Abstract

Fourier-transform profilometry (FTP) and phase-shifting profilometry (PSP) are two mainstream fringe projection techniques widely used for three-dimensional (3D) shape measurement. The former is well known for its single-shot nature and the latter for its higher measurement resolution and precision. However, when it comes to measuring the dynamic objects, neither approach is able to produce high-resolution, high-accuracy measurement results that are free from any depth ambiguities and motion-related artifacts. Furthermore, for scenes consisting of both static and dynamic objects, a trade-off between measurement precision and efficiency has to be made, suggesting that using a single approach can yield only suboptimal results. To this end, we propose a novel hybrid Fourier-transform phase-shifting profilometry method to integrate the advantages of both approaches. The motion vulnerability of multi-shot PSP can be overcome, or at least significantly alleviated, through the combination of single-shot FTP, while the high accuracy of PSP can also be preserved when the object is motionless. We design a phase-based, pixel-wise motion detection strategy that can accurately outline the moving object regions from their motionless counterparts. The final measurement result is obtained by fusing the determined regions where the PSP or FTP is applied correspondingly. To validate the proposed hybrid approach, we develop a real-time 3D shape measurement system for measuring multiple isolated moving objects. Experimental results demonstrate that our method achieves significantly higher precision and better robustness compared with conventional approaches where PSP or FTP is applied separately.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2018 (5)

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: A review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

C. Zuo, T. Tao, S. Feng, L. Huang, A. Asundi, and Q. Chen, “Micro fourier transform profilometry (μftp): 3d shape measurement at 10,000 frames per second,” Opt. Lasers Eng. 102, 70–91 (2018).
[Crossref]

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-d measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

Z. Liu, P. C. Zibley, and S. Zhang, “Motion-induced error compensation for phase shifting profilometry,” Opt. Express 26, 12632–12637 (2018).
[Crossref]

T. Tao, Q. Chen, S. Feng, J. Qian, Y. Hu, L. Huang, and C. Zuo, “High-speed real-time 3d shape measurement based on adaptive depth constraint,” Opt. Express 26, 22440–22456 (2018).
[Crossref]

2017 (3)

2016 (3)

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

S. Van der Jeught and J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).
[Crossref]

B. Li, Z. Liu, and S. Zhang, “Motion-induced error reduction by combining fourier transform profilometry with phase-shifting profilometry,” Opt. Express 24, 23289–23303 (2016).
[Crossref]

2015 (1)

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with fourier-assisted phase shifting,” IEEE J. Sel. Top. Signal Process. 9, 396–408 (2015).
[Crossref]

2014 (2)

S. Feng, Y. Zhang, Q. Chen, C. Zuo, R. Li, and G. Shen, “General solution for high dynamic range three-dimensional shape measurement using the fringe projection technique,” Opt. Lasers Eng. 59, 56–71 (2014).
[Crossref]

L. Lu, J. Xi, Y. Yu, and Q. Guo, “Improving the accuracy performance of phase-shifting profilometry for the measurement of objects in motion,” Opt. Lett. 39, 6715–6718 (2014).
[Crossref]

2013 (2)

Z. Li, K. Zhong, Y. F. Li, X. Zhou, and Y. Shi, “Multiview phase shifting: a full-resolution and high-speed 3d measurement framework for arbitrary shape dynamic objects,” Opt. Lett. 38, 1389–1391 (2013).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

2012 (3)

2011 (2)

S. Foix, G. Alenya, and C. Torras, “Lock-in time-of-flight (tof) cameras: A survey,” IEEE Sensors J. 11, 1917–1926 (2011).
[Crossref]

J. Geng, “Structured-light 3d surface imaging: a tutorial,” Adv. Opt. Photonics 3, 128–160 (2011).
[Crossref]

2010 (7)

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43, 2666–2680 (2010).
[Crossref]

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

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

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of fourier transform, windowed fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[Crossref]

X. Su and Q. Zhang, “Dynamic 3-d shape measurement method: a review,” Opt. Lasers Eng. 48, 191–204 (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]

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

2008 (1)

H. Guo and P. S. Huang, “3-d shape measurement by use of a modified fourier transform method,” Proc. SPIE 7066, 70660E (2008).

