## Embedded pupil function recovery for Fourier ptychographic microscopy |

Optics Express, Vol. 22, Issue 5, pp. 4960-4972 (2014)

http://dx.doi.org/10.1364/OE.22.004960

Acrobat PDF (2954 KB)

### Abstract

We develop and test a pupil function determination algorithm, termed embedded pupil function recovery (EPRY), which can be incorporated into the Fourier ptychographic microscopy (FPM) algorithm and recover both the Fourier spectrum of sample and the pupil function of imaging system simultaneously. This EPRY-FPM algorithm eliminates the requirement of the previous FPM algorithm for a priori knowledge of the aberration in the imaging system to reconstruct a high quality image. We experimentally demonstrate the effectiveness of this algorithm by reconstructing high resolution, large field-of-view images of biological samples. We also illustrate that the pupil function we retrieve can be used to study the spatially varying aberration of a large field-of-view imaging system. We believe that this algorithm adds more flexibility to FPM and can be a powerful tool for the characterization of an imaging system’s aberration.

© 2014 Optical Society of America

## 1. Introduction

1. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics **7**(9), 739–745 (2013). [CrossRef]

2. A. Lohmann, R. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A **13**(3), 470–473 (1996). [CrossRef]

1. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics **7**(9), 739–745 (2013). [CrossRef]

*et al.*[1

1. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics **7**(9), 739–745 (2013). [CrossRef]

3. G. Zheng, X. Ou, R. Horstmeyer, and C. Yang, “Characterization of spatially varying aberrations for wide field-of-view microscopy,” Opt. Express **21**(13), 15131–15143 (2013). [CrossRef] [PubMed]

^{2}) microscope with a final SBP of ~1 gigapixel [1

**7**(9), 739–745 (2013). [CrossRef]

3. G. Zheng, X. Ou, R. Horstmeyer, and C. Yang, “Characterization of spatially varying aberrations for wide field-of-view microscopy,” Opt. Express **21**(13), 15131–15143 (2013). [CrossRef] [PubMed]

6. Z. Bian, S. Dong, and G. Zheng, “Adaptive system correction for robust Fourier ptychographic imaging,” Opt. Express **21**(26), 32400–32410 (2013). [CrossRef] [PubMed]

## 2. Reconstruction algorithm

**7**(9), 739–745 (2013). [CrossRef]

7. X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. **38**(22), 4845–4848 (2013). [CrossRef] [PubMed]

**7**(9), 739–745 (2013). [CrossRef]

3. G. Zheng, X. Ou, R. Horstmeyer, and C. Yang, “Characterization of spatially varying aberrations for wide field-of-view microscopy,” Opt. Express **21**(13), 15131–15143 (2013). [CrossRef] [PubMed]

8. W. Hoppe, “Diffraction in inhomogeneous primary wave fields. 1. Principle of phase determination from electron diffraction interference,” Acta Crystallogr. A **25**, 495–501 (1969). [CrossRef]

10. J. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron Phys. **150**, 87–184 (2008). [CrossRef]

11. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**(15), 2758–2769 (1982). [CrossRef] [PubMed]

13. F. Hüe, J. Rodenburg, A. Maiden, F. Sweeney, and P. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B **82**(12), 121415 (2010). [CrossRef]

14. H. M. L. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy **103**(2), 153–164 (2005). [CrossRef] [PubMed]

15. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express **16**(10), 7264–7278 (2008). [CrossRef] [PubMed]

*et al.*[16

16. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science **321**(5887), 379–382 (2008). [CrossRef] [PubMed]

17. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy **109**(4), 338–343 (2009). [CrossRef] [PubMed]

18. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy **109**(10), 1256–1262 (2009). [CrossRef] [PubMed]

18. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy **109**(10), 1256–1262 (2009). [CrossRef] [PubMed]

20. J. Marrison, L. Räty, P. Marriott, and P. O’Toole, “Ptychography--a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. **3**, 2369 (2013). [CrossRef] [PubMed]

*n*th loop, with the knowledge of reconstructed

18. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy **109**(10), 1256–1262 (2009). [CrossRef] [PubMed]

**7**(9), 739–745 (2013). [CrossRef]

*n*raw pixels. For each loop, the exit wave simulation, inverse Fourier transform, intensity constraint and Fourier transform process has computational cost of

*n*,

*n*log(

*n*),

*n*and

*n*log(

*n*) respectively. The sample spectrum update, pupil function update and pupil function constraint has computational cost of 3

*n*, 3

*n*and

*n*respectively. So the computational cost of the original FPM algorithm is 5

*n*+ 2

*n*log(

*n*) for each loop, and the computational cost of the EPRY-FPM algorithm is 9

*n*+ 2

*n*log(

*n*). Generally, the raw pixel count

*n*is in the order of a million, so the incremental computational cost of 4

*n*is ignorable compare to 2

*n*log(

*n*).

