Abstract

Wigner distribution deconvolution (WDD) is a decades-old method for recovering phase from intensity measurements. Although the technique offers an elegant linear solution to the quadratic phase retrieval problem, it has seen limited adoption due to its high computational/memory requirements and the fact that the technique often exhibits high noise sensitivity. Here, we propose a method for noise suppression in WDD via low-rank noisy matrix completion. Our technique exploits the redundancy of an object’s phase space to denoise its WDD reconstruction. We show in model calculations that our technique outperforms other WDD algorithms as well as modern iterative methods for phase retrieval such as ptychography. Our results suggest that a class of phase retrieval techniques relying on regularized direct inversion of ptychographic datasets (instead of iterative reconstruction techniques) can provide accurate quantitative phase information in the presence of high levels of noise.

© 2016 Optical Society of America

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References

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  1. R.H.T. Bates and J.M. Rodenburg, “Sub-Angstrom transmission microscopy: a Fourier transform algorithm for microdiffraction plane intensity information,” Ultramicroscopy 31, 303–307 (1989).
    [Crossref]
  2. J.M. Rodenburg and R.H.T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Philos. T. R. Soc. Lond. 339, 521–553 (1992).
    [Crossref]
  3. B.C. McCallum and J.M. Rodenburg, “2-dimensional optical demonstration of Wigner phase retrieval microscopy in the STEM configuration,” Ultramicroscopy 45, 371–380 (1992).
    [Crossref]
  4. H.N. Chapman, “Phase-retrieval X-Ray microscopy by Wigner distribution deconvolution,” Ultramicroscopy 66, 153–172 (1996).
    [Crossref]
  5. J.M. Rodenburg, B.C. McCallum, and P.D. Nellist, “Experimental tests on double resolution coherent imaging via STEM,” Ultramicroscopy 48, 304–314 (1993).
    [Crossref]
  6. P. Li, T.B. Edo, and J.M. Rodenburg, “Ptychographic inversion via Wigner distribution deconvolution: noise suppression and probe design,” Ultramicroscopy 147, 106–113 (2014).
    [Crossref] [PubMed]
  7. A.M. Maiden, M.J. Humphrey, and J.M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A 28, 604–612 (2011).
    [Crossref]
  8. A.M. Maiden, G.R. Morrison, B. Kaulich, A. Gianocelli, and J.M. Rodenburg, “Soft X-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013).
    [Crossref] [PubMed]
  9. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
    [Crossref]
  10. J. Cai, E. Candès, and Z. Shen., “A singular value thresholding algorithm for matrix completion,” SIAM J. Optimiz. 20(4),1956–1982 (2010).
    [Crossref]
  11. E. Candès and Y. Plan., “Matrix completion with noise,” Proc. IEEE 98(6), 925–936 (2010).
    [Crossref]
  12. R.H. Keshavan, A. Montanari, and S. Oh., “Matrix completion from noisy entries,” http//:arXiv:0906.2027 (2009).
  13. R.H. Keshavan, A. Montanari, and S. Oh., “Low-rank matrix completion with noisy observations: a quantitative comparison,” http//:arXiv:0910.0921 (2009).
  14. K. Toh and S. Yun., “An accelerated proximal gradient algorithm for nuclear norm regularized least square problems,” Pac. J. Optim. 6, 615–640 (2010).
  15. S. Ma, D. Goldfarb, and L. Chen., “Fixed point and Bregman iterative methods for matrix rank minimization,” http//:arXiv:0905.1643 (2009).
  16. K. Lee and Y. Bresler., “Admira: atomic decomposition for minimum rank approximation,” http//:arXiv:0905.0044 (2009).
  17. E. Candès and B. Recht., “Exact matrix completion via complex optimization,” Found. Comput. Math 9, 717–772 (2009).
    [Crossref]
  18. E. Candès, T. Strohmer, and V. Voroninsky., “PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming,” http//:arXiv:1109.4499 (2011).
  19. L. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015)
    [Crossref]
  20. M. Guizar-Sicairos and J. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
    [Crossref] [PubMed]
  21. P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
    [Crossref]
  22. J. Cesar da Silva and A. Menzel, “Elementary signals in ptychography,” Opt. Express 23, 33812–33821 (2015)
    [Crossref]
  23. R. Horstmeyer and et al., “Solving ptychography with a convex relaxation,” http//:arXiv:1412.1209 (2014)
  24. P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012)
    [Crossref] [PubMed]

