Abstract

We present a new method using Bayesian probability theory and neural networks for the evaluation of speckle interference patterns for an automated analysis of deformation and erosion measurements. The method is applied to the fringe pattern reconstruction of speckle measurements with a Twyman-Green interferometer. Given a binary speckle image, the method returns the fringe pattern without noise, thus removing the need for smoothing and allowing a straightforward unwrapping procedure and determination of the surface shape. Because no parameters have to be adjusted, the method is especially suited for continuous and automated monitoring of surface changes.

© 2004 Optical Society of America

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  1. G. Janeschitz, “Plasma-wall interaction issues in ITER,” J. Nucl. Mater. 290–293, 1–11 (2001).
    [CrossRef]
  2. R. A. Zuhr, J. Roth, W. Eckstein, U. V. Toussaint, J. Luthin, “Implantation, erosion, and retention of tungsten in carbon,” J. Nucl. Mater. 290–293, 162–165 (2001).
    [CrossRef]
  3. E. Gauthier, G. Roupillard, “Speckle interferometry diagnostic for erosion/redeposition measurements in tokamaks,” J. Nucl. Mater. 313–316, 701–705 (2003).
    [CrossRef]
  4. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  5. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  6. J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
    [CrossRef] [PubMed]
  7. J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Anal. 33, 105–125 (1998).
    [CrossRef]
  8. R. Seara, A. A. Goncalves, P. B. Uliana, “Filtering algorithm for noise reduction in phase images with 2π phase jumps,” Appl. Opt. 37, 2046–2050 (1998).
    [CrossRef]
  9. D. C. Ghiglia, G. A. Mastin, L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–280 (1987).
    [CrossRef]
  10. D. J. Tipper, D. R. Burton, M. J. Lalor, “A neural network approach to the phase unwrapping problem in fringe analysis,” Nondestr. Test. Eval. 12, 391–400 (1996).
    [CrossRef]
  11. P. G. Charette, I. W. Hunter, “Robust phase-unwrapping method for phase images with high noise content,” Appl. Opt. 35, 3506–3513 (1996).
    [CrossRef] [PubMed]
  12. M. A. Herraez, M. A. Gdeisat, D. R. Burton, M. J. Lalor, “Robust, fast, and effective two-dimensional automatic phase unwrapping algorithm based on image decomposition,” Appl. Opt. 41, 7445–7455 (2002).
    [CrossRef] [PubMed]
  13. J. Arines, “Least-squares model estimation of wrapped phases: application to phase unwrapping,” Appl. Opt. 42, 3373–3378 (2003).
    [CrossRef] [PubMed]
  14. O. Marklund, “Noise-insensitive two-dimensional phase unwrapping method,” J. Opt. Soc. Am. A 15, 42–60 (1998).
    [CrossRef]
  15. X. Y. He, X. Kang, C. J. Tay, C. Quan, H. M. Shang, “Proposed algorithm for phase unwrapping,” Appl. Opt. 41, 7422–7428 (2002).
    [CrossRef] [PubMed]
  16. B. V. Dorrio, J. L. Fernandez, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33–R55 (1999).
    [CrossRef]
  17. E. Berger, W. von der Linden, V. Dose, M. Ruprecht, A. Koch, “Approach for the evaluation of speckle deformation measurements by application of the wavelet transformation,” Appl. Opt. 36, 7455–7460 (1997).