## 3. Simulation results

**7**(9), 739–745 (2013). [CrossRef]

**7**(9), 739–745 (2013). [CrossRef]

**7**(9), 739–745 (2013). [CrossRef]

21. J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. **36**(32), 8352–8357 (1997). [CrossRef] [PubMed]

*m*iterations.

## 4. Experimental results

**7**(9), 739–745 (2013). [CrossRef]

23. V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. **33**(34), 8121–8124 (1994). [CrossRef] [PubMed]

## 5. EPRY-FPM for large FOV, high resolution image reconstruction

**21**(13), 15131–15143 (2013). [CrossRef] [PubMed]

24. G. Zheng, X. Ou, and C. Yang, “0.5 gigapixel microscopy using a flatbed scanner,” Biomed. Opt. Express **5**(1), 1–8 (2014). [CrossRef] [PubMed]

**7**(9), 739–745 (2013). [CrossRef]

**21**(13), 15131–15143 (2013). [CrossRef] [PubMed]

25. M. Watanabe and S. K. Nayar, “Telecentric optics for focus analysis,” IEEE Trans. Pattern Anal. Mach. Intell. **19**(12), 1360–1365 (1997). [CrossRef]

## 5. Comparison with original phase retrieval algorithm

**21**(13), 15131–15143 (2013). [CrossRef] [PubMed]

**21**(13), 15131–15143 (2013). [CrossRef] [PubMed]

^{N}times that of the original 3 aberration optimization. Second, the image sets that are used for aberration characterization only contain low resolution images captured by the NA = 0.08 objective. The high order aberration information is easily overwhelmed by the noise of the imaging system, resulting in an imprecise measurement of the high order aberration. Third, the characterized high order aberration information can be volatile. This is because such aberration is highly sensitive to mechanical or optical system drifts. In conclusion, method 2 is impractical to correct higher order aberration.

## 6. Conclusion

## Acknowledgments

## References and links

1. | G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics |

2. | A. Lohmann, R. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A |

3. | G. Zheng, X. Ou, R. Horstmeyer, and C. Yang, “Characterization of spatially varying aberrations for wide field-of-view microscopy,” Opt. Express |

4. | H. Nomura and T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed patterns,” Appl. Opt. |

5. | J. Wesner, J. Heil, and Th. Sure, “Reconstructing the pupil function of microscope objectives from the intensity PSF,” in Current Developments in Lens Design and Optical Engineering III, R. E. Fischer, W. J. Smith, and R. B. Johnson, eds., Proc. SPIE |

6. | Z. Bian, S. Dong, and G. Zheng, “Adaptive system correction for robust Fourier ptychographic imaging,” Opt. Express |

7. | X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. |

8. | W. Hoppe, “Diffraction in inhomogeneous primary wave fields. 1. Principle of phase determination from electron diffraction interference,” Acta Crystallogr. A |

9. | J. Rodenburg and R. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Philos. Trans. R. Soc. Lond. A |

10. | J. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron Phys. |

11. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

12. | J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. |

13. | F. Hüe, J. Rodenburg, A. Maiden, F. Sweeney, and P. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B |

14. | H. M. L. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy |

15. | M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express |

16. | P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science |

17. | P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy |

18. | A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy |

19. | A. Maiden, J. Rodenburg, and M. Humphry, “A new method of high resolution, quantitative phase scanning microscopy,” in: M.T. Postek, D.E. Newbury, S.F. Platek, D.C. Joy (Eds.), SPIE Proceedings of Scanning Microscopy, |

20. | J. Marrison, L. Räty, P. Marriott, and P. O’Toole, “Ptychography--a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. |

