Abstract

A miniaturized three-dimensional endoscopic imaging system is presented. The system consists of two imaging channels that can be used to obtain an image from an object of interest and to project a structured light onto the imaged object to measure the surface topology. The structured light was generated with a collimated monochromatic light source and a holographic binary phase grating. The imaging and projection channels were calibrated by use of a modified pinhole camera. The surface profile was extracted by use of triangulation between the projected feature points and the two channels of the endoscope. The imaging system was evaluated in three-dimensional measurements of several objects with known geometries. The results show that surface profiles of the objects with different surfaces and dimensions can be obtained at high accuracy. The in vivo measurements at tissue sites of human skin and an oral cavity demonstrated the potential of the technique for clinical applications.

© 2003 Optical Society of America

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References

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2002 (2)

F. Bernardini, H. Rushmeier, “The 3D model acquisition pipeline,” Comput. Graph. Forum 21, 149–172 (2002).
[CrossRef]

K. Yao, T. Matsui, H. Furukawa, T. Yao, T. Sakurai, T. Mitsuyasu, “A new stereoscopic endoscopy system: accurate 3-dimensional measurement in vitro and in vivo with distortion-correction function,” Gastrointest. Endosc. 55, 412–420 (2002).
[CrossRef] [PubMed]

2000 (1)

J. Heikkila, “Geometric camera calibration using circular control points,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1066–1077 (2000).
[CrossRef]

1999 (1)

J. Batista, H. Araujo, A. T. de Almeida, “Iterative multistep explicit camera calibration,” IEEE Trans. Rob. Autom. 15, 897–917 (1999).
[CrossRef]

1998 (2)

C. J. Calvano, M. E. Moran, L. D. Tackett, P. P. Reddy, M. M. Pankratov, “New visualization techniques for in utero surgery: amnioscopy with a three-dimensional head-mounted display and a computer-controlled endoscope,” J. Endourol. 12, 407–410 (1998).
[CrossRef] [PubMed]

J. Batlle, E. Mouaddib, J. Salvi, “Recent progress in coded structured light as a technique to solve the correspondence problem: a survey,” Pattern Recogn. 31, 963–982 (1998).
[CrossRef]

1997 (1)

C. Chatterjee, V. P. Roychowdhury, E. K. P. Chong, “A nonlinear Gauss–Seidel algorithm for noncoplanar and coplanar camera calibration with convergence analysis,” Comput. Vision Image Understand. 67, 58–80 (1997).
[CrossRef]

1995 (1)

1993 (2)

J. Strutz, “3-D endoscopy,” HNO 41, 128–130 (1993).
[PubMed]

A. Blake, D. McCowen, H. R. Lo, P. J. Lindsey, “Trinocular active range-sensing,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 477–483 (1993).
[CrossRef]

1992 (1)

J. Weng, P. Cohen, M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–980 (1992).
[CrossRef]

1989 (1)

G. Hu, G. Stockman, “3-D surface solution using structured light and constraint propagation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 390–402 (1989).
[CrossRef]

1988 (1)

Y. F. Wang, J. K. Aggarwal, “An overview of geometric modeling using active sensing,” IEEE Control Systems Mag. 8, 5–13 (1988).
[CrossRef]

1987 (1)

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

1982 (2)

S. T. Barnard, M. A. Fischler, “Computational stereo,” ACM (Assoc. Comput. Math) Computing Surveys 14, 553–572 (1982).
[CrossRef]

E. Hall, J. B. K. Tio, C. A. McPherson, F. A. Sadjadi, “Measuring curved surfaces for robot vision,” Computer 15, 42–54 (1982).
[CrossRef]

Aggarwal, J. K.

Y. F. Wang, J. K. Aggarwal, “An overview of geometric modeling using active sensing,” IEEE Control Systems Mag. 8, 5–13 (1988).
[CrossRef]

Araujo, H.

