## A three-dimensional point spread function for phase retrieval and deconvolution |

Optics Express, Vol. 20, Issue 14, pp. 15392-15405 (2012)

http://dx.doi.org/10.1364/OE.20.015392

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### Abstract

We present a formulation of optical point spread function based on a scaled three-dimensional Fourier transform expression of focal field distribution and the expansion of generalized aperture function. It provides an equivalent but more flexible representation compared with the analytic expression of the extended Nijboer-Zernike approach. A phase diversity algorithm combined with an appropriate regularization strategy is derived and analyzed to demonstrate the effectiveness of the presented formulation for phase retrieval and deconvolution. Experimental results validate the performance of presented algorithm.

© 2012 OSA

## 1. Introduction

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

6. G. Chenegros, L. M. Mugnier, F. Lacombe, and M. Glanc, “3D phase diversity: a myopic deconvolution method for short-exposure images: application to retinal imaging,” J. Opt. Soc. Am. A **24**(5), 1349–1357 (2007). [CrossRef] [PubMed]

7. B. M. Hanser, M. G. L. 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]

8. A. J. E. M. Janssen, “Extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A **19**(5), 849–857 (2002). [CrossRef] [PubMed]

12. C. W. McCutchen, “Generalized Aperture and the Three-Dimensional Diffraction Image,” J. Opt. Soc. Am. **54**(2), 240–242 (1964). [CrossRef]

13. E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. **39**(4), 205–210 (1981). [CrossRef]

14. J. Lin, X.-C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett. **36**(8), 1341–1343 (2011). [CrossRef] [PubMed]

## 2. The three-dimensional PSF

### 2.1 Three-dimensional focal field distribution

*x,y,z*). A circular aperture

*W*lies in the (

*u,v*) plane that is parallel to the (

*x,y*) plane at a distance

*d*from the focus

*F*which is defined as the origin of the coordinates. When a convergent spherical wave passes through the aperture and propagates to the focal region, the complex amplitude at

*P*is given by the Huygens-Fresnel principle [15]where

*λ*is the wavelength of the monochromatic light and

*k*is the wavenumber. By using a paraxial approximation with the cos

*θ*=

*d*/

*f*, which is equivalent to the Debye approximation [10,12

12. C. W. McCutchen, “Generalized Aperture and the Three-Dimensional Diffraction Image,” J. Opt. Soc. Am. **54**(2), 240–242 (1964). [CrossRef]

**(**

*A**u,v,w*) is the three-dimensional aperture function and is nonzero only on the spherical cap

*W*, which can be defined bywhere

*δ*(●) is the Kronecker delta function. Like the two-dimensional case, we can define the generalized aperture function as

*λ*/

*π*(NA)

^{2}and lateral unit

*λ*/2

*π*(NA) respectively as

18. C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A **20**(11), 2156–2162 (2003). [CrossRef] [PubMed]

14. J. Lin, X.-C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett. **36**(8), 1341–1343 (2011). [CrossRef] [PubMed]

**(**

*g**x,y,z*) in the following derivation. With the correcting coordinate transformation, the Debye approximation expressed as a scaled three-dimensional Fourier transform can be effectively extended to the cases of low NA and small FN.

### 2.2 Formulation of three-dimensional PSF

*F*^{3}[●] denotes the three-dimensional Fourier transform. Following the treatment of the ENZ approach [10], we can expand the generalized aperture function

**(**

*P**u,v,w*) with the Zernike polynomialswhere

*β*are the complex Zernike coefficients, and

**(**

*Z'**u,v,w*) are the three-dimensional Zernike polynomials on the spherical cap, which can be defined aswhere

**(**

*Z**u,v*) are the ordinary two-dimensional Zernike polynomials. With this expansion, we can express the focal field distribution aswhere

**(**

*V'**x,y,z*) are the three-dimensional basis functions for the expansion of focal field distribution, which are defined by

**(**

*h**r,φ,z*) is the normalizing factor, and the two-dimensional basis functions

**(**

*V**r,z*) can be approximated analytically, which are defined aswhere

**(●) are the radial parts of the Zernike polynomials and**

*R***(●) are the Bessel functions of the first kind. It is obvious that the two representations are equivalent, and the relationship between the two sets of basis functions can be expressed explicitly**

*J*19. S. M. Jefferies, M. Lloyd-Hart, E. K. Hege, and J. Georges, “Sensing wave-front amplitude and phase with phase diversity,” Appl. Opt. **41**(11), 2095–2102 (2002). [CrossRef] [PubMed]

## 3. The phase diversity algorithm

### 3.1 Description of the algorithm

**(**

*I**x,y,z*) is the image,

**(**

*N**x,y,z*) is the noise,

**(**

*O**x,y,z*) is the object, and ⊗ denotes the three-dimensional convolution operator. In this paper, we will use the regularized least-square approach for the optical inverse problem of phase diversity like in [20

