Abstract

Phase shifting profilometry can achieve high accuracy for the 3D shape measurement of static object. Errors will be introduced when the object is moved during the movement. The fundamental reason causing the above issue is: PSP requires multiple fringe patterns but the reconstruction model does not include the object movement information. This paper proposes a new method to automatically measure the 3D shape of the rigid object with arbitrary 2D movement. Firstly, the object movement is tracked by the SIFT algorithm and the rotation matrix and translation vector describing the movement are estimated. Then, with the reconstruction model including movement information, a least-square algorithm is applied to retrieve the correct phase value. The proposed method can significantly reduce the errors caused by the object movement. The whole reconstruction process does not need human intervention and the proposed method has high potential to be applied in industrial applications. Experiments are presented to verify the effectiveness.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. M. Zhang, Q. Chen, T. Tao, S. Feng, Y. Hu, H. Li, and C. Zuo, “Robust and efficient multi-frequency temporal phase unwrapping: optimal fringe frequency and pattern sequence selection,” Opt. Express 25(17), 20381–20400 (2017).
    [Crossref] [PubMed]
  2. S. Zhang, D. Van Der Weide, and J. Oliver, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express 18(9), 9684–9689 (2010).
    [Crossref] [PubMed]
  3. Y. Xing, C. Quan, and C. Tay, “A modified phase-coding method for absolute phase retrieval,” Opt. Lasers Eng. 87(1), 97–102 (2016).
    [Crossref]
  4. H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602 (2011).
    [Crossref]
  5. Y. Ding, K. Peng, L. Lu, K. Zhong, and Z. Zhu, “Simplified fringe order correction for absolute phase maps recovered with multiple-spatial-frequency fringe projections,” Meas. Sci. Technol. 28(2), 025203 (2017).
    [Crossref]
  6. Z. Zhang, S. Huang, S. Meng, F. Gao, and X. Jiang, “A simple, flexible and automatic 3D calibration method for a phase calculation-based fringe projection imaging system,” Opt. Express 21(10), 12218–12227 (2013).
    [Crossref] [PubMed]
  7. X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36(1), 49–64 (2001).
    [Crossref]
  8. S. Zhang, “Recent progresses on real-time 3-D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
    [Crossref]
  9. E. Hu and Y. He, “Surface profile measurement of moving objects by using an improved π phase-shifting Fourier transform profilometry,” Opt. Lasers Eng. 47(1), 57–61 (2009).
    [Crossref]
  10. S. Zhang and S. Yau, “High-speed three-dimensional shape measurement system using a modified two-plus-one phase-shifting algorithm,” Opt. Eng. 46(11), 113603 (2007).
    [Crossref]
  11. Y. Chen, Y. Cao, H. Yuan and Y. Wan, “A stroboscopic online three-dimensional measurement for fast rotating object with binary dithered patterns,” T. I. Meas. Control, 1–9 (2017).
  12. Y. Wang, S. Zhang, and J. H. Oliver, “3D shape measurement technique for multiple rapidly moving objects,” Opt. Express 19(9), 8539–8545 (2011).
    [Crossref] [PubMed]
  13. L. Lu, J. Xi, Y. Yu, and Q. Guo, “New approach to improve the accuracy of 3-D shape measurement of moving object using phase shifting profilometry,” Opt. Express 21(25), 30610–30622 (2013).
    [Crossref] [PubMed]
  14. Y. Hu, J. Xi, J. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas. 58(9), 3305–3314 (2009).
    [Crossref]
  15. D. Lowe, “Distinctive Image Features from Scale-Invariant Keypoints,” Int. J. Comput. Vis. 60(2), 91–110 (2004).
    [Crossref]
  16. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).
    [Crossref]
  17. X. Hu, Y. Tang, and Z. Zhang, “Video object matching based on SIFT algorithm,” in Proceedings of International Conference on Neural Networks and Signal Processing (Academic, 2008), pp. 412–415.
  18. K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-Squares Fitting of Two 3-D Point Sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9(5), 698–700 (1987).
    [Crossref] [PubMed]
  19. K. Wang, B. Cheng, and T. Long, “An Improved SIFT Feature Matching Algorithm Based on Maximizing Minimum Distance Cluster,” in Proceedings of International Conference on Computer Science and Information Technology (Academic, 2011), pp. 255–259.

