Abstract

We present an efficient phase retrieval approach for imaging systems with high numerical aperture based on the vectorial model of the point spread function. The algorithm is in the class of alternating minimization methods and can be adjusted for applications with either known or unknown amplitude of the field in the pupil. The algorithm outperforms existing solutions for high-numerical-aperture phase retrieval: (1) the generalization of the method of Hanser et al., based on extension of the scalar diffraction theory by representing the out-of-focus diversity applied to the image by a spherical cap, and (2) the method of Braat et al., which assumes through the use of extended Nijboer–Zernike expansion the phase to be smooth. The former is limited in terms of accuracy due to model deviations, while the latter is of high computational complexity and excludes phase retrieval problems where the phase is discontinuous or sparse. Extensive numerical results demonstrate the efficiency, robustness, and practicability of the proposed algorithm in various practically relevant simulations.

© 2019 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  15. H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
    [Crossref]
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    [Crossref]
  17. E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imaging Sci. 6, 199–225 (2013).
    [Crossref]
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    [Crossref]
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    [Crossref]
  20. Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32, 87–109 (2015).
    [Crossref]
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    [Crossref]
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    [Crossref]
  30. S. van Haver, J. Braat, D. Peter, and A. Janssen, “High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory,” J. Eur. Math. Soc. 1, 06004 (2006).
    [Crossref]
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    [Crossref]
  32. O. Raz, N. Dudovich, and B. Nadler, “Vectorial phase retrieval of 1D signals,” IEEE Trans. Signal Process. 61, 1632–1643 (2013).
    [Crossref]
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    [Crossref]
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    [Crossref]
  36. F. Soulez, É. Thiébaut, A. Schutz, A. Ferrari, F. Courbin, and M. Unser, “Proximity operators for phase retrieval,” Appl. Opt. 55, 7412–7421 (2016).
    [Crossref]
  37. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
    [Crossref]
  38. J.-S. Pang, “Error bounds in mathematical programming,” Math. Programming, Ser. B 79, 299–332 (1997). Lectures on Mathematical Programming (ISMP97) (Lausanne, 1997).
    [Crossref]
  39. D. R. Luke, N. H. Thao, and M. K. Tam, “Quantitative convergence analysis of iterated expansive, set-valued mappings,” Math. Oper. Res. 43, 1143–1176 (2018).
    [Crossref]
  40. N. H. Thao, R. Luke, O. Soloviev, and M. Verhaegen, “Phase retrieval with sparse phase constraint,” arXiv:1804.01878v2 (2018).
  41. R. Doelman, N. H. Thao, and M. Verhaegen, “Solving large-scale general phase retrieval problems via a sequence of convex relaxations,” J. Opt. Soc. Am. A 35, 1410–1419 (2018).
    [Crossref]

2019 (1)

R. Luke, S. Sabach, and M. Teboulle, “Optimization on spheres: models and proximal algorithms with computational performance comparisons,” SIAM J. Math. Data Sci. 1, 408–445 (2019).
[Crossref]

2018 (2)

D. R. Luke, N. H. Thao, and M. K. Tam, “Quantitative convergence analysis of iterated expansive, set-valued mappings,” Math. Oper. Res. 43, 1143–1176 (2018).
[Crossref]

R. Doelman, N. H. Thao, and M. Verhaegen, “Solving large-scale general phase retrieval problems via a sequence of convex relaxations,” J. Opt. Soc. Am. A 35, 1410–1419 (2018).
[Crossref]

2017 (1)

D. R. Luke, “Phase retrieval, what’s new?” SIAG/OPT Views News 25, 1–6 (2017).

2016 (3)

2015 (2)

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

J. Antonello and M. Verhaegen, “Modal-based phase retrieval for adaptive optics,” J. Opt. Soc. Am. A 32, 1160–1170 (2015).
[Crossref]

2013 (4)

C. C. de Visser and M. Verhaegen, “Wavefront reconstruction in adaptive optics systems using nonlinear multivariate splines,” J. Opt. Soc. Am. A 30, 82–95 (2013).
[Crossref]

J. R. Fienup, “Phase retrieval algorithms: a personal tour,” Appl. Opt. 52, 45–56 (2013).
[Crossref]

O. Raz, N. Dudovich, and B. Nadler, “Vectorial phase retrieval of 1D signals,” IEEE Trans. Signal Process. 61, 1632–1643 (2013).
[Crossref]

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imaging Sci. 6, 199–225 (2013).
[Crossref]

2012 (1)

