Abstract

Reduction of image speckle noise with the use of an integrative synthetic aperture imaging technique is studied. It is found that the Fourier inversion of the mutual intensity estimate [ Appl. Opt. 30, 206– 213 ( 1991)] yields an image intensity that corresponds exactly to the illumination of the object with partially coherent light from source optics imaging a delta-function incoherent source. An expression for the signal-to-noise ratio at an image point is derived for a large rough object with delta-function correlated amplitude reflection. A synthetic aperture system receiver of sufficiently small diameter yields image speckle with a signal-to-noise ratio (SNR) equal to 1. When the receiver and the transmitter diameters are equal, the SNR is 2 for linearly polarized speckle. The SNR continues to increase with receiver size and is linear in the diameter for large receiver-to-transmitter diameter ratios.

© 1992 Optical Society of America

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References

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  1. L. Sica, “Estimator and signal-to-noise ratio for an integrative synthetic aperture imaging technique,” Appl. Opt. 30, 206–213 (1991).
    [CrossRef] [PubMed]
  2. N. D. Ustinov, A. V. Anufriev, A. L. Vol’pov, Yu. A. Zimin, A. I. Tolmachev, “Active aperture synthesis in observation of objects via distorting media,” Sov. J. Quantum Electron. 17, 108–110 (1987).
    [CrossRef]
  3. C. C. Aleksoff, “Synthetic interferometric imaging technique for moving objects,” Appl. Opt. 15, 1923–1929 (1976).
    [CrossRef] [PubMed]
  4. A. Kozma, C. R. Christensen, “Effects of speckle on resolution,” J. Opt. Soc. Am. 66, 1257–1260 (1976).
    [CrossRef]
  5. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1984), p. 38.
  6. M. Born, E. Wolf, Principles of Optics, 5th ed., (Pergamon, Oxford, 1975) p. 529.
  7. G. O. Reynolds, D. J. Cronin, “Imaging with optical synthetic apertures (Mills–Cross analog),” J. Opt. Soc. Am. 60, 634–640 (1970).
    [CrossRef]
  8. M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), p. 35.
  9. H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London Ser. A 208, 263–277 (1951).
    [CrossRef]
  10. A. Papoulis, Probability Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 325.
  11. Ref. 5, p. 41. Analogously, the far-field statistics of speckle at the receiver depend on the intensity reflectivity and shape of the object. See L. I. Goldfischer, “Autocorrelation function and power spectral density of laser-produced speckle patterns,” J. Opt. Soc. Am. 55, 247–253 (1965).
    [CrossRef]
  12. J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970). For reviews of various speckle reduction methods see G. Parry, “Speckle patterns in partially coherent light,” and T. S. McKechnie, “Speckle reduction,” both in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1984).
    [CrossRef]
  13. The square aperture is also of interest because it is advantageous to know the object Fourier transform at points on a square grid for purposes of eventual Fourier inversion by the FFT algorithm.
  14. This plot has the same qualitative character as that given in Dainty’s paper in Ref. 12.

1991

1987

N. D. Ustinov, A. V. Anufriev, A. L. Vol’pov, Yu. A. Zimin, A. I. Tolmachev, “Active aperture synthesis in observation of objects via distorting media,” Sov. J. Quantum Electron. 17, 108–110 (1987).
[CrossRef]

1976

1970

G. O. Reynolds, D. J. Cronin, “Imaging with optical synthetic apertures (Mills–Cross analog),” J. Opt. Soc. Am. 60, 634–640 (1970).
[CrossRef]

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970). For reviews of various speckle reduction methods see G. Parry, “Speckle patterns in partially coherent light,” and T. S. McKechnie, “Speckle reduction,” both in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1984).
[CrossRef]

1965

Ref. 5, p. 41. Analogously, the far-field statistics of speckle at the receiver depend on the intensity reflectivity and shape of the object. See L. I. Goldfischer, “Autocorrelation function and power spectral density of laser-produced speckle patterns,” J. Opt. Soc. Am. 55, 247–253 (1965).
[CrossRef]

1951

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London Ser. A 208, 263–277 (1951).
[CrossRef]

Aleksoff, C. C.

Anufriev, A. V.