2017 (2)

M. Zhang, Q. Chen, T. Tao, S. Feng, Y. Hu, H. Li, and C. Zuo, “Robust and efficient multi-frequency temporal phase unwrapping: optimal fringe frequency and pattern sequence selection,” Opt. Express 25(17), 20381–20400 (2017).
[Crossref] [PubMed]

Y. Ding, K. Peng, L. Lu, K. Zhong, and Z. Zhu, “Simplified fringe order correction for absolute phase maps recovered with multiple-spatial-frequency fringe projections,” Meas. Sci. Technol. 28(2), 025203 (2017).
[Crossref]

2016 (1)

Y. Xing, C. Quan, and C. Tay, “A modified phase-coding method for absolute phase retrieval,” Opt. Lasers Eng. 87(1), 97–102 (2016).
[Crossref]

2013 (2)

2011 (2)

Y. Wang, S. Zhang, and J. H. Oliver, “3D shape measurement technique for multiple rapidly moving objects,” Opt. Express 19(9), 8539–8545 (2011).
[Crossref] [PubMed]

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602 (2011).
[Crossref]

2010 (2)

S. Zhang, D. Van Der Weide, and J. Oliver, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express 18(9), 9684–9689 (2010).
[Crossref] [PubMed]

S. Zhang, “Recent progresses on real-time 3-D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

2009 (2)

E. Hu and Y. He, “Surface profile measurement of moving objects by using an improved π phase-shifting Fourier transform profilometry,” Opt. Lasers Eng. 47(1), 57–61 (2009).
[Crossref]

Y. Hu, J. Xi, J. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas. 58(9), 3305–3314 (2009).
[Crossref]

2007 (1)

S. Zhang and S. Yau, “High-speed three-dimensional shape measurement system using a modified two-plus-one phase-shifting algorithm,” Opt. Eng. 46(11), 113603 (2007).
[Crossref]

2004 (1)

D. Lowe, “Distinctive Image Features from Scale-Invariant Keypoints,” Int. J. Comput. Vis. 60(2), 91–110 (2004).
[Crossref]

2001 (1)

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36(1), 49–64 (2001).
[Crossref]

1987 (1)

K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-Squares Fitting of Two 3-D Point Sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9(5), 698–700 (1987).
[Crossref] [PubMed]

1982 (1)

Arun, K. S.

K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-Squares Fitting of Two 3-D Point Sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9(5), 698–700 (1987).
[Crossref] [PubMed]

Blostein, S. D.

K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-Squares Fitting of Two 3-D Point Sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9(5), 698–700 (1987).
[Crossref] [PubMed]

Chao, Y.

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36(1), 49–64 (2001).
[Crossref]

Chen, Q.

Chen, W.

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36(1), 49–64 (2001).
[Crossref]

Cheng, B.

K. Wang, B. Cheng, and T. Long, “An Improved SIFT Feature Matching Algorithm Based on Maximizing Minimum Distance Cluster,” in Proceedings of International Conference on Computer Science and Information Technology (Academic, 2011), pp. 255–259.

Cheng, W.

Y. Hu, J. Xi, J. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas. 58(9), 3305–3314 (2009).
[Crossref]

Cheng, X.

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602 (2011).
[Crossref]

Chicharo, J.

Y. Hu, J. Xi, J. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas. 58(9), 3305–3314 (2009).
[Crossref]

Cui, H.

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602 (2011).
[Crossref]

Dai, N.

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602 (2011).
[Crossref]

Ding, Y.

Y. Ding, K. Peng, L. Lu, K. Zhong, and Z. Zhu, “Simplified fringe order correction for absolute phase maps recovered with multiple-spatial-frequency fringe projections,” Meas. Sci. Technol. 28(2), 025203 (2017).
[Crossref]

Feng, S.

