Abstract

The use of a spatial light modulator for implementing a digital phase-shifting (PS) point diffraction interferometer (PDI) allows tunability in fringe spacing and in achieving PS without the need for mechanically moving parts. However, a small amount of detector or scatter noise could affect the accuracy of wavefront sensing. Here, a novel method of wavefront reconstruction incorporating a virtual Hartmann-Shack (HS) wavefront sensor is proposed that allows easy tuning of several wavefront sensor parameters. The proposed method was tested and compared with a Fourier unwrapping method implemented on a digital PS PDI. The rewrapping of the Fourier reconstructed wavefronts resulted in phase maps that matched well the original wrapped phase and the performance was found to be more stable and accurate than conventional methods. Through simulation studies, the superiority of the proposed virtual HS phase unwrapping method is shown in comparison with the Fourier unwrapping method in the presence of noise. Further, combining the two methods could improve accuracy when the signal-to-noise ratio is sufficiently high.

© 2015 Optical Society of America

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References

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2015 (1)

A. R. Jewel, V. Akondi, and B. Vohnsen, “Optimization of sensing parameters for a confocal signal-based wave-front corrector in microscopy,” J. Mod. Opt. 62(10), 786–792 (2015).
[Crossref]

2014 (6)

R. M. Basavaraju, V. Akondi, S. J. Weddell, and R. P. Budihal, “Myopic aberrations: Simulation based comparison of curvature and Hartmann Shack wavefront sensors,” Opt. Commun. 312, 23–30 (2014).
[Crossref]

V. Akondi, S. Castillo, and B. Vohnsen, “Multi-faceted digital pyramid wavefront sensor,” Opt. Commun. 323, 77–86 (2014).
[Crossref]

V. Akondi, A. R. Jewel, and B. Vohnsen, “Closed-loop adaptive optics using a spatial light modulator for sensing and compensating of optical aberrations in ophthalmic applications,” J. Biomed. Opt. 19(9), 096014 (2014).
[Crossref]

S. Heshmat, S. Tomioka, and S. Nishiyama, “Performance evaluation of phase unwrapping algorithms for noisy phase measurements,” Int. J. Optomechatronics,  8(4), 260–274 (2014).
[Crossref]

V. Akondi, A. R. Jewel, and B. Vohnsen, “Digital phase-shifting point diffraction interferometer,” Opt. Lett. 39(6), 1641–1644 (2014).
[Crossref] [PubMed]

I. Iglesias, “Phase estimation from digital holograms without unwrapping,” Opt. Express 22(18), 21340–21346 (2014).
[Crossref] [PubMed]

2013 (5)

V. Akondi, S. Castillo, and B. Vohnsen, “Digital pyramid wavefront sensor with tunable modulation,” Opt. Express 21(15), 18261–18272 (2013).
[Crossref] [PubMed]

C. Falldorf, C. v. Kopylow, and R. B. Bergmann, “Wave field sensing by means of computational shear interferometry,” J. Opt. Soc. Am. A 30(10), 1905–1912 (2013).
[Crossref]

V. Akondi and B. Vohnsen, “Myopic aberrations: impact of centroiding noise in Hartmann Shack wavefront sensing,” Ophthalmic Physiol. Opt. 33(4), 434–443 (2013).
[Crossref] [PubMed]

F. Bai, X. Wang, K. Huang, N. Gu, S. Gan, and F. Tian, “Analysis of spatial resolution and pinhole size for single-shot point-diffraction interferometer using in closed-loop adaptive optics,” Opt. Commun. 297, 27–31 (2013).
[Crossref]

A. R. Jewel, V. Akondi, and B. Vohnsen, “A direct comparison between a MEMS deformable mirror and a liquid crystal spatial light modulator in signal-based wavefront sensing,” J. Eur. Opt. Soc. Rapid Pub. 8, 13073 (2013).
[Crossref]

2010 (3)

2003 (2)

2001 (1)

1999 (1)

1998 (1)

J-S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36(5), 1456–1465 (1998).
[Crossref]

1996 (2)

1995 (1)

1994 (2)

J. A. Quiroga and E. Bernabeu, “Phase-unwrapping algorithm for noisy phase-map processing,” Appl. Opt. 33(29), 6725–6731 (1994).
[Crossref] [PubMed]

M. D. Pritt and J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens 32(3), 706–708 (1994).
[Crossref]

1993 (1)

1989 (1)

1988 (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

1987 (2)

1984 (1)

1983 (1)

1982 (1)

C. C. Paige and M. A. Saunders, “LSQR: An algorithm for sparse linear equations And sparse least squares,” ACM Trans. Math. Soft. 8, 43–71 (1982).
[Crossref]

1980 (1)

1975 (1)

R. N. Smartt and W. H. Steel, “Theory and application of point-diffraction interferometers,” J. Appl. Phys. 14(141), 351–356 (1975).
[Crossref]

Acosta, E.