N. D. Ustinov, A. V. Anufriev, A. L. Vol’pov, Yu. A. Zimin, A. I. Tolmachev, “Active aperture synthesis in observation of objects via distorting media,” Sov. J. Quantum Electron. 17, 108–110 (1987).
[CrossRef]

Beran, M. J.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), p. 35.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed., (Pergamon, Oxford, 1975) p. 529.

Christensen, C. R.

Cronin, D. J.

Dainty, J. C.

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970). For reviews of various speckle reduction methods see G. Parry, “Speckle patterns in partially coherent light,” and T. S. McKechnie, “Speckle reduction,” both in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1984).
[CrossRef]

Goldfischer, L. I.

Ref. 5, p. 41. Analogously, the far-field statistics of speckle at the receiver depend on the intensity reflectivity and shape of the object. See L. I. Goldfischer, “Autocorrelation function and power spectral density of laser-produced speckle patterns,” J. Opt. Soc. Am. 55, 247–253 (1965).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1984), p. 38.

Hopkins, H. H.

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London Ser. A 208, 263–277 (1951).
[CrossRef]

Kozma, A.

Papoulis, A.

A. Papoulis, Probability Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 325.

Parrent, G. B.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), p. 35.

Reynolds, G. O.

Sica, L.

Tolmachev, A. I.

N. D. Ustinov, A. V. Anufriev, A. L. Vol’pov, Yu. A. Zimin, A. I. Tolmachev, “Active aperture synthesis in observation of objects via distorting media,” Sov. J. Quantum Electron. 17, 108–110 (1987).
[CrossRef]

Ustinov, N. D.

N. D. Ustinov, A. V. Anufriev, A. L. Vol’pov, Yu. A. Zimin, A. I. Tolmachev, “Active aperture synthesis in observation of objects via distorting media,” Sov. J. Quantum Electron. 17, 108–110 (1987).
[CrossRef]

Vol’pov, A. L.

N. D. Ustinov, A. V. Anufriev, A. L. Vol’pov, Yu. A. Zimin, A. I. Tolmachev, “Active aperture synthesis in observation of objects via distorting media,” Sov. J. Quantum Electron. 17, 108–110 (1987).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed., (Pergamon, Oxford, 1975) p. 529.

Zimin, Yu. A.

N. D. Ustinov, A. V. Anufriev, A. L. Vol’pov, Yu. A. Zimin, A. I. Tolmachev, “Active aperture synthesis in observation of objects via distorting media,” Sov. J. Quantum Electron. 17, 108–110 (1987).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

G. O. Reynolds, D. J. Cronin, “Imaging with optical synthetic apertures (Mills–Cross analog),” J. Opt. Soc. Am. 60, 634–640 (1970).
[CrossRef]

A. Kozma, C. R. Christensen, “Effects of speckle on resolution,” J. Opt. Soc. Am. 66, 1257–1260 (1976).
[CrossRef]

Ref. 5, p. 41. Analogously, the far-field statistics of speckle at the receiver depend on the intensity reflectivity and shape of the object. See L. I. Goldfischer, “Autocorrelation function and power spectral density of laser-produced speckle patterns,” J. Opt. Soc. Am. 55, 247–253 (1965).
[CrossRef]

Opt. Acta

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970). For reviews of various speckle reduction methods see G. Parry, “Speckle patterns in partially coherent light,” and T. S. McKechnie, “Speckle reduction,” both in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1984).
[CrossRef]

Proc. R. Soc. London Ser. A

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London Ser. A 208, 263–277 (1951).
[CrossRef]

Sov. J. Quantum Electron.

N. D. Ustinov, A. V. Anufriev, A. L. Vol’pov, Yu. A. Zimin, A. I. Tolmachev, “Active aperture synthesis in observation of objects via distorting media,” Sov. J. Quantum Electron. 17, 108–110 (1987).
[CrossRef]

Other

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), p. 35.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1984), p. 38.

M. Born, E. Wolf, Principles of Optics, 5th ed., (Pergamon, Oxford, 1975) p. 529.

A. Papoulis, Probability Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 325.

The square aperture is also of interest because it is advantageous to know the object Fourier transform at points on a square grid for purposes of eventual Fourier inversion by the FFT algorithm.

This plot has the same qualitative character as that given in Dainty’s paper in Ref. 12.

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Figures (4)

Fig. 1
Fig. 1

Example of a possible system architecture. A transmitter of the sparse aperture type, a Mills-Cross, is embedded in the receiver plane. Mutually coherent beams illuminating the object from any two points on the cross result in interference fringes across the object.

