Abstract

Optical speckle is commonly observed in measurements using coherent radiation. While lacking experimental validation, previous work has often assumed that speckle’s random spatial pattern follows a Markov process. Here, we present a derivation and experimental confirmation of conditions under which this assumption holds true. We demonstrate that a detected speckle field can be designed to obey the first-order Markov property by using a Cauchy attenuation mask to modulate scattered light. Creating Markov speckle enables the development of more accurate and efficient image post-processing algorithms, with applications including improved de-noising, segmentation and super-resolution. To show its versatility, we use the Cauchy mask to maximize the entropy of a detected speckle field with fixed average speckle size, allowing cryptographic applications to extract a maximum number of useful random bits from speckle images.

© 2012 OSA

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References

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  1. P. A. Kelly, H. Derin, and K. D. Hartt, “Adaptive segmentation of speckle images using a hierarchical random field model,” IEEE Trans. Acoust., Speech Sig. Process.36(10), 1628–1640 (1988).
    [CrossRef]
  2. B. Skoric, “On the entropy of keys derived from laser speckle: statistical properties of Gabor-transformed speckle,” J. Opt. A: Pure Appl. Opt10, 055304 (2008).
    [CrossRef]
  3. H. J. Rabal and R. A. Braga, Dynamic Laser Speckle and Applications (CRC Press, 2009).
  4. R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science297, 1074376 (2002).
    [CrossRef]
  5. Y. M. Wang, B. Judkewitz, C. DiMarzio, and C. Yang,“Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nature Commun.3, 928 (2012).
    [CrossRef]
  6. D. P. Kelly, J. E. Ward, U. Gopinathan, and J. T. Sheridan, “Controlling speckle using lenses and free space,” Opt. Lett.32, 3394–3396 (2007).
    [CrossRef]
  7. E. Mundry, K. Belkebir, J. Girard, J. Savatier, E. Moal, C. Nocoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics6, 312–315 (2012).
    [CrossRef]
  8. O. Lankoande, M. M. Hayat, and B. Santhanam, “Scene estimation from speckled synthetic aperture radar imagery: Markov random-field approach,” J. Opt. Soc. Am. A23, 1269–1272 (2006).
    [CrossRef]
  9. R. T. Frankot and R. Chellappa, “Lognormal random-field models and their applications to radar image synthesis,” IEEE Trans. Geosci. Remote Sens.25, 2196–2212 (2002).
  10. H. Xie, L. E. Pierce, and F. T. Ulaby, “SAR speckle reduction using wavelet denoising and Markov random field modeling,” IEEE Trans. Geosci. Remote Sens.40, 195–208 (1987).
  11. J. Goodman, Speckle Phenomena in Optics (Ben Roberts and Company, 2007).
  12. J. C. Dainty, Topics in Applied Physics: Laser Speckle and Related Phenomena (Springer-Verlag, 1984).
  13. J. Grimmett and D. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford University Press, 2001).
  14. H. Derin and P. A. Kelly, “Discrete-index Markov-type random processes,” Proc. IEEE77, 1485–1510 (1989).
    [CrossRef]
  15. H. Rue and L. Held, Gaussian Markov Random Fields: Theory and Applications (Chapman and Hall, 2005).
    [CrossRef]
  16. H. Derin, P. A. Kelly, G. Veniza, and S. G. Labitt, “Modeling and segmentation of speckle images using complex data,” IEEE Trans. Geosci. Remote Sens.40(1), 76–87 (1990).
    [CrossRef]
  17. Y. Ait-Sahalia, “Do interest rates really follow continuous-time Markov diffusions?,” Tech Rep., University of Chicago (1997).
  18. A. de Matos and M. Fernandes, “Testing the Markov property with high frequency data,” J. Econometrics141, 44–64 (2007).
    [CrossRef]
  19. S. Park and V. S. Pande, “Validation of Markov state models using Shannon’s entropy,” J. Chem. Phys124, 054118 (2006).
    [CrossRef] [PubMed]
  20. B. Chen and Y. Hong, “Testing for the Markov property in time series,” Econ. Theory28, 130–178 (2012).
    [CrossRef]
  21. T. W. Anderson and L. A. Goodman, “Statistical inference about Markov chains,” Ann. Math. Statist.28(1), 89–110 (1957).
    [CrossRef]
  22. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett.22(16), 1268–1270 (1997).
    [CrossRef] [PubMed]
  23. M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys.71, 313–369 (1999).
    [CrossRef]
  24. T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley and Sons, Inc., 1991), chap. 11.
    [CrossRef]
  25. W. C. Swope, J. W. Pitera, and F. Suits, “Describing protein folding kinetics by molecular dynamics simulations 1. theory,” J. Phys. Chem. B108, 6571–6581 (2004).
    [CrossRef]
  26. A. W. Marshall and I. Olkin, “A multivariate exponential distribution” J. Amer. Statist. Assoc.62, 30–44 (1967).
    [CrossRef]

2012 (3)

Y. M. Wang, B. Judkewitz, C. DiMarzio, and C. Yang,“Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nature Commun.3, 928 (2012).
[CrossRef]

E. Mundry, K. Belkebir, J. Girard, J. Savatier, E. Moal, C. Nocoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics6, 312–315 (2012).
[CrossRef]

B. Chen and Y. Hong, “Testing for the Markov property in time series,” Econ. Theory28, 130–178 (2012).
[CrossRef]

2008 (1)

B. Skoric, “On the entropy of keys derived from laser speckle: statistical properties of Gabor-transformed speckle,” J. Opt. A: Pure Appl. Opt10, 055304 (2008).
[CrossRef]

2007 (2)

D. P. Kelly, J. E. Ward, U. Gopinathan, and J. T. Sheridan, “Controlling speckle using lenses and free space,” Opt. Lett.32, 3394–3396 (2007).
[CrossRef]

A. de Matos and M. Fernandes, “Testing the Markov property with high frequency data,” J. Econometrics141, 44–64 (2007).
[CrossRef]

2006 (2)

2004 (1)

W. C. Swope, J. W. Pitera, and F. Suits, “Describing protein folding kinetics by molecular dynamics simulations 1. theory,” J. Phys. Chem. B108, 6571–6581 (2004).
[CrossRef]

2002 (2)

R. T. Frankot and R. Chellappa, “Lognormal random-field models and their applications to radar image synthesis,” IEEE Trans. Geosci. Remote Sens.25, 2196–2212 (2002).

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science297, 1074376 (2002).
[CrossRef]

1999 (1)

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys.71, 313–369 (1999).
[CrossRef]

1997 (1)

1990 (1)

H. Derin, P. A. Kelly, G. Veniza, and S. G. Labitt, “Modeling and segmentation of speckle images using complex data,” IEEE Trans. Geosci. Remote Sens.40(1), 76–87 (1990).
[CrossRef]

1989 (1)

H. Derin and P. A. Kelly, “Discrete-index Markov-type random processes,” Proc. IEEE77, 1485–1510 (1989).
[CrossRef]

1988 (1)

P. A. Kelly, H. Derin, and K. D. Hartt, “Adaptive segmentation of speckle images using a hierarchical random field model,” IEEE Trans. Acoust., Speech Sig. Process.36(10), 1628–1640 (1988).
[CrossRef]

1987 (1)

H. Xie, L. E. Pierce, and F. T. Ulaby, “SAR speckle reduction using wavelet denoising and Markov random field modeling,” IEEE Trans. Geosci. Remote Sens.40, 195–208 (1987).

1967 (1)

A. W. Marshall and I. Olkin, “A multivariate exponential distribution” J. Amer. Statist. Assoc.62, 30–44 (1967).
[CrossRef]

1957 (1)

T. W. Anderson and L. A. Goodman, “Statistical inference about Markov chains,” Ann. Math. Statist.28(1), 89–110 (1957).
[CrossRef]

Ait-Sahalia, Y.

Y. Ait-Sahalia, “Do interest rates really follow continuous-time Markov diffusions?,” Tech Rep., University of Chicago (1997).

Allain, M.

E. Mundry, K. Belkebir, J. Girard, J. Savatier, E. Moal, C. Nocoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics6, 312–315 (2012).
[CrossRef]

Anderson, T. W.

T. W. Anderson and L. A. Goodman, “Statistical inference about Markov chains,” Ann. Math. Statist.28(1), 89–110 (1957).
[CrossRef]

Belkebir, K.