    [CrossRef]
  18. C. M. Bishop, Neural Networks for Pattern Recognition (Oxford U. Press, Oxford, UK, 1995).
  19. A. R. Barron, “Universal approximation bounds for superposition of a sigmoidal function,” IEEE Trans. Inf. Theory 39, 930–945 (1993).
    [CrossRef]
  20. D. Sivia, Data Analysis: A Bayesian Tutorial (Oxford U. Press, Oxford, UK, 1996).
  21. E. T. Jaynes, “Prior probabilities,” in Papers on Probability, Statistics and Statistical Physics, R. Rosenkrantz, ed. (Reidel, Dordrecht, The Netherlands, 1983).
  22. V. Dose, “Hyperplane priors,” in Bayesian Inference and Maximum Entropy Methods in Science and Engineering: 23rd International Workshop, C. J. Williams, ed. (American Institute of Physics, Melville, N.Y., 2003), pp. 350–357.
  23. B. Buck, V. A. Macaulay, Maximum Entropy in Action (Oxford U. Press, Oxford, UK, 1991).
  24. U. V. Toussaint, S. Gori, V. Dose, “A Bayesian neural network,” Neural Networks, submitted for publication.
  25. R. M. Neal, “Bayesian learning for neural networks,” in Lecture Notes in Statistics, P. Bickel, P. Diggle, S. Fienberg, U. Gather, I. Olkin, S. Zeger, eds. (Springer, New York, 1996), Vol. 118.
    [CrossRef]
  26. E. Berger, W. von der Linden, V. Dose, M. Jakobi, A. W. Koch, “Reconstruction of surfaces from phase-shifting speckle interferometry,” Appl. Opt. 38, 4997–5003 (1999).
    [CrossRef]
  27. Optimization routine nag_nlp_sol, mark18 from Numerical Algorithms Group Ltd., Oxford, OX2 8DR, UK, http://www.nag.co.uk .
  28. H. H. Thodberg, “A review of Bayesian neural networks with an application to near infrared spectroscopy,” IEEE Trans. Neural Networks 7, 56–72 (1995).
    [CrossRef]
  29. D. Kerr, G. H. Kaufmann, G. E. Galizzi, “Unwrapping of interferometric phase-fringe maps by discrete cosine transform,” Appl. Opt. 35, 810–816 (1996).
    [CrossRef] [PubMed]
  30. G. H. Kaufmann, G. E. Galizzi, P. D. Ruiz, “Evaluation of a preconditioned conjugate-gradient algorithm for weighted least-squares unwrapping of digital speckle-pattern interferometry phase maps,” Appl. Opt. 37, 3076–3084 (1998).
    [CrossRef]
  31. T. R. Crimmins, “Geometric filter for speckle reduction,” Appl. Opt. 24, 1438–1443 (1985).
    [CrossRef] [PubMed]
  32. T. R. Crimmins, “Geometric filter for reducing speckle,” Opt. Eng. 25, 651–654 (1986).
    [CrossRef]
  33. J. S. Lee, I. Jurkevich, “Speckle filtering of synthetic aperture radar images: a review,” Remote Sens. Rev. 8, 313–340 (1994).
    [CrossRef]
  34. J. S. Lee, “A simple speckle smoothing algorithm for synthetic aperture radar images,” IEEE Trans. Syst. Man. Cybern. SMC-13, 85–89 (1983).
    [CrossRef]
  35. D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. 35, 373–383 (1987).
    [CrossRef]
  36. A. Lopes, R. Touzi, E. Nezry, “Adaptive speckle filters and scene heterogeneity,” IEEE Trans. Geosci. Remote Sens. 28, 992–1000 (1990).
    [CrossRef]
  37. J. S. Lee, “Digital image noise smoothing and the sigma filter,” Comput. Vision Graph. Image Process. 24, 255–269 (1983).
    [CrossRef]

