## Characterization of spatially varying aberrations for wide field-of-view microscopy |

Optics Express, Vol. 21, Issue 13, pp. 15131-15143 (2013)

http://dx.doi.org/10.1364/OE.21.015131

Acrobat PDF (2543 KB)

### Abstract

We describe a simple and robust approach for characterizing the spatially varying pupil aberrations of microscopy systems. In our demonstration with a standard microscope, we derive the location-dependent pupil transfer functions by first capturing multiple intensity images at different defocus settings. Next, a generalized pattern search algorithm is applied to recover the complex pupil functions at ~350 different spatial locations over the entire field-of-view. Parameter fitting transforms these pupil functions into accurate 2D aberration maps. We further demonstrate how these aberration maps can be applied in a phase-retrieval based microscopy setup to compensate for spatially varying aberrations and to achieve diffraction-limited performance over the entire field-of-view. We believe that this easy-to-use spatially-varying pupil characterization method may facilitate new optical imaging strategies for a variety of wide field-of-view imaging platforms.

© 2013 OSA

## 1. Introduction

2. O. S. Cossairt, D. Miau, and S. K. Nayar, “Scaling law for computational imaging using spherical optics,” J. Opt. Soc. Am. A **28**(12), 2540–2553 (2011). [CrossRef] [PubMed]

2. O. S. Cossairt, D. Miau, and S. K. Nayar, “Scaling law for computational imaging using spherical optics,” J. Opt. Soc. Am. A **28**(12), 2540–2553 (2011). [CrossRef] [PubMed]

3. D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography,” Nature **486**(7403), 386–389 (2012). [CrossRef] [PubMed]

4. A. W. Lohmann, “Scaling laws for lens systems,” Appl. Opt. **28**(23), 4996–4998 (1989). [CrossRef] [PubMed]

17. M. J. Booth, “Adaptive optics in microscopy,” Philos Trans A Math Phys Eng Sci **365**(1861), 2829–2843 (2007). [CrossRef] [PubMed]

13. L. Seifert, J. Liesener, and H. J. Tiziani, “The adaptive Shack–Hartmann sensor,” Opt. Commun. **216**(4-6), 313–319 (2003). [CrossRef]

18. R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” JOSA A **9**(7), 1072–1085 (1992). [CrossRef]

24. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. **21**(5), 215829 (1982). [CrossRef]

18. R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” JOSA A **9**(7), 1072–1085 (1992). [CrossRef]

24. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. **21**(5), 215829 (1982). [CrossRef]

25. L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express **18**(12), 12552–12561 (2010). [CrossRef] [PubMed]

28. S. S. Kou, L. Waller, G. Barbastathis, and C. J. Sheppard, “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. **35**(3), 447–449 (2010). [CrossRef] [PubMed]

29. L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. **199**(1-4), 65–75 (2001). [CrossRef]

30. Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Reconstruction of in-line digital holograms from two intensity measurements,” Opt. Lett. **29**(15), 1787–1789 (2004). [CrossRef] [PubMed]

31. B. Das and C. S. Yelleswarapu, “Dual plane in-line digital holographic microscopy,” Opt. Lett. **35**(20), 3426–3428 (2010). [CrossRef] [PubMed]

## 2. Overview of phase retrieval and spatially varying pupil aberrations

### 2.1 Phase retrieval and defocus diversity

29. L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. **199**(1-4), 65–75 (2001). [CrossRef]

37. J. Fienup and C. Wackerman, “Phase-retrieval stagnation problems and solutions,” JOSA A **3**(11), 1897–1907 (1986). [CrossRef]

*I*(s) (s = −2, −1, 0, 1, 2 in Fig. 1(a)) at different defocus planes, we follow the multi-plane iterative algorithm outlined in Fig. 1(b) [29

29. L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. **199**(1-4), 65–75 (2001). [CrossRef]

*I*(s), while the phase is kept unchanged. Such a propagate-and-replace process is repeated until the complex solution converges (see Section 4 for implementation details).

