Abstract

Quasi-monochromatic light will form laser speckle upon reflection from a rough object. This laser speckle provides information about the shape of the illuminated object. Further information can be obtained if two colors of coherent light are used, provided that the colors are sufficiently close in wavelength that the interference is also measurable. It is shown that no more than two intensities of two speckle patterns and their interference are required to produce an unambiguous band-limited image of an object, to within an overall spatial translation of the image, in the absence of measurement errors and in the case where all roots of both fields and their complex conjugates are distinct. This result is proven with a root-matching technique, which treats the electric fields as polynomials in the pupil plane, the coefficients of which form the desired complex object. Several root-matching algorithms are developed and tested. These algorithms are generally slow and sensitive to noise. So motivated, several other techniques are applied to the problem, including phase retrieval, expectation maximization, and probability maximization in a sequel paper [J. Opt. Soc. Am. A 19, 458 (2002)]. The phase-retrieval and expectation-maximization techniques proved to be most effective for reconstructions of complex objects larger than 10 pixels across.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. I. Tatarskii, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman, (McGraw-Hill, New York, 1961).
  2. K. T. Knox, B. J. Thompson, “Recovery of images from astronomically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
    [Crossref]
  3. A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [Crossref] [PubMed]
  4. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  5. A. C. S. Readhead, T. S. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction-limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
    [Crossref]
  6. J. Hardy, J. Lefebvre, C. Koliopoulis, “Real time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
    [Crossref]
  7. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–68.
  8. Paul S. Idell, J. R. Fienup, Ron S. Goodman, “Image synthesis from nonimaged laser-speckle patterns,” Opt. Lett. 12, 858–860 (1987).
    [Crossref] [PubMed]
  9. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), p. 356.
  10. R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lensless imaging,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 161–168 (1993).
    [Crossref]
  11. S. M. Stahl, R. M. Kremer, P. W. Fairchild, K. Hughes, B. A. Spivey, R. Stagat, “Sheared-beam coherent image reconstruction,” in Applications of Digital Image Processing XIX, A. G. Tescher, ed., Proc. SPIE2847, 150–158 (1996).
    [Crossref]
  12. M. Born, E. Wolf, Principles of Optics, 7th ed., pp. 572–577 (Cambridge U. Press, Cambridge, UK, 1999).
  13. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.
  14. Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [Crossref]
  15. H. B. Deighton, M. S. Scivier, M. A. Fiddy, “Solution of the two-dimensional phase-retrieval problem,” Opt. Lett. 10, 250–251 (1985).
    [Crossref] [PubMed]
  16. R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
    [Crossref]
  17. D. Israelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
    [Crossref]
  18. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [Crossref]
  19. R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
    [Crossref]
  20. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [Crossref]
  21. C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros to overcome a phase retrieval stagnation,” J. Opt. Soc. Am. A 8, 1898–1904 (1991).
    [Crossref]
  22. C. C. Wackerman, A. E. Yagle, “Phase retrieval and estimation with use of real-plane zeros,” J. Opt. Soc. Am. A 11, 2016–2026 (1994).
    [Crossref]
  23. P. J. Bones, C. R. Parker, B. L. Satherley, R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842–1857 (1995).
    [Crossref]
  24. T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
    [Crossref]
  25. R. H. T. Bates, B. K. Quek, C. R. Parker, “Some implications of zero sheets for blind deconvolution and phase retrieval,” J. Opt. Soc. Am. A 7, 468–479 (1990).
    [Crossref]
  26. P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampled data,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 14–22 (1993).
    [Crossref]
  27. D. C. Ghiglia, L. A. Romero, G. A. Mastin, “Systematic approach to two-dimensional blind deconvolution by zero-sheet separation,” J. Opt. Soc. Am. A 10, 1024–1036 (1993).
    [Crossref]
  28. B. R. Hunt, T. L. Overman, P. Gough, “Image reconstruction from pairs of Fourier transform magnitude,” Opt. Lett. 23, 1123–1125 (1998).
    [Crossref]
  29. B. Ya Zeldovich, Principles of Phase Conjugation (Springer-Verlag, New York, 1985), Chap. 3.
  30. M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
    [Crossref]
  31. E. P. Wallner, “Optimal wave-front correction using slope measurements,” J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [Crossref]
  32. J. D. Downie, J. W. Goodman, “Optimal wave-front correction with segmented mirrors,” Appl. Opt. 28, 5326–5332 (1989).
    [Crossref] [PubMed]
  33. R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
    [Crossref]
  34. R. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: iterative reconstructors,” J. Opt. Soc. Am. A 19, 458–471 (2002).
    [Crossref]
  35. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plan pictures,” Optik 35, 237–246 (1972).
  36. V. S. R. Gudimetla, J. F. Holmes, “Probability density function of the intensity for a laser-generated speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 72, 1213–1218 (1982), and references therein.
    [Crossref]
  37. M. H. Lee, J. F. Holmes, J. R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 66, 1164–1172 (1976).
    [Crossref]
  38. G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Eq. 3.19.
  39. P. S. Idell, A. 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).
    [Crossref]
  40. R. B. Holmes, B. Spivey, A. Smith, “Recovery of images from two-color, pupil-plane speckle data using object-plane root-matching and pupil-plane error minimiziation,” in Digital Image Reconstruction and Synthesis IV, P. S. Idell, T. J. Schulz, eds., Proc. SPIE3815, 70–89 (1999).
    [Crossref]