2003 (2)

E. Gauthier, G. Roupillard, “Speckle interferometry diagnostic for erosion/redeposition measurements in tokamaks,” J. Nucl. Mater. 313–316, 701–705 (2003).
[CrossRef]

J. Arines, “Least-squares model estimation of wrapped phases: application to phase unwrapping,” Appl. Opt. 42, 3373–3378 (2003).
[CrossRef] [PubMed]

2002 (2)

2001 (2)

G. Janeschitz, “Plasma-wall interaction issues in ITER,” J. Nucl. Mater. 290–293, 1–11 (2001).
[CrossRef]

R. A. Zuhr, J. Roth, W. Eckstein, U. V. Toussaint, J. Luthin, “Implantation, erosion, and retention of tungsten in carbon,” J. Nucl. Mater. 290–293, 162–165 (2001).
[CrossRef]

1999 (2)

B. V. Dorrio, J. L. Fernandez, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33–R55 (1999).
[CrossRef]

E. Berger, W. von der Linden, V. Dose, M. Jakobi, A. W. Koch, “Reconstruction of surfaces from phase-shifting speckle interferometry,” Appl. Opt. 38, 4997–5003 (1999).
[CrossRef]

1998 (4)

1997 (1)

1996 (3)

1995 (2)

H. H. Thodberg, “A review of Bayesian neural networks with an application to near infrared spectroscopy,” IEEE Trans. Neural Networks 7, 56–72 (1995).
[CrossRef]

J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
[CrossRef] [PubMed]

1994 (1)

J. S. Lee, I. Jurkevich, “Speckle filtering of synthetic aperture radar images: a review,” Remote Sens. Rev. 8, 313–340 (1994).
[CrossRef]

1993 (1)

A. R. Barron, “Universal approximation bounds for superposition of a sigmoidal function,” IEEE Trans. Inf. Theory 39, 930–945 (1993).
[CrossRef]

1990 (1)

A. Lopes, R. Touzi, E. Nezry, “Adaptive speckle filters and scene heterogeneity,” IEEE Trans. Geosci. Remote Sens. 28, 992–1000 (1990).
[CrossRef]

1989 (1)

1988 (1)

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

1987 (2)

D. C. Ghiglia, G. A. Mastin, L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–280 (1987).
[CrossRef]

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. 35, 373–383 (1987).
[CrossRef]

1986 (1)

T. R. Crimmins, “Geometric filter for reducing speckle,” Opt. Eng. 25, 651–654 (1986).
[CrossRef]

1985 (1)

1983 (2)

J. S. Lee, “Digital image noise smoothing and the sigma filter,” Comput. Vision Graph. Image Process. 24, 255–269 (1983).
[CrossRef]

J. S. Lee, “A simple speckle smoothing algorithm for synthetic aperture radar images,” IEEE Trans. Syst. Man. Cybern. SMC-13, 85–89 (1983).
[CrossRef]

Arines, J.

Barron, A. R.

A. R. Barron, “Universal approximation bounds for superposition of a sigmoidal function,” IEEE Trans. Inf. Theory 39, 930–945 (1993).
[CrossRef]

Berger, E.

Bishop, C. M.

C. M. Bishop, Neural Networks for Pattern Recognition (Oxford U. Press, Oxford, UK, 1995).

Buck, B.

B. Buck, V. A. Macaulay, Maximum Entropy in Action (Oxford U. Press, Oxford, UK, 1991).

Buckland, J. R.

Burton, D. R.

M. A. Herraez, M. A. Gdeisat, D. R. Burton, M. J. Lalor, “Robust, fast, and effective two-dimensional automatic phase unwrapping algorithm based on image decomposition,” Appl. Opt. 41, 7445–7455 (2002).
[CrossRef] [PubMed]

D. J. Tipper, D. R. Burton, M. J. Lalor, “A neural network approach to the phase unwrapping problem in fringe analysis,” Nondestr. Test. Eval. 12, 391–400 (1996).
[CrossRef]

Charette, P. G.

Chavel, P.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. 35, 373–383 (1987).
[CrossRef]

Crimmins, T. R.

T. R. Crimmins, “Geometric filter for reducing speckle,” Opt. Eng. 25, 651–654 (1986).
[CrossRef]

T. R. Crimmins, “Geometric filter for speckle reduction,” Appl. Opt. 24, 1438–1443 (1985).
[CrossRef] [PubMed]

Dorrio, B. V.

B. V. Dorrio, J. L. Fernandez, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33–R55 (1999).
[CrossRef]

Dose, V.

E. Berger, W. von der Linden, V. Dose, M. Jakobi, A. W. Koch, “Reconstruction of surfaces from phase-shifting speckle interferometry,” Appl. Opt. 38, 4997–5003 (1999).
[CrossRef]

E. Berger, W. von der Linden, V. Dose, M. Ruprecht, A. Koch, “Approach for the evaluation of speckle deformation measurements by application of the wavelet transformation,” Appl. Opt. 36, 7455–7460 (1997).
[CrossRef]

V. Dose, “Hyperplane priors,” in Bayesian Inference and Maximum Entropy Methods in Science and Engineering: 23rd International Workshop, C. J. Williams, ed. (American Institute of Physics, Melville, N.Y., 2003), pp. 350–357.

