deformation. The data are processed by a wavelet transformation, which analyzes the image structures on different length scales. Thus it is possible to separate the interference fringes from the noise. From the locations of the interference fringes, the deformation of the surface can be reconstructed by means of probability theory.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, UK, 1989).
    [CrossRef]
  2. M. Françon, Laser Speckle and Applications in Optics (Academic, New York, 1979).
  3. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1989).
  4. A. W. Koch, M. Ruprecht, R. Wilhelm, “Laser speckle techniques for in-situ monitoring of erosion and redeposition at inner walls in large experimental fusion devices,” 4/271 (Max-Planck-Institut für Plasmaphysik, Garching bei München, November1995).
  5. H. Schwieger, R. Streubel, “Speckle-Interferometrie, eine einfache Methode zur Verformungsanalyse,” Materialprüfung 23, 105–113 (1983).
  6. I. V. Volkov, I. S. Klimenko, “Production and interpretation of speckle interferograms of deformable objects,” Sov. Phys. Technol. 25, 626–635 (1980).
  7. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (Elsevier, New York, 1988), pp. 350–393.
  8. A. J. P. van Haasteren, H. J. Frankena, “Real-time displacement measurement using a multicamera phase-stepping speckle interferometer,” Appl. Opt. 33, 4137–4142 (1994).
    [CrossRef]
  9. W. Osten, Digitale Verarbeitung und Auswertung von Interferenzbildern (Akademie Verlag, Berlin, 1991).
  10. A. Davila, D. Kerr, G. H. Kaufmann, “Digital processing of electronic speckle pattern interferometry addition fringes,” Appl. Opt. 33, 5964–5969 (1994).
    [CrossRef] [PubMed]
  11. 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.
  12. A. W. Koch, M. Ruprecht, “Material surface testing using laser speckle techniques,” in Proceedings Mechanik und Optik [Institut Franco-Allemand de Recherches de Saint-Louis (ISL), Saint-Louis, France, 1995], pp. 269–280.
  13. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
    [CrossRef]
  14. G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, Boston, Mass., 1994).
  15. 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.
  16. R. D. Rosenkrantz, ed., E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics (Reidel, Dordrecht, The Netherlands, 1983).
    [CrossRef]
  17. E. T. Jaynes, “Marginalization and prior probabilities,” in E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics, R. Rosenkrantz, ed. (Reidel, Dordrecht, The Netherlands, 1983), p. 337.

1994

1983

H. Schwieger, R. Streubel, “Speckle-Interferometrie, eine einfache Methode zur Verformungsanalyse,” Materialprüfung 23, 105–113 (1983).

1980

I. V. Volkov, I. S. Klimenko, “Production and interpretation of speckle interferograms of deformable objects,” Sov. Phys. Technol. 25, 626–635 (1980).

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (Elsevier, New York, 1988), pp. 350–393.

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]

Davila, A.

Françon, M.

M. Françon, Laser Speckle and Applications in Optics (Academic, New York, 1979).

Frankena, H. J.

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.

Jaynes, E. T.

E. T. Jaynes, “Marginalization and prior probabilities,” in E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics, R. Rosenkrantz, ed. (Reidel, Dordrecht, The Netherlands, 1983), p. 337.

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).

Kaufmann, G. H.

Kerr, D.

Klimenko, I. S.

I. V. Volkov, I. S. Klimenko, “Production and interpretation of speckle interferograms of deformable objects,” Sov. Phys. Technol. 25, 626–635 (1980).

Koch, A. W.

A. W. Koch, M. Ruprecht, R. Wilhelm, “Laser speckle techniques for in-situ monitoring of erosion and redeposition at inner walls in large experimental fusion devices,” 4/271 (Max-Planck-Institut für Plasmaphysik, Garching bei München, November1995).

A. W. Koch, M. Ruprecht, “Material surface testing using laser speckle techniques,” in Proceedings Mechanik und Optik [Institut Franco-Allemand de Recherches de Saint-Louis (ISL), Saint-Louis, France, 1995], pp. 269–280.

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.

Osten, W.

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

Ruprecht, M.

A. W. Koch, M. Ruprecht, R. Wilhelm, “Laser speckle techniques for in-situ monitoring of erosion and redeposition at inner walls in large experimental fusion devices,” 4/271 (Max-Planck-Institut für Plasmaphysik, Garching bei München, November1995).

A. W. Koch, M. Ruprecht, “Material surface testing using laser speckle techniques,” in Proceedings Mechanik und Optik [Institut Franco-Allemand de Recherches de Saint-Louis (ISL), Saint-Louis, France, 1995], pp. 269–280.

Schwieger, H.

H. Schwieger, R. Streubel, “Speckle-Interferometrie, eine einfache Methode zur Verformungsanalyse,” Materialprüfung 23, 105–113 (1983).

Streubel, R.

H. Schwieger, R. Streubel, “Speckle-Interferometrie, eine einfache Methode zur Verformungsanalyse,” Materialprüfung 23, 105–113 (1983).

van Haasteren, A. J. P.