2002 (1)

1998 (1)

1995 (1)

1994 (1)

1993 (2)

1992 (2)

1991 (1)

1990 (1)

1989 (1)

1988 (1)

A. C. S. Readhead, T. S. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction-limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[Crossref]

1987 (5)

Paul S. Idell, J. R. Fienup, Ron S. Goodman, “Image synthesis from nonimaged laser-speckle patterns,” Opt. Lett. 12, 858–860 (1987).
[Crossref] [PubMed]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[Crossref]

D. Israelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[Crossref]

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
[Crossref]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[Crossref]

1986 (1)

1985 (2)

1983 (2)

1982 (2)

1979 (1)

Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[Crossref]

1977 (1)

1976 (1)

1974 (1)

K. T. Knox, B. J. Thompson, “Recovery of images from astronomically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[Crossref]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plan pictures,” Optik 35, 237–246 (1972).

Bates, R. H. T.

Bones, P. J.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed., pp. 572–577 (Cambridge U. Press, Cambridge, UK, 1999).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), p. 356.

Bruck, Yu. M.

Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[Crossref]

Chen, P.

P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampled data,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 14–22 (1993).
[Crossref]

Deighton, H. B.

Downie, J. D.

Fairchild, P.

Fairchild, P. W.

S. M. Stahl, R. M. Kremer, P. W. Fairchild, K. Hughes, B. A. Spivey, R. Stagat, “Sheared-beam coherent image reconstruction,” in Applications of Digital Image Processing XIX, A. G. Tescher, ed., Proc. SPIE2847, 150–158 (1996).
[Crossref]

Fiddy, M. A.

Fienup, J. R.

Fright, W. R.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plan pictures,” Optik 35, 237–246 (1972).

Ghiglia, D. C.

Goodman, J. W.

J. D. Downie, J. W. Goodman, “Optimal wave-front correction with segmented mirrors,” Appl. Opt. 28, 5326–5332 (1989).
[Crossref] [PubMed]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–68.

Goodman, Ron S.

Gough, P.

Greenaway, A. H.

P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampled data,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 14–22 (1993).
[Crossref]

Gudimetla, V. S. R.

Hardy, J.

Holmes, J. F.

Holmes, R.

Holmes, R. B.

R. B. Holmes, B. Spivey, A. Smith, “Recovery of images from two-color, pupil-plane speckle data using object-plane root-matching and pupil-plane error minimiziation,” in Digital Image Reconstruction and Synthesis IV, P. S. Idell, T. J. Schulz, eds., Proc. SPIE3815, 70–89 (1999).
[Crossref]

Hughes, K.

R. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: iterative reconstructors,” J. Opt. Soc. Am. A 19, 458–471 (2002).
[Crossref]

S. M. Stahl, R. M. Kremer, P. W. Fairchild, K. Hughes, B. A. Spivey, R. Stagat, “Sheared-beam coherent image reconstruction,” in Applications of Digital Image Processing XIX, A. G. Tescher, ed., Proc. SPIE2847, 150–158 (1996).
[Crossref]

Hunt, B. R.

Hutchin, R. A.

R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lensless imaging,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 161–168 (1993).
[Crossref]

Idell, P. S.

Idell, Paul S.

Israelevitz, D.

D. Israelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[Crossref]

Kerr, J. R.

Knox, K. T.

K. T. Knox, B. J. Thompson, “Recovery of images from astronomically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[Crossref]

Koliopoulis, C.

Kremer, R. M.

S. M. Stahl, R. M. Kremer, P. W. Fairchild, K. Hughes, B. A. Spivey, R. Stagat, “Sheared-beam coherent image reconstruction,” in Applications of Digital Image Processing XIX, A. G. Tescher, ed., Proc. SPIE2847, 150–158 (1996).
[Crossref]

Lane, R. G.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[Crossref]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[Crossref]

Lee, M. H.

Lefebvre, J.

Lim, J. S.

D. Israelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[Crossref]

Lohmann, A. W.

Mastin, G. A.

Nakajima, T. S.

A. C. S. Readhead, T. S. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction-limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[Crossref]

Neugebauer, G.

A. C. S. Readhead, T. S. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction-limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[Crossref]

Oke, J. B.

A. C. S. Readhead, T. S. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction-limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[Crossref]

Overman, T. L.

Parker, C. R.

Parry, G.

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Eq. 3.19.

Paxman, R. G.

Pearson, T. J.

A. C. S. Readhead, T. S. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction-limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[Crossref]

Quek, B. K.

Readhead, A. C. S.

A. C. S. Readhead, T. S. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction-limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[Crossref]

Romero, L. A.

Sargent, W. L. W.

A. C. S. Readhead, T. S. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction-limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[Crossref]

Satherley, B. L.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plan pictures,” Optik 35, 237–246 (1972).

Schulz, T. J.

Scivier, M. S.

Smith, A.

R. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: iterative reconstructors,” J. Opt. Soc. Am. A 19, 458–471 (2002).
[Crossref]

R. B. Holmes, B. Spivey, A. Smith, “Recovery of images from two-color, pupil-plane speckle data using object-plane root-matching and pupil-plane error minimiziation,” in Digital Image Reconstruction and Synthesis IV, P. S. Idell, T. J. Schulz, eds., Proc. SPIE3815, 70–89 (1999).
[Crossref]

Sodin, L. G.

Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[Crossref]

Spivey, B.

R. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: iterative reconstructors,” J. Opt. Soc. Am. A 19, 458–471 (2002).
[Crossref]

R. B. Holmes, B. Spivey, A. Smith, “Recovery of images from two-color, pupil-plane speckle data using object-plane root-matching and pupil-plane error minimiziation,” in Digital Image Reconstruction and Synthesis IV, P. S. Idell, T. J. Schulz, eds., Proc. SPIE3815, 70–89 (1999).
[Crossref]

Spivey, B. A.

S. M. Stahl, R. M. Kremer, P. W. Fairchild, K. Hughes, B. A. Spivey, R. Stagat, “Sheared-beam coherent image reconstruction,” in Applications of Digital Image Processing XIX, A. G. Tescher, ed., Proc. SPIE2847, 150–158 (1996).
[Crossref]

Stagat, R.

S. M. Stahl, R. M. Kremer, P. W. Fairchild, K. Hughes, B. A. Spivey, R. Stagat, “Sheared-beam coherent image reconstruction,” in Applications of Digital Image Processing XIX, A. G. Tescher, ed., Proc. SPIE2847, 150–158 (1996).
[Crossref]

Stahl, S. M.

S. M. Stahl, R. M. Kremer, P. W. Fairchild, K. Hughes, B. A. Spivey, R. Stagat, “Sheared-beam coherent image reconstruction,” in Applications of Digital Image Processing XIX, A. G. Tescher, ed., Proc. SPIE2847, 150–158 (1996).
[Crossref]

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman, (McGraw-Hill, New York, 1961).

