Abstract

This paper proposes a novel phase-shifting method for fast, accurate, and unambiguous 3D shape measurement. The basic idea is embedding a speckle-like signal in three sinusoidal fringe patterns to eliminate the phase ambiguity, but without reducing the fringe amplitude or frequency. The absolute depth is then recovered through a robust region-wise voting strategy relying on the embedded signal. Using the theoretical minimum of only three images, the proposed method greatly facilitates the application of phase shifting in time-critical conditions. Moreover, the proposed method is resistant to the global illumination effects, as the fringe patterns used are with a single high frequency. Based on the proposed method, we further demonstrate a real-time, high-precision 3D scanning system with an off-the-shelf projector and a commodity camera.

© 2013 Optical Society of America

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    [CrossRef]
  2. S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Laser Eng. 48, 133–140 (2010).
    [CrossRef]
  3. H. Zhao, W. Chen, and Y. Tan, “Phase-unwrapping algorithm for the measurement of three-dimensional object shapes,” Appl. Opt. 33, 4497–4500 (1994).
    [CrossRef]
  4. G. Sansoni, S. Corini, S. Lazzari, R. Rodella, and F. Docchio, “Three-dimensional imaging based on gray-code light projection: characterization of the measuring algorithm and development of a measuring system for industrial applications,” Appl. Opt. 36, 4463–4472 (1997).
    [CrossRef]
  5. S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45, 123601 (2006).
  6. T. Weise, B. Leibe, and L. Van Gool, “Fast 3D scanning with automatic motion compensation,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2007), pp. 1–8.
  7. Y. Wang, K. Liu, Q. Hao, D. L. Lau, and L. G. Hassebrook, “Period coded phase shifting strategy for realtime 3D structured light illumination,” IEEE Trans. Image Process. 20, 3001–3013 (2011).
    [CrossRef]
  8. P. Wissmann, R. Schmitt, and F. Forster, “Fast and accurate 3D scanning using coded phase shifting and high speed pattern projection,” in International Conference on 3D Imaging, Modeling, Processing, Visualization and Transmission (IEEE, 2011), pp. 108–115.
  9. K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express 18, 5229–5244 (2010).
    [CrossRef]
  10. Y. Wang and S. Zhang, “Novel phase-coding method for absolute phase retrieval,” Opt. Lett. 37, 2067–2069 (2012).
    [CrossRef]
  11. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).
  12. S. K. Nayar, G. Krishnan, M. D. Grossberg, and R. Raskar, “Fast separation of direct and global components of a scene using high frequency illumination,” in ACM SIGGRAPH (ACM, 2006), pp. 935–944.
  13. S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Laser Eng. 48, 149–158 (2010).
    [CrossRef]
  14. X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Laser Eng. 42, 245–261 (2004).
    [CrossRef]
  15. S. Zhang, X. Li, and S.-T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46, 50–57 (2007).
    [CrossRef]
  16. C. E. Towers, D. P. Towers, and J. D. C. Jones, “Time efficient Chinese remainder theorem algorithm for full-field fringe phase analysis in multi-wavelength interferometry,” Opt. Express 12, 1136–1143 (2004).
    [CrossRef]
  17. J. Li, L. G. Hassebrook, and C. Guan, “Optimized two-frequency phase-measuring-profilometry light-sensor temporal-noise sensitivity,” J. Opt. Soc. Am. A 20, 106–115 (2003).
  18. R. R. Garcia and A. Zakhor, “Consistent stereo-assisted absolute phase unwrapping methods for structured light systems,” IEEE J. Sel. Top. Signal Process. 6, 411–424 (2012).
    [CrossRef]
  19. T. Chen, H. P. Seidel, and H. Lensch, “Modulated phase-shifting for 3D scanning,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.
  20. J. Gu, T. Kobayashi, M. Gupta, and S. K. Nayar, “Multiplexed illumination for scene recovery in the presence of global illumination,” in IEEE International Conference on Computer Vision (IEEE, 2011), pp. 691–698.
  21. M. Gupta and S. K. Nayar, “Micro phase shifting,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2012), pp. 813–820.
  22. B. Pan, “Recent progress in digital image correlation,” Exp. Mech. 51, 1223–1235 (2011).
    [CrossRef]
  23. Y. H. Huang, S. Y. Hung, F. J. Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Laser Eng. 50, 534–539 (2012).
    [CrossRef]

2012 (3)

R. R. Garcia and A. Zakhor, “Consistent stereo-assisted absolute phase unwrapping methods for structured light systems,” IEEE J. Sel. Top. Signal Process. 6, 411–424 (2012).
[CrossRef]

Y. H. Huang, S. Y. Hung, F. J. Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Laser Eng. 50, 534–539 (2012).
[CrossRef]

Y. Wang and S. Zhang, “Novel phase-coding method for absolute phase retrieval,” Opt. Lett. 37, 2067–2069 (2012).
[CrossRef]

2011 (2)

B. Pan, “Recent progress in digital image correlation,” Exp. Mech. 51, 1223–1235 (2011).
[CrossRef]

Y. Wang, K. Liu, Q. Hao, D. L. Lau, and L. G. Hassebrook, “Period coded phase shifting strategy for realtime 3D structured light illumination,” IEEE Trans. Image Process. 20, 3001–3013 (2011).
[CrossRef]

2010 (3)

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

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

K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express 18, 5229–5244 (2010).
[CrossRef]

2007 (1)

2006 (1)

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45, 123601 (2006).

