Abstract

In coherent lensless imaging, the presence of image sidelobes, which arise as a natural consequence of the finite nature of the detector array, was early recognized as a convergence issue for phase retrieval algorithms that rely on an object support constraint. To mitigate the problem of truncated far-field measurement, a controlled analytic continuation by means of an iterative transform algorithm with weighted projections is proposed and tested. This approach avoids the use of sidelobe reduction windows and achieves full-resolution reconstructions.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. 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]
  2. P. S. Idell, J. R. Fienup, and R. S. Goodman, “Image synthesis from nonimaged laser-speckle patterns,” Opt. Lett. 12, 858-860 (1987).
    [CrossRef] [PubMed]
  3. J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrieval from experimental far-field speckle data,” Opt. Lett. 13, 619-621 (1988).
    [CrossRef] [PubMed]
  4. J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express 14, 498-508 (2006).
    [CrossRef] [PubMed]
  5. S. G. Podorov, K. M. Pavlov, and D. M. Paganin, “A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging,” Opt. Express 15, 9954-9962 (2007).
    [CrossRef] [PubMed]
  6. M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express 15, 17592-17612 (2007).
    [CrossRef] [PubMed]
  7. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123-1130 (1962).
    [CrossRef]
  8. H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179-1200 (2006).
    [CrossRef]
  9. R. G. Paxman, J. R. Fienup, and J. T. Clinthorne, “The effects of tapered illumination and Fourier intensity errors on phase retrieval,” Proc. SPIE 828, 184-189 (1987).
  10. J. R. Fienup and A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A 7, 450-458 (1990).
    [CrossRef]
  11. J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60-69 (1999).
    [CrossRef]
  12. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef] [PubMed]
  13. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897-1907 (1986).
    [CrossRef]
  14. M. Guizar-Sicairos and J. R. Fienup, “Complex valued object reconstruction from extrapolated intensity measurements,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2006), paper FMI6.
  15. From the ERIM (now General Dynamics, Ypsilanti Michigan) X-band DCS radar.
  16. T. R. Crimmins, J. R. Fienup, and B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 3-13 (1990).
    [CrossRef]
  17. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
    [CrossRef]
  18. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156-158 (2008).
    [CrossRef] [PubMed]
  19. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737-1746 (1993).
    [CrossRef] [PubMed]
  20. A. Levi and H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H.Stark, ed. (Academic, 1987), pp. 277-320.
  21. J. R. Fienup and J. D. Gorman, “Image reconstruction for an aberrated amplitude interferometer with a partially-filled aperture,” in NOAO-ESO Conference on High-Resolution Imaging by Interferometry: Ground-Based Interferometry at Visible and Infrared Wavelengths, F.Merkle, ed. (1988), pp. 293-301.
  22. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709-720 (1974).
    [CrossRef]
  23. L. S. Joyce and W. L. Root, “Precision bounds in superresolution processing,” J. Opt. Soc. Am. A 1, 149-168 (1984).
    [CrossRef]
  24. J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. 36, 8352-8357 (1997).
    [CrossRef]
  25. M. Guizar-Sicairos and J. R. Fienup, “Iterative extrapolation for image reconstruction by phase retrieval from truncated far field intensity measurements,” in Coherence 2007: International Workshop on Phase Retrieval and Coherent Scattering (ALS Communications, 2007), poster P10.

2008 (1)

2007 (2)

2006 (2)

2003 (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

1999 (1)

J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60-69 (1999).
[CrossRef]

1997 (1)

1993 (1)

1990 (2)

1988 (1)

1987 (3)

1986 (1)

1984 (1)

1982 (1)

1974 (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709-720 (1974).
[CrossRef]

1962 (1)

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Acta (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709-720 (1974).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. B (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[CrossRef]

Proc. SPIE (2)

R. G. Paxman, J. R. Fienup, and J. T. Clinthorne, “The effects of tapered illumination and Fourier intensity errors on phase retrieval,” Proc. SPIE 828, 184-189 (1987).

J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60-69 (1999).
[CrossRef]

Other (5)

M. Guizar-Sicairos and J. R. Fienup, “Complex valued object reconstruction from extrapolated intensity measurements,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2006), paper FMI6.

From the ERIM (now General Dynamics, Ypsilanti Michigan) X-band DCS radar.