U. V. Toussaint, S. Gori, V. Dose, “A Bayesian neural network,” Neural Networks, submitted for publication.

Eckstein, W.

R. A. Zuhr, J. Roth, W. Eckstein, U. V. Toussaint, J. Luthin, “Implantation, erosion, and retention of tungsten in carbon,” J. Nucl. Mater. 290–293, 162–165 (2001).
[CrossRef]

Fernandez, J. L.

B. V. Dorrio, J. L. Fernandez, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33–R55 (1999).
[CrossRef]

Galizzi, G. E.

Gauthier, E.

E. Gauthier, G. Roupillard, “Speckle interferometry diagnostic for erosion/redeposition measurements in tokamaks,” J. Nucl. Mater. 313–316, 701–705 (2003).
[CrossRef]

Gdeisat, M. A.

Ghiglia, D. C.

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Goncalves, A. A.

Gori, S.

U. V. Toussaint, S. Gori, V. Dose, “A Bayesian neural network,” Neural Networks, submitted for publication.

He, X. Y.

Herraez, M. A.

Hunter, I. W.

Huntley, J. M.

Jakobi, M.

Janeschitz, G.

G. Janeschitz, “Plasma-wall interaction issues in ITER,” J. Nucl. Mater. 290–293, 1–11 (2001).
[CrossRef]

Jaynes, E. T.

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

Jurkevich, I.

J. S. Lee, I. Jurkevich, “Speckle filtering of synthetic aperture radar images: a review,” Remote Sens. Rev. 8, 313–340 (1994).
[CrossRef]

Kang, X.

Kaufmann, G. H.

Kerr, D.

Koch, A.

Koch, A. W.

Kuan, D. T.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. 35, 373–383 (1987).
[CrossRef]

Lalor, M. J.

M. A. Herraez, M. A. Gdeisat, D. R. Burton, M. J. Lalor, “Robust, fast, and effective two-dimensional automatic phase unwrapping algorithm based on image decomposition,” Appl. Opt. 41, 7445–7455 (2002).
[CrossRef] [PubMed]

D. J. Tipper, D. R. Burton, M. J. Lalor, “A neural network approach to the phase unwrapping problem in fringe analysis,” Nondestr. Test. Eval. 12, 391–400 (1996).
[CrossRef]

Lee, J. S.

J. S. Lee, I. Jurkevich, “Speckle filtering of synthetic aperture radar images: a review,” Remote Sens. Rev. 8, 313–340 (1994).
[CrossRef]

J. S. Lee, “A simple speckle smoothing algorithm for synthetic aperture radar images,” IEEE Trans. Syst. Man. Cybern. SMC-13, 85–89 (1983).
[CrossRef]

J. S. Lee, “Digital image noise smoothing and the sigma filter,” Comput. Vision Graph. Image Process. 24, 255–269 (1983).
[CrossRef]

Lopes, A.

A. Lopes, R. Touzi, E. Nezry, “Adaptive speckle filters and scene heterogeneity,” IEEE Trans. Geosci. Remote Sens. 28, 992–1000 (1990).
[CrossRef]

Luthin, J.

R. A. Zuhr, J. Roth, W. Eckstein, U. V. Toussaint, J. Luthin, “Implantation, erosion, and retention of tungsten in carbon,” J. Nucl. Mater. 290–293, 162–165 (2001).
[CrossRef]

Macaulay, V. A.

B. Buck, V. A. Macaulay, Maximum Entropy in Action (Oxford U. Press, Oxford, UK, 1991).

Marklund, O.

Mastin, G. A.

Neal, R. M.

R. M. Neal, “Bayesian learning for neural networks,” in Lecture Notes in Statistics, P. Bickel, P. Diggle, S. Fienberg, U. Gather, I. Olkin, S. Zeger, eds. (Springer, New York, 1996), Vol. 118.
[CrossRef]

Nezry, E.

A. Lopes, R. Touzi, E. Nezry, “Adaptive speckle filters and scene heterogeneity,” IEEE Trans. Geosci. Remote Sens. 28, 992–1000 (1990).
[CrossRef]

Quan, C.