21. | J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. |

22. | J. W. Goodman, |

23. | V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. |

24. | G. Zheng, X. Ou, and C. Yang, “0.5 gigapixel microscopy using a flatbed scanner,” Biomed. Opt. Express |

25. | M. Watanabe and S. K. Nayar, “Telecentric optics for focus analysis,” IEEE Trans. Pattern Anal. Mach. Intell. |

**OCIS Codes**

(100.0100) Image processing : Image processing

(180.0180) Microscopy : Microscopy

**ToC Category:**

Image Processing

**History**

Original Manuscript: December 26, 2013

Revised Manuscript: February 12, 2014

Manuscript Accepted: February 17, 2014

Published: February 24, 2014

**Virtual Issues**

Vol. 9, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Xiaoze Ou, Guoan Zheng, and Changhuei Yang, "Embedded pupil function recovery for Fourier ptychographic microscopy," Opt. Express **22**, 4960-4972 (2014)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-22-5-4960

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### References

- G. Zheng, R. Horstmeyer, C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013). [CrossRef]
- A. Lohmann, R. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13(3), 470–473 (1996). [CrossRef]
- G. Zheng, X. Ou, R. Horstmeyer, C. Yang, “Characterization of spatially varying aberrations for wide field-of-view microscopy,” Opt. Express 21(13), 15131–15143 (2013). [CrossRef] [PubMed]
- H. Nomura, T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed patterns,” Appl. Opt. 38(13), 2800–2807 (1999). [CrossRef] [PubMed]
- J. Wesner, J. Heil, and Th. Sure, “Reconstructing the pupil function of microscope objectives from the intensity PSF,” in Current Developments in Lens Design and Optical Engineering III, R. E. Fischer, W. J. Smith, and R. B. Johnson, eds., Proc. SPIE 4767, 32–43 (2002).
- Z. Bian, S. Dong, G. Zheng, “Adaptive system correction for robust Fourier ptychographic imaging,” Opt. Express 21(26), 32400–32410 (2013). [CrossRef] [PubMed]
- X. Ou, R. Horstmeyer, C. Yang, G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(22), 4845–4848 (2013). [CrossRef] [PubMed]
- W. Hoppe, “Diffraction in inhomogeneous primary wave fields. 1. Principle of phase determination from electron diffraction interference,” Acta Crystallogr. A 25, 495–501 (1969). [CrossRef]
- J. Rodenburg, R. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Philos. Trans. R. Soc. Lond. A 339(1655), 521–553 (1992). [CrossRef]
- J. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron Phys. 150, 87–184 (2008). [CrossRef]
- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982). [CrossRef] [PubMed]
- J. M. Rodenburg, H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795–4797 (2004). [CrossRef]
- F. Hüe, J. Rodenburg, A. Maiden, F. Sweeney, P. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82(12), 121415 (2010). [CrossRef]
- H. M. L. Faulkner, J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy 103(2), 153–164 (2005). [CrossRef] [PubMed]
- M. Guizar-Sicairos, J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16(10), 7264–7278 (2008). [CrossRef] [PubMed]
- P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008). [CrossRef] [PubMed]
- P. Thibault, M. Dierolf, O. Bunk, A. Menzel, F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009). [CrossRef] [PubMed]
- A. M. Maiden, J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009). [CrossRef] [PubMed]
- A. Maiden, J. Rodenburg, and M. Humphry, “A new method of high resolution, quantitative phase scanning microscopy,” in: M.T. Postek, D.E. Newbury, S.F. Platek, D.C. Joy (Eds.), SPIE Proceedings of Scanning Microscopy, 7729, 2010. [CrossRef]
- J. Marrison, L. Räty, P. Marriott, P. O’Toole, “Ptychography--a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. 3, 2369 (2013). [CrossRef] [PubMed]
- J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. 36(32), 8352–8357 (1997). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).
- V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. 33(34), 8121–8124 (1994). [CrossRef] [PubMed]
- G. Zheng, X. Ou, C. Yang, “0.5 gigapixel microscopy using a flatbed scanner,” Biomed. Opt. Express 5(1), 1–8 (2014). [CrossRef] [PubMed]
- M. Watanabe, S. K. Nayar, “Telecentric optics for focus analysis,” IEEE Trans. Pattern Anal. Mach. Intell. 19(12), 1360–1365 (1997). [CrossRef]

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