Abstract

We analyze the characteristics of noise-induced phase inconsistencies, or residues, in a wrapped phase map. Because residues are the potential source of phase-error propagation, it is essential to filter them before two-dimensional phase unwrapping. We propose an unsupervised-clustering-driven noise-residue filter, and apply it as a preprocessing procedure of phase unwrapping. The filter is based on the fact that most residues are present in the form of adjacency caused by noisy wrapped phases. These noisy phases differ from the other correct ones numerically within the local k1×k2 window containing the adjacent residues, and it is possible to group the correct and noisy wrapped phases into different clusters. The window size is determined adaptively according to the local noise level. The proposed procedure avoids constructing branch cuts, and converts path-following unwrapping to path independence, which improves the operating speed of phase unwrapping significantly. The tests performed on simulated and real projected fringe patterns confirm the validity of our approach in residue reduction, fringe preservation, and rapidity.

© 2010 Optical Society of America

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References

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  1. K. Itoh, “Analysis of the phase unwrapping problem,” Appl. Opt. 21, 2470 (1982).
    [CrossRef] [PubMed]
  2. K. A. Stetson, J. Wahid, and P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 36, 4830-4838 (1997).
    [CrossRef] [PubMed]
  3. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
    [CrossRef]
  4. R. Cusack, J. M. Huntley, and H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781-789 (1995).
    [CrossRef] [PubMed]
  5. M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813-821 (1998).
    [CrossRef]
  6. C. W. Chen and H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A 17, 401-414 (2000).
    [CrossRef]
  7. A. Baldi, “Phase unwrapping by region growing,” Appl. Opt. 42, 2498-2505 (2003).
    [CrossRef] [PubMed]
  8. X. Y. Su and W. J. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245-261 (2004).
    [CrossRef]
  9. D. C. Ghiglia and L. A. Romero, “Minimum lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999-2013 (1996).
    [CrossRef]
  10. M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728-739 (1996).
    [CrossRef]
  11. J. Strand, T. Taxt, and A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375-386 (1999).
    [CrossRef]
  12. J. Strand and T. Taxt, “Two-dimensional phase unwrapping using robust derivative estimation and adaptive integration,” IEEE Trans. Image Process. 11, 1192-1200 (2002).
    [CrossRef]
  13. B. Friedlander and J. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999-3007 (1996).
    [CrossRef]
  14. Z. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893-897 (1996).
    [CrossRef] [PubMed]
  15. J. M. N. Leitao and M. A. T. Figueiredo, “Absolute phase image reconstruction: a stochastic nonlinear filtering approach,” IEEE Trans. Image Process. 7, 868-882 (1998).
    [CrossRef]
  16. G. Nico, G. Palubinskas, and M. Datcu, “Bayesian approaches to phase unwrapping: theoretical study,” IEEE Trans. Signal Process. 48, 2545-2556 (2000).
    [CrossRef]
  17. A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466-2472 (1997).
    [CrossRef]
  18. J. S. Lee, K. P. Papathanassiou, and T. L. Ainsworth, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1173(1998).
    [CrossRef]
  19. G. Nico, “Noise-residue filtering of interferometric phase images,” J. Opt. Soc. Am. A 17, 1962-1974 (2000).
    [CrossRef]
  20. K. M. Qian, W. J. Gao, and H. X. Wang, “Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm,” Appl. Opt. 47, 5420-5428 (2008).
    [CrossRef]
  21. G. Fornaro, A. Pauciullo, and E. Sansosti, “Phase difference-based multichannel phase unwrapping,” IEEE Trans. Image Process. 14, 960-972 (2005).
    [CrossRef] [PubMed]
  22. D. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627-3632 (1991).
    [CrossRef] [PubMed]
  23. S. A. Karout, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Residue vector, an approach to branch-cut placement in phase unwrapping: theoretical study,” Appl. Opt. 46, 4712-4727 (2007).
    [CrossRef] [PubMed]
  24. J. Jiang and J. Cheng, “Noise-residue filtering based on unsupervised clustering for phase unwrapping,” presented at the Fifth International Symposium on Visual Computing, Las Vegas, Nev., USA, 30 Nov.-2 Dec. 2009.
  25. P. Berkhin, Survey Of Clustering Data Mining Techniques (Springer, 2002).
  26. J. C. Bezdek and N. R. Pal, “Some new indexes of cluster validity,” IEEE Trans. Syst. Man Cybern. B 28, 301-315(1998).
    [CrossRef]
  27. X. Su, G. Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150(1993).
    [CrossRef]
  28. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

