Abstract

This paper discusses the possibility of evaluating the shape of a free-form object in comparison with its shape prescribed by a CAD model. Measurements are made based on a single-shot recording using dual-wavelength holography with a synthetic wavelength of 1.4 mm. Each hologram is numerically propagated to different focus planes and correlated. The result is a vector field of speckle displacements that is linearly dependent on the local distance between the measured surface and the focus plane. From these speckle displacements, a gradient field of the measured surface is extracted through a proportional relationship. The gradient field obtained from the measurement is then aligned to the shape of the CAD model using the iterative closest point (ICP) algorithm and regularization. Deviations between the measured shape and the CAD model are found from the phase difference field, giving a high precision shape evaluation. The phase differences and the CAD model are also used to find a representation of the measured shape. The standard deviation of the measured shape relative the CAD model varies between 7 and 19 μm, depending on the slope.

© 2013 Optical Society of America

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References

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  1. Y. Li and P. Gu, “Free-form surface inspection techniques state of the art review,” Comput. Aided Des. 36, 1395–1417 (2004).
    [CrossRef]
  2. M.-W. Cho, H. Lee, G.-S. Yoon, and J. Choi, “A feature-based inspection planning system for coordinate measuring machines,” Int. J. Adv. Manuf. Tech. 26, 1078–1087 (2005).
    [CrossRef]
  3. F. Zhao, X. Xu, and S. Q. Xie, “Survey paper: computer-aided inspection planning-the state of the art,” Comput. Ind. 60, 453–466 (2009).
  4. F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
    [CrossRef]
  5. K. J. Gåsvik, Optical Metrology (Wiley, 1995).
  6. T. Kreis, Holographic Interferometry Principles and Methods (Akademie Verlag, 1996).
  7. C. J. Mann, P. R. Bingham, V. C. Paquit, and K. W. Tobin, “Quantitative phase imaging by three-wavelength digital holography,” Opt. Express 16, 9753–9764 (2008).
    [CrossRef]
  8. Y. Fu, G. Pedrini, B. M. Hennelly, R. M. Groves, and W. Osten, “Dual-wavelength image-plane digital holography for dynamic measurement,” Opt. Laser. Eng. 47, 552–557 (2009).
    [CrossRef]
  9. A. Wada, M. Kato, and Y. Ishii, “Large step-height measurements using multiple-wavelength holographic interferometry with tunable laser diodes,” J. Opt. Soc. Am. A 25, 3013–3020 (2008).
    [CrossRef]
  10. P. Bergström, S. Rosendahl, P. Gren, and M. Sjödahl, “Shape verification using dual-wavelength holographic interferometry,” Opt. Eng. 50, 101503 (2011).
    [CrossRef]
  11. M. Sjödahl, E. Hällstig, and D. Khodadad, “Multi-spectral speckles: theory and applications,” Proc. SPIE 8413, 841306 (2012).
  12. M. Sjödahl, “Dynamic properties of multispectral speckles in digital holography and image correlation,” Opt. Eng. 52, 101908 (2013).
    [CrossRef]
  13. D. Khodadad, E. Hällstig, and M. Sjödahl, “Shape reconstruction using dual wavelength digital holography and speckle movements,” Proc. SPIE 8788, 878801 (2013).
  14. D. Khodadad, E. Hällstig, and M. Sjödahl, “Dual-wavelength digital holographic shape measurement using speckle movements and phase gradients,” Opt. Eng. 52, 101912 (2013).
    [CrossRef]
  15. M. Sjödahl, “Electronic speckle photography: increased accuracy by nonintegral pixel shifting,” Appl. Opt. 33, 6667–6673 (1994).
    [CrossRef]
  16. E.-L. Johansson, L. Benckert, and M. Sjödahl, “Phase object data obtained from defocused laser speckle displacement,” Appl. Opt. 43, 3229–3234 (2004).
    [CrossRef]
  17. M. Sjödahl, E. Olsson, E. Amer, and P. Gren, “Depth-resolved measurement of phase gradients in a transient phase object field using pulsed digital holography,” Appl. Opt. 48, H31–H39 (2009).
    [CrossRef]
  18. J.-D. Durou, J.-F. Aujol, and F. Courteille, “Integrating the normal field of a surface in the presence of discontinuities,” in Energy Minimization Methods in Computer Vision and Pattern Recognition, Lecture Notes in Computer Science (Springer-Verlag, 2009), pp. 261–273.
  19. A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?” in Computer Vision ECCV 2006, Lecture Notes in Computer Science (Springer-Verlag, 2006), pp. 578–591.
  20. R. Klette and K. Schlüns, “Height data from gradient fields,” Proc. SPIE 2908, 204–215 (1996).
  21. T. Wei and R. Klette, “Depth recovery from noisy gradient vector fields using regularization,” in Computer Analysis of Images and Patterns, Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 116–123.
  22. T. Wei and R. Klette, “Height from gradient using surface curvature and area constraints,” in 3rd Indian Conference on Computer Vision, Graphics and Image Processing (Allied Publishers, 2002), pp. 1–6.
  23. P. J. Besl and N. D. McKay, “A method for registration of 3D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992).
    [CrossRef]
  24. M. Sjödahl, “Accuracy in electronic speckle photography,” Appl. Opt. 36, 2875–2885 (1997).
    [CrossRef]

