Abstract

Three-dimensional (3D) shape measurement of an aspheric mirror with fringe reflection photogrammetry involves three steps: correspondence matching, triangulation, and bundle adjustment. Correspondence matching is realized by absolute phase tracking and triangulation is computed by the intersection of reflection and incidence rays. The main contribution in this paper is constraint bundle adjustment for carefully dealing with lens distortion in the process of ray intersection, as compared to the well-known grating reflection photogrammetry. Additionally, a free frame is proposed to alleviate troublesome system geometrical calibration, and constraint bundle adjustment is operated in the free frame to refine the 3D shape. Simulation and experiment demonstrate that constraint bundle adjustment can improve absolute measurement accuracy of aspheric mirrors.

© 2012 Optical Society of America

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References

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  1. T. Luhmann, “Close range photogrammetry for industrial applications,” J. Photogramm. Remote Sens. 65, 558–569 (2010).
    [CrossRef]
  2. R. Tutsch, M. Petz, and M. Fischer, “Optical three-dimensional metrology with structured illumination,” Opt. Eng. 50, 101507 (2011).
    [CrossRef]
  3. X. Su and Q. Zhang, “Dynamic 3-D shape measurement method: a review,” Opt. Lasers Eng. 48, 191–204 (2010).
    [CrossRef]
  4. S. Zhang, “Recent progress on real-time 3D shape measurement using digital fringe projection technique,” Opt. Lasers Eng. 48, 149–158 (2010).
    [CrossRef]
  5. M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring deflectometry: a new approach to measure specular freeform surfaces,” Proc. SPIE 5457, 366–376 (2004).
    [CrossRef]
  6. T. Bothe, W. Li, C. V. Kopylow, and W. Juptner, “High resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
    [CrossRef]
  7. Y. Tang, X. Su, Y. Liu, and H. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express 16, 15090–15096 (2008).
    [CrossRef]
  8. Y. Tang, X. Su, F. Wu, and Y. Liu, “A novel phase measuring deflectometry for aspheric mirror test,” Opt. Express 17, 19778–19784 (2009).
    [CrossRef]
  9. W. Zhao, X. Su, Y. Liu, and Q. Zhang, “Testing an aspheric mirror based on phase measuring deflectometry,” Opt. Eng. 48, 103603 (2009).
    [CrossRef]
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    [CrossRef]
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  12. J. Balzer and S. Werling, “Principles of shape from specular reflection,” Measurement 43, 1305–1317 (2010).
    [CrossRef]
  13. M. Petz and R. Tutsch, “Reflection grating photogrammetry: a technique for absolute shape measurement of specular free-form surfaces,” Proc. SPIE 5869, 58691D (2005).
    [CrossRef]
  14. H. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. 48, 166–171 (2010).
    [CrossRef]
  15. T. Bothe, W. Li, M. Schulte, C. Von Kopylow, R. B. Bergmann, and W. P. P. Juptner, “Vision ray calibration for the quantitative geometric description of general imaging and projection optics in metrology,” Applied Optics 49, 5851–5860 (2010).
    [CrossRef]
  16. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Machine Intell. 22, 1330–1334 (2002).
    [CrossRef]
  17. Y. L. Xiao, X. Su, and W. Chen, “Fringe inverse videogrammetry based on global pose estimation,” Appl. Opt. 50, 5630–5638 (2011).
    [CrossRef]
  18. S. Zhang and S. T. Yau, “High-resolution, real-time 3D absolute coordinate measurement based on a phase- shifting method,” Opt. Express 14, 2644–2649 (2006).
    [CrossRef]
  19. X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
    [CrossRef]
  20. B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle adjustment—a modern synthesis,” in Vision Algorithm: Theory and Practice, Vol. 1883 Lecture Notes in Computer Science (2000), pp. 153–177.
  21. J. Heikkila and O. Silven, “A four-step camera calibration procedure with implicit image correction,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 1997), pp. 1106–1112.
  22. K. Madsen, H. B. Nielsen, and O. Tingleff, “Methods for non-least least squares problems,” in Informatics and Mathematical Modeling (Technical University of Denmark, 2004), Chap. 3, p. 24.
  23. H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36, 2770–2775 (1997).
    [CrossRef]
  24. K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency scheme for high-speed 3-D shape measurement,” Opt. Express 18, 5229–5244 (2010).
    [CrossRef]
  25. Y. Ding, J. Xi, Y. Yu, and J. Chicharo, “Recovering the absolute phase maps of two fringe patterns with selected frequencies,” Opt. Lett. 36, 2518–2520 (2011).
    [CrossRef]