Agour, M.

Ainsworth, T. L.

J-S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36(5), 1456–1465 (1998).
[Crossref]

Akondi, V.

A. R. Jewel, V. Akondi, and B. Vohnsen, “Optimization of sensing parameters for a confocal signal-based wave-front corrector in microscopy,” J. Mod. Opt. 62(10), 786–792 (2015).
[Crossref]

V. Akondi, S. Castillo, and B. Vohnsen, “Multi-faceted digital pyramid wavefront sensor,” Opt. Commun. 323, 77–86 (2014).
[Crossref]

R. M. Basavaraju, V. Akondi, S. J. Weddell, and R. P. Budihal, “Myopic aberrations: Simulation based comparison of curvature and Hartmann Shack wavefront sensors,” Opt. Commun. 312, 23–30 (2014).
[Crossref]

V. Akondi, A. R. Jewel, and B. Vohnsen, “Digital phase-shifting point diffraction interferometer,” Opt. Lett. 39(6), 1641–1644 (2014).
[Crossref] [PubMed]

V. Akondi, A. R. Jewel, and B. Vohnsen, “Closed-loop adaptive optics using a spatial light modulator for sensing and compensating of optical aberrations in ophthalmic applications,” J. Biomed. Opt. 19(9), 096014 (2014).
[Crossref]

V. Akondi, S. Castillo, and B. Vohnsen, “Digital pyramid wavefront sensor with tunable modulation,” Opt. Express 21(15), 18261–18272 (2013).
[Crossref] [PubMed]

A. R. Jewel, V. Akondi, and B. Vohnsen, “A direct comparison between a MEMS deformable mirror and a liquid crystal spatial light modulator in signal-based wavefront sensing,” J. Eur. Opt. Soc. Rapid Pub. 8, 13073 (2013).
[Crossref]

V. Akondi and B. Vohnsen, “Myopic aberrations: impact of centroiding noise in Hartmann Shack wavefront sensing,” Ophthalmic Physiol. Opt. 33(4), 434–443 (2013).
[Crossref] [PubMed]

V. Akondi, R. M. Basavaraju, and R. P. Budihal, “Centroid detection by Gaussian pattern matching in adaptive optics,” Int. J. Comp. 26(1), 30–35 (2010).

Arines, J

Artal, P.

Asakura, T.

Bai, F.

F. Bai, X. Wang, K. Huang, N. Gu, S. Gan, and F. Tian, “Analysis of spatial resolution and pinhole size for single-shot point-diffraction interferometer using in closed-loop adaptive optics,” Opt. Commun. 297, 27–31 (2013).
[Crossref]

Baranova, N. B.

Basavaraju, R. M.

R. M. Basavaraju, V. Akondi, S. J. Weddell, and R. P. Budihal, “Myopic aberrations: Simulation based comparison of curvature and Hartmann Shack wavefront sensors,” Opt. Commun. 312, 23–30 (2014).
[Crossref]

V. Akondi, R. M. Basavaraju, and R. P. Budihal, “Centroid detection by Gaussian pattern matching in adaptive optics,” Int. J. Comp. 26(1), 30–35 (2010).

Bergmann, R. B.

Bernabeu, E.

Budihal, R. P.

R. M. Basavaraju, V. Akondi, S. J. Weddell, and R. P. Budihal, “Myopic aberrations: Simulation based comparison of curvature and Hartmann Shack wavefront sensors,” Opt. Commun. 312, 23–30 (2014).
[Crossref]

V. Akondi, R. M. Basavaraju, and R. P. Budihal, “Centroid detection by Gaussian pattern matching in adaptive optics,” Int. J. Comp. 26(1), 30–35 (2010).

Bueno, J. M.

Burton, D. R.

Castillo, S.

V. Akondi, S. Castillo, and B. Vohnsen, “Multi-faceted digital pyramid wavefront sensor,” Opt. Commun. 323, 77–86 (2014).
[Crossref]

V. Akondi, S. Castillo, and B. Vohnsen, “Digital pyramid wavefront sensor with tunable modulation,” Opt. Express 21(15), 18261–18272 (2013).
[Crossref] [PubMed]

Clegg, D. B.

Creath, K.

Cusack, R.

Elster, C.