### 2.2 Spatially varying pupil aberrations

*x, y*) and (

*k*,

_{x}*k*), respectively, with

_{y}*k*and

_{x}*k*as the wave number in the

_{y}*x*and

*y*directions. Due to such a Fourier relationship, aberrations of an imaging platform are often characterized at the pupil plane for simplicity [42]. Different types of aberrations can be quantified as different Zernike modes at the pupil plane. For example, defocus aberration can be modeled as a phase factor

*p*, where

_{5Z20(kx,ky)}*p*denotes the amount of defocus aberration (subscript ‘5’ indicates the fifth Zernike mode).

_{5}*W(*k

_{x,}k

_{y}), whose phase factor is a summation of different Zernike modes with different aberration coefficients

*p*

_{m}(

*p*

_{m}denotes the amount of

*m*

^{th}Zernike mode; refer to Eq. (1) in Section 4). If the imaging platform is shift-invariant, each aberration coefficient

*p*

_{m}is constant over the entire imaging FOV and the generalized pupil function

*W(*k

_{x}, k

_{y}) is independent of spatial coordinates

*x*and

*y*. However, as noted above, recent extreme-FOV computational imaging platforms push beyond the limits of conventional lens design and thus invalidate this shift-invariant assumption. Aberration coefficients

*p*

_{m}s are 2D functions of x and y in this case, and thus, the generalized pupil function can be expressed as a function of both k

_{x}, k

_{y}and x, y, i.e.

*W(*k

_{x}, k

_{y}, x, y). Our goal here is to characterize the aberration parameters

*p*

_{m}(m = 1, 2, …) as a function of spatial coordinates x and y. Based on

*p*

_{m}(x, y), we can derive the generalized pupil function

*W(*k

_{x}, k

_{y}, x, y) at any given spatial location (Section 5) and accurately perform post-detection image deconvolution (Section 6).

## 3. Experimental setup and sample preparation

32. Y. Kawano, C. Higgins, Y. Yamamoto, J. Nyhus, A. Bernard, H.-W. Dong, H. J. Karten, and T. Schilling, “Darkfield adapter for whole slide imaging: Adapting a darkfield internal reflection illumination system to extend wsi applications,” PLoS ONE **8**(3), e58344 (2013). [CrossRef] [PubMed]

4. A. W. Lohmann, “Scaling laws for lens systems,” Appl. Opt. **28**(23), 4996–4998 (1989). [CrossRef] [PubMed]

## 4. Off-axis pupil function recovery

*Phase retrieval*. Following the general procedure outlined in Section 2, we displace the microscope stage from the focal plane at

*δ*= 50 µm increments in either defocus direction, capturing a total of 17 images of the microsphere calibration target

*I*(

*s*), where

*s*= (−8,…0,…8). The maximum defocus distance with such a scheme is 400 µm in either direction. For each image, the microsphere target is illuminated with a quasi-monochromatic collimated plane wave (632 nm).

^{2}-pixel cropped image set

*I*

_{c}(

*s*) that contains one microsphere at the center FOV (see Fig. 2, left). We recover the complex profile of this centered microsphere using the multi-plane phase retrieval algorithm [29

**199**(1-4), 65–75 (2001). [CrossRef]

*I*(0). Second, this complex field estimate is Fourier transformed and multiplied by a quadratic phase factor exp(

_{c}*ik*), describing defocus of the field by axial distance

_{z}z*z = s∙δ*. To begin, we set

*s*= 1, corresponding to

*z =*+ 50 µm of defocus. Third, after digitally defocusing, we again replace the amplitude values of the complex field estimate with the square root of the intensity data from recorded image,

*I*(

_{c}*s*). Beginning with

*s*= 1, we first use the intensity values

*I*(1) captured at z = +50 µm for amplitude value replacement, while the estimate’s phase values remain unchanged. This digital propagate-and-replace process is repeated for all values of

