Abstract

The incremental gradient approaches, such as PIE and ePIE, are widely used in the field of ptychographic imaging due to their great flexibility and computational efficiency. Nevertheless, their stability and reconstruction quality may be significantly degraded when non-negligible noise is present in the image. Though this problem is often attributed to the non-convex nature of phase retrieval, we found the reason for this is more closely related to the choice of the step-size, which needs to be gradually diminishing for convergence even in the convex case. To this end, we introduce an adaptive step-size strategy that decreases the step-size whenever sufficient progress is not made. The synthetic and real experiments on Fourier ptychographic microscopy show that the adaptive step-size strategy significantly improves the stability and robustness of the reconstruction towards noise yet retains the fast initial convergence speed of PIE and ePIE. More importantly, the proposed approach is simple, nonparametric, and does not require any preknowledge about the noise statistics. The great performance and limited computational complexity make it a very attractive and promising technique for robust Fourier ptychographic microscopy under noisy conditions.

© 2016 Optical Society of America

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References

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2016 (4)

2015 (6)

2014 (4)

2013 (4)

2012 (4)

A. Maiden, M. Humphry, M. Sarahan, B. Kraus, and J. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[Crossref] [PubMed]

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

A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29, 1606–1614 (2012).
[Crossref]

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]

2011 (1)

D. P. Bertsekas, “Incremental gradient, subgradient, and proximal methods for convex optimization: A survey,” Optimization for Machine Learning 2010, 1–38 (2011).

2009 (2)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

2008 (2)

2007 (1)

J. Rodenburg, A. Hurst, and A. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy 107, 227–231 (2007).
[Crossref]

2006 (1)

2005 (1)

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

2004 (2)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phy. Lett. 85, 4795–4797 (2004).
[Crossref]

2003 (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[Crossref]

2002 (1)

2001 (1)

A. Nedic and D. P. Bertsekas, “Incremental subgradient methods for nondifferentiable optimization,” SIAM J. Optim. 12, 109–138 (2001).
[Crossref]

1993 (1)

R. Correa and C. Lemaréchal, “Convergence of some algorithms for convex minimization,” Math. Program. 62, 261–275 (1993).
[Crossref]

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–249 (1972).

Allain, M.

Ames, B.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New Journal of Physics 17, 053044 (2015).
[Crossref] [PubMed]

Batey, D. J.

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref] [PubMed]

Bauschke, H. H.

Bean, R.

Berenguer, F.

Bertsekas, D. P.

D. P. Bertsekas, “Incremental gradient, subgradient, and proximal methods for convex optimization: A survey,” Optimization for Machine Learning 2010, 1–38 (2011).

A. Nedic and D. P. Bertsekas, “Incremental subgradient methods for nondifferentiable optimization,” SIAM J. Optim. 12, 109–138 (2001).
[Crossref]

D. P. Bertsekas and D. P. Bertsekas, Nonlinear Programming (Athena Scientific, 1999), 2nd ed.

D. P. Bertsekas and D. P. Bertsekas, Nonlinear Programming (Athena Scientific, 1999), 2nd ed.

Bian, L.

Bunk, O.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

Chamard, V.

Chapman, H. N.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[Crossref]

Chen, B.

Chen, F.

Chen, M.

Chen, Q.

Chen, R. Y.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New Journal of Physics 17, 053044 (2015).
[Crossref] [PubMed]

Claus, D.

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref] [PubMed]

Claverley, J. D.

Combettes, P. L.

Correa, R.

R. Correa and C. Lemaréchal, “Convergence of some algorithms for convex minimization,” Math. Program. 62, 261–275 (1993).
[Crossref]

Cullis, A.

J. Rodenburg, A. Hurst, and A. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy 107, 227–231 (2007).
[Crossref]

Dai, Q.

Diaz, A.

Dierolf, M.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

Diniz, P. S.

P. S. Diniz, Adaptive filtering” (Springer, 1997).
[Crossref]

Dong, J.

Dong, S.

Faulkner, H. M. L.

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phy. Lett. 85, 4795–4797 (2004).
[Crossref]

Fienup, J. R.

Gao, P.

García, J.

García-Martínez, P.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–249 (1972).

Godard, P.

Godden, T. M.

Guizar-Sicairos, M.

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

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

Guo, K.

Hau-Riege, S. P.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[Crossref]

He, H.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[Crossref]

Horstmeyer, R.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New Journal of Physics 17, 053044 (2015).
[Crossref] [PubMed]

X. Ou, R. Horstmeyer, G. Zheng, and C. Yang, “High numerical aperture Fourier ptychography: principle, implementation and characterization,” Opt. Express 23, 3472–3491 (2015).
[Crossref] [PubMed]

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

Howells, M. R.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[Crossref]

Humphry, M.

A. Maiden, M. Humphry, M. Sarahan, B. Kraus, and J. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[Crossref] [PubMed]

Humphry, M. J.

Hurst, A.

J. Rodenburg, A. Hurst, and A. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy 107, 227–231 (2007).
[Crossref]

Kraus, B.

A. Maiden, M. Humphry, M. Sarahan, B. Kraus, and J. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[Crossref] [PubMed]

Lemaréchal, C.

R. Correa and C. Lemaréchal, “Convergence of some algorithms for convex minimization,” Math. Program. 62, 261–275 (1993).
[Crossref]

Li, X.

Liu, H.

Liu, Z.

Luke, D. R.

Maia, F.

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv preprint arXiv:1105.5628 (2011).

Maiden, A.

A. Maiden, M. Humphry, M. Sarahan, B. Kraus, and J. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[Crossref] [PubMed]

Maiden, A. M.

A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29, 1606–1614 (2012).
[Crossref]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

Marchesini, S.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[Crossref]

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv preprint arXiv:1105.5628 (2011).

Menzel, A.

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013). .
[Crossref] [PubMed]

F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. K. Robinson, and J. M. Rodenburg, “Translation position determination in ptychographic coherent diffraction imaging,” Opt. Express 21, 13592–13606 (2013).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

Mico, V.

Muñiz-Piniella, A.

Nanda, P.

Nedic, A.

A. Nedic and D. P. Bertsekas, “Incremental subgradient methods for nondifferentiable optimization,” SIAM J. Optim. 12, 109–138 (2001).
[Crossref]

Noy, A.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[Crossref]

Osten, W.

Ou, X.

Pedrini, G.

Peterson, I.

Pfeiffer, F.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

Qian, J.

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv preprint arXiv:1105.5628 (2011).

Ramchandran, K.

Robinson, I. K.

Rodenburg, J.

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]

A. Maiden, M. Humphry, M. Sarahan, B. Kraus, and J. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[Crossref] [PubMed]

J. Rodenburg, A. Hurst, and A. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy 107, 227–231 (2007).
[Crossref]

Rodenburg, J. M.

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref] [PubMed]

F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. K. Robinson, and J. M. Rodenburg, “Translation position determination in ptychographic coherent diffraction imaging,” Opt. Express 21, 13592–13606 (2013).
[Crossref] [PubMed]

A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29, 1606–1614 (2012).
[Crossref]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phy. Lett. 85, 4795–4797 (2004).
[Crossref]

Sarahan, M.

A. Maiden, M. Humphry, M. Sarahan, B. Kraus, and J. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[Crossref] [PubMed]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–249 (1972).

Schirotzek, A.

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv preprint arXiv:1105.5628 (2011).

Shiradkar, R.

Soltanolkotabi, M.

Spence, J. C. H.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[Crossref]

Sun, J.

Suo, J.

Tang, G.

Thibault, P.

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013). .
[Crossref] [PubMed]

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

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Supplementary Material (2)

NameDescription
» Visualization 1: MOV (5923 KB)      The evolution of the amplitude reconstructions of an USAF resolution target with an adaptive step-size.
» Visualization 2: MOV (1133 KB)      The evolution of the amplitude and phase reconstructions of a pathological section of human kidney with an adaptive step-size.

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Figures (11)

Fig. 1
Fig. 1

Simulated amplitude and phase images for FPM reconstruction. (a) Amplitude; (b) Phase; (c) Fourier spectrum; the red and blue circles indicate the captured subspectrums under orthogonal and oblique illuminations; (d)–(i) low-resolution raw images with different levels of Gaussian noise.

Fig. 2
Fig. 2

Comparison of amplitude (a) and phase (b) reconstruction accuracy versus intensity noise for the adaptive step-size and fixed step-sizes α=1, 0.5, and 0.05. (a1)–(a4) and (b1)–(b4) are recovered amplitudes and phases when the noise level is 30% and 40%, respectively.

Fig. 3
Fig. 3

Error trajectories for the adaptive step-size and fixed step-sizes α=1, 0.5, and 0.05. (a) Phase error curves under 40% Gaussian noise with last 10 iterations (shaded region) magnified in (b). Note each iteration comprises 225 sub-iterations carried on each low-resolution image.

Fig. 4
Fig. 4

Comparison of amplitude (a) and phase (b) reconstruction accuracy under Poisson noise for the adaptive step-size and fixed step-sizes α=1, 0.5, and 0.05.

Fig. 5
Fig. 5

Comparison of convergence speed and reconstruction accuracy of the adaptive step-size approach and four state-of-the-art FPM reconstruction algorithms under 50% Gaussian noise.

