Abstract

A new fringe projection method for surface-shape measurement was developed using four high-frequency phase-shifted background modulation fringe patterns. The pattern frequency is determined using a new fringe-wavelength geometry-constraint model that allows only two corresponding-point candidates in the measurement volume. The correct corresponding point is selected with high reliability using a binary pattern computed from intensity background encoded in the fringe patterns. Equations of geometry-constraint parameters permit parameter calculation prior to measurement, thus reducing measurement computational cost. Experiments demonstrated the ability of the method to perform 3D shape measurement for a surface with geometric discontinuity, and for spatially isolated objects.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  27. K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Gamma model and its analysis for phase measuring profilometry,” J. Opt. Soc. Am. A 27(3), 553–562 (2010).
    [Crossref] [PubMed]

2017 (2)

X. Liu and J. Kofman, “Background and amplitude encoded fringe patterns for 3D surface-shape measurement,” Opt. Lasers Eng. 94, 63–69 (2017).
[Crossref]

C. Jiang, B. Li, and S. Zhang, “Pixel-by-pixel absolute phase retrieval using three phase-shifted fringe patterns without markers,” Opt. Lasers Eng. 91, 232–241 (2017).
[Crossref]

2016 (4)

2015 (2)

K. Zhong, Z. Li, X. Zhou, Y. Li, Y. Shi, and C. Wang, “Enhanced phase measurement profilometry for industrial 3D inspection automation,” Int. J. Adv. Manuf. Technol. 76(9-12), 1563–1574 (2015).
[Crossref]

B. Wei, J. Liang, J. Li, and M. Ren, “Rapid three-dimensional chromoscan system of body surface based on digital fringe projection,” Proc. SPIE 9576, 95760P (2015).
[Crossref]

2014 (6)

2013 (3)

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51(11), 1213–1222 (2013).
[Crossref]

C. Bräuer-Burchardt, M. Möller, C. Munkelt, M. Heinze, P. Kühmstedt, and G. Notni, “On the accuracy of point correspondence methods in three-dimensional measurement systems using fringe projection,” Opt. Eng. 52(6), 063601 (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(8), 953–960 (2013).
[Crossref]

2012 (2)

D. Zheng and F. Da, “Self-correction phase unwrapping method based on Gray-code light,” Opt. Lasers Eng. 50(8), 1130–1139 (2012).
[Crossref]

Y. Wang and S. Zhang, “Novel phase-coding method for absolute phase retrieval,” Opt. Lett. 37(11), 2067–2069 (2012).
[Crossref] [PubMed]

2010 (4)

S. Gai and F. Da, “Fringe image analysis based on the amplitude modulation method,” Opt. Express 18(10), 10704–10719 (2010).
[Crossref] [PubMed]

Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

C. L. Heike, K. Upson, E. Stuhaug, and S. M. Weinberg, “3D digital stereophotogrammetry: a practical guide to facial image acquisition,” Head Face Med. 6(1), 18 (2010).
[Crossref] [PubMed]

K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Gamma model and its analysis for phase measuring profilometry,” J. Opt. Soc. Am. A 27(3), 553–562 (2010).
[Crossref] [PubMed]

2008 (1)

Z. Li, “Accurate calibration method for a structured light system,” Opt. Eng. 47(5), 053604 (2008).
[Crossref]

2004 (1)

G. Sansoni and F. Docchio, “Three-dimensional optical measurements and reverse engineering for automotive applications,” Robot. Comput.-Integr. Manuf. 20(5), 359–367 (2004).
[Crossref]

1998 (1)

An, Y.

Asundi, A.

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]

A. Asundi and Z. Wensen, “Fast phase-unwrapping algorithm based on a gray-scale mask and flood fill,” Appl. Opt. 37(23), 5416–5420 (1998).
[Crossref] [PubMed]

Bao, Q.

Barnes, J. C.

Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

Bräuer-Burchardt, C.

C. Bräuer-Burchardt, M. Möller, C. Munkelt, M. Heinze, P. Kühmstedt, and G. Notni, “On the accuracy of point correspondence methods in three-dimensional measurement systems using fringe projection,” Opt. Eng. 52(6), 063601 (2013).
[Crossref]

Cao, C.

C. Cao, Y. Weng, S. Zhou, Y. Tong, and K. Zhou, “FaceWarehouse: A 3D facial expression database for visual computing,” IEEE Trans. Vis. Comput. Graph. 20(3), 413–425 (2014).
[Crossref] [PubMed]

Chang, Y.