2007 (1)

Q. Kemao, “Two-dimensional windowed fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[Crossref]

2006 (1)

2005 (1)

2004 (2)

2003 (1)

J. Li, L. G. Hassebrook, and C. Guan, “Optimized two-frequency phase-measuring-profilometry light-sensor temporal-noise sensitivity,” JOSA A 20, 106–115 (2003).
[Crossref]

1999 (1)

1993 (1)

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

1992 (1)

S. D. Cochran and G. Medioni, “3-d surface description from binocular stereo,” IEEE Trans. Pattern Anal. Mach. Intell 10, 981–994 (1992).
[Crossref]

1990 (1)

J. Li, X. Su, and L. Guo, “Improved fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1445 (1990).
[Crossref]

1987 (1)

K. L. Boyer and A. C. Kak, “Color-encoded structured light for rapid active ranging,” IEEE Trans. Pattern Anal. Mach. Intell 1, 14–28 (1987).
[Crossref] [PubMed]

1984 (1)

1982 (1)

1979 (1)

D. Marr and T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. Lond. B 204, 301–328 (1979).
[Crossref] [PubMed]

Alenya, G.

S. Foix, G. Alenya, and C. Torras, “Lock-in time-of-flight (tof) cameras: A survey,” IEEE Sensors J. 11, 1917–1926 (2011).
[Crossref]

Asundi, A.

C. Zuo, T. Tao, S. Feng, L. Huang, A. Asundi, and Q. Chen, “Micro fourier transform profilometry (μftp): 3d shape measurement at 10,000 frames per second,” Opt. Lasers Eng. 102, 70–91 (2018).
[Crossref]

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

Asundi, A. K.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of fourier transform, windowed fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[Crossref]

Boyer, K. L.

K. L. Boyer and A. C. Kak, “Color-encoded structured light for rapid active ranging,” IEEE Trans. Pattern Anal. Mach. Intell 1, 14–28 (1987).
[Crossref] [PubMed]

Bräuer-Burchardt, C.

C. Bräuer-Burchardt, C. Munkelt, M. Heinze, P. Kühmstedt, and G. Notni, “Using geometric constraints to solve the point correspondence problem in fringe projection based 3d measuring systems,” in International Conference on Image Analysis and Processing, (2011), pp. 265–274.

Carocci, M.

Chan, D.

Y. Cui, S. Schuon, D. Chan, S. Thrun, and C. Theobalt, “3d shape scanning with a time-of-flight camera,” in 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2010), pp. 1173–1180.

Chen, K.

Chen, Q.

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: A review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

C. Zuo, T. Tao, S. Feng, L. Huang, A. Asundi, and Q. Chen, “Micro fourier transform profilometry (μftp): 3d shape measurement at 10,000 frames per second,” Opt. Lasers Eng. 102, 70–91 (2018).
[Crossref]

T. Tao, Q. Chen, S. Feng, J. Qian, Y. Hu, L. Huang, and C. Zuo, “High-speed real-time 3d shape measurement based on adaptive depth constraint,” Opt. Express 26, 22440–22456 (2018).
[Crossref]

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-d measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

T. Tao, Q. Chen, S. Feng, Y. Hu, M. Zhang, and C. Zuo, “High-precision real-time 3d shape measurement based on a quad-camera system,” J. Opt. 20, 014009 (2017).
[Crossref]

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

S. Feng, Y. Zhang, Q. Chen, C. Zuo, R. Li, and G. Shen, “General solution for high dynamic range three-dimensional shape measurement using the fringe projection technique,” Opt. Lasers Eng. 59, 56–71 (2014).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, and F. Feng, “High-speed three-dimensional profilometry for multiple objects with complex shapes,” Opt. Express 20, 19493–19510 (2012).
[Crossref]

Cochran, S. D.

S. D. Cochran and G. Medioni, “3-d surface description from binocular stereo,” IEEE Trans. Pattern Anal. Mach. Intell 10, 981–994 (1992).
[Crossref]

Cong, P.

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with fourier-assisted phase shifting,” IEEE J. Sel. Top. Signal Process. 9, 396–408 (2015).
[Crossref]

Cui, Y.

Y. Cui, S. Schuon, D. Chan, S. Thrun, and C. Theobalt, “3d shape scanning with a time-of-flight camera,” in 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2010), pp. 1173–1180.

Curless, B.

L. Zhang, B. Curless, and S. M. Seitz, “Rapid shape acquisition using color structured light and multi-pass dynamic programming,” in Proceedings. First International Symposium on 3D Data Processing Visualization and Transmission, (2002), pp. 24–36.

Dirckx, J. J.

S. Van der Jeught and J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).
[Crossref]

Feng, F.

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, and F. Feng, “High-speed three-dimensional profilometry for multiple objects with complex shapes,” Opt. Express 20, 19493–19510 (2012).
[Crossref]

Feng, S.

T. Tao, Q. Chen, S. Feng, J. Qian, Y. Hu, L. Huang, and C. Zuo, “High-speed real-time 3d shape measurement based on adaptive depth constraint,” Opt. Express 26, 22440–22456 (2018).
[Crossref]

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-d measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: A review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

C. Zuo, T. Tao, S. Feng, L. Huang, A. Asundi, and Q. Chen, “Micro fourier transform profilometry (μftp): 3d shape measurement at 10,000 frames per second,” Opt. Lasers Eng. 102, 70–91 (2018).
[Crossref]

T. Tao, Q. Chen, S. Feng, Y. Hu, M. Zhang, and C. Zuo, “High-precision real-time 3d shape measurement based on a quad-camera system,” J. Opt. 20, 014009 (2017).
[Crossref]

S. Feng, Y. Zhang, Q. Chen, C. Zuo, R. Li, and G. Shen, “General solution for high dynamic range three-dimensional shape measurement using the fringe projection technique,” Opt. Lasers Eng. 59, 56–71 (2014).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, and F. Feng, “High-speed three-dimensional profilometry for multiple objects with complex shapes,” Opt. Express 20, 19493–19510 (2012).
[Crossref]

Fernandez, S.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43, 2666–2680 (2010).
[Crossref]

Foix, S.

S. Foix, G. Alenya, and C. Torras, “Lock-in time-of-flight (tof) cameras: A survey,” IEEE Sensors J. 11, 1917–1926 (2011).
[Crossref]

Gao, B. Z.

Geng, J.

J. Geng, “Structured-light 3d surface imaging: a tutorial,” Adv. Opt. Photonics 3, 128–160 (2011).
[Crossref]

Gool, L. V.

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

Gorthi, S. S.

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

Gu, G.

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-d measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, and F. Feng, “High-speed three-dimensional profilometry for multiple objects with complex shapes,” Opt. Express 20, 19493–19510 (2012).
[Crossref]

Guan, C.

J. Li, L. G. Hassebrook, and C. Guan, “Optimized two-frequency phase-measuring-profilometry light-sensor temporal-noise sensitivity,” JOSA A 20, 106–115 (2003).
[Crossref]

Guan, Y.

Guo, H.

H. Guo and P. S. Huang, “3-d shape measurement by use of a modified fourier transform method,” Proc. SPIE 7066, 70660E (2008).

Guo, L.

J. Li, X. Su, and L. Guo, “Improved fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1445 (1990).
[Crossref]

Guo, Q.

Halioua, M.

Hao, Q.

Hassebrook, L. G.

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]

J. Li, L. G. Hassebrook, and C. Guan, “Optimized two-frequency phase-measuring-profilometry light-sensor temporal-noise sensitivity,” JOSA A 20, 106–115 (2003).
[Crossref]

Heinze, M.

C. Bräuer-Burchardt, C. Munkelt, M. Heinze, P. Kühmstedt, and G. Notni, “Using geometric constraints to solve the point correspondence problem in fringe projection based 3d measuring systems,” in International Conference on Image Analysis and Processing, (2011), pp. 265–274.

Hu, Y.

T. Tao, Q. Chen, S. Feng, J. Qian, Y. Hu, L. Huang, and C. Zuo, “High-speed real-time 3d shape measurement based on adaptive depth constraint,” Opt. Express 26, 22440–22456 (2018).
[Crossref]

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-d measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

T. Tao, Q. Chen, S. Feng, Y. Hu, M. Zhang, and C. Zuo, “High-precision real-time 3d shape measurement based on a quad-camera system,” J. Opt. 20, 014009 (2017).
[Crossref]

Huang, L.

T. Tao, Q. Chen, S. Feng, J. Qian, Y. Hu, L. Huang, and C. Zuo, “High-speed real-time 3d shape measurement based on adaptive depth constraint,” Opt. Express 26, 22440–22456 (2018).
[Crossref]

C. Zuo, T. Tao, S. Feng, L. Huang, A. Asundi, and Q. Chen, “Micro fourier transform profilometry (μftp): 3d shape measurement at 10,000 frames per second,” Opt. Lasers Eng. 102, 70–91 (2018).
[Crossref]

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: A review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of fourier transform, windowed fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[Crossref]

Huang, P. S.

