Abstract

We present two robust algorithms for fringe pattern analysis with partial-field and closed fringes. The algorithm for partial-field fringe patterns is presented as a refinement method for precomputed coarse phases. Such an algorithm consists of the minimization of a regularized cost function that incorporates an outlier rejection strategy, which causes the algorithm to become robust. On the basis of the phase refinement method, we propose a propagative scheme for phase retrieval from closed-fringe interferograms. The algorithm performance is demonstrated by demodulating closed-fringe interferograms with complex spatial distribution of stationary points and gradients in the illumination components.

© 2005 Optical Society of America

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References

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  1. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 4, 156–160 (1982).
    [CrossRef]
  2. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. (Bellingham) 23, 391–395 (1984).
    [CrossRef]
  3. D. W. Robinson and G. T. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, UK, 1993).
  4. J. L. Marroquin, J. E. Figueroa, M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A 14, 779–791 (1997).
    [CrossRef]
  5. J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A 14, 1742–1753 (1997).
    [CrossRef]
  6. 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]
  7. J. L. Marroquin, R. Rodriguez-Vera, M. Servin, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A 15, 1536–1544 (1998)
    [CrossRef]
  8. J. Villa, J. A. Quiroga, M. Servin, “Improved regularized phase-tracking technique for the processing of squared-grating deflectograms” Appl. Opt. 39, 502–508 (2000).
    [CrossRef]
  9. M. Servin, J. L. Marroquin, F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18, 689–695 (2001).
    [CrossRef]
  10. K. G. Larkin, D. Bone, M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18, 1862–1870 (2001).
    [CrossRef]
  11. K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18, 1871–1881 (2001).
    [CrossRef]
  12. R. Legarda-Saenz, 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]
  13. M. Servin, J. A. Quiroga, J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20, 925–934 (2003)
    [CrossRef]
  14. M. Servin, J. L. Marroquin, J. A. Quiroga, “Regularized quadrature and phase tracking from a single closed-fringe interferogram,” J. Opt. Soc. Am. A 21, 411–419 (2004).
    [CrossRef]
  15. D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Image Process. 14, 367–383 (1992).
  16. M. J. Black, 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]
  17. P. Charbonnier, L. Blanc-Féraud, G. Aubert, M. Barlaud, “Deterministic edge-preserving regularization in computer imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
    [CrossRef]
  18. M. Rivera, J. L. Marroquin, “Adaptive rest condition potentials: Second order edge-preserving regularization,” Comput. Vis. Image Underst. 88, 76–93 (2002).
    [CrossRef]
  19. M. Rivera, J. L. Marroquin, “Efficient half-quadratic regularization with granularity control,” Image Vis. Comput. 21, 345–357 (2003).
    [CrossRef]
  20. M. Rivera, J. L. Marroquin, “Half-quadratic cost functions for phase unwrapping,” Opt. Lett. 29, 504–506 (2004).
    [CrossRef] [PubMed]
  21. B. Jahne, Digital Image Processing, 5th ed., (Springer-Verlag, Berlin, 2002).
    [CrossRef]

2004

2003

M. Rivera, J. L. Marroquin, “Efficient half-quadratic regularization with granularity control,” Image Vis. Comput. 21, 345–357 (2003).
[CrossRef]

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

2002

R. Legarda-Saenz, 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]

M. Rivera, J. L. Marroquin, “Adaptive rest condition potentials: Second order edge-preserving regularization,” Comput. Vis. Image Underst. 88, 76–93 (2002).
[CrossRef]

2001

2000

1998

1997

1996

M. J. Black, 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]

1992

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Image Process. 14, 367–383 (1992).

1984

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. (Bellingham) 23, 391–395 (1984).
[CrossRef]

1982

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 4, 156–160 (1982).
[CrossRef]

Aubert, G.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, M. Barlaud, “Deterministic edge-preserving regularization in computer imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef]

Barlaud, M.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, M. Barlaud, “Deterministic edge-preserving regularization in computer imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef]

Black, M. J.

M. J. Black, 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-Féraud, L.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, M. Barlaud, “Deterministic edge-preserving regularization in computer imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef]

Bone, D.

Charbonnier, P.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, M. Barlaud, “Deterministic edge-preserving regularization in computer imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef]

Cuevas, F. J.

Figueroa, J. E.

Geman, D.

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Image Process. 14, 367–383 (1992).

Ina, H.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 4, 156–160 (1982).
[CrossRef]

Jahne, B.

