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. In a prior paper [J. Opt. Soc. Am. A 19, 444 (2002)], it was shown that two intensities of two speckle patterns and their interference are sufficient to produce an unambiguous (except for object translation) band-limited image of the object, based on a root-matching technique described therein, in the absence of measurement error and in the case of distinct roots of the field polynomials and their complex conjugates. On the other hand, algorithms based on the root-matching technique are found to be slow and sensitive to noise. So motivated, several other techniques are applied to the problem, including phase retrieval, expectation maximization, and statistical maximization. The phase-retrieval and expectation-maximization techniques proved to be most effective for reconstructions of complex objects larger than 10 pixels across, and high-quality images were formed by using three independent sets of two-field data (three frames of two-wavelength data), each comprising two speckle intensity patterns and their interference. Two additional results of note are reported. First, the expectation-maximization algorithm produced relatively good images when three or more frames each of only one speckle intensity pattern (data at just one wavelength) were used and second, the phase-retrieval algorithm when only the object autocorrelation was used also produced relatively good images for the chosen test object.

© 2002 Optical Society of America

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References

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  1. 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.
  2. 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]
  3. S. M. Ebstein, “High-light-level variance of estimators for intensity interferometry and fourth-order correlation interferometry,” J. Opt. Soc. Am. A 8, 1450–1456 (1991).
    [CrossRef]
  4. P. S. Idell, J. R. Fienup, R. S. Goodman, “Image synthesis from nonimaged laser-speckle patterns,” Opt. Lett. 12, 858–860 (1987).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. M. H. Lee, J. F. Holmes, R. Kerr, “Statistics of speckle propagation through the turbulent atmosphere,” J. Opt. Soc. Am. 66, 1164–1172 (1976).
    [CrossRef]
  7. B. R. Hunt, T. L. Overman, P. Gough, “Image reconstruction from pairs of Fourier transform magnitude,” Opt. Lett. 23, 1123–1125 (1998).
    [CrossRef]
  8. 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 minimization,” in Digital Image Reconstruction and Synthesis IV, P. S. Idell, T. J. Schulz, eds., Proc. SPIE3815, 70–89 (1999).
    [CrossRef]
  9. R. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: root reconstructors,” J. Opt. Soc. Am. A 19, 444–457 (2002).
    [CrossRef]
  10. G. R. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
    [CrossRef]
  11. 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]
  12. T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum likelihood approach,” J. Opt. Soc. Am. A 9, 1052–1061 (1992).
    [CrossRef] [PubMed]
  13. T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
    [CrossRef]
  14. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  15. C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros to overcome phase retrieval stagnation,” J. Opt. Soc. Am. A 8, 1898–1904 (1991).
    [CrossRef]
  16. 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]
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  18. K. E. Atkinson, An Introduction to Numerical Analysis (Wiley, New York, 1978), pp. 478–486.
  19. 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]

2002 (1)

1998 (1)

1994 (1)

1993 (1)

1992 (3)

1991 (2)

1989 (1)

1988 (1)

1987 (2)

1986 (1)

1976 (1)

Atkinson, K. E.

K. E. Atkinson, An Introduction to Numerical Analysis (Wiley, New York, 1978), pp. 478–486.

Ayers, G. R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U.P. Press, Cambridge, UK, 1999), p. 520.

Dainty, J. C.

Ebstein, S. M.

Fairchild, P.

Fienup, J. R.

Gonglewski, J. D.

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, 1975), pp. 9–68.

Goodman, R. S.

Gough, P.

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 minimization,” in Digital Image Reconstruction and Synthesis IV, P. S. Idell, T. J. Schulz, eds., Proc. SPIE3815, 70–89 (1999).
[CrossRef]

Holmes, T. J.

Hughes, K.

Hunt, B. R.

Idell, P. S.

Kerr, R.

Knopp, J.

Lee, M. H.

Overman, T. L.

Paxman, R. G.

Schulz, T. J.

Smith, A.

R. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: root reconstructors,” J. Opt. Soc. Am. A 19, 444–457 (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 minimization,” in Digital Image Reconstruction and Synthesis IV, P. S. Idell, T. J. Schulz, eds., Proc. SPIE3815, 70–89 (1999).
[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: root reconstructors,” J. Opt. Soc. Am. A 19, 444–457 (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 minimization,” in Digital Image Reconstruction and Synthesis IV, P. S. Idell, T. J. Schulz, eds., Proc. SPIE3815, 70–89 (1999).
[CrossRef]

Voelz, D. G.

Wackerman, C. C.

Webster, A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U.P. Press, Cambridge, UK, 1999), p. 520.

Yagle, A. E.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (10)

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: root reconstructors,” J. Opt. Soc. Am. A 19, 444–457 (2002).
[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]

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

S. M. Ebstein, “High-light-level variance of estimators for intensity interferometry and fourth-order correlation interferometry,” J. Opt. Soc. Am. A 8, 1450–1456 (1991).
[CrossRef]

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

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]

T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum likelihood approach,” J. Opt. Soc. Am. A 9, 1052–1061 (1992).
[CrossRef] [PubMed]

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

Opt. Lett. (4)

Other (4)

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.