H. Guo and P. S. Huang, “3-d shape measurement by use of a modified fourier transform method,” Proc. SPIE 7066, 70660E (2008).

Ina, H.

Kak, A. C.

K. L. Boyer and A. C. Kak, “Color-encoded structured light for rapid active ranging,” IEEE Trans. Pattern Anal. Mach. Intell 1, 14–28 (1987).
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L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of fourier transform, windowed fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
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Q. Kemao, “Windowed fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
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Kofman, J.

Kühmstedt, P.

C. Bräuer-Burchardt, C. Munkelt, M. Heinze, P. Kühmstedt, and G. Notni, “Using geometric constraints to solve the point correspondence problem in fringe projection based 3d measuring systems,” in International Conference on Image Analysis and Processing, (2011), pp. 265–274.

Lau, D. L.

Leibe, B.

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

Li, A.

Li, B.

Li, J.

J. Li, L. G. Hassebrook, and C. Guan, “Optimized two-frequency phase-measuring-profilometry light-sensor temporal-noise sensitivity,” JOSA A 20, 106–115 (2003).
[Crossref]

J. Li, X. Su, and L. Guo, “Improved fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1445 (1990).
[Crossref]

Li, R.

S. Feng, Y. Zhang, Q. Chen, C. Zuo, R. Li, and G. Shen, “General solution for high dynamic range three-dimensional shape measurement using the fringe projection technique,” Opt. Lasers Eng. 59, 56–71 (2014).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

Li, Y. F.

Li, Z.

Liu, H.-C.

Liu, K.

Liu, X.

Liu, Z.

Llado, X.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43, 2666–2680 (2010).
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S. D. Cochran and G. Medioni, “3-d surface description from binocular stereo,” IEEE Trans. Pattern Anal. Mach. Intell 10, 981–994 (1992).
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Munkelt, C.

C. Bräuer-Burchardt, C. Munkelt, M. Heinze, P. Kühmstedt, and G. Notni, “Using geometric constraints to solve the point correspondence problem in fringe projection based 3d measuring systems,” in International Conference on Image Analysis and Processing, (2011), pp. 265–274.

Notni, G.

C. Bräuer-Burchardt, C. Munkelt, M. Heinze, P. Kühmstedt, and G. Notni, “Using geometric constraints to solve the point correspondence problem in fringe projection based 3d measuring systems,” in International Conference on Image Analysis and Processing, (2011), pp. 265–274.

Oliver, J.

Pan, B.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of fourier transform, windowed fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
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Poggio, T.

D. Marr and T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. Lond. B 204, 301–328 (1979).
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Pribanic, T.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43, 2666–2680 (2010).
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Qian, J.

Rastogi, P.

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

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43, 2666–2680 (2010).
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Scharstein, D.

D. Scharstein and R. Szeliski, “High-accuracy stereo depth maps using structured light,” in 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings, vol. 1 (2003), pp. 195–202.

Schuon, S.

Y. Cui, S. Schuon, D. Chan, S. Thrun, and C. Theobalt, “3d shape scanning with a time-of-flight camera,” in 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2010), pp. 1173–1180.

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. First International Symposium on 3D Data Processing Visualization and Transmission, (2002), pp. 24–36.

Shen, G.

S. Feng, Y. Zhang, Q. Chen, C. Zuo, R. Li, and G. Shen, “General solution for high dynamic range three-dimensional shape measurement using the fringe projection technique,” Opt. Lasers Eng. 59, 56–71 (2014).
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C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
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Srinivasan, V.

Su, X.

X. Su and Q. Zhang, “Dynamic 3-d shape measurement method: a review,” Opt. Lasers Eng. 48, 191–204 (2010).
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Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express 13, 3110–3116 (2005).
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J. Li, X. Su, and L. Guo, “Improved fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1445 (1990).
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Su, X.-Y.

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

D. Scharstein and R. Szeliski, “High-accuracy stereo depth maps using structured light,” in 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings, vol. 1 (2003), pp. 195–202.

Takeda, M.

Tao, T.