Fig. 2
Fig. 2

Coordinates used in Eq. (1).

Fig. 3
Fig. 3

Equivalent optical system for the interpretation of Eq. (7). S is a delta-function incoherent source. The source optics, consisting of lens L1 and square mask M1 form an image on the x1 plane, where a partially transparent object is located. The intensity image of this object formed by lens L2 is observed in the x plane.

Fig. 4
Fig. 4

Plot of the SNR (inverse speckle contrast) versus the number of coherence patches within the point-spread-function area, L/Lτ.

Equations (32)

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J ˆ ( u 1 , u 2 ) = 1 ( 2 L ) 2 L L A ˜ ( u 1 + v λ R ) A ˜ * ( u 2 + v λ R ) d v .
J ˆ ( u 1 , u 2 ) = 1 ( 2 L ) 2 L L A ˜ ( u 1 + v λ R ) A ˜ * ( u 2 + v λ R ) d v ,
J ˆ ( u 1 , u 2 ) = 1 ( 2 L ) 2 L L J ( u 1 u 2 λ R ) d v = J ( u 1 u 2 λ R ) .
I ˆ ( x ) = P ( u 1 ) P * ( u 2 ) J ˆ ( u 1 , u 2 ) exp [ 2 π i x ( u 1 u 2 ) λ R ] d u 1 d u 2 ,
I ( x ) = P ( u 1 ) P * ( u 2 ) J ( u 1 u 2 ) exp [ 2 π i x ( u 1 u 2 ) λ R ] d u 1 d u 2 .
u 1 = u 1 1 + u 2 , d u 1 = d u 1 1 ,
I ( x ) = P ( u 1 1 + u 2 ) P * ( u 2 ) J ( u 1 1 ) exp [ 2 π i x · u 1 1 λ R ] d u 1 1 d u 2 .
I ( x ) = τ ( u 1 1 ) J ( u 1 1 ) exp [ 2 π i x · u 1 1 λ R ] d u 1 1 ,
A ˜ ( u i + v λ R ) = A ( x 1 ) exp [ 2 π i x 1 ( u i + v ) λ R ] d x 1 , ( i = 1 , 2 )
I ˆ ( x ) = P ( u 1 ) P ( u 2 ) exp [ 2 π i u 1 ( x 1 + x ) λ R ] × exp [ 2 π i u 2 ( x 2 + x ) λ R ] d u 1 d u 2 × A ( x 1 ) A * ( x 2 ) 1 ( 2 L ) 2 L L exp [ 2 π i v ( x 1 x 2 ) λ R ] d v d x 1 d x 2 .
I ˆ ( x ) = K ( x x 1 1 λ R ) K * ( x x 2 1 λ R ) A ( x 1 1 ) A * ( x 2 1 ) × sinc [ 2 π L ( x 1 1 x 2 1 ) λ R ] sinc [ 2 π L ( y 1 1 y 2 1 ) λ R ] d x 1 1 d x 2 1 .
I ˆ ( x ) 2 = P ( u 1 ) P * ( u 2 ) P ( u 3 ) P * ( u 4 ) J ˆ ( u 1 , u 2 ) J ˆ ( u 3 , u 4 ) × exp [ 2 π i x ( u 1 u 2 ) λ R ] exp [ 2 π i x ( u 3 u 4 ) λ R ] d u 1 d u 2 d u 3 d u 4 ,
J ˆ ( u 1 , u 2 ) J ˆ ( u 3 , u 4 ) = 1 ( 2 L ) 4 L L A ˜ ( u 1 + u 1 λ R ) A ˜ * ( u 2 + u 1 λ R ) × A ˜ * ( u 4 + u 11 λ R ) A ˜ ( u 3 + u 11 λ R ) d u 1 d u 11 .