Falldorf, C.

Gan, S.

F. Bai, X. Wang, K. Huang, N. Gu, S. Gan, and F. Tian, “Analysis of spatial resolution and pinhole size for single-shot point-diffraction interferometer using in closed-loop adaptive optics,” Opt. Commun. 297, 27–31 (2013).
[Crossref]

Goldrein, H. T.

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

Grunes, M. R.

J-S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36(5), 1456–1465 (1998).
[Crossref]

Gu, N.

F. Bai, X. Wang, K. Huang, N. Gu, S. Gan, and F. Tian, “Analysis of spatial resolution and pinhole size for single-shot point-diffraction interferometer using in closed-loop adaptive optics,” Opt. Commun. 297, 27–31 (2013).
[Crossref]

Hariharan, P.

Herráez, M. A.

Heshmat, S.

S. Heshmat, S. Tomioka, and S. Nishiyama, “Performance evaluation of phase unwrapping algorithms for noisy phase measurements,” Int. J. Optomechatronics,  8(4), 260–274 (2014).
[Crossref]

Huang, K.

F. Bai, X. Wang, K. Huang, N. Gu, S. Gan, and F. Tian, “Analysis of spatial resolution and pinhole size for single-shot point-diffraction interferometer using in closed-loop adaptive optics,” Opt. Commun. 297, 27–31 (2013).
[Crossref]

Huntley, J. M.

Iglesias, I.

Jewel, A. R.

A. R. Jewel, V. Akondi, and B. Vohnsen, “Optimization of sensing parameters for a confocal signal-based wave-front corrector in microscopy,” J. Mod. Opt. 62(10), 786–792 (2015).
[Crossref]

V. Akondi, A. R. Jewel, and B. Vohnsen, “Closed-loop adaptive optics using a spatial light modulator for sensing and compensating of optical aberrations in ophthalmic applications,” J. Biomed. Opt. 19(9), 096014 (2014).
[Crossref]

V. Akondi, A. R. Jewel, and B. Vohnsen, “Digital phase-shifting point diffraction interferometer,” Opt. Lett. 39(6), 1641–1644 (2014).
[Crossref] [PubMed]

A. R. Jewel, V. Akondi, and B. Vohnsen, “A direct comparison between a MEMS deformable mirror and a liquid crystal spatial light modulator in signal-based wavefront sensing,” J. Eur. Opt. Soc. Rapid Pub. 8, 13073 (2013).
[Crossref]

Jüptner, W.

U. Schnars, C. Falldorf, J. Watson, and W. Jüptner, Digital Holography and Wavefront Sensing (Springer, 2015).

Kadono, H.

Kopylow, C. v.

Kwon, O. Y.

Lalor, M. J.

Lee, J-S.

J-S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36(5), 1456–1465 (1998).
[Crossref]

Mamaev, A. V.

Mercer, C. R.

Nishiyama, S.

S. Heshmat, S. Tomioka, and S. Nishiyama, “Performance evaluation of phase unwrapping algorithms for noisy phase measurements,” Int. J. Optomechatronics,  8(4), 260–274 (2014).
[Crossref]

Paige, C. C.

C. C. Paige and M. A. Saunders, “LSQR: An algorithm for sparse linear equations And sparse least squares,” ACM Trans. Math. Soft. 8, 43–71 (1982).
[Crossref]

Papathanassiou, K. P.

J-S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36(5), 1456–1465 (1998).
[Crossref]

Pilipetsky, N. F.

Pritt, M. D.

M. D. Pritt and J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens 32(3), 706–708 (1994).
[Crossref]

Quiroga, J. A.

Reigber, A.

J-S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36(5), 1456–1465 (1998).
[Crossref]

Saldner, H.

Saunders, M. A.

C. C. Paige and M. A. Saunders, “LSQR: An algorithm for sparse linear equations And sparse least squares,” ACM Trans. Math. Soft. 8, 43–71 (1982).
[Crossref]

Schnars, U.

U. Schnars, C. Falldorf, J. Watson, and W. Jüptner, Digital Holography and Wavefront Sensing (Springer, 2015).

Schwarz, C.

Shipman, J. S.

M. D. Pritt and J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens 32(3), 706–708 (1994).
[Crossref]

Shkunov, V. V.

Smartt, R. N.

R. N. Smartt and W. H. Steel, “Theory and application of point-diffraction interferometers,” J. Appl. Phys. 14(141), 351–356 (1975).
[Crossref]

Southwell, W. H.

Steel, W. H.

