Abstract

In this paper, a structured-light-based highly dense and robust 3D reconstruction method is proposed by combining a Gray code and region-shifting pattern. The region-shifting pattern is transformed to the trapezoidal and triangle wave shifting pattern by combining all frames of the region-shifting pattern, and then the boundary of the trapezoidal wave shifting pattern and the peak and phase of the triangle wave shifting pattern are estimated. Through this technique, the spatial resolution is increased about three times. Consequently, the 3D points are reconstructed with a resolution much higher than a camera image resolution. Moreover, as the proposed method measures the boundary and the peak with all frames, it increases the signal-to-noise ratio and is more robust than the conventional methods that use only one or two frames to detect them.

© 2013 Optical Society of America

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  1. L. Zhang, B. Curless, and S. M. Seitz, “Rapid shape acquisition using color structured light and multi-pass dynamic programming,” in Proceedings of the IEEE International Symposium on 3D Data Processing, Visualization, and Transmission (IEEE, 2002).
  2. H. Li, R. Straub, and H. Prautzsch, “Structured light based reconstruction under local spatial coherence assumption,” in Proceedings of the 3rd IEEE International Symposium on 3D Data Processing, Visualization and Transmission (IEEE, 2006), pp. 575–582.
  3. J. Pages and J. Salvi, “Optimised de Bruijn patterns for one-shot shape acquisition,” Image Vis. Comput. 23, 707–720(2005).
    [CrossRef]
  4. T. P. Koninckx and L. V. Gool, “Real-time range acquisition by adaptive structured light,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 432–445 (2006).
    [CrossRef]
  5. S. Y. Chen, Y. F. Li, and J. Zhang, “Vision processing for realtime 3-D data acquisition based on coded structured light,” IEEE Trans. Image Process. 17, 167–176 (2008).
    [CrossRef]
  6. L. Zhang, B. Curless, and S. Seitz, “Spacetime faces: high resolution capture for modeling and animation,” in Proceedings of ACM SIGGRAPH (ACM, 2004), pp. 548–558.
  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.
  8. P. Jia, J. Kofman, and C. English, “Two-step triangular-pattern phase-shifting method for three dimensional object-shape measurement,” Opt. Eng. 46, 083201 (2007).
    [CrossRef]
  9. M. Trobina, “Error model of a coded-light range sensor,” Tech. Rep. BIWI-TR-164 (Communication Technology Laboratory, ETH Zentrum, 1995).
  10. D. Kim, M. Ryu, and S. Lee, “Antipodal gray codes for structured light,” in International Conference on Robotics and Automation (IEEE, 2008), pp. 3016–3021.
  11. S. Rusinkiewicz, O. Hall-Holt, and M. Levoy, “Real-time 3D model acquisition,” in Proceedings of ACM SIGGRAPH (ACM, 2002), pp. 438–446.
  12. O. Hall-Holt and S. Rusinkiewicz, “Stripe boundary codes for realtime structured-light range scanning of moving objects,” in Proceedings of 8th IEEE ICCV (IEEE, 2001), pp. 359–366.
  13. P. S. Huang, S. Zhang, and F. P. Chiang, “Trapezoidal phase-shifting method for 3-D shape measurement,” Proc. SPIE 5606, 142 (2004).
    [CrossRef]
  14. S. Zhang, D. Royer, and S. T. Yau, “High-resolution, real-time-geometry video acquisition system,” in Proceedings of ACM SIGGRAPH (ACM, 2006).
  15. G. Sansoni, M. Carocci, and R. Rodella, “3D vision based on the combination of gray code and phase-shift light projection: analysis and compensation of the systematic errors,” Appl. Opt. 38, 6565–6573 (1999).
    [CrossRef]
  16. J. Guhring, “Dense 3-D surface acquisition by structured light using off-the-shelf components,” Proc. SPIE 4309, 220–231 (2001).
  17. D. Kim, S. Lee, H. Kim, and S. Lee, “Wide-angle laser structured light system calibration with a planar object,” in International Conference on Control Automation and Systems (IEEE, 2010), pp. 1879–1882.

2008 (1)

S. Y. Chen, Y. F. Li, and J. Zhang, “Vision processing for realtime 3-D data acquisition based on coded structured light,” IEEE Trans. Image Process. 17, 167–176 (2008).
[CrossRef]

2007 (1)

P. Jia, J. Kofman, and C. English, “Two-step triangular-pattern phase-shifting method for three dimensional object-shape measurement,” Opt. Eng. 46, 083201 (2007).
[CrossRef]

2006 (1)

T. P. Koninckx and L. V. Gool, “Real-time range acquisition by adaptive structured light,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 432–445 (2006).
[CrossRef]

2005 (1)

J. Pages and J. Salvi, “Optimised de Bruijn patterns for one-shot shape acquisition,” Image Vis. Comput. 23, 707–720(2005).
[CrossRef]

2004 (1)

P. S. Huang, S. Zhang, and F. P. Chiang, “Trapezoidal phase-shifting method for 3-D shape measurement,” Proc. SPIE 5606, 142 (2004).
[CrossRef]

2001 (1)

J. Guhring, “Dense 3-D surface acquisition by structured light using off-the-shelf components,” Proc. SPIE 4309, 220–231 (2001).

1999 (1)

Carocci, M.

Chen, S. Y.

S. Y. Chen, Y. F. Li, and J. Zhang, “Vision processing for realtime 3-D data acquisition based on coded structured light,” IEEE Trans. Image Process. 17, 167–176 (2008).
[CrossRef]

Chiang, F. P.

P. S. Huang, S. Zhang, and F. P. Chiang, “Trapezoidal phase-shifting method for 3-D shape measurement,” Proc. SPIE 5606, 142 (2004).
[CrossRef]

Curless, B.

L. Zhang, B. Curless, and S. Seitz, “Spacetime faces: high resolution capture for modeling and animation,” in Proceedings of ACM SIGGRAPH (ACM, 2004), pp. 548–558.

L. Zhang, B. Curless, and S. M. Seitz, “Rapid shape acquisition using color structured light and multi-pass dynamic programming,” in Proceedings of the IEEE International Symposium on 3D Data Processing, Visualization, and Transmission (IEEE, 2002).

English, C.

P. Jia, J. Kofman, and C. English, “Two-step triangular-pattern phase-shifting method for three dimensional object-shape measurement,” Opt. Eng. 46, 083201 (2007).
[CrossRef]

Gool, L. V.

T. P. Koninckx and L. V. Gool, “Real-time range acquisition by adaptive structured light,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 432–445 (2006).
[CrossRef]

Guhring, J.

J. Guhring, “Dense 3-D surface acquisition by structured light using off-the-shelf components,” Proc. SPIE 4309, 220–231 (2001).

Hall-Holt, O.

S. Rusinkiewicz, O. Hall-Holt, and M. Levoy, “Real-time 3D model acquisition,” in Proceedings of ACM SIGGRAPH (ACM, 2002), pp. 438–446.

O. Hall-Holt and S. Rusinkiewicz, “Stripe boundary codes for realtime structured-light range scanning of moving objects,” in Proceedings of 8th IEEE ICCV (IEEE, 2001), pp. 359–366.

Huang, P. S.

P. S. Huang, S. Zhang, and F. P. Chiang, “Trapezoidal phase-shifting method for 3-D shape measurement,” Proc. SPIE 5606, 142 (2004).
[CrossRef]

Jia, P.

P. Jia, J. Kofman, and C. English, “Two-step triangular-pattern phase-shifting method for three dimensional object-shape measurement,” Opt. Eng. 46, 083201 (2007).
[CrossRef]

Kim, D.

D. Kim, M. Ryu, and S. Lee, “Antipodal gray codes for structured light,” in International Conference on Robotics and Automation (IEEE, 2008), pp. 3016–3021.

D. Kim, S. Lee, H. Kim, and S. Lee, “Wide-angle laser structured light system calibration with a planar object,” in International Conference on Control Automation and Systems (IEEE, 2010), pp. 1879–1882.

Kim, H.

