Abstract

Fourier Ptychography is a new computational microscopy technique that achieves gigapixel images with both wide field of view and high resolution in both phase and amplitude. The hardware setup involves a simple replacement of the microscope’s illumination unit with a programmable LED array, allowing one to flexibly pattern illumination angles without any moving parts. In previous work, a series of low-resolution images was taken by sequentially turning on each single LED in the array, and the data were then combined to recover a bandwidth much higher than the one allowed by the original imaging system. Here, we demonstrate a multiplexed illumination strategy in which multiple randomly selected LEDs are turned on for each image. Since each LED corresponds to a different area of Fourier space, the total number of images can be significantly reduced, without sacrificing image quality. We demonstrate this method experimentally in a modified commercial microscope. Compared to sequential scanning, our multiplexed strategy achieves similar results with approximately an order of magnitude reduction in both acquisition time and data capture requirements.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Zheng, C. Kolner, and C. Yang, “Microscopy refocusing and dark-field imaging by using a simple LED array,” Opt. Lett. 36, 3987–3989 (2011).
    [Crossref] [PubMed]
  2. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier Ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
    [Crossref]
  3. L. Tian, J. Wang, and L. Waller, “3D differential phase-contrast microscopy with computational illumination using an LED array,” Opt. Lett. 39, 1326–1329 (2014).
    [Crossref] [PubMed]
  4. D. Hamilton and C. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microscopy 133, 27–39 (1984).
    [Crossref]
  5. T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nature Methods 9, 1195–1197 (2012).
    [Crossref] [PubMed]
  6. M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microscopy 235, 144–162 (2009).
    [Crossref]
  7. L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
    [Crossref]
  8. X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
    [Crossref] [PubMed]
  9. L. Pantanowitz, “Digital images and the future of digital pathology,” J. Pathology Informatics 1, 15 (2010).
    [Crossref]
  10. X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via fourier ptychographic microscopy,” Opt. Lett. 38, 4845–4848 (2013).
    [Crossref] [PubMed]
  11. S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture fourier holographic optical microscopy,” Phys. Rev. Lett. 97, 168102 (2006).
    [Crossref] [PubMed]
  12. T. R. Hillman, T. Gutzler, S. A. Alexandrov, and D. D. Sampson, “High-resolution, wide-field object reconstruction with synthetic aperturefourier holographic optical microscopy,” Opt. Express 17, 7873–7892 (2009).
    [Crossref] [PubMed]
  13. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  14. J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
    [Crossref]
  15. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
    [Crossref] [PubMed]
  16. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
    [Crossref] [PubMed]
  17. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
    [Crossref] [PubMed]
  18. O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
    [Crossref]
  19. Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, “Multiplexing for optimal lighting,” IEEE Trans. Pattern Analysis Machine Intelligence 29, 1339–1354 (2007).
    [Crossref]
  20. N. Ratner and Y. Y. Schechner, “Illumination multiplexing within fundamental limits,” in “Computer Vision and Pattern Recognition” (IEEE, 2007), pp. 1–8.
  21. R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).
  22. C. Rydberg, J. Bengtsson, and et al., “Numerical algorithm for the retrieval of spatial coherence properties of partially coherent beams from transverse intensity measurements,” Opt. Express 15, 13613–13623 (2007).
    [Crossref] [PubMed]
  23. P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
    [Crossref] [PubMed]
  24. D. Dominguez, L. Molina, D. B. Desai, T. O’Loughlin, A. A. Bernussi, and L. G. de Peralta, “Hemispherical digital optical condensers with no lenses, mirrors, or moving parts,” Opt. Express 22, 6948–6957 (2014).
    [Crossref] [PubMed]
  25. L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, “Experimental compressive phase space tomography,” Opt. Express 20, 8296–8308 (2012).
    [Crossref] [PubMed]
  26. Y. Shechtman, A. Beck, and Y. Eldar, “GESPAR: Efficient phase retrieval of sparse signals,” IEEE Trans. Sig. Processing 62, 928–938 (2014).
    [Crossref]

2014 (4)

2013 (3)

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref] [PubMed]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier Ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
[Crossref]

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via fourier ptychographic microscopy,” Opt. Lett. 38, 4845–4848 (2013).
[Crossref] [PubMed]

2012 (3)

T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nature Methods 9, 1195–1197 (2012).
[Crossref] [PubMed]

L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, “Experimental compressive phase space tomography,” Opt. Express 20, 8296–8308 (2012).
[Crossref] [PubMed]

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

2011 (1)

2010 (1)

L. Pantanowitz, “Digital images and the future of digital pathology,” J. Pathology Informatics 1, 15 (2010).
[Crossref]

2009 (4)

M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microscopy 235, 144–162 (2009).
[Crossref]

T. R. Hillman, T. Gutzler, S. A. Alexandrov, and D. D. Sampson, “High-resolution, wide-field object reconstruction with synthetic aperturefourier holographic optical microscopy,” Opt. Express 17, 7873–7892 (2009).
[Crossref] [PubMed]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

2008 (2)

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[Crossref]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
[Crossref] [PubMed]

2007 (2)

C. Rydberg, J. Bengtsson, and et al., “Numerical algorithm for the retrieval of spatial coherence properties of partially coherent beams from transverse intensity measurements,” Opt. Express 15, 13613–13623 (2007).
[Crossref] [PubMed]

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, “Multiplexing for optimal lighting,” IEEE Trans. Pattern Analysis Machine Intelligence 29, 1339–1354 (2007).
[Crossref]

2006 (1)

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture fourier holographic optical microscopy,” Phys. Rev. Lett. 97, 168102 (2006).
[Crossref] [PubMed]

2004 (1)

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

1984 (1)

D. Hamilton and C. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microscopy 133, 27–39 (1984).
[Crossref]

1982 (1)

1971 (1)

R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Alexandrov, S. A.

Barbastathis, G.

Beck, A.

Y. Shechtman, A. Beck, and Y. Eldar, “GESPAR: Efficient phase retrieval of sparse signals,” IEEE Trans. Sig. Processing 62, 928–938 (2014).
[Crossref]

Belhumeur, P. N.

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, “Multiplexing for optimal lighting,” IEEE Trans. Pattern Analysis Machine Intelligence 29, 1339–1354 (2007).
[Crossref]

Bengtsson, J.

Bernussi, A. A.

Bunk, O.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[Crossref]

Chu, K. K.

T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nature Methods 9, 1195–1197 (2012).
[Crossref] [PubMed]

de Peralta, L. G.

Desai, D. B.

Dierolf, M.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[Crossref]

Dominguez, D.

Eldar, Y.

Y. Shechtman, A. Beck, and Y. Eldar, “GESPAR: Efficient phase retrieval of sparse signals,” IEEE Trans. Sig. Processing 62, 928–938 (2014).
[Crossref]

Faulkner, H. M.

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

Fienup, J. R.

Fleischer, J.

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

Ford, T. N.

T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nature Methods 9, 1195–1197 (2012).
[Crossref] [PubMed]

Gerchberg, R.

R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Guizar-Sicairos, M.

Gutzler, T.

Hamilton, D.

D. Hamilton and C. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microscopy 133, 27–39 (1984).
[Crossref]

Hillman, T. R.

Horstmeyer, R.

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier Ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
[Crossref]

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via fourier ptychographic microscopy,” Opt. Lett. 38, 4845–4848 (2013).
[Crossref] [PubMed]

Johnson, I.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[Crossref]

Kolner, C.

Kynde, S.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[Crossref]

Lee, J.

Levoy, M.

M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microscopy 235, 144–162 (2009).
[Crossref]

Maiden, A. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

Marti, O.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[Crossref]

McDowall, I.

M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microscopy 235, 144–162 (2009).
[Crossref]

Menzel, A.

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

Mertz, J.

T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nature Methods 9, 1195–1197 (2012).
[Crossref] [PubMed]

Molina, L.

Nayar, S. K.

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, “Multiplexing for optimal lighting,” IEEE Trans. Pattern Analysis Machine Intelligence 29, 1339–1354 (2007).
[Crossref]

O’Loughlin, T.

