Abstract

We present a method for determining the demodulated phase from a single fringe pattern. This method, based on a correlation technique, searches in a zone of interest for the degree of similarity between a real fringe pattern and a mathematical model. This method, named modulated phase correlation, is tested with different examples.

© 2004 Optical Society of America

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References

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  1. G. Mauvoisin, F. Bremand, A. Lagarde, “Three-dimensional shape reconstruction by phase-shifting shadow moiré,” Appl. Opt. 33, 2163–2169 (1994).
    [CrossRef] [PubMed]
  2. J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow moiré to micro deformation measurements of curved surfaces,” Opt. Lasers Eng. 36, 29–40 (2001).
    [CrossRef]
  3. J. C. Dupre, A. Lagarde, “Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing,” Exp. Mech. 37, 393–397 (1997).
    [CrossRef]
  4. J. Villa, J. A. Quiroga, J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Lasers Eng. 41, 127–137 (2004).
    [CrossRef]
  5. H. Aben, L. Ainola, “Isochromatic fringes in photoelasticity,” J. Opt. Soc. Am. A 13, 750–755 (2000).
    [CrossRef]
  6. R. A. Tomlinson, E. A. Patterson, “The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
    [CrossRef]
  7. Y. Morimoto, M. Fujisaa, “Fringe pattern analysis by a phase shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
    [CrossRef]
  8. M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use of a two dimensional regularized phase tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
    [CrossRef] [PubMed]
  9. M. Servin, J. L. Marroquin, J. A. Quiroga, “Regularized quadrature and phase tracking (RQPT) from a single closed-fringe interferogram,” Reportes Tecnicos del Cimat 2001, Guanajuto, GTO, Mexico, 22August, 2003).
  10. R. Legarda-Sáenz, W. Osten, W. Jüptner, “Improvement of the regularized phase tracking technique for the processing of nonnormalized fringe patterns,” Appl. Opt. 41, 5519–5526 (2002).
    [CrossRef] [PubMed]
  11. M. A. Gdeisat, D. R. Burton, M. J. Lalor, “Fringe pattern demodulation with a two-dimensional digital phase locked loop algorithm,” Appl. Opt. 41, 5479–5487 (2002).
    [CrossRef] [PubMed]
  12. M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
    [CrossRef]
  13. H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newtown–Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
    [CrossRef]
  14. M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, S. R. Mcneill, “Full-field speckle pattern image correlation with B-spline deformation function,” Exp. Mech. 42, 344–352 (2002).
    [CrossRef]
  15. H. W. Schreier, M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape function,” Exp. Mech. 42, 303–310 (2000).
    [CrossRef]
  16. H. Weber, R. Lichtenberger, “The combination of speckle correlation and fringe projection for the measurement of dynamic 3-D deformation of airbag caps,” in International Union of Theoretical and Applied Mechanics Symposium on Advanced Optical Methods and Application in Solid Mechanics, A. Lagarde, ed., (Kluwer Academic, Dordrecht, The Netherlands, 1998), pp. 619–626.
  17. F. Bremand, “A phase unwrapping technique for object relief determination,” Opt. Lasers Eng. 21, 49–60 (1994).
    [CrossRef]
  18. B. Gutmann, H. Weber, “Phase unwrapping with the branch cut method: role of phase field direction,” Appl. Opt. 39, 4802–4816 (2000).
    [CrossRef]
  19. L. Humbert, V. Valle, M. Cottron, “Experimental determination and empirical representation of out-of-plane displacements in a cracked elastic plate loaded in mode I,” Int. J. Solids Struct. 37, 5493–5504 (2000).
    [CrossRef]

2004

J. Villa, J. A. Quiroga, J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Lasers Eng. 41, 127–137 (2004).
[CrossRef]

2002

R. A. Tomlinson, E. A. Patterson, “The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
[CrossRef]

M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, S. R. Mcneill, “Full-field speckle pattern image correlation with B-spline deformation function,” Exp. Mech. 42, 344–352 (2002).
[CrossRef]

M. A. Gdeisat, D. R. Burton, M. J. Lalor, “Fringe pattern demodulation with a two-dimensional digital phase locked loop algorithm,” Appl. Opt. 41, 5479–5487 (2002).
[CrossRef] [PubMed]

R. Legarda-Sáenz, W. Osten, W. Jüptner, “Improvement of the regularized phase tracking technique for the processing of nonnormalized fringe patterns,” Appl. Opt. 41, 5519–5526 (2002).
[CrossRef] [PubMed]

2001

J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow moiré to micro deformation measurements of curved surfaces,” Opt. Lasers Eng. 36, 29–40 (2001).
[CrossRef]