_{c}*s*(all 17 cropped intensity measurements from the captured focal stack). Finally, we iterate the entire phase retrieval loop approximately 10 times. The final recovered complex image, denoted as

*Off-axis pupil function estimation*. Next, we select a microsphere at a position (

*x*,

_{0}*y*) off the optical axis and generate a new 64

_{0}^{2}-pixel cropped image set

*I*(

_{d}*s*) from our initial measurements, centered at (

*x*,

_{0}*y*) (see Fig. 2). We also initialize an estimate of the unknown location-dependent pupil function for this position,

_{0}*W*(

*k*,

_{x}*k*,

_{y}*x*,

_{0}*y*). For simplicity, we approximate the unknown pupil function

_{0}*W*(

*k*,

_{x}*k*,

_{y}*x*,

_{0}*y*) with 8 Zernike modes,

_{0}*x*-tilt,

*y*-tilt,

*x*-astigmatism,

*y*-astigmatism, defocus, x-coma, y-coma and spherical aberration, respectively [1]. The point-spread function at the selected off-axis microsphere location (

*x*,

_{0}*y*) may be uniquely influenced by each mode above. We denote the coefficient for each Zernike mode with

_{0}*p*(

_{m}*x*,

_{0}*y*), where the subscript ‘

_{0}*m*’ stands for the mode’s polynomial expansion order (in our case,

*m*= 1, 2…8). With this notation, our unknown pupil function estimate

*W*(

*k*,

_{x}*k*,

_{y}*x*,

_{0}*y*) can be expressed as,

_{0}*p*(

_{m}*x*,

_{0}*y*) is a space-dependent function evaluated at (

_{0}*x*=

*x*,

_{0}*y*=

*y*), allowing the pupil function

_{0}*W*to model spatially varying aberrations. This pupil function estimate is then used along with the “ground truth” complex field of the centered microsphere found in step 2 to generate a set of aberrated intensity images,

*I*(

_{a}*s*), as follows:where F is the Fourier transform operator and the term

*e*represents defocus of the ground truth microsphere field to plane

^{ikz δs }*s*. We then adjust the values of the 8 unknown Zernike coefficients

*p*comprising the pupil function

_{m}*W*to minimize the difference between this modeled set of aberrated intensity images

*I*(

_{a}*s*) and the actual set intensity measurements of the selected off-axis microsphere,

*I*(s). The corresponding pupil function described by 8 Zernike coefficients is recovered when the mean-squared error difference is minimized. We apply a Generalized Pattern Search (GPS) algorithm [43

_{d}43. C. Audet and J. E. Dennis Jr., “Analysis of generalized pattern searches,” SIAM J. Optim. **13**(3), 889–903 (2002). [CrossRef]

## 5. Spatially varying aberration characterization over the entire FOV

44. X. Yang, H. Li, and X. Zhou, “Nuclei segmentation using marker-controlled watershed, tracking using mean-shift, and kalman filter in time-lapse microscopy,” Circuits and Systems I: Regular Papers, IEEE Transactions on **53**, 2405–2414 (2006). [CrossRef]

*W*recovered following Eq. (3) at position (

*x*,

_{1}*y*), the center of the black square in Fig. 3(a). Figures 3(c1)-(c5) are 5 of the 17 intensity measurements of the microsphere at position (

_{1}*x*,

_{1}*y*) under different amounts of defocus:

_{1}*I*(s = 0),

_{d}*I*(s = ± 3), and

_{d}*I*(s = ± 6). Figures 3(d1)-(d5) display the corresponding aberrated image estimates

_{d}*I*(s) generated by the recovered pupil function in Fig. 3(b). Following the convex form of Eq. (3), the applied GPS algorithm successfully minimizes the mean-squared error difference between the measurements

_{a}*I*(s) and the estimates

_{d}*I*(s).