Oh, S. B.

Ou, X.

Pantanowitz, L.

L. Pantanowitz, “Digital images and the future of digital pathology,” J. Pathology Informatics 1, 15 (2010).
[Crossref]

Pfeiffer, F.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[Crossref]

Ratner, N.

N. Ratner and Y. Y. Schechner, “Illumination multiplexing within fundamental limits,” in “Computer Vision and Pattern Recognition” (IEEE, 2007), pp. 1–8.

Rodenburg, J. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

Rydberg, C.

Sampson, D. D.

Saxton, W.

R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Schechner, Y. Y.

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, “Multiplexing for optimal lighting,” IEEE Trans. Pattern Analysis Machine Intelligence 29, 1339–1354 (2007).
[Crossref]

N. Ratner and Y. Y. Schechner, “Illumination multiplexing within fundamental limits,” in “Computer Vision and Pattern Recognition” (IEEE, 2007), pp. 1–8.

Shechtman, Y.

Y. Shechtman, A. Beck, and Y. Eldar, “GESPAR: Efficient phase retrieval of sparse signals,” IEEE Trans. Sig. Processing 62, 928–938 (2014).
[Crossref]

Sheppard, C.

D. Hamilton and C. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microscopy 133, 27–39 (1984).
[Crossref]

Situ, G.

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

Thibault, P.

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

Tian, L.

Waller, L.

L. Tian, J. Wang, and L. Waller, “3D differential phase-contrast microscopy with computational illumination using an LED array,” Opt. Lett. 39, 1326–1329 (2014).
[Crossref] [PubMed]

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

Wang, J.

Yang, C.

Zhang, Z.

M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microscopy 235, 144–162 (2009).
[Crossref]

Zheng, G.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

IEEE Trans. Pattern Analysis Machine Intelligence (1)

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, “Multiplexing for optimal lighting,” IEEE Trans. Pattern Analysis Machine Intelligence 29, 1339–1354 (2007).
[Crossref]

IEEE Trans. Sig. Processing (1)

Y. Shechtman, A. Beck, and Y. Eldar, “GESPAR: Efficient phase retrieval of sparse signals,” IEEE Trans. Sig. Processing 62, 928–938 (2014).
[Crossref]

J. Microscopy (2)

D. Hamilton and C. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microscopy 133, 27–39 (1984).
[Crossref]

M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microscopy 235, 144–162 (2009).
[Crossref]

J. Pathology Informatics (1)

L. Pantanowitz, “Digital images and the future of digital pathology,” J. Pathology Informatics 1, 15 (2010).
[Crossref]

Nat. Photonics (2)

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier Ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
[Crossref]

Nature (1)

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref] [PubMed]

Nature Methods (1)

T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nature Methods 9, 1195–1197 (2012).
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (3)

Optik (1)

R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Phys. Rev. Lett. (1)

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture fourier holographic optical microscopy,” Phys. Rev. Lett. 97, 168102 (2006).
[Crossref] [PubMed]

Ultramicroscopy (3)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[Crossref]

Other (1)

N. Ratner and Y. Y. Schechner, “Illumination multiplexing within fundamental limits,” in “Computer Vision and Pattern Recognition” (IEEE, 2007), pp. 1–8.

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 (5)

Fig. 1
Fig. 1

Summary of Fourier Ptychography (FP) in an LED array microscope. (a) A sample is illuminated from different angles by turning on different LEDs of an array. (b) Our experimental setup on a Nikon TE300 microscope. (c) Images taken with different LEDs contain information from different spatial frequency areas of the sample. The central (brightfield) LED fills an area defined by the NA (0.1) of the objective. Images taken with top and left (dark field) LEDs result in accentuated edges along the corresponding orientations. (d) FP reconstructs a high resolution image from many LEDs, while simultaneously estimating aberrations.