Romero, L. A.

Roth, J.

R. A. Zuhr, J. Roth, W. Eckstein, U. V. Toussaint, J. Luthin, “Implantation, erosion, and retention of tungsten in carbon,” J. Nucl. Mater. 290–293, 162–165 (2001).
[CrossRef]

Roupillard, G.

E. Gauthier, G. Roupillard, “Speckle interferometry diagnostic for erosion/redeposition measurements in tokamaks,” J. Nucl. Mater. 313–316, 701–705 (2003).
[CrossRef]

Ruiz, P. D.

Ruprecht, M.

Sawchuk, A. A.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. 35, 373–383 (1987).
[CrossRef]

Seara, R.

Shang, H. M.

Sivia, D.

D. Sivia, Data Analysis: A Bayesian Tutorial (Oxford U. Press, Oxford, UK, 1996).

Strand, T. C.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. 35, 373–383 (1987).
[CrossRef]

Tay, C. J.

Thodberg, H. H.

H. H. Thodberg, “A review of Bayesian neural networks with an application to near infrared spectroscopy,” IEEE Trans. Neural Networks 7, 56–72 (1995).
[CrossRef]

Tipper, D. J.

D. J. Tipper, D. R. Burton, M. J. Lalor, “A neural network approach to the phase unwrapping problem in fringe analysis,” Nondestr. Test. Eval. 12, 391–400 (1996).
[CrossRef]

Toussaint, U. V.

R. A. Zuhr, J. Roth, W. Eckstein, U. V. Toussaint, J. Luthin, “Implantation, erosion, and retention of tungsten in carbon,” J. Nucl. Mater. 290–293, 162–165 (2001).
[CrossRef]

U. V. Toussaint, S. Gori, V. Dose, “A Bayesian neural network,” Neural Networks, submitted for publication.

Touzi, R.

A. Lopes, R. Touzi, E. Nezry, “Adaptive speckle filters and scene heterogeneity,” IEEE Trans. Geosci. Remote Sens. 28, 992–1000 (1990).
[CrossRef]

Turner, S. R. E.

Uliana, P. B.

von der Linden, W.

Werner, C. L.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Zuhr, R. A.

R. A. Zuhr, J. Roth, W. Eckstein, U. V. Toussaint, J. Luthin, “Implantation, erosion, and retention of tungsten in carbon,” J. Nucl. Mater. 290–293, 162–165 (2001).
[CrossRef]

Appl. Opt. (12)

T. R. Crimmins, “Geometric filter for speckle reduction,” Appl. Opt. 24, 1438–1443 (1985).
[CrossRef] [PubMed]

J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
[CrossRef] [PubMed]

R. Seara, A. A. Goncalves, P. B. Uliana, “Filtering algorithm for noise reduction in phase images with 2π phase jumps,” Appl. Opt. 37, 2046–2050 (1998).
[CrossRef]

J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
[CrossRef] [PubMed]

D. Kerr, G. H. Kaufmann, G. E. Galizzi, “Unwrapping of interferometric phase-fringe maps by discrete cosine transform,” Appl. Opt. 35, 810–816 (1996).
[CrossRef] [PubMed]

P. G. Charette, I. W. Hunter, “Robust phase-unwrapping method for phase images with high noise content,” Appl. Opt. 35, 3506–3513 (1996).
[CrossRef] [PubMed]

E. Berger, W. von der Linden, V. Dose, M. Ruprecht, A. Koch, “Approach for the evaluation of speckle deformation measurements by application of the wavelet transformation,” Appl. Opt. 36, 7455–7460 (1997).
[CrossRef]

G. H. Kaufmann, G. E. Galizzi, P. D. Ruiz, “Evaluation of a preconditioned conjugate-gradient algorithm for weighted least-squares unwrapping of digital speckle-pattern interferometry phase maps,” Appl. Opt. 37, 3076–3084 (1998).
[CrossRef]

E. Berger, W. von der Linden, V. Dose, M. Jakobi, A. W. Koch, “Reconstruction of surfaces from phase-shifting speckle interferometry,” Appl. Opt. 38, 4997–5003 (1999).
[CrossRef]

X. Y. He, X. Kang, C. J. Tay, C. Quan, H. M. Shang, “Proposed algorithm for phase unwrapping,” Appl. Opt. 41, 7422–7428 (2002).
[CrossRef] [PubMed]

M. A. Herraez, M. A. Gdeisat, D. R. Burton, M. J. Lalor, “Robust, fast, and effective two-dimensional automatic phase unwrapping algorithm based on image decomposition,” Appl. Opt. 41, 7445–7455 (2002).
[CrossRef] [PubMed]

J. Arines, “Least-squares model estimation of wrapped phases: application to phase unwrapping,” Appl. Opt. 42, 3373–3378 (2003).
[CrossRef] [PubMed]

Comput. Vision Graph. Image Process. (1)

J. S. Lee, “Digital image noise smoothing and the sigma filter,” Comput. Vision Graph. Image Process. 24, 255–269 (1983).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. 35, 373–383 (1987).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

A. Lopes, R. Touzi, E. Nezry, “Adaptive speckle filters and scene heterogeneity,” IEEE Trans. Geosci. Remote Sens. 28, 992–1000 (1990).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. R. Barron, “Universal approximation bounds for superposition of a sigmoidal function,” IEEE Trans. Inf. Theory 39, 930–945 (1993).
[CrossRef]

IEEE Trans. Neural Networks (1)

H. H. Thodberg, “A review of Bayesian neural networks with an application to near infrared spectroscopy,” IEEE Trans. Neural Networks 7, 56–72 (1995).
[CrossRef]

IEEE Trans. Syst. Man. Cybern. (1)

J. S. Lee, “A simple speckle smoothing algorithm for synthetic aperture radar images,” IEEE Trans. Syst. Man. Cybern. SMC-13, 85–89 (1983).
[CrossRef]

J. Nucl. Mater. (3)

G. Janeschitz, “Plasma-wall interaction issues in ITER,” J. Nucl. Mater. 290–293, 1–11 (2001).
[CrossRef]

R. A. Zuhr, J. Roth, W. Eckstein, U. V. Toussaint, J. Luthin, “Implantation, erosion, and retention of tungsten in carbon,” J. Nucl. Mater. 290–293, 162–165 (2001).
[CrossRef]

E. Gauthier, G. Roupillard, “Speckle interferometry diagnostic for erosion/redeposition measurements in tokamaks,” J. Nucl. Mater. 313–316, 701–705 (2003).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Strain Anal. (1)

J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Anal. 33, 105–125 (1998).
[CrossRef]

Meas. Sci. Technol. (1)

B. V. Dorrio, J. L. Fernandez, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33–R55 (1999).
[CrossRef]

Nondestr. Test. Eval. (1)

D. J. Tipper, D. R. Burton, M. J. Lalor, “A neural network approach to the phase unwrapping problem in fringe analysis,” Nondestr. Test. Eval. 12, 391–400 (1996).
[CrossRef]

Opt. Eng. (1)

T. R. Crimmins, “Geometric filter for reducing speckle,” Opt. Eng. 25, 651–654 (1986).
[CrossRef]

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Remote Sens. Rev. (1)

J. S. Lee, I. Jurkevich, “Speckle filtering of synthetic aperture radar images: a review,” Remote Sens. Rev. 8, 313–340 (1994).
[CrossRef]

Other (8)

C. M. Bishop, Neural Networks for Pattern Recognition (Oxford U. Press, Oxford, UK, 1995).

Optimization routine nag_nlp_sol, mark18 from Numerical Algorithms Group Ltd., Oxford, OX2 8DR, UK, http://www.nag.co.uk .

D. Sivia, Data Analysis: A Bayesian Tutorial (Oxford U. Press, Oxford, UK, 1996).

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

V. Dose, “Hyperplane priors,” in Bayesian Inference and Maximum Entropy Methods in Science and Engineering: 23rd International Workshop, C. J. Williams, ed. (American Institute of Physics, Melville, N.Y., 2003), pp. 350–357.

