Abstract

Iterative phase retrieval methods based on the Gerchberg–Saxton (GS) or Fienup algorithm typically show stagnation artifacts even after a large number of iterations. We introduce a complexity parameter ζ that can be computed directly from the Fourier magnitude data and provides a measure of fluctuations in the desired phase retrieval solution. It is observed that when initiated with a constant or a uniformly random phase map, the complexity of the Fienup solution containing stagnation artifacts stabilizes at a numerical value that is higher than ζ. We propose a modified Fienup algorithm that uses a controlled sparsity-enhancing step such that in every iteration the complexity of the resulting guess solution is explicitly made close to ζ. This approach, which we refer to as complexity-guided phase retrieval, is seen to provide an artifact-free phase retrieval solution within a few hundred iterations. Numerical illustrations are provided for both amplitude as well as phase objects with and without Poisson noise introduced in the Fourier intensity data. The complexity-guidance concept may potentially be combined with a variety of phase retrieval algorithms and can enable several practical applications.

© 2019 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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2018 (2)

T. Latychevskaia, “Iterative phase retrieval in coherent diffractive imaging: practical issues,” Appl. Opt. 25, 7187–7197 (2018).
[Crossref]

S. Rajora, M. Butola, and K. Khare, “Calculating numerical derivatives using Fourier transform: some pitfalls and how to avoid them,” Eur. J. Phys. 39, 065806 (2018).
[Crossref]

2015 (6)

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with applications to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

C. Guo, S. Liu, and J. T. Sheridan, “Iterative phase retrieval algorithms I: optimization,” Appl. Opt. 54, 4698–4708 (2015).
[Crossref]

C. Gaur, B. Mohan, and K. Khare, “Sparsity-assisted solution to the twin image problem in phase retrieval,” J. Opt. Soc. Am. A 32, 1922–1927 (2015).
[Crossref]

K. He, M. K. Sharma, and O. Cossairt, “High dynamic range coherent imaging with compressed sensing,” Opt. Express 23, 30904–30916 (2015).
[Crossref]

2014 (1)

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62, 928–938 (2014).
[Crossref]

2013 (1)

2012 (1)

2010 (2)

K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59, 1–99 (2010).
[Crossref]

H. N. Chapman and K. A. Nugent, “Coherent lensless x-ray imaging,” Nat. Photonics 4, 833–839 (2010).
[Crossref]

2008 (1)

J. Miao, T. Ishikawa, Q. Shen, and T. Earnest, “Extending x-ray crystallography to allow the imaging of noncrystalline materials, cells, and single protein complexes,” Annu. Rev. Phys. Chem. 59, 387–410 (2008).
[Crossref]

2007 (2)

C. Chen, J. Miao, C. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

J. A. Marozas, “Fourier transform-based continuous phase-plate design technique: a high-pass phase-plate design as an application for OMEGA and the National Ignition Facility,” J. Opt. Soc. Am. A 24, 74–83 (2007).
[Crossref]

2006 (2)

J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express 14, 498–508 (2006).
[Crossref]

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[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]

D. R. Luke, “Relaxed averaged alternating reflections for diffraction,” Inverse Prob. 21, 37–50 (2004).
[Crossref]

2003 (2)

V. Elser, “Random projections and optimization of an algorithm for phase retrieval,” J. Phys. A 36, 2995–3007 (2003).
[Crossref]

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Hybrid projection reflection method for phase retrieval,” J. Opt. Soc. Am. A 20, 1025–1034 (2003).
[Crossref]

1999 (1)

1997 (1)

1996 (1)

1986 (1)

1978 (1)

1972 (1)

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

1952 (1)

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

Bauschke, H. H.

Beck, A.

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62, 928–938 (2014).
[Crossref]

Brasher, J. D.

Butola, M.

S. Rajora, M. Butola, and K. Khare, “Calculating numerical derivatives using Fourier transform: some pitfalls and how to avoid them,” Eur. J. Phys. 39, 065806 (2018).
[Crossref]

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Candes, E. J.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with applications to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

H. N. Chapman and K. A. Nugent, “Coherent lensless x-ray imaging,” Nat. Photonics 4, 833–839 (2010).
[Crossref]

Chen, C.

C. Chen, J. Miao, C. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with applications to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Combettes, P. L.

Cossairt, O.

Dainty, J. C.

J. C. Dainty and J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, 1987).

Earnest, T.

J. Miao, T. Ishikawa, Q. Shen, and T. Earnest, “Extending x-ray crystallography to allow the imaging of noncrystalline materials, cells, and single protein complexes,” Annu. Rev. Phys. Chem. 59, 387–410 (2008).
[Crossref]

Eldar, Y. C.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with applications to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62, 928–938 (2014).
[Crossref]

Elser, V.

V. Elser, “Random projections and optimization of an algorithm for phase retrieval,” J. Phys. A 36, 2995–3007 (2003).
[Crossref]

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]

Feinup, J. R.

Fienup, J. R.

Gaur, C.

Gerchberg, R. W.

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

Guizar-Sicairos, M.

Guo, C.

He, K.

Ishikawa, T.

J. Miao, T. Ishikawa, Q. Shen, and T. Earnest, “Extending x-ray crystallography to allow the imaging of noncrystalline materials, cells, and single protein complexes,” Annu. Rev. Phys. Chem. 59, 387–410 (2008).
[Crossref]

Ivankovski, Y.

Johnson, E. G.

Khare, K.

S. Rajora, M. Butola, and K. Khare, “Calculating numerical derivatives using Fourier transform: some pitfalls and how to avoid them,” Eur. J. Phys. 39, 065806 (2018).
[Crossref]

C. Gaur, B. Mohan, and K. Khare, “Sparsity-assisted solution to the twin image problem in phase retrieval,” J. Opt. Soc. Am. A 32, 1922–1927 (2015).
[Crossref]

Latychevskaia, T.

T. Latychevskaia, “Iterative phase retrieval in coherent diffractive imaging: practical issues,” Appl. Opt. 25, 7187–7197 (2018).
[Crossref]

Lee, T. K.

C. Chen, J. Miao, C. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

Li, X.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

Liu, S.

Luke, D. R.

Marozas, J. A.

Mendlovic, D.

Miao, J.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with applications to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

J. Miao, T. Ishikawa, Q. Shen, and T. Earnest, “Extending x-ray crystallography to allow the imaging of noncrystalline materials, cells, and single protein complexes,” Annu. Rev. Phys. Chem. 59, 387–410 (2008).
[Crossref]

C. Chen, J. Miao, C. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

Mohan, B.

Nugent, K. A.

H. N. Chapman and K. A. Nugent, “Coherent lensless x-ray imaging,” Nat. Photonics 4, 833–839 (2010).
[Crossref]

K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59, 1–99 (2010).
[Crossref]

Rajora, S.

S. Rajora, M. Butola, and K. Khare, “Calculating numerical derivatives using Fourier transform: some pitfalls and how to avoid them,” Eur. J. Phys. 39, 065806 (2018).
[Crossref]

Rodenburg, J. M.

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]

Romberg, J.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for determination of the 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. 5, 843 (1952).
[Crossref]

Segev, M.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with applications to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Sharma, M. K.

Shechtman, Y.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with applications to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62, 928–938 (2014).
[Crossref]

Shen, Q.

J. Miao, T. Ishikawa, Q. Shen, and T. Earnest, “Extending x-ray crystallography to allow the imaging of noncrystalline materials, cells, and single protein complexes,” Annu. Rev. Phys. Chem. 59, 387–410 (2008).
[Crossref]

Sheridan, J. T.

Soltanolkotabi, M.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
[Crossref]

Tao, T.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Wackerman, C. C.

Wang, C.

C. Chen, J. Miao, C. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

Acta Crystallogr. (1)

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

Adv. Phys. (1)

K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59, 1–99 (2010).
[Crossref]

Annu. Rev. Phys. Chem. (1)

J. Miao, T. Ishikawa, Q. Shen, and T. Earnest, “Extending x-ray crystallography to allow the imaging of noncrystalline materials, cells, and single protein complexes,” Annu. Rev. Phys. Chem. 59, 387–410 (2008).
[Crossref]

Appl. Comput. Harmon. Anal. (1)

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
[Crossref]

Appl. Opt. (5)

Eur. J. Phys. (1)

S. Rajora, M. Butola, and K. Khare, “Calculating numerical derivatives using Fourier transform: some pitfalls and how to avoid them,” Eur. J. Phys. 39, 065806 (2018).
[Crossref]

IEEE Signal Process. Mag. (1)

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with applications to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

IEEE Trans. Inf. Theory (2)

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

IEEE Trans. Signal Process. (1)

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62, 928–938 (2014).
[Crossref]

Inverse Prob. (1)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction,” Inverse Prob. 21, 37–50 (2004).
[Crossref]

J. Opt. Soc. Am. A (5)

J. Phys. A (1)

V. Elser, “Random projections and optimization of an algorithm for phase retrieval,” J. Phys. A 36, 2995–3007 (2003).
[Crossref]

Nat. Photonics (1)

H. N. Chapman and K. A. Nugent, “Coherent lensless x-ray imaging,” Nat. Photonics 4, 833–839 (2010).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Optik (1)

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

Phys. Rev. B (1)

C. Chen, J. Miao, C. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

Phys. Rev. Lett. (1)

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]

Other (2)

P. Ferraro, A. Wax, and Z. Zalevsky, eds., Coherent Light Microscopy: Imaging and Quantitative Phase Analysis (Springer, 2011).

J. C. Dainty and J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, 1987).

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Figures (8)

Fig. 1.
Fig. 1. Test images g(x,y) used for illustration: (a) cameraman phase object with phase in the range [0,2π/3], (b) Fourier transform magnitude corresponding to the phase object, (c) penguin amplitude object with amplitude values distributed in the range [0,255], (d) Fourier transform magnitude corresponding to the amplitude object. Both the objects are defined over a support window of size 280×280, which is embedded in a computational window of size 600×600 in order to make sure that the Fourier transform intensity is adequately sampled. The Fourier magnitude is shown as |G(fx,fy)|0.25 to suit the display.
Fig. 2.
Fig. 2. Reconstruction of the cameraman phase object shown in Fig. 1(a) and penguin amplitude object shown in Fig. 1(c) after 500 iterations of the HIO algorithm. (a), (c) A constant phase map as the initial guess and (b), (d) a random phase map as the initial guess.
Fig. 3.
Fig. 3. Behavior of complexity of the solution as a function of number of HIO iterations for (a) the cameraman phase object and (b) the penguin amplitude object. The red and blue curves represent complexity values for random and constant initial guesses, respectively. The desired complexity ζ obtained from Fourier magnitude data is shown as a constant black colored solid line. The inset shows the behavior of the complexity curves for the first 20 iterations.
Fig. 4.
Fig. 4. Reconstruction of the cameraman phase object when initiated with a constant guess after 500 iterations of the following cases: (a) HIO update alone, (b) HIO update accompanied by single TV-reduction step (TV1) in each iteration, (c) HIO update accompanied by 20 TV-reduction steps (TV20) in each iteration, and (d) the CGPR algorithm. The insets in (c) and (d) clearly show that the features of the cameraman phase object are over-smoothened by TV20, whereas these features are recovered correctly by the CGPR algorithm, which maintains the solution complexity near the desired value ζ.
Fig. 5.
Fig. 5. (a) Variation of the complexity parameter versus iteration number for four different cases (i.e., HIO, TV1, TV20, and CGPR) for the cameraman phase object; (b) the corresponding error metrics plotted against the number of iterations. The complexity plots show that as the solution progresses, its complexity remains highest for HIO alone, while with TV1 and TV20, the complexity value remains above and below ζ, respectively. The error plots indicate that in the case of CGPR, where complexity remains close to the desired complexity ζ, shows the best error performance among all four methodologies.
Fig. 6.
Fig. 6. Reconstruction of the penguin amplitude object when initiated with a constant guess after 500 iterations of the following cases: (a) HIO update alone, (b) HIO update accompanied by single TV-reduction step (TV1) in each iteration, (c) HIO update accompanied by 20 TV-reduction steps (TV20), and (d) the CGPR algorithm. The insets in (c) and (d) clearly show that the features of the penguin are over-smoothened by TV20, whereas these features are recovered correctly by the CGPR algorithm which maintains the solution complexity near the desired value ζ.
Fig. 7.
Fig. 7. (a) Variation of the complexity parameter versus iteration number for four different cases (i.e., HIO, TV1, TV20, and CGPR) for the penguin amplitude object, (b) the corresponding error metrics plotted against the number of iterations. The complexity plots show that as the solution progresses, its complexity remains highest for HIO alone, while with TV1 and TV20, the complexity value remains above and below ζ, respectively. The error plots indicate that in the case in CGPR, where complexity remains close to the desired complexity ζ, shows the best error performance among all four methodologies.
Fig. 8.
Fig. 8. Phase reconstruction for the Fourier intensity data with Poisson noise corresponding to the light level of 103photons/pixels for the cameraman phase object after 500 iterations of (a) HIO and (b) CGPR. The corresponding reconstruction results for the penguin amplitude object are shown in (c), (d) respectively.

Tables (1)

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Table 1. Error Metric [Eq. (15)] Performance of HIO Alone and CGPR for Fourier Intensity Data with Poisson Noise After 500 Iterationsa

Equations (15)

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ζ=m=all pixels(|xgm|2+|ygm|2),
(|xg|2+|yg|2)dxdy=(|i2πfxG|2+|i2πfyG|2)dfxdfy.
xg(x,y)=g(x+Δx,y)g(xΔx,y)2Δx.
F{xg(x,y)}=G(fx,fy)exp(i2πfxΔx)exp(i2πfxΔx)2Δx,
=isin(2πfxΔx)ΔxG(fx,fy).
ζ=n=all pixels[sin2(2πfxnΔx)Δx2+sin2(2πfynΔy)Δy2]|G(fxn,fyn)|2,
TV(g,g*)=m=all pixels|xgm|2+|ygm|2.
g*TV(g,g*)=12·(g|g|).
u^=g*TV(g,g*)g*TV(g,g*)2.
G^n=|G|exp(i2πϕn(fx,fy)).
g^n+1=g^n,(x,y)C,
=gnβg^n,(x,y)C,
g^n+1k+1=g^n+1ktg^n+1k2[u^]g=g^n+1k.
E2=min(E2(gn(x,y)),E2(gn*(x,y))),
E2(gn(x,y))=|gn|2+|g|22corr(gn,g)|g|2.

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