Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Nonlinear self-filtering of noisy images via dynamical stochastic resonance

Nature Photonics volume 4, pages 323–328 (2010)Cite this article

Subjects

Abstract

From night vision and objects overwhelmed by sunlight to jammed signals and those that are purposely encrypted, detecting low-level or hidden signals is a fundamental problem in imaging. Here, we develop and exploit a new type of stochastic resonance, in which nonlinear coupling allows signals to grow at the expense of noise, to recover noise-hidden images propagating in a self-focusing medium. The growth rate is derived analytically by treating the signal–noise interaction as a photonic beam–plasma instability and matches experimentally measured resonances in coupling strength, noise statistics and modal content of the signal. This is the first observation of nonlinear intensity exchange between coherent and spatially incoherent light and the first demonstration of spatial coherence resonance for a dynamically evolving signal. The results suggest a general method of reconstructing images through seeded instability and confirm information limits predicted, but not yet observed, in nonlinear communications systems.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Experimental set-up.
Figure 2: Nonlinear self-filtering as a function of signal–noise coupling.
Figure 3: Nonlinear self-filtering as a function of noise.
Figure 4: Measurement versus theory of dynamical stochastic resonance.

Similar content being viewed by others

References

  1. Gammaitoni, L., Hanggi, P., Jung, P. & Marchesoni, F. Stochastic resonance. Rev. Mod. Phys. 70, 223–287 (1998).

    Article  ADS  Google Scholar 

  2. Benzi, R., Parisi, G., Sutera, A. & Vulpiani, A. Stochastic resonance in climatic-change. Tellus 34, 10–16 (1982).

    Article  ADS  Google Scholar 

  3. Nicolis, C. Stochastic aspects of climatic transitions—response to a periodic forcing. Tellus 34, 1–9 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  4. Fauve, S. & Heslot, F. Stochastic resonance in a bistable system. Phys. Lett. A 97, 5–7 (1983).

    Article  ADS  Google Scholar 

  5. Bezrukov, S. M. & Vodyanoy, I. Stochastic resonance in non-dynamical systems without response thresholds. Nature 385, 319–321 (1997).

    Article  ADS  Google Scholar 

  6. Douglass, J. K., Wilkens, L., Pantazelou, E. & Moss, F. Noise enhancement of information-transfer in crayfish mechanoreceptors by stochastic resonance. Nature 365, 337–340 (1993).

    Article  ADS  Google Scholar 

  7. Bulsara, A. R., Elston, T. C., Doering, C. R., Lowen, S. B. & Lindenberg, K. Cooperative behavior in periodically driven noisy integrate-fire models of neuronal dynamics. Phys. Rev. E 53, 3958–3969 (1996).

    Article  ADS  Google Scholar 

  8. Simonotto, E. et al. Visual perception of stochastic resonance. Phys. Rev. Lett. 78, 1186–1189 (1997).

    Article  ADS  Google Scholar 

  9. Vaudelle, F., Gazengel, J., Rivoire, G., Godivier, X. & Chapeau-Blondeau, F. Stochastic resonance and noise-enhanced transmission of spatial signals in optics: the case of scattering. J. Opt. Soc. Am. B 15, 2674–2680 (1998).

    Article  ADS  Google Scholar 

  10. Blanchard, S., Rousseau, D., Gindre, D. & Chapeau-Blondeau, F. Constructive action of the speckle noise in a coherent imaging system. Opt. Lett. 32, 1983–1985 (2007).

    Article  ADS  Google Scholar 

  11. Mitchell, M., Chen, Z. G., Shih, M. F. & Segev, M. Self-trapping of partially spatially incoherent light. Phys. Rev. Lett. 77, 490–493 (1996).

    Article  ADS  Google Scholar 

  12. Christodoulides, D. N., Coskun, T. H., Mitchell, M. & Segev, M. Theory of incoherent self-focusing in biased photorefractive media. Phys. Rev. Lett. 78, 646–649 (1997).

    Article  ADS  Google Scholar 

  13. Shkunov, V. V. & Anderson, D. Z. Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media. Phys. Rev. Lett. 81, 2683–2686 (1998).

    Article  ADS  Google Scholar 

  14. Sukhorukov, A. A. & Akhmediev, N. N. Coherent and incoherent contributions to multisoliton complexes. Phys. Rev. Lett. 83, 4736–4739 (1999).

    Article  ADS  Google Scholar 

  15. Mendonca, J. T. & Tsintsadze, N. L. Analog of the Wigner–Moyal equation for the electromagnetic field. Phys. Rev. E 62, 4276–4282 (2000).

    Article  ADS  Google Scholar 

  16. Soljacic, M., Segev, M., Coskun, T., Christodoulides, D. N. & Vishwanath, A. Modulation instability of incoherent beams in noninstantaneous nonlinear media. Phys. Rev. Lett. 84, 467–470 (2000).

    Article  ADS  Google Scholar 

  17. Kip, D., Soljacic, M., Segev, M., Eugenieva, E. & Christodoulides, D. N. Modulation instability and pattern formation in spatially incoherent light beams. Science 290, 495–498 (2000).

    Article  ADS  Google Scholar 

  18. Coskun, T. H., Grandpierre, A. G., Christodoulides, D. N. & Segev, M. Coherence enhancement of spatially incoherent light beams through soliton interactions. Opt. Lett. 25, 826–828 (2000).

    Article  ADS  Google Scholar 

  19. Coskun, T. H. et al. Bright spatial solitons on a partially incoherent background. Phys. Rev. Lett. 84, 2374–2377 (2000).

    Article  ADS  Google Scholar 

  20. Fedele, R. & Anderson, D. A quantum-like Landau damping of an electromagnetic wavepacket. J. Opt. B 2, 207–213 (2000).

    Article  ADS  Google Scholar 

  21. Cohen, O. et al. Observation of random-phase lattice solitons. Nature 433, 500–503 (2005).

    Article  ADS  Google Scholar 

  22. Dylov, D. V. & Fleischer, J. W. Observation of all-optical bump-on-tail instability. Phys. Rev. Lett. 100, 103903 (2008).

    Article  ADS  Google Scholar 

  23. Dylov, D. V. & Fleischer, J. W. Multiple-stream instabilities and soliton turbulence in photonic plasma. Phys. Rev. A 78, 061804 (2008).

    Article  ADS  Google Scholar 

  24. Mcnamara, B., Wiesenfeld, K. & Roy, R. Observation of stochastic resonance in a ring laser. Phys. Rev. Lett. 60, 2626–2629 (1988).

    Article  ADS  Google Scholar 

  25. Martienssen, W. & Spiller, E. Coherence + fluctuations in light beams. Am. J. Phys. 32, 919–926 (1964).

    Article  ADS  Google Scholar 

  26. Bulsara, A. R. & Gammaitoni, L. Tuning in to noise. Phys. Today 49, 39–45 (1996).

    Article  Google Scholar 

  27. Mitra, P. P. & Stark, J. B. Nonlinear limits to the information capacity of optical fibre communications. Nature 411, 1027–1030 (2001).

    Article  ADS  Google Scholar 

  28. Essiambre, R. J., Foschini, G. J., Kramer, G. & Winzer, P. J. Capacity limits of information transport in fiber-optic networks. Phys. Rev. Lett. 101, 163901 (2008).

    Article  ADS  Google Scholar 

  29. Jia, S., Wan, W. & Fleischer, J. W. Forward four-wave mixing with defocusing nonlinearity. Opt. Lett. 32, 1668–1670 (2007).

    Article  ADS  Google Scholar 

  30. Tsang, M., Psaltis, D. & Omenetto, F. G. Reverse propagation of femtosecond pulses in optical fibers. Opt. Lett. 28, 1873–1875 (2003).

    Article  ADS  Google Scholar 

  31. Barsi, C., Wan, W. & Fleischer, J. W. Imaging through nonlinear media using digital holography. Nature Photon. 3, 211–215 (2009).

    Article  ADS  Google Scholar 

  32. Weilnau, C. & Denz, C. Solitary beam formation with partially coherent light in an anisotropic photorefractive medium. J. Opt. A 5, S529–S535 (2003).

    Article  ADS  Google Scholar 

  33. Vedenov, A. A. & Rudakov, L. I. Wave interaction in continuous media. Doklady Akademii Nauk Sssr 159, 767–770 (1964).

    Google Scholar 

  34. O'Neil, T. Collisionless damping of nonlinear plasma oscillations. Phys. Fluids 8, 2255–2262 (1965).

    Article  ADS  MathSciNet  Google Scholar 

  35. Chen, Z., Klinger, J. & Christodoulides, D. N. Induced modulation instability of partially spatially incoherent light with varying perturbation periods. Phys. Rev. E 66, 066601 (2002).

    Article  ADS  Google Scholar 

  36. Sun, C., Dylov, D. V. & Fleischer, J. W. Nonlinear focusing and defocusing of partially-coherent spatial beams. Opt. Lett. 34, 3003–3005 (2009).

    Article  ADS  Google Scholar 

  37. Cross, M. C. & Hohenberg, P. C. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993).

    Article  ADS  Google Scholar 

  38. Park, K., Lai, Y. C., Liu, Z. H. & Nachman, A. Aperiodic stochastic resonance and phase synchronization. Phys. Lett. A 326, 391–396 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  39. Gilbreath, G. C. & Reintjes, J. F. Photorefractive Fourier image amplification for low-light-level image detection. Microwave Opt. Technol. Lett. 12, 119–123 (1996).

    Article  Google Scholar 

  40. Heebner, J. E. & Boyd, R. W. Photorefractive optical recycling for contrast enhancement. Opt. Commun. 182, 243–247 (2000).

    Article  ADS  Google Scholar 

  41. Breugnot, S., Rajbenbach, H., Defour, M. & Huignard, J. P. Low-noise photorefractive amplification and detection of very weak signal beams. Opt. Lett. 20, 447–449 (1995).

    Article  ADS  Google Scholar 

  42. Shiratori, A. & Obara, M. Photorefractive coherence-gated interferometry. Rev. Sci. Instrum. 69, 3741–3745 (1998).

    Article  ADS  Google Scholar 

  43. Carrillo, O., Santos, M. A., Garcia-Ojalvo, J. & Sancho, J. M. Spatial coherence resonance near pattern-forming instabilities. Europhys. Lett. 65, 452–458 (2004).

    Article  ADS  Google Scholar 

  44. Freund, J. A., Schimansky-Geier, L. & Hanggi, P. Frequency and phase synchronization in stochastic systems. Chaos 13, 225–238 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  45. Giacomelli, G., Giudici, M., Balle, S. & Tredicce, J. R. Experimental evidence of coherence resonance in an optical system. Phys. Rev. Lett. 84, 3298–3301 (2000).

    Article  ADS  Google Scholar 

  46. Wu, B. B., Prucnal, P. R. & Narimanov, E. E. Secure transmission over an existing public WDM lightwave network. IEEE Photon. Technol. Lett. 18, 1870–1872 (2006).

    Article  ADS  Google Scholar 

  47. Yang, Y. B., Jiang, Z. P., Xu, B. H. & Repperger, D. W. An investigation of two-dimensional parameter-induced stochastic resonance and applications in nonlinear image processing. J. Phys. A 42, 145207 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  48. Deco, G. & Schurmann, B. Stochastic resonance in the mutual information between input and output spike trains of noisy central neurons. Physica D 117, 276–282 (1998).

    Article  ADS  Google Scholar 

  49. Fraser, A. M. Reconstructing attractors from scalar time-series—a comparison of singular system and redundancy criteria. Physica D 34, 391–404 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  50. García-Ojalvo, J. & Sancho, J. M. Noise in Spatially Extended Systems (Springer, 1999).

    Book  Google Scholar 

Download references

Acknowledgements

The authors would like to thank S. Verdú, E.E. Narimanov and P.R. Prucnal for valuable discussions. This work was supported by the National Science Foundation, the Department of Energy, and the Air Force Office of Scientific Research.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering and Program in Applied and Computational Mathematics, Princeton University, Princeton, 08544, New Jersey, USA

    Dmitry V. Dylov & Jason W. Fleischer

Authors
  1. Dmitry V. Dylov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jason W. Fleischer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jason W. Fleischer.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dylov, D., Fleischer, J. Nonlinear self-filtering of noisy images via dynamical stochastic resonance. Nature Photon 4, 323–328 (2010). https://doi.org/10.1038/nphoton.2010.31

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphoton.2010.31

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing