Abstract

In this paper, a simple isoclinic phase map unwrapping method is proposed to retrieve map with ambiguities at photoelastic isotropic points. Regional phase unwrapping method is also utilized to enhance the retrieving efficiency after all phase inconsistencies have been fully detected and branch cutting works have been properly done to ensure blockings of all the paths which could cause incorrect integrations while involuntarily crossing them. The correctly retrieved isoclinic data are then fed into isochromatic formulation, and as a consequence an inconsistency free isochromatic phase map will be obtained. This map can be unwrapped by any simple and fast unwrapping algorithm accurately and effectively. Circular disk and ring under diametric compression samples are both applied for the verification of the proposed algorithm. The experimental results show the proposed algorithm can successfully solve the annoying problems occurred at photoelastic isotropic points with a processing time of roughly 2 seconds for a 420 x 420 pixels map by a general personal computer.

©2010 Optical Society of America

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References

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  1. S. K. Mangal and K. Ramesh, “Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique,” Opt. Lasers Eng. 31(4), 263–278 (1999).
    [Crossref]
  2. D. R. Petersen, R. E. Link, M. N. Pacey, S. J. Haake, and E. A. Patterson, “A novel instrumentation for automated principal strain separation in reflection photoelasticity,” J. Test. Eval. 28(4), 229–235 (2000).
    [Crossref]
  3. G. Petrucci, “Full-field automatic evaluation of an isoclinic parameter in white light,” Exp. Mech. 37(4), 420–426 (1997).
    [Crossref]
  4. M. J. Ekman and A. D. Nurse, “Completely automated determination of two-dimensional photoelastic parameters using load stepping,” Opt. Eng. 37(6), 1845–1851 (1998).
    [Crossref]
  5. T. Y. Chen and C. H. Lin, “Whole-field digital measurement of principal stress directions in photoelasticity,” Opt. Lasers Eng. 30(6), 527–537 (1998).
    [Crossref]
  6. A. D. Nurse, “Automated photoelasticity: weighted least-squares determination of field stresses,” Opt. Lasers Eng. 31(5), 353–370 (1999).
    [Crossref]
  7. T. Liu, A. Asundi, and C. G. Boay, “Full field automated photoelasticity using two-load-step method,” Opt. Eng. 40(8), 1629–1635 (2001).
    [Crossref]
  8. D. Zhang, Y. Han, B. Zhang, and D. Arola, “Automated determination of parameters in photoelasticity,” Opt. Lasers Eng. 45(8), 860–867 (2007).
    [Crossref]
  9. Z. F. Wang and E. A. Patterson, “Use of phase-stepping with demodulation and fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22(2), 91–104 (1995).
    [Crossref]
  10. V. Prasad, K. Madhu, and K. Ramesh, “Towards effective phase unwrapping in digital photoelasticity,” Opt. Lasers Eng. 42(4), 421–436 (2004).
    [Crossref]
  11. P. Siegmann, D. Backman, and E. A. Patterson, “A robust approach to demodulating and unwrapping phase-stepped photoelastic data,” Exp. Mech. 45(3), 278–289 (2005).
    [Crossref]
  12. K. Ashokan and K. Ramesh, “A novel approach for ambiguity removal in isochromatic phasemap in digital phtoelasticity,” Meas. Sci. Technol. 17(11), 2891–2896 (2006).
    [Crossref]
  13. P. Pinit and E. Umezaki, “Digitally whole-field analysis of isoclinic parameter in photoelasticity for four-step color phase-shifting technique,” Opt. Lasers Eng. 45(7), 795–807 (2007).
    [Crossref]
  14. A. Baldi, F. Bertolino, and F. Ginesu, “A temporal phase unwrapping algorithm for photoelastic stress analysis,” Opt. Lasers Eng. 45(5), 612–617 (2007).
    [Crossref]
  15. J. Villa, J. A. Quiroga, and E. Pascual, “Determination of isoclinics in phtoelasticity with a fast regularized estimator,” Opt. Lasers Eng. 46(3), 236–242 (2008).
    [Crossref]
  16. T. Y. Chen, “A simple method for the determination of the photoelastic fringe order,” Exp. Mech. 40(3), 256–260 (2000).
    [Crossref]
  17. T. Y. Chen, “Digital determination of phtoelastic birefringence using two wavelengths,” Exp. Mech. 37(3), 232–236 (1997).
    [Crossref]
  18. D. C. Ghiglia, and M. D. Pritt, Two-dimensional phase unwrapping: theory, algorithms, and software (John Wiley & Sons, Inc., New York, 1998).
  19. M. J. Huang, and B. C. Song, “Ambiguity elimination of photoelastic phase map,” presented at the 2009 SEM Annual Conference and Exposition of Experimental and Applied Mechanics, Albuquerque, New Mexico, USA, 1–4 Jun. 2009.
  20. J. J. Gierloff, “Phase unwrapping by regions,” Proc. SPIE 818, 2–9 (1987).
  21. D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellur-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4(1), 267–280 (1987).
    [Crossref]