_{a}*p*(

_{m}*x*,

*y*), allowing us to accurately recover the pupil function at any location across the image plane

**(**curved surfaces in Fig. 4). The order of each polynomial function can be predicted via aberration theory for a conventional imaging platform [1]. The aberrations of increasingly unconventional optical designs in computational imaging systems may not follow such predictable trends, which we may account for with alternative fitting models and/or recovering coefficients at more than 350 unique spatial locations.

## 6. Image deconvolution using the recovered aberration parameters

*Full-FOV phase retrieval.*We use the multi-plane phase retrieval algorithm described in Section 2 to recover the amplitude and phase of a sample over the microscope’s entire FOV. This complex image contains the objective lens’s spatially varying aberrations.

*Segment decomposition and shift-invariant image deconvolution.*We then divide the full-FOV complex image into smaller 128 x 128 pixel image segments, denoted by

*I*

_{seg}(

*n*) (

*n*= 1, 2,… 1600 for our employed detector). Aberrations within each small segment are treated as shift-invariant, a common strategy for wide FOV imaging processing [45]. The pupil function

*W*(

*k*,

_{x}*k*,

_{y}*x*(

_{c}*n*),

*y*(

_{c}*n*)) is then calculated for each small segment following Eq. (1), where (

*x*(

_{c}*n*),

*y*(

_{c}*n*)) represents the central spatial location of the

*n*

^{th}segment. We then perform image deconvolution to recover the corrected image segment

*I*

_{cor}(

*n*) as follows:where

*I*directly captured using the aberrated objective lens, while Fig. 6(b2)-(d2) are the corresponding processed images

_{seg}*I*using Eq. (4). From Fig. 6(b2)-(d2), Group 7, element 1 (line width of 3.9 µm) of the USAF target can be resolved, in a good agreement with the Abbe diffraction limit of 3.94 µm of our 0.08 NA objective lens. This simple experiment indicates our aberration correction scheme can correct this particular objective’s aberration blur to yield diffraction-limited performance across its entire image FOV.

_{cor}*I*

_{cor}(

*n*) to form a correct full FOV image. Figure 7 and Fig. 8 show the results of a second experiment, where the entire FOV of images of two samples are corrected. An alpha blending algorithm [46] is used to remove edge artifacts at the segment boundary. Specifically, we cut away 2 pixels at the edge of each segment and use another 5 pixels to overlap with the adjacent portions. This blending comes at a small computational cost of redundantly processing the regions of overlap twice.

## 7. Conclusion

## Acknowledgments

## References and links

1. | H. Gross, W. Singer, M. Totzeck, F. Blechinger, and B. Achtner, |

2. | O. S. Cossairt, D. Miau, and S. K. Nayar, “Scaling law for computational imaging using spherical optics,” J. Opt. Soc. Am. A |

3. | D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography,” Nature |

4. | A. W. Lohmann, “Scaling laws for lens systems,” Appl. Opt. |

5. | F. Berny and S. Slansky, “Wavefront determination resulting from Foucault test as applied to the human eye and visual instruments,” in |

6. | S. Yokozeki and K. Ohnishi, “Spherical aberration measurement with shearing interferometer using Fourier imaging and moiré method,” Appl. Opt. |

7. | M. Ma, X. Wang, and F. Wang, “Aberration measurement of projection optics in lithographic tools based on two-beam interference theory,” Appl. Opt. |

8. | M. Takeda and S. Kobayashi, “Lateral aberration measurements with a digital Talbot interferometer,” Appl. Opt. |

9. | J. Sung, M. Pitchumani, and E. G. Johnson, “Aberration measurement of photolithographic lenses by use of hybrid diffractive photomasks,” Appl. Opt. |

10. | Q. Gong and S. S. Hsu, “Aberration measurement using axial intensity,” Opt. Eng. |

11. | L. N. Thibos, “Principles of hartmann-shack aberrometry,” in |

12. | J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three - dimensional microscope point spread function using a Shack - Hartmann wavefront sensor,” J. Microsc. |

