Abstract

We introduce a method based on the minimization of a total variation regularization cost function for computing discontinuous phase maps from fringe patterns. The performance of the method is demonstrated by numerical experiments with both synthetic and real data.

© 2014 Optical Society of America

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References

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  1. D. W. Robinson and G. T. Reid, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Taylor & Francis, 1993).
  2. G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50, iii–x (2012).
    [CrossRef]
  3. 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]
  4. J. Villa, J. A. Quiroga, and M. Servin, “Improved regularized phase-tracking technique for the processing of squared-grating deflectograms,” Appl. Opt. 39, 502–508 (2000).
    [CrossRef]
  5. R. Legarda-Saenz, W. Osten, and W. P. Juptner, “Improvement of the regularized phase tracking technique for the processing of nonnormalized fringe patterns,” Appl. Opt. 41, 5519–5526 (2002).
    [CrossRef]
  6. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A 22, 1170–1175 (2005).
    [CrossRef]
  7. C. Galvan and M. Rivera, “Second-order robust regularization cost function for detecting and reconstructing phase discontinuities,” Appl. Opt. 45, 353–359 (2006).
    [CrossRef]
  8. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
    [CrossRef]
  9. R. Acar and C. R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Probl. 10, 1217–1229 (1994).
    [CrossRef]
  10. P. Getreuer, “Rudin-Osher-Fatemi total variation denoising using split Bregman,” IPOL 2012, 1–20 (2012).
    [CrossRef]
  11. J. Nocedal and S. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).
  12. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).
  13. H. Liao, F. Li, and M. K. Ng, “Selection of regularization parameter in total variation image restoration,” J. Opt. Soc. Am. A 26, 2311–2320 (2009).
    [CrossRef]
  14. T. Kreis, Holographic Interferometry: Principles and Methods (Wiley-VCH, 1996).
  15. M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20, 925–934 (2003).
    [CrossRef]
  16. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).
  17. K. J. Gasvik, Optical Metrology, 3rd ed. (Wiley, 2002).
  18. D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165–S187 (2003).
    [CrossRef]

2012

G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50, iii–x (2012).
[CrossRef]

P. Getreuer, “Rudin-Osher-Fatemi total variation denoising using split Bregman,” IPOL 2012, 1–20 (2012).
[CrossRef]

2009

2006

2005

2003

M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20, 925–934 (2003).
[CrossRef]

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165–S187 (2003).
[CrossRef]

2002

2000

1999

1994

R. Acar and C. R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Probl. 10, 1217–1229 (1994).
[CrossRef]

1992

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[CrossRef]

Acar, R.

R. Acar and C. R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Probl. 10, 1217–1229 (1994).
[CrossRef]

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).

Botello, S.

Chan, T.

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165–S187 (2003).
[CrossRef]

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[CrossRef]

Galvan, C.

Gasvik, K. J.

K. J. Gasvik, Optical Metrology, 3rd ed. (Wiley, 2002).

Getreuer, P.

P. Getreuer, “Rudin-Osher-Fatemi total variation denoising using split Bregman,” IPOL 2012, 1–20 (2012).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).

Juptner, W. P.

Kreis, T.

T. Kreis, Holographic Interferometry: Principles and Methods (Wiley-VCH, 1996).

Legarda-Saenz, R.

Li, F.

Liao, H.

Marroquin, J. L.

Ng, M. K.

Nocedal, J.

J. Nocedal and S. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[CrossRef]

Osten, W.

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).

Quiroga, J. A.

Rajshekhar, G.

G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50, iii–x (2012).
[CrossRef]

Rastogi, P.

G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50, iii–x (2012).
[CrossRef]

Reid, G. T.

D. W. Robinson and G. T. Reid, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Taylor & Francis, 1993).

Rivera, M.

Robinson, D. W.

D. W. Robinson and G. T. Reid, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Taylor & Francis, 1993).

Rodriguez-Vera, R.