R. N. Smartt and W. H. Steel, “Theory and application of point-diffraction interferometers,” J. Appl. Phys. 14(141), 351–356 (1975).
[Crossref]

Takai, N.

Tian, F.

F. Bai, X. Wang, K. Huang, N. Gu, S. Gan, and F. Tian, “Analysis of spatial resolution and pinhole size for single-shot point-diffraction interferometer using in closed-loop adaptive optics,” Opt. Commun. 297, 27–31 (2013).
[Crossref]

Tomioka, S.

S. Heshmat, S. Tomioka, and S. Nishiyama, “Performance evaluation of phase unwrapping algorithms for noisy phase measurements,” Int. J. Optomechatronics,  8(4), 260–274 (2014).
[Crossref]

Vohnsen, B.

A. R. Jewel, V. Akondi, and B. Vohnsen, “Optimization of sensing parameters for a confocal signal-based wave-front corrector in microscopy,” J. Mod. Opt. 62(10), 786–792 (2015).
[Crossref]

V. Akondi, S. Castillo, and B. Vohnsen, “Multi-faceted digital pyramid wavefront sensor,” Opt. Commun. 323, 77–86 (2014).
[Crossref]

V. Akondi, A. R. Jewel, and B. Vohnsen, “Digital phase-shifting point diffraction interferometer,” Opt. Lett. 39(6), 1641–1644 (2014).
[Crossref] [PubMed]

V. Akondi, A. R. Jewel, and B. Vohnsen, “Closed-loop adaptive optics using a spatial light modulator for sensing and compensating of optical aberrations in ophthalmic applications,” J. Biomed. Opt. 19(9), 096014 (2014).
[Crossref]

V. Akondi, S. Castillo, and B. Vohnsen, “Digital pyramid wavefront sensor with tunable modulation,” Opt. Express 21(15), 18261–18272 (2013).
[Crossref] [PubMed]

A. R. Jewel, V. Akondi, and B. Vohnsen, “A direct comparison between a MEMS deformable mirror and a liquid crystal spatial light modulator in signal-based wavefront sensing,” J. Eur. Opt. Soc. Rapid Pub. 8, 13073 (2013).
[Crossref]

V. Akondi and B. Vohnsen, “Myopic aberrations: impact of centroiding noise in Hartmann Shack wavefront sensing,” Ophthalmic Physiol. Opt. 33(4), 434–443 (2013).
[Crossref] [PubMed]

Volkov, V. V.

Wang, X.

F. Bai, X. Wang, K. Huang, N. Gu, S. Gan, and F. Tian, “Analysis of spatial resolution and pinhole size for single-shot point-diffraction interferometer using in closed-loop adaptive optics,” Opt. Commun. 297, 27–31 (2013).
[Crossref]

Watson, J.

U. Schnars, C. Falldorf, J. Watson, and W. Jüptner, Digital Holography and Wavefront Sensing (Springer, 2015).

Weddell, S. J.

R. M. Basavaraju, V. Akondi, S. J. Weddell, and R. P. Budihal, “Myopic aberrations: Simulation based comparison of curvature and Hartmann Shack wavefront sensors,” Opt. Commun. 312, 23–30 (2014).
[Crossref]

Weingärtner, I.

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

Zel’dovich, B. Y.

Zhu, Y.

ACM Trans. Math. Soft. (1)

C. C. Paige and M. A. Saunders, “LSQR: An algorithm for sparse linear equations And sparse least squares,” ACM Trans. Math. Soft. 8, 43–71 (1982).
[Crossref]

Appl. Opt. (13)

J. M. Huntley, “Three-dimensional noise-immune phase unwrapping algorithm,” Appl. Opt. 40(23), 3901–3908 (2001).
[Crossref]

J Arines, “Least-squares modal estimation of wrapped phases: application to phase unwrapping,” Appl. Opt. 42(17), 3373–3378 (2003).
[Crossref] [PubMed]

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Figures (7)

Fig. 1
Fig. 1 Illustration of the digital PS PDI. All lenses (L1–4) are achomats. BS1,2 are pellicle beam splitters.
Fig. 2
Fig. 2 (a) Measurements of induced wave aberrations (from top to bottom: defocus, astigmatism, coma, secondary astigmatism and a combination of defocus and astigmatism) using a commercial Hartmann-Shack instrument; (b) Wrapped phase calculated using Eq. (2) from the interferograms recorded by the digital PS PDI; (c) Reconstructed wavefronts estimated by the virtual HS; (d) Reconstructed wavefront in (c) decomposed into first 21 Zernike polynomials excluding piston and the two tilt terms; (e) Rewrapped phase obtained by wrapping (c); (f) Reconstructed wavefronts estimated by Fourier unwrapping; (g) Reconstructed wavefront in (f) decomposed into first 21 Zernike polynomials excluding piston and the two tilt terms; (h) Rewrapped phase obtained by wrapping (f). The wavefront maps in (a), (c), (d), (f) and (g) are in μm and the phase maps in (b), (e) and (h) are in radians.
Fig. 3
Fig. 3 An observed increasing trend of the Strehl ratio while increasing wavefront slope sampling with the virtual HS in the case of (a) defocus, (b) secondary astigmatism and (c) for a combination of defocus and astigmatism. Here, the Strehl ratio was evaluated from the residual wavefront error, φ.
Fig. 4
Fig. 4 Comparison of the HS measured Zernike coefficients with those estimated by the digital PS PDI using virtual HS and Fourier unwrapping methods.
Fig. 5
Fig. 5 Simulations: (a) Induced wave aberrations (from top to bottom: defocus, astigmatism, coma, secondary astigmatism and a combination of defocus and astigmatism); (b) Wrapped phase calculated using Eq. (2) from the interferograms determined by Eq. (11); (c) Reconstructed wavefronts estimated by the virtual HS; (d) Reconstructed wavefront in (c) decomposed into first 21 Zernike polynomials excluding piston and the two tilt terms; (e) Rewrapped phase obtained by wrapping (c); (f) Reconstructed wavefronts estimated by Fourier unwrapping; (g) Reconstructed wavefront in (f) decomposed into first 21 Zernike polynomials excluding piston and the two tilt terms; (h) Rewrapped phase obtained by wrapping (f). The wavefront maps in (a), (c), (d), (f) and (g) are in μm and the phase maps in (b), (e) and (h) are in radians.
Fig. 6
Fig. 6 Simulations: On effect of wavefront sampling in a digital PS-PDI + virtual HS unwrapping. Here, S = 5 and S = 10 are compared. The error bars represent the standard deviation for 100 randomly generated wavefronts. Strehl ratio and RMS wavefront error were evaluated from the residual wavefront error.
Fig. 7
Fig. 7 Simulations: Evaluation of virtual HS (VHS) and Fourier unwrapping methods (a) without and (b) with noise (SNR is shown in decibels). Here, the Strehl ratio was evaluated from the residual wavefront error.

Equations (12)

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W ( x , y ) = j a j Z j ( x , y )
ϕ w ( x , y ) = tan 1 [ 2 I 2 ( x , y ) I 3 ( x , y ) I 1 ( x , y ) I 3 ( x , y ) I 1 ( x , y ) ]
J n = | F [ p ( ξ , η ) e i ϕ n w ] | 2
L ( f ) = Δ f x ( x , y ) Δ ϕ x ( x , y ) 2 + Δ f y ( x , y ) Δ ϕ y ( x , y ) 2 + γ f ( x , y ) 2 .
L ( f ) = ( f h x ) ( x , y ) Δ ϕ x ( x , y ) 2 , ( f h y ) ( x , y ) Δ ϕ y ( x , y ) 2 , γ f ( x , y ) 2 .
L ( f ^ ) = f ^ ( v , η ) h ^ x ( v , η ) Δ ϕ x ^ ( v , η ) 2 + f ^ ( v , η ) h ^ y ( v , η ) Δ ϕ y ^ ( v , η ) 2 + γ f ^ ( v , η ) 2
L f ^ = L f ^ R + i L f ^ I = 2 h ^ x * ( f ^ h ^ x Δ ϕ x ^ ) + 2 h ^ y * ( f ^ h ^ y Δ ϕ y ^ ) + 2 γ f ^ = 0 ,
f ^ ( v , η ) = Δ ϕ x ^ ( v , η ) h ^ x * ( v , η ) + Δ ϕ y ^ ( v , η ) h ^ y * ( v , η ) | h ^ x ( v , η ) | 2 + | h ^ y ( v , η ) | 2 + γ .
ϕ ( x , y ) = f ( x , y ) + 2 π round ( f ( x , y ) ϕ w ( x , y ) 2 π ) .
RMS = ( < φ 2 > < φ > 2 )
I ( x , y ) = | F [ F ( P ( x , y ) e i ϕ ( x , y ) ) K ( X , Y ) ] | 2
K ( X , Y ) = { e i θ 1 , if L / 2 X L / 2 , and L / 2 Y L / 2 e i θ 2 , if h l / 2 X h + l / 2 , and l / 2 Y l / 2 , c , otherwise .

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