13. | L. Seifert, J. Liesener, and H. J. Tiziani, “The adaptive Shack–Hartmann sensor,” Opt. Commun. |

14. | R. G. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. |

15. | D. Debarre, M. J. Booth, and T. Wilson, “Image based adaptive optics through optimisation of low spatial frequencies,” Opt. Express |

16. | T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics |

17. | M. J. Booth, “Adaptive optics in microscopy,” Philos Trans A Math Phys Eng Sci |

18. | R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” JOSA A |

19. | B. M. Hanser, M. G. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase retrieval for high-numerical-aperture optical systems,” Opt. Lett. |

20. | B. M. Hanser, M. G. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase - retrieved pupil functions in wide - field fluorescence microscopy,” J. Microsc. |

21. | J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. |

22. | J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. |

23. | G. R. Brady and J. R. Fienup, “Nonlinear optimization algorithm for retrieving the full complex pupil function,” Opt. Express |

24. | R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. |

25. | L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express |

26. | N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. |

27. | T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. |

28. | S. S. Kou, L. Waller, G. Barbastathis, and C. J. Sheppard, “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. |

29. | L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. |

30. | Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Reconstruction of in-line digital holograms from two intensity measurements,” Opt. Lett. |

31. | B. Das and C. S. Yelleswarapu, “Dual plane in-line digital holographic microscopy,” Opt. Lett. |

32. | Y. Kawano, C. Higgins, Y. Yamamoto, J. Nyhus, A. Bernard, H.-W. Dong, H. J. Karten, and T. Schilling, “Darkfield adapter for whole slide imaging: Adapting a darkfield internal reflection illumination system to extend wsi applications,” PLoS ONE |

33. | H. Nomura, K. Tawarayama, and T. Kohno, “Aberration measurement from specific photolithographic images: a different approach,” Appl. Opt. |

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

35. | R. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) |

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

37. | J. Fienup and C. Wackerman, “Phase-retrieval stagnation problems and solutions,” JOSA A |

38. | J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” JOSA A |

39. | M. R. Bolcar and J. R. Fienup, “Sub-aperture piston phase diversity for segmented and multi-aperture systems,” Appl. Opt. |

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

41. | B. H. Dean and C. W. Bowers, “Diversity selection for phase-diverse phase retrieval,” J. Opt. Soc. Am. A |

42. | J. W. Goodman, |

43. | C. Audet and J. E. Dennis Jr., “Analysis of generalized pattern searches,” SIAM J. Optim. |

44. | X. Yang, H. Li, and X. Zhou, “Nuclei segmentation using marker-controlled watershed, tracking using mean-shift, and kalman filter in time-lapse microscopy,” Circuits and Systems I: Regular Papers, IEEE Transactions on |

45. | B. K. Gunturk and X. Li, |

46. | T. McReynolds and D. Blythe, |

**OCIS Codes**

(100.0100) Image processing : Image processing

(170.0180) Medical optics and biotechnology : Microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: May 15, 2013

Revised Manuscript: May 31, 2013

Manuscript Accepted: June 10, 2013

Published: June 17, 2013

**Virtual Issues**

Vol. 8, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Guoan Zheng, Xiaoze Ou, Roarke Horstmeyer, and Changhuei Yang, "Characterization of spatially varying aberrations for wide field-of-view microscopy," Opt. Express **21**, 15131-15143 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-13-15131

