Abstract

In this paper, we propose the use of Gaussian radial basis functions (GRBFs) to model the generalized pupil function for phase retrieval. The selection of the GRBF hyper-parameters is analyzed to achieve an increased accuracy of approximation. The performance of the GRBF-based method is compared in a simulation study with another modal-based approach considering extended Nijboer–Zernike (ENZ) polynomials. The almost local character of the GRBFs makes them a much more flexible basis with respect to the pupil geometry. It has been shown that for aberrations containing higher spatial frequencies, the GRBFs outperform ENZ polynomials significantly, even on a circular pupil. Moreover, the flexibility has been demonstrated by considering the phase retrieval problem on an annular pupil.

© 2018 Optical Society of America

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References

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  1. 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(3), 87–109 (2015).
    [Crossref]
  2. D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
    [Crossref]
  3. R. W. Gerchberg, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  4. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [Crossref]
  5. E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Rev. 57, 225–251 (2015).
    [Crossref]
  6. Y. Chen and E. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” in Advances in Neural Information Processing Systems (2015), pp. 739–747.
  7. I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. 149, 47–81 (2015).
    [Crossref]
  8. R. Doelman, H. T. Nguyen, and M. Verhaegen, “Solving large-scale general phase retrieval problems via a sequence of convex relaxations,” arXiv:1803.02652 (2018).
  9. 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]
  10. J. Antonello and M. Verhaegen, “Modal-based phase retrieval for adaptive optics,” J. Opt. Soc. Am. A 32, 1160–1170 (2015).
    [Crossref]
  11. A. J. Janssen, “Extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 849–857 (2002).
    [Crossref]
  12. S. Van Haver, “The extended Nijboer–Zernike diffraction theory and its applications,” Ph.D. thesis (Delft University of Technology, 2010).
  13. A. Martinez-Finkelshtein, D. Ramos-Lopez, and D. Iskander, “Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions,” Appl. Comput. Harmon. Anal. 43, 424–448 (2017).
    [Crossref]
  14. M. Maksimovic, “Optical design and tolerancing of freeform surfaces using anisotropic radial basis functions,” Opt. Eng. 55, 071203 (2016).
    [Crossref]
  15. B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33, 869–892 (2011).
    [Crossref]
  16. Boston Micromachines Corporation, 2018, http://www.bmc.bostonmicromachines.com/pdf/Kilo-DM.pdf .
  17. M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).
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    [Crossref]
  19. M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437–7444 (2002).
    [Crossref]
  20. L. Kocis and W. J. Whiten, “Computational investigations of low-discrepancy sequences,” ACM Trans. Math. Softw. 23, 266–294 (1997).
    [Crossref]

2017 (1)

A. Martinez-Finkelshtein, D. Ramos-Lopez, and D. Iskander, “Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions,” Appl. Comput. Harmon. Anal. 43, 424–448 (2017).
[Crossref]

2016 (1)

M. Maksimovic, “Optical design and tolerancing of freeform surfaces using anisotropic radial basis functions,” Opt. Eng. 55, 071203 (2016).
[Crossref]

2015 (4)

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(3), 87–109 (2015).
[Crossref]

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Rev. 57, 225–251 (2015).
[Crossref]

I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. 149, 47–81 (2015).
[Crossref]

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

2011 (1)

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33, 869–892 (2011).
[Crossref]

2002 (4)

1997 (1)

L. Kocis and W. J. Whiten, “Computational investigations of low-discrepancy sequences,” ACM Trans. Math. Softw. 23, 266–294 (1997).
[Crossref]

1991 (1)

1978 (1)

1972 (1)

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

Antonello, J.

Bauschke, H. H.

Burke, J. V.

D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[Crossref]

Burton, D. R.

Candes, E.

Y. Chen and E. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” in Advances in Neural Information Processing Systems (2015), pp. 739–747.

Candes, E. J.

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Rev. 57, 225–251 (2015).
[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(3), 87–109 (2015).
[Crossref]

Chen, Y.

Y. Chen and E. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” in Advances in Neural Information Processing Systems (2015), pp. 739–747.

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(3), 87–109 (2015).
[Crossref]

Combettes, P. L.

d’Aspremont, A.

I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. 149, 47–81 (2015).
[Crossref]

Doelman, R.

R. Doelman, H. T. Nguyen, and M. Verhaegen, “Solving large-scale general phase retrieval problems via a sequence of convex relaxations,” arXiv:1803.02652 (2018).

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(3), 87–109 (2015).
[Crossref]

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Rev. 57, 225–251 (2015).
[Crossref]

Fienup, J. R.

Flyer, N.

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33, 869–892 (2011).
[Crossref]

Fornberg, B.

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33, 869–892 (2011).
[Crossref]

Gdeisat, M. A.