C. Zuo, T. Tao, S. Feng, L. Huang, A. Asundi, and Q. Chen, “Micro fourier transform profilometry (μftp): 3d shape measurement at 10,000 frames per second,” Opt. Lasers Eng. 102, 70–91 (2018).
[Crossref]

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: A review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

T. Tao, Q. Chen, S. Feng, J. Qian, Y. Hu, L. Huang, and C. Zuo, “High-speed real-time 3d shape measurement based on adaptive depth constraint,” Opt. Express 26, 22440–22456 (2018).
[Crossref]

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-d measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
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T. Tao, Q. Chen, S. Feng, Y. Hu, M. Zhang, and C. Zuo, “High-precision real-time 3d shape measurement based on a quad-camera system,” J. Opt. 20, 014009 (2017).
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Y. Cui, S. Schuon, D. Chan, S. Thrun, and C. Theobalt, “3d shape scanning with a time-of-flight camera,” in 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2010), pp. 1173–1180.

Thrun, S.

Y. Cui, S. Schuon, D. Chan, S. Thrun, and C. Theobalt, “3d shape scanning with a time-of-flight camera,” in 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2010), pp. 1173–1180.

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S. Foix, G. Alenya, and C. Torras, “Lock-in time-of-flight (tof) cameras: A survey,” IEEE Sensors J. 11, 1917–1926 (2011).
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Towers, C. E.

Towers, D. P.

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S. Van der Jeught and J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).
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Van Der Weide, D.

Von Bally, G.

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

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

Weise, T.

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

Weng, J.

Wu, F.

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with fourier-assisted phase shifting,” IEEE J. Sel. Top. Signal Process. 9, 396–408 (2015).
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Z. Yang, Z. Xiong, Y. Zhang, J. Wang, and F. Wu, “Depth acquisition from density modulated binary patterns,” in Proceedings of the IEEE conference on computer vision and pattern recognition, (2013), pp. 25–32.

Y. Zhang, Z. Xiong, and F. Wu, “Hybrid structured light for scalable depth sensing,” 2012 19th IEEE Int. Conf. on Image Process. pp. 17–20 (2012).

Xi, J.

Xiong, Z.

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with fourier-assisted phase shifting,” IEEE J. Sel. Top. Signal Process. 9, 396–408 (2015).
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Y. Zhang, Z. Xiong, and F. Wu, “Hybrid structured light for scalable depth sensing,” 2012 19th IEEE Int. Conf. on Image Process. pp. 17–20 (2012).

Z. Yang, Z. Xiong, Y. Zhang, J. Wang, and F. Wu, “Depth acquisition from density modulated binary patterns,” in Proceedings of the IEEE conference on computer vision and pattern recognition, (2013), pp. 25–32.

Xu, J.

Yang, Z.

Z. Yang, Z. Xiong, Y. Zhang, J. Wang, and F. Wu, “Depth acquisition from density modulated binary patterns,” in Proceedings of the IEEE conference on computer vision and pattern recognition, (2013), pp. 25–32.

Yin, W.

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: A review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

Yin, Y.

Yu, Y.

Zhang, G.

Zhang, L.

L. Zhang, B. Curless, and S. M. Seitz, “Rapid shape acquisition using color structured light and multi-pass dynamic programming,” in Proceedings. First International Symposium on 3D Data Processing Visualization and Transmission, (2002), pp. 24–36.

Zhang, M.

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-d measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

T. Tao, Q. Chen, S. Feng, Y. Hu, M. Zhang, and C. Zuo, “High-precision real-time 3d shape measurement based on a quad-camera system,” J. Opt. 20, 014009 (2017).
[Crossref]

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

Zhang, Q.

X. Su and Q. Zhang, “Dynamic 3-d shape measurement method: a review,” Opt. Lasers Eng. 48, 191–204 (2010).
[Crossref]

Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express 13, 3110–3116 (2005).
[Crossref]

Zhang, S.

Zhang, Y.

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with fourier-assisted phase shifting,” IEEE J. Sel. Top. Signal Process. 9, 396–408 (2015).
[Crossref]

S. Feng, Y. Zhang, Q. Chen, C. Zuo, R. Li, and G. Shen, “General solution for high dynamic range three-dimensional shape measurement using the fringe projection technique,” Opt. Lasers Eng. 59, 56–71 (2014).
[Crossref]

Z. Yang, Z. Xiong, Y. Zhang, J. Wang, and F. Wu, “Depth acquisition from density modulated binary patterns,” in Proceedings of the IEEE conference on computer vision and pattern recognition, (2013), pp. 25–32.