2000

H. Aben, L. Ainola, “Isochromatic fringes in photoelasticity,” J. Opt. Soc. Am. A 13, 750–755 (2000).
[CrossRef]

B. Gutmann, H. Weber, “Phase unwrapping with the branch cut method: role of phase field direction,” Appl. Opt. 39, 4802–4816 (2000).
[CrossRef]

H. W. Schreier, M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape function,” Exp. Mech. 42, 303–310 (2000).
[CrossRef]

L. Humbert, V. Valle, M. Cottron, “Experimental determination and empirical representation of out-of-plane displacements in a cracked elastic plate loaded in mode I,” Int. J. Solids Struct. 37, 5493–5504 (2000).
[CrossRef]

1997

M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use of a two dimensional regularized phase tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef] [PubMed]

J. C. Dupre, A. Lagarde, “Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing,” Exp. Mech. 37, 393–397 (1997).
[CrossRef]

1995

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

1994

Y. Morimoto, M. Fujisaa, “Fringe pattern analysis by a phase shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

G. Mauvoisin, F. Bremand, A. Lagarde, “Three-dimensional shape reconstruction by phase-shifting shadow moiré,” Appl. Opt. 33, 2163–2169 (1994).
[CrossRef] [PubMed]

F. Bremand, “A phase unwrapping technique for object relief determination,” Opt. Lasers Eng. 21, 49–60 (1994).
[CrossRef]

1989

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newtown–Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
[CrossRef]

Aben, H.

H. Aben, L. Ainola, “Isochromatic fringes in photoelasticity,” J. Opt. Soc. Am. A 13, 750–755 (2000).
[CrossRef]

Ainola, L.

H. Aben, L. Ainola, “Isochromatic fringes in photoelasticity,” J. Opt. Soc. Am. A 13, 750–755 (2000).
[CrossRef]

Boone, P.

J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow moiré to micro deformation measurements of curved surfaces,” Opt. Lasers Eng. 36, 29–40 (2001).
[CrossRef]

Bremand, F.

Bruck, H. A.

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newtown–Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
[CrossRef]

Burton, D. R.

Cottron, M.

L. Humbert, V. Valle, M. Cottron, “Experimental determination and empirical representation of out-of-plane displacements in a cracked elastic plate loaded in mode I,” Int. J. Solids Struct. 37, 5493–5504 (2000).
[CrossRef]

Cuevas, F. J.

Degrieck, J.

J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow moiré to micro deformation measurements of curved surfaces,” Opt. Lasers Eng. 36, 29–40 (2001).
[CrossRef]

Dupre, J. C.

J. C. Dupre, A. Lagarde, “Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing,” Exp. Mech. 37, 393–397 (1997).
[CrossRef]

Fujisaa, M.

Y. Morimoto, M. Fujisaa, “Fringe pattern analysis by a phase shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

Gdeisat, M. A.

Gómez-Pedrero, J. A.

J. Villa, J. A. Quiroga, J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Lasers Eng. 41, 127–137 (2004).
[CrossRef]

Gutmann, B.

Humbert, L.

L. Humbert, V. Valle, M. Cottron, “Experimental determination and empirical representation of out-of-plane displacements in a cracked elastic plate loaded in mode I,” Int. J. Solids Struct. 37, 5493–5504 (2000).
[CrossRef]

Jüptner, W.

Lagarde, A.

J. C. Dupre, A. Lagarde, “Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing,” Exp. Mech. 37, 393–397 (1997).
[CrossRef]

G. Mauvoisin, F. Bremand, A. Lagarde, “Three-dimensional shape reconstruction by phase-shifting shadow moiré,” Appl. Opt. 33, 2163–2169 (1994).
[CrossRef] [PubMed]

Lalor, M. J.

Legarda-Sáenz, R.

Lichtenberger, R.

H. Weber, R. Lichtenberger, “The combination of speckle correlation and fringe projection for the measurement of dynamic 3-D deformation of airbag caps,” in International Union of Theoretical and Applied Mechanics Symposium on Advanced Optical Methods and Application in Solid Mechanics, A. Lagarde, ed., (Kluwer Academic, Dordrecht, The Netherlands, 1998), pp. 619–626.

Malacara, D.

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

Marroquin, J. L.

M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use of a two dimensional regularized phase tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef] [PubMed]

M. Servin, J. L. Marroquin, J. A. Quiroga, “Regularized quadrature and phase tracking (RQPT) from a single closed-fringe interferogram,” Reportes Tecnicos del Cimat 2001, Guanajuto, GTO, Mexico, 22August, 2003).

Mauvoisin, G.

Mcneill, S. R.

