Abstract

A new framework for phase recovery from a single fringe pattern with closed fringes is proposed. Our algorithm constructs an unwrapped phase from previously computed phases with a simple open-fringe-analysis algorithm, twice applied for analyzing horizontal and vertical oriented fringes, respectively. That reduces the closed-fringe-analysis problem to that of choosing the better phase between the two oriented computed phases and then of estimating the local sign. By propagating the phase sign [and a tilewise constant (DC) term] by regions [here named tiles] instead of a pixelwise phase propagation, our analysis of closed-fringe patterns becomes more robust and faster. Additionally, we propose a multigrid refinement for improving the final computed phase.

© 2008 Optical Society of America

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    [CrossRef]
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2007 (3)

2006 (2)

2005 (3)

2004 (3)

2003 (1)

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

2001 (3)

2000 (1)

H. Knutsson and M. Andersson, “Implications of invariance and uncertainty for local structure analysis filter sets,” Signal Process. Image Commun. 20, 569-581 (2000).
[CrossRef]

1998 (2)

1997 (2)

1992 (1)

T. R. Judge, C. Quan, and P. J. Bryanston Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. (Bellingham) 31, 533-543 (1992).
[CrossRef]

1986 (1)

1984 (2)

K. H. Womak, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. (Bellingham) 23, 391-395 (1984).

J. J. Koenderink, “The structure of images,” Biol. Cybern. 50, 363-370 (1984).
[CrossRef] [PubMed]

1983 (1)

1982 (1)

Andersson, M.

H. Knutsson and M. Andersson, “Implications of invariance and uncertainty for local structure analysis filter sets,” Signal Process. Image Commun. 20, 569-581 (2000).
[CrossRef]

Bizuet, R.

Bone, D.

Botello, S.

Bowler, I. W.

I. W. Bowler and K. Paler, “A Gabor filter approach to fringe analysis, in Proceedings of the International Conference on Pattern Recognition (ICPR'86) (ICPR, 1986), pp. 558-560.

Briggs, W. L.

W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, 2nd ed. (SIAM, 2000).
[CrossRef]

Bryanston Cross, P. J.

T. R. Judge, C. Quan, and P. J. Bryanston Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. (Bellingham) 31, 533-543 (1992).
[CrossRef]

Cuevas, F.

Dardyk, G.

G. Dardyk and I. Yavneh, “A multigrid approach to two-dimensional phase unwrapping,” Numer. Linear Algebra Appl. 11, 241-259 (2004).
[CrossRef]

Estrada, J. C.

Felsberg, M.

M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136-3144 (2001).
[CrossRef]

Fleet, D. J.

D. J. Fleet and Y. Weiss, “Optical flow estimation,” in Mathematical Models for Computer Vision: The Handbook, N.Paragios, Y.Chen, and O.Faugeras, eds. (Springer, 2005), Chap. 4.

Fu, S.

X. Yang, Q. Yu, and S. Fu, “An algorithm for estimating both fringe orientation and fringe density,” Opt. Commun. 274, 286-292 (2007).
[CrossRef]

Granlund, G. H.

L. Haglund, H. Knutsson, and G. H. Granlund, “Scale and orientation adaptive filtering,” in Proceedings of the 8th Scandinavian Conference on Image Analysis (NOBIM, May 1993, Report LiTH-ISY-I-1527, Computer Vision Laboratory, Linköping University, 1993), pp. 1-26.

Guerrero, J. A.

Haglund, L.

L. Haglund, H. Knutsson, and G. H. Granlund, “Scale and orientation adaptive filtering,” in Proceedings of the 8th Scandinavian Conference on Image Analysis (NOBIM, May 1993, Report LiTH-ISY-I-1527, Computer Vision Laboratory, Linköping University, 1993), pp. 1-26.

Henson, V. E.

W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, 2nd ed. (SIAM, 2000).
[CrossRef]

Ina, H.

Jähne, B.

B. Jähne, Digital Image Processing; Concepts, Algorithms, and Scientific Applications, 2nd ed. (Springer-Verlag, 1991).
[PubMed]

Judge, T. R.

