OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 9, Iss. 1 — Jan. 1, 1992
  • pp: 43–56

Resolution limits for coherent optical imaging: signal-to-noise analysis in the spatial-frequency domain

Paul S. Idell and Arthur Webster  »View Author Affiliations

JOSA A, Vol. 9, Issue 1, pp. 43-56 (1992)

View Full Text Article

Enhanced HTML    Acrobat PDF (2630 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

Original Manuscript: January 17, 1991
Revised Manuscript: July 2, 1991
Manuscript Accepted: July 4, 1991
Published: January 1, 1992

Paul S. Idell and Arthur Webster, "Resolution limits for coherent optical imaging: signal-to-noise analysis in the spatial-frequency domain," J. Opt. Soc. Am. A 9, 43-56 (1992)

Sort:  Author  |  Year  |  Journal  |  Reset  


  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

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited