Abstract

We investigate the way in which errors arise in photocount-limited, coherent imaging systems and how such errors fundamentally limit the quality of images formed. To reflect the best possible imaging performance with a given optical system, we utilize a continuous-photodetection model to describe the operation of the image-recording mechanism, in which the image-plane camera records the exact x and y positions of each photodetection event produced by the detected coherent field intensity. Using this continuous-detection model and well-known statistical properties of. laser-speckle patterns, we compute the signal-to-noise ratio of the complex Fourier amplitudes estimated by the detected coherent image. With the help of computer-simulated coherent imagery, we illustrate how this expression can be used to characterize the effective resolving power of multiple-snapshot coherent imaging systems.

© 1992 Optical Society of America

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References

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  1. A. Kozma, C. R. Christensen, “Effects of speckle on resolution,”J. Opt. Soc. Am. 66, 1257–1260 (1976).
    [CrossRef]
  2. N. George, C. R. Christensen, J. S. Bennett, B. D. Guenther, “Speckle noise in displays,”J. Opt. Soc. Am. 66, 1282–1290 (1976).
    [CrossRef]
  3. H. H. Arsenault, G. V. April, “Information content of images degraded by speckle noise,” Opt. Eng. 25, 662–667 (1986).
    [CrossRef]
  4. J. W. Goodman, “Statistical properties of laser speckle,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), pp. 9–75.
  5. S. Lowenthal, H. Arsenault, “Image formation for coherent diffuse objects: statistical properties,”J. Opt. Soc. Am. 60, 1478–1483 (1970).
    [CrossRef]
  6. V. S. Frost, K. S. Shanmugan, “The information content of synthetic aperture radar images of terrain,”IEEE Trans. Aerosp. Electron. Syst. AES-19, 768–775 (1983).
    [CrossRef]
  7. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 85–98.
  8. Ref. 7, pp. 511–512.
  9. J. W. Goodman, J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976). See also Ref. 7, pp. 512–515.
    [CrossRef]
  10. J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,”J. Opt. Soc. Am. 69, 786–790 (1979).
    [CrossRef]
  11. G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
    [CrossRef]
  12. J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), pp. 255–320.
  13. L. J. Porcello, N. G. Massey, R. B. Innes, J. M. Marks, “Speckle reduction in synthetic aperture radars,”J. Opt. Soc. Am. 66, 1305–1311 (1976).
    [CrossRef]
  14. K. Kondo, Y. Ichioka, T. Suzuki, “Image restoration by Wiener filtering in the presence of signal-dependent noise,” Appl. Opt. 16, 2554–2558 (1977).
    [CrossRef] [PubMed]
  15. B. E. A. Saleh, M. Rabbani, “Linear filtering of speckled images,” Opt. Commun. 35, 327–331 (1980).
    [CrossRef]
  16. R. G. Morton, W. J. Connally, K. Avicola, D. Monjo, T. Olson, C. Buczek, “Coherent sub-aperture ultraviolet imagery,” in Laser Radar IV, R. J. Becherer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1103, 207–218 (1989).
    [CrossRef]
  17. P. S. Idell, J. D. Gonglewski, “Image synthesis from wavefront measurements of a coherent diffraction field,” Opt. Lett. 15, 1309–1311 (1990). See also P. S. Idell, J. D. Gonglewski, “Coherent image synthesis from wavefront-slope measurements of a nonimaged laser-speckle field,” in Signal Recovery and Synthesis III, Vol. 15 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 160–163.
    [CrossRef] [PubMed]
  18. T. Mavroidis, J. C. Dainty, M. J. Northcott, “Imaging coherently illuminated objects through turbulence: plane wave illumination,” J. Opt. Soc. Am. A. 7, 348–355 (1990).
    [CrossRef]
  19. While a large SNR [such as that defined in Eq. (1)] is a necessary condition for good image fidelity, it is not a sufficient condition. As is well known (see, for example, Ref. 20, pp. 113 – 117 ), spatial-frequency variations in the phase transfer function of an imaging system can cause severe image distortion. While a large SNR offers the possibility that the image has good fidelity, spatial frequencies for which the SNR is poor will surely guarantee poor image fidelity, since the spectral components of the image signal at those spatial frequencies will be indiscernible from random (noiselike) fluctuations in the signal. The utility of the frequency-domain SNR expression, therefore, lies in identifying those spatial frequencies for which the frequency-domain image information is effectively lost because of noise effects.
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 116.
  21. G. April, H. H. Arsenault, “Nonstationary image-plane speckle statistics,” J. Opt. Soc. Am. A 1, 738–741 (1984).
    [CrossRef]
  22. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 177.
  23. An alternative definition for κIcan be obtained by applying Rayleigh’s theorem (see, for example, Ref. 24, p. 112) to Eq. (45), which enables us to writeκI2=∫∣i¯(r)2d2r|∫i¯(r)d2r|2,where the integral over ris performed over the entire image plane. Here ī(r) denotes the ensemble-averaged, coherent image intensity, which is equal to the (incoherent) object brightness function convolved with the point-spread function of the imaging lens. From this expression, we see that [κI]−2is a measure of the size of the object field (i.e., the effective, solid-angle field of view).
  24. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1968), p. 115.
  25. See, for example, T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Chap. 4, pp. 123–170.
  26. For this object scene, the tribar patterns corresponding to the spatial frequencies for which the target SNR(N)=10 was established are not all clearly discernable. For example, the largest bar pattern in column 1 (f0,x= 13 cycles per frame) is discernable, but the next-smaller bar pattern in column 2 (f0,x= 16) is not. In general the simple fact that a single-frequency SNR value exceeds some threshold (10 or any other) will not guarantee that object-feature characteristic of that spatial frequency will be discernible in all images, since these image-domain features are the Fourier synthesis of the entire object spectrum. As a consequence, if one is interested in discriminating specific object features in a given image scene, one must study the applicability of single-frequency SNR thresholds for that specific image scene. Regardless of this limitation, however, we believe that the results of this example illustrate the usefulness of the frequency-domain SNR measure in comparing the overall expected imaging performance for different coherent optical imaging configurations.s