T. R. Judge, C. Quan, and P. J. Bryanston Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. (Bellingham) 31, 533-543 (1992).
[CrossRef]

Kemao, Q.

Knutsson, H.

H. Knutsson and M. Andersson, “Implications of invariance and uncertainty for local structure analysis filter sets,” Signal Process. Image Commun. 20, 569-581 (2000).
[CrossRef]

H. Knutsson, “Representing local structure using tensors,” in The 6th Scandinavian Conference on Image Analysis (Report LiTH-ISY-I-1019, Computer Vision Laboratory, Linköping University, 1989), pp. 244-251.

H. Knutsson, “A tensor representation of 3-D structure,” in 5th IEEE-ASSP and EURASIP Workshop on Multidimensional Signal Processing (IEEE, 1987), poster presentation.

L. Haglund, H. Knutsson, and G. H. Granlund, “Scale and orientation adaptive filtering,” in Proceedings of the 8th Scandinavian Conference on Image Analysis (NOBIM, May 1993, Report LiTH-ISY-I-1527, Computer Vision Laboratory, Linköping University, 1993), pp. 1-26.

Kobayashi, S.

Koenderink, J. J.

J. J. Koenderink, “The structure of images,” Biol. Cybern. 50, 363-370 (1984).
[CrossRef] [PubMed]

Kreis, T.

Larkin, K. G.

Legarda-Saenz, R.

Macy, W. W.

Marroquin, J.

Marroquin, J. L.

Martínez, A.

McCormick, S. F.

W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, 2nd ed. (SIAM, 2000).
[CrossRef]

Oldfield, M. A.

Paler, K.

I. W. Bowler and K. Paler, “A Gabor filter approach to fringe analysis, in Proceedings of the International Conference on Pattern Recognition (ICPR'86) (ICPR, 1986), pp. 558-560.

Pritt, M. D.

M. D. Pritt, “Multigrid phase unwrapping for interferometric SAR,” in Proceedings of IGARSS (IGARSS, 1995), Vol. 1, pp. 562-564.

Quan, C.

T. R. Judge, C. Quan, and P. J. Bryanston Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. (Bellingham) 31, 533-543 (1992).
[CrossRef]

Quiroga, J. A.

Rayas, J. A.

Rivera, M.

Rodriguez-Vera, R.

Servin, M.

Sommer, G.

M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136-3144 (2001).
[CrossRef]

Soon, H.

Takeda, M.

Weiss, Y.

D. J. Fleet and Y. Weiss, “Optical flow estimation,” in Mathematical Models for Computer Vision: The Handbook, N.Paragios, Y.Chen, and O.Faugeras, eds. (Springer, 2005), Chap. 4.

Witkin, A. P.

A. P. Witkin, “Scale-space filtering,” in Proceedings of the 8th International Joint Conference on Artificial Intelligence (Morgan Kaufmann, 1983), 1019-1021.

Womak, K. H.

K. H. Womak, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. (Bellingham) 23, 391-395 (1984).

Yang, X.

X. Yang, Q. Yu, and S. Fu, “An algorithm for estimating both fringe orientation and fringe density,” Opt. Commun. 274, 286-292 (2007).
[CrossRef]

Yavneh, I.

G. Dardyk and I. Yavneh, “A multigrid approach to two-dimensional phase unwrapping,” Numer. Linear Algebra Appl. 11, 241-259 (2004).
[CrossRef]

Yu, Q.

X. Yang, Q. Yu, and S. Fu, “An algorithm for estimating both fringe orientation and fringe density,” Opt. Commun. 274, 286-292 (2007).
[CrossRef]

Appl. Opt. (3)

Biol. Cybern. (1)

J. J. Koenderink, “The structure of images,” Biol. Cybern. 50, 363-370 (1984).
[CrossRef] [PubMed]

IEEE Trans. Signal Process. (1)

M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136-3144 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

Numer. Linear Algebra Appl. (1)

G. Dardyk and I. Yavneh, “A multigrid approach to two-dimensional phase unwrapping,” Numer. Linear Algebra Appl. 11, 241-259 (2004).
[CrossRef]

Opt. Commun. (2)

X. Yang, Q. Yu, and S. Fu, “An algorithm for estimating both fringe orientation and fringe density,” Opt. Commun. 274, 286-292 (2007).
[CrossRef]

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

Opt. Eng. (Bellingham) (2)

K. H. Womak, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. (Bellingham) 23, 391-395 (1984).

T. R. Judge, C. Quan, and P. J. Bryanston Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. (Bellingham) 31, 533-543 (1992).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Signal Process. Image Commun. (1)

H. Knutsson and M. Andersson, “Implications of invariance and uncertainty for local structure analysis filter sets,” Signal Process. Image Commun. 20, 569-581 (2000).
[CrossRef]

Other (9)

B. Jähne, Digital Image Processing; Concepts, Algorithms, and Scientific Applications, 2nd ed. (Springer-Verlag, 1991).
[PubMed]

L. Haglund, H. Knutsson, and G. H. Granlund, “Scale and orientation adaptive filtering,” in Proceedings of the 8th Scandinavian Conference on Image Analysis (NOBIM, May 1993, Report LiTH-ISY-I-1527, Computer Vision Laboratory, Linköping University, 1993), pp. 1-26.

I. W. Bowler and K. Paler, “A Gabor filter approach to fringe analysis, in Proceedings of the International Conference on Pattern Recognition (ICPR'86) (ICPR, 1986), pp. 558-560.

H. Knutsson, “Representing local structure using tensors,” in The 6th Scandinavian Conference on Image Analysis (Report LiTH-ISY-I-1019, Computer Vision Laboratory, Linköping University, 1989), pp. 244-251.

W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, 2nd ed. (SIAM, 2000).
[CrossRef]

M. D. Pritt, “Multigrid phase unwrapping for interferometric SAR,” in Proceedings of IGARSS (IGARSS, 1995), Vol. 1, pp. 562-564.

D. J. Fleet and Y. Weiss, “Optical flow estimation,” in Mathematical Models for Computer Vision: The Handbook, N.Paragios, Y.Chen, and O.Faugeras, eds. (Springer, 2005), Chap. 4.

A. P. Witkin, “Scale-space filtering,” in Proceedings of the 8th International Joint Conference on Artificial Intelligence (Morgan Kaufmann, 1983), 1019-1021.

H. Knutsson, “A tensor representation of 3-D structure,” in 5th IEEE-ASSP and EURASIP Workshop on Multidimensional Signal Processing (IEEE, 1987), poster presentation.

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Figures (9)

Fig. 1
Fig. 1

(a) Level set of the tensors’ quadratic form superimposed on the FP, I, of Fig. 2b; see text. (b) Coherence map, C . (c) Quality measure map, B . (d) Propagation map, U.

Fig. 2
Fig. 2

(a) FP with open fringes, (b) FP with closed fringes, (c) and (d) phases computed with a horizontally oriented quadrature filter, ϕ v . (e) and (f) Phases computed with a vertically oriented quadrature filter, ϕ u .

Fig. 3
Fig. 3

(a) Selection map: in white if the phase is taken from ϕ v and in black if the phase is taken from ϕ u . (b) Unwrapped phase by tiles, ϕ ̃ .

Fig. 4
Fig. 4

Illustration of the tiles' coupling band.

Fig. 5
Fig. 5

Illustration of the multigrid refinement process.

Fig. 6
Fig. 6

Phase recovered from single closed-fringe patterns with the proposed method. Fringe pattern, I (first column). Resultant estimated phase of the propagation stage, Φ (middle column). Refined phase, Φ * , that results from the complete proposed procedure (last column).

Fig. 7
Fig. 7

Phases (rewrapped for illustration purposes) computed with RPT algorithms from the FP in Fig. 6 (row 6). The phases correspond to the propagation step results.

Fig. 8
Fig. 8

Phase results computed by partitioning the fringe pattern in tiles with size of n × n pixels.

Fig. 9
Fig. 9

From top down: Refinement evolution of the weights and the phase; the phase is wrapped for illustration purposes in the refinement iterations (see text).

Tables (4)

Tables Icon

Table 1 Algorithm 1: Computation of the Propagation List, U.

Tables Icon

Table 2 Algorithm 2: Multigrid Phase Refinement (MPR).

Tables Icon

Table 1 Algorithm Performance on Real and Synthetic Fringe Patterns

Tables Icon

Table 2 Propagation Time for RPT Algorithms: Linearized (LRPT) and Nonlinear (bRPT) Versions

Equations (26)

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

I ( r ) = a ( r ) + b ( r ) cos [ f ( r ) ] + η ( r ) , r L ,
W P W Q = 0 , P Q
P W P = L .
T P = 1 P ( r P I x 2 r P I x I y r P I x I y r P I y 2 ) ,
O P = [ T 11 P T 22 P 2 T 12 P ] .
C P = ( T 11 P T 22 P ) 2 + 4 ( T 12 P ) 2 ( T 11 P + T 22 P ) 2 .
B P = λ 1 P C P .
ϕ ̃ P = { W 1 ( ϕ u P ) if T 11 P T 22 P W 1 ( ϕ v P ) otherwise ,
N U = { Q P \ U : ( r , s ) , s Q , r P , P U , r s = 1 } .
Φ P b = σ P b ϕ ̃ P b + δ P b ;
δ i * = arg min δ R F ̃ i ( δ ) ,
i * = arg min i { 1 , 2 } F ̃ i ( δ i * ) .
F i ( δ ) = r P a P b 3 t 3 [ σ i ( r ) S ( r ) + δ ( r ) ] 2 d r ,
δ ( r ) = d e f { δ if r P b 0 if r P a ,
σ i ( r ) = d e f { ( 1 ) i + 1 if r P b 1 if r P a .
F ̃ 1 ( δ ) = i = 0 m 1 { { s [ i ] [ 0 ] 3 s [ i ] [ 1 ] + 3 s [ i ] [ 2 ] ( s [ i ] [ 3 ] + δ ) } 2 + { s [ i ] [ 1 ] 3 s [ i ] [ 2 ] + 3 ( s [ i ] [ 3 ] + δ ) ( s [ i ] [ 4 ] + δ ) } 2 + { s [ i ] [ 2 ] 3 ( s [ i ] [ 3 ] + δ ) + 3 ( s [ i ] [ 4 ] + δ ) ( s [ i ] [ 5 ] + δ ) } 2 } ,
F ̃ 2 ( δ ) = i = 0 m 1 { { s [ i ] [ 0 ] 3 s [ i ] [ 1 ] + 3 s [ i ] [ 2 ] + ( s [ i ] [ 3 ] δ ) } 2 + { s [ i ] [ 1 ] 3 s [ i ] [ 2 ] 3 ( s [ i ] [ 3 ] δ ) + ( s [ i ] [ 4 ] δ ) } 2 + { s [ i ] [ 2 ] + 3 ( s [ i ] [ 3 ] δ ) 3 ( s [ i ] [ 4 ] δ ) + ( s [ i ] [ 5 ] δ ) } 2 } .
δ 1 * = i = 0 m 1 j = 0 5 ( 1 ) j ( 5 j ) s [ i ] [ j ] 6 m ,
δ 2 * = i = 0 m 1 j = 0 5 ( 1 ) j + δ b ( 5 j ) s [ i ] [ j ] 6 m ,
δ b = d e f { 1 if j > 2 0 otherwise .
E ( ψ , r ) = I ( r ) a ̂ ( r ) b ̂ ( r ) [ cos Φ ( r ) ψ ( r ) sin Φ ( r ) ] 0 .
U ( ψ , ω ) = r R [ ω r 2 E 2 ( ψ , r ) + μ ( 1 ω r ) 2 + γ ψ 2 ( r ) ] + λ q , r , s R [ Φ ( q ) + ψ ( q ) 2 ( Φ ( r ) + ψ ( r ) ) + Φ ( s ) + ψ ( s ) ] 2 ,
R μ , λ , I Φ = d e f Φ + ψ * ,
Φ 1 = D 2 Φ 0 .
Φ ̃ 0 = E 2 R μ , λ 4 , D 2 I Φ 1 .
Φ 0 * = R μ , λ , I Φ ̃ 0 .

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