20. C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Proc. SPIE **3353**, 994–1005 (1998). [CrossRef]

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

6. G. Chenegros, L. M. Mugnier, F. Lacombe, and M. Glanc, “3D phase diversity: a myopic deconvolution method for short-exposure images: application to retinal imaging,” J. Opt. Soc. Am. A **24**(5), 1349–1357 (2007). [CrossRef] [PubMed]

*P**(*

_{0}*u,v,w*)| is the initial estimation of aperture amplitude according to the prior knowledge, and

*γ*and

*μ*are the object and amplitude regularization parameters, respectively. This is a general model for three-dimensional phase diversity, additional constraints can also be incorporated according to the prior knowledge such as the Z support constraint [6

6. G. Chenegros, L. M. Mugnier, F. Lacombe, and M. Glanc, “3D phase diversity: a myopic deconvolution method for short-exposure images: application to retinal imaging,” J. Opt. Soc. Am. A **24**(5), 1349–1357 (2007). [CrossRef] [PubMed]

*z*= 0. Therefore, another error metric can be obtained for the two-dimensional opaque objects aswhere

*S**is the two-dimensional PSF in a plane of defocused distance*

_{z}*z*. Since the above error metrics are the quadratic convex functions of the object spectrum, they can be reduced to the object-independent forms by the minimization with respect to the object spectrum as and the minimizers are expressed by

*W**are the basis functions for the expansion of three-dimensional PSF. With this expression, the Fourier spectrum of both three-dimensional and two-dimensional PSF can be represented as*

_{kl}*β*can be obtained straightforwardly

**|**

*P*^{2}can be represented aswhere

*X**are the basis functions for the expansion of |*

_{kl}**|**

*P*^{2}. The gradient and Hessian of |

**|**

*P*^{2}with respect to the real and imaginary parts can be obtained accordingly

**|**

*T*^{2},

**and**

*D*

*Q*^{2}can be expressed using those of

**and |**

*S***|**

*P*^{2}as

*β*coefficients, the generalized aperture function can be obtained. Then the phase is retrieved by unwrapping the aperture phase and the object is reconstructed by using Eq. (24).

*W**and*

_{kl}

*X**and as well as the Hessian of*

_{kl}**and |**

*S***|**

*P*^{2}can be precomputed. In addition, the regular structural properties of the optimization problem expression can also be exploited to speed the optimization procedure.

### 3.2 Implementation and analysis

23. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express **14**(23), 11277–11291 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11277. [CrossRef] [PubMed]

*λ*/(NA)

^{2}and lateral unit

*λ*/(NA), respectively. Both the circular and annular apertures are used with the initial estimation of aperture amplitude is set to the aperture function. In the nonlinear optimization, the initial

*β*coefficients are set to zero except that

*β*= 1, and the regularization parameters are tuned empirically.

_{0}*μ*= 0) are also given. In the simulations, we find that the amplitude regularization plays an important role for the accuracy of presented algorithm even without the amplitude variations, which can be seen obviously from Fig. 2. In addition, it also shows that the cases of annular aperture can be accurately approximated with the corresponding annular Zernike polynomials.

**24**(5), 1349–1357 (2007). [CrossRef] [PubMed]

*β*)≈

*α*between the complex Zernike coefficients

*β*scaled by

*β*and the ordinary Zernike coefficients

_{0}*α*[11]. This is also justified in the simulations, which suggests that using certain terms of

*β*comparable to that of

*α*should accurately represent the generalized aperture function. But for large phase aberrations, we find that the above relationship does not hold. As shown in Fig. 4 , by direct decomposition of a generalized aperture function with uniform amplitude and large phase aberrations, whose phase is generated with the random low-order

*α*coefficients, the scaled

*β*coefficients distribute over a rather broad range and the above relationship is not valid anymore. Therefore, enough expansion terms must be used for adequate approximation of large aberrations.

## 4. Experimental setup and results

*λ*= 632.8nm emitted from a ZYGO laser interferometer illuminates a reflective liquid crystal spatial light modulator (LC-SLM) from BNS through a beam splitter. The LC-SLM with 256 × 256 pixels and 6.14mm × 6.14mm array size can accurately generate low-order phase aberrations. Large phase aberrations can also be generated by means of the phase wrapping method. For the phase-only modulation, the LC-SLM is adjusted to align the vertical axis to the polarization direction of the incident beam. A circular aperture stop of diameter 6mm is positioned in front of the LC-SLM, and the reflected beam through the aperture stop is sent to the interferometer for phase measurement and to a camera for PSF measurement. The camera is an ANDOR DU-888 EMCCD camera with 14 bit depth and 13μm × 13μm pixel size, which is mounted on a motorized translate stage for the defocus movement.

*λ*RMS, where the residual phase errors can be kept almost within the diffraction limit. The experimental results can also demonstrate the effectiveness of the formulation of three-dimensional PSF for phase retrieval and deconvolution. More appropriate object and noise models should be considered to improve the algorithmic performance especially under the low signal-noise-ratio conditions, and further experimental validations will be carried out for the practical applications.