Thompson, B. J.

K. T. Knox, B. J. Thompson, “Recovery of images from astronomically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[Crossref]

Wackerman, C. C.

Wallner, E. P.

Wang, Y.

P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampled data,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 14–22 (1993).
[Crossref]

Watson, R. W.

Webster, A.

Weigelt, G.

Wirnitzer, B.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed., pp. 572–577 (Cambridge U. Press, Cambridge, UK, 1999).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), p. 356.

Ya Zeldovich, B.

B. Ya Zeldovich, Principles of Phase Conjugation (Springer-Verlag, New York, 1985), Chap. 3.

Yagle, A. E.

Appl. Opt. (3)

Astron. J. (1)

A. C. S. Readhead, T. S. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction-limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[Crossref]

Astrophys. J. Lett. (1)

K. T. Knox, B. J. Thompson, “Recovery of images from astronomically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[Crossref]

IEEE Trans. Acoust. Speech Signal Process. (2)

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[Crossref]

D. Israelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[Crossref]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (13)

P. S. Idell, A. 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).
[Crossref]

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[Crossref]

R. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: iterative reconstructors,” J. Opt. Soc. Am. A 19, 458–471 (2002).
[Crossref]

M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
[Crossref]

D. C. Ghiglia, L. A. Romero, G. A. Mastin, “Systematic approach to two-dimensional blind deconvolution by zero-sheet separation,” J. Opt. Soc. Am. A 10, 1024–1036 (1993).
[Crossref]

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
[Crossref]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[Crossref]

J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[Crossref]

C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros to overcome a phase retrieval stagnation,” J. Opt. Soc. Am. A 8, 1898–1904 (1991).
[Crossref]

C. C. Wackerman, A. E. Yagle, “Phase retrieval and estimation with use of real-plane zeros,” J. Opt. Soc. Am. A 11, 2016–2026 (1994).
[Crossref]

P. J. Bones, C. R. Parker, B. L. Satherley, R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842–1857 (1995).
[Crossref]

T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
[Crossref]

R. H. T. Bates, B. K. Quek, C. R. Parker, “Some implications of zero sheets for blind deconvolution and phase retrieval,” J. Opt. Soc. Am. A 7, 468–479 (1990).
[Crossref]

Opt. Commun. (1)

Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[Crossref]

Opt. Lett. (3)

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plan pictures,” Optik 35, 237–246 (1972).

Other (11)

R. B. Holmes, B. Spivey, A. Smith, “Recovery of images from two-color, pupil-plane speckle data using object-plane root-matching and pupil-plane error minimiziation,” in Digital Image Reconstruction and Synthesis IV, P. S. Idell, T. J. Schulz, eds., Proc. SPIE3815, 70–89 (1999).
[Crossref]

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Eq. 3.19.

B. Ya Zeldovich, Principles of Phase Conjugation (Springer-Verlag, New York, 1985), Chap. 3.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium, translated by R. S. Silverman, (McGraw-Hill, New York, 1961).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), p. 356.

R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lensless imaging,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 161–168 (1993).
[Crossref]

S. M. Stahl, R. M. Kremer, P. W. Fairchild, K. Hughes, B. A. Spivey, R. Stagat, “Sheared-beam coherent image reconstruction,” in Applications of Digital Image Processing XIX, A. G. Tescher, ed., Proc. SPIE2847, 150–158 (1996).
[Crossref]

M. Born, E. Wolf, Principles of Optics, 7th ed., pp. 572–577 (Cambridge U. Press, Cambridge, UK, 1999).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–68.

P. Chen, M. A. Fiddy, A. H. Greenaway, Y. Wang, “Zero estimation for blind deconvolution from noisy sampled data,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 14–22 (1993).
[Crossref]

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 (16)

Fig. 1
Fig. 1

Schematic of image-forming apparatus. Two colors of quasi-monochromatic light are transmitted to a distant target. The reflected, speckled light is sampled at a (nonimaged) pupil plane.

Fig. 2
Fig. 2

Pristine image used as basis for comparison of algorithms, shown in both inverted gray-scale and hue-saturation-value color maps.

Fig. 3
Fig. 3

Aperture over which data are sampled for two-dimensional reconstructions.

Fig. 4
Fig. 4

Speckle-limited image of the pristine object given the aperture of Fig. 3.

Fig. 5
Fig. 5

Relationship between measured quantities in the pupil plane and fields in the target plane.

Fig. 6
Fig. 6

Image reconstruction process. (a) object, (b) pupil-plane intensities: field 1 (blue), field 2 (green); (c) object autocorrelation from pupil intensities: field 1 (blue), field 2 (green); (d) roots of intensity 1 (blue crosses), intensity 2 (green pluses), and field cross product (red circles); (e) reconstructed object: field 1 (blue), field 2 (green); (f) recreated pupil plane intensities: field 1 (blue), field 2 (green).

Fig. 7
Fig. 7

Flow diagram for simple one-dimensional root reconstructor.

Fig. 8
Fig. 8

Sample reconstructed images versus SNR for a random one-dimensional object. (a) SNR=106, resulting field Strehl = 1.00; (b) SNR = 100, resulting field Strehl = 0.995; (c) SNR = 10, resulting field Strehl = 0.425; (d) SNR = 3, resulting field Strehl = 0.062. Solid curve, field 1; dotted curve, field 2.

Fig. 9
Fig. 9

Field Strehl for 20 random one-dimensional objects versus noise-to-signal ratio, for various grid sizes.

Fig. 10
Fig. 10

Field Strehl for 20 random one-dimensional objects versus noise-to-signal ratio, for various sampling densities. Nyquist sampling corresponds to an object filling 0.5× the initial numerical grid.

Fig. 11
Fig. 11

Reconstruction of a simple two-dimensional simulated object. (a) pupil map, (b) pristine coherent object, (c) coherent image before weave, (d) weave-reconstructed coherent image, Strehl = 0.24, image Strehl = 0.48.

Fig. 12
Fig. 12

Field 1 of a weave reconstruction using perfect-strip reconstructions.

Fig. 13
Fig. 13

Field 1 of a weave reconstruction using strip reconstructions from the irregular aperture.

Fig. 14
Fig. 14

Initial fit of the intensity data and the consistency constraint. The curves are indicated as follows: x’s, I1; circles, I2; pluses, Re(I12); squares, Im(I12). The constraint curve, Eq. (8), is marked by C.

Fig. 15
Fig. 15

Simplified flow diagram for minimum-error root reconstructor.

Fig. 16
Fig. 16

Pristine and reconstructed image—no atmosphere, no measurement noise. Only the noise expected from speckle is present.

Tables (1)

Tables Icon

Table 1 Median Field Strehls versus SNR and Patch Size for the Minimum-Error Version of the Root Reconstructor

Equations (17)

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

I(x, t)=|E1(x)exp(iω1t)+E2(x)exp(iω2t)|2.
I1(x)=|E1(x)|2+n1(x),
I2(x)=|E2(x)|2+n2(x),
Ir(x)=Re[E1(x)E2*(x)]+nr(x),
Ii(x)=Im[E1(x)E2*(x)]+ni(x).
I12(x)=Ir(x)+iIi(x)=E1(x)E2*(x)+n12(x).
D=Σj,k,n,m|exp[i(ϕn+Δϕnj)]E1x,n(j)-exp[i(ϕm+Δϕmk)]E1y,m(k)|2.
I1I2-|I12|2=0,
P(x)=ΣiviTi(x),
I1f(xj)=ΣiviTc(i, j).
Q=|vd1-Tdv1|2.
v1=pinv(Td)vd1,
I2ΔI1+I1ΔI2-I12ΔI12*-I12*ΔI12=-I1I2+|I12|2.
diag(I2)TcΔv1+diag(I1)TcΔv2-2[diag(Ir)Re(Tc)
+diag(Ii)Im(Tc)]Re(Δv12)+2[diag(Ir)Im(Tc)
-diag(Ii)Re(Tc)]Im(Δv12)=-I1I2+|I12|2,
Δv1=pinv(M)diag(-I1I2+|I12|2),

Metrics