E. Mundry, K. Belkebir, J. Girard, J. Savatier, E. Moal, C. Nocoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics6, 312–315 (2012).
[CrossRef]

Braga, R. A.

H. J. Rabal and R. A. Braga, Dynamic Laser Speckle and Applications (CRC Press, 2009).

Chellappa, R.

R. T. Frankot and R. Chellappa, “Lognormal random-field models and their applications to radar image synthesis,” IEEE Trans. Geosci. Remote Sens.25, 2196–2212 (2002).

Chen, B.

B. Chen and Y. Hong, “Testing for the Markov property in time series,” Econ. Theory28, 130–178 (2012).
[CrossRef]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley and Sons, Inc., 1991), chap. 11.
[CrossRef]

Dainty, J. C.

J. C. Dainty, Topics in Applied Physics: Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

de Matos, A.

A. de Matos and M. Fernandes, “Testing the Markov property with high frequency data,” J. Econometrics141, 44–64 (2007).
[CrossRef]

Derin, H.

H. Derin, P. A. Kelly, G. Veniza, and S. G. Labitt, “Modeling and segmentation of speckle images using complex data,” IEEE Trans. Geosci. Remote Sens.40(1), 76–87 (1990).
[CrossRef]

H. Derin and P. A. Kelly, “Discrete-index Markov-type random processes,” Proc. IEEE77, 1485–1510 (1989).
[CrossRef]

P. A. Kelly, H. Derin, and K. D. Hartt, “Adaptive segmentation of speckle images using a hierarchical random field model,” IEEE Trans. Acoust., Speech Sig. Process.36(10), 1628–1640 (1988).
[CrossRef]

DiMarzio, C.

Y. M. Wang, B. Judkewitz, C. DiMarzio, and C. Yang,“Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nature Commun.3, 928 (2012).
[CrossRef]

Fernandes, M.

A. de Matos and M. Fernandes, “Testing the Markov property with high frequency data,” J. Econometrics141, 44–64 (2007).
[CrossRef]

Frankot, R. T.

R. T. Frankot and R. Chellappa, “Lognormal random-field models and their applications to radar image synthesis,” IEEE Trans. Geosci. Remote Sens.25, 2196–2212 (2002).

Gershenfeld, N.

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science297, 1074376 (2002).
[CrossRef]

Girard, J.

E. Mundry, K. Belkebir, J. Girard, J. Savatier, E. Moal, C. Nocoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics6, 312–315 (2012).
[CrossRef]

Goodman, J.

J. Goodman, Speckle Phenomena in Optics (Ben Roberts and Company, 2007).

Goodman, L. A.

T. W. Anderson and L. A. Goodman, “Statistical inference about Markov chains,” Ann. Math. Statist.28(1), 89–110 (1957).
[CrossRef]

Gopinathan, U.

Grimmett, J.

J. Grimmett and D. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford University Press, 2001).

Hartt, K. D.

P. A. Kelly, H. Derin, and K. D. Hartt, “Adaptive segmentation of speckle images using a hierarchical random field model,” IEEE Trans. Acoust., Speech Sig. Process.36(10), 1628–1640 (1988).
[CrossRef]

Hayat, M. M.

Held, L.

H. Rue and L. Held, Gaussian Markov Random Fields: Theory and Applications (Chapman and Hall, 2005).
[CrossRef]

Hong, Y.

B. Chen and Y. Hong, “Testing for the Markov property in time series,” Econ. Theory28, 130–178 (2012).
[CrossRef]

Judkewitz, B.

Y. M. Wang, B. Judkewitz, C. DiMarzio, and C. Yang,“Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nature Commun.3, 928 (2012).
[CrossRef]

Kelly, D. P.

Kelly, P. A.

H. Derin, P. A. Kelly, G. Veniza, and S. G. Labitt, “Modeling and segmentation of speckle images using complex data,” IEEE Trans. Geosci. Remote Sens.40(1), 76–87 (1990).
[CrossRef]

H. Derin and P. A. Kelly, “Discrete-index Markov-type random processes,” Proc. IEEE77, 1485–1510 (1989).
[CrossRef]

P. A. Kelly, H. Derin, and K. D. Hartt, “Adaptive segmentation of speckle images using a hierarchical random field model,” IEEE Trans. Acoust., Speech Sig. Process.36(10), 1628–1640 (1988).
[CrossRef]

Labitt, S. G.