Volkov, I. V.

I. V. Volkov, I. S. Klimenko, “Production and interpretation of speckle interferograms of deformable objects,” Sov. Phys. Technol. 25, 626–635 (1980).

Wilhelm, R.

A. W. Koch, M. Ruprecht, R. Wilhelm, “Laser speckle techniques for in-situ monitoring of erosion and redeposition at inner walls in large experimental fusion devices,” 4/271 (Max-Planck-Institut für Plasmaphysik, Garching bei München, November1995).

Wykes, C.

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

Appl. Opt.

Materialprüfung

H. Schwieger, R. Streubel, “Speckle-Interferometrie, eine einfache Methode zur Verformungsanalyse,” Materialprüfung 23, 105–113 (1983).

Sov. Phys. Technol.

I. V. Volkov, I. S. Klimenko, “Production and interpretation of speckle interferograms of deformable objects,” Sov. Phys. Technol. 25, 626–635 (1980).

Other

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (Elsevier, New York, 1988), pp. 350–393.

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

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

M. Françon, Laser Speckle and Applications in Optics (Academic, New York, 1979).

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

A. W. Koch, M. Ruprecht, R. Wilhelm, “Laser speckle techniques for in-situ monitoring of erosion and redeposition at inner walls in large experimental fusion devices,” 4/271 (Max-Planck-Institut für Plasmaphysik, Garching bei München, November1995).

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.

A. W. Koch, M. Ruprecht, “Material surface testing using laser speckle techniques,” in Proceedings Mechanik und Optik [Institut Franco-Allemand de Recherches de Saint-Louis (ISL), Saint-Louis, France, 1995], pp. 269–280.

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).
[CrossRef]

E. T. Jaynes, “Marginalization and prior probabilities,” in E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics, R. Rosenkrantz, ed. (Reidel, Dordrecht, The Netherlands, 1983), p. 337.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Twyman–Green interferometer: LA, laser; BS, beam splitter; L, lens; Smes, surface to be measured; Sref, reference surface; dr and dm, distances of the reference and measured surfaces from the beam splitter, respectively; Δd, surface deformation; MS, micrometer screw; T, telescope optics; CCD, CCD camera.

Fig. 2
Fig. 2

Example of an interference pattern resulting from the subtraction of the images taken before and after the deformation.

Fig. 3
Fig. 3

(a) Cut through the interference pattern shown in Fig. 2. The cut is done horizontally through the middle of the image. (b) Cut through the smoothed image at the same location as for (a). (c) Cut through the spline at the same location as for (a). In the ideal case a surface deformation would result in a cosinelike structure for the intensity difference ΔI (dashed curve).

Fig. 4
Fig. 4

One particular component of the wavelet family ψ (x, y). In one direction it is a Gaussian, in the other the first derivative of a Gaussian.

Fig. 5
Fig. 5

Reconstructed surface deformation: The vertical scale indicates deformation in units of λ/2. Results from the minima search by means of the wavelet transform, e.g., the data entering Eq. (23), are displayed in the base plane of the figure. Please note the residual noise distortion.

Equations (30)

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

Iij=Rm2Em2+Rr2Er2+2RmRrEmEr cosϕm-ϕr,
Iijd=Rmd2Em2+Rr2Er2+2RmdRrEmEr×cosϕm-ϕr+Δϕx, y;
Δϕx, y=k·Δdx, y,
ΔIij=Iij-Iijd=Rm2Em2-Rmd2Em2+2RrErRmEm cosϕm-ϕr-RmdEm cosϕm-ϕr+Δϕx, y.
Δh=λ2.
f˜s, x=1s-ψ*x-xsfxdx.
ψx, y=gx, y,
gx, y=12πexp-x2+y22.
gsx, y=12πexp-12x2+y2s2,
ψsx, y=gsx, y.
f˜sx, y=f*ψsx, y.
f˜sx, y=sf*gsx, y.
f˜sx, y=f˜s1x, y2+f˜s2x, y21/2
αsx, y=arctanf˜s2x, yf˜s1x, y.
pρ|dI=pd|ρIpρ|Ipd|I,
pρ|μI=exp-μΦρZμ.
Zμ= Dρ exp-μΦρ.
Φρ=i,jρxx2xi, yj+ρyy2xi, yj+2ρxy2xi, yj
pρ|I=0dμpρ|μIpμ|I.
pμ|I1μ.
pρ|Iexp-Npixel2ln Φ,
pd|σρI=2πσ2-Ndata2 exp-12 χ2,
χ2=l=1Ndatadxl-ρxl2σ2.
pd|ρI=0dσpd|σρIpσ|I.
pσ|I1σ.
pd|ρIexp-Ndata2 ln χ2.
pρ|dIexp-Npixel2 ln Φexp-Ndata2 ln χ2=F1ΦF2χ2,
pm=maxρF1ΦF2χ2.
pm=maxSF2SmaxρF1Φ|χ2=S.
pm=maxSF2SF1Φ*S.

Metrics