2004 (2)

2003 (1)

1997 (1)

1994 (1)

1984 (1)

Chen, T.

T. Chen, H. P. Seidel, and H. Lensch, “Modulated phase-shifting for 3D scanning,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.

Chen, W.

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

H. Zhao, W. Chen, and Y. Tan, “Phase-unwrapping algorithm for the measurement of three-dimensional object shapes,” Appl. Opt. 33, 4497–4500 (1994).
[CrossRef]

Corini, S.

Docchio, F.

Forster, F.

P. Wissmann, R. Schmitt, and F. Forster, “Fast and accurate 3D scanning using coded phase shifting and high speed pattern projection,” in International Conference on 3D Imaging, Modeling, Processing, Visualization and Transmission (IEEE, 2011), pp. 108–115.

Garcia, R. R.

R. R. Garcia and A. Zakhor, “Consistent stereo-assisted absolute phase unwrapping methods for structured light systems,” IEEE J. Sel. Top. Signal Process. 6, 411–424 (2012).
[CrossRef]

Ghiglia, D. C.

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. Laser Eng. 48, 133–140 (2010).
[CrossRef]

Grossberg, M. D.

S. K. Nayar, G. Krishnan, M. D. Grossberg, and R. Raskar, “Fast separation of direct and global components of a scene using high frequency illumination,” in ACM SIGGRAPH (ACM, 2006), pp. 935–944.

Gu, J.

J. Gu, T. Kobayashi, M. Gupta, and S. K. Nayar, “Multiplexed illumination for scene recovery in the presence of global illumination,” in IEEE International Conference on Computer Vision (IEEE, 2011), pp. 691–698.

Guan, C.

Gupta, M.

M. Gupta and S. K. Nayar, “Micro phase shifting,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2012), pp. 813–820.

J. Gu, T. Kobayashi, M. Gupta, and S. K. Nayar, “Multiplexed illumination for scene recovery in the presence of global illumination,” in IEEE International Conference on Computer Vision (IEEE, 2011), pp. 691–698.

Halioua, M.

Hao, Q.

Y. Wang, K. Liu, Q. Hao, D. L. Lau, and L. G. Hassebrook, “Period coded phase shifting strategy for realtime 3D structured light illumination,” IEEE Trans. Image Process. 20, 3001–3013 (2011).
[CrossRef]

K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express 18, 5229–5244 (2010).
[CrossRef]

Hassebrook, L. G.

Huang, P. S.

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45, 123601 (2006).

Huang, Y. H.

Y. H. Huang, S. Y. Hung, F. J. Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Laser Eng. 50, 534–539 (2012).
[CrossRef]

Hung, S. Y.

Y. H. Huang, S. Y. Hung, F. J. Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Laser Eng. 50, 534–539 (2012).
[CrossRef]

Jones, J. D. C.

Kobayashi, T.

J. Gu, T. Kobayashi, M. Gupta, and S. K. Nayar, “Multiplexed illumination for scene recovery in the presence of global illumination,” in IEEE International Conference on Computer Vision (IEEE, 2011), pp. 691–698.

Krishnan, G.

S. K. Nayar, G. Krishnan, M. D. Grossberg, and R. Raskar, “Fast separation of direct and global components of a scene using high frequency illumination,” in ACM SIGGRAPH (ACM, 2006), pp. 935–944.

Lau, D. L.

Y. Wang, K. Liu, Q. Hao, D. L. Lau, and L. G. Hassebrook, “Period coded phase shifting strategy for realtime 3D structured light illumination,” IEEE Trans. Image Process. 20, 3001–3013 (2011).
[CrossRef]

K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express 18, 5229–5244 (2010).
[CrossRef]

Lazzari, S.

Leibe, B.

T. Weise, B. Leibe, and L. Van Gool, “Fast 3D scanning with automatic motion compensation,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2007), pp. 1–8.

Lensch, H.

T. Chen, H. P. Seidel, and H. Lensch, “Modulated phase-shifting for 3D scanning,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.

Li, J.

Li, X.

Liu, H. C.

Liu, K.