2011

2010

K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency scheme for high-speed 3-D shape measurement,” Opt. Express 18, 5229–5244 (2010).
[CrossRef]

J. Balzer and S. Werling, “Principles of shape from specular reflection,” Measurement 43, 1305–1317 (2010).
[CrossRef]

H. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. 48, 166–171 (2010).
[CrossRef]

T. Bothe, W. Li, M. Schulte, C. Von Kopylow, R. B. Bergmann, and W. P. P. Juptner, “Vision ray calibration for the quantitative geometric description of general imaging and projection optics in metrology,” Applied Optics 49, 5851–5860 (2010).
[CrossRef]

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

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

T. Luhmann, “Close range photogrammetry for industrial applications,” J. Photogramm. Remote Sens. 65, 558–569 (2010).
[CrossRef]

2009

Y. Tang, X. Su, F. Wu, and Y. Liu, “A novel phase measuring deflectometry for aspheric mirror test,” Opt. Express 17, 19778–19784 (2009).
[CrossRef]

W. Zhao, X. Su, Y. Liu, and Q. Zhang, “Testing an aspheric mirror based on phase measuring deflectometry,” Opt. Eng. 48, 103603 (2009).
[CrossRef]

2008

2006

2005

M. Petz and R. Tutsch, “Reflection grating photogrammetry: a technique for absolute shape measurement of specular free-form surfaces,” Proc. SPIE 5869, 58691D (2005).
[CrossRef]

2004

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
[CrossRef]

M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring deflectometry: a new approach to measure specular freeform surfaces,” Proc. SPIE 5457, 366–376 (2004).
[CrossRef]

T. Bothe, W. Li, C. V. Kopylow, and W. Juptner, “High resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

2002

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Machine Intell. 22, 1330–1334 (2002).
[CrossRef]

1997

Agrawal, A.

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?” in European Conference on Computer Vision (2006) Vol. 3951, pp. 578–591.

Asundi, A.

Balzer, J.

J. Balzer and S. Werling, “Principles of shape from specular reflection,” Measurement 43, 1305–1317 (2010).
[CrossRef]

Bergmann, R. B.

T. Bothe, W. Li, M. Schulte, C. Von Kopylow, R. B. Bergmann, and W. P. P. Juptner, “Vision ray calibration for the quantitative geometric description of general imaging and projection optics in metrology,” Applied Optics 49, 5851–5860 (2010).
[CrossRef]

Bothe, T.

T. Bothe, W. Li, M. Schulte, C. Von Kopylow, R. B. Bergmann, and W. P. P. Juptner, “Vision ray calibration for the quantitative geometric description of general imaging and projection optics in metrology,” Applied Optics 49, 5851–5860 (2010).
[CrossRef]

T. Bothe, W. Li, C. V. Kopylow, and W. Juptner, “High resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Chellappa, R.

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?” in European Conference on Computer Vision (2006) Vol. 3951, pp. 578–591.

Chen, W.

Y. L. Xiao, X. Su, and W. Chen, “Fringe inverse videogrammetry based on global pose estimation,” Appl. Opt. 50, 5630–5638 (2011).
[CrossRef]

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
[CrossRef]

Chicharo, J.

Ding, Y.

Feng, P.

H. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. 48, 166–171 (2010).
[CrossRef]

Fischer, M.

R. Tutsch, M. Petz, and M. Fischer, “Optical three-dimensional metrology with structured illumination,” Opt. Eng. 50, 101507 (2011).
[CrossRef]

Fitzgibbon, A. W.

B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle adjustment—a modern synthesis,” in Vision Algorithm: Theory and Practice, Vol. 1883 Lecture Notes in Computer Science (2000), pp. 153–177.

Guo, H.

H. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. 48, 166–171 (2010).
[CrossRef]

Hao, Q.

Hartley, R. I.

B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle adjustment—a modern synthesis,” in Vision Algorithm: Theory and Practice, Vol. 1883 Lecture Notes in Computer Science (2000), pp. 153–177.

Hassebrook, L. G.

Hausler, G.

M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring deflectometry: a new approach to measure specular freeform surfaces,” Proc. SPIE 5457, 366–376 (2004).
[CrossRef]

Heikkila, J.

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

Huang, L.

Huntley, J. M.

Jing, H.

Juptner, W.

T. Bothe, W. Li, C. V. Kopylow, and W. Juptner, “High resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Juptner, W. P. P.

T. Bothe, W. Li, M. Schulte, C. Von Kopylow, R. B. Bergmann, and W. P. P. Juptner, “Vision ray calibration for the quantitative geometric description of general imaging and projection optics in metrology,” Applied Optics 49, 5851–5860 (2010).
[CrossRef]

Kaminski, J.

M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring deflectometry: a new approach to measure specular freeform surfaces,” Proc. SPIE 5457, 366–376 (2004).
[CrossRef]

Knauer, M. C.

M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring deflectometry: a new approach to measure specular freeform surfaces,” Proc. SPIE 5457, 366–376 (2004).
[CrossRef]

Kopylow, C. V.

T. Bothe, W. Li, C. V. Kopylow, and W. Juptner, “High resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Lau, D. L.

Li, W.

T. Bothe, W. Li, M. Schulte, C. Von Kopylow, R. B. Bergmann, and W. P. P. Juptner, “Vision ray calibration for the quantitative geometric description of general imaging and projection optics in metrology,” Applied Optics 49, 5851–5860 (2010).
[CrossRef]

T. Bothe, W. Li, C. V. Kopylow, and W. Juptner, “High resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Liu, K.

Liu, Y.

Luhmann, T.

T. Luhmann, “Close range photogrammetry for industrial applications,” J. Photogramm. Remote Sens. 65, 558–569 (2010).
[CrossRef]

Madsen, K.

K. Madsen, H. B. Nielsen, and O. Tingleff, “Methods for non-least least squares problems,” in Informatics and Mathematical Modeling (Technical University of Denmark, 2004), Chap. 3, p. 24.

McLauchlan, P. F.

B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle adjustment—a modern synthesis,” in Vision Algorithm: Theory and Practice, Vol. 1883 Lecture Notes in Computer Science (2000), pp. 153–177.

Ng, C.

Nielsen, H. B.

K. Madsen, H. B. Nielsen, and O. Tingleff, “Methods for non-least least squares problems,” in Informatics and Mathematical Modeling (Technical University of Denmark, 2004), Chap. 3, p. 24.

Petz, M.

R. Tutsch, M. Petz, and M. Fischer, “Optical three-dimensional metrology with structured illumination,” Opt. Eng. 50, 101507 (2011).
[CrossRef]

M. Petz and R. Tutsch, “Reflection grating photogrammetry: a technique for absolute shape measurement of specular free-form surfaces,” Proc. SPIE 5869, 58691D (2005).
[CrossRef]

Raskar, R.

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?” in European Conference on Computer Vision (2006) Vol. 3951, pp. 578–591.

Saldner, H. O.

Schulte, M.

T. Bothe, W. Li, M. Schulte, C. Von Kopylow, R. B. Bergmann, and W. P. P. Juptner, “Vision ray calibration for the quantitative geometric description of general imaging and projection optics in metrology,” Applied Optics 49, 5851–5860 (2010).
[CrossRef]

Silven, O.