Gao, F.

Guo, Q.

He, Y.

E. Hu and Y. He, “Surface profile measurement of moving objects by using an improved π phase-shifting Fourier transform profilometry,” Opt. Lasers Eng. 47(1), 57–61 (2009).
[Crossref]

Hu, E.

E. Hu and Y. He, “Surface profile measurement of moving objects by using an improved π phase-shifting Fourier transform profilometry,” Opt. Lasers Eng. 47(1), 57–61 (2009).
[Crossref]

Hu, X.

X. Hu, Y. Tang, and Z. Zhang, “Video object matching based on SIFT algorithm,” in Proceedings of International Conference on Neural Networks and Signal Processing (Academic, 2008), pp. 412–415.

Hu, Y.

M. Zhang, Q. Chen, T. Tao, S. Feng, Y. Hu, H. Li, and C. Zuo, “Robust and efficient multi-frequency temporal phase unwrapping: optimal fringe frequency and pattern sequence selection,” Opt. Express 25(17), 20381–20400 (2017).
[Crossref] [PubMed]

Y. Hu, J. Xi, J. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas. 58(9), 3305–3314 (2009).
[Crossref]

Huang, S.

Huang, T. S.

K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-Squares Fitting of Two 3-D Point Sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9(5), 698–700 (1987).
[Crossref] [PubMed]

Ina, H.

Jiang, X.

Kobayashi, S.

Li, H.

Liao, W.

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602 (2011).
[Crossref]

Long, T.

K. Wang, B. Cheng, and T. Long, “An Improved SIFT Feature Matching Algorithm Based on Maximizing Minimum Distance Cluster,” in Proceedings of International Conference on Computer Science and Information Technology (Academic, 2011), pp. 255–259.

Lowe, D.

D. Lowe, “Distinctive Image Features from Scale-Invariant Keypoints,” Int. J. Comput. Vis. 60(2), 91–110 (2004).
[Crossref]

Lu, L.

Y. Ding, K. Peng, L. Lu, K. Zhong, and Z. Zhu, “Simplified fringe order correction for absolute phase maps recovered with multiple-spatial-frequency fringe projections,” Meas. Sci. Technol. 28(2), 025203 (2017).
[Crossref]

L. Lu, J. Xi, Y. Yu, and Q. Guo, “New approach to improve the accuracy of 3-D shape measurement of moving object using phase shifting profilometry,” Opt. Express 21(25), 30610–30622 (2013).
[Crossref] [PubMed]

Meng, S.

Oliver, J.

Oliver, J. H.

Peng, K.

Y. Ding, K. Peng, L. Lu, K. Zhong, and Z. Zhu, “Simplified fringe order correction for absolute phase maps recovered with multiple-spatial-frequency fringe projections,” Meas. Sci. Technol. 28(2), 025203 (2017).
[Crossref]

Quan, C.

Y. Xing, C. Quan, and C. Tay, “A modified phase-coding method for absolute phase retrieval,” Opt. Lasers Eng. 87(1), 97–102 (2016).
[Crossref]

Su, X.

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36(1), 49–64 (2001).
[Crossref]

Takeda, M.

Tang, Y.

X. Hu, Y. Tang, and Z. Zhang, “Video object matching based on SIFT algorithm,” in Proceedings of International Conference on Neural Networks and Signal Processing (Academic, 2008), pp. 412–415.

Tao, T.

Tay, C.

Y. Xing, C. Quan, and C. Tay, “A modified phase-coding method for absolute phase retrieval,” Opt. Lasers Eng. 87(1), 97–102 (2016).
[Crossref]

Van Der Weide, D.

Wang, K.

K. Wang, B. Cheng, and T. Long, “An Improved SIFT Feature Matching Algorithm Based on Maximizing Minimum Distance Cluster,” in Proceedings of International Conference on Computer Science and Information Technology (Academic, 2011), pp. 255–259.

Wang, Y.

Xi, J.