B. Jahne, Digital Image Processing, 5th ed., (Springer-Verlag, Berlin, 2002).
[CrossRef]

Jüptner, W.

Kobayashi, S.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 4, 156–160 (1982).
[CrossRef]

Larkin, K. G.

Legarda-Saenz, R.

Marroquin, J. L.

M. Servin, J. L. Marroquin, J. A. Quiroga, “Regularized quadrature and phase tracking from a single closed-fringe interferogram,” J. Opt. Soc. Am. A 21, 411–419 (2004).
[CrossRef]

M. Rivera, J. L. Marroquin, “Half-quadratic cost functions for phase unwrapping,” Opt. Lett. 29, 504–506 (2004).
[CrossRef] [PubMed]

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

M. Rivera, J. L. Marroquin, “Efficient half-quadratic regularization with granularity control,” Image Vis. Comput. 21, 345–357 (2003).
[CrossRef]

M. Rivera, J. L. Marroquin, “Adaptive rest condition potentials: Second order edge-preserving regularization,” Comput. Vis. Image Underst. 88, 76–93 (2002).
[CrossRef]

M. Servin, J. L. Marroquin, F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18, 689–695 (2001).
[CrossRef]

J. L. Marroquin, R. Rodriguez-Vera, M. Servin, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A 15, 1536–1544 (1998)
[CrossRef]

J. L. Marroquin, J. E. Figueroa, M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A 14, 779–791 (1997).
[CrossRef]

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. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A 14, 1742–1753 (1997).
[CrossRef]

Oldfield, M. A.

Osten, W.

Quiroga, J. A.

Rangarajan, A.

M. J. Black, 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, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Image Process. 14, 367–383 (1992).

Rivera, M.

M. Rivera, J. L. Marroquin, “Half-quadratic cost functions for phase unwrapping,” Opt. Lett. 29, 504–506 (2004).
[CrossRef] [PubMed]

M. Rivera, J. L. Marroquin, “Efficient half-quadratic regularization with granularity control,” Image Vis. Comput. 21, 345–357 (2003).
[CrossRef]

M. Rivera, J. L. Marroquin, “Adaptive rest condition potentials: Second order edge-preserving regularization,” Comput. Vis. Image Underst. 88, 76–93 (2002).
[CrossRef]

Rodriguez-Vera, R.

Servin, M.

Takeda, M.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 4, 156–160 (1982).
[CrossRef]

Villa, J.

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. (Bellingham) 23, 391–395 (1984).
[CrossRef]

Appl. Opt.

Comput. Vis. Image Underst.

M. Rivera, J. L. Marroquin, “Adaptive rest condition potentials: Second order edge-preserving regularization,” Comput. Vis. Image Underst. 88, 76–93 (2002).
[CrossRef]

IEEE Trans. Image Process.

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Image Process. 14, 367–383 (1992).

P. Charbonnier, L. Blanc-Féraud, G. Aubert, M. Barlaud, “Deterministic edge-preserving regularization in computer imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef]

Image Vis. Comput.

M. Rivera, J. L. Marroquin, “Efficient half-quadratic regularization with granularity control,” Image Vis. Comput. 21, 345–357 (2003).
[CrossRef]

Int. J. Comput. Vis.

M. J. Black, 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.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 4, 156–160 (1982).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng. (Bellingham)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. (Bellingham) 23, 391–395 (1984).
[CrossRef]

Opt. Lett.

Other

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

B. Jahne, Digital Image Processing, 5th ed., (Springer-Verlag, Berlin, 2002).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Cliques with triads of pixels q , r , s .

Fig. 2
Fig. 2

Phase refinement. (a) Fringe pattern. (b) Approximated phase. (c) Mask. (d) Computed refined phase. The phases in (b) and (d) are rewrapped for display purposes.

Fig. 3
Fig. 3

Details of the phase refinement. The illustrated region corresponds to a rectangular region ( 80 × 80 pixels) located at the lower left of Figs. 2b, 2c, 2d. (a) Approximated phase, (b) mask, (c) refined phase.

Fig. 4
Fig. 4

Synthetic fringe pattern (top left) and a computed phase propagation sequence, rewrapped for visualization purposes.

Fig. 5
Fig. 5

Closed-fringe analysis. (a) Fringe pattern, (b) gray-scale plots of the row (top) and the column (bottom) that cross at the fringe center, (c) binary map of fringes, (d) coherency map, (e) computed phase, (f) rewrapped phase, for display purposes.

Fig. 6
Fig. 6

Phase-tracker reconstructions with (a) RPT reported in Ref. [9]. (b) RPT with joint estimation of the phase, f, and the contrast, b.[12]

Tables (2)

Tables Icon

Table 1 Algorithm 1: Phase Refinement

Tables Icon

Table 2 Algorithm 2: Closed-Fringe Analysis

Equations (26)

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g r = a r + b r cos ( f r ) + η r ,
g ̂ r = b ̂ r cos f r ,
E ( ϕ r ; ψ r ) = def g ̂ r b ̂ r ( cos ψ r ϕ r sin ψ r ) 0 .
U 1 ( ϕ , ω ; ψ ) = r R [ ω r 2 E 2 ( ϕ r ; ψ r ) + μ ( 1 ω r ) 2 ] + λ q , r , s R [ ψ q + ϕ q 2 ( ψ r + ϕ r ) + ψ s + ϕ s ] 2 ,
ω r = μ [ μ + E 2 ( ϕ r ; ψ r ) ] .
g ̂ b ̂ . cos ψ g ̂ b ̂ . ( cos ψ ϕ sin ψ ) g ̂ b ̂ . cos ( ψ + ϕ ) 0
S = { s T s R , r R , r s < d } .
U 2 ( ψ ) = q , r , s : { q , r , s } S ( ψ q 2 ψ r + ψ s ) 2 ,
U 3 ( ω ̂ ) = r S { [ ω ̂ r 2 E 2 ( ψ r ) + μ ( 1 ω ̂ r ) 2 ] + λ ̂ s N r d r s ( ω ̂ r ω ̂ s ) 2 } ,
d r s = exp [ ( r s ) T J r ( r s ) ] ,
J = trace [ ( τ τ T ) α ] I ( τ τ T ) α
R n + 1 = R n { r S c r ω ̂ r > θ , card ( N r R n ) 2 } ,
c r = ( λ 1 r λ 2 r ) ( λ 1 r + λ 2 r )
ϕ i j = ω i j 2 b ̂ i j sin ψ i j ( b ̂ i j cos ψ i j g i j ) + λ V ( i , j ) ω i j 2 b ̂ i j 2 sin 2 ψ i j + λ W ( i , j ) ,
V ( i , j ) = def c = 1 1 [ P 1 ( i c , j ) + P 2 ( i , j c ) + P 3 ( i c , j c ) + P 4 ( i c , j + c ) ] ,
W ( i , j ) = def c = 1 1 [ N 1 2 ( i c , j ) + N 2 2 ( i , j c ) + N 3 2 ( i c , j c ) + N 4 2 ( i c , j + c ) ] ,
P m ( k , l ) = def ϕ i j N m 2 ( k , l ) Q m ( k , l ) N m ( k , l ) ,
N m ( k , l ) = def A m ( k , l ) ϕ i j Q m ( ϕ , k , l ) ,
A 1 ( k , l ) = { 1 if ( k 1 , l ) , ( k , l ) , ( k + 1 , l ) R 0 otherwise ,
A 2 ( k , l ) = { 1 if ( k , l 1 ) , ( k , l ) , ( k , l + 1 ) R 0 otherwise ,
A 3 ( k , l ) = { 1 if ( k 1 , l l ) , ( k , l ) , ( k + 1 , l + 1 ) R 0 otherwise ,
A 4 ( k , l ) = { 1 if ( k 1 , l + 1 ) , ( k , l ) , ( k + 1 , l 1 ) R 0 otherwise ,
Q 1 ( k , l ) = def ψ k 1 , l + ϕ k 1 , l 2 ( ψ k , l + ϕ k , l ) + ψ k + 1 , l + ϕ k + 1 , l ,
Q 2 ( k , l ) = def ψ k , l 1 + ϕ k , l 1 2 ( ψ k , l + ϕ k , l ) + ψ k , l + 1 + ϕ k , l + 1 ,
Q 3 ( k , l ) = def ψ k 1 , l 1 + ϕ k 1 , l 1 2 ( ψ k , l + ϕ k , l ) + ψ k + 1 , l + 1 + ϕ k + 1 , l + 1 ,
Q 4 ( k , l ) = def ψ k 1 , l + 1 + ϕ k 1 , l + 1 2 ( ψ k , l + ϕ k , l ) + ψ k + 1 , l 1 + ϕ k + 1 , l 1 .

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