1990 (2)

1988 (1)

1986 (1)

H. H. Arsenault, G. V. April, “Information content of images degraded by speckle noise,” Opt. Eng. 25, 662–667 (1986).
[CrossRef]

1984 (1)

1983 (1)

V. S. Frost, K. S. Shanmugan, “The information content of synthetic aperture radar images of terrain,”IEEE Trans. Aerosp. Electron. Syst. AES-19, 768–775 (1983).
[CrossRef]

1980 (1)

B. E. A. Saleh, M. Rabbani, “Linear filtering of speckled images,” Opt. Commun. 35, 327–331 (1980).
[CrossRef]

1979 (1)

1977 (1)

1976 (3)

1970 (1)

April, G.

April, G. V.

H. H. Arsenault, G. V. April, “Information content of images degraded by speckle noise,” Opt. Eng. 25, 662–667 (1986).
[CrossRef]

Arsenault, H.

Arsenault, H. H.

H. H. Arsenault, G. V. April, “Information content of images degraded by speckle noise,” Opt. Eng. 25, 662–667 (1986).
[CrossRef]

G. April, H. H. Arsenault, “Nonstationary image-plane speckle statistics,” J. Opt. Soc. Am. A 1, 738–741 (1984).
[CrossRef]

Avicola, K.

R. G. Morton, W. J. Connally, K. Avicola, D. Monjo, T. Olson, C. Buczek, “Coherent sub-aperture ultraviolet imagery,” in Laser Radar IV, R. J. Becherer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1103, 207–218 (1989).
[CrossRef]

Ayers, G. R.

Belsher, J. F.

J. W. Goodman, J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976). See also Ref. 7, pp. 512–515.
[CrossRef]

Bennett, J. S.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1968), p. 115.

Buczek, C.

R. G. Morton, W. J. Connally, K. Avicola, D. Monjo, T. Olson, C. Buczek, “Coherent sub-aperture ultraviolet imagery,” in Laser Radar IV, R. J. Becherer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1103, 207–218 (1989).
[CrossRef]

Christensen, C. R.

Connally, W. J.

R. G. Morton, W. J. Connally, K. Avicola, D. Monjo, T. Olson, C. Buczek, “Coherent sub-aperture ultraviolet imagery,” in Laser Radar IV, R. J. Becherer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1103, 207–218 (1989).
[CrossRef]

Dainty, J. C.

T. Mavroidis, J. C. Dainty, M. J. Northcott, “Imaging coherently illuminated objects through turbulence: plane wave illumination,” J. Opt. Soc. Am. A. 7, 348–355 (1990).
[CrossRef]

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
[CrossRef]

J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,”J. Opt. Soc. Am. 69, 786–790 (1979).
[CrossRef]

J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), pp. 255–320.