2015 (2)

2014 (1)

P. Li, T.B. Edo, and J.M. Rodenburg, “Ptychographic inversion via Wigner distribution deconvolution: noise suppression and probe design,” Ultramicroscopy 147, 106–113 (2014).
[Crossref] [PubMed]

2013 (2)

A.M. Maiden, G.R. Morrison, B. Kaulich, A. Gianocelli, and J.M. Rodenburg, “Soft X-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013).
[Crossref] [PubMed]

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

2012 (2)

P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012)
[Crossref] [PubMed]

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
[Crossref]

2011 (1)

2010 (3)

J. Cai, E. Candès, and Z. Shen., “A singular value thresholding algorithm for matrix completion,” SIAM J. Optimiz. 20(4),1956–1982 (2010).
[Crossref]

E. Candès and Y. Plan., “Matrix completion with noise,” Proc. IEEE 98(6), 925–936 (2010).
[Crossref]

K. Toh and S. Yun., “An accelerated proximal gradient algorithm for nuclear norm regularized least square problems,” Pac. J. Optim. 6, 615–640 (2010).

2009 (1)

E. Candès and B. Recht., “Exact matrix completion via complex optimization,” Found. Comput. Math 9, 717–772 (2009).
[Crossref]

2008 (1)

1996 (1)

H.N. Chapman, “Phase-retrieval X-Ray microscopy by Wigner distribution deconvolution,” Ultramicroscopy 66, 153–172 (1996).
[Crossref]

1993 (1)

J.M. Rodenburg, B.C. McCallum, and P.D. Nellist, “Experimental tests on double resolution coherent imaging via STEM,” Ultramicroscopy 48, 304–314 (1993).
[Crossref]

1992 (2)

J.M. Rodenburg and R.H.T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Philos. T. R. Soc. Lond. 339, 521–553 (1992).
[Crossref]

B.C. McCallum and J.M. Rodenburg, “2-dimensional optical demonstration of Wigner phase retrieval microscopy in the STEM configuration,” Ultramicroscopy 45, 371–380 (1992).
[Crossref]

1989 (1)

R.H.T. Bates and J.M. Rodenburg, “Sub-Angstrom transmission microscopy: a Fourier transform algorithm for microdiffraction plane intensity information,” Ultramicroscopy 31, 303–307 (1989).
[Crossref]

Allain, M.

Bates, R.H.T.

J.M. Rodenburg and R.H.T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Philos. T. R. Soc. Lond. 339, 521–553 (1992).
[Crossref]

R.H.T. Bates and J.M. Rodenburg, “Sub-Angstrom transmission microscopy: a Fourier transform algorithm for microdiffraction plane intensity information,” Ultramicroscopy 31, 303–307 (1989).
[Crossref]

Cai, J.

J. Cai, E. Candès, and Z. Shen., “A singular value thresholding algorithm for matrix completion,” SIAM J. Optimiz. 20(4),1956–1982 (2010).
[Crossref]

Candès, E.

J. Cai, E. Candès, and Z. Shen., “A singular value thresholding algorithm for matrix completion,” SIAM J. Optimiz. 20(4),1956–1982 (2010).
[Crossref]

E. Candès and Y. Plan., “Matrix completion with noise,” Proc. IEEE 98(6), 925–936 (2010).
[Crossref]

E. Candès and B. Recht., “Exact matrix completion via complex optimization,” Found. Comput. Math 9, 717–772 (2009).
[Crossref]

Cesar da Silva, J.

Chamard, V.

Chapman, H.N.

H.N. Chapman, “Phase-retrieval X-Ray microscopy by Wigner distribution deconvolution,” Ultramicroscopy 66, 153–172 (1996).
[Crossref]

Chen, M.