J. Batista, H. Araujo, A. T. de Almeida, “Iterative multistep explicit camera calibration,” IEEE Trans. Rob. Autom. 15, 897–917 (1999).
[CrossRef]

Ayache, N.

N. Ayache, P. T. Sander, Artificial Vision for Mobile Robots: Stereo Vision and Multisensory Perception (MIT, Cambridge, Mass., 1991).

Baker, H. H.

H. H. Baker, “Depth from edge and intensity based stereo,” in Stanford Artificial Intelligence Laboratory Tech. Rep. AIM-347 (Stanford University, Stanford, Calif., 1982).

Barnard, S. T.

S. T. Barnard, M. A. Fischler, “Computational stereo,” ACM (Assoc. Comput. Math) Computing Surveys 14, 553–572 (1982).
[CrossRef]

Batista, J.

J. Batista, H. Araujo, A. T. de Almeida, “Iterative multistep explicit camera calibration,” IEEE Trans. Rob. Autom. 15, 897–917 (1999).
[CrossRef]

Batlle, J.

J. Batlle, E. Mouaddib, J. Salvi, “Recent progress in coded structured light as a technique to solve the correspondence problem: a survey,” Pattern Recogn. 31, 963–982 (1998).
[CrossRef]

Bernardini, F.

F. Bernardini, H. Rushmeier, “The 3D model acquisition pipeline,” Comput. Graph. Forum 21, 149–172 (2002).
[CrossRef]

Blake, A.

A. Blake, D. McCowen, H. R. Lo, P. J. Lindsey, “Trinocular active range-sensing,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 477–483 (1993).
[CrossRef]

Calvano, C. J.

C. J. Calvano, M. E. Moran, L. D. Tackett, P. P. Reddy, M. M. Pankratov, “New visualization techniques for in utero surgery: amnioscopy with a three-dimensional head-mounted display and a computer-controlled endoscope,” J. Endourol. 12, 407–410 (1998).
[CrossRef] [PubMed]

Chatterjee, C.

C. Chatterjee, V. P. Roychowdhury, E. K. P. Chong, “A nonlinear Gauss–Seidel algorithm for noncoplanar and coplanar camera calibration with convergence analysis,” Comput. Vision Image Understand. 67, 58–80 (1997).
[CrossRef]

Chong, E. K. P.

C. Chatterjee, V. P. Roychowdhury, E. K. P. Chong, “A nonlinear Gauss–Seidel algorithm for noncoplanar and coplanar camera calibration with convergence analysis,” Comput. Vision Image Understand. 67, 58–80 (1997).
[CrossRef]

Cohen, P.

J. Weng, P. Cohen, M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–980 (1992).
[CrossRef]

de Almeida, A. T.

J. Batista, H. Araujo, A. T. de Almeida, “Iterative multistep explicit camera calibration,” IEEE Trans. Rob. Autom. 15, 897–917 (1999).
[CrossRef]

Faugeras, O.

O. Faugeras, Three-Dimensional Computer Vision: a Geometric Viewpoint (MIT, Cambridge, Mass., 1993).

Fischler, M. A.

S. T. Barnard, M. A. Fischler, “Computational stereo,” ACM (Assoc. Comput. Math) Computing Surveys 14, 553–572 (1982).
[CrossRef]

Furukawa, H.

K. Yao, T. Matsui, H. Furukawa, T. Yao, T. Sakurai, T. Mitsuyasu, “A new stereoscopic endoscopy system: accurate 3-dimensional measurement in vitro and in vivo with distortion-correction function,” Gastrointest. Endosc. 55, 412–420 (2002).
[CrossRef] [PubMed]

Gonzalez, R. C.

R. C. Gonzalez, R. C. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).

Hall, E.

E. Hall, J. B. K. Tio, C. A. McPherson, F. A. Sadjadi, “Measuring curved surfaces for robot vision,” Computer 15, 42–54 (1982).
[CrossRef]

Hasegawa, K.