M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, S. R. Mcneill, “Full-field speckle pattern image correlation with B-spline deformation function,” Exp. Mech. 42, 344–352 (2002).
[CrossRef]

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newtown–Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
[CrossRef]

Morimoto, Y.

Y. Morimoto, M. Fujisaa, “Fringe pattern analysis by a phase shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

Osten, W.

Patterson, E. A.

R. A. Tomlinson, E. A. Patterson, “The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
[CrossRef]

Peng Cheng, M. A.

M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, S. R. Mcneill, “Full-field speckle pattern image correlation with B-spline deformation function,” Exp. Mech. 42, 344–352 (2002).
[CrossRef]

Peters, W. H.

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newtown–Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
[CrossRef]

Quiroga, J. A.

J. Villa, J. A. Quiroga, J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Lasers Eng. 41, 127–137 (2004).
[CrossRef]

M. Servin, J. L. Marroquin, J. A. Quiroga, “Regularized quadrature and phase tracking (RQPT) from a single closed-fringe interferogram,” Reportes Tecnicos del Cimat 2001, Guanajuto, GTO, Mexico, 22August, 2003).

Rodriguez-Vera, R.

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

Schreier, H. W.

M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, S. R. Mcneill, “Full-field speckle pattern image correlation with B-spline deformation function,” Exp. Mech. 42, 344–352 (2002).
[CrossRef]

H. W. Schreier, M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape function,” Exp. Mech. 42, 303–310 (2000).
[CrossRef]

Servin, M.

M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use of a two dimensional regularized phase tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef] [PubMed]

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

M. Servin, J. L. Marroquin, J. A. Quiroga, “Regularized quadrature and phase tracking (RQPT) from a single closed-fringe interferogram,” Reportes Tecnicos del Cimat 2001, Guanajuto, GTO, Mexico, 22August, 2003).

Sutton, M. A.

M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, S. R. Mcneill, “Full-field speckle pattern image correlation with B-spline deformation function,” Exp. Mech. 42, 344–352 (2002).
[CrossRef]

H. W. Schreier, M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape function,” Exp. Mech. 42, 303–310 (2000).
[CrossRef]

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newtown–Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
[CrossRef]

Tomlinson, R. A.

R. A. Tomlinson, E. A. Patterson, “The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
[CrossRef]

Valle, V.

L. Humbert, V. Valle, M. Cottron, “Experimental determination and empirical representation of out-of-plane displacements in a cracked elastic plate loaded in mode I,” Int. J. Solids Struct. 37, 5493–5504 (2000).
[CrossRef]

Van Paepegem, W.

J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow moiré to micro deformation measurements of curved surfaces,” Opt. Lasers Eng. 36, 29–40 (2001).
[CrossRef]

Villa, J.

J. Villa, J. A. Quiroga, J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Lasers Eng. 41, 127–137 (2004).
[CrossRef]

Weber, H.

B. Gutmann, H. Weber, “Phase unwrapping with the branch cut method: role of phase field direction,” Appl. Opt. 39, 4802–4816 (2000).
[CrossRef]

H. Weber, R. Lichtenberger, “The combination of speckle correlation and fringe projection for the measurement of dynamic 3-D deformation of airbag caps,” in International Union of Theoretical and Applied Mechanics Symposium on Advanced Optical Methods and Application in Solid Mechanics, A. Lagarde, ed., (Kluwer Academic, Dordrecht, The Netherlands, 1998), pp. 619–626.

Appl. Opt.

Exp. Mech.

R. A. Tomlinson, E. A. Patterson, “The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
[CrossRef]

J. C. Dupre, A. Lagarde, “Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing,” Exp. Mech. 37, 393–397 (1997).
[CrossRef]

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newtown–Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
[CrossRef]

M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, S. R. Mcneill, “Full-field speckle pattern image correlation with B-spline deformation function,” Exp. Mech. 42, 344–352 (2002).
[CrossRef]

H. W. Schreier, M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape function,” Exp. Mech. 42, 303–310 (2000).
[CrossRef]

Int. J. Solids Struct.

L. Humbert, V. Valle, M. Cottron, “Experimental determination and empirical representation of out-of-plane displacements in a cracked elastic plate loaded in mode I,” Int. J. Solids Struct. 37, 5493–5504 (2000).
[CrossRef]

J. Opt. Soc. Am. A

H. Aben, L. Ainola, “Isochromatic fringes in photoelasticity,” J. Opt. Soc. Am. A 13, 750–755 (2000).
[CrossRef]

Opt. Eng.

Y. Morimoto, M. Fujisaa, “Fringe pattern analysis by a phase shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

Opt. Lasers Eng.

