Abstract

Phase diversity is a method of image-based wavefront sensing that simultaneously estimates the unknown phase aberrations of an imaging system along with an image of the object. To perform this estimation a series of images differing by a known aberration, typically defocus, are used. In this paper we present a new method of introducing the diversity unique to segmented and multi-aperture systems in which individual segments or sub-apertures are pistoned with respect to one another. We compare this new diversity with the conventional focus diversity.

© 2008 Optical Society of America

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References

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  1. TPF Science Working Group, Terrestrial Planet Finder Interferometer: Science Working Group Report, P. R. Lawson, O. P. Lay, K. J. Johnston, and C. A. Beichman, eds., JPL Publication 07-01 (Jet Propulsion Laboratory, 2007).
  2. J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).
  3. R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32-39 (1979).
  4. R. G. Paxman and J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914-923 (1988).
  5. R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072-1085 (1992).
  6. R. G. Paxman and J. H. Seldin, “Fine-resolution astronomical imaging with phase-diverse speckle,” Proc. SPIE 2029, 287-298 (1993).
  7. M. R. Bolcar and J. R. Fienup, “Phase diversity with broadband illumination,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JTuA6.
  8. M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE 5896, 58960G (2005).
  9. S. T. Thurman and J. R. Fienup, “Multi-aperture Fourier transform imaging spectroscopy: theory and imaging properties,” Opt. Express 13, 2160-2175 (2005).
  10. J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2004).
  11. D. J. Lee, M. C. Roggemann, B. M. Welsh, and E. R. Crosby, “Evaluation of least-squares phase-diversity techniques for space telescope wave-front sensing,” Appl. Opt. 36, 9186-9197 (1997).
  12. O. M. Bucci, A. Capozzoli, and G. D'Elia, “Regularizing strategy for image restoration and wave-front sensing by phase diversity,” J. Opt. Soc. Am. A 16, 1759-1768 (1999).
    [CrossRef]
  13. M. G. Löfdahl and G. B. Scharmer, “Phase diverse speckle inversion applied to data from the Swedish 1-meter solar telescope,” Proc. SPIE 4853, 567-575 (2002).
  14. A. Blanc, L. Mugnier, and J. Idier, “Marginal estimation of aberrations and image restoration by use of phase diversity,” J. Opt. Soc. Am. A 20, 1035-1045 (2003).
    [CrossRef]
  15. J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, multiple telescope array using broad-band phase diversity,” Proc. SPIE 4091, 48-63 (2000).
  16. Provided through the courtesy of Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California (http://aviris.jpl.nasa.gov/).
  17. R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R. J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express 15, 15935-15951 (2007).
  18. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156-158 (2008).
    [CrossRef]
  19. L. P. Yaroslavsky and H. J. Caulfield, “Deconvolution of multiple images of the same object,” Appl. Opt. 33, 2157-2162 (1994).

2008

2007

2005

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE 5896, 58960G (2005).

S. T. Thurman and J. R. Fienup, “Multi-aperture Fourier transform imaging spectroscopy: theory and imaging properties,” Opt. Express 13, 2160-2175 (2005).

2004

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

2003

2002

M. G. Löfdahl and G. B. Scharmer, “Phase diverse speckle inversion applied to data from the Swedish 1-meter solar telescope,” Proc. SPIE 4853, 567-575 (2002).

2000

J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, multiple telescope array using broad-band phase diversity,” Proc. SPIE 4091, 48-63 (2000).

1999

1997

1994

1993

R. G. Paxman and J. H. Seldin, “Fine-resolution astronomical imaging with phase-diverse speckle,” Proc. SPIE 2029, 287-298 (1993).

1992

1988

1979

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32-39 (1979).

Benson, L.

J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, multiple telescope array using broad-band phase diversity,” Proc. SPIE 4091, 48-63 (2000).

Bierhaus, E.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Blanc, A.

Bolcar, M. R.

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE 5896, 58960G (2005).

M. R. Bolcar and J. R. Fienup, “Phase diversity with broadband illumination,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JTuA6.

Bucci, O. M.

Capozzoli, A.

Caulfield, H. J.

Chidlaw, R.

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32-39 (1979).

Crosby, E. R.

Dalton, J. B.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

de Pater, I.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

D'Elia, G.

Delory, G.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Duncan, A.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Fienup, J. R.