K. Hasegawa, K. Noda, Y. Sato, “Electronic endoscope system for shape measurement,” in Proceedings of the 16th International Conference on Pattern Recognition (Publications Office, IEEE Computer Society, P.O. Box 3014, Los Alamitos, Calif. 90720-1264, 2002), Vol. 2, pp. 761–764.

Heikkila, J.

J. Heikkila, “Geometric camera calibration using circular control points,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1066–1077 (2000).
[CrossRef]

Herniou, M.

J. Weng, P. Cohen, M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–980 (1992).
[CrossRef]

Hu, G.

G. Hu, G. Stockman, “3-D surface solution using structured light and constraint propagation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 390–402 (1989).
[CrossRef]

Lindsey, P. J.

A. Blake, D. McCowen, H. R. Lo, P. J. Lindsey, “Trinocular active range-sensing,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 477–483 (1993).
[CrossRef]

Liu, L.

Lo, H. R.

A. Blake, D. McCowen, H. R. Lo, P. J. Lindsey, “Trinocular active range-sensing,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 477–483 (1993).
[CrossRef]

Matsui, T.

K. Yao, T. Matsui, H. Furukawa, T. Yao, T. Sakurai, T. Mitsuyasu, “A new stereoscopic endoscopy system: accurate 3-dimensional measurement in vitro and in vivo with distortion-correction function,” Gastrointest. Endosc. 55, 412–420 (2002).
[CrossRef] [PubMed]

McCowen, D.

A. Blake, D. McCowen, H. R. Lo, P. J. Lindsey, “Trinocular active range-sensing,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 477–483 (1993).
[CrossRef]

McPherson, C. A.

E. Hall, J. B. K. Tio, C. A. McPherson, F. A. Sadjadi, “Measuring curved surfaces for robot vision,” Computer 15, 42–54 (1982).
[CrossRef]

Mitsuyasu, T.

K. Yao, T. Matsui, H. Furukawa, T. Yao, T. Sakurai, T. Mitsuyasu, “A new stereoscopic endoscopy system: accurate 3-dimensional measurement in vitro and in vivo with distortion-correction function,” Gastrointest. Endosc. 55, 412–420 (2002).
[CrossRef] [PubMed]

Moran, M. E.

C. J. Calvano, M. E. Moran, L. D. Tackett, P. P. Reddy, M. M. Pankratov, “New visualization techniques for in utero surgery: amnioscopy with a three-dimensional head-mounted display and a computer-controlled endoscope,” J. Endourol. 12, 407–410 (1998).
[CrossRef] [PubMed]

Mouaddib, E.

J. Batlle, E. Mouaddib, J. Salvi, “Recent progress in coded structured light as a technique to solve the correspondence problem: a survey,” Pattern Recogn. 31, 963–982 (1998).
[CrossRef]

Noda, K.

K. Hasegawa, K. Noda, Y. Sato, “Electronic endoscope system for shape measurement,” in Proceedings of the 16th International Conference on Pattern Recognition (Publications Office, IEEE Computer Society, P.O. Box 3014, Los Alamitos, Calif. 90720-1264, 2002), Vol. 2, pp. 761–764.

Pankratov, M. M.

C. J. Calvano, M. E. Moran, L. D. Tackett, P. P. Reddy, M. M. Pankratov, “New visualization techniques for in utero surgery: amnioscopy with a three-dimensional head-mounted display and a computer-controlled endoscope,” J. Endourol. 12, 407–410 (1998).
[CrossRef] [PubMed]

Reddy, P. P.

C. J. Calvano, M. E. Moran, L. D. Tackett, P. P. Reddy, M. M. Pankratov, “New visualization techniques for in utero surgery: amnioscopy with a three-dimensional head-mounted display and a computer-controlled endoscope,” J. Endourol. 12, 407–410 (1998).
[CrossRef] [PubMed]

Roychowdhury, V. P.

C. Chatterjee, V. P. Roychowdhury, E. K. P. Chong, “A nonlinear Gauss–Seidel algorithm for noncoplanar and coplanar camera calibration with convergence analysis,” Comput. Vision Image Understand. 67, 58–80 (1997).
[CrossRef]

Rushmeier, H.

F. Bernardini, H. Rushmeier, “The 3D model acquisition pipeline,” Comput. Graph. Forum 21, 149–172 (2002).
[CrossRef]

Sadjadi, F. A.

E. Hall, J. B. K. Tio, C. A. McPherson, F. A. Sadjadi, “Measuring curved surfaces for robot vision,” Computer 15, 42–54 (1982).
[CrossRef]

Sakurai, T.

K. Yao, T. Matsui, H. Furukawa, T. Yao, T. Sakurai, T. Mitsuyasu, “A new stereoscopic endoscopy system: accurate 3-dimensional measurement in vitro and in vivo with distortion-correction function,” Gastrointest. Endosc. 55, 412–420 (2002).
[CrossRef] [PubMed]

Salvi, J.

J. Batlle, E. Mouaddib, J. Salvi, “Recent progress in coded structured light as a technique to solve the correspondence problem: a survey,” Pattern Recogn. 31, 963–982 (1998).
[CrossRef]

Sander, P. T.

N. Ayache, P. T. Sander, Artificial Vision for Mobile Robots: Stereo Vision and Multisensory Perception (MIT, Cambridge, Mass., 1991).

Sato, Y.

K. Hasegawa, K. Noda, Y. Sato, “Electronic endoscope system for shape measurement,” in Proceedings of the 16th International Conference on Pattern Recognition (Publications Office, IEEE Computer Society, P.O. Box 3014, Los Alamitos, Calif. 90720-1264, 2002), Vol. 2, pp. 761–764.

Stockman, G.

G. Hu, G. Stockman, “3-D surface solution using structured light and constraint propagation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 390–402 (1989).
[CrossRef]

Strutz, J.

J. Strutz, “3-D endoscopy,” HNO 41, 128–130 (1993).
[PubMed]

Tackett, L. D.

C. J. Calvano, M. E. Moran, L. D. Tackett, P. P. Reddy, M. M. Pankratov, “New visualization techniques for in utero surgery: amnioscopy with a three-dimensional head-mounted display and a computer-controlled endoscope,” J. Endourol. 12, 407–410 (1998).
[CrossRef] [PubMed]

Tio, J. B. K.

E. Hall, J. B. K. Tio, C. A. McPherson, F. A. Sadjadi, “Measuring curved surfaces for robot vision,” Computer 15, 42–54 (1982).
[CrossRef]

Trucco, E.

E. Trucco, A. Verri, Introductory Techniques for 3-D Computer Vision (Prentice-Hall, Upper Saddle River, N.J., 1998).

Tsai, R. Y.

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

Verri, A.

E. Trucco, A. Verri, Introductory Techniques for 3-D Computer Vision (Prentice-Hall, Upper Saddle River, N.J., 1998).

Wang, Y. F.

Y. F. Wang, J. K. Aggarwal, “An overview of geometric modeling using active sensing,” IEEE Control Systems Mag. 8, 5–13 (1988).
[CrossRef]

Weng, J.

J. Weng, P. Cohen, M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–980 (1992).
[CrossRef]

Woods, R. C.

R. C. Gonzalez, R. C. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).

Yao, K.

K. Yao, T. Matsui, H. Furukawa, T. Yao, T. Sakurai, T. Mitsuyasu, “A new stereoscopic endoscopy system: accurate 3-dimensional measurement in vitro and in vivo with distortion-correction function,” Gastrointest. Endosc. 55, 412–420 (2002).
[CrossRef] [PubMed]

Yao, T.

K. Yao, T. Matsui, H. Furukawa, T. Yao, T. Sakurai, T. Mitsuyasu, “A new stereoscopic endoscopy system: accurate 3-dimensional measurement in vitro and in vivo with distortion-correction function,” Gastrointest. Endosc. 55, 412–420 (2002).
[CrossRef] [PubMed]

Zhou, C.

ACM (Assoc. Comput. Math) Computing Surveys (1)

S. T. Barnard, M. A. Fischler, “Computational stereo,” ACM (Assoc. Comput. Math) Computing Surveys 14, 553–572 (1982).
[CrossRef]

Appl. Opt. (1)

Comput. Graph. Forum (1)

F. Bernardini, H. Rushmeier, “The 3D model acquisition pipeline,” Comput. Graph. Forum 21, 149–172 (2002).
[CrossRef]

Comput. Vision Image Understand. (1)

C. Chatterjee, V. P. Roychowdhury, E. K. P. Chong, “A nonlinear Gauss–Seidel algorithm for noncoplanar and coplanar camera calibration with convergence analysis,” Comput. Vision Image Understand. 67, 58–80 (1997).
[CrossRef]

Computer (1)

E. Hall, J. B. K. Tio, C. A. McPherson, F. A. Sadjadi, “Measuring curved surfaces for robot vision,” Computer 15, 42–54 (1982).
[CrossRef]

Gastrointest. Endosc. (1)

K. Yao, T. Matsui, H. Furukawa, T. Yao, T. Sakurai, T. Mitsuyasu, “A new stereoscopic endoscopy system: accurate 3-dimensional measurement in vitro and in vivo with distortion-correction function,” Gastrointest. Endosc. 55, 412–420 (2002).
[CrossRef] [PubMed]

HNO (1)

J. Strutz, “3-D endoscopy,” HNO 41, 128–130 (1993).
[PubMed]

IEEE Control Systems Mag. (1)

Y. F. Wang, J. K. Aggarwal, “An overview of geometric modeling using active sensing,” IEEE Control Systems Mag. 8, 5–13 (1988).
[CrossRef]

IEEE J. Rob. Autom. (1)

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

IEEE Trans. Pattern Anal. Mach. Intell. (4)

J. Heikkila, “Geometric camera calibration using circular control points,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1066–1077 (2000).
[CrossRef]

J. Weng, P. Cohen, M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–980 (1992).
[CrossRef]

G. Hu, G. Stockman, “3-D surface solution using structured light and constraint propagation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 390–402 (1989).
[CrossRef]

A. Blake, D. McCowen, H. R. Lo, P. J. Lindsey, “Trinocular active range-sensing,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 477–483 (1993).
[CrossRef]

IEEE Trans. Rob. Autom. (1)

J. Batista, H. Araujo, A. T. de Almeida, “Iterative multistep explicit camera calibration,” IEEE Trans. Rob. Autom. 15, 897–917 (1999).
[CrossRef]

J. Endourol. (1)

C. J. Calvano, M. E. Moran, L. D. Tackett, P. P. Reddy, M. M. Pankratov, “New visualization techniques for in utero surgery: amnioscopy with a three-dimensional head-mounted display and a computer-controlled endoscope,” J. Endourol. 12, 407–410 (1998).
[CrossRef] [PubMed]

Pattern Recogn. (1)

J. Batlle, E. Mouaddib, J. Salvi, “Recent progress in coded structured light as a technique to solve the correspondence problem: a survey,” Pattern Recogn. 31, 963–982 (1998).
[CrossRef]

Other (6)

N. Ayache, P. T. Sander, Artificial Vision for Mobile Robots: Stereo Vision and Multisensory Perception (MIT, Cambridge, Mass., 1991).

H. H. Baker, “Depth from edge and intensity based stereo,” in Stanford Artificial Intelligence Laboratory Tech. Rep. AIM-347 (Stanford University, Stanford, Calif., 1982).

O. Faugeras, Three-Dimensional Computer Vision: a Geometric Viewpoint (MIT, Cambridge, Mass., 1993).

E. Trucco, A. Verri, Introductory Techniques for 3-D Computer Vision (Prentice-Hall, Upper Saddle River, N.J., 1998).

K. Hasegawa, K. Noda, Y. Sato, “Electronic endoscope system for shape measurement,” in Proceedings of the 16th International Conference on Pattern Recognition (Publications Office, IEEE Computer Society, P.O. Box 3014, Los Alamitos, Calif. 90720-1264, 2002), Vol. 2, pp. 761–764.

R. C. Gonzalez, R. C. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).

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Figures (15)

Fig. 1
Fig. 1

Schematic of the dual-channel endoscopic imaging system.

Fig. 2
Fig. 2

Dot matrices produced by the gratings taken in the focal plane of a lens: (a) pattern generated by a 16 × 8 grating and (b) pattern generated by a 64 × 64 grating.

Fig. 3
Fig. 3

Quarter of a dot matrix generated by a 64 × 64 grating taken through the imaging channel of an endoscope.

Fig. 4
Fig. 4

Checkerboard pattern used for calibration of the imaging channel: (a) image captured at 19.75 mm from the endoscope and (b) image captured at 24.75 mm from the endoscope.

Fig. 5
Fig. 5

Positions of true feature points and reprojected feature points in the calibration planes at calibration distances of (a) 19.75 mm and (b) 24.75 mm.

Fig. 6
Fig. 6

Extraction of feature points by use of the blob detection algorithm. The cross points represent the gravity centers of the dots: left, image captured by the imaging channel; right, enlarged image from the frame shown at the right-hand side.

Fig. 7
Fig. 7

Reconstruction of the surface of a flat target at different distances from the endoscope tip: (a) 3-D displayed results and (b) projection to the XZ plane.

Fig. 8
Fig. 8

(a) Structure of the step target that was used to evaluate the reconstruction of the flat surface at different view angles and (b) image of the step target taken at the initial angle.

Fig. 9
Fig. 9

(a) 3-D measurements taken at the initial angle and (b) projection of the reconstructed 3-D surface to the XZ plane.

Fig. 10
Fig. 10

(a) Image of the step target taken in increments of 20 deg, (b) 3-D measurements taken in increments of 20 deg with respect to the initial angle, (c) projection of the reconstructed 3-D surface to the XZ plane.

Fig. 11
Fig. 11

Reconstruction of the curved surface of a target with two cylinders attached to each other: (a) image of the target taken with the endoscope close to the large cylinder, (b) 3-D measurement, (c) projection of the reconstructed 3-D surface to the XZ plane.

Fig. 12
Fig. 12

3-D measurements of skin tissue with a flat surface: (a) image taken at the initial angle and a distance away from the endoscope, (b) projections of the reconstructed surfaces to the XZ plane at the initial distance and the distances in increments of 5 and 10 mm, (c) projections of the reconstructed surfaces to the XZ plane at the initial angle and at angles in increments of 10 and 20 deg.

Fig. 13
Fig. 13

Reconstructed surfaces of (a) fingers and (b) fingers projected by a dot matrix pattern. (c) Illustration of the reconstructed 3-D surface of fingers.

Fig. 14
Fig. 14

(a) Projection of the 3-D surfaces to the X–Z plane. (b) Image of an oral cavity site for 3-D measurements and (c) image of the tissue site projected with feature points.

Fig. 15
Fig. 15

(a) Illustration of the reconstructed 3-D surface of part of the oral cavity and (b) projection of the 3-D surfaces to the XZ plane.

Tables (1)

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Table 1 Summary of the 3-D Measurements with the Step Targeta

Equations (5)

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xcyczc = RTxwywzw+T,
xu=f xczc=f nˆxTM+TxnˆzTM+Tz, yu=f yczc=f nˆyTM+TynˆzTM+Tz,
xd=1+κ1r2+κ2r4+κ3r6+κ4r8xu, yd=1+κ1r2+κ2r4+κ3r6+κ4r8yu,
xf=sxdx-1xd+Cx, yf=sxdy-1yd+Cy,
%=i=1Nxi-xiri+yi-yiri2N,

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