Gerchberg, R. W.

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

Herráez, M. A.

Iskander, D.

A. Martinez-Finkelshtein, D. Ramos-Lopez, and D. Iskander, “Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions,” Appl. Comput. Harmon. Anal. 43, 424–448 (2017).
[Crossref]

Janssen, A. J.

Kocis, L.

L. Kocis and W. J. Whiten, “Computational investigations of low-discrepancy sequences,” ACM Trans. Math. Softw. 23, 266–294 (1997).
[Crossref]

Lalor, M. J.

Larsson, E.

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33, 869–892 (2011).
[Crossref]

Luke, D. R.

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]

D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[Crossref]

Lyon, R. G.

D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[Crossref]

Maksimovic, M.

M. Maksimovic, “Optical design and tolerancing of freeform surfaces using anisotropic radial basis functions,” Opt. Eng. 55, 071203 (2016).
[Crossref]

Mallat, S.

I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. 149, 47–81 (2015).
[Crossref]

Martinez-Finkelshtein, A.

A. Martinez-Finkelshtein, D. Ramos-Lopez, and D. Iskander, “Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions,” Appl. Comput. Harmon. Anal. 43, 424–448 (2017).
[Crossref]

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(3), 87–109 (2015).
[Crossref]

Nguyen, H. T.

R. Doelman, H. T. Nguyen, and M. Verhaegen, “Solving large-scale general phase retrieval problems via a sequence of convex relaxations,” arXiv:1803.02652 (2018).

Ramos-Lopez, D.

A. Martinez-Finkelshtein, D. Ramos-Lopez, and D. Iskander, “Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions,” Appl. Comput. Harmon. Anal. 43, 424–448 (2017).
[Crossref]

Roddier, C.

Roddier, F.

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(3), 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(3), 87–109 (2015).
[Crossref]

Strohmer, T.

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Rev. 57, 225–251 (2015).
[Crossref]

Van Haver, S.

S. Van Haver, “The extended Nijboer–Zernike diffraction theory and its applications,” Ph.D. thesis (Delft University of Technology, 2010).

Verdult, V.

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).

Verhaegen, M.

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

R. Doelman, H. T. Nguyen, and M. Verhaegen, “Solving large-scale general phase retrieval problems via a sequence of convex relaxations,” arXiv:1803.02652 (2018).

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).

Voroninski, V.

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Rev. 57, 225–251 (2015).
[Crossref]

Waldspurger, I.

I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. 149, 47–81 (2015).
[Crossref]

Whiten, W. J.

L. Kocis and W. J. Whiten, “Computational investigations of low-discrepancy sequences,” ACM Trans. Math. Softw. 23, 266–294 (1997).
[Crossref]

ACM Trans. Math. Softw. (1)

L. Kocis and W. J. Whiten, “Computational investigations of low-discrepancy sequences,” ACM Trans. Math. Softw. 23, 266–294 (1997).
[Crossref]

Appl. Comput. Harmon. Anal. (1)

A. Martinez-Finkelshtein, D. Ramos-Lopez, and D. Iskander, “Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions,” Appl. Comput. Harmon. Anal. 43, 424–448 (2017).
[Crossref]

Appl. Opt. (2)

IEEE Signal Process. Mag. (1)

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(3), 87–109 (2015).
[Crossref]

J. Opt. Soc. Am. A (3)

Math. Program. (1)

I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. 149, 47–81 (2015).
[Crossref]

Opt. Eng. (1)

M. Maksimovic, “Optical design and tolerancing of freeform surfaces using anisotropic radial basis functions,” Opt. Eng. 55, 071203 (2016).
[Crossref]

Opt. Lett. (1)

Optik (1)

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

SIAM J. Sci. Comput. (1)

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33, 869–892 (2011).
[Crossref]

SIAM Rev. (2)

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Rev. 57, 225–251 (2015).
[Crossref]

D. R. Luke, J. V. Burke, and R. G. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[Crossref]

Other (5)

Y. Chen and E. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” in Advances in Neural Information Processing Systems (2015), pp. 739–747.

S. Van Haver, “The extended Nijboer–Zernike diffraction theory and its applications,” Ph.D. thesis (Delft University of Technology, 2010).

Boston Micromachines Corporation, 2018, http://www.bmc.bostonmicromachines.com/pdf/Kilo-DM.pdf .

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).

R. Doelman, H. T. Nguyen, and M. Verhaegen, “Solving large-scale general phase retrieval problems via a sequence of convex relaxations,” arXiv:1803.02652 (2018).