Y. Zhang, Z. Xiong, and F. Wu, “Hybrid structured light for scalable depth sensing,” 2012 19th IEEE Int. Conf. on Image Process. pp. 17–20 (2012).

Zhang, Z.

Z. Zhang, “Review of single-shot 3d shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50, 1097–1106 (2012).
[Crossref]

Z. Zhang, C. E. Towers, and D. P. Towers, “Time efficient color fringe projection system for 3d shape and color using optimum 3-frequency selection,” Opt. Express 14, 6444–6455 (2006).
[Crossref]

Zhao, S.

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with fourier-assisted phase shifting,” IEEE J. Sel. Top. Signal Process. 9, 396–408 (2015).
[Crossref]

Zhong, J.

Zhong, K.

Zhou, X.

Zibley, P. C.

Zuo, C.

C. Zuo, T. Tao, S. Feng, L. Huang, A. Asundi, and Q. Chen, “Micro fourier transform profilometry (μftp): 3d shape measurement at 10,000 frames per second,” Opt. Lasers Eng. 102, 70–91 (2018).
[Crossref]

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: A review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-d measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

T. Tao, Q. Chen, S. Feng, J. Qian, Y. Hu, L. Huang, and C. Zuo, “High-speed real-time 3d shape measurement based on adaptive depth constraint,” Opt. Express 26, 22440–22456 (2018).
[Crossref]

T. Tao, Q. Chen, S. Feng, Y. Hu, M. Zhang, and C. Zuo, “High-precision real-time 3d shape measurement based on a quad-camera system,” J. Opt. 20, 014009 (2017).
[Crossref]

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

S. Feng, Y. Zhang, Q. Chen, C. Zuo, R. Li, and G. Shen, “General solution for high dynamic range three-dimensional shape measurement using the fringe projection technique,” Opt. Lasers Eng. 59, 56–71 (2014).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, and F. Feng, “High-speed three-dimensional profilometry for multiple objects with complex shapes,” Opt. Express 20, 19493–19510 (2012).
[Crossref]

Adv. Opt. Photonics (1)

J. Geng, “Structured-light 3d surface imaging: a tutorial,” Adv. Opt. Photonics 3, 128–160 (2011).
[Crossref]

Appl. Opt. (5)

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

P. Cong, Z. Xiong, Y. Zhang, S. Zhao, and F. Wu, “Accurate dynamic 3d sensing with fourier-assisted phase shifting,” IEEE J. Sel. Top. Signal Process. 9, 396–408 (2015).
[Crossref]

IEEE Sensors J. (1)

S. Foix, G. Alenya, and C. Torras, “Lock-in time-of-flight (tof) cameras: A survey,” IEEE Sensors J. 11, 1917–1926 (2011).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell (2)

K. L. Boyer and A. C. Kak, “Color-encoded structured light for rapid active ranging,” IEEE Trans. Pattern Anal. Mach. Intell 1, 14–28 (1987).
[Crossref] [PubMed]

S. D. Cochran and G. Medioni, “3-d surface description from binocular stereo,” IEEE Trans. Pattern Anal. Mach. Intell 10, 981–994 (1992).
[Crossref]

J. Opt. (1)

T. Tao, Q. Chen, S. Feng, Y. Hu, M. Zhang, and C. Zuo, “High-precision real-time 3d shape measurement based on a quad-camera system,” J. Opt. 20, 014009 (2017).
[Crossref]

J. Opt. Soc. Am. (1)

JOSA A (1)

J. Li, L. G. Hassebrook, and C. Guan, “Optimized two-frequency phase-measuring-profilometry light-sensor temporal-noise sensitivity,” JOSA A 20, 106–115 (2003).
[Crossref]

Opt. Commun. (1)

X.-Y. Su, G. Von 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)

J. Li, X. Su, and L. Guo, “Improved fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1445 (1990).
[Crossref]

Opt. Express (9)

C. Zuo, Q. Chen, G. Gu, S. Feng, and F. Feng, “High-speed three-dimensional profilometry for multiple objects with complex shapes,” Opt. Express 20, 19493–19510 (2012).
[Crossref]

Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express 13, 3110–3116 (2005).
[Crossref]

Z. Zhang, C. E. Towers, and D. P. Towers, “Time efficient color fringe projection system for 3d shape and color using optimum 3-frequency selection,” Opt. Express 14, 6444–6455 (2006).
[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]

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

X. Liu and J. Kofman, “High-frequency background modulation fringe patterns based on a fringe-wavelength geometry-constraint model for 3d surface-shape measurement,” Opt. Express 25, 16618–16628 (2017).
[Crossref] [PubMed]

Z. Liu, P. C. Zibley, and S. Zhang, “Motion-induced error compensation for phase shifting profilometry,” Opt. Express 26, 12632–12637 (2018).
[Crossref]

T. Tao, Q. Chen, S. Feng, J. Qian, Y. Hu, L. Huang, and C. Zuo, “High-speed real-time 3d shape measurement based on adaptive depth constraint,” Opt. Express 26, 22440–22456 (2018).
[Crossref]

B. Li, Z. Liu, and S. Zhang, “Motion-induced error reduction by combining fourier transform profilometry with phase-shifting profilometry,” Opt. Express 24, 23289–23303 (2016).
[Crossref]

Opt. Lasers Eng. (13)

Z. Zhang, “Review of single-shot 3d shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50, 1097–1106 (2012).
[Crossref]

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of fourier transform, windowed fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51, 953–960 (2013).
[Crossref]

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-d measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

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

C. Zuo, T. Tao, S. Feng, L. Huang, A. Asundi, and Q. Chen, “Micro fourier transform profilometry (μftp): 3d shape measurement at 10,000 frames per second,” Opt. Lasers Eng. 102, 70–91 (2018).
[Crossref]

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

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Supplementary Material (7)

NameDescription
» Visualization 1       The motion regions determined by FFDM and PFDM.
» Visualization 2       Measurement results in the first complex scene
» Visualization 3       Measurement results in the second complex scene.
» Visualization 4       The measurement results of complex rigid measures.
» Visualization 5       The measurement results of complex non-rigid measures.
» Visualization 6       The real-time measurement processes and results based on PSP.
» Visualization 7       The real-time measurement processes and results based on our method.

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Figures (16)

Fig. 1
Fig. 1 The principle of SPU.
Fig. 2
Fig. 2 The phases solved by PSP and the phases solved by FTP of the three three-step phase-shift fringe patterns in the case where the object moves in translation.
Fig. 3
Fig. 3 The phases solved by PSP and the phases solved by FTP of the three three-step phase-shift fringe patterns in the case where the object changes direction of motion.
Fig. 4
Fig. 4 Fusion algorithm flow diagram (The dark blue region indicates the result of the PSP, the dark red region indicates the result of the FTP and the dark green region indicates the combined result).
Fig. 5
Fig. 5 Difference between PFDM by two FTP phases and PFDM by two PSP phases.
Fig. 6
Fig. 6 Threshold determination process.
Fig. 7
Fig. 7 Simulation results of FFDM. (a) Intensity changes of the first image of the three-step phase-shifting fringe images in a noise-free environment when the phase changes at a speed of π/150 per frame. (b) Intensity changes of three fringe images in a noise-free environment when the phase changes at a speed of π/150 per frame. (c) The intensity changes in a noisy environment when the phase changes at a speed of π/150 per frame. (d) Another perspective of (c). (e) The relationship between the accuracy of motion judgment and the speed of phase change under different noise.
Fig. 8
Fig. 8 Simulation results of PFDM. (a) The changes of the PSP phase in a noise-free environment when the phase changes at a speed of π/150 per frame. (b) The changes of the FTP phase in a noise-free environment when the phase changes at a speed of π/150 per frame. (c) The changes of the PSP phase in a noisy environment when the phase changes at a speed of π/150 per frame. (d) The changes of the FTP phase in a noisey environment when the phase changes at a speed of π/150 per frame. (e) The relationship between the accuracy of motion judgment of PFDM using PSP phases and the speed of phase change under different noise. (f) The relationship between the accuracy of motion judgment of PFDM using FTP phases and the speed of phase change under different noise.
Fig. 9
Fig. 9 The quad-camera color real-time 3D imaging system.
Fig. 10
Fig. 10 The motion regions determined by FFDM and PFDM (See Visualization 1 for the whole results). (a) The first measurement scene (A flat plate in translational motion). (b) The result of FFDM of the first scene. (c) The result of PFDM of the first scene. (d) The second measurement scene (A flat plate in rotational motion). (e) The result of FFDM of the second scene. (f) The result of PFDM of the second scene. (g) The third measurement scene. (h) The result of FFDM of the third scene (A complex object in translational motion). (i) The result of PFDM of the third scene. (j) The fourth measurement scene (A hand in arbitrary motion). (k) The result of FFDM of the fourth scene. (l) The result of PFDM of the fourth scene.
Fig. 11
Fig. 11 The accuracy of motion judgment of FFPM and PFDM. (a) The result of the first scene. (b) The result of the second scene. (c) The result of the third scene. (d) The result of the fourth scene.
Fig. 12
Fig. 12 Measurement results in the first complex scene (See Visualization 2 for the whole results). (a) The captured all-white map. (b) The detected motion areas. (c) The results measured by the conventional PSP. (d) The results measured by our method. (e) The error distribution of the precision ball data in (c). (g) The error distribution of the flat plate data in (c). (i) The error distribution of the data of the precision ball in (d). (k) The error distribution of the flat plate data in (d). (f), (h), (j), (l) The histograms of (e), (g), (i), (k).
Fig. 13
Fig. 13 Measurement results in the second complex scenario (see Visualization 3 for the whole results). (a) The background map collected. (b) The detected motion areas. (c) The results measured by the conventional PSP. (e) The error distribution of the flat plate data of (c). (g)The histogram of (e). (d) The results measured by our method. (f) The error distribution of the flat plate data in (e). (h) The histogram of (f).
Fig. 14
Fig. 14 The measurement results of the first scene (see Visualization 4 for the whole results). (a) The measurement results of PSP. (b) The motion areas determined by PFDM. (c) The measurement results of our method.
Fig. 15
Fig. 15 The measurement results of the second scene (see Visualization 5 for the whole results). (a) The measurement results of PSP. (b) The motion areas determined by PFDM. (c) The measurement results of our method.
Fig. 16
Fig. 16 The real-time measurement processes and results based on (a) PSP (see Visualization 6 for the whole process) and (b) our method (see Visualization 7 for the whole process).

Equations (16)

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I 1 c ( u c , v c ) = A c ( u c , v c ) + B c ( u c , v c ) cos ( Φ c ( u c , v c ) ) ,
I 2 c ( u c , v c ) = A c ( u c , v c ) + B c ( u c , v c ) cos ( Φ c ( u c , v c ) + 2 π 3 ) ,
I 3 c ( u c , v c ) = A c ( u c , v c ) + B c ( u c , v c ) cos ( Φ c ( u c , v c ) + 4 π 3 ) ,
ϕ c ( u c , v c ) = arctan ( 3 ( I 2 c ( u c , v c ) I 3 c ( u c , v c ) ) 2 I 1 c ( u c , v c ) I 2 c ( u c , v c ) I 3 c ( u c , v c ) ) ,
Φ c ( u c , v c ) = ϕ c ( u c , v c ) + 2 k c ( u c , v c ) π , k c ( u c , v c ) [ 0 , N 1 ] ,
I 0 c ( u c , v c ) = A c ( u c , v c ) .
I n c ( u c , v c ) = I 2 c ( u c , v c ) I 0 c ( u c , v c ) I 0 c ( u c , v c ) + b ,
I m 1 c ( u c , v c ) = A c ( u c , v c ) + B c ( u c , v c ) cos ( Φ c ( u c , v c ) + Δ Φ 1 c ( u c , v c ) ) ,
I m 2 c ( u c , v c ) = A c ( u c , v c ) + B c ( u c , v c ) cos ( Φ c ( u c , v c ) + 2 π 3 ) ,
I m 3 c ( u c , v c ) = A c ( u c , v c ) + B c ( u c , v c ) cos ( Φ c ( u c , v c ) + 4 π 3 + Δ Φ 2 c ( u c , v c ) ) ,
Δ Φ 1 c ( u c , v c ) > 0 > Δ Φ 2 c ( u c , v c ) ,
Δ Φ 1 c ( u c , v c ) < 0 < Δ Φ 2 c ( u c , v c ) .
Δ Φ 1 c ( u c , v c ) > 0 , Δ Φ 2 c ( u c , v c ) > 0 ,
Δ Φ 1 c ( u c , v c ) < 0 , Δ Φ 2 c ( u c , v c ) < 0 .
Δ Φ c ( u c , v c , t ) = | Φ c ( u c , v c , t ) Φ c ( u c , v c , t Δ t ) | = | d Φ c ( u c , v c , t ) d t Δ t | ,
flag c ( u c , v c , t ) = { 0 , Δ Φ c ( u c , v c , t ) < Th 1 , Δ Φ c ( u c , v c , t ) Th ,

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