D. Kim, S. Lee, H. Kim, and S. Lee, “Wide-angle laser structured light system calibration with a planar object,” in International Conference on Control Automation and Systems (IEEE, 2010), pp. 1879–1882.

Kofman, J.

P. Jia, J. Kofman, and C. English, “Two-step triangular-pattern phase-shifting method for three dimensional object-shape measurement,” Opt. Eng. 46, 083201 (2007).
[CrossRef]

Koninckx, T. P.

T. P. Koninckx and L. V. Gool, “Real-time range acquisition by adaptive structured light,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 432–445 (2006).
[CrossRef]

Lee, S.

D. Kim, S. Lee, H. Kim, and S. Lee, “Wide-angle laser structured light system calibration with a planar object,” in International Conference on Control Automation and Systems (IEEE, 2010), pp. 1879–1882.

D. Kim, S. Lee, H. Kim, and S. Lee, “Wide-angle laser structured light system calibration with a planar object,” in International Conference on Control Automation and Systems (IEEE, 2010), pp. 1879–1882.

D. Kim, M. Ryu, and S. Lee, “Antipodal gray codes for structured light,” in International Conference on Robotics and Automation (IEEE, 2008), pp. 3016–3021.

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.

Levoy, M.

S. Rusinkiewicz, O. Hall-Holt, and M. Levoy, “Real-time 3D model acquisition,” in Proceedings of ACM SIGGRAPH (ACM, 2002), pp. 438–446.

Li, H.

H. Li, R. Straub, and H. Prautzsch, “Structured light based reconstruction under local spatial coherence assumption,” in Proceedings of the 3rd IEEE International Symposium on 3D Data Processing, Visualization and Transmission (IEEE, 2006), pp. 575–582.

Li, Y. F.

S. Y. Chen, Y. F. Li, and J. Zhang, “Vision processing for realtime 3-D data acquisition based on coded structured light,” IEEE Trans. Image Process. 17, 167–176 (2008).
[CrossRef]

Pages, J.

J. Pages and J. Salvi, “Optimised de Bruijn patterns for one-shot shape acquisition,” Image Vis. Comput. 23, 707–720(2005).
[CrossRef]

Prautzsch, H.

H. Li, R. Straub, and H. Prautzsch, “Structured light based reconstruction under local spatial coherence assumption,” in Proceedings of the 3rd IEEE International Symposium on 3D Data Processing, Visualization and Transmission (IEEE, 2006), pp. 575–582.

Rodella, R.

Royer, D.

S. Zhang, D. Royer, and S. T. Yau, “High-resolution, real-time-geometry video acquisition system,” in Proceedings of ACM SIGGRAPH (ACM, 2006).

Rusinkiewicz, S.

O. Hall-Holt and S. Rusinkiewicz, “Stripe boundary codes for realtime structured-light range scanning of moving objects,” in Proceedings of 8th IEEE ICCV (IEEE, 2001), pp. 359–366.

S. Rusinkiewicz, O. Hall-Holt, and M. Levoy, “Real-time 3D model acquisition,” in Proceedings of ACM SIGGRAPH (ACM, 2002), pp. 438–446.

Ryu, M.

D. Kim, M. Ryu, and S. Lee, “Antipodal gray codes for structured light,” in International Conference on Robotics and Automation (IEEE, 2008), pp. 3016–3021.

Salvi, J.

J. Pages and J. Salvi, “Optimised de Bruijn patterns for one-shot shape acquisition,” Image Vis. Comput. 23, 707–720(2005).
[CrossRef]

Sansoni, G.

Seitz, S.

L. Zhang, B. Curless, and S. Seitz, “Spacetime faces: high resolution capture for modeling and animation,” in Proceedings of ACM SIGGRAPH (ACM, 2004), pp. 548–558.

Seitz, S. M.

L. Zhang, B. Curless, and S. M. Seitz, “Rapid shape acquisition using color structured light and multi-pass dynamic programming,” in Proceedings of the IEEE International Symposium on 3D Data Processing, Visualization, and Transmission (IEEE, 2002).

Straub, R.