Y. Wang, K. Liu, Q. Hao, D. L. Lau, and L. G. Hassebrook, “Period coded phase shifting strategy for realtime 3D structured light illumination,” IEEE Trans. Image Process. 20, 3001–3013 (2011).
[CrossRef]

K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express 18, 5229–5244 (2010).
[CrossRef]

Liu, Y. S.

Y. H. Huang, S. Y. Hung, F. J. Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Laser Eng. 50, 534–539 (2012).
[CrossRef]

Nayar, S. K.

S. K. Nayar, G. Krishnan, M. D. Grossberg, and R. Raskar, “Fast separation of direct and global components of a scene using high frequency illumination,” in ACM SIGGRAPH (ACM, 2006), pp. 935–944.

J. Gu, T. Kobayashi, M. Gupta, and S. K. Nayar, “Multiplexed illumination for scene recovery in the presence of global illumination,” in IEEE International Conference on Computer Vision (IEEE, 2011), pp. 691–698.

M. Gupta and S. K. Nayar, “Micro phase shifting,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2012), pp. 813–820.

Pan, B.

B. Pan, “Recent progress in digital image correlation,” Exp. Mech. 51, 1223–1235 (2011).
[CrossRef]

Pritt, M. D.

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

Raskar, R.

S. K. Nayar, G. Krishnan, M. D. Grossberg, and R. Raskar, “Fast separation of direct and global components of a scene using high frequency illumination,” in ACM SIGGRAPH (ACM, 2006), pp. 935–944.

Rastogi, P.

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

Rodella, R.

Sansoni, G.

Schmitt, R.

P. Wissmann, R. Schmitt, and F. Forster, “Fast and accurate 3D scanning using coded phase shifting and high speed pattern projection,” in International Conference on 3D Imaging, Modeling, Processing, Visualization and Transmission (IEEE, 2011), pp. 108–115.

Seidel, H. P.

T. Chen, H. P. Seidel, and H. Lensch, “Modulated phase-shifting for 3D scanning,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.

Sharifi, F. J.

Y. H. Huang, S. Y. Hung, F. J. Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Laser Eng. 50, 534–539 (2012).
[CrossRef]

Srinivasan, V.

Su, X.

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

Tan, Y.

Towers, C. E.

Towers, D. P.

Van Gool, L.

T. Weise, B. Leibe, and L. Van Gool, “Fast 3D scanning with automatic motion compensation,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2007), pp. 1–8.

Wang, W.

Y. H. Huang, S. Y. Hung, F. J. Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Laser Eng. 50, 534–539 (2012).
[CrossRef]

Wang, Y.

Weise, T.

T. Weise, B. Leibe, and L. Van Gool, “Fast 3D scanning with automatic motion compensation,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2007), pp. 1–8.

Wissmann, P.

P. Wissmann, R. Schmitt, and F. Forster, “Fast and accurate 3D scanning using coded phase shifting and high speed pattern projection,” in International Conference on 3D Imaging, Modeling, Processing, Visualization and Transmission (IEEE, 2011), pp. 108–115.

Yau, S.-T.

Zakhor, A.

R. R. Garcia and A. Zakhor, “Consistent stereo-assisted absolute phase unwrapping methods for structured light systems,” IEEE J. Sel. Top. Signal Process. 6, 411–424 (2012).
[CrossRef]

Zhang, S.

Y. Wang and S. Zhang, “Novel phase-coding method for absolute phase retrieval,” Opt. Lett. 37, 2067–2069 (2012).
[CrossRef]

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

S. Zhang, X. Li, and S.-T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46, 50–57 (2007).
[CrossRef]

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45, 123601 (2006).

Zhao, H.

Appl. Opt. (4)

Exp. Mech. (1)

B. Pan, “Recent progress in digital image correlation,” Exp. Mech. 51, 1223–1235 (2011).
[CrossRef]

IEEE J. Sel. Top. Signal Process. (1)

R. R. Garcia and A. Zakhor, “Consistent stereo-assisted absolute phase unwrapping methods for structured light systems,” IEEE J. Sel. Top. Signal Process. 6, 411–424 (2012).
[CrossRef]

IEEE Trans. Image Process. (1)

Y. Wang, K. Liu, Q. Hao, D. L. Lau, and L. G. Hassebrook, “Period coded phase shifting strategy for realtime 3D structured light illumination,” IEEE Trans. Image Process. 20, 3001–3013 (2011).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45, 123601 (2006).

Opt. Express (2)

Opt. Laser Eng. (4)

Y. H. Huang, S. Y. Hung, F. J. Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Laser Eng. 50, 534–539 (2012).
[CrossRef]

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

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

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

Opt. Lett. (1)

Other (7)

T. Weise, B. Leibe, and L. Van Gool, “Fast 3D scanning with automatic motion compensation,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2007), pp. 1–8.

