Abstract

A theory is presented which relates the minimum detectable contrast level for an object in the presence of noise to the statistics of the speckle. Consideration is given to smoothing of the noise by multiple looks and by area. Measurements of the minimum detectable contrast are made for two types of speckle noise. First, a coherent, plane wave is added to an ideal diffuse wave and the threshold of detection is established as a function of the beam ratio. Secondly, these results are compared to the technique of speckle smoothing using an N-fold intensity superposition of fully developed speckle pattens. Good agreement of experiments with theory is observed.

© 1976 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. S. McKechnie, “Speckle Reduction,” in Laser Speckle, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), p. 123.
    [Crossref]
  2. J. C. Dainty, “Detection of images immersed in speckle noise,” Opt. Acta 18, 327–339 (1971).
    [Crossref]
  3. A. Kozma and C. R. Christensen, “The effects of speckle on resolution,” J. Opt. Soc. Am. 66, 1257–1260 (1976) (this issue).
    [Crossref]
  4. Albert Rose, Vision, Human and Electronic (PlenumNew York, 1974).
  5. N. George and A. Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201–212 (1974).
    [Crossref]
  6. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Ref. 1, p. 9.
  7. J. M. Burch, “Interferometry with Scattered Light,” in Optical Instruments and Techniques, edited by J. Home Dickson (Oriel, Newcastle upon Tyne, England, 1970).
  8. J. D. Briers, “A note on the statistics of laser speckle patterns,” Opt. Quantum Electron. 7, 422–424 (1975).
    [Crossref]
  9. A. Papoulis, “Probability, Random Variables, and Stochastic Processes,” (McGraw-Hill, New York, 1965).
  10. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, (Academic, New York, 1965).
  11. N. George, A. Jain, and R. D. S. Melville, “Experiments on the Space and Wavelength Dependence of Speckle,” Appl. Phys. 7, 157–169 (1975).
    [Crossref]
  12. POTA is composed of 1-phenyl-3-pyrazolidone, 1.5 g; sodium sulfite, 30 g; and cold water (25 °C) to make 1000 cm3.
  13. S. Lowenthal and D. Joyeux, “Speckle removal by a slowly moving diffuser associated with a motionless diffuser,” J. Opt. Soc. Am. 61, 847–851 (1971).
    [Crossref]
  14. E. G. Rawson, A. B. Nafarrate, R. E. Norton, and J. W. Goodman, “Speckle-free rear-projection screen using two close screens with slow relative motion,” J. Opt. Soc. Am. 66, 1290–1294 (1976) (following article).
    [Crossref]
  15. C. R. Christensen, Nicholas George, B. D. Guenther, and J. S. Bennett, “Noise in Coherent Optical Systems: Minimum Detectable Object Contrast and Speckle Smoothing,” , Redstone Arsenal, 1976.

1976 (2)

1975 (2)

J. D. Briers, “A note on the statistics of laser speckle patterns,” Opt. Quantum Electron. 7, 422–424 (1975).
[Crossref]

N. George, A. Jain, and R. D. S. Melville, “Experiments on the Space and Wavelength Dependence of Speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

1974 (1)

N. George and A. Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

1971 (2)

Bennett, J. S.

C. R. Christensen, Nicholas George, B. D. Guenther, and J. S. Bennett, “Noise in Coherent Optical Systems: Minimum Detectable Object Contrast and Speckle Smoothing,” , Redstone Arsenal, 1976.

Briers, J. D.

J. D. Briers, “A note on the statistics of laser speckle patterns,” Opt. Quantum Electron. 7, 422–424 (1975).
[Crossref]

Burch, J. M.

J. M. Burch, “Interferometry with Scattered Light,” in Optical Instruments and Techniques, edited by J. Home Dickson (Oriel, Newcastle upon Tyne, England, 1970).

Christensen, C. R.

A. Kozma and C. R. Christensen, “The effects of speckle on resolution,” J. Opt. Soc. Am. 66, 1257–1260 (1976) (this issue).
[Crossref]

C. R. Christensen, Nicholas George, B. D. Guenther, and J. S. Bennett, “Noise in Coherent Optical Systems: Minimum Detectable Object Contrast and Speckle Smoothing,” , Redstone Arsenal, 1976.

Dainty, J. C.

J. C. Dainty, “Detection of images immersed in speckle noise,” Opt. Acta 18, 327–339 (1971).
[Crossref]

George, N.

N. George, A. Jain, and R. D. S. Melville, “Experiments on the Space and Wavelength Dependence of Speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

N. George and A. Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

George, Nicholas

C. R. Christensen, Nicholas George, B. D. Guenther, and J. S. Bennett, “Noise in Coherent Optical Systems: Minimum Detectable Object Contrast and Speckle Smoothing,” , Redstone Arsenal, 1976.

Goodman, J. W.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, (Academic, New York, 1965).

Guenther, B. D.

C. R. Christensen, Nicholas George, B. D. Guenther, and J. S. Bennett, “Noise in Coherent Optical Systems: Minimum Detectable Object Contrast and Speckle Smoothing,” , Redstone Arsenal, 1976.

Jain, A.

N. George, A. Jain, and R. D. S. Melville, “Experiments on the Space and Wavelength Dependence of Speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

N. George and A. Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

Joyeux, D.

Kozma, A.

Lowenthal, S.

McKechnie, T. S.

T. S. McKechnie, “Speckle Reduction,” in Laser Speckle, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), p. 123.
[Crossref]

Melville, R. D. S.