A. Levi and H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H.Stark, ed. (Academic, 1987), pp. 277-320.

J. R. Fienup and J. D. Gorman, “Image reconstruction for an aberrated amplitude interferometer with a partially-filled aperture,” in NOAO-ESO Conference on High-Resolution Imaging by Interferometry: Ground-Based Interferometry at Visible and Infrared Wavelengths, F.Merkle, ed. (1988), pp. 293-301.

M. Guizar-Sicairos and J. R. Fienup, “Iterative extrapolation for image reconstruction by phase retrieval from truncated far field intensity measurements,” in Coherence 2007: International Workshop on Phase Retrieval and Coherent Scattering (ALS Communications, 2007), poster P10.

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 (7)

Fig. 1
Fig. 1

Setup for coherent lensless imaging by phase retrieval.

Fig. 2
Fig. 2

(a) Original SAR image. (b) Hard-edge object support. (c) 256 × 256 finite-support object and (d) its Fourier magnitude. Nonphysical wrap-around effects are present in (d) at the edge of the computational window. (e) 128 × 128 central portion of (d), which is used by the phase retrieval algorithms. (f) Diffraction-limited image magnitude; the 128 × 128 image was upsampled by 2 × , and only a 160 × 160 pixel inset is displayed. Due to their large dynamic ranges, the square root of the magnitude is displayed in (a), (c), and (f).

Fig. 3
Fig. 3

Block diagram of the iterative transform algorithm.

Fig. 4
Fig. 4

(a) Final reconstruction using the far-field magnitude shown in Fig. 2e. (b) Far-field phase error Δ ϕ ( u ) of (a). (c) Low-pass diffraction-limited image and (d) its Fourier magnitude. (e) Reconstruction from the low-pass far-field magnitude shown in (d). (f) Far-field phase error of (e). Phase is shown from π to π in (b) and (f).

Fig. 5
Fig. 5

(a) 256 × 256 weighting function W ( u ) . (b) Fourth root of the far-field magnitude of the extrapolated reconstruction. (c) Reconstruction with the ITA with weighted projections with the far-field extrapolated data included. (d) 256 × 256 far-field phase error Δ ϕ ( u ) of (c) with respect to object Fourier transform [Fig. 2d]. (e) 40 × 80 inset of (d); dashed line shows the edge of the measurement window. (f) Same as (c) but zeroing the far-field extrapolated data. (g) Far-field phase error within the original 128 × 128 measurement area. Phase is shown from π to π in (d), (e), and (g).

Fig. 6
Fig. 6

(a) Cut through the weighting function W ( u ) (solid curve) and W η ( u ) for the weight modulation approach used for SNR = 5 and η = 0.1 (dashed curve). (b) NRMSE, ε, versus iteration number for the fully resolved (with and without use of extrapolation by weighted projections) and the low-pass reconstructions. (c) Object support error, E, versus iteration number for the reconstruction with the ITA with extrapolation by weighted projections. E was computed with and without the far-field extrapolated data for the same reconstruction. The support error, E = 0.139 , for the diffraction-limited image is shown for comparison purposes (horizontal dashed–dotted line).

Fig. 7
Fig. 7

Reconstruction with the ITA with weighted projections for an average SNR of (a) 100, (b) 10, and (c) 5. (d) Reconstruction for SNR = 5 using the alternative weighting for noisy data ( η = 0.1 ) . (e) NRMSE, ε, versus iteration number for reconstructions from noisy data.

Equations (9)

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

F ( u ) = F ( u ) exp [ i ϕ ( u ) ] = f ( x ) exp ( i 2 π u x ) d x ,
G k ( u ) = G k ( u ) exp [ i θ k ( u ) ] .
G k ( u ) = F ( u ) exp [ i θ k ( u ) ] ,
E 2 = x S g ( x ) 2 x g ( x ) 2 ,
g k + 1 ( x ) = { g k ( x ) , if x S 0 if x S ,
g k + 1 ( x ) = { g k ( x ) , if x S g k ( x ) β g k ( x ) , if x S ,
G k ( u ) = W ( u ) F ( u ) exp [ i θ k ( u ) ] + [ 1 W ( u ) ] G k ( u ) ,
ε k 2 = min α , x 0 [ x f ( x ) α g k ( x x 0 ) 2 x f ( x ) 2 ] ,
W η ( u ) = 1 η + η F ( u ) max u F ( u )

Metrics