H. Li, R. Straub, and H. Prautzsch, “Structured light based reconstruction under local spatial coherence assumption,” in Proceedings of the 3rd IEEE International Symposium on 3D Data Processing, Visualization and Transmission (IEEE, 2006), pp. 575–582.

Trobina, M.

M. Trobina, “Error model of a coded-light range sensor,” Tech. Rep. BIWI-TR-164 (Communication Technology Laboratory, ETH Zentrum, 1995).

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.

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.

Yau, S. T.

S. Zhang, D. Royer, and S. T. Yau, “High-resolution, real-time-geometry video acquisition system,” in Proceedings of ACM SIGGRAPH (ACM, 2006).

Zhang, J.

S. Y. Chen, Y. F. Li, and J. Zhang, “Vision processing for realtime 3-D data acquisition based on coded structured light,” IEEE Trans. Image Process. 17, 167–176 (2008).
[CrossRef]

Zhang, L.

L. Zhang, B. Curless, and S. M. Seitz, “Rapid shape acquisition using color structured light and multi-pass dynamic programming,” in Proceedings of the IEEE International Symposium on 3D Data Processing, Visualization, and Transmission (IEEE, 2002).

L. Zhang, B. Curless, and S. Seitz, “Spacetime faces: high resolution capture for modeling and animation,” in Proceedings of ACM SIGGRAPH (ACM, 2004), pp. 548–558.

Zhang, S.

P. S. Huang, S. Zhang, and F. P. Chiang, “Trapezoidal phase-shifting method for 3-D shape measurement,” Proc. SPIE 5606, 142 (2004).
[CrossRef]

S. Zhang, D. Royer, and S. T. Yau, “High-resolution, real-time-geometry video acquisition system,” in Proceedings of ACM SIGGRAPH (ACM, 2006).

Appl. Opt. (1)

IEEE Trans. Image Process. (1)

S. Y. Chen, Y. F. Li, and J. Zhang, “Vision processing for realtime 3-D data acquisition based on coded structured light,” IEEE Trans. Image Process. 17, 167–176 (2008).
[CrossRef]

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

T. P. Koninckx and L. V. Gool, “Real-time range acquisition by adaptive structured light,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 432–445 (2006).
[CrossRef]

Image Vis. Comput. (1)

J. Pages and J. Salvi, “Optimised de Bruijn patterns for one-shot shape acquisition,” Image Vis. Comput. 23, 707–720(2005).
[CrossRef]

Opt. Eng. (1)

P. Jia, J. Kofman, and C. English, “Two-step triangular-pattern phase-shifting method for three dimensional object-shape measurement,” Opt. Eng. 46, 083201 (2007).
[CrossRef]

Proc. SPIE (2)

P. S. Huang, S. Zhang, and F. P. Chiang, “Trapezoidal phase-shifting method for 3-D shape measurement,” Proc. SPIE 5606, 142 (2004).
[CrossRef]

J. Guhring, “Dense 3-D surface acquisition by structured light using off-the-shelf components,” Proc. SPIE 4309, 220–231 (2001).

Other (10)

D. Kim, S. Lee, H. Kim, and S. Lee, “Wide-angle laser structured light system calibration with a planar object,” in International Conference on Control Automation and Systems (IEEE, 2010), pp. 1879–1882.

S. Zhang, D. Royer, and S. T. Yau, “High-resolution, real-time-geometry video acquisition system,” in Proceedings of ACM SIGGRAPH (ACM, 2006).

L. Zhang, B. Curless, and S. Seitz, “Spacetime faces: high resolution capture for modeling and animation,” in Proceedings of ACM SIGGRAPH (ACM, 2004), pp. 548–558.

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.

M. Trobina, “Error model of a coded-light range sensor,” Tech. Rep. BIWI-TR-164 (Communication Technology Laboratory, ETH Zentrum, 1995).

D. Kim, M. Ryu, and S. Lee, “Antipodal gray codes for structured light,” in International Conference on Robotics and Automation (IEEE, 2008), pp. 3016–3021.

S. Rusinkiewicz, O. Hall-Holt, and M. Levoy, “Real-time 3D model acquisition,” in Proceedings of ACM SIGGRAPH (ACM, 2002), pp. 438–446.