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156-158 (2008).
[CrossRef]

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE 5896, 58960G (2005).

S. T. Thurman and J. R. Fienup, “Multi-aperture Fourier transform imaging spectroscopy: theory and imaging properties,” Opt. Express 13, 2160-2175 (2005).

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

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

R. G. Paxman and J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914-923 (1988).

M. R. Bolcar and J. R. Fienup, “Phase diversity with broadband illumination,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JTuA6.

Gonsalves, R. A.

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32-39 (1979).

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2004).

Graham, J. R.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Guizar-Sicairos, M.

Idier, J.

Kendrick, R. L.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Lee, D. J.

Lipps, J. H.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Löfdahl, M. G.

M. G. Löfdahl and G. B. Scharmer, “Phase diverse speckle inversion applied to data from the Swedish 1-meter solar telescope,” Proc. SPIE 4853, 567-575 (2002).

Manga, M.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Marcus, P.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Mason, J. E.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Mugnier, L.

Paxman, R. G.

J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, multiple telescope array using broad-band phase diversity,” Proc. SPIE 4091, 48-63 (2000).

R. G. Paxman and J. H. Seldin, “Fine-resolution astronomical imaging with phase-diverse speckle,” Proc. SPIE 2029, 287-298 (1993).

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

R. G. Paxman and J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914-923 (1988).

Pitman, J. T.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Pueyo, L.

Reiboldt, S.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Roggemann, M. C.

Scharmer, G. B.

M. G. Löfdahl and G. B. Scharmer, “Phase diverse speckle inversion applied to data from the Swedish 1-meter solar telescope,” Proc. SPIE 4853, 567-575 (2002).

Schulz, T. J.

Seldin, J. H.

J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, multiple telescope array using broad-band phase diversity,” Proc. SPIE 4091, 48-63 (2000).

R. G. Paxman and J. H. Seldin, “Fine-resolution astronomical imaging with phase-diverse speckle,” Proc. SPIE 2029, 287-298 (1993).

Sigler, R. D.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Sivaramakrishnan, A.

Smith, E. H.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Soummer, R.

Stone, R. E.

J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, multiple telescope array using broad-band phase diversity,” Proc. SPIE 4091, 48-63 (2000).

Stubbs, D.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Thurman, S. T.

Vanderbei, R. J.

Welsh, B. M.

Yaroslavsky, L. P.

Yu, J. W.

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

Zarifis, V. G.

J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, multiple telescope array using broad-band phase diversity,” Proc. SPIE 4091, 48-63 (2000).

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Proc. SPIE

J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, multiple telescope array using broad-band phase diversity,” Proc. SPIE 4091, 48-63 (2000).

M. G. Löfdahl and G. B. Scharmer, “Phase diverse speckle inversion applied to data from the Swedish 1-meter solar telescope,” Proc. SPIE 4853, 567-575 (2002).

R. G. Paxman and J. H. Seldin, “Fine-resolution astronomical imaging with phase-diverse speckle,” Proc. SPIE 2029, 287-298 (1993).

J. T. Pitman, A. Duncan, D. Stubbs, R. D. Sigler, R. L. Kendrick, E. H. Smith, J. E. Mason, G. Delory, J. H. Lipps, M. Manga, J. R. Graham, I. de Pater, S. Reiboldt, P. Marcus, E. Bierhaus, J. B. Dalton, J. R. Fienup, and J. W. Yu, “Multiple Instrument Distributed Aperture Sensor (MIDAS) for planetary remote sensing,” Proc. SPIE 5660, 168-180(2004).

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32-39 (1979).

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE 5896, 58960G (2005).

Other

TPF Science Working Group, Terrestrial Planet Finder Interferometer: Science Working Group Report, P. R. Lawson, O. P. Lay, K. J. Johnston, and C. A. Beichman, eds., JPL Publication 07-01 (Jet Propulsion Laboratory, 2007).

M. R. Bolcar and J. R. Fienup, “Phase diversity with broadband illumination,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JTuA6.

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2004).

Provided through the courtesy of Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California (http://aviris.jpl.nasa.gov/).

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Figures (7)

Fig. 1
Fig. 1

Example of phase diversity implementation: (a) sub- aperture piston diversity, (b) focus diversity. The scale has units of waves.