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 minimization,” in Digital Image Reconstruction and Synthesis IV, P. S. Idell, T. J. Schulz, eds., Proc. SPIE3815, 70–89 (1999).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U.P. Press, Cambridge, UK, 1999), p. 520.

K. E. Atkinson, An Introduction to Numerical Analysis (Wiley, New York, 1978), pp. 478–486.

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Figures (21)

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

Fig. 3
Fig. 3

Speckle-limited image of the pristine object.

Fig. 4
Fig. 4

Pupil of the simulated receiver aperture.

Fig. 5
Fig. 5

Flow diagram for computation of the autocorrelation of the object for fields 1 and 2.

Fig. 6
Fig. 6

Flow diagram for the statistical reconstructor.

Fig. 7
Fig. 7

Image reconstruction of the comparison object with use of the statistical formulation, assuming that true field autocorrelations are available. (a) Pristine image at system resolution, (b) typical speckle field in pupil plane, (c) three-frame-average reconstructed image, and (d) three-frame-average reconstructed image, with square-root intensity compression.

Fig. 8
Fig. 8

Field Strehls of eight reconstructed fields as a function of SNR per subaperture with use of known field autocorrelation.

Fig. 9
Fig. 9

Image of a single field selected by the statistical reconstructor.

Fig. 10
Fig. 10

Best image for a single field at an intermediate iteration for statistical reconstruction.

Fig. 11
Fig. 11

Flow diagram for the phase-retrieval algorithm.

Fig. 12
Fig. 12

Image generated by using autocorrelation processing and phase retrieval.

Fig. 13
Fig. 13

Image of a single field at best intermediate iteration.

Fig. 14
Fig. 14

Image of a single field selected by the phase-retrieval algorithm.

Fig. 15
Fig. 15

Sum of three frames.

Fig. 16
Fig. 16

Typical single image for laboratory data.

Fig. 17
Fig. 17

Flow diagram for computation of the autocorrelation of the object for fields 1 and 2.

Fig. 18
Fig. 18

Reconstructed image after 9 iterations.

Fig. 19
Fig. 19

Reconstructed image after 100 iterations.

Fig. 20
Fig. 20

Reconstructed image after 8 iterations using only first field of each of 3 frames.

Fig. 21
Fig. 21

Reconstructed image after 100 iterations using only the first field of each of 3 frames.

Tables (2)

Tables Icon

Table 1 Image Strehl Results for Statistical Reconstructor for the Comparison Object

Tables Icon

Table 2 Image Strehl Results for Phase Retrieval for the Comparison Object

Equations (19)

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

Strehl=maxΔx|ΣxG1(x)G2*(x+Δx)|2/Σx|G1(x)|2Σx|G2(x)|2,
I1i(x)=|E1i|2+n1i,I2i(x)=|E2i|2+n2i,
I12i(x)=E1iE2i*+n12i
L=-(1/2){ΣiE1i*Me11-1E1i+ΣiE2i*Me22-1E2i+Σi(E1i*Me12-1E2i+c.c.)+Σi,x[I1i(x)-|E1i|2]2/σ12+[I2i(x)-|E2i|2]2/σ22+|I12i(x)-E1iE2i*|2/σ122},
Me11-1Me12-1Me12-1*TMe22-1
E1(x)E1(x+Δx)*E1(x)E2(x+Δx)*E2(x)E1(x+Δx)*E2(x)E2(x+Δx)*-1,
OAC12(Δx)ΣO1(x)O2(x+Δx).
OAC12(Δx)=min[|OAC12(Δx)|,×|OAC11(Δx)OAC22(Δx)|1/2]×exp{i arg[OAC12(Δx)]}.
(1+Me11|E2i(x)|2/σ122)E1i(x)̲
=-(Me11Me12-1)E2i(x)+(2/σ12)Me11(I1(x)-|E1i(x)|2)E1i(x)+(1/σ122)Me11I12(x)E2i(x)̲,
(1+Me22|E1i(x)|2/σ122)E2i(x)̲
=-(Me22Me12-1)E2i(x)+(2/σ22)Me22(I2(x)-|E2i(x)|2)E2i(x)+(1/σ122)Me22I12(x)*E1i(x)̲.
|E2i|=I2i(x)1/2,
|E1i|=I1i(x)1/2,
arg[E1i(x)]=arg[Me11I12i(x)E2i(x)],
arg[E2i(x)]=arg[Me22I12i(x)*E1i(x)].
1/2{[Σj,k|E1(j, k, n+1)-E1(j, k, n)|2/
Σj,k|E1(j, k, n+1)|2]1/2+[Σj,k|E2(j, k, n+1)
-E2(j, k, n)|2/Σj,k|E2(j, k, n+1)|2]1/2},

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