H. Derin, P. A. Kelly, G. Veniza, and S. G. Labitt, “Modeling and segmentation of speckle images using complex data,” IEEE Trans. Geosci. Remote Sens.40(1), 76–87 (1990).
[CrossRef]

Lankoande, O.

Marshall, A. W.

A. W. Marshall and I. Olkin, “A multivariate exponential distribution” J. Amer. Statist. Assoc.62, 30–44 (1967).
[CrossRef]

Moal, E.

E. Mundry, K. Belkebir, J. Girard, J. Savatier, E. Moal, C. Nocoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics6, 312–315 (2012).
[CrossRef]

Mundry, E.

E. Mundry, K. Belkebir, J. Girard, J. Savatier, E. Moal, C. Nocoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics6, 312–315 (2012).
[CrossRef]

Nieuwenhuizen, T. M.

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys.71, 313–369 (1999).
[CrossRef]

Nocoletti, C.

E. Mundry, K. Belkebir, J. Girard, J. Savatier, E. Moal, C. Nocoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics6, 312–315 (2012).
[CrossRef]

Olkin, I.

A. W. Marshall and I. Olkin, “A multivariate exponential distribution” J. Amer. Statist. Assoc.62, 30–44 (1967).
[CrossRef]

Pande, V. S.

S. Park and V. S. Pande, “Validation of Markov state models using Shannon’s entropy,” J. Chem. Phys124, 054118 (2006).
[CrossRef] [PubMed]

Pappu, R.

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science297, 1074376 (2002).
[CrossRef]

Park, S.

S. Park and V. S. Pande, “Validation of Markov state models using Shannon’s entropy,” J. Chem. Phys124, 054118 (2006).
[CrossRef] [PubMed]

Pierce, L. E.

H. Xie, L. E. Pierce, and F. T. Ulaby, “SAR speckle reduction using wavelet denoising and Markov random field modeling,” IEEE Trans. Geosci. Remote Sens.40, 195–208 (1987).

Pitera, J. W.

W. C. Swope, J. W. Pitera, and F. Suits, “Describing protein folding kinetics by molecular dynamics simulations 1. theory,” J. Phys. Chem. B108, 6571–6581 (2004).
[CrossRef]

Rabal, H. J.

H. J. Rabal and R. A. Braga, Dynamic Laser Speckle and Applications (CRC Press, 2009).

Recht, B.

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science297, 1074376 (2002).
[CrossRef]

Rue, H.

H. Rue and L. Held, Gaussian Markov Random Fields: Theory and Applications (Chapman and Hall, 2005).
[CrossRef]

Santhanam, B.

Savatier, J.

E. Mundry, K. Belkebir, J. Girard, J. Savatier, E. Moal, C. Nocoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics6, 312–315 (2012).
[CrossRef]

Sentenac, A.

E. Mundry, K. Belkebir, J. Girard, J. Savatier, E. Moal, C. Nocoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics6, 312–315 (2012).
[CrossRef]

Sheridan, J. T.

Skoric, B.

B. Skoric, “On the entropy of keys derived from laser speckle: statistical properties of Gabor-transformed speckle,” J. Opt. A: Pure Appl. Opt10, 055304 (2008).
[CrossRef]

Stirzaker, D.

J. Grimmett and D. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford University Press, 2001).

Suits, F.

W. C. Swope, J. W. Pitera, and F. Suits, “Describing protein folding kinetics by molecular dynamics simulations 1. theory,” J. Phys. Chem. B108, 6571–6581 (2004).
[CrossRef]

Swope, W. C.

W. C. Swope, J. W. Pitera, and F. Suits, “Describing protein folding kinetics by molecular dynamics simulations 1. theory,” J. Phys. Chem. B108, 6571–6581 (2004).
[CrossRef]

Taylor, J.

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science297, 1074376 (2002).
[CrossRef]

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley and Sons, Inc., 1991), chap. 11.
[CrossRef]

Ulaby, F. T.