B. Buck, V. A. Macaulay, Maximum Entropy in Action (Oxford U. Press, Oxford, UK, 1991).

U. V. Toussaint, S. Gori, V. Dose, “A Bayesian neural network,” Neural Networks, submitted for publication.

R. M. Neal, “Bayesian learning for neural networks,” in Lecture Notes in Statistics, P. Bickel, P. Diggle, S. Fienberg, U. Gather, I. Olkin, S. Zeger, eds. (Springer, New York, 1996), Vol. 118.
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Figures (5)

Fig. 1
Fig. 1

Example of a multilayer feed-forward NN, in this case having two input units (N = 2), three units in the hidden layer (K = 3), and one output unit (M = 1).

Fig. 2
Fig. 2

Binary [0, 1] speckle image with 512 × 512 pixels obtained by subtraction of two phase-shifting images with a bias step of π.

Fig. 3
Fig. 3

Misfit of the NN on the data set is monotonically decreasing with an increasing number of neurons. The evidence exhibits a strong decrease left of the maximum, indicating that the NN does not have enough hidden neurons to fit essential structures. The slow decrease to the right side shows that Ockham’s razor that penalizes the increased model complexity is no longer compensated by the higher likelihood.

Fig. 4
Fig. 4

(a) Result of a wavelet-based segmentation.26 The speckle noise is suppressed, but artifacts remain and some of the contour lines are disrupted, requiring additional postprocessing. (b) Joined result of the NNs. The obtained fringe pattern is displayed. Note the absence of artifacts and that even the hardly visible structures at the right edge of the image are resolved.

Fig. 5
Fig. 5

(a) Original binary image. (b) Noisy binary test image by use of a binomial probability distribution with p(black pixel|black) = p(white pixel|white) = 0.54. (c) Result of median filtering with a filter size of 5 × 5 pixels. (d) Result of median filtering with a filter size of 21 × 21 pixels. (e) Result of median filtering with an increased filter size of 41 × 41 pixels. (f) Result of the Bayesian neural-network approach.

Equations (23)

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zk=fn=1N w˜nkxn+bk, k=1,, K.
ymx1,, xN =k=1K w˜kmfn=1N w˜nkxn+bk+bm, m=1,, M.
PD, w|H, I=Pw|H, IPD|w, H, I=PD|H, IPw|D, H, I.
Pw|D, H, I=Pw|H, IPD|w, H, IPD|H, I.
PD|H, I= dwPw, D|H, I= dwPw|H, IPD|w, H, I,
PHi|D, I  PD|Hi, IPHi|I.
z=fb+n=1N w˜nxn,
b+n=1N w˜nxn=b1+w1x1+w2x2++wNxN=0
pwdw1dw2dwN=pwdw1dw2dwN,
pw=pwdetwiwk.
pw=pTwdetTwwk.
pTwdetTwwk=0=0.
1+x1w1+x2w2++xNwN=0,
pw1,, wN|I=ΓN/2r0N2πN/2 ×1w12++wN2N+1/2,w2r0N>0.
g=g1+g2 bw1wN
pb|w, λ, I=λ n=1N wn22π1/2 exp-λ2 b2n=1N wn2,
pλ|I  1/λ=limc0ccΓc λc-1 exp-cλ.
pb|w, I=limc0ccΓc0dλk=1K12πn=1N wnk21/2×λK2+c-1 exp-cλ-λ2 B=limc0ccΓck=1K12πn=1N wnk21/2×ΓK2+cc+12 BK2+cΨ.
pb, w|I=ΓN2r0Nπ N+12 2KΓK2k=1Kn=1N wnk2N/2×1k=1K bk2n=1N wnk2K/2.
pH|D, I=-dbdwpD|b, w, H, Ipb, w|Iexp-ϕ.
pH|D, IpD|b*, w*, H, Ipb*, w*|I×2πE/2detH1/2,
pD|q, I=i,jNx=512,Ny=512 qijDij1-qij1-Dij,
Qij=-dwpw|D, Hi1/2pw|D, Hj1/2=2E/2det Hi1/2det Hj1/2detHi+Hj1/2exp-14wi*Hiwi*+wj*Hjwj*exp14Hiwi*+Hjwj*T×Hi+Hj-1Hiwi*+Hjwj*.

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