2013 (3)

M. Sjödahl, “Dynamic properties of multispectral speckles in digital holography and image correlation,” Opt. Eng. 52, 101908 (2013).
[CrossRef]

D. Khodadad, E. Hällstig, and M. Sjödahl, “Shape reconstruction using dual wavelength digital holography and speckle movements,” Proc. SPIE 8788, 878801 (2013).

D. Khodadad, E. Hällstig, and M. Sjödahl, “Dual-wavelength digital holographic shape measurement using speckle movements and phase gradients,” Opt. Eng. 52, 101912 (2013).
[CrossRef]

2012 (1)

M. Sjödahl, E. Hällstig, and D. Khodadad, “Multi-spectral speckles: theory and applications,” Proc. SPIE 8413, 841306 (2012).

2011 (1)

P. Bergström, S. Rosendahl, P. Gren, and M. Sjödahl, “Shape verification using dual-wavelength holographic interferometry,” Opt. Eng. 50, 101503 (2011).
[CrossRef]

2009 (3)

F. Zhao, X. Xu, and S. Q. Xie, “Survey paper: computer-aided inspection planning-the state of the art,” Comput. Ind. 60, 453–466 (2009).

Y. Fu, G. Pedrini, B. M. Hennelly, R. M. Groves, and W. Osten, “Dual-wavelength image-plane digital holography for dynamic measurement,” Opt. Laser. Eng. 47, 552–557 (2009).
[CrossRef]

M. Sjödahl, E. Olsson, E. Amer, and P. Gren, “Depth-resolved measurement of phase gradients in a transient phase object field using pulsed digital holography,” Appl. Opt. 48, H31–H39 (2009).
[CrossRef]

2008 (2)

2005 (1)

M.-W. Cho, H. Lee, G.-S. Yoon, and J. Choi, “A feature-based inspection planning system for coordinate measuring machines,” Int. J. Adv. Manuf. Tech. 26, 1078–1087 (2005).
[CrossRef]

2004 (2)

E.-L. Johansson, L. Benckert, and M. Sjödahl, “Phase object data obtained from defocused laser speckle displacement,” Appl. Opt. 43, 3229–3234 (2004).
[CrossRef]

Y. Li and P. Gu, “Free-form surface inspection techniques state of the art review,” Comput. Aided Des. 36, 1395–1417 (2004).
[CrossRef]

2000 (1)

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

1997 (1)

1996 (1)

R. Klette and K. Schlüns, “Height data from gradient fields,” Proc. SPIE 2908, 204–215 (1996).

1994 (1)

1992 (1)

P. J. Besl and N. D. McKay, “A method for registration of 3D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992).
[CrossRef]

Agrawal, A.

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?” in Computer Vision ECCV 2006, Lecture Notes in Computer Science (Springer-Verlag, 2006), pp. 578–591.

Amer, E.

Aujol, J.-F.

J.-D. Durou, J.-F. Aujol, and F. Courteille, “Integrating the normal field of a surface in the presence of discontinuities,” in Energy Minimization Methods in Computer Vision and Pattern Recognition, Lecture Notes in Computer Science (Springer-Verlag, 2009), pp. 261–273.

Benckert, L.

Bergström, P.

P. Bergström, S. Rosendahl, P. Gren, and M. Sjödahl, “Shape verification using dual-wavelength holographic interferometry,” Opt. Eng. 50, 101503 (2011).
[CrossRef]

Besl, P. J.

P. J. Besl and N. D. McKay, “A method for registration of 3D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992).
[CrossRef]

Bingham, P. R.

Brown, G. M.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

Chellappa, R.

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?” in Computer Vision ECCV 2006, Lecture Notes in Computer Science (Springer-Verlag, 2006), pp. 578–591.

Chen, F.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

Cho, M.-W.