## 5. Conclusion

## References and links

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

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

3. | R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations using phase diversity,” J. Opt. Soc. Am. A |

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

5. | J. B. Sibarita, “Deconvolution Microscopy,” Adv. Biochem. Eng. Biotechnol. |

6. | G. Chenegros, L. M. Mugnier, F. Lacombe, and M. Glanc, “3D phase diversity: a myopic deconvolution method for short-exposure images: application to retinal imaging,” J. Opt. Soc. Am. A |

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

8. | A. J. E. M. Janssen, “Extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A |

9. | J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A |

10. | J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in |

11. | A. A. Ramos and A. L. Ariste, “Image reconstruction with analytical point spread functions,” Astron. Astrophys . |

12. | C. W. McCutchen, “Generalized Aperture and the Three-Dimensional Diffraction Image,” J. Opt. Soc. Am. |

13. | E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. |

14. | J. Lin, X.-C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett. |

15. | M. Born and E. Wolf, |

16. | S. van Haver, “The Extended Nijboer-Zernike diffraction theory and its applications,” PhD Dissertation, Delft University of Technology (2010). |

17. | J. J. M. Braat, S. van Haver, and S. F. Pereira, “Microlens quality assessment using the Extended Nijboer-Zernike (ENZ) diffraction theory,” presented at EOS Optical Microsystems, Capri, Italy, 27–30 Sept. 2009. |

18. | C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A |

19. | S. M. Jefferies, M. Lloyd-Hart, E. K. Hege, and J. Georges, “Sensing wave-front amplitude and phase with phase diversity,” Appl. Opt. |

20. | C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Proc. SPIE |

21. | J. Nocedal and S. J. Wright, |

22. | D. C. Ghiglia and M. D. Pritt, |

23. | M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express |

**OCIS Codes**

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

(100.1830) Image processing : Deconvolution

(100.5070) Image processing : Phase retrieval

(100.6890) Image processing : Three-dimensional image processing

**ToC Category:**

Image Processing

**History**

Original Manuscript: May 9, 2012

Revised Manuscript: June 16, 2012

Manuscript Accepted: June 18, 2012

Published: June 25, 2012

**Citation**

Xinyue Liu, Liang Wang, Jianli Wang, and Haoran Meng, "A three-dimensional point spread function for phase retrieval and deconvolution," Opt. Express **20**, 15392-15405 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15392

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### References

- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21(15), 2758–2769 (1982). [CrossRef] [PubMed]
- R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng.21, 829–832 (1982).
- R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations using phase diversity,” J. Opt. Soc. Am. A9(7), 1072–1085 (1992). [CrossRef]
- J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt.32(10), 1737–1746 (1993). [CrossRef] [PubMed]
- J. B. Sibarita, “Deconvolution Microscopy,” Adv. Biochem. Eng. Biotechnol.95, 201–243 (2005). [PubMed]
- G. Chenegros, L. M. Mugnier, F. Lacombe, and M. Glanc, “3D phase diversity: a myopic deconvolution method for short-exposure images: application to retinal imaging,” J. Opt. Soc. Am. A24(5), 1349–1357 (2007). [CrossRef] [PubMed]
- B. M. Hanser, M. G. L. 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]
- A. J. E. M. Janssen, “Extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A19(5), 849–857 (2002). [CrossRef] [PubMed]
- J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A19(5), 858–870 (2002). [CrossRef] [PubMed]
- J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed., (Elsevier, 2008), 51, 349–468.
- A. A. Ramos and A. L. Ariste, “Image reconstruction with analytical point spread functions,” Astron. Astrophys. 518, paper A6 (2010).
- C. W. McCutchen, “Generalized Aperture and the Three-Dimensional Diffraction Image,” J. Opt. Soc. Am.54(2), 240–242 (1964). [CrossRef]
- E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun.39(4), 205–210 (1981). [CrossRef]
- J. Lin, X.-C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett.36(8), 1341–1343 (2011). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999).
- S. van Haver, “The Extended Nijboer-Zernike diffraction theory and its applications,” PhD Dissertation, Delft University of Technology (2010).
- J. J. M. Braat, S. van Haver, and S. F. Pereira, “Microlens quality assessment using the Extended Nijboer-Zernike (ENZ) diffraction theory,” presented at EOS Optical Microsystems, Capri, Italy, 27–30 Sept. 2009.
- C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A20(11), 2156–2162 (2003). [CrossRef] [PubMed]
- S. M. Jefferies, M. Lloyd-Hart, E. K. Hege, and J. Georges, “Sensing wave-front amplitude and phase with phase diversity,” Appl. Opt.41(11), 2095–2102 (2002). [CrossRef] [PubMed]
- C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Proc. SPIE3353, 994–1005 (1998). [CrossRef]
- J. Nocedal and S. J. Wright, Numerical Optimization 2nd ed. (Springer, 2006).
- D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).
- M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express14(23), 11277–11291 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11277 . [CrossRef] [PubMed]

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