Abstract

The profile measurement system is widely used in industrial quality control, and phase unwrapping (PU) is a key technique. An algorithm-driven PU is often used to reduce the impact of noise-induced residues to retrieve the most reliable solution. However, measuring speed is lowered due to the searching of optimal integration paths or correcting of phase gradients. From the viewpoint of the rapidity of the system, this paper characterizes the noise-induced residues, and it proposes a clustering-driven residue filter based on a set of directional windows. The proposed procedure makes the wrapped phases included in the filtering window have more similar values, and it groups the correct and noisy phases into individual clusters along the local fringe direction adaptively. It is effective for the tightly packed fringes, and it converts the algorithm-driven PU to the residue-filtering-driven one. This improves the operating speed of the 3D reconstruction significantly. The tests performed on simulated and real projected fringes confirm the validity of our approach.

© 2011 Optical Society of America

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References

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  1. F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  4. J. Jiang, J. Cheng, and B. Luong, “Unsupervised-clustering-driven noise-residue filter for phase images,” Appl. Opt. 49, 2143–2150 (2010).
    [CrossRef] [PubMed]
  5. H. A. Zebker and Y. P. Lu, “Phase unwrapping algorithms for radar interferometry: residue-cut, least-squares, and synthesis algorithms,” J. Opt. Soc. Am. A 15, 586–598 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sensing 36, 813–821 (1998).
    [CrossRef]
  9. M. Hubig, S. Suchandt, and N. Adam, “A class of solution-invariant transformations of cost functions for minimum cost flow phase unwrapping,” J. Opt. Soc. Am. A 21, 1975–1987(2004).
    [CrossRef]
  10. Z. P. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893–897 (1996).
    [CrossRef] [PubMed]
  11. J. M. N. Leitão and M. A. T. Figueiredo, “Absolute phase image reconstruction: a stochastic nonlinear filtering approach,” IEEE Trans. Image Process. 7, 868–882 (1998).
    [CrossRef]
  12. G. Nico, G. Palubinskas, and M. Datcu, “Bayesian approaches to phase unwrapping: theoretical study,” IEEE Trans. Signal Process. 48, 2545–2556 (2000).
    [CrossRef]
  13. L. An, Q. S. Xiang, and S. Chavez, “A fast implementation of the minimum spanning tree method for phase unwrapping,” IEEE Trans. Med. Imaging 19, 805–808 (2000).
    [CrossRef] [PubMed]
  14. 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]
  15. J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sensing 36, 1173 (1998).
    [CrossRef]
  16. G. Nico, “Noise-residue filtering of interferometric phase images,” J. Opt. Soc. Am. A. 17, 1962–1974 (2000).
    [CrossRef]
  17. G. Fornaro, A. Pauciullo, and E. Sansosti, “Phase difference-based multichannel phase unwrapping,” IEEE Trans. Image Process. 14, 960–972 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  21. 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]
  22. P. Berkhin, Survey of Clustering Data Mining Techniques (Springer, 2002).
  23. cmin is the cluster in which the number of members is smallest.
  24. V. Chandola, A. Banerjee, and V. Kumar, “Anomaly detection: a survey,” ACM Comput. Surv. 41, 1–58 (2009).
    [CrossRef]
  25. cmax is the cluster in which the number of members is biggest.
  26. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

2010

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

J. Jiang, J. Cheng, and B. Luong, “Unsupervised-clustering-driven noise-residue filter for phase images,” Appl. Opt. 49, 2143–2150 (2010).
[CrossRef] [PubMed]

2009

V. Chandola, A. Banerjee, and V. Kumar, “Anomaly detection: a survey,” ACM Comput. Surv. 41, 1–58 (2009).
[CrossRef]

2008

2007

2005

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

2004

2003

2000

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

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

L. An, Q. S. Xiang, and S. Chavez, “A fast implementation of the minimum spanning tree method for phase unwrapping,” IEEE Trans. Med. Imaging 19, 805–808 (2000).
[CrossRef] [PubMed]

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

1998

J. M. N. Leitão 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, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sensing 36, 1173 (1998).
[CrossRef]

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

H. A. Zebker and Y. P. Lu, “Phase unwrapping algorithms for radar interferometry: residue-cut, least-squares, and synthesis algorithms,” J. Opt. Soc. Am. A 15, 586–598 (1998).
[CrossRef]

1997

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

Z. P. 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]

1982

Adam, N.

Ainsworth, T. L.

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

An, L.

L. An, Q. S. Xiang, and S. Chavez, “A fast implementation of the minimum spanning tree method for phase unwrapping,” IEEE Trans. Med. Imaging 19, 805–808 (2000).
[CrossRef] [PubMed]

Baldi, A.

Banerjee, A.

V. Chandola, A. Banerjee, and V. Kumar, “Anomaly detection: a survey,” ACM Comput. Surv. 41, 1–58 (2009).
[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]

Brown, G. M.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

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]

Chandola, V.

V. Chandola, A. Banerjee, and V. Kumar, “Anomaly detection: a survey,” ACM Comput. Surv. 41, 1–58 (2009).
[CrossRef]

Chavez, S.

L. An, Q. S. Xiang, and S. Chavez, “A fast implementation of the minimum spanning tree method for phase unwrapping,” IEEE Trans. Med. Imaging 19, 805–808 (2000).
[CrossRef] [PubMed]

Chen, F.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

Cheng, J.

J. Jiang, J. Cheng, and B. Luong, “Unsupervised-clustering-driven noise-residue filter for phase images,” Appl. Opt. 49, 2143–2150 (2010).
[CrossRef] [PubMed]

J. Jiang and J. Cheng, “Noise-residue filtering based on unsupervised clustering for phase unwrapping,” in Proceedings of 5th International Symposium on Visual Computing, M.Z.Nashed, ed. (Springer, 2009), pp. 719–727.

Costantini, M.

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

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]

Du, H.

Figueiredo, M. A. T.

J. M. N. Leitão 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]

Gao, W.

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).

Gorthi, S. S.

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

Grunes, M. R.

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

Hubig, M.

Itoh, K.

Jiang, J.

J. Jiang, J. Cheng, and B. Luong, “Unsupervised-clustering-driven noise-residue filter for phase images,” Appl. Opt. 49, 2143–2150 (2010).
[CrossRef] [PubMed]

J. Jiang and J. Cheng, “Noise-residue filtering based on unsupervised clustering for phase unwrapping,” in Proceedings of 5th International Symposium on Visual Computing, M.Z.Nashed, ed. (Springer, 2009), pp. 719–727.

Karout, S. A.

Kumar, V.

V. Chandola, A. Banerjee, and V. Kumar, “Anomaly detection: a survey,” ACM Comput. Surv. 41, 1–58 (2009).
[CrossRef]

Lalor, M. J.

Lee, J. S.

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

Leitão, J. M. N.

J. M. N. Leitão 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. P.

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

Lu, Y. P.

Luong, B.

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]

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, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sensing 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.

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

Qian, K. M.

Rastogi, P.

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

Reigber, A.

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

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]

Song, M.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

Suchandt, S.

Wang, H.

Wang, Z. Y.

Xiang, Q. S.

L. An, Q. S. Xiang, and S. Chavez, “A fast implementation of the minimum spanning tree method for phase unwrapping,” IEEE Trans. Med. Imaging 19, 805–808 (2000).
[CrossRef] [PubMed]

Zebker, H. A.

ACM Comput. Surv.

V. Chandola, A. Banerjee, and V. Kumar, “Anomaly detection: a survey,” ACM Comput. Surv. 41, 1–58 (2009).
[CrossRef]

Appl. Opt.

IEEE Trans. Geosci. Remote Sensing

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

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

IEEE Trans. Image Process.

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

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

IEEE Trans. Med. Imaging

L. An, Q. S. Xiang, and S. Chavez, “A fast implementation of the minimum spanning tree method for phase unwrapping,” IEEE Trans. Med. Imaging 19, 805–808 (2000).
[CrossRef] [PubMed]

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

IEEE Trans. Signal Process.

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

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A.

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

Opt. Eng.

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]

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

Opt. Lasers Eng.

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

Opt. Lett.

Other

J. Jiang and J. Cheng, “Noise-residue filtering based on unsupervised clustering for phase unwrapping,” in Proceedings of 5th International Symposium on Visual Computing, M.Z.Nashed, ed. (Springer, 2009), pp. 719–727.

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

cmin is the cluster in which the number of members is smallest.

cmax is the cluster in which the number of members is biggest.

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

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Figures (12)

Fig. 1
Fig. 1

Work flow in FPP: (a) projection of sinusoidal fringes, (b) image acquisition, (c) fringe analysis and PU, and (d) phase-to-height conversion.

Fig. 2
Fig. 2

Optical geometry for fringe analysis: the reference plane o x y , the projection plane o p x p y p , and the imaging plane o c x c y c are arbitrarily arranged; Q represents an arbitrary point on the object; A and B indicate the original fringe point projected at Q and the imaging point of Q respectively; O p and O c denote the lens centers of the projector and the camera, respectively.

Fig. 3
Fig. 3

(a) 256 × 256 phase image and (b) corresponding residue image of (a); the white and black flags denote positive and negative residues, respectively.

Fig. 4
Fig. 4

Configurations of residues: (a) a couple of adjacent opposite-sign residues in the form of up–down, (b) a couple of adjacent opposite-sign residues in the form of diagonal, (c) a couple of disjoint opposite-sign residues, and (d) two couples of adjacent opposite- sign residues joined together. g ( · ) denotes one pixel in the adjacent residue area.

Fig. 5
Fig. 5

This is a clustering example. A local window contains a pair of adjacent residues in the form of the diagonal in a wrapped phase image. The seven phases in this pair of residues are shown by a bigger and bold style. The seven phases can be classified into three groups easily based on the knowledge of clustering. In fact, 0.0008 is the noisy phase inducing this pair of residues.

Fig. 6
Fig. 6

Sixteen directional windows for the filtering of noisy phases. Only the white pixels are included in the computation.

Fig. 7
Fig. 7

(a) Corresponding phase image of a set of four 256 × 256 fringe images contaminated by random noise, (b) the rewrapped phase fringe based on the proposed filter, (c) the rewrapped phase reconstruction using complex average filter with area A blurred, and (d) the rewrapped phase reconstruction using local histogram based filter with areas B and C blurred.

Fig. 8
Fig. 8

Schematic diagram of our FPP system. There are four modules: fringe projection, image acquisition, fringe analysis, and high-precision motion.

Fig. 9
Fig. 9

Actual size of the observed surface of a container.

Fig. 10
Fig. 10

Process of 3D reconstruction: (a) one fringe image of a partial surface of the container, (b) the wrapped phase image of (a), (c) the residue image corresponding to (b), and (d) the reconstruction result using PU based on the proposed filter.

Fig. 11
Fig. 11

Projected fringe image on the two gauge blocks.

Fig. 12
Fig. 12

Height map of a partial surface of the measured container.

Tables (3)

Tables Icon

Table 1 Comparison of Rapidity between the Proposed Algorithm and Several Famous Phase Unwrappings

Tables Icon

Table 2 Comparison of Computational Complexity between the Proposed Algorithm and Several Famous Phase Unwrappings

Tables Icon

Table 3 Analysis of the Measured Height Data (Unit: mm)

Equations (13)

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

ψ = W ( ϕ ) = ϕ 2 π ϕ 2 π ,
I ( x , y ) = A ( x , y ) + B ( x , y ) cos [ 2 n ( x , y ) π + ψ ( x , y ) + φ ] ,
ψ ( i , j ) = arctan k = 1 N I k ( i , j ) sin ( 2 k π N ) k = 1 N I k ( i , j ) cos ( 2 k π N ) ,
ϕ ( i , j ) = ϕ ( 1 , 1 ) + i = 1 i 1 Δ 1 ϕ ( i , 1 ) + j = 1 j 1 Δ 2 ϕ ( i , j ) ,
Δ 1 ϕ ( i , j ) = ψ 1 ( i , j ) = Δ 1 ψ ( i , j ) + 2 π n 1 ( i , j ) , Δ 2 ϕ ( i , j ) = ψ 2 ( i , j ) = Δ 2 ψ ( i , j ) + 2 π n 2 ( i , j ) ,
h = z Q = 1 + c 1 ϕ B + ( c 2 + c 3 ϕ B ) i B + ( c 4 + c 5 ϕ B ) j B d o + d 1 ϕ B + ( d 2 + d 3 ϕ B ) i B + ( d 4 + d 5 ϕ B ) j B ,
E = k = 1 m [ 1 + c 1 ϕ k + ( c 2 + c 3 ϕ k ) i k + ( c 4 + c 5 ϕ k ) j k d o + d 1 ϕ k + ( d 2 + d 3 ϕ k ) i k + ( d 4 + d 5 ϕ k ) j k z k g ] 2 ,
Δ ( i , j ) = ( 1 / 2 π ) [ ψ 1 ( i , j + 1 ) ψ 1 ( i , j ) ψ 2 ( i + 1 , j ) + ψ 2 ( i , j ) ] ,
v = 1 N i = 1 N ( x i x ¯ ) 2 ,
x ¯ = 1 N i = 1 N x i ,
ϕ 1 ϕ m i 1 i m ϕ 1 i 1 ϕ m i m j 1 j m ϕ 1 j 1 ϕ m j m z 1 g z m g ϕ 1 z 1 g ϕ m z m g i 1 z 1 g i m z m g ϕ 1 i 1 z 1 g ϕ m i m z m g j 1 z 1 g j m z m g ϕ 1 j 1 z 1 g ϕ m j m z m g T c 1 c 2 c 3 c 4 c 5 d 0 d 1 d 2 d 3 d 4 d 5 = 1 1 1 1 1 1 1 1 1 1 1 .
Ax = b .
x = ( A T A ) 1 A T b .

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