D. R. Luke, “Local linear convergence of approximate projections onto regularized sets,” Nonlinear Anal. 75, 1531–1546 (2012).
[Crossref]

2009 (1)

2006 (1)

S. van Haver, J. Braat, D. Peter, and A. Janssen, “High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory,” J. Eur. Math. Soc. 1, 06004 (2006).
[Crossref]

2005 (2)

2004 (1)

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, 32–48 (2004).
[Crossref]

2003 (2)

2002 (2)

2000 (1)

J. W. Hardy and L. Thompson, “Adaptive optics for astronomical telescopes,” Phys. Today 53(4), 69 (2000).
[Crossref]

1999 (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[Crossref]

1997 (1)

J.-S. Pang, “Error bounds in mathematical programming,” Math. Programming, Ser. B 79, 299–332 (1997). Lectures on Mathematical Programming (ISMP97) (Lausanne, 1997).
[Crossref]

1993 (1)

1990 (1)

1987 (1)

J. C. Dainty and J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” Image Recovery: Theory Appl. 13, 231–275 (1987).

1986 (1)

H. Hauptman, “The direct methods of x-ray crystallography,” Science 233, 178–183 (1986).
[Crossref]

1982 (2)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[Crossref]

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

1977 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1964 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

1952 (1)

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. Online 5, 843 (1952).
[Crossref]

Agard, D. A.

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, 32–48 (2004).
[Crossref]

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase retrieval for high-numerical-aperture optical systems,” Opt. Lett. 28, 801–803 (2003).
[Crossref]

Antonello, J.

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[Crossref]

Bauschke, H. H.

Blanc, A.

L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” in Advances in Imaging and Electron Physics, P. Hawkes, ed. (Elsevier, 2006), Vol. 141, pp. 1–76.

Braat, J.

S. van Haver, J. Braat, D. Peter, and A. Janssen, “High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory,” J. Eur. Math. Soc. 1, 06004 (2006).
[Crossref]

Braat, J. J.

Braat, J. J. M.

Brunner, E.

Candès, E. J.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imaging Sci. 6, 199–225 (2013).
[Crossref]

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

Combettes, P. L.

Courbin, F.

Dainty, J. C.

J. C. Dainty and J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” Image Recovery: Theory Appl. 13, 231–275 (1987).

de Visser, C. C.

Dirksen, P.

Doelman, R.

Dudovich, N.

O. Raz, N. Dudovich, and B. Nadler, “Vectorial phase retrieval of 1D signals,” IEEE Trans. Signal Process. 61, 1632–1643 (2013).
[Crossref]

Eldar, Y. C.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imaging Sci. 6, 199–225 (2013).
[Crossref]

Ferrari, A.

Fienup, J. R.

J. R. Fienup, “Phase retrieval algorithms: a personal tour,” Appl. Opt. 52, 45–56 (2013).
[Crossref]

J. C. Dainty and J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” Image Recovery: Theory Appl. 13, 231–275 (1987).

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

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goddard, L. L.

T. Kim, R. Zhou, L. L. Goddard, and G. Popescu, “Solving inverse scattering problems in biological samples by quantitative phase imaging,” Laser Photon. Rev. 10, 13–39 (2016).
[Crossref]

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2005).

Gustafsson, M. G. L.

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, 32–48 (2004).
[Crossref]

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase retrieval for high-numerical-aperture optical systems,” Opt. Lett. 28, 801–803 (2003).
[Crossref]

Hanser, B. M.

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, 32–48 (2004).
[Crossref]

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase retrieval for high-numerical-aperture optical systems,” Opt. Lett. 28, 801–803 (2003).
[Crossref]

Hardy, J. W.

J. W. Hardy and L. Thompson, “Adaptive optics for astronomical telescopes,” Phys. Today 53(4), 69 (2000).
[Crossref]

Harrison, R. W.

Hauptman, H.

H. Hauptman, “The direct methods of x-ray crystallography,” Science 233, 178–183 (1986).
[Crossref]

Idier, J.

L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” in Advances in Imaging and Electron Physics, P. Hawkes, ed. (Elsevier, 2006), Vol. 141, pp. 1–76.

Janssen, A.

S. van Haver, J. Braat, D. Peter, and A. Janssen, “High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory,” J. Eur. Math. Soc. 1, 06004 (2006).
[Crossref]

Janssen, A. J.

Janssen, A. J. E. M.

Kim, T.

T. Kim, R. Zhou, L. L. Goddard, and G. Popescu, “Solving inverse scattering problems in biological samples by quantitative phase imaging,” Laser Photon. Rev. 10, 13–39 (2016).
[Crossref]

Luke, D. R.