Dong, J.

Edo, T.B.

P. Li, T.B. Edo, and J.M. Rodenburg, “Ptychographic inversion via Wigner distribution deconvolution: noise suppression and probe design,” Ultramicroscopy 147, 106–113 (2014).
[Crossref] [PubMed]

Fienup, J.

Gianocelli, A.

A.M. Maiden, G.R. Morrison, B. Kaulich, A. Gianocelli, and J.M. Rodenburg, “Soft X-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013).
[Crossref] [PubMed]

Godard, P.

Guizar-Sicairos, M.

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
[Crossref]

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

Horstmeyer, R.

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

Humphrey, M.J.

Kaulich, B.

A.M. Maiden, G.R. Morrison, B. Kaulich, A. Gianocelli, and J.M. Rodenburg, “Soft X-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013).
[Crossref] [PubMed]

Li, P.

P. Li, T.B. Edo, and J.M. Rodenburg, “Ptychographic inversion via Wigner distribution deconvolution: noise suppression and probe design,” Ultramicroscopy 147, 106–113 (2014).
[Crossref] [PubMed]

Maiden, A.M.

A.M. Maiden, G.R. Morrison, B. Kaulich, A. Gianocelli, and J.M. Rodenburg, “Soft X-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013).
[Crossref] [PubMed]

A.M. Maiden, M.J. Humphrey, and J.M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A 28, 604–612 (2011).
[Crossref]

McCallum, B.C.

J.M. Rodenburg, B.C. McCallum, and P.D. Nellist, “Experimental tests on double resolution coherent imaging via STEM,” Ultramicroscopy 48, 304–314 (1993).
[Crossref]

B.C. McCallum and J.M. Rodenburg, “2-dimensional optical demonstration of Wigner phase retrieval microscopy in the STEM configuration,” Ultramicroscopy 45, 371–380 (1992).
[Crossref]

Menzel, A.

Morrison, G.R.

A.M. Maiden, G.R. Morrison, B. Kaulich, A. Gianocelli, and J.M. Rodenburg, “Soft X-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013).
[Crossref] [PubMed]

Nellist, P.D.

J.M. Rodenburg, B.C. McCallum, and P.D. Nellist, “Experimental tests on double resolution coherent imaging via STEM,” Ultramicroscopy 48, 304–314 (1993).
[Crossref]

Plan., Y.

E. Candès and Y. Plan., “Matrix completion with noise,” Proc. IEEE 98(6), 925–936 (2010).
[Crossref]

Recht., B.

E. Candès and B. Recht., “Exact matrix completion via complex optimization,” Found. Comput. Math 9, 717–772 (2009).
[Crossref]

Rodenburg, J.

Rodenburg, J.M.

P. Li, T.B. Edo, and J.M. Rodenburg, “Ptychographic inversion via Wigner distribution deconvolution: noise suppression and probe design,” Ultramicroscopy 147, 106–113 (2014).
[Crossref] [PubMed]

A.M. Maiden, G.R. Morrison, B. Kaulich, A. Gianocelli, and J.M. Rodenburg, “Soft X-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013).
[Crossref] [PubMed]

A.M. Maiden, M.J. Humphrey, and J.M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A 28, 604–612 (2011).
[Crossref]

J.M. Rodenburg, B.C. McCallum, and P.D. Nellist, “Experimental tests on double resolution coherent imaging via STEM,” Ultramicroscopy 48, 304–314 (1993).
[Crossref]

B.C. McCallum and J.M. Rodenburg, “2-dimensional optical demonstration of Wigner phase retrieval microscopy in the STEM configuration,” Ultramicroscopy 45, 371–380 (1992).
[Crossref]

J.M. Rodenburg and R.H.T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Philos. T. R. Soc. Lond. 339, 521–553 (1992).
[Crossref]

R.H.T. Bates and J.M. Rodenburg, “Sub-Angstrom transmission microscopy: a Fourier transform algorithm for microdiffraction plane intensity information,” Ultramicroscopy 31, 303–307 (1989).
[Crossref]

Shen., Z.