2008 (1)

2007 (1)

2005 (1)

G. Fornaro, A. Pauciullo, and E. Sansosti, “Phase difference-based multichannel phase unwrapping,” IEEE Trans. Image Process. 14, 960-972 (2005).
[CrossRef] [PubMed]

2004 (1)

X. Y. Su and W. J. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

2003 (1)

2002 (1)

J. Strand and T. Taxt, “Two-dimensional phase unwrapping using robust derivative estimation and adaptive integration,” IEEE Trans. Image Process. 11, 1192-1200 (2002).
[CrossRef]

2000 (3)

1999 (1)

J. Strand, T. Taxt, and A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375-386 (1999).
[CrossRef]

1998 (4)

J. M. N. Leitao and M. A. T. Figueiredo, “Absolute phase image reconstruction: a stochastic nonlinear filtering approach,” IEEE Trans. Image Process. 7, 868-882 (1998).
[CrossRef]

J. S. Lee, K. P. Papathanassiou, and T. L. Ainsworth, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1173(1998).
[CrossRef]

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813-821 (1998).
[CrossRef]

J. C. Bezdek and N. R. Pal, “Some new indexes of cluster validity,” IEEE Trans. Syst. Man Cybern. B 28, 301-315(1998).
[CrossRef]

1997 (2)

K. A. Stetson, J. Wahid, and P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 36, 4830-4838 (1997).
[CrossRef] [PubMed]

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

1996 (4)

B. Friedlander and J. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999-3007 (1996).
[CrossRef]

Z. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893-897 (1996).
[CrossRef] [PubMed]

D. C. Ghiglia and L. A. Romero, “Minimum lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999-2013 (1996).
[CrossRef]

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728-739 (1996).
[CrossRef]

1995 (1)

1993 (1)

X. Su, G. Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150(1993).
[CrossRef]

1991 (1)

1988 (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

1982 (1)

Ainsworth, T. L.

J. S. Lee, K. P. Papathanassiou, and T. L. Ainsworth, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1173(1998).
[CrossRef]

Baldi, A.

Bally, G.

X. Su, G. Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150(1993).
[CrossRef]

Berkhin, P.

P. Berkhin, Survey Of Clustering Data Mining Techniques (Springer, 2002).

Bertani, D.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Bezdek, J. C.

J. C. Bezdek and N. R. Pal, “Some new indexes of cluster validity,” IEEE Trans. Syst. Man Cybern. B 28, 301-315(1998).
[CrossRef]

Bone, D.

Burton, D. R.

Capanni, A.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Cetica, M.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Chen, C. W.

Chen, W. J.

X. Y. Su and W. J. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

Cheng, J.

J. Jiang and J. Cheng, “Noise-residue filtering based on unsupervised clustering for phase unwrapping,” presented at the Fifth International Symposium on Visual Computing, Las Vegas, Nev., USA, 30 Nov.-2 Dec. 2009.

Costantini, M.

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813-821 (1998).
[CrossRef]

Cusack, R.

Datcu, M.

G. Nico, G. Palubinskas, and M. Datcu, “Bayesian approaches to phase unwrapping: theoretical study,” IEEE Trans. Signal Process. 48, 2545-2556 (2000).
[CrossRef]

Figueiredo, M. A. T.

J. M. N. Leitao and M. A. T. Figueiredo, “Absolute phase image reconstruction: a stochastic nonlinear filtering approach,” IEEE Trans. Image Process. 7, 868-882 (1998).
[CrossRef]

Fornaro, G.

G. Fornaro, A. Pauciullo, and E. Sansosti, “Phase difference-based multichannel phase unwrapping,” IEEE Trans. Image Process. 14, 960-972 (2005).
[CrossRef] [PubMed]

Francini, F.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Francos, J.

B. Friedlander and J. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999-3007 (1996).
[CrossRef]

Friedlander, B.

B. Friedlander and J. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999-3007 (1996).
[CrossRef]

Gao, W. J.

Gauthier, P.

Gdeisat, M. A.

Ghiglia, D. C.

D. C. Ghiglia and L. A. Romero, “Minimum lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999-2013 (1996).
[CrossRef]

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

Goldrein, H. T.

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

Huntley, J. M.

Itoh, K.

Jain, A. K.

J. Strand, T. Taxt, and A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375-386 (1999).
[CrossRef]

Jiang, J.

J. Jiang and J. Cheng, “Noise-residue filtering based on unsupervised clustering for phase unwrapping,” presented at the Fifth International Symposium on Visual Computing, Las Vegas, Nev., USA, 30 Nov.-2 Dec. 2009.

Karout, S. A.

Lalor, M. J.

Lee, J. S.

J. S. Lee, K. P. Papathanassiou, and T. L. Ainsworth, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1173(1998).
[CrossRef]

Leitao, J. M. N.

J. M. N. Leitao and M. A. T. Figueiredo, “Absolute phase image reconstruction: a stochastic nonlinear filtering approach,” IEEE Trans. Image Process. 7, 868-882 (1998).
[CrossRef]

Liang, Z.

Z. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893-897 (1996).
[CrossRef] [PubMed]

Nico, G.

G. Nico, G. Palubinskas, and M. Datcu, “Bayesian approaches to phase unwrapping: theoretical study,” IEEE Trans. Signal Process. 48, 2545-2556 (2000).
[CrossRef]

G. Nico, “Noise-residue filtering of interferometric phase images,” J. Opt. Soc. Am. A 17, 1962-1974 (2000).
[CrossRef]

Pal, N. R.

J. C. Bezdek and N. R. Pal, “Some new indexes of cluster validity,” IEEE Trans. Syst. Man Cybern. B 28, 301-315(1998).
[CrossRef]

Palubinskas, G.

G. Nico, G. Palubinskas, and M. Datcu, “Bayesian approaches to phase unwrapping: theoretical study,” IEEE Trans. Signal Process. 48, 2545-2556 (2000).
[CrossRef]

Papathanassiou, K. P.

J. S. Lee, K. P. Papathanassiou, and T. L. Ainsworth, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1173(1998).
[CrossRef]

Pauciullo, A.

G. Fornaro, A. Pauciullo, and E. Sansosti, “Phase difference-based multichannel phase unwrapping,” IEEE Trans. Image Process. 14, 960-972 (2005).
[CrossRef] [PubMed]

Pezzati, L.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Pritt, M. D.

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728-739 (1996).
[CrossRef]

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

Qian, K. M.

Romero, L. A.

Sansosti, E.

G. Fornaro, A. Pauciullo, and E. Sansosti, “Phase difference-based multichannel phase unwrapping,” IEEE Trans. Image Process. 14, 960-972 (2005).
[CrossRef] [PubMed]

Stetson, K. A.

Strand, J.

J. Strand and T. Taxt, “Two-dimensional phase unwrapping using robust derivative estimation and adaptive integration,” IEEE Trans. Image Process. 11, 1192-1200 (2002).
[CrossRef]

J. Strand, T. Taxt, and A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375-386 (1999).
[CrossRef]

Su, X.

X. Su, G. Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150(1993).
[CrossRef]

Su, X. Y.

X. Y. Su and W. J. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

Taxt, T.

J. Strand and T. Taxt, “Two-dimensional phase unwrapping using robust derivative estimation and adaptive integration,” IEEE Trans. Image Process. 11, 1192-1200 (2002).
[CrossRef]

J. Strand, T. Taxt, and A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375-386 (1999).
[CrossRef]

Vukicevic, D.

X. Su, G. Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150(1993).
[CrossRef]

Wahid, J.

Wang, H. X.

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

Zebker, H. A.

C. W. Chen and H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A 17, 401-414 (2000).
[CrossRef]

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

Appl. Opt. (7)

IEEE Trans. Geosci. Remote Sens. (3)

J. S. Lee, K. P. Papathanassiou, and T. L. Ainsworth, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1173(1998).
[CrossRef]

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813-821 (1998).
[CrossRef]

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728-739 (1996).
[CrossRef]

IEEE Trans. Image Process. (4)

J. Strand, T. Taxt, and A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375-386 (1999).
[CrossRef]

J. Strand and T. Taxt, “Two-dimensional phase unwrapping using robust derivative estimation and adaptive integration,” IEEE Trans. Image Process. 11, 1192-1200 (2002).
[CrossRef]

J. M. N. Leitao and M. A. T. Figueiredo, “Absolute phase image reconstruction: a stochastic nonlinear filtering approach,” IEEE Trans. Image Process. 7, 868-882 (1998).
[CrossRef]

G. Fornaro, A. Pauciullo, and E. Sansosti, “Phase difference-based multichannel phase unwrapping,” IEEE Trans. Image Process. 14, 960-972 (2005).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging (1)

Z. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893-897 (1996).
[CrossRef] [PubMed]

IEEE Trans. Signal Process. (2)

G. Nico, G. Palubinskas, and M. Datcu, “Bayesian approaches to phase unwrapping: theoretical study,” IEEE Trans. Signal Process. 48, 2545-2556 (2000).
[CrossRef]

B. Friedlander and J. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999-3007 (1996).
[CrossRef]

IEEE Trans. Syst. Man Cybern. B (1)

J. C. Bezdek and N. R. Pal, “Some new indexes of cluster validity,” IEEE Trans. Syst. Man Cybern. B 28, 301-315(1998).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

X. Su, G. Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150(1993).
[CrossRef]

Opt. Eng. (1)

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Opt. Lasers Eng. (1)

X. Y. Su and W. J. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

Other (3)

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

J. Jiang and J. Cheng, “Noise-residue filtering based on unsupervised clustering for phase unwrapping,” presented at the Fifth International Symposium on Visual Computing, Las Vegas, Nev., USA, 30 Nov.-2 Dec. 2009.

P. Berkhin, Survey Of Clustering Data Mining Techniques (Springer, 2002).

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Figures (9)

Fig. 1
Fig. 1

(a) Simulated 256 × 256 wrapped phase map. (b) Corresponding residue map of (a).

Fig. 2
Fig. 2

(a) Adjacent opposite-sign residues in the form of up–down, (b) adjacent opposite-sign residues in the form of diagonal, (c) disjoint opposite-sign residues, and (d) two couples of adjacent opposite-sign residues joined together.

Fig. 3
Fig. 3

Intercluster distance for SL.

Fig. 4
Fig. 4

(a) Example of two adjacent opposite-sign residues in the form of diagonal, and bold data should be classified into three groups; (b) SL dendogram of the bold data in (a).

Fig. 5
Fig. 5

Histogram analysis for different positions of a filtering window: (a) inside a fringe, (b) close to a fringe edge, and (c) across two adjacent fringes.

Fig. 6
Fig. 6

(a)  256 × 256 interferometric image with random salt and pepper noise. (b) Rewrapped phase construction based on the proposed filter.

Fig. 7
Fig. 7

(a) Performance of residue filtering on conditions of various noise levels and (b) the computing time at different noise levels corresponding to (a).

Fig. 8
Fig. 8

(a) Rewrapped phase reconstruction using the complex average filter with area A blurred and (b) the rewrapped phase reconstruction using local-histogram-based filter with areas B and C blurred.

Fig. 9
Fig. 9

(a) Real interferometric images of a metal part, (b) the wrapped phase map of (a), (c) the residue map corresponding to (b), and (d) the reconstruction result using PU based on the proposed filter.

Tables (2)

Tables Icon

Table 1 Comparison of Rapidity among the Proposed Algorithm and Several Famous Phase-Unwrapping Algorithms

Tables Icon

Table 2 Comparison of Time Complexity among the Proposed Algorithm and Several Famous Phase-Unwrapping Algorithms

Equations (7)

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

Ψ = W ( Φ ) = Φ 2 π Φ 2 π ,
π Δ s Φ ( i , j ) < π ,
Δ s Φ ( i , j ) = W ( Δ s Ψ ( i , j ) ) ,
Δ ( i , j ) = 1 2 π [ Ψ 1 ( i , j + 1 ) Ψ 1 ( i , j ) Ψ 2 ( i + 1 , j ) + Ψ 2 ( i , j ) ] ,
Ψ 1 ( i , j ) = Δ 1 Ψ ( i , j ) + 2 π n 1 ( i , j ) , Ψ 2 ( i , j ) = Δ 2 Ψ ( i , j ) + 2 π n 2 ( i , j ) ,
M ( i , j ) = [ ( n = 1 N I n ( i , j ) sin ( 2 π n / N ) ) 2 + ( n = 1 N I n ( i , j ) cos ( 2 π n / N ) ) 2 ] 1 / 2 ,
Φ ( i , j ) = Φ ( 1 , 1 ) + i = 1 i 1 Δ 1 Φ ( i , 1 ) + j = 1 j 1 Δ 2 Φ ( i , j ) ,

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