L. Lu, J. Xi, Y. Yu, and Q. Guo, “New approach to improve the accuracy of 3-D shape measurement of moving object using phase shifting profilometry,” Opt. Express 21(25), 30610–30622 (2013).
[Crossref] [PubMed]

Y. Hu, J. Xi, J. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas. 58(9), 3305–3314 (2009).
[Crossref]

Xing, Y.

Y. Xing, C. Quan, and C. Tay, “A modified phase-coding method for absolute phase retrieval,” Opt. Lasers Eng. 87(1), 97–102 (2016).
[Crossref]

Yang, Z.

Y. Hu, J. Xi, J. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas. 58(9), 3305–3314 (2009).
[Crossref]

Yau, S.

S. Zhang and S. Yau, “High-speed three-dimensional shape measurement system using a modified two-plus-one phase-shifting algorithm,” Opt. Eng. 46(11), 113603 (2007).
[Crossref]

Yu, Y.

Zhang, M.

Zhang, Q.

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36(1), 49–64 (2001).
[Crossref]

Zhang, S.

Y. Wang, S. Zhang, and J. H. Oliver, “3D shape measurement technique for multiple rapidly moving objects,” Opt. Express 19(9), 8539–8545 (2011).
[Crossref] [PubMed]

S. Zhang, “Recent progresses on real-time 3-D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

S. Zhang, D. Van Der Weide, and J. Oliver, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express 18(9), 9684–9689 (2010).
[Crossref] [PubMed]

S. Zhang and S. Yau, “High-speed three-dimensional shape measurement system using a modified two-plus-one phase-shifting algorithm,” Opt. Eng. 46(11), 113603 (2007).
[Crossref]

Zhang, Z.

Z. Zhang, S. Huang, S. Meng, F. Gao, and X. Jiang, “A simple, flexible and automatic 3D calibration method for a phase calculation-based fringe projection imaging system,” Opt. Express 21(10), 12218–12227 (2013).
[Crossref] [PubMed]

X. Hu, Y. Tang, and Z. Zhang, “Video object matching based on SIFT algorithm,” in Proceedings of International Conference on Neural Networks and Signal Processing (Academic, 2008), pp. 412–415.

Zhong, K.

Y. Ding, K. Peng, L. Lu, K. Zhong, and Z. Zhu, “Simplified fringe order correction for absolute phase maps recovered with multiple-spatial-frequency fringe projections,” Meas. Sci. Technol. 28(2), 025203 (2017).
[Crossref]

Zhu, Z.

Y. Ding, K. Peng, L. Lu, K. Zhong, and Z. Zhu, “Simplified fringe order correction for absolute phase maps recovered with multiple-spatial-frequency fringe projections,” Meas. Sci. Technol. 28(2), 025203 (2017).
[Crossref]

Zuo, C.

IEEE Trans. Instrum. Meas. (1)

Y. Hu, J. Xi, J. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas. 58(9), 3305–3314 (2009).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-Squares Fitting of Two 3-D Point Sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9(5), 698–700 (1987).
[Crossref] [PubMed]

Int. J. Comput. Vis. (1)

D. Lowe, “Distinctive Image Features from Scale-Invariant Keypoints,” Int. J. Comput. Vis. 60(2), 91–110 (2004).
[Crossref]

J. Opt. Soc. Am. (1)

Meas. Sci. Technol. (1)

Y. Ding, K. Peng, L. Lu, K. Zhong, and Z. Zhu, “Simplified fringe order correction for absolute phase maps recovered with multiple-spatial-frequency fringe projections,” Meas. Sci. Technol. 28(2), 025203 (2017).
[Crossref]

Opt. Eng. (2)

S. Zhang and S. Yau, “High-speed three-dimensional shape measurement system using a modified two-plus-one phase-shifting algorithm,” Opt. Eng. 46(11), 113603 (2007).
[Crossref]

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602 (2011).
[Crossref]

Opt. Express (5)

Opt. Lasers Eng. (4)

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36(1), 49–64 (2001).
[Crossref]

S. Zhang, “Recent progresses on real-time 3-D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

E. Hu and Y. He, “Surface profile measurement of moving objects by using an improved π phase-shifting Fourier transform profilometry,” Opt. Lasers Eng. 47(1), 57–61 (2009).
[Crossref]

Y. Xing, C. Quan, and C. Tay, “A modified phase-coding method for absolute phase retrieval,” Opt. Lasers Eng. 87(1), 97–102 (2016).
[Crossref]

Other (3)

Y. Chen, Y. Cao, H. Yuan and Y. Wan, “A stroboscopic online three-dimensional measurement for fast rotating object with binary dithered patterns,” T. I. Meas. Control, 1–9 (2017).