J. Cai, E. Candès, and Z. Shen., “A singular value thresholding algorithm for matrix completion,” SIAM J. Optimiz. 20(4),1956–1982 (2010).
[Crossref]

Soltanolkotabi, M.

Tang, G.

Thibault, P.

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
[Crossref]

Tian, L.

Toh, K.

K. Toh and S. Yun., “An accelerated proximal gradient algorithm for nuclear norm regularized least square problems,” Pac. J. Optim. 6, 615–640 (2010).

Waller, L.

Yang, C.

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

Yeh, L.

Yun., S.

K. Toh and S. Yun., “An accelerated proximal gradient algorithm for nuclear norm regularized least square problems,” Pac. J. Optim. 6, 615–640 (2010).

Zheng, G.

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

Zhong, J.

Found. Comput. Math (1)

E. Candès and B. Recht., “Exact matrix completion via complex optimization,” Found. Comput. Math 9, 717–772 (2009).
[Crossref]

J. Opt. Soc. Am. A (1)

Nat. Commun. (1)

A.M. Maiden, G.R. Morrison, B. Kaulich, A. Gianocelli, and J.M. Rodenburg, “Soft X-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013).
[Crossref] [PubMed]

Nat. Photonics (1)

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

New J. Phys. (1)

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
[Crossref]

Opt. Express (4)

Pac. J. Optim. (1)

K. Toh and S. Yun., “An accelerated proximal gradient algorithm for nuclear norm regularized least square problems,” Pac. J. Optim. 6, 615–640 (2010).

Philos. T. R. Soc. Lond. (1)

J.M. Rodenburg and R.H.T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Philos. T. R. Soc. Lond. 339, 521–553 (1992).
[Crossref]

Proc. IEEE (1)

E. Candès and Y. Plan., “Matrix completion with noise,” Proc. IEEE 98(6), 925–936 (2010).
[Crossref]

SIAM J. Optimiz. (1)

J. Cai, E. Candès, and Z. Shen., “A singular value thresholding algorithm for matrix completion,” SIAM J. Optimiz. 20(4),1956–1982 (2010).
[Crossref]

Ultramicroscopy (5)

B.C. McCallum and J.M. Rodenburg, “2-dimensional optical demonstration of Wigner phase retrieval microscopy in the STEM configuration,” Ultramicroscopy 45, 371–380 (1992).
[Crossref]

H.N. Chapman, “Phase-retrieval X-Ray microscopy by Wigner distribution deconvolution,” Ultramicroscopy 66, 153–172 (1996).
[Crossref]

J.M. Rodenburg, B.C. McCallum, and P.D. Nellist, “Experimental tests on double resolution coherent imaging via STEM,” Ultramicroscopy 48, 304–314 (1993).
[Crossref]

P. Li, T.B. Edo, and J.M. Rodenburg, “Ptychographic inversion via Wigner distribution deconvolution: noise suppression and probe design,” Ultramicroscopy 147, 106–113 (2014).
[Crossref] [PubMed]

R.H.T. Bates and J.M. Rodenburg, “Sub-Angstrom transmission microscopy: a Fourier transform algorithm for microdiffraction plane intensity information,” Ultramicroscopy 31, 303–307 (1989).
[Crossref]

Other (6)

R. Horstmeyer and et al., “Solving ptychography with a convex relaxation,” http//:arXiv:1412.1209 (2014)

R.H. Keshavan, A. Montanari, and S. Oh., “Matrix completion from noisy entries,” http//:arXiv:0906.2027 (2009).

R.H. Keshavan, A. Montanari, and S. Oh., “Low-rank matrix completion with noisy observations: a quantitative comparison,” http//:arXiv:0910.0921 (2009).

S. Ma, D. Goldfarb, and L. Chen., “Fixed point and Bregman iterative methods for matrix rank minimization,” http//:arXiv:0905.1643 (2009).