O. Hall-Holt and S. Rusinkiewicz, “Stripe boundary codes for realtime structured-light range scanning of moving objects,” in Proceedings of 8th IEEE ICCV (IEEE, 2001), pp. 359–366.

L. Zhang, B. Curless, and S. M. Seitz, “Rapid shape acquisition using color structured light and multi-pass dynamic programming,” in Proceedings of the IEEE International Symposium on 3D Data Processing, Visualization, and Transmission (IEEE, 2002).

H. Li, R. Straub, and H. Prautzsch, “Structured light based reconstruction under local spatial coherence assumption,” in Proceedings of the 3rd IEEE International Symposium on 3D Data Processing, Visualization and Transmission (IEEE, 2006), pp. 575–582.

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Figures (17)

Fig. 1.
Fig. 1.

(a) Projected black-and-white regions with various widths in the projector. (b) Captured image in the camera. (c) Intensity profile of the center row in the captured image.

Fig. 2.
Fig. 2.

(a) 6-frame binary code pattern. (b) 6-frame Gray code pattern.

Fig. 3.
Fig. 3.

(a) Combination of a 4-frame Gray code and an 8-frame line-shifting pattern. (b) Combination of a 4-frame Gray code and a 4-frame (8-pixel-period) phase-shifting pattern.

Fig. 4.
Fig. 4.

Combination of a 4-frame Gray code, a 4-frame inverted Gray code, and an 8-frame region-shifting pattern.

Fig. 5.
Fig. 5.

(a) The boundary in the region-shifting pattern can be transformed to the trapezoidal wave shifting pattern. (b) Profile of the trapezoidal wave shifting pattern at the 0th frame. (c) Profile of the trapezoidal wave shifting pattern computed with derivative of Gaussian at the 0th frame.

Fig. 6.
Fig. 6.

(a) Each pixel in the region is amplified spatially and temporally. (b) Profile of the triangle wave pattern at the 0th frame. (c) Triangle wave shifting pattern approximated by the Fourier series with three harmonics. (d) Triangle wave shifting pattern approximated by the Fourier series with one harmonic.

Fig. 7.
Fig. 7.

(a) Non-normalized trapezoidal wave shifting pattern. (b) Normalized trapezoidal wave shifting pattern. (c) The profile of the non-normalized trapezoidal wave pattern at the 0th frame. (d) The profile of the normalized trapezoidal wave pattern at the 0th frame.

Fig. 8.
Fig. 8.

(a) Non-normalized triangle wave shifting pattern. (b) Normalized triangle wave shifting pattern. (c) Profile of the non-normalized triangle wave pattern at the 0th frame. (d) Profile of the normalized triangle wave pattern at the 0th frame.

Fig. 9.
Fig. 9.

Corresponding points can be obtained by detecting the boundary and peak of the pattern at the subpixel location, and by measuring the phase at each pixel in the camera.

Fig. 10.
Fig. 10.

Peak detection in the line-shifting pattern.

Fig. 11.
Fig. 11.

Phase measurement in the sine wave shifting pattern.

Fig. 12.
Fig. 12.

Boundary detection in the region-shifting pattern.

Fig. 13.
Fig. 13.

Phase measurement in the triangle wave shifting pattern.

Fig. 14.
Fig. 14.

Peak detection in the triangle wave shifting pattern.

Fig. 15.
Fig. 15.

Boundary detection in the trapezoidal wave shifting pattern.

Fig. 16.
Fig. 16.

Cross comparison.

Fig. 17.
Fig. 17.

3D reconstruction, rendering and mesh generation results. (a), (b) Results of combining a Gray code and phase-shifting pattern. (c), (d) Results of the proposed method.