M.-W. Cho, H. Lee, G.-S. Yoon, and J. Choi, “A feature-based inspection planning system for coordinate measuring machines,” Int. J. Adv. Manuf. Tech. 26, 1078–1087 (2005).
[CrossRef]

Choi, J.

M.-W. Cho, H. Lee, G.-S. Yoon, and J. Choi, “A feature-based inspection planning system for coordinate measuring machines,” Int. J. Adv. Manuf. Tech. 26, 1078–1087 (2005).
[CrossRef]

Courteille, F.

J.-D. Durou, J.-F. Aujol, and F. Courteille, “Integrating the normal field of a surface in the presence of discontinuities,” in Energy Minimization Methods in Computer Vision and Pattern Recognition, Lecture Notes in Computer Science (Springer-Verlag, 2009), pp. 261–273.

Durou, J.-D.

J.-D. Durou, J.-F. Aujol, and F. Courteille, “Integrating the normal field of a surface in the presence of discontinuities,” in Energy Minimization Methods in Computer Vision and Pattern Recognition, Lecture Notes in Computer Science (Springer-Verlag, 2009), pp. 261–273.

Fu, Y.

Y. Fu, G. Pedrini, B. M. Hennelly, R. M. Groves, and W. Osten, “Dual-wavelength image-plane digital holography for dynamic measurement,” Opt. Laser. Eng. 47, 552–557 (2009).
[CrossRef]

Gåsvik, K. J.

K. J. Gåsvik, Optical Metrology (Wiley, 1995).

Gren, P.

P. Bergström, S. Rosendahl, P. Gren, and M. Sjödahl, “Shape verification using dual-wavelength holographic interferometry,” Opt. Eng. 50, 101503 (2011).
[CrossRef]

M. Sjödahl, E. Olsson, E. Amer, and P. Gren, “Depth-resolved measurement of phase gradients in a transient phase object field using pulsed digital holography,” Appl. Opt. 48, H31–H39 (2009).
[CrossRef]

Groves, R. M.

Y. Fu, G. Pedrini, B. M. Hennelly, R. M. Groves, and W. Osten, “Dual-wavelength image-plane digital holography for dynamic measurement,” Opt. Laser. Eng. 47, 552–557 (2009).
[CrossRef]

Gu, P.

Y. Li and P. Gu, “Free-form surface inspection techniques state of the art review,” Comput. Aided Des. 36, 1395–1417 (2004).
[CrossRef]

Hällstig, E.

D. Khodadad, E. Hällstig, and M. Sjödahl, “Shape reconstruction using dual wavelength digital holography and speckle movements,” Proc. SPIE 8788, 878801 (2013).

D. Khodadad, E. Hällstig, and M. Sjödahl, “Dual-wavelength digital holographic shape measurement using speckle movements and phase gradients,” Opt. Eng. 52, 101912 (2013).
[CrossRef]

M. Sjödahl, E. Hällstig, and D. Khodadad, “Multi-spectral speckles: theory and applications,” Proc. SPIE 8413, 841306 (2012).

Hennelly, B. M.

Y. Fu, G. Pedrini, B. M. Hennelly, R. M. Groves, and W. Osten, “Dual-wavelength image-plane digital holography for dynamic measurement,” Opt. Laser. Eng. 47, 552–557 (2009).
[CrossRef]

Ishii, Y.

Johansson, E.-L.

Kato, M.

Khodadad, D.

D. Khodadad, E. Hällstig, and M. Sjödahl, “Shape reconstruction using dual wavelength digital holography and speckle movements,” Proc. SPIE 8788, 878801 (2013).

D. Khodadad, E. Hällstig, and M. Sjödahl, “Dual-wavelength digital holographic shape measurement using speckle movements and phase gradients,” Opt. Eng. 52, 101912 (2013).
[CrossRef]

M. Sjödahl, E. Hällstig, and D. Khodadad, “Multi-spectral speckles: theory and applications,” Proc. SPIE 8413, 841306 (2012).

Klette, R.

R. Klette and K. Schlüns, “Height data from gradient fields,” Proc. SPIE 2908, 204–215 (1996).

T. Wei and R. Klette, “Height from gradient using surface curvature and area constraints,” in 3rd Indian Conference on Computer Vision, Graphics and Image Processing (Allied Publishers, 2002), pp. 1–6.

T. Wei and R. Klette, “Depth recovery from noisy gradient vector fields using regularization,” in Computer Analysis of Images and Patterns, Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 116–123.

Kreis, T.

T. Kreis, Holographic Interferometry Principles and Methods (Akademie Verlag, 1996).

Lee, H.