N. George, A. Jain, and R. D. S. Melville, “Experiments on the Space and Wavelength Dependence of Speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

Nafarrate, A. B.

Norton, R. E.

Papoulis, A.

A. Papoulis, “Probability, Random Variables, and Stochastic Processes,” (McGraw-Hill, New York, 1965).

Rawson, E. G.

Rose, Albert

Albert Rose, Vision, Human and Electronic (PlenumNew York, 1974).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, (Academic, New York, 1965).

Appl. Phys. (2)

N. George and A. Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201–212 (1974).
[Crossref]

N. George, A. Jain, and R. D. S. Melville, “Experiments on the Space and Wavelength Dependence of Speckle,” Appl. Phys. 7, 157–169 (1975).
[Crossref]

J. Opt. Soc. Am. (3)

Opt. Acta (1)

J. C. Dainty, “Detection of images immersed in speckle noise,” Opt. Acta 18, 327–339 (1971).
[Crossref]

Opt. Quantum Electron. (1)

J. D. Briers, “A note on the statistics of laser speckle patterns,” Opt. Quantum Electron. 7, 422–424 (1975).
[Crossref]

Other (8)

A. Papoulis, “Probability, Random Variables, and Stochastic Processes,” (McGraw-Hill, New York, 1965).

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, (Academic, New York, 1965).

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Ref. 1, p. 9.

J. M. Burch, “Interferometry with Scattered Light,” in Optical Instruments and Techniques, edited by J. Home Dickson (Oriel, Newcastle upon Tyne, England, 1970).

T. S. McKechnie, “Speckle Reduction,” in Laser Speckle, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), p. 123.
[Crossref]

Albert Rose, Vision, Human and Electronic (PlenumNew York, 1974).

C. R. Christensen, Nicholas George, B. D. Guenther, and J. S. Bennett, “Noise in Coherent Optical Systems: Minimum Detectable Object Contrast and Speckle Smoothing,” , Redstone Arsenal, 1976.

POTA is composed of 1-phenyl-3-pyrazolidone, 1.5 g; sodium sulfite, 30 g; and cold water (25 °C) to make 1000 cm3.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

FIG. 1
FIG. 1

Setup for measuring thresholds of object contrast for speckled images. The test object is imaged at unity magnification with a small pupil of diameter D to control the size of the speckle.

FIG. 2
FIG. 2

Test object with intensity transmittances and disk sizes as labeled.

FIG. 3
FIG. 3

Model for area averaging of speckle. The signal T1 is a dark spot in a bright background T2 ≈ 1. The average length of a speckle is s and the signal extent is X.

FIG. 4
FIG. 4

Superposition of N independent speckle patterns using an ensemble of diffusers as described in Sec. IVB1.

FIG. 5
FIG. 5

Contrast required to detect a disk of a given diameter as a function of the number of independent speckle pattern superpositions used to record the image.

FIG. 6
FIG. 6

Test pattern images formed by superposition of N-independent speckle fields compared with images formed when the illumination is a plane wave and a diffuse beam.

FIG. 7
FIG. 7

Contrast required to detect a disk of a given diameter as a function of the ratio R of the plane wave beam intensity to the diffuse beam intensity.

Tables (1)

Tables Icon

TABLE I Density function and contrast ratio for speckled illumination in the image plane. The brightness limited case is included for white light. Tabulation is for f(u) when u ≥ 0; f(u) = 0 when u < 0. n0 is the number of photons crossing per unit area; u0 is the plane wave intensity; us is the speckle beam, average intensity; and R =u0/us.

Equations (24)

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

I ( x ) = T 1 u ( x ) .
u = - u f ( u ) d u ,
u 2 = - u 2 f ( u ) d u ,
σ u 2 = u 2 - u 2 ,
F ( η ) = - f ( u ) e i η a d u .
σ 1 = T 1 σ u .
( T 2 - T 1 ) u 1 2 σ u ( T 1 + T 2 ) .
( T 1 - T 2 ) u = O s n σ u T 12 ,
N = ( A / s 2 ) M .
( T 2 - T 1 ) u A = O s n σ u T 12 A [ ( A / s 2 ) M ] 1 / 2 .
T 2 - T 1 T 12 = O s n σ u u × 1 ( A M / s 2 ) 1 / 2 .
f ( u ) = ( e - u / a / a ) H ( u ) ,
H ( u ) = { 1 , u 0 , 0 , u < 0.
u = ( 1 / N ) ( u 1 + u 2 + + u N ) .
exp ( i η u ) = exp ( i η u 1 / N ) exp ( i η u N / N ) .
f ( u k ) = ( e - u k / a / a ) H ( u ) ;
F k ( η N ) = - f ( u k ) e i η u k / N H ( u k ) d u k , F k ( η / N ) = ( 1 - i η a / N ) - 1 .
F ( η ) = [ ( 1 - i η a / N ) - 1 ] N .
f ( u ) = [ ( N u ) N - 1 e - N u / a / a N ( N - 1 ) ! ] H ( u ) .
u = a ,             u 2 = ( 1 + 1 / N ) a 2 ,             σ 2 = a 2 / N .
f ( u ) = exp [ - ( u + u 0 ) / u s ] I 0 [ 2 ( u u 0 ) 1 / 2 / u s ] H ( u ) / u s ,
R = u 0 / u s .
1 / N 1 / 2 = ( 1 + 2 R ) 1 / 2 / ( 1 + R ) .
R = N - 1 + [ N ( N - 1 ) ] 1 / 2 ,             R 2 N - 3 2 .