Abstract

Considering the measuring range limitation of a single sensor system, multi-sensor system has become essential in obtaining complete image information of the object in the field of 3D image reconstruction. However, for the traditional multi-sensors worked independently in its system, there was some point in calibrating each sensor system separately. And the calibration between all single sensor systems was complicated and required a long time. In this paper, we present a flexible 3D reconstruction method based on phase-matching in multi-sensor system. While calibrating each sensor, it realizes the data registration of multi-sensor system in a unified coordinate system simultaneously. After all sensors are calibrated, the whole 3D image data directly exist in the unified coordinate system, and there is no need to calibrate the positions between sensors any more. Experimental results prove that the method is simple in operation, accurate in measurement, and fast in 3D image reconstruction.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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  28. J. H. Huang and Q. Y. Wu, “A new reconstruction method based on fringe projection of three-dimensional measuring system,” Opt. Lasers Eng. 52, 115–122 (2014).
    [Crossref]

2014 (3)

J. F. Dai, G. Chen, and S. Zhang, “Three-dimensional shape measurement with dual reference phase maps,” Opt. Eng. 53(1), 014102 (2014).
[Crossref]

W. Lohry, V. Chen, and S. Zhang, “Absolute three-dimensional shape measurement using coded fringe patterns without phase unwrapping or projector calibration,” Opt. Express 22(2), 1287–1301 (2014).
[Crossref] [PubMed]

J. H. Huang and Q. Y. Wu, “A new reconstruction method based on fringe projection of three-dimensional measuring system,” Opt. Lasers Eng. 52, 115–122 (2014).
[Crossref]

2013 (2)

J. Chen, X. J. Wu, M. Y. Wang, and X. F. Li, “3D shape modeling using a self-developed hand-held 3D laser scanner and an efficient HT-ICP point cloud registration algorithm,” Opt. Lasers Eng. 45, 414–423 (2013).
[Crossref]

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

2012 (3)

I. Léandry, C. Brèque, and V. Vallea, “Calibration of a structured-light projection system: Development to large dimension objects,” Opt. Lasers Eng. 50(3), 373–379 (2012).
[Crossref]

Q. C. Zhang, X. Y. Su, L. Q. Xiang, and X. Z. Sun, “3-D shape measurement based on complementary Gray-code light,” Opt. Lasers Eng. 50(4), 574–579 (2012).
[Crossref]

I. Léandry, C. Brèque, and V. Valle, “Calibration of a structured-light projection system: Development to large dimension objects,” Opt. Lasers Eng. 50(3), 373–379 (2012).
[Crossref]

2010 (4)

Y. Li, C. Zhao, Y. Qian, H. Wang, and H. Jin, “High-speed and dense three-dimensional surface acquisition using defocused binary patterns for spatially isolated objects,” Opt. Express 18(21), 21628–21635 (2010).
[Crossref] [PubMed]

S. Zhang, “Flexible 3D shape measurement using projector defocusing: extended measurement range,” Opt. Lett. 35(7), 934–936 (2010).
[Crossref] [PubMed]

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

X. Y. Su and Q. C. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
[Crossref]

2009 (1)

G. Sansoni, M. Trebeschi, and F. Docchio, “State-of-The-Art and Applications of 3D Imaging Sensors in Industry, Cultural Heritage, Medicine, and Criminal Investigation,” Sensors (Basel) 9(1), 568–601 (2009).
[Crossref] [PubMed]

2008 (1)

W. Gao, L. Wang, and Z. Y. Hu, “Flexible Calibration of a Portable Structured Light System through Surface Plane,” Acta Automatica Sinica 34(11), 1358–1362 (2008).
[Crossref]

2006 (1)

Y. Li and X. Y. Su, “New method for system calibration in phase measurement profilometry with large view field,” Acta Opt. Sin. 26(8), 1162–1166 (2006).

2005 (1)

J. S. Park, “Interactive 3D reconstruction from multiple images: A primitive-based approach,” Pattern Recognit. Lett. 26(16), 2558–2571 (2005).
[Crossref]

2001 (1)

Y. Q. Cheng, X. G. Wang, R. T. Collins, E. M. Riseman, and A. R. Hanson, “Three-dimensional reconstruction of points and lines with unknown correspondence across images,” Int. J. Comput. Vis. 45(2), 129–156 (2001).
[Crossref]

2000 (1)

Q. Li and J. G. Griffiths, “Iterative closest geometric objects registration,” Comput. Math. Appl. 40(10), 1171–1188 (2000).
[Crossref]

1998 (1)

T. A. Clarke and J. F. Fryer, “The development of camera calibration methods and models,” Photogramm. Rec. 16(91), 51–66 (1998).
[Crossref]

1991 (1)

1989 (1)

X. Y. Su, X. X. Chen, and L. R. Guo, “An automated method for 360° surface measurement of 3D objects,” Acta Opt. Sin. 9(7), 670–672 (1989).

1987 (1)

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. RA-3(4), 323–344 (1987).
[Crossref]

Blais, G.

G. Blais and M. D. Levine, “Registering Multiview Range Data to Create 3D Computer Objects,” in Proceedings of IEEE Transactions on Pattern Analysis and Maching Intelligence (IEEE, 1995), pp. 820–824.
[Crossref]

Brèque, C.

I. Léandry, C. Brèque, and V. Vallea, “Calibration of a structured-light projection system: Development to large dimension objects,” Opt. Lasers Eng. 50(3), 373–379 (2012).
[Crossref]

I. Léandry, C. Brèque, and V. Valle, “Calibration of a structured-light projection system: Development to large dimension objects,” Opt. Lasers Eng. 50(3), 373–379 (2012).
[Crossref]

Chen, G.

J. F. Dai, G. Chen, and S. Zhang, “Three-dimensional shape measurement with dual reference phase maps,” Opt. Eng. 53(1), 014102 (2014).
[Crossref]

Chen, J.

J. Chen, X. J. Wu, M. Y. Wang, and X. F. Li, “3D shape modeling using a self-developed hand-held 3D laser scanner and an efficient HT-ICP point cloud registration algorithm,” Opt. Lasers Eng. 45, 414–423 (2013).
[Crossref]

Chen, V.

Chen, X. X.

X. X. Chen, X. Y. Su, and L. R. Guo, “An automated method for 360°profilometry of 3-D diffuse objects,” Appl. Opt. 30(10), 1274–1278 (1991).
[Crossref] [PubMed]

X. Y. Su, X. X. Chen, and L. R. Guo, “An automated method for 360° surface measurement of 3D objects,” Acta Opt. Sin. 9(7), 670–672 (1989).

Chen, Y.

Y. Chen and G. Medioni, “Object Modeling by Registration of Multiple Range Images,” inProceedings of IEEE International Conference on Robotics and Automation (IEEE, 1991), pp. 2724–2729.
[Crossref]

Cheng, Y. Q.

Y. Q. Cheng, X. G. Wang, R. T. Collins, E. M. Riseman, and A. R. Hanson, “Three-dimensional reconstruction of points and lines with unknown correspondence across images,” Int. J. Comput. Vis. 45(2), 129–156 (2001).
[Crossref]

Clarke, T. A.

T. A. Clarke and J. F. Fryer, “The development of camera calibration methods and models,” Photogramm. Rec. 16(91), 51–66 (1998).
[Crossref]

Collins, R. T.

Y. Q. Cheng, X. G. Wang, R. T. Collins, E. M. Riseman, and A. R. Hanson, “Three-dimensional reconstruction of points and lines with unknown correspondence across images,” Int. J. Comput. Vis. 45(2), 129–156 (2001).
[Crossref]

Dai, J. F.

J. F. Dai, G. Chen, and S. Zhang, “Three-dimensional shape measurement with dual reference phase maps,” Opt. Eng. 53(1), 014102 (2014).
[Crossref]

Docchio, F.

G. Sansoni, M. Trebeschi, and F. Docchio, “State-of-The-Art and Applications of 3D Imaging Sensors in Industry, Cultural Heritage, Medicine, and Criminal Investigation,” Sensors (Basel) 9(1), 568–601 (2009).
[Crossref] [PubMed]

Fryer, J. F.

T. A. Clarke and J. F. Fryer, “The development of camera calibration methods and models,” Photogramm. Rec. 16(91), 51–66 (1998).
[Crossref]

Gao, F.

Gao, W.

W. Gao, L. Wang, and Z. Y. Hu, “Flexible Calibration of a Portable Structured Light System through Surface Plane,” Acta Automatica Sinica 34(11), 1358–1362 (2008).
[Crossref]

Griffiths, J. G.

Q. Li and J. G. Griffiths, “Iterative closest geometric objects registration,” Comput. Math. Appl. 40(10), 1171–1188 (2000).
[Crossref]

Guo, L. R.

X. X. Chen, X. Y. Su, and L. R. Guo, “An automated method for 360°profilometry of 3-D diffuse objects,” Appl. Opt. 30(10), 1274–1278 (1991).
[Crossref] [PubMed]

X. Y. Su, X. X. Chen, and L. R. Guo, “An automated method for 360° surface measurement of 3D objects,” Acta Opt. Sin. 9(7), 670–672 (1989).

Hanson, A. R.

Y. Q. Cheng, X. G. Wang, R. T. Collins, E. M. Riseman, and A. R. Hanson, “Three-dimensional reconstruction of points and lines with unknown correspondence across images,” Int. J. Comput. Vis. 45(2), 129–156 (2001).
[Crossref]

Heikkila, J.

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

Hu, Z. Y.

W. Gao, L. Wang, and Z. Y. Hu, “Flexible Calibration of a Portable Structured Light System through Surface Plane,” Acta Automatica Sinica 34(11), 1358–1362 (2008).
[Crossref]

Huang, J. H.

J. H. Huang and Q. Y. Wu, “A new reconstruction method based on fringe projection of three-dimensional measuring system,” Opt. Lasers Eng. 52, 115–122 (2014).
[Crossref]

Huang, S.

Jiang, X.

Jin, H.

Léandry, I.

I. Léandry, C. Brèque, and V. Valle, “Calibration of a structured-light projection system: Development to large dimension objects,” Opt. Lasers Eng. 50(3), 373–379 (2012).
[Crossref]

I. Léandry, C. Brèque, and V. Vallea, “Calibration of a structured-light projection system: Development to large dimension objects,” Opt. Lasers Eng. 50(3), 373–379 (2012).
[Crossref]

Levine, M. D.

G. Blais and M. D. Levine, “Registering Multiview Range Data to Create 3D Computer Objects,” in Proceedings of IEEE Transactions on Pattern Analysis and Maching Intelligence (IEEE, 1995), pp. 820–824.
[Crossref]

Li, Q.

Q. Li and J. G. Griffiths, “Iterative closest geometric objects registration,” Comput. Math. Appl. 40(10), 1171–1188 (2000).
[Crossref]

Li, X. F.

J. Chen, X. J. Wu, M. Y. Wang, and X. F. Li, “3D shape modeling using a self-developed hand-held 3D laser scanner and an efficient HT-ICP point cloud registration algorithm,” Opt. Lasers Eng. 45, 414–423 (2013).
[Crossref]

Li, Y.

Y. Li, C. Zhao, Y. Qian, H. Wang, and H. Jin, “High-speed and dense three-dimensional surface acquisition using defocused binary patterns for spatially isolated objects,” Opt. Express 18(21), 21628–21635 (2010).
[Crossref] [PubMed]

Y. Li and X. Y. Su, “New method for system calibration in phase measurement profilometry with large view field,” Acta Opt. Sin. 26(8), 1162–1166 (2006).

Lohry, W.

Medioni, G.

Y. Chen and G. Medioni, “Object Modeling by Registration of Multiple Range Images,” inProceedings of IEEE International Conference on Robotics and Automation (IEEE, 1991), pp. 2724–2729.
[Crossref]

Meng, S.

Park, J. S.

J. S. Park, “Interactive 3D reconstruction from multiple images: A primitive-based approach,” Pattern Recognit. Lett. 26(16), 2558–2571 (2005).
[Crossref]

Qian, Y.

Riseman, E. M.

Y. Q. Cheng, X. G. Wang, R. T. Collins, E. M. Riseman, and A. R. Hanson, “Three-dimensional reconstruction of points and lines with unknown correspondence across images,” Int. J. Comput. Vis. 45(2), 129–156 (2001).
[Crossref]

Sansoni, G.

G. Sansoni, M. Trebeschi, and F. Docchio, “State-of-The-Art and Applications of 3D Imaging Sensors in Industry, Cultural Heritage, Medicine, and Criminal Investigation,” Sensors (Basel) 9(1), 568–601 (2009).
[Crossref] [PubMed]

Silvén, O.

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

Su, X. Y.

Q. C. Zhang, X. Y. Su, L. Q. Xiang, and X. Z. Sun, “3-D shape measurement based on complementary Gray-code light,” Opt. Lasers Eng. 50(4), 574–579 (2012).
[Crossref]

X. Y. Su and Q. C. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
[Crossref]

Y. Li and X. Y. Su, “New method for system calibration in phase measurement profilometry with large view field,” Acta Opt. Sin. 26(8), 1162–1166 (2006).

X. X. Chen, X. Y. Su, and L. R. Guo, “An automated method for 360°profilometry of 3-D diffuse objects,” Appl. Opt. 30(10), 1274–1278 (1991).
[Crossref] [PubMed]

X. Y. Su, X. X. Chen, and L. R. Guo, “An automated method for 360° surface measurement of 3D objects,” Acta Opt. Sin. 9(7), 670–672 (1989).

Sun, X. Z.

Q. C. Zhang, X. Y. Su, L. Q. Xiang, and X. Z. Sun, “3-D shape measurement based on complementary Gray-code light,” Opt. Lasers Eng. 50(4), 574–579 (2012).
[Crossref]

Trebeschi, M.

G. Sansoni, M. Trebeschi, and F. Docchio, “State-of-The-Art and Applications of 3D Imaging Sensors in Industry, Cultural Heritage, Medicine, and Criminal Investigation,” Sensors (Basel) 9(1), 568–601 (2009).
[Crossref] [PubMed]

Tsai, R. Y.

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. RA-3(4), 323–344 (1987).
[Crossref]

Valle, V.

I. Léandry, C. Brèque, and V. Valle, “Calibration of a structured-light projection system: Development to large dimension objects,” Opt. Lasers Eng. 50(3), 373–379 (2012).
[Crossref]

Vallea, V.

I. Léandry, C. Brèque, and V. Vallea, “Calibration of a structured-light projection system: Development to large dimension objects,” Opt. Lasers Eng. 50(3), 373–379 (2012).
[Crossref]

Wang, H.

Wang, L.

W. Gao, L. Wang, and Z. Y. Hu, “Flexible Calibration of a Portable Structured Light System through Surface Plane,” Acta Automatica Sinica 34(11), 1358–1362 (2008).
[Crossref]

Wang, M. Y.

J. Chen, X. J. Wu, M. Y. Wang, and X. F. Li, “3D shape modeling using a self-developed hand-held 3D laser scanner and an efficient HT-ICP point cloud registration algorithm,” Opt. Lasers Eng. 45, 414–423 (2013).
[Crossref]

Wang, X. G.

Y. Q. Cheng, X. G. Wang, R. T. Collins, E. M. Riseman, and A. R. Hanson, “Three-dimensional reconstruction of points and lines with unknown correspondence across images,” Int. J. Comput. Vis. 45(2), 129–156 (2001).
[Crossref]

Wu, Q. Y.

J. H. Huang and Q. Y. Wu, “A new reconstruction method based on fringe projection of three-dimensional measuring system,” Opt. Lasers Eng. 52, 115–122 (2014).
[Crossref]

Wu, X. J.

J. Chen, X. J. Wu, M. Y. Wang, and X. F. Li, “3D shape modeling using a self-developed hand-held 3D laser scanner and an efficient HT-ICP point cloud registration algorithm,” Opt. Lasers Eng. 45, 414–423 (2013).
[Crossref]

Xiang, L. Q.

Q. C. Zhang, X. Y. Su, L. Q. Xiang, and X. Z. Sun, “3-D shape measurement based on complementary Gray-code light,” Opt. Lasers Eng. 50(4), 574–579 (2012).
[Crossref]

Zhang, Q. C.

Q. C. Zhang, X. Y. Su, L. Q. Xiang, and X. Z. Sun, “3-D shape measurement based on complementary Gray-code light,” Opt. Lasers Eng. 50(4), 574–579 (2012).
[Crossref]

X. Y. Su and Q. C. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
[Crossref]

Zhang, S.

J. F. Dai, G. Chen, and S. Zhang, “Three-dimensional shape measurement with dual reference phase maps,” Opt. Eng. 53(1), 014102 (2014).
[Crossref]

W. Lohry, V. Chen, and S. Zhang, “Absolute three-dimensional shape measurement using coded fringe patterns without phase unwrapping or projector calibration,” Opt. Express 22(2), 1287–1301 (2014).
[Crossref] [PubMed]

S. Zhang, “Flexible 3D shape measurement using projector defocusing: extended measurement range,” Opt. Lett. 35(7), 934–936 (2010).
[Crossref] [PubMed]

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

Zhang, Z.

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

Z. Zhang, “A flexible new technique for camera calibration,” in Proceedings of IEEE Transactions on Pattern Analysis and Machine Intelligence (IEEE, 2000), pp. 1330–1334.
[Crossref]

Zhao, C.

Acta Automatica Sinica (1)

W. Gao, L. Wang, and Z. Y. Hu, “Flexible Calibration of a Portable Structured Light System through Surface Plane,” Acta Automatica Sinica 34(11), 1358–1362 (2008).
[Crossref]

Acta Opt. Sin. (2)

Y. Li and X. Y. Su, “New method for system calibration in phase measurement profilometry with large view field,” Acta Opt. Sin. 26(8), 1162–1166 (2006).

X. Y. Su, X. X. Chen, and L. R. Guo, “An automated method for 360° surface measurement of 3D objects,” Acta Opt. Sin. 9(7), 670–672 (1989).

Appl. Opt. (1)

Comput. Math. Appl. (1)

Q. Li and J. G. Griffiths, “Iterative closest geometric objects registration,” Comput. Math. Appl. 40(10), 1171–1188 (2000).
[Crossref]

IEEE J. Robot. Autom. (1)

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. RA-3(4), 323–344 (1987).
[Crossref]

Int. J. Comput. Vis. (1)

Y. Q. Cheng, X. G. Wang, R. T. Collins, E. M. Riseman, and A. R. Hanson, “Three-dimensional reconstruction of points and lines with unknown correspondence across images,” Int. J. Comput. Vis. 45(2), 129–156 (2001).
[Crossref]

Opt. Eng. (1)

J. F. Dai, G. Chen, and S. Zhang, “Three-dimensional shape measurement with dual reference phase maps,” Opt. Eng. 53(1), 014102 (2014).
[Crossref]

Opt. Express (3)

Opt. Lasers Eng. (7)

Q. C. Zhang, X. Y. Su, L. Q. Xiang, and X. Z. Sun, “3-D shape measurement based on complementary Gray-code light,” Opt. Lasers Eng. 50(4), 574–579 (2012).
[Crossref]

I. Léandry, C. Brèque, and V. Valle, “Calibration of a structured-light projection system: Development to large dimension objects,” Opt. Lasers Eng. 50(3), 373–379 (2012).
[Crossref]

J. H. Huang and Q. Y. Wu, “A new reconstruction method based on fringe projection of three-dimensional measuring system,” Opt. Lasers Eng. 52, 115–122 (2014).
[Crossref]

I. Léandry, C. Brèque, and V. Vallea, “Calibration of a structured-light projection system: Development to large dimension objects,” Opt. Lasers Eng. 50(3), 373–379 (2012).
[Crossref]

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Figures (24)

Fig. 1
Fig. 1

The schematic measurement setup.

Fig. 2
Fig. 2

Linear interpolation schematic diagram.

Fig. 3
Fig. 3

X p is the point of the first camera to be matched; X p is the point of the second camera, and its unwrapped phase value is equal to point X p .

Fig. 4
Fig. 4

The calibration diagram.

Fig. 5
Fig. 5

The closed-loop calibration diagram.

Fig. 6
Fig. 6

The projection and collection system.

Fig. 7
Fig. 7

The three-dimensional space model of the calibration.

Fig. 8
Fig. 8

The image above is one group of collected images. (a) is the horizontal fringe image, which is collected by the second camera; (b) is the vertical fringe image, which is collected by the second camera; (c) is the horizontal fringe image, which is collected by the first camera; (b) is the vertical fringe image, which is collected by the first camera.

Fig. 9
Fig. 9

In the vertical phase diagram, the Red Cross is the image coordinates. (a) is selected coordinates of (u,v) in the first images. (b) is obtained coordinates of ( u , v ) in the second images by the phase equality.

Fig. 10
Fig. 10

The Red Cross points are the selected points in vertical phase diagram. (a) the image with the first projector. (b) the image with the second projector.

Fig. 11
Fig. 11

(a) is Roman images, which are collected by three measurement systems. (b) is standard block images.

Fig. 12
Fig. 12

The unwrapped phase images of the young Roman, which are obtained by three measurement systems.

Fig. 13
Fig. 13

Three shapes are separately shown in Geomagic software.

Fig. 14
Fig. 14

Three shapes are shown in the global coordinate system.

Fig. 15
Fig. 15

(a) selected the boundary area with the red lines. (b) the horizontal contour curves of the shape, and on the left are the eyes, nose and lips, on the right is vertical facial contour curves.

Fig. 16
Fig. 16

(a) the fusion shape of Roman by ICP algorithm. (b) the selecting the boundary area with the red lines. (c) the horizontal contour curves of point cloud, and on the left are the eyes, nose and lips, on the right is vertical facial contour curves.

Fig. 17
Fig. 17

(a) the standard block on the bracket. (b) the shape of the standard block in Geomagic software.

Fig. 18
Fig. 18

(a), (b) and (c) are one surface of the standard block with different colors. (d) the two colors of point cloud.

Fig. 19
Fig. 19

(a) is the color vector diagram of the surface. (b) is measuring angle of two surfaces.

Fig. 20
Fig. 20

The fusion result of the standard block by the traditional method.

Fig. 21
Fig. 21

(a) is the color vector diagram of the surface. (b) is measuring angle of two surfaces.

Fig. 22
Fig. 22

The schematic diagram of the calibration process, blue arrow represents the forward calibration, and the red arrows represent the reverse calibration.

Fig. 23
Fig. 23

(a) is the color vector diagram of the surface. (b) is measuring angle of two surfaces.

Fig. 24
Fig. 24

(a) is the color vector diagram of the surface. (b) is measuring angle of two surfaces.

Tables (1)

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Table 1 The internal parameters of the first measurement system

Equations (19)

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I m (i,j)=A(i,j)+B(i,j)cos(φ(i,j)m π 2 ),m=0,1,2,3
ϕ(i,j)= tan 1 I 1 (i,j) I 3 (i,j) I 0 (i,j) I 2 (i,j)
X p = X r + ϕ( X p )ϕ( X r ) ϕ( X r+1 )ϕ( X r ) ,ϕ( X r+1 )ϕ( X r ).
X p = X r ,ϕ( X r+1 )=ϕ( X r ).
s×[ u v 1 ]=I×[ R T 0 T 1 ]×[ X w Y w Z w 1 ].
δ ur =x( k 1 r 2 + k 2 r 4 + k 3 6 +...).
δ uv =y( k 1 r 2 + k 2 r 4 + k 3 6 +...).
δ ut =2 p 1 xy+ p 2 ( r 2 +2 x 2 ).
δ vt =2 p 2 xy+ p 1 ( r 2 +2 y 2 ).
X d = X u + δ ur + δ ut .
Y d = Y u + δ vr + δ vt .
( X w , Y w , Z w )=f(u,v, ϕ v ).
{ X w = a 1 u 3 + a 2 v 3 + a 3 ϕ 3 + a 4 u 2 v+ a 5 u 2 ϕ+ a 6 v 2 u+ a 7 v 2 ϕ+ a 8 φ 2 u+ a 9 ϕ 2 v+ a 10 u 2 + a 11 v 2 + a 12 ϕ 2 + a 13 uv+ a 14 uϕ+ a 15 vϕ+ a 16 uvϕ+ a 17 u+ a 18 v+ a 19 ϕ+ a 20 Y w = b 1 u 3 + b 2 v 3 + b 3 ϕ 3 + b 4 u 2 v+ b 5 u 2 ϕ+ b 6 v 2 u+ b 7 v 2 ϕ+ b 8 ϕ 2 u+ b 9 ϕ 2 v+ b 10 u 2 + b 11 v 2 + b 12 ϕ 2 + b 13 uv+ b 14 uϕ+ b 15 vϕ+ b 16 uvϕ+ b 17 u+ b 18 v+ b 19 ϕ+ b 20 Z w = c 1 u 3 + c 2 v 3 + c 3 ϕ 3 + c 4 u 2 v+ c 5 u 2 ϕ+ c 6 v 2 u+ c 7 v 2 ϕ+ c 8 ϕ 2 u+ c 9 ϕ 2 v+ c 10 u 2 + c 11 v 2 + c 12 ϕ 2 + c 13 uv+ c 14 uϕ+ c 15 vϕ+ c 16 uvϕ+ c 17 u+ c 18 v+ c 19 ϕ+ c 20
( X w , Y w , Z w )= ρ 1 f(u,v,ϕ)+ ρ 2 f ( u , v , ϕ ).
ρ 1 = Nn+1 N .
ρ 2 = n1 N ,(n=2,3,....N).
( X w n , Y w n , Z w n )= Nn+1 N f n ( u n , v n , ϕ n )+ n1 N f n ( u n , v n , ϕ n ).
( X w 4 , Y w 4 , Z w 4 )= 4 6 f 2 ( u 2 , v 2 , ϕ 2 )+ 2 6 f 4 ( u 4 , v 4 , ϕ 4 ).
( X w 5 , Y w 5 , Z w 5 )= 5 6 f 1 ( u 1 , v 1 , ϕ 1 )+ 1 6 f 5 ( u 5 , v 5 , ϕ 5 ).

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