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

Su, X.

Y. L. Xiao, X. Su, and W. Chen, “Fringe inverse videogrammetry based on global pose estimation,” Appl. Opt. 50, 5630–5638 (2011).
[CrossRef]

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

W. Zhao, X. Su, Y. Liu, and Q. Zhang, “Testing an aspheric mirror based on phase measuring deflectometry,” Opt. Eng. 48, 103603 (2009).
[CrossRef]

Y. Tang, X. Su, F. Wu, and Y. Liu, “A novel phase measuring deflectometry for aspheric mirror test,” Opt. Express 17, 19778–19784 (2009).
[CrossRef]

Y. Tang, X. Su, Y. Liu, and H. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express 16, 15090–15096 (2008).
[CrossRef]

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
[CrossRef]

Tang, Y.

Tao, T.

H. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. 48, 166–171 (2010).
[CrossRef]

Tingleff, O.

K. Madsen, H. B. Nielsen, and O. Tingleff, “Methods for non-least least squares problems,” in Informatics and Mathematical Modeling (Technical University of Denmark, 2004), Chap. 3, p. 24.

Triggs, B.

B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle adjustment—a modern synthesis,” in Vision Algorithm: Theory and Practice, Vol. 1883 Lecture Notes in Computer Science (2000), pp. 153–177.

Tutsch, R.

R. Tutsch, M. Petz, and M. Fischer, “Optical three-dimensional metrology with structured illumination,” Opt. Eng. 50, 101507 (2011).
[CrossRef]

M. Petz and R. Tutsch, “Reflection grating photogrammetry: a technique for absolute shape measurement of specular free-form surfaces,” Proc. SPIE 5869, 58691D (2005).
[CrossRef]

Von Kopylow, C.

T. Bothe, W. Li, M. Schulte, C. Von Kopylow, R. B. Bergmann, and W. P. P. Juptner, “Vision ray calibration for the quantitative geometric description of general imaging and projection optics in metrology,” Applied Optics 49, 5851–5860 (2010).
[CrossRef]

Wang, Y.

Werling, S.

J. Balzer and S. Werling, “Principles of shape from specular reflection,” Measurement 43, 1305–1317 (2010).
[CrossRef]

Wu, F.

Xi, J.

Xiao, Y. L.

Yau, S. T.

Yu, Y.

Zhang, Q.

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

W. Zhao, X. Su, Y. Liu, and Q. Zhang, “Testing an aspheric mirror based on phase measuring deflectometry,” Opt. Eng. 48, 103603 (2009).
[CrossRef]

Zhang, S.

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

S. Zhang and S. T. Yau, “High-resolution, real-time 3D absolute coordinate measurement based on a phase- shifting method,” Opt. Express 14, 2644–2649 (2006).
[CrossRef]

Zhang, Z.

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Machine Intell. 22, 1330–1334 (2002).
[CrossRef]

Zhao, W.

W. Zhao, X. Su, Y. Liu, and Q. Zhang, “Testing an aspheric mirror based on phase measuring deflectometry,” Opt. Eng. 48, 103603 (2009).
[CrossRef]

Appl. Opt.

Applied Optics

T. Bothe, W. Li, M. Schulte, C. Von Kopylow, R. B. Bergmann, and W. P. P. Juptner, “Vision ray calibration for the quantitative geometric description of general imaging and projection optics in metrology,” Applied Optics 49, 5851–5860 (2010).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell.

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Machine Intell. 22, 1330–1334 (2002).
[CrossRef]

J. Photogramm. Remote Sens.

T. Luhmann, “Close range photogrammetry for industrial applications,” J. Photogramm. Remote Sens. 65, 558–569 (2010).
[CrossRef]

Measurement

J. Balzer and S. Werling, “Principles of shape from specular reflection,” Measurement 43, 1305–1317 (2010).
[CrossRef]

Opt. Eng.

R. Tutsch, M. Petz, and M. Fischer, “Optical three-dimensional metrology with structured illumination,” Opt. Eng. 50, 101507 (2011).
[CrossRef]

W. Zhao, X. Su, Y. Liu, and Q. Zhang, “Testing an aspheric mirror based on phase measuring deflectometry,” Opt. Eng. 48, 103603 (2009).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

H. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. 48, 166–171 (2010).
[CrossRef]

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261 (2004).
[CrossRef]

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

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

Opt. Lett.

Proc. SPIE

M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring deflectometry: a new approach to measure specular freeform surfaces,” Proc. SPIE 5457, 366–376 (2004).
[CrossRef]

T. Bothe, W. Li, C. V. Kopylow, and W. Juptner, “High resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

M. Petz and R. Tutsch, “Reflection grating photogrammetry: a technique for absolute shape measurement of specular free-form surfaces,” Proc. SPIE 5869, 58691D (2005).
[CrossRef]

Other

B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle adjustment—a modern synthesis,” in Vision Algorithm: Theory and Practice, Vol. 1883 Lecture Notes in Computer Science (2000), pp. 153–177.

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

K. Madsen, H. B. Nielsen, and O. Tingleff, “Methods for non-least least squares problems,” in Informatics and Mathematical Modeling (Technical University of Denmark, 2004), Chap. 3, p. 24.

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?” in European Conference on Computer Vision (2006) Vol. 3951, pp. 578–591.

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Figures (11)

Fig. 1.
Fig. 1.

Schematic diagram of fringe reflection photogrammetry.

Fig. 2.
Fig. 2.

Schematic diagram of flipping deviation in one direction.

Fig. 3.
Fig. 3.

Imaging model of central perspective projection.

Fig. 4.
Fig. 4.

Ideal paraboloid shape.

Fig. 5.
Fig. 5.

Comparison of bundle adjustment.

Fig. 6.
Fig. 6.

Coefficient of normal equation. (a) Without constraint bundle adjustment. (b) With constraint bundle adjustment.

Fig. 7.
Fig. 7.

Preliminary experiment device.

Fig. 8.
Fig. 8.

References used for camera pose estimation.

Fig. 9.
Fig. 9.

Absolute phase distribution of (a) horizontal and (b) vertical deform fringes.

Fig. 10.
Fig. 10.

(a) Shape of aspheric mirror by triangulation. (b) Error distribution.

Fig. 11.
Fig. 11.

(a) Shape of aspheric mirror after constraint bundle adjustment. (b) Error distribution.

Tables (1)

Tables Icon

Table 1. Camera Calibration Results (Pixel)

Equations (13)

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

I(x,y)=a+bcos[φ(x,y)],
ϕwrap(u,v)=tan1[3(I1I3)2I2I1I3].
[0,0,0]=Rro+T.
[uiCxFxf,uiCyFyf,f]T=Rri+T.
λ[xy1]=K[RT][XY1],
K=[Fx0Cx0FyCy001].
δx=k1rd2+k2rd4+k5rd6+2k3xdyd+k4(rd2+2xd2),δy=k1rd2+k2rd4+k5rd6+k3(rd2+2yd2)+2k4xdyd
Δix=R(Fx,Fy,Cx,Cy,k1,k2,k3,k4,k5,Xiw)xreal,Δiy=R(Fx,Fy,Cx,Cy,k1,k2,k3,k4,k5,Xiw)yreal.
ρi(Xiw)=Xiw[γri1+(ri2ri1)],
{min[i(Δix2+Δiy2)]subject toρi(Xiw)=Xiw[γri1+(ri2ri1)].
αi=[giFx,giFy,giCx,giCy,gik1,gik2,gik3,gik4,gik5,giX1w,giX2wgiX3w],
[αTαβTβ0][ΔϖΔκ]=[gi(ϖn)ρi(κn)].
R=[0.00340.99530.09680.99940.000310.03220.03210.096820.9948],T=[53.569,46.431,345.274]mm.

Metrics