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Fig. 3. Input–output image rms error E SY of Eq. (7) as a function of the rms amplitude 2 N of the speckle noise Nu,v for various values of the input image contrast R 1/R 0. Solid lines stand for the theoretical expression of Eq. (7). The table gives the speckle noise optimal rms amplitude of Eq. (8). The discrete data sets (circles) are obtained by injecting in Eq. (1) real speckle images collected from the experimental setup of Fig. 1. The other parameters are =0.75, p 1=0.6, R 1=1. Nopt = R 1 − R 0 R 0R 1 lnK a , K a = R 1 R 0 1−p 1 p 1 . 8 As seen in Eq. (8), there exist domains where the optimal speckle noise level Nopt is nonzero and positive when 0, p 11, R 0R 1 if K a1. One can check in Fig. 3 that the positions of the optimal speckle noise level 2 Nopt given by Eq. (8) show an exact agreement with the numerical calculations. Finally, a visual appreciation of the cooperative effect of the speckle noise quantitatively illustrated in Fig. 3 is also presented in Fig. 2, where the multiplicative speckle noise injected in Eq. (1) comes from real speckle images collected from the experimental setup of Fig. 1. We have demonstrated, theoretically and experimentally, the possibility of a constructive action of the multiplicative speckle noise in the transmission of an image in a coherent imaging system. For what July 15, 2007 / Vol. 32, No. 14 / OPTICS LETTERS 1985 we believe is a first report of this effect, the models for the input image, for the speckle noise, and for the imaging sensor have been purposely taken in their most simple forms. As a result, we obtained a theoretical prediction of the constructive role of the speckle noise, through an explicit theoretical analysis of the behavior of a relevant input–output similarity measure. The theoretical predictions displayed close agreement with experiment. A noticeable feature, in particular, is that our theoretical model authorizes an explicit derivation, without approximation, of an analytical expression (an outcome rarely accessible in studies of stochastic resonance in nonlinear systems) for the optimal level of the noise maximizing the performance in given conditions. The present demonstration of the feasibility of a constructive action of speckle noise in coherent imaging can be extended in various directions. More sophisticated images (with distributed gray levels, for example) could be considered, as well as other types of speckle noise, such as the one appearing in polarimetric imaging [10]. The simple threshold detector chosen here could be replaced by a multilevel quantizer or a linear sensor with saturation, closely matching attributes of digital cameras. It would then be interesting to confront, as done here, experiment and theoretical modeling, and examine how the phenomenon of improvement by noise evolves in these other conditions. References 1. L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998). 2. F. Chapeau-Blondeau and D. Rousseau, Fluct. Noise Lett. 2, 221 (2002). 3. B. M. Jost and B. E. A. Saleh, Opt. Lett. 21, 287 (1996). 4. F. Vaudelle, J. Gazengel, G. Rivoire, X. Godivier, and F. Chapeau-Blondeau, J. Opt. Soc. Am. B 13, 2674 (1998). 5. F. Moss, L. M. Ward, and W. G. Sannita, Clin. Neurophysiol. 115, 267 (2004). 6. R. Etchnique and J. Aliaga, Am. J. Phys. 72, 159 (2004). 7. A. Histace and D. Rousseau, Electron. Lett. 42, 393 (2006). 8. Y. Jia, S. N. Yu, and J.-R. Li, Phys. Rev. E 62, 1869 (2000). 9. J. W. Goodman, Statistical Optics (Wiley, 1985). 10. F. Goudail and P. Réfrégier, Opt. Lett. 26, 644 (2001). 133/197
F. CHAPEAU-BLONDEAU, D. ROUSSEAU, S. BLANCHARD, and D. GINDRE. Optimizing the speckle noise for maximum efficacy of data acquisition in coherent imaging. Journal of the Optical Society of America A, vol. 25:1287–1292, 2008. 134/197
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