D. R. Luke, N. H. Thao, and M. K. Tam, “Quantitative convergence analysis of iterated expansive, set-valued mappings,” Math. Oper. Res. 43, 1143–1176 (2018).
[Crossref]

D. R. Luke, “Phase retrieval, what’s new?” SIAG/OPT Views News 25, 1–6 (2017).

D. R. Luke, “Local linear convergence of approximate projections onto regularized sets,” Nonlinear Anal. 75, 1531–1546 (2012).
[Crossref]

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2005).
[Crossref]

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
[Crossref]

Luke, R.

R. Luke, S. Sabach, and M. Teboulle, “Optimization on spheres: models and proximal algorithms with computational performance comparisons,” SIAM J. Math. Data Sci. 1, 408–445 (2019).
[Crossref]

N. H. Thao, R. Luke, O. Soloviev, and M. Verhaegen, “Phase retrieval with sparse phase constraint,” arXiv:1804.01878v2 (2018).

Mansuripur, M.

M. Mansuripur, Classical Optics and Its Applications (Cambridge University, 2009).

McCutchen, C. W.

Miao, J.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

Millane, R. P.

Mugnier, L. M.

L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” in Advances in Imaging and Electron Physics, P. Hawkes, ed. (Elsevier, 2006), Vol. 141, pp. 1–76.

Nadler, B.

O. Raz, N. Dudovich, and B. Nadler, “Vectorial phase retrieval of 1D signals,” IEEE Trans. Signal Process. 61, 1632–1643 (2013).
[Crossref]

Nakajima, N.

Pang, J.-S.

J.-S. Pang, “Error bounds in mathematical programming,” Math. Programming, Ser. B 79, 299–332 (1997). Lectures on Mathematical Programming (ISMP97) (Lausanne, 1997).
[Crossref]

Peter, D.

S. van Haver, J. Braat, D. Peter, and A. Janssen, “High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory,” J. Eur. Math. Soc. 1, 06004 (2006).
[Crossref]

Popescu, G.

T. Kim, R. Zhou, L. L. Goddard, and G. Popescu, “Solving inverse scattering problems in biological samples by quantitative phase imaging,” Laser Photon. Rev. 10, 13–39 (2016).
[Crossref]

Raz, O.

O. Raz, N. Dudovich, and B. Nadler, “Vectorial phase retrieval of 1D signals,” IEEE Trans. Signal Process. 61, 1632–1643 (2013).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

Sabach, S.

R. Luke, S. Sabach, and M. Teboulle, “Optimization on spheres: models and proximal algorithms with computational performance comparisons,” SIAM J. Math. Data Sci. 1, 408–445 (2019).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Sayre, D.

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. Online 5, 843 (1952).
[Crossref]

Schutz, A.

Sedat, J. W.

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, 32–48 (2004).
[Crossref]

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase retrieval for high-numerical-aperture optical systems,” Opt. Lett. 28, 801–803 (2003).
[Crossref]

Segev, M.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

Shechtman, Y.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

Soloviev, O.

N. H. Thao, R. Luke, O. Soloviev, and M. Verhaegen, “Phase retrieval with sparse phase constraint,” arXiv:1804.01878v2 (2018).

Soulez, F.

Southwell, W. H.

Strohmer, T.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imaging Sci. 6, 199–225 (2013).
[Crossref]

Tam, M. K.

D. R. Luke, N. H. Thao, and M. K. Tam, “Quantitative convergence analysis of iterated expansive, set-valued mappings,” Math. Oper. Res. 43, 1143–1176 (2018).
[Crossref]

Teboulle, M.

R. Luke, S. Sabach, and M. Teboulle, “Optimization on spheres: models and proximal algorithms with computational performance comparisons,” SIAM J. Math. Data Sci. 1, 408–445 (2019).
[Crossref]

Thao, N. H.

D. R. Luke, N. H. Thao, and M. K. Tam, “Quantitative convergence analysis of iterated expansive, set-valued mappings,” Math. Oper. Res. 43, 1143–1176 (2018).
[Crossref]

R. Doelman, N. H. Thao, and M. Verhaegen, “Solving large-scale general phase retrieval problems via a sequence of convex relaxations,” J. Opt. Soc. Am. A 35, 1410–1419 (2018).
[Crossref]

N. H. Thao, R. Luke, O. Soloviev, and M. Verhaegen, “Phase retrieval with sparse phase constraint,” arXiv:1804.01878v2 (2018).

Thiébaut, É.

Thompson, L.

J. W. Hardy and L. Thompson, “Adaptive optics for astronomical telescopes,” Phys. Today 53(4), 69 (2000).
[Crossref]

Unser, M.

van de Nes, A. S.

van Haver, S.

S. van Haver, J. Braat, D. Peter, and A. Janssen, “High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory,” J. Eur. Math. Soc. 1, 06004 (2006).
[Crossref]

J. J. Braat, P. Dirksen, A. J. Janssen, S. van Haver, and A. S. van de Nes, “Extended Nijboer-Zernike approach to aberration and birefringence retrieval in a high-numerical-aperture optical system,” J. Opt. Soc. Am. A 22, 2635–2650 (2005).
[Crossref]

Verhaegen, M.

Voroninski, V.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imaging Sci. 6, 199–225 (2013).
[Crossref]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

Zhou, R.

T. Kim, R. Zhou, L. L. Goddard, and G. Popescu, “Solving inverse scattering problems in biological samples by quantitative phase imaging,” Laser Photon. Rev. 10, 13–39 (2016).
[Crossref]

Acta Crystallogr. Online (1)

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Figures (17)

Fig. 1.
Fig. 1. Schematic diagram depicts the vectorial PSF model and the setup of phase retrieval given several out-of-focus measurements. A collimated beam with (possibly unknown) amplitude $ \chi $ at the entrance pupil plane is focused by an aplanatic system at plane $ z = {z_0} $, not necessarily on axis. Several out-of-focus PSFs are registered in $ z $ planes with known displacements from the focal plane. For high values of NA, the bending of rays introduced by the lens produces a significant $ z $ component of the electrical field (here shown on example of $ x $-polarization component $ {E_x} $), which should be taken into account when calculating the intensity in the imaging planes.
Fig. 2.
Fig. 2. Experiments with noise show the advantages of VAM and $ {\text{VAM}_ + } $ over SAM for various NA values. The relative RMS errors of the restored wavefronts compared to the correct solution are presented for 75 wavefront realizations and six different NA values ranging from 0.7 to 0.95. The performance of VAM (red) and $ {\text{VAM}_ + } $ (blue) is consistent for all experiments. Suffering model deviations, SAM (black) is significantly outperformed by the others in terms of both accuracy and consistency. $ {\text{VAM}_ + } $ slightly outperforms VAM thanks to the additional information of $ \chi $.
Fig. 3.
Fig. 3. Errors of phase retrieval over 75 experiments are presented for six different NA values. The relative errors of VAM (red curve) and $ {\text{VAM}_ + } $ (blue curve) are consistent at about 2%, while that of SAM (black curve) increases from 5.5% for $ \text{NA} = 0.7 $ up to 8.5% for $ \text{NA} = 0.95 $. The error of phase retrieval by SAM is approximately proportional to the NA value as indicated by the upward black curve.
Fig. 4.
Fig. 4. Experiments demonstrate the robustness of VAM and $ {\text{VAM}_ + } $ against noise. The two algorithms are consistent and reliable for SNR from 35 dB as demonstrated by small variation ranges of errors. The performance of SAM (black) is almost unaffected by noise with SNR from 30 dB; however, due to model deviations, it is clearly outperformed by VAM (red) and $ {\text{VAM}_ + } $ (blue).
Fig. 5.
Fig. 5. Average of the relative RMS errors of phase retrieval over 75 experiments is presented for seven different levels of Gaussian noise. $ {\text{VAM}_ + } $ (blue curve) is clearly superior to VAM (red curve) and SAM (black curve). The quality of phase retrieval by VAM and SAM for SNR 25 dB is at the same level, but the difference becomes more substantial for higher SNR. The relative RMS error of $ {\text{VAM}_ + } $ and VAM sharply decreases for higher SNR, while that of SAM remains high (above 8.5%) for all SNR due to model deviations.
Fig. 6.
Fig. 6. Experiments demonstrate descent property of VAM and $ {\text{VAM}_ + } $. In the noise-free setting (solid curves), the objective value converges to zero, and with the additional information of $ \chi $, $ {\text{VAM}_ + } $ (blue) reaches the optimal value faster than VAM (red). In the presence of noise (dashed curves), the objective value reaches an objective gap in about 50 iterations, and we note that VAM optimizing over both $ \chi $ and $ \Phi $ yields a solution better fitting to the noisy data (smaller objective gap) than $ {\text{VAM}_ + } $ optimizing over only $ \Phi $.
Fig. 7.
Fig. 7. Experiments show the feasibility gap versus the number of iterations of VAM and $ {\text{VAM}_ + } $ compared to SAM. Regardless of the presence of noise, the performance of VAM (red curves) and $ {\text{VAM}_ + } $ (blue curves) is consistent, and the latter one is favorable due to the additional information of $ \chi $. Model deviations make SAM (black curves) much less accurate than VAM and $ {\text{VAM}_ + } $.
Fig. 8.
Fig. 8. Experiments demonstrate the convergence of VAM and $ {\text{VAM}_ + } $ in terms of the change in the estimated phase in iteration. In both scenarios of noise, the iterative change of the estimate decreases to zero in a stable and consistent manner. Due to the additional knowledge of $ \chi $, $ {\text{VAM}_ + } $ (blue curves) converges slightly faster than VAM (red curves).
Fig. 9.
Fig. 9. Realization of phase retrieval by VAM (bottom-left), $ {\text{VAM}_ + } $ (bottom-right) and SAM (top-right).
Fig. 10.
Fig. 10. Four noiseless out-of-focus PSFs corresponding to the phase presented in Fig. 9: the central part of each image with size $ 48 \times 48 $ pixels is shown.
Fig. 11.
Fig. 11. Experiments show that more input images lead to more accuracy of phase retrieval by VAM and $ {\text{VAM}_ + } $.
Fig. 12.
Fig. 12. Experiments show that more input images lead to faster convergence of VAM and $ {\text{VAM}_ + } $.
Fig. 13.
Fig. 13. Experiment with noise demonstrates the solvability of VAM and $ {\text{VAM}_ + } $ for phase retrieval with discontinuous phase. $ {\text{VAM}_ + } $ (bottom-right) outperforms VAM (bottom-left) due to the knowledge of $ \chi $, while SAM (top-right) is more erroneous than the others due to model deviations with relative RMS error 9.4% compared to 5% and 4.3% of VAM and $ {\text{VAM}_ + } $.
Fig. 14.
Fig. 14. Experiment with noise demonstrates the solvability of VAM and $ {\text{VAM}_ + } $ for phase retrieval with sparse phase constraint. The main features observed here resemble those in Fig. 13.
Fig. 15.
Fig. 15. Amplitude restoration by VAM for the experiment with continuous phase in Fig. 10: correct amplitude (left) and its residual (right) relative to the restoration (error 3.3%).
Fig. 16.
Fig. 16. Amplitude restoration by VAM for the experiment with discontinuous phase in Fig. 13: restoration (left) and its residual (right) relative to the data (error 1.1%).
Fig. 17.
Fig. 17. Amplitude restoration by VAM for the experiment with sparse phase shown in Fig. 14: restoration (left) and its residual (right) relative to the data (error 1.1%).

Tables (3)

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Algorithm 1. Vectorial PSF model-based alternating minimization with amplitude constraint ($ {\text{VAM}_ + } $)

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Algorithm 2. Vectorial PSF model-based alternating minimization (VAM)

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Algorithm 3. Scalar PSF model-based alternating minimization (SAM)

Equations (16)

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I ( u ) = | F ( χ ( x ) e j Φ ( x ) ) | 2 ,
ϕ d ( x ) = π 2 λ NA 2 z d Z 2 0 ( x ) .
I ( ϕ d ) = | F ( χ e j ( Φ + ϕ d ) ) | 2 .
ϕ d = 2 π λ z d k z ,
k z = 1 k x 2 k y 2 ,
k x 2 + k y 2 NA 2 .
1 / cos θ = k z 1 / 2 ,
E x x = 1 k x 2 1 + k z , E y x = k x k y 1 + k z , E z x = k x , E x y = k y k x 1 + k z , E y y = 1 k y 2 1 + k z , E z y = k y .
p c c = | F ( E c c χ e j Φ ) | 2 .
I = c c p c c .
I ( ϕ d ) = c c | F ( E c c χ e j ( Φ + ϕ d ) ) | 2 .
r d = c c | F ( E c c χ e j ( Φ + ϕ d ) ) | 2 + w d ( d = 1 , , m ) ,
I d [ χ , Φ ] := c c | F ( E c c χ e j ( Φ + ϕ d ) ) | 2 ( d = 1 , , m )
L ( w ) := log P ( I [ χ , Φ ] | r ) .
min χ R + n × n , Φ R n × n f ( χ , Φ ) := pixels L ( r I [ χ , Φ ] ) .
min χ R + n × n , Φ R n × n f ( χ , Φ ) := d = 1 m r d I d [ χ , Φ ] F 2 .

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