Abstract

Although one of the simplest and powerful approaches for the demodulation of a single fringe pattern with closed fringes is the regularized phase-tracking (RPT) technique, this technique has two important drawbacks: its sensibility at the fringe-pattern modulation and the time employed in the estimation. We present modifications to the RPT technique that consist of the inclusion of a rough estimate of the fringe-pattern modulation and the linearization of the fringe-pattern model that allows the minimization of the cost function through stable numerical linear techniques. With these changes, the demodulation of nonnormalized fringe patterns is made with a significant reduction in the processing time, preserving the demodulation accuracy of the original RPT method.

© 2006 Optical Society of America

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References

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  1. D.W.Robinson and G.T.Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, 1993).
  2. J. L. Marroquin, M. Rivera, S. Botello, R. Rodriguez-Vera, and M. Servin, "Regularization methods for processing fringe-pattern images," Appl. Opt. 38, 788-794 (1999).
    [CrossRef]
  3. M. Servin, J. L. Marroquin, and 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]
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    [CrossRef]
  5. M. Servin, J. L. Marroquin, and F. J. Cuevas, "Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms," J. Opt. Soc. Am. A 18, 689-695 (2001).
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  6. R. Legarda-Sáenz, W. Osten, and W. Jüptner, "Improvement of the regularized phase tracking technique for the processing of nonnormalized fringe patterns," Appl. Opt. 41, 5519-5526 (2002).
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  7. M. Rivera, "Robust phase demodulation of interferograms with open or closed fringes," J. Opt. Soc. Am. A 22, 1170-1175 (2005).
    [CrossRef]
  8. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998). Sec. 5.6.
    [CrossRef]
  9. J. A. Quiroga and M. Servin, "Isotropic n-dimensional fringe pattern normalization," Opt. Commun. 224, 221-227 (2003).
    [CrossRef]
  10. T. Kreis, Holographic Interferometry. Principles and Methods (Akademie Verlag, 1996). Sec. 4.6.
  11. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1999). Sec. 10.7.
  12. J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, 1999).
    [CrossRef]
  13. D. Geman and G. Reynolds, "Constrained restoration and the recovery of discontinuities," IEEE Trans. Pattern Anal. Mach. Intell. 14, 367-383 (1997).
    [CrossRef]
  14. M. J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vis. 19, 57-91 (1996).
    [CrossRef]
  15. P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, "Deterministic edge preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997).
    [CrossRef] [PubMed]
  16. G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1996), Chap. 10.
  17. B. Strobel, "Processing of interferometric phase maps as complex-valued phasor images," Appl. Opt. 35, 2192-2198 (1996).
    [CrossRef] [PubMed]

2005 (1)

2003 (1)

J. A. Quiroga and M. Servin, "Isotropic n-dimensional fringe pattern normalization," Opt. Commun. 224, 221-227 (2003).
[CrossRef]

2002 (1)

2001 (1)

2000 (1)

1999 (1)

1997 (3)

M. Servin, J. L. Marroquin, and 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]

D. Geman and G. Reynolds, "Constrained restoration and the recovery of discontinuities," IEEE Trans. Pattern Anal. Mach. Intell. 14, 367-383 (1997).
[CrossRef]

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, "Deterministic edge preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef] [PubMed]

1996 (2)

B. Strobel, "Processing of interferometric phase maps as complex-valued phasor images," Appl. Opt. 35, 2192-2198 (1996).
[CrossRef] [PubMed]

M. J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vis. 19, 57-91 (1996).
[CrossRef]

Aubert, G.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, "Deterministic edge preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef] [PubMed]

Barlaud, M.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, "Deterministic edge preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef] [PubMed]

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998). Sec. 5.6.
[CrossRef]

Black, M. J.

M. J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vis. 19, 57-91 (1996).
[CrossRef]

Blanc-Feraud, L.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, "Deterministic edge preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef] [PubMed]

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998). Sec. 5.6.
[CrossRef]

Botello, S.

Charbonnier, P.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, "Deterministic edge preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef] [PubMed]

Cuevas, F. J.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1999). Sec. 10.7.

Geman, D.

D. Geman and G. Reynolds, "Constrained restoration and the recovery of discontinuities," IEEE Trans. Pattern Anal. Mach. Intell. 14, 367-383 (1997).
[CrossRef]

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1996), Chap. 10.

Jüptner, W.

Kreis, T.

T. Kreis, Holographic Interferometry. Principles and Methods (Akademie Verlag, 1996). Sec. 4.6.

Legarda-Sáenz, R.

Marroquin, J. L.

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, 1999).
[CrossRef]

Osten, W.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1999). Sec. 10.7.

Quiroga, J. A.

Rangarajan, A.

M. J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vis. 19, 57-91 (1996).
[CrossRef]

Reynolds, G.

D. Geman and G. Reynolds, "Constrained restoration and the recovery of discontinuities," IEEE Trans. Pattern Anal. Mach. Intell. 14, 367-383 (1997).
[CrossRef]

Rivera, M.

Rodriguez-Vera, R.

Servin, M.

Strobel, B.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1999). Sec. 10.7.

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1996), Chap. 10.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1999). Sec. 10.7.

Villa, J.

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, 1999).
[CrossRef]

Appl. Opt. (5)

IEEE Trans. Image Process. (1)

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, "Deterministic edge preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

D. Geman and G. Reynolds, "Constrained restoration and the recovery of discontinuities," IEEE Trans. Pattern Anal. Mach. Intell. 14, 367-383 (1997).
[CrossRef]

Int. J. Comput. Vis. (1)

M. J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vis. 19, 57-91 (1996).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

J. A. Quiroga and M. Servin, "Isotropic n-dimensional fringe pattern normalization," Opt. Commun. 224, 221-227 (2003).
[CrossRef]

Other (6)

T. Kreis, Holographic Interferometry. Principles and Methods (Akademie Verlag, 1996). Sec. 4.6.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1999). Sec. 10.7.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, 1999).
[CrossRef]

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998). Sec. 5.6.
[CrossRef]

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1996), Chap. 10.

D.W.Robinson and G.T.Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, 1993).

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Figures (5)

Fig. 1
Fig. 1

(a) Synthetic fringe pattern, (b) modulation, and (c) phase fields used to generate the fringe pattern. The phase field is wrapped for purposes of illustration.

Fig. 2
Fig. 2

Resultant phase estimate using (a) Eq. (2), (b) method from Ref. [6], (c) Eq. (4), (d) Eq. (13). Phase estimates are wrapped for purposes of illustration.

Fig. 3
Fig. 3

(a) Synthetic fringe pattern. (b) Modulation field used to generate the fringe pattern. Resultant phase estimates using (c) method from Ref. [6], (d) Eq. (4), (e) Eq. (13). The phase field is wrapped for purposes of illustration.

Fig. 4
Fig. 4

(a)–(c) Synthetic fringe patterns with different levels of modulation. (d)–(f) Estimated modulation from fringe patterns shown in (a)–(c), respectively. (g)–(i) Resultant phase estimates using the fringe pattern from (a)–(c) and the modulation from (d)–(f), respectively. The phase field is wrapped for purposes of illustration.

Fig. 5
Fig. 5

(a) Experimental fringe pattern. Resultant phase estimate using (b) method from Ref. [6], (c) Eq. (4), (d) Eq. (13). Phase estimates are wrapped for purposes of illustration.

Tables (3)

Tables Icon

Table 1 Time Employed to Demodulate the Fringe Pattern Shown in Fig. 1

Tables Icon

Table 2 Time Employed to Demodulate the Fringe Pattern Shown in Fig. 3

Tables Icon

Table 3 Time Employed to Demodulate the Fringe Pattern Shown in Fig. 5

Equations (32)

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I r = a r + b r cos ( f r ) + η r ,
U ( p r ) = s N r ( { I s h cos [ p r ( s ) ] } 2 + { I s h cos [ p r ( s ) + α ] } 2 + λ [ f s p r ( s ) ] 2 m s ) ,
p r ( s ) = f r + ω r T ( r s )
U ( p r ) = s N r ( { g s b ̂ s cos [ p r ( s ) ] } 2 + { g s b ̂ s cos [ p r ( s ) + α ] } 2 + λ [ f s p r ( s ) ] 2 m s ) ,
g r b r cos ( f r ) ,
f = f ̂ + f ̃ ,
E ( f ̃ r ; f ̂ r ) = g r b ̂ r [ cos ( f ̂ r ) f ̃ r sin ( f ̂ r ) ] 0 .
ω = ω ̂ + ω ̃ ,
p ̂ r ( s ) = f ̂ r + ω ̂ r T ( r s ) ,
p ̃ r ( s ) = f ̃ r + ω ̃ r T ( r s ) ,
b ̂ s cos [ p r ( s ) ] = b ̂ s cos [ p ̂ r ( s ) + p ̃ r ( s ) ] b ̂ s { cos [ ( p ̂ r ( s ) ) p ̃ r ( s ) sin ( p ̂ r ( s ) ) ] } = b ̂ s { cos [ f ̂ r + ω ̂ r T ( r s ) ] [ f ̃ r + ω ̃ r T ( r s ) ] sin [ f ̂ r + ω ̂ r T ( r s ) ] } .
E ( f ̃ r , ω ̃ r ; f ̂ r , ω ̂ r ) = g r b ̂ r [ cos ( p ̂ r ( s ) ) p ̃ r ( s ) sin ( p ̂ r ( s ) ) ] 0 .
U ( f ̂ r , ω ̃ r , l s ) = 1 2 s N r { l s 2 { g s b ̂ s cos [ p ̂ r ( s ) ] + p ̃ r ( s ) b ̂ s sin [ p ̂ r ( s ) ] } 2 + l s 2 { g s b ̂ s cos [ p ̂ r ( s ) + α ] + p ̃ r ( s ) b ̂ s sin [ p ̂ r ( s ) + α ] } 2 + λ [ f s p ̂ r ( s ) p ̃ r ( s ) ] 2 m s + μ ( 1 l s ) 2 } ,
l s = μ μ + E 0 2 [ p r ( s ) ] + E 1 2 [ p r ( s ) ] ,
g r b ̂ r cos ( p ̂ r ( s ) ) g r b ̂ r { cos [ p ̂ r ( s ) ] p ̃ r ( s ) sin [ p ̂ r ( s ) ] } g r b ̂ r cos [ p ̂ r ( s ) + p ̃ r ( s ) ] .
A r s = def l s 2 b ̂ s 2 { sin 2 [ p ̂ r ( s ) ] + sin 2 [ p ̂ r ( s ) + α ] } + λ m s ,
B r s = def l s 2 { g s b ̂ s cos [ p ̂ r ( s ) ] } b ̂ s sin [ p ̂ r ( s ) ] + l s 2 { g s b ̂ s cos [ p ̂ r ( s ) + α ] } b ̂ s sin ( p ̂ r ( s ) + α ) λ m s [ f s p ̂ r ( s ) ] .
U [ p ̃ r ( s ) ] f ̃ r = U [ p ̃ r ( s ) ] p ̃ r ( s ) p ̃ r ( s ) f ̃ r ,
U [ p ̃ r ( s ) ] ω ̃ r = U [ p ̃ r ( s ) ] p ̃ r ( s ) p ̃ r ( s ) ω ̃ r .
U [ p ̃ r ( s ) ] p ̃ r ( s ) = s N r [ f ̃ r + ω ̃ r T ( r s ) ] A r s + B r s ,
f ̃ r = s N r [ ω ̃ r T ( r s ) A r s + B r s ] s N r A r s .
w r p ̃ r ( s ) = [ p ̃ r ( s ) u ̃ r , p ̃ r ( s ) v ̃ r ] T = [ r x s x , r y s y ] T .
u ̃ r = s N r { [ f ̃ r + v ̃ r ( r y s y ) ] A r s + B r s } ( r x s x ) s N r A r s ( r x s x ) 2 ,
v ̃ r = s N r { [ f ̃ r + u ̃ r ( r x s x ) ] A r s + B r s } ( r y s y ) s N r A r s ( r y s y ) 2 .
l s = μ μ + E 0 2 [ p r ( s ) ] + E 1 2 [ p r ( s ) ] ,
E 0 [ p r ( s ) ] = def g s b ̂ s cos [ p ̂ r ( s ) ] + p ̂ r ( s ) b ̂ s sin [ p ̂ r ( s ) ] ,
E 1 [ p r ( s ) ] = def g s b ̂ s cos [ p ̂ r ( s ) + α ] + p ̃ r ( s ) b ̂ s sin [ p ̂ r ( s ) + α ] .
l s = μ μ + [ g s b ̂ s cos p ̂ r ( s ) ] 2 + { g s b ̂ s cos [ p ̂ r ( s ) + α ] } 2
for all s N r .
f ̃ r = s N r B r s s N r A r s .
u ̃ r = s N r B r s ( r x s x ) s N r A r s ( r x s x ) 2 .
v ̃ r = s N r B r s ( r y s y ) s N r A r s ( r y s y ) 2 .

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