2008 (1)

J. Villa, J. A. Quiroga, and E. Pascual, “Determination of isoclinics in phtoelasticity with a fast regularized estimator,” Opt. Lasers Eng. 46(3), 236–242 (2008).
[Crossref]

2007 (3)

D. Zhang, Y. Han, B. Zhang, and D. Arola, “Automated determination of parameters in photoelasticity,” Opt. Lasers Eng. 45(8), 860–867 (2007).
[Crossref]

P. Pinit and E. Umezaki, “Digitally whole-field analysis of isoclinic parameter in photoelasticity for four-step color phase-shifting technique,” Opt. Lasers Eng. 45(7), 795–807 (2007).
[Crossref]

A. Baldi, F. Bertolino, and F. Ginesu, “A temporal phase unwrapping algorithm for photoelastic stress analysis,” Opt. Lasers Eng. 45(5), 612–617 (2007).
[Crossref]

2006 (1)

K. Ashokan and K. Ramesh, “A novel approach for ambiguity removal in isochromatic phasemap in digital phtoelasticity,” Meas. Sci. Technol. 17(11), 2891–2896 (2006).
[Crossref]

2005 (1)

P. Siegmann, D. Backman, and E. A. Patterson, “A robust approach to demodulating and unwrapping phase-stepped photoelastic data,” Exp. Mech. 45(3), 278–289 (2005).
[Crossref]

2004 (1)

V. Prasad, K. Madhu, and K. Ramesh, “Towards effective phase unwrapping in digital photoelasticity,” Opt. Lasers Eng. 42(4), 421–436 (2004).
[Crossref]

2001 (1)

T. Liu, A. Asundi, and C. G. Boay, “Full field automated photoelasticity using two-load-step method,” Opt. Eng. 40(8), 1629–1635 (2001).
[Crossref]

2000 (2)

T. Y. Chen, “A simple method for the determination of the photoelastic fringe order,” Exp. Mech. 40(3), 256–260 (2000).
[Crossref]

D. R. Petersen, R. E. Link, M. N. Pacey, S. J. Haake, and E. A. Patterson, “A novel instrumentation for automated principal strain separation in reflection photoelasticity,” J. Test. Eval. 28(4), 229–235 (2000).
[Crossref]

1999 (2)

S. K. Mangal and K. Ramesh, “Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique,” Opt. Lasers Eng. 31(4), 263–278 (1999).
[Crossref]

A. D. Nurse, “Automated photoelasticity: weighted least-squares determination of field stresses,” Opt. Lasers Eng. 31(5), 353–370 (1999).
[Crossref]

1998 (2)

M. J. Ekman and A. D. Nurse, “Completely automated determination of two-dimensional photoelastic parameters using load stepping,” Opt. Eng. 37(6), 1845–1851 (1998).
[Crossref]

T. Y. Chen and C. H. Lin, “Whole-field digital measurement of principal stress directions in photoelasticity,” Opt. Lasers Eng. 30(6), 527–537 (1998).
[Crossref]

1997 (2)

G. Petrucci, “Full-field automatic evaluation of an isoclinic parameter in white light,” Exp. Mech. 37(4), 420–426 (1997).
[Crossref]

T. Y. Chen, “Digital determination of phtoelastic birefringence using two wavelengths,” Exp. Mech. 37(3), 232–236 (1997).
[Crossref]

1995 (1)

Z. F. Wang and E. A. Patterson, “Use of phase-stepping with demodulation and fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22(2), 91–104 (1995).
[Crossref]

1987 (2)

Arola, D.

D. Zhang, Y. Han, B. Zhang, and D. Arola, “Automated determination of parameters in photoelasticity,” Opt. Lasers Eng. 45(8), 860–867 (2007).
[Crossref]

Ashokan, K.

K. Ashokan and K. Ramesh, “A novel approach for ambiguity removal in isochromatic phasemap in digital phtoelasticity,” Meas. Sci. Technol. 17(11), 2891–2896 (2006).
[Crossref]

Asundi, A.

T. Liu, A. Asundi, and C. G. Boay, “Full field automated photoelasticity using two-load-step method,” Opt. Eng. 40(8), 1629–1635 (2001).
[Crossref]

Backman, D.

P. Siegmann, D. Backman, and E. A. Patterson, “A robust approach to demodulating and unwrapping phase-stepped photoelastic data,” Exp. Mech. 45(3), 278–289 (2005).
[Crossref]

Baldi, A.

A. Baldi, F. Bertolino, and F. Ginesu, “A temporal phase unwrapping algorithm for photoelastic stress analysis,” Opt. Lasers Eng. 45(5), 612–617 (2007).
[Crossref]

Bertolino, F.

A. Baldi, F. Bertolino, and F. Ginesu, “A temporal phase unwrapping algorithm for photoelastic stress analysis,” Opt. Lasers Eng. 45(5), 612–617 (2007).
[Crossref]

Boay, C. G.

T. Liu, A. Asundi, and C. G. Boay, “Full field automated photoelasticity using two-load-step method,” Opt. Eng. 40(8), 1629–1635 (2001).
[Crossref]

Chen, T. Y.

T. Y. Chen, “A simple method for the determination of the photoelastic fringe order,” Exp. Mech. 40(3), 256–260 (2000).
[Crossref]

T. Y. Chen and C. H. Lin, “Whole-field digital measurement of principal stress directions in photoelasticity,” Opt. Lasers Eng. 30(6), 527–537 (1998).
[Crossref]

T. Y. Chen, “Digital determination of phtoelastic birefringence using two wavelengths,” Exp. Mech. 37(3), 232–236 (1997).
[Crossref]

Ekman, M. J.

M. J. Ekman and A. D. Nurse, “Completely automated determination of two-dimensional photoelastic parameters using load stepping,” Opt. Eng. 37(6), 1845–1851 (1998).
[Crossref]

Ghiglia, D. C.

Gierloff, J. J.

J. J. Gierloff, “Phase unwrapping by regions,” Proc. SPIE 818, 2–9 (1987).

Ginesu, F.

A. Baldi, F. Bertolino, and F. Ginesu, “A temporal phase unwrapping algorithm for photoelastic stress analysis,” Opt. Lasers Eng. 45(5), 612–617 (2007).
[Crossref]

Haake, S. J.

D. R. Petersen, R. E. Link, M. N. Pacey, S. J. Haake, and E. A. Patterson, “A novel instrumentation for automated principal strain separation in reflection photoelasticity,” J. Test. Eval. 28(4), 229–235 (2000).
[Crossref]

Han, Y.

D. Zhang, Y. Han, B. Zhang, and D. Arola, “Automated determination of parameters in photoelasticity,” Opt. Lasers Eng. 45(8), 860–867 (2007).
[Crossref]

Lin, C. H.

T. Y. Chen and C. H. Lin, “Whole-field digital measurement of principal stress directions in photoelasticity,” Opt. Lasers Eng. 30(6), 527–537 (1998).
[Crossref]

Link, R. E.

D. R. Petersen, R. E. Link, M. N. Pacey, S. J. Haake, and E. A. Patterson, “A novel instrumentation for automated principal strain separation in reflection photoelasticity,” J. Test. Eval. 28(4), 229–235 (2000).
[Crossref]

Liu, T.

T. Liu, A. Asundi, and C. G. Boay, “Full field automated photoelasticity using two-load-step method,” Opt. Eng. 40(8), 1629–1635 (2001).
[Crossref]

Madhu, K.