H. Xie, L. E. Pierce, and F. T. Ulaby, “SAR speckle reduction using wavelet denoising and Markov random field modeling,” IEEE Trans. Geosci. Remote Sens.40, 195–208 (1987).

van Rossum, M. C. W.

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys.71, 313–369 (1999).
[CrossRef]

Veniza, G.

H. Derin, P. A. Kelly, G. Veniza, and S. G. Labitt, “Modeling and segmentation of speckle images using complex data,” IEEE Trans. Geosci. Remote Sens.40(1), 76–87 (1990).
[CrossRef]

Wang, Y. M.

Y. M. Wang, B. Judkewitz, C. DiMarzio, and C. Yang,“Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nature Commun.3, 928 (2012).
[CrossRef]

Ward, J. E.

Xie, H.

H. Xie, L. E. Pierce, and F. T. Ulaby, “SAR speckle reduction using wavelet denoising and Markov random field modeling,” IEEE Trans. Geosci. Remote Sens.40, 195–208 (1987).

Yamaguchi, I.

Yang, C.

Y. M. Wang, B. Judkewitz, C. DiMarzio, and C. Yang,“Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nature Commun.3, 928 (2012).
[CrossRef]

Zhang, T.

Ann. Math. Statist. (1)

T. W. Anderson and L. A. Goodman, “Statistical inference about Markov chains,” Ann. Math. Statist.28(1), 89–110 (1957).
[CrossRef]

Econ. Theory (1)

B. Chen and Y. Hong, “Testing for the Markov property in time series,” Econ. Theory28, 130–178 (2012).
[CrossRef]

IEEE Trans. Acoust., Speech Sig. Process. (1)

P. A. Kelly, H. Derin, and K. D. Hartt, “Adaptive segmentation of speckle images using a hierarchical random field model,” IEEE Trans. Acoust., Speech Sig. Process.36(10), 1628–1640 (1988).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (3)

R. T. Frankot and R. Chellappa, “Lognormal random-field models and their applications to radar image synthesis,” IEEE Trans. Geosci. Remote Sens.25, 2196–2212 (2002).

H. Xie, L. E. Pierce, and F. T. Ulaby, “SAR speckle reduction using wavelet denoising and Markov random field modeling,” IEEE Trans. Geosci. Remote Sens.40, 195–208 (1987).

H. Derin, P. A. Kelly, G. Veniza, and S. G. Labitt, “Modeling and segmentation of speckle images using complex data,” IEEE Trans. Geosci. Remote Sens.40(1), 76–87 (1990).
[CrossRef]

J. Amer. Statist. Assoc. (1)

A. W. Marshall and I. Olkin, “A multivariate exponential distribution” J. Amer. Statist. Assoc.62, 30–44 (1967).
[CrossRef]

J. Chem. Phys (1)

S. Park and V. S. Pande, “Validation of Markov state models using Shannon’s entropy,” J. Chem. Phys124, 054118 (2006).
[CrossRef] [PubMed]

J. Econometrics (1)

A. de Matos and M. Fernandes, “Testing the Markov property with high frequency data,” J. Econometrics141, 44–64 (2007).
[CrossRef]

J. Opt. A: Pure Appl. Opt (1)

B. Skoric, “On the entropy of keys derived from laser speckle: statistical properties of Gabor-transformed speckle,” J. Opt. A: Pure Appl. Opt10, 055304 (2008).
[CrossRef]

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

J. Phys. Chem. B (1)

W. C. Swope, J. W. Pitera, and F. Suits, “Describing protein folding kinetics by molecular dynamics simulations 1. theory,” J. Phys. Chem. B108, 6571–6581 (2004).
[CrossRef]

Nat. Photonics (1)

E. Mundry, K. Belkebir, J. Girard, J. Savatier, E. Moal, C. Nocoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics6, 312–315 (2012).
[CrossRef]

Nature Commun. (1)

Y. M. Wang, B. Judkewitz, C. DiMarzio, and C. Yang,“Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nature Commun.3, 928 (2012).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (1)

H. Derin and P. A. Kelly, “Discrete-index Markov-type random processes,” Proc. IEEE77, 1485–1510 (1989).
[CrossRef]

Rev. Mod. Phys. (1)

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys.71, 313–369 (1999).
[CrossRef]

Science (1)

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science297, 1074376 (2002).
[CrossRef]

Other (7)

H. Rue and L. Held, Gaussian Markov Random Fields: Theory and Applications (Chapman and Hall, 2005).
[CrossRef]

Y. Ait-Sahalia, “Do interest rates really follow continuous-time Markov diffusions?,” Tech Rep., University of Chicago (1997).

H. J. Rabal and R. A. Braga, Dynamic Laser Speckle and Applications (CRC Press, 2009).

J. Goodman, Speckle Phenomena in Optics (Ben Roberts and Company, 2007).

J. C. Dainty, Topics in Applied Physics: Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

J. Grimmett and D. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford University Press, 2001).

T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley and Sons, Inc., 1991), chap. 11.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Outline of an optical speckle field as a Markov process. (a) A 2D complex speckle field (phase in color) is examined along one dimension. If the speckle field obeys the first-order Markov property, the conditional dependence of the field at pixel n (blue dot) will only depend on its immediate neighbors (red and green dots), and no other pixels. (b) This relationship can be visualized as a transition process between field values in complex space, or through transitions over an undirected graph.

Fig. 2
Fig. 2

Speckle is designed to follow a first-order Markov process using a Cauchy-distributed apodizing mask M placed either (a) directly at the scatterer surface (z = 0) or (b) in the aperture plane of an imaging lens with focal length f.

Fig. 3
Fig. 3

Speckle as a second-order Markov process in 2D with a neighborhood defined over 8 pixels (here only speckle amplitude is displayed). Independent of average speckle size, the conditional probability of each pixel in this Markov speckle field only depends on these 8 neighbors.

Fig. 4
Fig. 4

Aperture windowing of the Cauchy mask function M(η) slightly shifts the desired autocorrelation function Jx) from an ideal exponential curve.

Fig. 5
Fig. 5

Diagram of the experimental setup used to generate correlated speckle field measurements. (a) K random speckle fields are generated via phase-shifting holography by imaging K random SLM patterns onto a volumetric scatterer. (b) Estimates of the three transition matrices in Eq. (22) are formed by vectorizing and processing the detected speckle fields.

Fig. 6
Fig. 6

An example set of transition matrices for the case of speckle modulated by a square aperture mask (unmodified speckle). (a) Transition matrices generated through simulation of square-masked speckle. (b) Transition matrices found experimentally. (c) Speckle field autocorrelation functions for simulated and experimental data. (d) Example square-masked speckle field data used for these plots.

Fig. 7
Fig. 7

An example set of transition matrices for the case of speckle modulated by a Cauchy function mask in the same layout as Fig. 6. Note the R32R21 and R31 matrices match more closely in width and slant than those generated by the square mask in Fig. 6, leading to difference matrices ΔP31 and ΔR31 that are closer to 0.

Fig. 8
Fig. 8

Plots comparing Markov TV error vs. speckle size for a Cauchy, Gaussian and square apodizing mask. Speckle generated via the Cauchy mask exhibits a lower TV error and thus is in closer agreement with a Markov process.

Fig. 9
Fig. 9

Given a fixed desired speckle size, the entropy of a detected speckle field can be maximized using a Cauchy mask with an easily determined autocorrelation parameter ρ. Table 1 offers calculated field entropies h(A) for several different autocorrelation functions assuming N = 100 and ρ = .9. *Note since the sinc autocorrelation’s covariance matrix J is singular, its entropy is calculated using only J’s nonzero eigenvalues.

Equations (36)

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

p ( A ) = 1 2 π σ 0 2 e | A | 2 / 2 σ 0 2 ,
p ( A 1 , , A N ) = p ( A ) = 1 π N det ( J ) e A * J 1 A ,
J ( x 1 , x 2 ) = k λ 2 z 2 | M ( η ) | 2 exp ( 2 π j λ z ( η ( x 2 x 2 ) ) ) d η ,
J ( Δ x ) = k λ 2 z 2 Δ x λ z , η [ | M ( η ) | 2 ] ,
( A n | A n 1 , , A 1 ) = ( A n | A n 1 ) .
( A n | A n 1 , , A 1 , A n + 1 , , A N ) = ( A n | A n 1 , A n + 1 ) .
P m n = P m v P v n .
𝔼 ^ [ A n | A n 1 , , A n p , A n + 1 , , A n + q ] = 𝔼 ^ [ A n | A n 1 , A n + 1 ] ,
A n = k = 1 , 1 ρ k A n k + U n ,
p ( A n | A m , n m ) = J n n 1 π exp ( J n n 1 | A n + m n J n m 1 J n n 1 A m | 2 ) .
J = σ 0 2 I J 1 = I / σ 0 2 ,
p ( A n | A m , n m ) = J n n 1 π exp ( J n n 1 | A n + B 1 A n 1 + B 2 A n + 1 | 2 ) = p ( A n | A n 1 , A n + 1 ) ,
J e = σ 2 ( 1 ρ ρ 2 ρ N 1 ρ 1 ρ ρ N 2 ρ 2 ρ 1 ρ N 3 ρ N 1 ρ 2 ρ 1 ) .
J e 1 = 1 σ 2 ( 1 ρ 2 ) ( 1 ρ 0 0 ρ 1 + ρ 2 ρ 0 0 ρ 1 + ρ 2 0 0 0 ρ 1 ) .
J e ( Δ x ) = σ 2 ρ | Δ x | = σ 2 e γ o | Δ x | ,
| M ( η ) | 2 = ( σ λ z ) 2 k γ 2 η 2 + γ 2 ,
( A i , j | A k , l , ( k , l ) Ω ) = ( A i , j | A k , l , ( k , l ) Ψ i , j ) .
A i , j = ( k , k ) Ψ i , j ρ i k , j l A k , l + U i , j .
A i , j = ρ ( A i 1 , j + A i + 1 , j + A i , j 1 + A i , j + 1 ) + ρ 2 ( A i 1 , j 1 + A i 1 , j + 1 + A i + 1 , j 1 + A i + 1 , j + 1 ) + U i , j .
| M ( η , ξ ) | 2 = ( σ λ z ) 4 k 2 ( γ 2 η 2 + γ 2 ) ( γ 2 ξ 2 + γ 2 ) ,
( A n | A n + 1 , A n 1 , A n m ) = ( A n | A n + 1 , A n 1 )
P n , n 2 = P n , n 1 P n 1 , n 2
r = 1 2 | S | α , β | P ^ n , n 2 ( α , β ) ( P ^ n , n 1 P ^ n 1 , n 2 ) ( α , β ) | ,
1 2 [ Re ( J ) Im ( J ) Im ( J ) Re ( J ) ] = 1 2 [ J 0 0 J ] .
h ( A ) = h ( X , Y ) = h ( X ) + h ( Y ) = log [ ( 2 π e ) N det ( J / 2 ) ] .
ρ 1 = sinc ( δ / l c ) = sinc ( w δ / λ z ) .
A 0 ( x ) = A ( x ) Π ( x δ x ) d x = 1 δ A ( x ) rect ( x δ x δ ) d x ,
A ^ 0 ( ν x ) = A ^ ( ν x ) Π ^ ( δ ν x ) = A ^ ( ν x ) sinc ( δ ν x ) ,
J ^ 0 ( ν x ) = [ J 0 ( Δ x ) ] = [ ( A 0 ( x ) A 0 * ( x ) ) Δ x ]
= [ ( A ( x ) Π ( x ) ) ( A * ( x ) Π * ( x ) ) ] = J ^ ( ν x ) sinc 2 ( δ ν x ) .
p ( A ) = p ( A 1 , A 2 , , A n ) = p ( A 1 ) p ( A 2 ) p ( A n ) ,
( A n | A n + 1 , A n 1 , A n 2 ) = ( A n | A n + 1 , A n 1 ) ,
( A n | A n 1 , A n 2 ) = ( A n | A n 1 ) .
( A n , A n 1 , A n 2 ) = ( A n 2 ) ( A n 1 | A n 2 ) ( A n | A n 1 ) .
( A n | A n 2 ) = a n 1 ( A n 1 | A n 2 ) ( A n | A n 1 ) .
p ( I 1 , , I n ) = exp [ 1 n ω i I i i < j ω i j max ( I i , I j ) i < j < k ω i j k max ( I i , I j , I k ) ω 12 n max ( I 1 , I 2 , , I n ) ] ,

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