Fig. 2
Fig. 2

Panchromatic representation of multi-spectral object. The object consists of 11 spectral bands, centered about 1 μm and spanning 96 nm .

Fig. 3
Fig. 3

Example phase realization composed of up to 6th order Zernike terms on the global aperture and up to 2nd order Zernike terms on each sub-aperture. The scale has units of waves.

Fig. 4
Fig. 4

Phase estimation results in terms of Strehl ratio: (a) average pixel SNR of 20, (b) SNR 74, (c) SNR 170. Vertical axis shows Strehl ratio, horizontal axis is the peak-to-valley amount of diversity in units of waves. Each data point is an average of 25 trials ( 5   phase realizations × 5   noise realizations ); error bars show a single standard deviation.

Fig. 5
Fig. 5

Number of iterations before algorithm reaches exit criteria: (a) average pixel SNR of 20, (b) SNR 74, (c) SNR 170. Vertical axis shows the number of iterations, horizontal axis is the peak-to-valley amount of diversity in units of waves. Each data point is an average of 25 trials; error bars show a single standard deviation.

Fig. 6
Fig. 6

Normalized RMS error between reconstructed aberrated object and reconstructed diffraction-limited object: (a) average pixel SNR of 20, (b) SNR 74, (c) SNR 170. Vertical axis shows NRMSE, horizontal axis is the peak-to-valley amount of diversity in waves. Each data point is an average of 25 trials; error bars show a single standard deviation.

Fig. 7
Fig. 7

Example reconstructed images: (a) reconstructed gray-world image using phase estimate from phase diversity algorithm, (b) reconstructed gray-world diffraction-limited image, (c) original aberrated, zero-diversity image used as input to phase diversity algorithm. All images are shown on same color scale; SNR 74.

Equations (14)

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

d k ( u , v ) = λ f λ ( u , v ) * s k , λ ( u , v ) + n k ( u , v ) ,
s k , λ ( u , v ) = | h k , λ ( u , v ) | 2 ,
h k , λ ( u , v ) = exp [ i π D k λ B k ( u 2 + v 2 ) ] × P k , λ ( x , y ) exp [ i π A k λ B k ( x 2 + y 2 ) ] × exp [ i 2 π λ B k ( x u + y v ) ] d x d y ,
P k , λ ( x , y ) = q = 1 Q P q , k , λ ( x , y ) = q = 1 Q | P q ( x , y ) | exp { i 2 π λ [ W q ( x , y ) + W q , k div ( x , y ) ] } ,
W q ( x , y ) = m , n α q , m , n δ ( x m Δ x , y n Δ y ) ,
W q ( x , y ) = j = 1 J α q , j Z q , j ( x , y ) ,
L [ { d k ( u , v ) } ; f , α ] = k = 1 K f u , f v | D k ( f u , f v ) λ F λ ( f u , f v ) S k , λ ( f u , f v ) | 2 ,
f λ ( u , v ) = Φ λ f ( u , v ) ,
L RG [ { d k ( u , v ) } ; α ] = f u , f v χ { k = 1 K | D k | 2 | j = 1 K D j λ Φ λ S j , λ * ( α ) | 2 m = 1 K | λ Φ λ S m , λ ( α ) | 2 } ,
L R G α ξ , j = 8 π Im { k = 1 K λ Φ λ λ f u , f v Z ξ , j ( f u , f v ) P ξ , k , λ ( f u , f v ) exp [ i π A k λ B k ( f u 2 + f v 2 ) ] × f u , f v χ Y k * ( f u , f v ) H k , λ * ( f u f u , f v f v ) } ,
H k , λ ( f u , f v ) = P k , λ ( f u , f v ) exp [ i π A k λ B k ( f u 2 + f v 2 ) ] ,
Y k ( f u , f v ) = m = 1 K | λ Φ λ S m , λ | 2 ( = 1 K D λ Φ λ S , λ * ) D k * | = 1 K D λ Φ λ S , λ * | 2 λ Φ λ S k , λ * ( m = 1 K | λ Φ λ S m , λ | 2 ) 2 .
S . R . = max u , v [ s res ( u , v ) ] s per ( 0 , 0 ) ,
d DL ( u , v ) = [ λ f λ ( u , v ) ] * [ λ s DL , λ ( u , v ) ] + n ( u , v ) ,

Metrics