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[CrossRef]

Servin, M.

Strong, D.

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165–S187 (2003).
[CrossRef]

Villa, J.

Vogel, C. R.

R. Acar and C. R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Probl. 10, 1217–1229 (1994).
[CrossRef]

Wright, S.

J. Nocedal and S. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

Appl. Opt.

Inverse Probl.

R. Acar and C. R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Probl. 10, 1217–1229 (1994).
[CrossRef]

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165–S187 (2003).
[CrossRef]

IPOL

P. Getreuer, “Rudin-Osher-Fatemi total variation denoising using split Bregman,” IPOL 2012, 1–20 (2012).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lasers Eng.

G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50, iii–x (2012).
[CrossRef]

Phys. D

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[CrossRef]

Other

D. W. Robinson and G. T. Reid, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Taylor & Francis, 1993).

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).

K. J. Gasvik, Optical Metrology, 3rd ed. (Wiley, 2002).

T. Kreis, Holographic Interferometry: Principles and Methods (Wiley-VCH, 1996).

J. Nocedal and S. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).

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Figures (7)

Fig. 1.
Fig. 1.

(a) Synthetic fringe pattern. (b) Synthetic phase term ϕx.

Fig. 2.
Fig. 2.

Estimated phase terms using (a) Eq. (2) and (b) method from Ref. [7].

Fig. 3.
Fig. 3.

(a) Experimental fringe pattern. (b) Demodulated phase term.

Fig. 4.
Fig. 4.

Estimated phase terms using (a) Eq. (2) and (b) Schwider–Hariharan (4+1) algorithm [17].

Fig. 5.
Fig. 5.

Estimated phase terms using Eq. (2).

Fig. 6.
Fig. 6.

Estimated phase terms using Schwider–Hariharan (4+1) algorithm [17].

Fig. 7.
Fig. 7.

Columns of the estimated phase terms using Eq. (2) and Schwider–Hariharan (4+1) algorithm [17].

Equations (15)

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Ix=ax+bxcos(φx+ϕx),
minϕx,ax,bxE(ϕx,ax,bx){λ2Ω[Ixgx]2dx+Ω|ϕx|dx+Ω|ax|dx+Ω|bx|dx},
ϕxt=·ϕx|ϕx|+λ(Ixgx)Ixϕx,axt=·ax|ax|+λ(Ixgx)Ixax,bxt=·bx|bx|+λ(Ixgx)Ixbx,
ϕxν=0,axν=0,andbxν=0,
(x1,x2)=((2i1)hx12,(2j1)hx22)
δx1±ϕi,j=±ϕi±1,jϕi,jhx1
δx2±ϕi,j=±ϕi,j±1ϕi,jhx2.
·Vi,j=δx1Vi,j1+δx2Vi,j2,
Vi,j=(Vi,j1,Vi,j2)=ϕi,j|ϕ|i,j,ϕi,j=(δx1+ϕi,j,δx2+ϕi,j),|ϕ|i,j=(δx1+ϕi,j)2+(δx2+ϕi,j)2.
ϕi,0=ϕi,1,ϕi,n+1=ϕi,n,ϕ0,j=ϕ1,j,ϕm+1,j=ϕm,j,ϕ0,0=ϕ1,1,ϕ0,n+1=ϕ1,n,ϕm+1,0=ϕm,1,ϕm+1,n+1=ϕm,n.
Ix={ax=0.6,bx=0.5,if90(x1,x2)160ϕx=1.2+0.005xax=1.0,bx=0.9,otherwiseϕx=0.2+0.005x,φx=2πPx,
F=λ2Ω[Ixgx]2dx.=Ω[ax+bxcos(φx+ϕx)gx]2dx.
d2Fdax2=2Ωdx>0,
d2Fdbx2=2Ωcos2(φx+ϕx)dx0,
d2Fdϕx2=2Ω(gxax)cos(φx+ϕx)bxcos(2(φx+ϕx))dx.

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