Chen, H.

Chen, Q.

T. Tao, Q. Chen, J. Da, S. Feng, Y. Hu, and C. Zuo, “Real-time 3-D shape measurement with composite phase-shifting fringes and multi-view system,” Opt. Express 24(18), 20253–20269 (2016).
[Crossref] [PubMed]

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]

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(8), 953–960 (2013).
[Crossref]

Chen, V.

Da, F.

D. Zheng and F. Da, “Self-correction phase unwrapping method based on Gray-code light,” Opt. Lasers Eng. 50(8), 1130–1139 (2012).
[Crossref]

S. Gai and F. Da, “Fringe image analysis based on the amplitude modulation method,” Opt. Express 18(10), 10704–10719 (2010).
[Crossref] [PubMed]

Da, J.

Docchio, F.

G. Sansoni and F. Docchio, “Three-dimensional optical measurements and reverse engineering for automotive applications,” Robot. Comput.-Integr. Manuf. 20(5), 359–367 (2004).
[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(8), 953–960 (2013).
[Crossref]

Feng, S.

T. Tao, Q. Chen, J. Da, S. Feng, Y. Hu, and C. Zuo, “Real-time 3-D shape measurement with composite phase-shifting fringes and multi-view system,” Opt. Express 24(18), 20253–20269 (2016).
[Crossref] [PubMed]

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(8), 953–960 (2013).
[Crossref]

Gai, S.

Gu, G.

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(8), 953–960 (2013).
[Crossref]

Hao, Q.

Hassebrook, L. G.

Heike, C. L.

C. L. Heike, K. Upson, E. Stuhaug, and S. M. Weinberg, “3D digital stereophotogrammetry: a practical guide to facial image acquisition,” Head Face Med. 6(1), 18 (2010).
[Crossref] [PubMed]

Heikkila, J.

J. Heikkila and O. Silven, “A four-step camera calibration procedure with implicit image correction,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 1997), pp. 1106–1112.
[Crossref]

Heinze, M.

C. Bräuer-Burchardt, M. Möller, C. Munkelt, M. Heinze, P. Kühmstedt, and G. Notni, “On the accuracy of point correspondence methods in three-dimensional measurement systems using fringe projection,” Opt. Eng. 52(6), 063601 (2013).
[Crossref]

Heist, S.

S. Heist, A. Mann, P. Kühmstedt, P. Schreiber, and G. Notni, “Array projection of aperiodic sinusoidal fringes for high-speed three-dimensional shape measurement,” Opt. Eng. 53(11), 112208 (2014).
[Crossref]

Hu, Y.

Huang, L.

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]

Hyun, J.-S.

Jia, S.

Jiang, C.

C. Jiang, B. Li, and S. Zhang, “Pixel-by-pixel absolute phase retrieval using three phase-shifted fringe patterns without markers,” Opt. Lasers Eng. 91, 232–241 (2017).
[Crossref]

Kao, F. J.

Kofman, J.

X. Liu and J. Kofman, “Background and amplitude encoded fringe patterns for 3D surface-shape measurement,” Opt. Lasers Eng. 94, 63–69 (2017).
[Crossref]

Kühmstedt, P.

S. Heist, A. Mann, P. Kühmstedt, P. Schreiber, and G. Notni, “Array projection of aperiodic sinusoidal fringes for high-speed three-dimensional shape measurement,” Opt. Eng. 53(11), 112208 (2014).
[Crossref]

C. Bräuer-Burchardt, M. Möller, C. Munkelt, M. Heinze, P. Kühmstedt, and G. Notni, “On the accuracy of point correspondence methods in three-dimensional measurement systems using fringe projection,” Opt. Eng. 52(6), 063601 (2013).
[Crossref]

Kuo, C. Y.

Lau, D. L.

Lei, Y.

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51(11), 1213–1222 (2013).
[Crossref]

Li, B.

C. Jiang, B. Li, and S. Zhang, “Pixel-by-pixel absolute phase retrieval using three phase-shifted fringe patterns without markers,” Opt. Lasers Eng. 91, 232–241 (2017).
[Crossref]

Li, J.

B. Wei, J. Liang, J. Li, and M. Ren, “Rapid three-dimensional chromoscan system of body surface based on digital fringe projection,” Proc. SPIE 9576, 95760P (2015).
[Crossref]

Li, R.

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(8), 953–960 (2013).
[Crossref]

Li, Y.

K. Zhong, Z. Li, X. Zhou, Y. Li, Y. Shi, and C. Wang, “Enhanced phase measurement profilometry for industrial 3D inspection automation,” Int. J. Adv. Manuf. Technol. 76(9-12), 1563–1574 (2015).
[Crossref]

Li, Z.

K. Zhong, Z. Li, X. Zhou, Y. Li, Y. Shi, and C. Wang, “Enhanced phase measurement profilometry for industrial 3D inspection automation,” Int. J. Adv. Manuf. Technol. 76(9-12), 1563–1574 (2015).
[Crossref]

L. Song, Y. Chang, Z. Li, P. Wang, G. Xing, and J. Xi, “Application of global phase filtering method in multi frequency measurement,” Opt. Express 22(11), 13641–13647 (2014).
[Crossref] [PubMed]

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51(11), 1213–1222 (2013).
[Crossref]

Z. Li, “Accurate calibration method for a structured light system,” Opt. Eng. 47(5), 053604 (2008).
[Crossref]

Liang, J.

B. Wei, J. Liang, J. Li, and M. Ren, “Rapid three-dimensional chromoscan system of body surface based on digital fringe projection,” Proc. SPIE 9576, 95760P (2015).
[Crossref]

Liu, K.

Liu, X.

X. Liu and J. Kofman, “Background and amplitude encoded fringe patterns for 3D surface-shape measurement,” Opt. Lasers Eng. 94, 63–69 (2017).
[Crossref]

Lohry, W.

Mann, A.

S. Heist, A. Mann, P. Kühmstedt, P. Schreiber, and G. Notni, “Array projection of aperiodic sinusoidal fringes for high-speed three-dimensional shape measurement,” Opt. Eng. 53(11), 112208 (2014).
[Crossref]

Möller, M.

C. Bräuer-Burchardt, M. Möller, C. Munkelt, M. Heinze, P. Kühmstedt, and G. Notni, “On the accuracy of point correspondence methods in three-dimensional measurement systems using fringe projection,” Opt. Eng. 52(6), 063601 (2013).
[Crossref]

Munkelt, C.

C. Bräuer-Burchardt, M. Möller, C. Munkelt, M. Heinze, P. Kühmstedt, and G. Notni, “On the accuracy of point correspondence methods in three-dimensional measurement systems using fringe projection,” Opt. Eng. 52(6), 063601 (2013).
[Crossref]

Nguyen, D. A.

Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

Notni, G.

S. Heist, A. Mann, P. Kühmstedt, P. Schreiber, and G. Notni, “Array projection of aperiodic sinusoidal fringes for high-speed three-dimensional shape measurement,” Opt. Eng. 53(11), 112208 (2014).
[Crossref]

C. Bräuer-Burchardt, M. Möller, C. Munkelt, M. Heinze, P. Kühmstedt, and G. Notni, “On the accuracy of point correspondence methods in three-dimensional measurement systems using fringe projection,” Opt. Eng. 52(6), 063601 (2013).
[Crossref]

Ren, M.

B. Wei, J. Liang, J. Li, and M. Ren, “Rapid three-dimensional chromoscan system of body surface based on digital fringe projection,” Proc. SPIE 9576, 95760P (2015).
[Crossref]

Sansoni, G.

G. Sansoni and F. Docchio, “Three-dimensional optical measurements and reverse engineering for automotive applications,” Robot. Comput.-Integr. Manuf. 20(5), 359–367 (2004).
[Crossref]

Schreiber, P.

S. Heist, A. Mann, P. Kühmstedt, P. Schreiber, and G. Notni, “Array projection of aperiodic sinusoidal fringes for high-speed three-dimensional shape measurement,” Opt. Eng. 53(11), 112208 (2014).
[Crossref]

Shen, G.

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(8), 953–960 (2013).
[Crossref]

Shi, Y.

K. Zhong, Z. Li, X. Zhou, Y. Li, Y. Shi, and C. Wang, “Enhanced phase measurement profilometry for industrial 3D inspection automation,” Int. J. Adv. Manuf. Technol. 76(9-12), 1563–1574 (2015).
[Crossref]

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51(11), 1213–1222 (2013).
[Crossref]

Silven, O.

J. Heikkila and O. Silven, “A four-step camera calibration procedure with implicit image correction,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 1997), pp. 1106–1112.
[Crossref]

Song, L.

Stuhaug, E.

C. L. Heike, K. Upson, E. Stuhaug, and S. M. Weinberg, “3D digital stereophotogrammetry: a practical guide to facial image acquisition,” Head Face Med. 6(1), 18 (2010).
[Crossref] [PubMed]

Su, W. H.

Tao, T.

Tong, Y.

C. Cao, Y. Weng, S. Zhou, Y. Tong, and K. Zhou, “FaceWarehouse: A 3D facial expression database for visual computing,” IEEE Trans. Vis. Comput. Graph. 20(3), 413–425 (2014).
[Crossref] [PubMed]

Upson, K.

C. L. Heike, K. Upson, E. Stuhaug, and S. M. Weinberg, “3D digital stereophotogrammetry: a practical guide to facial image acquisition,” Head Face Med. 6(1), 18 (2010).
[Crossref] [PubMed]

Wang, C.

K. Zhong, Z. Li, X. Zhou, Y. Li, Y. Shi, and C. Wang, “Enhanced phase measurement profilometry for industrial 3D inspection automation,” Int. J. Adv. Manuf. Technol. 76(9-12), 1563–1574 (2015).
[Crossref]

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51(11), 1213–1222 (2013).
[Crossref]

Wang, P.

Wang, Y.

Wang, Z.

Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

Wei, B.

B. Wei, J. Liang, J. Li, and M. Ren, “Rapid three-dimensional chromoscan system of body surface based on digital fringe projection,” Proc. SPIE 9576, 95760P (2015).
[Crossref]

Weinberg, S. M.

C. L. Heike, K. Upson, E. Stuhaug, and S. M. Weinberg, “3D digital stereophotogrammetry: a practical guide to facial image acquisition,” Head Face Med. 6(1), 18 (2010).
[Crossref] [PubMed]

Weng, Y.

C. Cao, Y. Weng, S. Zhou, Y. Tong, and K. Zhou, “FaceWarehouse: A 3D facial expression database for visual computing,” IEEE Trans. Vis. Comput. Graph. 20(3), 413–425 (2014).
[Crossref] [PubMed]

Wensen, Z.

Xi, J.

Xing, G.

Xu, Y.

Yang, J.

Zhang, M.

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, S.

Zheng, D.

D. Zheng and F. Da, “Self-correction phase unwrapping method based on Gray-code light,” Opt. Lasers Eng. 50(8), 1130–1139 (2012).
[Crossref]

Zhong, K.

K. Zhong, Z. Li, X. Zhou, Y. Li, Y. Shi, and C. Wang, “Enhanced phase measurement profilometry for industrial 3D inspection automation,” Int. J. Adv. Manuf. Technol. 76(9-12), 1563–1574 (2015).
[Crossref]

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51(11), 1213–1222 (2013).
[Crossref]

Zhou, K.

C. Cao, Y. Weng, S. Zhou, Y. Tong, and K. Zhou, “FaceWarehouse: A 3D facial expression database for visual computing,” IEEE Trans. Vis. Comput. Graph. 20(3), 413–425 (2014).
[Crossref] [PubMed]

Zhou, S.

C. Cao, Y. Weng, S. Zhou, Y. Tong, and K. Zhou, “FaceWarehouse: A 3D facial expression database for visual computing,” IEEE Trans. Vis. Comput. Graph. 20(3), 413–425 (2014).
[Crossref] [PubMed]

Zhou, X.

K. Zhong, Z. Li, X. Zhou, Y. Li, Y. Shi, and C. Wang, “Enhanced phase measurement profilometry for industrial 3D inspection automation,” Int. J. Adv. Manuf. Technol. 76(9-12), 1563–1574 (2015).
[Crossref]

Zuo, C.

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]

T. Tao, Q. Chen, J. Da, S. Feng, Y. Hu, and C. Zuo, “Real-time 3-D shape measurement with composite phase-shifting fringes and multi-view system,” Opt. Express 24(18), 20253–20269 (2016).
[Crossref] [PubMed]

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Figures (5)

Fig. 1
Fig. 1 Examples of phase map values and their corresponding binary pattern values.
Fig. 2
Fig. 2 Diagram of geometry constraints and correspondence. (OLC, OP, and ORC, represent the left camera, projector, and right camera centers; eP and eRC are the projector and right camera epipolar lines; and Qk-1,… Qk+2 represent points on the line of sight).
Fig. 3
Fig. 3 Experimental result of measurement of double-hemisphere object: a) one of the captured images, b) wrapped phase map, c) computed binary pattern, and d) 3D reconstructed point cloud.
Fig. 4
Fig. 4 3D measurement of curling-stone handle: a) one of the captured images, b) computed wrapped phase map, c) computed binary pattern, and d) 3D reconstructed point cloud.
Fig. 5
Fig. 5 3D measurement of spatially isolated objects: a) mask and cylinder, and b) 3D reconstructed point cloud.

Equations (15)

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I 1 (x,y)=B(x,y)+A(x,y)cosφ(x,y)+ B 0 (x,y) I 2 (x,y)=B(x,y)+A(x,y)cos[ φ(x,y)π/2 ] I 3 (x,y)=B(x,y)+A(x,y)cos[ φ(x,y)π ]+ B 0 (x,y) , I 4 (x,y)=B(x,y)+A(x,y)cos[ φ(x,y) 3π/2 ]
φ(x,y)= tan 1 [ I 2 (x,y) I 4 (x,y) I 1 (x,y) I 3 (x,y) ] .
B 0 (x,y)= I 1 (x,y)+ I 3 (x,y) I 2 (x,y) I 4 (x,y) 2 .
[ X c Y c Z c ]=[ L T L ][ X w Y w Z w 1 ]=[ l 11 l 21 l 31 l 12 l 22 l 32 l 13 l 23 l 33 t 1 t 2 t 3 ][ X w Y w Z w 1 ] ,
[ x c y c ]=[ X c / Z c Y c / Z c ] .
X w = a 1 Z w + a 0 Y w = b 1 Z w + b 0 ,
a 1 = ( y c l 33 l 23 )( x c l 32 l 12 )( y c l 32 l 22 )( x c l 33 l 13 ) ( y c l 32 l 22 )( x c l 31 l 11 )( y c l 31 l 21 )( x c l 32 l 12 ) a 0 = ( y c t 3 t 2 )( x c l 32 l 12 )( y c l 32 l 22 )( x c t 3 t 1 ) ( y c l 32 l 22 )( x c l 31 l 11 )( y c l 31 l 21 )( x c l 32 l 12 ) . b 1 = ( y c l 33 l 23 )( x c l 31 l 11 )( y c l 31 l 21 )( x c l 33 l 13 ) ( y c l 31 l 21 )( x c l 32 l 12 )( y c l 32 l 22 )( x c l 31 l 11 ) b 0 = ( y c t 3 t 2 )( x c l 31 l 11 )( y c l 31 l 21 )( x c t 3 t 1 ) ( y c l 31 l 21 )( x c l 32 l 12 )( y c l 32 l 22 )( x c l 31 l 11 )
[ X p Y p Z p ]=[ M T M ][ X w Y w Z w 1 ]=[ m 11 m 21 m 31 m 12 m 22 m 32 m 13 m 23 m 33 t 4 t 5 t 6 ][ X w Y w Z w 1 ] ,
[ x p y p ]=[ X p / Z p Y p / Z p ]=[ p 1 Z w + p 0 s 1 Z w + s 0 q 1 Z w + q 0 s 1 Z w + s 0 ] ,
p 1 = m 11 a 1 + m 12 b 1 + m 13 p 0 = m 11 a 0 + m 12 b 0 + t 4 q 1 = m 21 a 1 + m 22 b 1 + m 23 . q 0 = m 21 a 0 + m 22 b 0 + t 5 s 1 = m 31 a 1 + m 32 b 1 + m 33 s 0 = m 31 a 0 + m 32 b 0 + t 6
( q 1 s 0 q 0 s 1 ) x p +( p 0 s 1 p 1 s 0 ) y p +( p 1 q 0 p 0 q 1 )=0 .
[ u p v p 1 ]=[ f u γ u 0 0 f v v 0 0 0 1 ][ x p y p 1 ] ,
u max p u min p = f u ( p 1 Z max + p 0 s 1 Z max + s 0 p 1 Z min + p 0 s 1 Z min + s 0 )=2 f u ( p 1 s 0 p 0 s 1 )d s 0 2 s 1 2 d 2 .
u max p u min p <2λ .
λ min = f u ( p 1 s 0 p 0 s 1 )d s 0 2 s 1 2 d 2 .

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