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### References

- H. Gross, W. Singer, M. Totzeck, F. Blechinger, and B. Achtner, Handbook of Optical Systems (Wiley Online Library, 2005), Vol. 2.
- O. S. Cossairt, D. Miau, and S. K. Nayar, “Scaling law for computational imaging using spherical optics,” J. Opt. Soc. Am. A28(12), 2540–2553 (2011). [CrossRef] [PubMed]
- D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Multiscale gigapixel photography,” Nature486(7403), 386–389 (2012). [CrossRef] [PubMed]
- A. W. Lohmann, “Scaling laws for lens systems,” Appl. Opt.28(23), 4996–4998 (1989). [CrossRef] [PubMed]
- F. Berny and S. Slansky, “Wavefront determination resulting from Foucault test as applied to the human eye and visual instruments,” in Optical Instruments and Techniques (Oriel, 1969), pp. 375–386.
- S. Yokozeki and K. Ohnishi, “Spherical aberration measurement with shearing interferometer using Fourier imaging and moiré method,” Appl. Opt.14(3), 623–627 (1975). [CrossRef] [PubMed]
- M. Ma, X. Wang, and F. Wang, “Aberration measurement of projection optics in lithographic tools based on two-beam interference theory,” Appl. Opt.45(32), 8200–8208 (2006). [CrossRef] [PubMed]
- M. Takeda and S. Kobayashi, “Lateral aberration measurements with a digital Talbot interferometer,” Appl. Opt.23(11), 1760–1764 (1984). [CrossRef] [PubMed]
- J. Sung, M. Pitchumani, and E. G. Johnson, “Aberration measurement of photolithographic lenses by use of hybrid diffractive photomasks,” Appl. Opt.42(11), 1987–1995 (2003). [CrossRef] [PubMed]
- Q. Gong and S. S. Hsu, “Aberration measurement using axial intensity,” Opt. Eng.33(4), 1176–1186 (1994). [CrossRef]
- L. N. Thibos, “Principles of hartmann-shack aberrometry,” in Vision Science and its Applications, (Optical Society of America, 2000)
- J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three - dimensional microscope point spread function using a Shack - Hartmann wavefront sensor,” J. Microsc.205(1), 61–75 (2002). [CrossRef] [PubMed]
- L. Seifert, J. Liesener, and H. J. Tiziani, “The adaptive Shack–Hartmann sensor,” Opt. Commun.216(4-6), 313–319 (2003). [CrossRef]
- R. G. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt.31(32), 6902–6908 (1992). [CrossRef] [PubMed]
- D. Debarre, M. J. Booth, and T. Wilson, “Image based adaptive optics through optimisation of low spatial frequencies,” Opt. Express15(13), 8176–8190 (2007). [CrossRef] [PubMed]
- T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics4(6), 388–394 (2010). [CrossRef]
- M. J. Booth, “Adaptive optics in microscopy,” Philos Trans A Math Phys Eng Sci365(1861), 2829–2843 (2007). [CrossRef] [PubMed]
- R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” JOSA A9(7), 1072–1085 (1992). [CrossRef]
- B. M. Hanser, M. G. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase retrieval for high-numerical-aperture optical systems,” Opt. Lett.28(10), 801–803 (2003). [CrossRef] [PubMed]
- B. M. Hanser, M. G. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase - retrieved pupil functions in wide - field fluorescence microscopy,” J. Microsc.216(1), 32–48 (2004). [CrossRef] [PubMed]
- J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt.32(10), 1737–1746 (1993). [CrossRef] [PubMed]
- J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt.32(10), 1747–1767 (1993). [CrossRef] [PubMed]
- G. R. Brady and J. R. Fienup, “Nonlinear optimization algorithm for retrieving the full complex pupil function,” Opt. Express14(2), 474–486 (2006). [CrossRef] [PubMed]
- R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng.21(5), 215829 (1982). [CrossRef]
- L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express18(12), 12552–12561 (2010). [CrossRef] [PubMed]
- N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49(1), 6–10 (1984). [CrossRef]
- T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun.133(1-6), 339–346 (1997). [CrossRef]
- S. S. Kou, L. Waller, G. Barbastathis, and C. J. Sheppard, “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett.35(3), 447–449 (2010). [CrossRef] [PubMed]
- L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun.199(1-4), 65–75 (2001). [CrossRef]
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