K. Lee and Y. Bresler., “Admira: atomic decomposition for minimum rank approximation,” http//:arXiv:0905.0044 (2009).

E. Candès, T. Strohmer, and V. Voroninsky., “PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming,” http//:arXiv:1109.4499 (2011).

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Figures (6)

Fig. 1
Fig. 1 Experimental setup for WDD. A shifted probe is imaged onto the object plane and the resultant field p(xsi) · o(x) is propagated to a Fourier plane where its intensity is measured.
Fig. 2
Fig. 2 Effects of a probe with limited spatial extent and limited bandwidth on object reconstruction in WDD. (Left) Actual Wigner distribution of object. (Top Center) Wigner distribution of a finite-extent probe. (Top Right) Wigner distribution of a band-limited probe. (Bottom Center) Recovered Wigner distribution of object using finite-extent probe. (Bottom Right) Recovered Wigner distribution of object using band-limited probe.
Fig. 3
Fig. 3 Effects of a probe with limited spatial extent and limited bandwidth on mutual intensity reconstruction in WDD. (1) Wigner distribution of object: (a) actual (b) retrieved using probe with limited spatial extent, (c) retrieved using probe with limited bandwidth. (2) sheared mutual intensity of object, o*(x1)o(x1 + x): (a) actual (b) retrieved if probe has limited spatial extent, (c) retrieved if probe has limited bandwidth, (3) sheared mutual intensity of object’s Fourier Transform, O(u)O*(us′): (a) actual (b) retrieved if probe has limited spatial extent, (c) retrieved if probe has limited bandwidth,
Fig. 4
Fig. 4 Flow chart of denoised WDD with LRMC using modified SVT. Application of our algorithm to a simulated noisy 1-D dataset with a band-limited probe yields a denoised rank-1 mutual intensity reconstruction similar to rightmost image (matrix is simultaneously completed and denoised).
Fig. 5
Fig. 5 Original object’s modulus (left) and phase (right)
Fig. 6
Fig. 6 Results of WDD using: LRMC, projection method from [6], and PIE at different noise levels. From left-to-right, the columns represent: moduli (1–3) of reconstructed object using LRMC, “projection method”, ptychography, and phase (4–6) of reconstructed object using LRMC, “projection method”, ptychography. The rows represent the total photon counts available to the detector in a shot-noise limited imaging system.

Tables (1)

Tables Icon

Table 1 Mean Squared Error of object reconstructions from datasets with different noise levels over 100 trials.

Equations (14)

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

I out ( u , s ) = | p ( x s ) o ( x ) e j 2 π x u d x | 2
H ( x , s ) = p * ( t ) p ( t + x ) o * ( x 1 ) o ( x 1 + x ) e j 2 π ( x 1 t ) s d x 1 d t
H ( x , s ) = p * ( t ) p ( t + x ) e j 2 π t s d t 𝒲 probe ( x , s ) o * ( x 1 ) o ( x 1 + x ) e j 2 π x 1 s d x 1 𝒲 object ( x , s )
𝒲 object ( x , s ) = 𝒲 probe * ( x , s ) H ( x , s ) | 𝒲 probe ( x , s ) | 2 +
I ( u , s ) + 𝒩 ( u , s ) 2 DDFT H ( x , s ) + 𝒩 ˜ ( x , s ) / . 𝒲 p ( x , s ) 𝒲 o ( x , s ) + 𝒩 ˜ ( x , s ) 𝒲 p ( x , s )
M = U Σ V T
M ˜ i j = M i j + Z i j
X i j minimizerank ( X ) subjectto 𝒫 Ω ( X ) = 𝒫 Ω ( M )
𝒫 Ω ( M ) i j = { M i j , if ( i , j ) E 0 , otherwise
X i j minimize X * subjectto 𝒫 Ω ( X ) = 𝒫 Ω ( M )
X i j minimize X * subjectto 𝒫 Ω ( X M ˜ ) F δ
minimize μ X * + 1 2 𝒫 Ω ( X M ˜ ) F 2
X i j minimize 𝒫 Ω ( X M ˜ ) F subjecttorank ( X ) r
X i j minimize X M ˜ F subjecttorank ( X ) r

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