V. Prasad, K. Madhu, and K. Ramesh, “Towards effective phase unwrapping in digital photoelasticity,” Opt. Lasers Eng. 42(4), 421–436 (2004).
[Crossref]

Mangal, S. K.

S. K. Mangal and K. Ramesh, “Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique,” Opt. Lasers Eng. 31(4), 263–278 (1999).
[Crossref]

Mastin, G. A.

Nurse, A. D.

A. D. Nurse, “Automated photoelasticity: weighted least-squares determination of field stresses,” Opt. Lasers Eng. 31(5), 353–370 (1999).
[Crossref]

M. J. Ekman and A. D. Nurse, “Completely automated determination of two-dimensional photoelastic parameters using load stepping,” Opt. Eng. 37(6), 1845–1851 (1998).
[Crossref]

Pacey, M. N.

D. R. Petersen, R. E. Link, M. N. Pacey, S. J. Haake, and E. A. Patterson, “A novel instrumentation for automated principal strain separation in reflection photoelasticity,” J. Test. Eval. 28(4), 229–235 (2000).
[Crossref]

Pascual, E.

J. Villa, J. A. Quiroga, and E. Pascual, “Determination of isoclinics in phtoelasticity with a fast regularized estimator,” Opt. Lasers Eng. 46(3), 236–242 (2008).
[Crossref]

Patterson, E. A.

P. Siegmann, D. Backman, and E. A. Patterson, “A robust approach to demodulating and unwrapping phase-stepped photoelastic data,” Exp. Mech. 45(3), 278–289 (2005).
[Crossref]

D. R. Petersen, R. E. Link, M. N. Pacey, S. J. Haake, and E. A. Patterson, “A novel instrumentation for automated principal strain separation in reflection photoelasticity,” J. Test. Eval. 28(4), 229–235 (2000).
[Crossref]

Z. F. Wang and E. A. Patterson, “Use of phase-stepping with demodulation and fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22(2), 91–104 (1995).
[Crossref]

Petersen, D. R.

D. R. Petersen, R. E. Link, M. N. Pacey, S. J. Haake, and E. A. Patterson, “A novel instrumentation for automated principal strain separation in reflection photoelasticity,” J. Test. Eval. 28(4), 229–235 (2000).
[Crossref]

Petrucci, G.

G. Petrucci, “Full-field automatic evaluation of an isoclinic parameter in white light,” Exp. Mech. 37(4), 420–426 (1997).
[Crossref]

Pinit, P.

P. Pinit and E. Umezaki, “Digitally whole-field analysis of isoclinic parameter in photoelasticity for four-step color phase-shifting technique,” Opt. Lasers Eng. 45(7), 795–807 (2007).
[Crossref]

Prasad, V.

V. Prasad, K. Madhu, and K. Ramesh, “Towards effective phase unwrapping in digital photoelasticity,” Opt. Lasers Eng. 42(4), 421–436 (2004).
[Crossref]

Quiroga, J. A.

J. Villa, J. A. Quiroga, and E. Pascual, “Determination of isoclinics in phtoelasticity with a fast regularized estimator,” Opt. Lasers Eng. 46(3), 236–242 (2008).
[Crossref]

Ramesh, K.

K. Ashokan and K. Ramesh, “A novel approach for ambiguity removal in isochromatic phasemap in digital phtoelasticity,” Meas. Sci. Technol. 17(11), 2891–2896 (2006).
[Crossref]

V. Prasad, K. Madhu, and K. Ramesh, “Towards effective phase unwrapping in digital photoelasticity,” Opt. Lasers Eng. 42(4), 421–436 (2004).
[Crossref]

S. K. Mangal and K. Ramesh, “Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique,” Opt. Lasers Eng. 31(4), 263–278 (1999).
[Crossref]

Romero, L. A.

Siegmann, P.

P. Siegmann, D. Backman, and E. A. Patterson, “A robust approach to demodulating and unwrapping phase-stepped photoelastic data,” Exp. Mech. 45(3), 278–289 (2005).
[Crossref]

Umezaki, E.