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Figures (9)

Fig. 1.
Fig. 1. Example of a low-order and high-order aberration generated as described in Sections 3.A and 3.B, respectively. The amplitude of the phase can be scaled to any desired value.
Fig. 2.
Fig. 2. Examples of node distributions on a 2D grid for a unit disk pupil aperture: (a) rectangular, (b) Halton, and (c) Fibonacci.
Fig. 3.
Fig. 3. Influence of λ on the mean value of ϵp,LS in a Monte Carlo simulation. The same simulation is performed on both low- and high-order aberration data discussed in Sections 3.A and 3.B for Np=64 on a circular aperture with a radius of 1.
Fig. 4.
Fig. 4. Comparison of the results of the phase retrieval simulation between lower- and higher-order aberrations for varying number of basis functions Nα. The boxplots show the normalized RMSE ϵy and ϵϕ [see Eqs. (19) and (20)] for a circular pupil with Np=64, and RMS(ϕ)0.75  rad. The boxes indicate the 25th and 75th percentile of the results in the Monte Carlo simulation. Lines are drawn through the medians of the data. Data outliers due to remaining phase ambiguities are discarded.
Fig. 5.
Fig. 5. Phase retrieval results for low-order and high-order aberrations for RMS(ϕ)1.5  rad. The presentation of the results is similar to Fig. 4.
Fig. 6.
Fig. 6. Example of the retrieved phase for a high-order aberration, Np=128, and Nα=377. From left to right, the first row shows the true phase aberration, the retrieved phase estimate using GRBF, and the retrieved estimate using ENZ. The second row shows the errors for GRBF (ϵϕ=0.42) and ENZ (ϵϕ=0.57), respectively. The figures are truncated to the color scale shown.
Fig. 7.
Fig. 7. Comparison of the results of the phase retrieval simulation between lower- and higher-order aberrations for varying number of basis functions Nα. The boxplots show the normalized RMSE ϵy and ϵϕ [see Eqs. (19) and (20)] for an annular pupil with Np=64, and RMS(ϕ)0.75  rad. The boxes indicate the 25th and 75th percentile of the results in the Monte Carlo simulation. Lines are drawn through the medians of the data. Data outliers due to remaining phase ambiguities are discarded.
Fig. 8.
Fig. 8. Example of the retrieved phase for a high-order aberration, Np=128 and Nα=377. From left to right, the first row shows the true phase aberration, the retrieved phase estimate using GRBF, and the retrieved estimate using ENZ. The second row shows the errors for GRBF (ϵϕ=0.27) and ENZ (ϵϕ=0.68), respectively. The figures are truncated to the color scale shown.
Fig. 9.
Fig. 9. Normalized RMS phase error ϵϕ [defined in Eq. (19)] as a function of the SNR for low-order aberrations with an average RMS(ϕ)0.75, using Nα=65 basis functions. The black dotted line shows the error without noise.

Tables (1)

Tables Icon

Table 1. Mean Values of ϵϕ,LS Using GRBF and ENZ Polynomials in the Monte Carlo Simulationa

Equations (25)

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

P(ρ,θ)=A(ρ,θ)exp(iϕ(ρ,θ)),
U(r,ϕ,f)=1π0102πexp(ifρ2)P(ρ,θ)×exp(i2πrρcos(θϕ))ρdρdθ,
y(r,ϕ,f)=|U(r,ϕ,f)|2.
P^E(ρ,θ)=n,mβnmNnm(ρ,θ).
P^R(ρ,θ)=A(ρ,θ)k=1NγγkΨk(ρ,θ),
Ψk(ρ,θ)=eλk(ρ2+ϱk22ρϱkcos(θϑk)),
P^(ρ,θ)=k=1NααkBk(ρ,θ)
P^=k=1NααkBk
p^=Bα,
U^d=k=1NααkCd,k,
u^d=Cdα
y^d=|Cdα|2.
minαCNαd=1Nfyd|Cdα|2,
ϕ(ρ,θ)=n,mζnmZnm(ρ,θ),
VAF=(1i=12000s˜iHu˜i22i=12000s˜i22)·100%
α^LS=argminαCNαpBα22,
ϵp,LS=pp^LS2p2.
ϵϕ,LS=ϕϕ^LS2ϕ2.
ϵϕ=ϕϕ^2ϕ2.
ϵy=i=1Nfyiy^i2i=1Nfyi2.
ϱk=ϱ0k1/2,ϑk=2πk/ϕ,
ϕ(ρ,θ)=n,mζnmZnm(ρ,θ),
Znm(ρ,θ)=cnmRn|m|Θnm(θ),
cnm={n+1m=02(n+1)m0,Θnm(θ)={cos(mθ)m0sin(mθ)m<0,Rnm(ρ)=s=0(nm)/2(1)s(ns)!s!(n+m2s)!(nm2s)!ρn2s.
Nnm(ρ,θ)=n+1Rn|m|(ρ)exp(imθ).