Fig. 2
Fig. 2

Sample datasets for multiplexed illumination coding in Fourier Ptychography. (Top) Four randomly chosen LEDs are turned on for each measurement. (Middle) The captured images corresponding to each LED pattern. (Bottom) Fourier coverage of the sample’s Fourier space for each of the LED patterns (drawn to scale). Turning on multiple well-separated LEDs allows information from multiple areas of Fourier space to pass through the system simultaneously. The center unshaded circle represents the NA of the objective lens.

Fig. 3
Fig. 3

Flow chart of the reconstruction algorithm.

Fig. 4
Fig. 4

Experimental results for multiplexed illumination of a resolution target. (a) The original low resolution image from a 4 × 0.1 NA objective taken with only the central LED on. (b) A zoom-in on the smallest features. (c) Reconstruction result from sequential FP with Nimg = 293 single LED images having a total acquisition time of T=586s. (d) Multiplexing 4 LEDs for each image while preserving the number of measurements Nimg = 293 reduces the total acquisition time to T=293s. (e) The multiplexed illumination also allows reduction of the number of measurements to Nimg = 74 with T=74s.

Fig. 5
Fig. 5

Experimental results for multiplexed illumination of a stained dog stomach cardiac region sample. (a) The original low resolution image from a 4× 0.1NA objective taken with only the central LED on. (b1, c1) Zoom-in of the regions denoted by red and green squares. (b2, c2) Amplitude and (b5, c5) phase reconstructions from sequential FP with Nimg = 293 single LED images and a total acquisition time of T=586s. (b3, c3) Amplitude and (b6, c6) phase reconstructions from multiplexing 8 LEDs with Nimg = 293 and T=293s. Amplitude (b4, c7) and phase (b4, c7) reconstructions for multiplexing 8 LEDs with Nimg = 40 and T=40s.

Equations (33)

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

i m ( r ) = | [ O ( k k m ) P ( k ) ] ( r ) | 2 ,
I p ( r ) = m p i m ( r ) = m p | [ O ( k k m ) P ( k ) ] ( r ) | 2 ,
A p , m = { 1 , m p 0 , otherwise , m = 1 , , N LED , p = 1 , , N img .
min O ( k ) , P ( k ) , { b p } p = 1 N img p = 1 N img r | I p ( r ) ( m p | [ O ( k k m ) P ( k ) ] ( r ) | 2 + b p ) | 2 .
I ^ p ( r ) = I p ( r ) b ^ p .
Ψ m ( k ) = O ( k k m ) P ( k ) ,
ψ m ( i ) ( r ) = [ Ψ m ( i ) ( k ) ] ( r ) , m p .
Φ m ( i ) ( k ) = [ ϕ m ( i ) ( r ) ] 1 ( k ) with ϕ m ( i ) ( r ) = I ^ p ( r ) m p | ψ m ( i ) ( r ) | 2 ψ m ( i ) ( r ) , m p .
O ( i + 1 ) ( k ) = O ( i ) ( k ) + m p | P ( i ) ( k + k m ) | [ P ( i ) ( k + k m ) ] * [ Φ m ( i ) ( k + k m ) O ( i ) ( k ) P ( i ) ( k + k m ) ] | P ( i ) ( k ) | max ( m p | P ( i ) ( k + k m ) | 2 + δ 1 )
P ( i + 1 ) ( k ) = P ( i ) ( k ) + m p | O ( i ) ( k k m ) | [ O ( i ) ( k k m ) ] * [ Φ m ( i ) ( k ) O ( i ) ( k k m ) P ( i ) ( k ) ] | O ( i ) ( k ) | max ( m p | O ( i ) ( k k m ) | 2 + δ 2 ) ,
𝒞 p , 1 { { Ψ m ( k ) } m p : I ^ p ( r ) = m p | [ Ψ m ( k ) ] ( r ) | 2 , r }
𝒞 p , 2 { { Ψ m ( k ) } m p : Ψ m ( k ) = O ( k k m ) P ( k ) , m p , O ( k ) , P ( k ) , k } .
i p mod N img .
{ Φ m ( i ) ( k ) } m p = arg min Φ m ( k ) m p k | Φ m ( k ) Ψ m ( i ) ( k ) | 2 , s . t . { Φ m ( k ) } m p 𝒞 p , 1 .
{ ϕ m ( i ) ( r ) } m p = arg min ϕ m ( r ) m p r | ϕ m ( r ) ψ m ( i ) ( r ) | 2 , s . t . I ^ p ( r ) = m p | ϕ m ( r ) | 2 ,
{ ϕ m ( i ) ( r ) } m p = arg min ϕ m ( r ) m p | ϕ m ( r ) ψ m ( i ) ( r ) | 2 , s . t . I ^ p ( r ) = m p | ϕ m ( r ) | 2 ,
{ ϕ m ( i ) ( r ) , λ } m p = arg min ϕ m ( r ) , λ m p | ϕ m ( r ) ψ m ( i ) ( r ) | 2 + λ ( I ^ p ( r ) m p | ϕ m ( r ) | 2 ) .
ϕ m ( i ) ( r ) = I ^ p ( r ) m p | ψ m ( i ) ( r ) | 2 ψ m ( i ) ( r ) , m p .
Φ m ( i ) ( k ) = [ ϕ m ( i ) ( r ) ] 1 ( k ) ,
Ψ m ( i + 1 ) ( k ) = O ( i + 1 ) ( k k m ) P ( i + 1 ) ( k ) , m p
{ O ( i + 1 ) ( k ) , P ( i + 1 ) ( k ) } = arg min O ( k ) , P ( k ) m p k | O ( k k m ) P ( k ) Φ m ( i ) ( k ) | 2 .
{ O ( i + 1 ) ( k ) , P ( i + 1 ) ( k ) } = arg min O ( k ) , P ( k ) m p | O ( k k m ) P ( k ) Φ m ( i ) ( k ) | 2 .
f ( O ( k k m ) , P ( k ) ) O ( k k m ) = 2 m p [ O ( k k m ) P ( k ) Φ m ( i ) ( k ) ] [ P ( k ) ] *
f ( O ( k k m ) , P ( k ) ) P ( k ) = 2 m p [ O ( k k m ) P ( k ) Φ m ( i ) ( k ) ] [ O ( k k m ) ] *
O ( i + 1 ) ( k ) = m p [ P ( i + 1 ) ( k + k m ) ] * Φ m ( i ) ( k + k m ) m p | P ( i + 1 ) ( k + k m ) | 2
P ( i + 1 ) ( k ) = m p [ O ( i + 1 ) ( k k m ) ] * Φ m ( i ) ( k ) m p | O ( i + 1 ) ( k k m ) | 2 .
2 f ( O ( k k m ) , P ( k ) ) O ( k k m ) 2 = 2 m p | P ( k ) | 2
2 f ( O ( k k m ) , P ( k ) ) P ( k ) 2 = 2 m p | O ( k k m ) | 2 .
O ( i , + 1 ) ( k ) = O ( i , ) ( k ) + step-size [ 2 f ( O ( k k m ) , P ( k ) ) O ( k k m ) 2 ] 1 f ( O ( k k m ) , P ( k ) ) O ( k )
P ( i , + 1 ) ( k ) = P ( i , ) ( k ) + step-size [ 2 f ( O ( k k m ) , P ( k ) ) P ( k ) 2 ] 1 f ( O ( k k m ) , P ( k ) ) P ( k ) .
O ( i , + 1 ) ( k ) = O ( i , ) ( k ) + α ( i , ) ( k + k m ) m p | P ( i , ) ( k + k m ) | 2 + δ 1 × m p [ P ( i , ) ( k + k m ) ] * [ Φ m ( i ) ( k + k m ) O ( i , ) ( k ) P ( i , ) ( k + k m ) ]
P ( i , + 1 ) ( k ) = P ( i , ) ( k ) + β ( i , ) ( k ) m p | O ( i , ) ( k k m ) | 2 + δ 2 × m p [ O ( i , ) ( k k m ) ] * [ Φ m ( i ) ( k ) O ( i , ) ( k k m ) P ( i , ) ( k ) ] .
{ O ( i + 1 ) ( k ) , P ( i + 1 ) ( k ) } = { O ( i , L ) ( k ) , P ( i , L ) ( k ) } .

Metrics