P. Wissmann, R. Schmitt, and F. Forster, “Fast and accurate 3D scanning using coded phase shifting and high speed pattern projection,” in International Conference on 3D Imaging, Modeling, Processing, Visualization and Transmission (IEEE, 2011), pp. 108–115.

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

S. K. Nayar, G. Krishnan, M. D. Grossberg, and R. Raskar, “Fast separation of direct and global components of a scene using high frequency illumination,” in ACM SIGGRAPH (ACM, 2006), pp. 935–944.

T. Chen, H. P. Seidel, and H. Lensch, “Modulated phase-shifting for 3D scanning,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.

J. Gu, T. Kobayashi, M. Gupta, and S. K. Nayar, “Multiplexed illumination for scene recovery in the presence of global illumination,” in IEEE International Conference on Computer Vision (IEEE, 2011), pp. 691–698.

M. Gupta and S. K. Nayar, “Micro phase shifting,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2012), pp. 813–820.

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Figures (9)

Fig. 1.
Fig. 1.

(a) Speckle-embedded fringe patterns Pk(k=1,2,3) (cropped). (b) Simulated binary speckle pattern Z (cropped). (c) Speckle intensity decision (the light red regions denote positive values and the light green regions denote negative values). (d) NCC statistics of a captured image of Pk at a reference plane. (e) NCC statistics of a captured image of Z at a reference plane.

Fig. 2.
Fig. 2.

(a) Flowchart of our method for absolute phase recovery. (b) One of the three captured images. (c) Wrapped phase map. (d) Spatially unwrapped phase map. (e) Continuous region mask (each color denotes a continuous surface). (f) Absolute depth map.

Fig. 3.
Fig. 3.

(a) Connectivity decision in a four-neighborhood. (b)–(d) Typical examples of continuous region detection.

Fig. 4.
Fig. 4.

Disparity voting results in five continuous regions. All peaks are consistent with the correct disparities.

Fig. 5.
Fig. 5.

(a) A slant planar surface with only direct illumination. (b) A V-groove with both direct and global illumination. (c) and (d) Disparity voting results from all locations on S1 for two different fringe frequencies. (e) Absolute depth difference averaged along each column on S1.

Fig. 6.
Fig. 6.

(a) Reconstruction error against depth (using fringe with 30 periods). (b) Reconstruction error against the period number of fringe (at 1300 mm depth).

Fig. 7.
Fig. 7.

3D reconstruction of several complex scenes in front, side, and top views. (The small color maps denote continuous region masks.)

Fig. 8.
Fig. 8.

3D reconstruction of scenes with global illumination. (a) Color scene image. (b) Proposed method using three images. (c) Three-frequency phase shifting using nine images.

Fig. 9.
Fig. 9.

Online 3D reconstruction of dynamic scenes at 15 fps. Top: hand gesture; bottom: face expression (with black eyeglasses).

Tables (1)

Tables Icon

Table 1. Execution Time of Main Procedures (ms)

Equations (17)

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

Pk(u,v)=B(u,v)Z(u,v)+C+Acos(vV2πN+2π3k),
B(u,v)=2Lmaxk=1,2,3{Fk(u,v)},
Fk(u,v)=C+Acos(vV2πN+2π3k).
B(u,v)={2LBmax2LBmaxBminBminotherwise,
Bmax=maxk=1,2,3{Fk(u,v)},Bmin=mink=1,2,3{Fk(u,v)}.
Ik(x,y)=IC(x,y)+IA(x,y)cos[ϕ(x,y)+2π3k],
IA(x,y)=α(x,y)A,
IC(x,y)=α(x,y)[B(x,y)+C+β(x,y)].
ϕ(x,y)=arctan{3[I2(x,y)I1(x,y)]2I3(x,y)I2(x,y)I1(x,y)}.
ϕa(x,y)=ϕ(x,y)+d(x,y)×2π,
γ(f,g)=x,y[f(x,y)f¯][g(x,y)g¯]x,y[f(x,y)f¯]2x,y[g(x,y)g¯]2,
d^(x0,y0)=arg maxd=0,,N1γ{p,q[x0,y0+δ(ϕ,d)]},
up:max{d(ϕ0,ϕ1),d(ϕ0,ϕ5),d(ϕ0,ϕ6)}<TV,
down:max{d(ϕ0,ϕ3),d(ϕ0,ϕ7),d(ϕ0,ϕ8)}<TV,
left:max{d(ϕ0,ϕ4),d(ϕ0,ϕ9)}<TH,
right:max{d(ϕ0,ϕ2),d(ϕ0,ϕ10)}<TH,
d(ϕi,ϕj)=min{|ϕiϕj|,|ϕiϕj+2π|,|ϕiϕj2π|},

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