M. Servin, R. Rodriguez-Vera, D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355–365 (1995).
[CrossRef]

F. Bremand, “A phase unwrapping technique for object relief determination,” Opt. Lasers Eng. 21, 49–60 (1994).
[CrossRef]

J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow moiré to micro deformation measurements of curved surfaces,” Opt. Lasers Eng. 36, 29–40 (2001).
[CrossRef]

J. Villa, J. A. Quiroga, J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Lasers Eng. 41, 127–137 (2004).
[CrossRef]

Other

H. Weber, R. Lichtenberger, “The combination of speckle correlation and fringe projection for the measurement of dynamic 3-D deformation of airbag caps,” in International Union of Theoretical and Applied Mechanics Symposium on Advanced Optical Methods and Application in Solid Mechanics, A. Lagarde, ed., (Kluwer Academic, Dordrecht, The Netherlands, 1998), pp. 619–626.

M. Servin, J. L. Marroquin, J. A. Quiroga, “Regularized quadrature and phase tracking (RQPT) from a single closed-fringe interferogram,” Reportes Tecnicos del Cimat 2001, Guanajuto, GTO, Mexico, 22August, 2003).

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Figures (9)

Fig. 1
Fig. 1

Interpolation of the correlation peak.

Fig. 2
Fig. 2

Example of analysis with a simulated fringe pattern: (a) The initial image gives maps of (b) parameter A, (c) parameter B, (d) pitch p, (e) inclinations α, and (f) phase φ.

Fig. 3
Fig. 3

Examples of determination of orientation with a simulated fringe pattern: We show maps of (a) orientations, (b) wrapped phase, and (c) the unwrapped phase.

Fig. 4
Fig. 4

Differences between the phase obtained by the MPC method and values of the mathematical phase: MPC used with a fringe pattern without noise.

Fig. 5
Fig. 5

(a) Results from a noisy fringe pattern, (b) map of the wrapped phase, and (c) map of the the unwrapped phase.

Fig. 6
Fig. 6

Differences between the phase obtained by the MPC method and values of the mathematical phase: MPC used with a noisy fringe pattern.

Fig. 7
Fig. 7

(a) Fringe pattern obtained from an interferometer and corresponding to an elastic cracked plate under loading. (b) The orientation and (c) the wrapped phase were calculated by the MPC method. (d) The unwrapped phase represents the field of the out-of-plane displacement.

Fig. 8
Fig. 8

(a) Fringe pattern from three-dimensional photoelasticimetry obtained from a bar in a torsion loading test. (b) The orientation and (c) the wrapped phase were calculated by the proposed method. (d) The unwrapped phase gives the field of the difference of principal stresses.

Fig. 9
Fig. 9

The shadow moire method applied to part of a telephone gives (a) a fringe pattern with which (b) the orientation and (c) the wrapped phase are calculated. (d) A relief of the telephone appears when the phase is unwrapped.

Tables (1)

Tables Icon

Table 1 Progressions Used for Parameters A, B, α, φ, and p

Equations (18)

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

Cφ=Dhξ-gξ, φ2dξ
hξ-gξ, φ2=h2ξ+g2ξ, φ-2hξgξ, φ.
Tφ=D hξgξ, φdξ.
fx, y=Ax, ycosθx, y+Bx, y,
fA, B, α, p, φ, ξ, γ=A cosθα, p, φ, ξ, γ+B,
θα, p, φ, ξ, γ=2πpcosαx+ξ+2πpsinαy+γ+φ,
CA, B, p, α, φ=Vξ,γfξ, γ, A, B, p, α, φ-Iξ, γ2dξdγ
CA*, B*, p*, α*, φ*=minA,B,p,α,φ CA, B, p, α, φ.
CA, B, p, α, φ=ξ,γNA cosθα, p, φ, ξ, γ+B-Iξ, γ2.
CA, B, p, α, φ=minA,B,p,α,φξ,γNA cosθα, p, φ, ξ, γ+B-Iξ, γ2,
A*=A+εA,B*=B+εB,p*=p+εp,α*=α+εα,φ*=φ+εφ.
C1=CA, B, p, α, φt-1,C2=CA, B, p, α, φt+1,Cm=CA, B, p, α, φt=φ.
φinterp*=12-φ2tC2+φ2tC1-φ2t-1Cm+φ2t+1Cm-φ2t+1C1-φ2t-1C2-φt-1Cm+φt-1φt+1-φtφt+1+φt+1Cm-φt+1C1+φtC1.
B+A<256, B-A0,
0<A<128, 0<B<256.
0α<π, 0φ<2π.
2p33;
-16ξ16, -16γ16.

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