M.-W. Cho, H. Lee, G.-S. Yoon, and J. Choi, “A feature-based inspection planning system for coordinate measuring machines,” Int. J. Adv. Manuf. Tech. 26, 1078–1087 (2005).
[CrossRef]

Li, Y.

Y. Li and P. Gu, “Free-form surface inspection techniques state of the art review,” Comput. Aided Des. 36, 1395–1417 (2004).
[CrossRef]

Mann, C. J.

McKay, N. D.

P. J. Besl and N. D. McKay, “A method for registration of 3D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992).
[CrossRef]

Olsson, E.

Osten, W.

Y. Fu, G. Pedrini, B. M. Hennelly, R. M. Groves, and W. Osten, “Dual-wavelength image-plane digital holography for dynamic measurement,” Opt. Laser. Eng. 47, 552–557 (2009).
[CrossRef]

Paquit, V. C.

Pedrini, G.

Y. Fu, G. Pedrini, B. M. Hennelly, R. M. Groves, and W. Osten, “Dual-wavelength image-plane digital holography for dynamic measurement,” Opt. Laser. Eng. 47, 552–557 (2009).
[CrossRef]

Raskar, R.

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?” in Computer Vision ECCV 2006, Lecture Notes in Computer Science (Springer-Verlag, 2006), pp. 578–591.

Rosendahl, S.

P. Bergström, S. Rosendahl, P. Gren, and M. Sjödahl, “Shape verification using dual-wavelength holographic interferometry,” Opt. Eng. 50, 101503 (2011).
[CrossRef]

Schlüns, K.

R. Klette and K. Schlüns, “Height data from gradient fields,” Proc. SPIE 2908, 204–215 (1996).

Sjödahl, M.

M. Sjödahl, “Dynamic properties of multispectral speckles in digital holography and image correlation,” Opt. Eng. 52, 101908 (2013).
[CrossRef]

D. Khodadad, E. Hällstig, and M. Sjödahl, “Shape reconstruction using dual wavelength digital holography and speckle movements,” Proc. SPIE 8788, 878801 (2013).

D. Khodadad, E. Hällstig, and M. Sjödahl, “Dual-wavelength digital holographic shape measurement using speckle movements and phase gradients,” Opt. Eng. 52, 101912 (2013).
[CrossRef]

M. Sjödahl, E. Hällstig, and D. Khodadad, “Multi-spectral speckles: theory and applications,” Proc. SPIE 8413, 841306 (2012).

P. Bergström, S. Rosendahl, P. Gren, and M. Sjödahl, “Shape verification using dual-wavelength holographic interferometry,” Opt. Eng. 50, 101503 (2011).
[CrossRef]

M. Sjödahl, E. Olsson, E. Amer, and P. Gren, “Depth-resolved measurement of phase gradients in a transient phase object field using pulsed digital holography,” Appl. Opt. 48, H31–H39 (2009).
[CrossRef]

E.-L. Johansson, L. Benckert, and M. Sjödahl, “Phase object data obtained from defocused laser speckle displacement,” Appl. Opt. 43, 3229–3234 (2004).
[CrossRef]

M. Sjödahl, “Accuracy in electronic speckle photography,” Appl. Opt. 36, 2875–2885 (1997).
[CrossRef]

M. Sjödahl, “Electronic speckle photography: increased accuracy by nonintegral pixel shifting,” Appl. Opt. 33, 6667–6673 (1994).
[CrossRef]

Song, M.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

Tobin, K. W.

Wada, A.

Wei, T.

T. Wei and R. Klette, “Height from gradient using surface curvature and area constraints,” in 3rd Indian Conference on Computer Vision, Graphics and Image Processing (Allied Publishers, 2002), pp. 1–6.

T. Wei and R. Klette, “Depth recovery from noisy gradient vector fields using regularization,” in Computer Analysis of Images and Patterns, Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 116–123.

Xie, S. Q.

F. Zhao, X. Xu, and S. Q. Xie, “Survey paper: computer-aided inspection planning-the state of the art,” Comput. Ind. 60, 453–466 (2009).

Xu, X.

F. Zhao, X. Xu, and S. Q. Xie, “Survey paper: computer-aided inspection planning-the state of the art,” Comput. Ind. 60, 453–466 (2009).

Yoon, G.-S.

M.-W. Cho, H. Lee, G.-S. Yoon, and J. Choi, “A feature-based inspection planning system for coordinate measuring machines,” Int. J. Adv. Manuf. Tech. 26, 1078–1087 (2005).
[CrossRef]

Zhao, F.

F. