Abstract

A coherent imaging system with speckle noise is devised and analyzed. This demonstrates the possibility of improving the nonlinear transmission of a coherent image by increasing the level of the multiplicative speckle noise. This noise-assisted image transmission is a novel instance of stochastic resonance phenomena by which nonlinear signal processing benefits from a constructive action of noise.

© 2007 Optical Society of America

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References

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  1. L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
    [CrossRef]
  2. F. Chapeau-Blondeau and D. Rousseau, Fluct. Noise Lett. 2, 221 (2002).
    [CrossRef]
  3. B. M. Jost and B. E. A. Saleh, Opt. Lett. 21, 287 (1996).
    [CrossRef] [PubMed]
  4. F. Vaudelle, J. Gazengel, G. Rivoire, X. Godivier, and F. Chapeau-Blondeau, J. Opt. Soc. Am. B 13, 2674 (1998).
    [CrossRef]
  5. F. Moss, L. M. Ward, and W. G. Sannita, Clin. Neurophysiol. 115, 267 (2004).
    [CrossRef] [PubMed]
  6. R. Etchnique and J. Aliaga, Am. J. Phys. 72, 159 (2004).
    [CrossRef]
  7. A. Histace and D. Rousseau, Electron. Lett. 42, 393 (2006).
    [CrossRef]
  8. Y. Jia, S. N. Yu, and J.-R. Li, Phys. Rev. E 62, 1869 (2000).
    [CrossRef]
  9. J. W. Goodman, Statistical Optics (Wiley, 1985).
  10. F. Goudail and P. Réfrégier, Opt. Lett. 26, 644 (2001).
    [CrossRef]

2006 (1)

A. Histace and D. Rousseau, Electron. Lett. 42, 393 (2006).
[CrossRef]

2004 (2)

F. Moss, L. M. Ward, and W. G. Sannita, Clin. Neurophysiol. 115, 267 (2004).
[CrossRef] [PubMed]

R. Etchnique and J. Aliaga, Am. J. Phys. 72, 159 (2004).
[CrossRef]

2002 (1)

F. Chapeau-Blondeau and D. Rousseau, Fluct. Noise Lett. 2, 221 (2002).
[CrossRef]

2001 (1)

2000 (1)

Y. Jia, S. N. Yu, and J.-R. Li, Phys. Rev. E 62, 1869 (2000).
[CrossRef]

1998 (2)

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
[CrossRef]

F. Vaudelle, J. Gazengel, G. Rivoire, X. Godivier, and F. Chapeau-Blondeau, J. Opt. Soc. Am. B 13, 2674 (1998).
[CrossRef]

1996 (1)

Aliaga, J.

R. Etchnique and J. Aliaga, Am. J. Phys. 72, 159 (2004).
[CrossRef]

Chapeau-Blondeau, F.

F. Chapeau-Blondeau and D. Rousseau, Fluct. Noise Lett. 2, 221 (2002).
[CrossRef]

F. Vaudelle, J. Gazengel, G. Rivoire, X. Godivier, and F. Chapeau-Blondeau, J. Opt. Soc. Am. B 13, 2674 (1998).
[CrossRef]

Etchnique, R.

R. Etchnique and J. Aliaga, Am. J. Phys. 72, 159 (2004).
[CrossRef]

Gammaitoni, L.

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
[CrossRef]

Gazengel, J.

F. Vaudelle, J. Gazengel, G. Rivoire, X. Godivier, and F. Chapeau-Blondeau, J. Opt. Soc. Am. B 13, 2674 (1998).
[CrossRef]

Godivier, X.

F. Vaudelle, J. Gazengel, G. Rivoire, X. Godivier, and F. Chapeau-Blondeau, J. Opt. Soc. Am. B 13, 2674 (1998).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Goudail, F.

Hänggi, P.

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
[CrossRef]

Histace, A.

A. Histace and D. Rousseau, Electron. Lett. 42, 393 (2006).
[CrossRef]

Jia, Y.

Y. Jia, S. N. Yu, and J.-R. Li, Phys. Rev. E 62, 1869 (2000).
[CrossRef]

Jost, B. M.

Jung, P.

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
[CrossRef]

Li, J.-R.

Y. Jia, S. N. Yu, and J.-R. Li, Phys. Rev. E 62, 1869 (2000).
[CrossRef]

Marchesoni, F.

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
[CrossRef]

Moss, F.

F. Moss, L. M. Ward, and W. G. Sannita, Clin. Neurophysiol. 115, 267 (2004).
[CrossRef] [PubMed]

Réfrégier, P.

Rivoire, G.

F. Vaudelle, J. Gazengel, G. Rivoire, X. Godivier, and F. Chapeau-Blondeau, J. Opt. Soc. Am. B 13, 2674 (1998).
[CrossRef]

Rousseau, D.

A. Histace and D. Rousseau, Electron. Lett. 42, 393 (2006).
[CrossRef]

F. Chapeau-Blondeau and D. Rousseau, Fluct. Noise Lett. 2, 221 (2002).
[CrossRef]

Saleh, B. E. A.

Sannita, W. G.

F. Moss, L. M. Ward, and W. G. Sannita, Clin. Neurophysiol. 115, 267 (2004).
[CrossRef] [PubMed]

Vaudelle, F.

F. Vaudelle, J. Gazengel, G. Rivoire, X. Godivier, and F. Chapeau-Blondeau, J. Opt. Soc. Am. B 13, 2674 (1998).
[CrossRef]

Ward, L. M.

F. Moss, L. M. Ward, and W. G. Sannita, Clin. Neurophysiol. 115, 267 (2004).
[CrossRef] [PubMed]

Yu, S. N.

Y. Jia, S. N. Yu, and J.-R. Li, Phys. Rev. E 62, 1869 (2000).
[CrossRef]

Am. J. Phys. (1)

R. Etchnique and J. Aliaga, Am. J. Phys. 72, 159 (2004).
[CrossRef]

Clin. Neurophysiol. (1)

F. Moss, L. M. Ward, and W. G. Sannita, Clin. Neurophysiol. 115, 267 (2004).
[CrossRef] [PubMed]

Electron. Lett. (1)

A. Histace and D. Rousseau, Electron. Lett. 42, 393 (2006).
[CrossRef]

Fluct. Noise Lett. (1)

F. Chapeau-Blondeau and D. Rousseau, Fluct. Noise Lett. 2, 221 (2002).
[CrossRef]

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

F. Vaudelle, J. Gazengel, G. Rivoire, X. Godivier, and F. Chapeau-Blondeau, J. Opt. Soc. Am. B 13, 2674 (1998).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. E (1)

Y. Jia, S. N. Yu, and J.-R. Li, Phys. Rev. E 62, 1869 (2000).
[CrossRef]

Rev. Mod. Phys. (1)

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
[CrossRef]

Other (1)

J. W. Goodman, Statistical Optics (Wiley, 1985).

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Figures (3)

Fig. 1
Fig. 1

Experimental setup producing an optical version of the theoretical coherent imaging process of Eqs. (1, 2, 3). The λ 2 plate in association with the Glan-Taylor polarizer are used to control the intensity of the incident coherent wave coming from the second harmonic generation ( 532 nm , 10 mW ) of a YAG:Nd compact laser. The spatial filter is used to obtain a uniform intensity on the static diffuser taken as a frosted glass. The first lens is adjusted with a micrometer-scale sensitivity linear stage to control the size of the speckle grain in the object plane. In Figs. 2, 3 the speckle grain size has been adjusted to be much larger than the pixel size (the domain of validity of our model) and much smaller than the CCD matrix size (to diminish fluctuations from one acquisition to another). The object, a slide with calibrated transparency levels carrying the contrast of the input image S ( u , v ) , is illuminated by the speckled wave field. The second lens images the object plane on the CCD matrix of the camera. Variations of the speckle noise level in Figs. 2, 3 are controlled by rotation of the λ 2 plate.

Fig. 2
Fig. 2

Output image Y ( u , v ) of the hard limiter of Eq. (6) for increasing rms amplitude 2 σ N of the speckle noise N ( u , v ) . From left to right 2 σ N = 0.28 , 0.84 (optimal value), 2.81 ; with threshold θ = 0.75 , p 1 = 0.27 and { R 0 = 1 2 , R 1 = 1 } .

Fig. 3
Fig. 3

Input–output image rms error E S Y of Eq. (7) as a function of the rms amplitude 2 σ N of the speckle noise N ( u , 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 θ = 0.75 , p 1 = 0.6 , R 1 = 1 .

Equations (8)

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X ( u , v ) = S ( u , v ) × N ( u , v ) ,
p N ( j ) = 1 σ N exp ( j σ N ) , j 0 ,
Y ( u , v ) = g [ X ( u , v ) ] ,
E S Y = ( S Y ) 2 = S 2 2 S Y + Y 2 ,
S Y = s d s s p S ( s ) n d n g ( s × n ) p N ( n ) ,
g [ X ( u , v ) ] = { 0 for X ( u , v ) θ 1 for X ( u , v ) > θ .
E S Y = p 1 + q 1 2 p 1 p 11 ,
σ N opt = R 1 R 0 R 0 R 1 θ ln ( K a ) , K a = R 1 R 0 1 p 1 p 1 .

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