K. Wang, B. Cheng, and T. Long, “An Improved SIFT Feature Matching Algorithm Based on Maximizing Minimum Distance Cluster,” in Proceedings of International Conference on Computer Science and Information Technology (Academic, 2011), pp. 255–259.

X. Hu, Y. Tang, and Z. Zhang, “Video object matching based on SIFT algorithm,” in Proceedings of International Conference on Neural Networks and Signal Processing (Academic, 2008), pp. 412–415.

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Figures (10)

Fig. 1
Fig. 1 The structure of the PSP system.
Fig. 2
Fig. 2 Remove the fringes by filter. (a) The original fringe pattern image; (b) The result after remove the fringes by filter.
Fig. 3
Fig. 3 The fringe pattern image in different components. (a) The captured object image with red fringe patterns; (b) The red component of Fig. 3(a); (c) The blue component of Fig. 3(a).
Fig. 4
Fig. 4 The detected feature points and the corresponding relationship.
Fig. 5
Fig. 5 The flow chart of feature points selecting.
Fig. 6
Fig. 6 Plastic mask used in the experiment. (a) The plastic mask used in the experiment; (b)–(d) The captured object fringe patterns for 3-step PSP with object movement.
Fig. 7
Fig. 7 Images in different components. (a) The captured image of the object; (b) The image of Fig. 7(a) in red component; (c) The image of Fig. 7(a) in blue component.
Fig. 8
Fig. 8 The result of the SIFT algorithm. (a) The feature points obtained by SIFT algorithm and the corresponding relationship; (b) The mosaic result for the images in Fig. 8(a).
Fig. 9
Fig. 9 The reconstructed results with the traditional PSP and the proposed algorithm. (a) The front view of the result with the traditional PSP; (b) The mesh display of Fig. 9(a); (c) The front view of the result with the proposed algorithm; (d) The mesh display of Fig. (c).
Fig. 10
Fig. 10 The comparison result between the proposed algorithm and FTP algorithm. (a) The front view of the result with FTP; (b) The mesh display of Fig. 10(a); (c) The cross section of the dash line in Fig. 10(a) where x = 135; (d) The cross section of Fig. 9(c) where x = 135.

Equations (16)

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

q i =R p i +T+ γ i
2 = i=1 N q i (R p i +T) 2
p = 1 N i=1 N p i , p i = p i p ,
q = 1 N i=1 N q i , q i = q i q .
2 = i=1 N q i R p i 2
H= i=1 N p i q i T
R ^ =V U T
T ^ = q R ^ p
d ˜ n (x,y)=a+bcos{ϕ[ f n (x,y), g n (x,y)]+Φ(x,y)+2π(n1)/N}
d ˜ n (x,y)=a+B(x,y)cosδ+C(x,y)sinδ
S(x,y)= n=1 N [ d ˜ n (x,y) d n m (x,y)] 2
X(x,y)= A 1 (x,y)B(x,y)
A(x,y)=[ N n=1 N cosδ n=1 N sinδ n=1 N cosδ n=1 N cos 2 δ 1 2 n=1 N sin2δ n=1 N sinδ 1 2 n=1 N sin2δ n=1 N sin 2 δ ],
X(x,y)= [ a B(x,y) C(x,y) ] T ,
B(x,y)= [ n=1 N d n m (x,y) n=1 N cosδ× d n m (x,y) n=1 N sinδ× d n m (x,y) ] T .
Φ(x,y)= tan 1 [C(x,y)/B(x,y)]

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