Fig. 6
Fig. 6

Comparison of convergence speed and reconstruction accuracy of the adaptive step-size approach and four state-of-the-art FPM reconstruction algorithms under 50% Poisson noise.

Fig. 7
Fig. 7

Experimental results of a resolution target. (a) The raw full FoV image. (b) and (c) are the corresponding magnified areas. (d)–(g) are the reconstruction results with fixed step-sizes α=1, 0.5, and 0.05 (each runs for 30 iterations) and the adaptive step-size (converges in 17 iterations). (h)–(k) shows the corresponding zoom-ins of the smallest features.

Fig. 8
Fig. 8

The progress of the error metric E for the fixed (α=1, 0.5, and 0.05) and adaptive step-sizes. Visualization 1 and images in the second row illustrate the evolution of the amplitude reconstruction of the adaptive step-size approach.

Fig. 9
Fig. 9

Comparison of reconstruction results and runtime of the adaptive step-size approach and four state-of-the-art FPM reconstruction algorithms.

Fig. 10
Fig. 10

Experimental results of a pathological section. (a) The full FoV image. (b–c) The corresponding region of interest. (d–e) The comparison of reconstructed amplitudes and phases of four state-of-the-arts and the adaptive step-size approach. Visualization 2 shows the evolution of amplitude and phase reconstructions with the adaptive step-size approach.

Fig. 11
Fig. 11

Comparison of convergence speed and reconstruction accuracy for joint object and pupil recovery under 50% Poisson noise. (a) Convergence curves for different fixed and the adaptive step-size approaches. (b) Convergence curves of adaptive step-size approaches with different initial values for pupil step-size. (c) The reconstructed amplitudes, phases, and pupil aberration functions with different choices of step-size.

Tables (1)

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Table 1 Runtime performance comparison of different algorithms

Equations (29)

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I i ( r ) = | 1 { P ( u ) O ( u u i ) } | 2 , i = 1 , 2 , , N
min O ( u ) ε = min O ( u ) i r | I i ( r ) | 1 { P ( u ) O ( u u i ) } | | 2
ε = i I i | F 1 P i O | 2 i I i | g i | 2
ε = i Ψ i P i O 2
Ψ i = Π m I i ( P i O )
ε = i [ P i * ( Ψ i P i O ) ]
O k + 1 = O k + α W ε ( O k ) = O i α W i [ P i * ( Ψ i k P i O k ) ]
W = i ( P i * P i ) 1
ε i = Ψ i P i O 2
ε i = P i * ( Ψ i P i O )
O i + 1 k = O i k + α k W i ε i ( O i k ) = O i k α k W i P i * ( Ψ i k P i O i k )
W i = Diag ( β m , i ( | P m , i | 2 + γ ) 1 )
β m , i = { | P m , i | / | P m , i | max PIE type | P m , i | 2 / | P m , i | max 2 ePIE type
O k + 1 = O k + α k i = 1 N W i ε i ( O i k ) = O k + α k W ( ε ( O k ) + e k ) ,
e k = i = 1 N ( ε i ( O k ) ε i ( O i k ) ) ,
α k = { α k 1 ( ε ( O k ) ε ( O k 1 ) ) ε ( O k 1 ) > η α k 1 2 otherwise
E = ε i I i 2 i I i | g i | 2 i I i 2
O k + 1 O W 1 2 O k O W 1 2 2 α k ( ε ( O k ) ε ( O ) ) + ( α k ) 2 C .
Q i + 1 k = Q i k + α k ξ i ( O i k ) .
ε ( O k ) ε ( O ) + 1 2 α k C + e ,
O k + 1 O W 1 2 O k O W 1 2 2 α k e ,
α k > 0 , lim k α k = 0 , and k = 0 α k = .
O k + 1 υ W 1 2 O k υ W 1 2 + α k ( α k C 2 δ ) O k υ W 1 2 α k δ
0 O k υ W 1 2 O k υ W 1 2 δ n = K k 1 α n .
α k > 0 , lim k α k = 0 , k = 0 α k = , and k = 0 ( α k ) 2 < .
O k + 1 υ W 1 2 O 0 υ W 1 2 2 n = 0 k α n ( ε ( O n ) ε ) + δ n = 0 k ( α n ) 2 C .
2 n = 0 k α n ( ε ( O n ) ε ) O 0 υ W 1 2 + n = 0 k ( α n ) 2 C < q < ,
O k + 1 O W 1 2 O k O W 1 2 + n = K k ( 2 α n ( ε ( O n ) ε ) + ( α n ) 2 C ) δ .
P i + 1 k = P i k β k W i O i * ( Ψ i k O i P i k )

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