P. Pinit and E. Umezaki, “Digitally whole-field analysis of isoclinic parameter in photoelasticity for four-step color phase-shifting technique,” Opt. Lasers Eng. 45(7), 795–807 (2007).
[Crossref]

Villa, J.

J. Villa, J. A. Quiroga, and E. Pascual, “Determination of isoclinics in phtoelasticity with a fast regularized estimator,” Opt. Lasers Eng. 46(3), 236–242 (2008).
[Crossref]

Wang, Z. F.

Z. F. Wang and E. A. Patterson, “Use of phase-stepping with demodulation and fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22(2), 91–104 (1995).
[Crossref]

Zhang, B.

D. Zhang, Y. Han, B. Zhang, and D. Arola, “Automated determination of parameters in photoelasticity,” Opt. Lasers Eng. 45(8), 860–867 (2007).
[Crossref]

Zhang, D.

D. Zhang, Y. Han, B. Zhang, and D. Arola, “Automated determination of parameters in photoelasticity,” Opt. Lasers Eng. 45(8), 860–867 (2007).
[Crossref]

Exp. Mech. (4)

G. Petrucci, “Full-field automatic evaluation of an isoclinic parameter in white light,” Exp. Mech. 37(4), 420–426 (1997).
[Crossref]

P. Siegmann, D. Backman, and E. A. Patterson, “A robust approach to demodulating and unwrapping phase-stepped photoelastic data,” Exp. Mech. 45(3), 278–289 (2005).
[Crossref]

T. Y. Chen, “A simple method for the determination of the photoelastic fringe order,” Exp. Mech. 40(3), 256–260 (2000).
[Crossref]

T. Y. Chen, “Digital determination of phtoelastic birefringence using two wavelengths,” Exp. Mech. 37(3), 232–236 (1997).
[Crossref]

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

J. Test. Eval. (1)

D. R. Petersen, R. E. Link, M. N. Pacey, S. J. Haake, and E. A. Patterson, “A novel instrumentation for automated principal strain separation in reflection photoelasticity,” J. Test. Eval. 28(4), 229–235 (2000).
[Crossref]

Meas. Sci. Technol. (1)

K. Ashokan and K. Ramesh, “A novel approach for ambiguity removal in isochromatic phasemap in digital phtoelasticity,” Meas. Sci. Technol. 17(11), 2891–2896 (2006).
[Crossref]

Opt. Eng. (2)

M. J. Ekman and A. D. Nurse, “Completely automated determination of two-dimensional photoelastic parameters using load stepping,” Opt. Eng. 37(6), 1845–1851 (1998).
[Crossref]

T. Liu, A. Asundi, and C. G. Boay, “Full field automated photoelasticity using two-load-step method,” Opt. Eng. 40(8), 1629–1635 (2001).
[Crossref]

Opt. Lasers Eng. (9)

D. Zhang, Y. Han, B. Zhang, and D. Arola, “Automated determination of parameters in photoelasticity,” Opt. Lasers Eng. 45(8), 860–867 (2007).
[Crossref]

Z. F. Wang and E. A. Patterson, “Use of phase-stepping with demodulation and fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22(2), 91–104 (1995).
[Crossref]

V. Prasad, K. Madhu, and K. Ramesh, “Towards effective phase unwrapping in digital photoelasticity,” Opt. Lasers Eng. 42(4), 421–436 (2004).
[Crossref]

T. Y. Chen and C. H. Lin, “Whole-field digital measurement of principal stress directions in photoelasticity,” Opt. Lasers Eng. 30(6), 527–537 (1998).
[Crossref]

A. D. Nurse, “Automated photoelasticity: weighted least-squares determination of field stresses,” Opt. Lasers Eng. 31(5), 353–370 (1999).
[Crossref]

P. Pinit and E. Umezaki, “Digitally whole-field analysis of isoclinic parameter in photoelasticity for four-step color phase-shifting technique,” Opt. Lasers Eng. 45(7), 795–807 (2007).
[Crossref]

A. Baldi, F. Bertolino, and F. Ginesu, “A temporal phase unwrapping algorithm for photoelastic stress analysis,” Opt. Lasers Eng. 45(5), 612–617 (2007).
[Crossref]

J. Villa, J. A. Quiroga, and E. Pascual, “Determination of isoclinics in phtoelasticity with a fast regularized estimator,” Opt. Lasers Eng. 46(3), 236–242 (2008).
[Crossref]

S. K. Mangal and K. Ramesh, “Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique,” Opt. Lasers Eng. 31(4), 263–278 (1999).
[Crossref]

Proc. SPIE (1)

J. J. Gierloff, “Phase unwrapping by regions,” Proc. SPIE 818, 2–9 (1987).

Other (2)

D. C. Ghiglia, and M. D. Pritt, Two-dimensional phase unwrapping: theory, algorithms, and software (John Wiley & Sons, Inc., New York, 1998).