Tables (2)

Equations (44)

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

Rjp(xp+0.5,yp)=Rjp(xp+1,yp)Rjp(xp,yp),
Rjp(xp+0.5,yp)=Rjp(xp+2,yp)+2Rjp(xp+1,yp)2Rjp(xp,yp)Rjp(xp1,yp).
Ijp(xp,yp)=t=t0t1Rtp(xp,yp)t=t2t3Rtp(xp,yp)+Rjp(xp,yp)+Rj+m2p(xp,yp),
t0=(j+1)modm,t1=(j+m21)modm,
t2=(j+m2+1)modm,t3=(j1)modm,
Jjp(xp,yp)=t=t0t1Rtp(xp,yp)t=t2t3Rtp(xp,yp),
t0=(jm2+1)modm,t1=j,
t2=(j+1)modm,t3=(j+m2)modm,
Jjp(xp,yp)=A(xp,yp)+B(xp,yp)k=0(1)ksin(2k+1)(xpΔj)(2k+1)2,
A(xp,yp)+B(xp,yp)(sin(xpΔj)19sin(3(xpΔj))+125sin(5(xpΔj))),
A(xp,yp)+B(xp,yp)(sin(xpΔj)19sin(3(xpΔj))),
A(xp,yp)+B(xp,yp)(sin(xpΔj)),
Pc(xc,yc)=γ(xc,yc)(Pp(xp,yp)+α(xc,yc)),
Wc(xc,yc)=γ(xc,yc)(Wp(xp,yp)+α(xc,yc)),Bc(xc,yc)=γ(xc,yc)(Bp(xp,yp)+α(xc,yc)).
Wc(xc,yc)Bc(xc,yc)=γ(xc,yc)Wp(xp,yp),
γ(xc,yc)=Wc(xc,yc)Bc(xc,yc)Wp(xp,yp).
Pc(xc,yc)Bc(xc,yc)=γ(xc,yc)Pp(xp,yp),
P¯c(xc,yc)=Pc(xc,yc)Bc(xc,yc)γ(xc,yc)=Pp(xp,yp),
G^ic(xc,yc)=G¯ic(xc,yc)H¯ic(xc,yc).
bi(xc,yc)={1,ifG^ic(xc,yc)00,otherwise.
g(xc,yc)=(b0(xc,yc)b1(xc,yc)bl(xc,yc))Gray.
R^jc(xc,yc)=R¯jc(xc,yc)R¯j+m2c(xc,yc),
R^jc(xc,yc)×R^jc(xc+1,yc)<0orR^jc(xc,yc)=0,
xzc=xcR^jc(x,y)R^jc(x+1,y)R^jc(x,y).
Ijc(xc,yc)=t=t0t1Rtc(xc,yc)t=t2t3Rtc(xc,yc)+Rjc(xc,yc)+Rj+m2c(xc,yc),
t0=(j+1)modm,t1=(j+m21)modm,
t2=(j+m2+1)modm,t3=(j1)modm,
Jjc(xc,yc)=t=t0t1R¯tc(xc,yc)t=t2t3R¯tc(xc,yc),
t0=(jm2+1)modm,t1=j,
t2=(j+1)modm,t3=(j+m2)modm,
Jjp(xc,yc)Ac(xc,yc)+Bc(xc,yc)(sin(xpΔj)19sin(3(xpΔj))+125sin(5(xpΔj))),
Jjc(xc,yc)Ac(xc,yc)+Bc(xc,yc)(sinxpcosΔjcosxpsinΔj19(sin3xpcos3Δjcos3xpsin3Δj)+125(sin5xpcos5Δjcos5xpsin5Δj))=a0+a1cosΔj+a2sinΔj+a3cos3Δj+a4sin3Δj+a5cos5Δj+a6sin5Δj,
a0=Ac(xc,yc),a1=Bc(xc,yc)sinxp,a2=Bc(xc,yc)cosxp,a3=(Bc(xc,yc)sin3xp)/9,a4=(Bc(xc,yc)cos3xp)/9,a5=(Bc(xc,yc)sin5xp)/25,a6=(Bc(xc,yc)cos5xp)/25.
xp=atan2(a1,a2),
x3p=atan2(a3,a4),
x5p=atan2(a5,a6),
x3p=3xpmod2π,
x5p=5xpmod2π.
3xp=x3p+(2π)t3,
5xp=x5p+(2π)t5.
t3=3xpx3p2π+0.5,
t5=5xpx5p2π+0.5.
w=2.5σandσ=w2.5,
w=1.5σandσ=w1.5,

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