J ˆ ( u 1 , u 2 ) J ˆ ( u 3 , u 4 ) = J ( u 1 u 2 λ R ) J ( u 3 u 4 λ R ) + 1 ( 2 L ) 4 L L × J ( u 1 u 4 + u 1 u 11 λ R ) J ( u 3 u 2 + u 11 u 1 λ R ) ] d u 1 d u 11 .
σ 2 = I ˆ 2 I ˆ 2 = P ( u 1 ) P * ( u 2 ) P ( u 3 ) P * ( u 4 ) × 1 ( 2 L ) 4 L L J ( u 1 u 4 + u 1 u 11 ) × J ( u 3 u 2 + u 11 u 1 ) d u 1 d u 11 × exp [ 2 π i x ( u 1 + u 3 u 2 u 4 ) λ R ] d u 1 d u 2 d u 3 d u 4 .
u 1 = u 1 1 + u 4 , d u 1 = d u 1 1 , u 3 = u 3 1 + u 2 , d u 3 = d u 3 1 ,
σ 2 = τ ( u 1 1 ) τ ( u 3 1 ) 1 ( 2 L ) 4 L L × J ( u 1 1 + u 1 u 11 ) J [ u 3 1 ( u 1 u 11 ) ] d u 1 d u 11 d u 1 1 d u 3 1
σ 2 = τ ( u 1 1 ) τ ( u 3 1 ) 1 ( 2 L ) 2 2 L 2 L ( 1 | V x | 2 L ) × ( 1 | V y | 2 L ) J ( u 1 1 + V ) J [ u 3 1 V ] d V d u 1 1 d u 3 1 .
σ 2 = 1 ( 2 L ) 2 2 L 2 L ( 1 | V x | 2 L ) ( 1 | V y | 2 L ) × τ ( u 1 1 ) J ( u 1 1 + V s ) d u 1 1 τ ( u 3 1 ) J [ u 3 1 V s ] d u 3 1 d V .
J ( u x 1 u x 2 , u y 1 u y 2 ) = I o sinc [ 2 π a ( u x 1 u x 2 ) λ R ] × sinc [ 2 π a ( u y 1 u y 2 ) λ R ] .
σ 2 1 ( 2 L ) 2 2 L 2 L ( 1 | V x | 2 L ) ( 1 | V y | 2 L ) τ ( V ) τ ( V ) d V × J ( u 1 1 + V s ) d u 1 1 J ( u 3 1 V s ) d u 3 1 ,
σ 2 = I 0 2 s 4 ( 2 L ) 2 2 L 2 L ( 1 | V x | 2 L ) ( 1 | V y | 2 L ) τ 2 ( V ) d V
I ( 0 ) = t ( u 1 1 ) J ( u 1 1 s ) d u 1 1 = τ ( 0 ) J ( u 1 1 s ) d u 1 1 = τ ( 0 ) I o s 2 .
I ( 0 ) σ = 2 L [ 2 L 2 L ( 1 | V x | 2 L ) ( 1 | V y | 2 L ) τ 2 ( V ) τ 2 ( 0 ) d V x d V y ] 1 / 2 .
I ( 0 ) σ = 2 L [ 2 L 2 L ( 1 | V x | 2 L ) ( 1 | V y | 2 L ) d V x d V y ] 1 / 2 = 2 L ( 4 L 2 ) 1 / 2 = 1 ,
I ( 0 ) σ = lim L 2 L [ 2 L 2 L ( 1 | V x | 2 L ) ( 1 | V y | 2 L ) τ 2 ( V ) τ 2 ( 0 ) d V x d V y ] 1 / 2 = 2 L [ τ 2 ( V ) τ 2 ( 0 ) d V x d V y ] 1 l 2 ·
τ ( V ) = ( 2 L τ ) 2 ( 1 | V | 2 L τ ) ( 2 L τ ) 2 ( 1 | V x | 2 L τ ) ( 1 | V y | 2 L τ ) { | V x | 2 L τ | V y | 2 L τ , τ ( V ) = 0 otherwise ,
I ( 0 ) σ = 2 L 2 L 2 L ( 1 | V | 2 L ) ( 1 | V | 2 L τ ) 2 d V ,
I ( 0 ) σ = L 0 2 L τ ( 1 | V | 2 L ) ( 1 | V | 2 L τ ) 2 d V = ( L L τ ) 2 2 L 3 L τ 1 6 , ( L L τ ) .
I ( 0 ) σ = L 0 2 L ( 1 | V | 2 L ) ( 1 | V | 2 L τ ) 2 d V = 1 1 2 L 3 L τ + 1 L 2 6 L τ 2 , ( L L τ ) .
I ( 0 ) σ = 2 , L = L τ .
I ( 0 ) σ = 3 L 2 L τ , ( L L τ ) .

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