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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 24 — Nov. 19, 2012
  • pp: 26394–26410

Markov speckle for efficient random bit generation

Roarke Horstmeyer, Richard Y. Chen, Benjamin Judkewitz, and Changhuei Yang  »View Author Affiliations

Optics Express, Vol. 20, Issue 24, pp. 26394-26410 (2012)

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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

OCIS Codes
(030.6140) Coherence and statistical optics : Speckle
(110.6150) Imaging systems : Speckle imaging

ToC Category:
Coherence and Statistical Optics

Original Manuscript: October 1, 2012
Revised Manuscript: October 30, 2012
Manuscript Accepted: October 31, 2012
Published: November 8, 2012

Roarke Horstmeyer, Richard Y. Chen, Benjamin Judkewitz, and Changhuei Yang, "Markov speckle for efficient random bit generation," Opt. Express 20, 26394-26410 (2012)

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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]

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