Frost, V. S.

V. S. Frost, K. S. Shanmugan, “The information content of synthetic aperture radar images of terrain,”IEEE Trans. Aerosp. Electron. Syst. AES-19, 768–775 (1983).
[CrossRef]

George, N.

Gonglewski, J. D.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), pp. 9–75.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 85–98.

J. W. Goodman, J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976). See also Ref. 7, pp. 512–515.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 116.

Greenaway, A. H.

Guenther, B. D.

Ichioka, Y.

Idell, P. S.

Innes, R. B.

Kondo, K.

Kozma, A.

Lowenthal, S.

Marks, J. M.

Massey, N. G.

Mavroidis, T.

T. Mavroidis, J. C. Dainty, M. J. Northcott, “Imaging coherently illuminated objects through turbulence: plane wave illumination,” J. Opt. Soc. Am. A. 7, 348–355 (1990).
[CrossRef]

McKechnie, T. S.

See, for example, T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Chap. 4, pp. 123–170.

Monjo, D.

R. G. Morton, W. J. Connally, K. Avicola, D. Monjo, T. Olson, C. Buczek, “Coherent sub-aperture ultraviolet imagery,” in Laser Radar IV, R. J. Becherer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1103, 207–218 (1989).
[CrossRef]

Morton, R. G.

R. G. Morton, W. J. Connally, K. Avicola, D. Monjo, T. Olson, C. Buczek, “Coherent sub-aperture ultraviolet imagery,” in Laser Radar IV, R. J. Becherer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1103, 207–218 (1989).
[CrossRef]

Northcott, M. J.

T. Mavroidis, J. C. Dainty, M. J. Northcott, “Imaging coherently illuminated objects through turbulence: plane wave illumination,” J. Opt. Soc. Am. A. 7, 348–355 (1990).
[CrossRef]

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
[CrossRef]

Olson, T.

R. G. Morton, W. J. Connally, K. Avicola, D. Monjo, T. Olson, C. Buczek, “Coherent sub-aperture ultraviolet imagery,” in Laser Radar IV, R. J. Becherer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1103, 207–218 (1989).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 177.

Porcello, L. J.

Rabbani, M.

B. E. A. Saleh, M. Rabbani, “Linear filtering of speckled images,” Opt. Commun. 35, 327–331 (1980).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh, M. Rabbani, “Linear filtering of speckled images,” Opt. Commun. 35, 327–331 (1980).
[CrossRef]

Shanmugan, K. S.

V. S. Frost, K. S. Shanmugan, “The information content of synthetic aperture radar images of terrain,”IEEE Trans. Aerosp. Electron. Syst. AES-19, 768–775 (1983).
[CrossRef]

Suzuki, T.

Appl. Opt. (1)

IEEE Trans. Aerosp. Electron. Syst. (1)

V. S. Frost, K. S. Shanmugan, “The information content of synthetic aperture radar images of terrain,”IEEE Trans. Aerosp. Electron. Syst. AES-19, 768–775 (1983).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (2)

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

T. Mavroidis, J. C. Dainty, M. J. Northcott, “Imaging coherently illuminated objects through turbulence: plane wave illumination,” J. Opt. Soc. Am. A. 7, 348–355 (1990).
[CrossRef]

Opt. Commun. (1)

B. E. A. Saleh, M. Rabbani, “Linear filtering of speckled images,” Opt. Commun. 35, 327–331 (1980).
[CrossRef]

Opt. Eng. (1)

H. H. Arsenault, G. V. April, “Information content of images degraded by speckle noise,” Opt. Eng. 25, 662–667 (1986).
[CrossRef]

Opt. Lett. (1)

Other (13)

J. W. Goodman, “Statistical properties of laser speckle,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), pp. 9–75.

R. G. Morton, W. J. Connally, K. Avicola, D. Monjo, T. Olson, C. Buczek, “Coherent sub-aperture ultraviolet imagery,” in Laser Radar IV, R. J. Becherer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1103, 207–218 (1989).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 85–98.

Ref. 7, pp. 511–512.

J. W. Goodman, J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976). See also Ref. 7, pp. 512–515.
[CrossRef]

J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), pp. 255–320.

While a large SNR [such as that defined in Eq. (1)] is a necessary condition for good image fidelity, it is not a sufficient condition. As is well known (see, for example, Ref. 20, pp. 113 – 117 ), spatial-frequency variations in the phase transfer function of an imaging system can cause severe image distortion. While a large SNR offers the possibility that the image has good fidelity, spatial frequencies for which the SNR is poor will surely guarantee poor image fidelity, since the spectral components of the image signal at those spatial frequencies will be indiscernible from random (noiselike) fluctuations in the signal. The utility of the frequency-domain SNR expression, therefore, lies in identifying those spatial frequencies for which the frequency-domain image information is effectively lost because of noise effects.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 116.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 177.

An alternative definition for κIcan be obtained by applying Rayleigh’s theorem (see, for example, Ref. 24, p. 112) to Eq. (45), which enables us to writeκI2=∫∣i¯(r)2d2r|∫i¯(r)d2r|2,where the integral over ris performed over the entire image plane. Here ī(r) denotes the ensemble-averaged, coherent image intensity, which is equal to the (incoherent) object brightness function convolved with the point-spread function of the imaging lens. From this expression, we see that [κI]−2is a measure of the size of the object field (i.e., the effective, solid-angle field of view).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1968), p. 115.

See, for example, T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, New York, 1984), Chap. 4, pp. 123–170.

For this object scene, the tribar patterns corresponding to the spatial frequencies for which the target SNR(N)=10 was established are not all clearly discernable. For example, the largest bar pattern in column 1 (f0,x= 13 cycles per frame) is discernable, but the next-smaller bar pattern in column 2 (f0,x= 16) is not. In general the simple fact that a single-frequency SNR value exceeds some threshold (10 or any other) will not guarantee that object-feature characteristic of that spatial frequency will be discernible in all images, since these image-domain features are the Fourier synthesis of the entire object spectrum. As a consequence, if one is interested in discriminating specific object features in a given image scene, one must study the applicability of single-frequency SNR thresholds for that specific image scene. Regardless of this limitation, however, we believe that the results of this example illustrate the usefulness of the frequency-domain SNR measure in comparing the overall expected imaging performance for different coherent optical imaging configurations.s

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Figures (6)

Fig. 1
Fig. 1

Coherent imaging configuration.

Fig. 2
Fig. 2

Representing the complex Fourier amplitude D(f) as the sum of a random number K unit-modulus phasors. Re D(f) and Im D(f) are the real and the imaginary parts of the complex phasor D(f), respectively, and σD is the standard deviation of the estimate for D(f), defined by the square root of the sum of the variances of the real and the imaginary parts of D(f) [see Eq. (4)].

Fig. 3
Fig. 3

Convolution operation in Eq. (44) that defines M(f). In this depiction we assume that the object scene is highly resolved so that the square modulus of the image spectrum |Ĩ(f)|2 is much narrower in spatial-frequency space than the transfer function T ˜ P ( 2 )(f). This figure shows a slice through the two-dimensional Fourier spectrum for which fy = 0.

Fig. 4
Fig. 4

Binary test object used in computer-simulation experiments. White pixels denote a pixel value of 1.0; black denotes a pixel value of 0.0. The tribar-array pattern is 63 pixels wide by 59 pixels tall, and is embedded in a 128-by-128 pixel array.

Fig. 5
Fig. 5

Images produced in coherent imaging simulations, corresponding to the number of coherent image frames N incoherently averaged together as specified in Table 2. Images in each column have the same SNR(N)D(f0) value for f0,x = 21 cycles per frame and f0,y = 0: column 1, SNR = 1.0; column 2, SNR = 3.0; column 3, SNR = 10; column 4, SNR = 10. Images in each row have the same average number of photocounts recorded per image frame: row 1, K ¯= 300; row 2, K ¯= 30; row 3, K ¯= 3, row 4, K ¯= 0.3 photocounts per frame. (See Table 2.)

Fig. 6
Fig. 6

Images produced in coherent imaging simulations, corresponding to the number of coherent image frames N incoherently averaged together as specified in Table 3. Images in each column have the same SNR ( N ) D ( f 0 ) = 10 value but at different spatial frequencies f0,x (with f0,y = 0 in each case): column 1, f0,x = 13; column 2, f0,x = 16; column 3, f0,x = 21; column 4, f0,x = 32 cycles per frame. Images in each row have the same average number of photocounts recorded per image frame: row 1, K ¯= 300; row 2, K ¯= 30; row 3, K ¯= 3; row 4, K ¯= 0.3 photocounts per frame. (See Table 3.)

Tables (3)

Tables Icon

Table 1 Object Scene and Optical-Transfer-Function Data for Computer-Simulation Experimentsa

Tables Icon

Table 2 Number of Image Frames N Needed to Achieve One of Four Specified SNR Levels When Each Frame Records an Average of K ¯ Detected Photocountsa

Tables Icon

Table 3 Number of Image Frames N Needed to Achieve SNR ( N ) D ( f 0 ) = 10 at Four Different Reference Frequencies When Each Frame Records an Average of K ¯ Detected Photoeventsa

Equations (59)

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SNR D ( f ) = E D ( f ) [ Var D ( f ) ] 1 / 2 ,
d ( r ) = k = 1 K δ ( r - r k ) ,
D ( f ) = k = 1 K δ ( r - r k ) exp ( - j 2 π f · r ) d 2 r = k = 1 K exp ( - j 2 π f · r k ) ,
Var D ( f ) = E D ( f ) 2 - E D ( f ) 2 ,
SNR ( N ) D ( f ) = N SNR D ( f ) .
p [ r k K , i ( r ) ] = λ ( r k ) λ ( r ) d 2 r ,
λ ( r ) = η T h ν 0 i ( r )
p [ K i ( r ) ] = [ λ ( r ) d 2 r ] K K ! exp [ - λ ( r ) d 2 r ] ,
E K i { K } = k = 1 K p [ K i ( r ) ] = λ ( r ) d 2 r ,
E K = E i { E K i { K } } = E i { λ ( r ) } d 2 r = η T h ν 0 i ¯ ( r ) d 2 r K ¯ ,
D ¯ ( f ) = E D ( f ) = E i { E K i { E r k K , i { D ( f ) } } } ,
E r k K , i { D ( f ) } = [ k = 1 K exp ( - j 2 π f · r k ) ] λ ( r k ) λ ( r ) d 2 r d 2 r k = k = 1 K λ ( r k ) λ ( r ) d 2 r exp ( - j 2 π f · r k ) d 2 r k = K Λ ( f ) Λ ( 0 ) ,
Λ ( f ) = λ ( x ) exp ( - j 2 π f · r ) d 2 r ,
E K i { K Λ ( f ) Λ ( 0 ) } = Λ ( f ) ,
E K i { K } = λ ( r ) d 2 r = Λ ( 0 ) .
D ¯ ( f ) = E i { Λ ( f ) } = Λ ¯ ( f ) = η T h ν 0 I ¯ ( f ) ,
K ¯ = η T h ν 0 I ¯ ( 0 ) = D ¯ ( 0 ) ,
D ¯ ( f ) = Λ ¯ ( f ) = K ¯ I ˜ ( f ) ,
I ¯ ( f ) = O ( f ) T P ( f ) ,
T P ( f ) = P * ( u ) P ( u + λ Z f ) d 2 u ,
T p ( 0 ) = P ( u ) 2 d 2 u A P ( 2 )
D ¯ ( f ) = K ¯ I ˜ ( f ) = K ¯ O ˜ ( f ) T ˜ P ( f ) ,
E D ( f ) 2 = K ¯ 2 I ˜ ( f ) 2 .
D ( f ) 2 = m = 1 K n = 1 K exp [ - j 2 π f · ( r m - r n ) ] ,
E D ( f ) 2 = E i { E K i { E r m , r n K , i { D ( f ) 2 } } } ,
E K i { E r m , r n K , i { D ( f ) 2 } } = Λ ( 0 ) + Λ ( f ) 2 ,
E D ( f ) 2 = E i { Λ ( 0 ) + Λ ( f ) 2 } .
E i { Λ ( 0 ) } = Λ ¯ ( 0 ) = K ¯ .
E i { Λ ( f ) 2 } = E i { λ ( r ) λ ( r ) } exp [ - j 2 π f · ( r - r ) ] d 2 r d 2 r .
E i { λ ( r ) λ ( r ) } = λ ¯ ( r ) λ ¯ ( r ) [ 1 + μ ( r - r ) 2 ] ,
μ ( r - r ) = j ( r - r ) [ i ( r ) i ( r ) ] 1 / 2 ,
E i { Λ ( f ) 2 } = K ¯ 2 I ˜ ( f ) 2 + K ¯ 2 [ M ( f ) ] - 1 ,
[ M ( f ) ] - 1 = K ¯ - 2 λ ¯ ( r ) λ ¯ ( r ) μ ( r - r ) 2 × exp [ - j 2 π f · ( r - r ) ] d 2 r d 2 r .
E { Λ ( f ) 2 } = K ¯ + K ¯ 2 I ˜ ( f ) 2 + K ¯ 2 [ M ( f ) ] - 1 .
Var D ( f ) = K ¯ + K ¯ 2 [ M ( f ) ] - 1 = K ¯ [ 1 + K ¯ M ( f ) ] .
E i { λ ( r ) λ ( r ) } = λ ¯ ( r ) λ ¯ ( r ) .
[ M ( f ) ] - 1 = K ¯ - 2 [ λ ¯ ( r + Δ r ) λ ¯ ( r ) d 2 r ] μ ( Δ r ) 2 × exp ( - j 2 π f · Δ r ) d 2 Δ r .
λ ¯ ( r + Δ r ) λ ¯ ( r ) d 2 r = Λ ¯ ( f ) 2 exp ( + j 2 π f · Δ r ) d 2 f .
[ M ( f ) ] - 1 = K ¯ - 2 Λ ¯ ( f ) 2 [ μ ( Δ r ) 2 × exp [ - j 2 π ( f - f ) · Δ r ] d 2 Δ r ] d 2 f .
μ ( Δ r ) 2 = | P ( u ) 2 exp [ j ( 2 π / λ Z ) u · Δ r ] d 2 u | 2 | P ( u ) | 2 d 2 u | 2 ,
μ ( Δ r ) 2 exp ( - j 2 π f · Δ r ) d 2 Δ r = A P ( 4 ) A P ( 2 ) 2 ( λ Z ) 2 T ˜ P ( 2 ) ( f ) ,
A P ( 4 ) P ( u ) 4 d 2 u = T P ( 2 ) ( 0 ) ;
T ˜ P ( 2 ) ( f ) = P ( u ) 2 P ( u + λ Z f ) 2 d 2 u P ( u ) 4 d 2 u = T P ( 2 ) ( f ) T P ( 2 ) ( 0 )
[ M ( f ) ] - 1 = K ¯ - 2 ( λ Z ) 2 A P ( 4 ) A P ( 2 ) 2 Λ ¯ ( f ) 2 T ˜ P ( 2 ) ( f - f ) d 2 f = ( λ Z ) 2 A P ( 4 ) A P ( 2 ) 2 I ˜ ( f ) 2 T ˜ P ( 2 ) ( f - f ) d 2 f ,
κ I 2 = I ˜ ( f ) 2 d 2 f
κ I D / λ Z ,
[ M ( f ) ] - 1 ( λ Z ) 2 A P ( 4 ) A P ( 2 ) 2 T ˜ P ( 2 ) ( f ) κ I 2 = T ˜ P ( 2 ) ( f ) M 0 ,
M 0 ( λ Z ) - 2 A P ( 2 ) 2 A P ( 4 ) κ I - 2 M ( f = 0 ) .
Var D ( f ) = K ¯ + K ¯ 2 M 0 T ˜ P ( 2 ) ( f ) = K ¯ [ 1 + K ¯ M 0 T ˜ P ( 2 ) ( f ) ] .
Var K = Var D ( 0 ) K + K ¯ 2 M 0 = K ¯ [ 1 + K ¯ M 0 ] .
SNR D ( f ) = K ¯ O ˜ ( f ) T ˜ P ( f ) { K ¯ + K ¯ 2 [ M ( f ) ] - 1 } 1 / 2 = O ˜ ( f ) T ˜ P ( f ) { K ¯ / [ 1 + K ¯ M ( f ) ] } 1 / 2 .
K ¯ M ( f ) K ¯ M 0 T ˜ P ( 2 ) ( f ) 1 ,
SNR D ( f ) = O ˜ ( f ) T ˜ P ( f ) K ¯ 1 / 2 .
K ¯ M ( f ) K ¯ M 0 T ˜ P ( 2 ) ( f ) 1 ,
SNR D ( f ) = O ˜ ( f ) T ˜ P ( f ) [ M ( f ) ] 1 / 2 O ˜ ( f ) T ˜ P ( f ) [ M 0 / T ˜ P ( 2 ) ( f ) ] 1 / 2 .
SNR     D ( f ) O ˜ ( f ) [ M 0 T ˜ P ( 2 ) ( f ) ] 1 / 2 ,
SNR ( N ) D ( f ) = O ˜ ( f ) T ˜ P ( f ) { N K ¯ / [ 1 + K ¯ M ( f ) ] } 1 / 2 ,
M 0 = κ I - 2 A P ( λ Z ) 2 600.
κI2=i¯(r)2d2r|i¯(r)d2r|2,

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