M. J. Huang, and B. C. Song, “Ambiguity elimination of photoelastic phase map,” presented at the 2009 SEM Annual Conference and Exposition of Experimental and Applied Mechanics, Albuquerque, New Mexico, USA, 1–4 Jun. 2009.

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Figures (7)

Fig. 1
Fig. 1 The plane polariscope used for the isoclinic calculation.
Fig. 2
Fig. 2 A 1D simulated isoclinic distribution with isotropic point being crossed through (wrapped and unwrapped).
Fig. 3
Fig. 3 The simulated study of a circular ring under diametric compression.
Fig. 4
Fig. 4 The experimental study of a circular ring under diametric compression.
Fig. 5
Fig. 5 The experimental study of a circular disk under diametric compression.
Fig. 6
Fig. 6 Verification between experimental and simulated results of ring sample. (a) The experimental result of isoclinic unwrapping. (b) and (c) represent respectively the horizontal data at 0.9 and 0.75 diameter height from the bottom (where red and green are experimental and simulated results, respectively)
Fig. 7
Fig. 7 Verification between experimental and simulated results of disk sample. (a) The experimental result of isoclinic unwrapping. (b) and (c) represent respectively the horizontal data at 0.9 and 0.75 diameter height from the bottom (where red and green are experimental and simulated results, respectively)

Equations (14)

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

φ w = 1 2 tan 1 ( I a sin Δ sin 2 φ I a sin Δ cos 2 φ ) ,
φ w = π 8 1 4 tan 1 ( I a sin 2 Δ 2 sin 4 φ I a sin 2 Δ 2 cos 4 φ ) ,
I 1 , i = I b , i + 1 2 I a , i sin 2 Δ 2 ( 1 cos 4 φ ) ,
I 2 , i = I b , i + 1 2 I a , i sin 2 Δ 2 ( 1 sin 4 φ ) ,
I 3 , i = I b , i + 1 2 I a , i sin 2 Δ 2 ( 1 + cos 4 φ ) ,
I 4 , i = I b , i + 1 2 I a , i sin 2 Δ 2 ( 1 + sin 4 φ ) .
    ϕ = 1 4 tan 1 { n < I 4 s I 0 > n < I 2 s I 0 > n < I 3 s I 0 > n < I 1 s I 0 > }           =   1 4 tan 1 { ( sin 2 Δ R 2 + sin 2 Δ G 2 + sin 2 Δ B 2 ) sin 4 φ ( sin 2 Δ R 2 + sin 2 Δ G 2 + sin 2 Δ B 2 ) cos 4 φ }           =   1 4 tan 1 { sin 4 φ cos 4 φ } ,
I 0 = 1 4 ( I 1 s + I 2 s + I 3 s + I 4 s ) ,
I j s = 1 3 ( I j , R + I j , G + I j , B ) , ˜ ^ ˜ ˜ ˜ ˜ f o r ˜ j = 1 ~ 4 ,
n < I > = I min ( I ) max ( I ) min ( I ) .
φ c = { φ w φ w + π 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ i f φ w > 0 i f φ w 0 ,
ϕ x ( m , n ) = ϕ ( m , n + 1 ) ϕ ( m , n ) ,
ϕ y ( m , n ) = ϕ ( m + 1 , n ) ϕ ( m , n ) ,
N ( m , n ) = int [ ϕ x ( m , n ) 2 π ] + int [ ϕ y ( m , n + 1 ) 2 π ] int [ ϕ x ( m + 1 , n ) 2 π ] int [ ϕ y ( m , n ) 2 π ] .

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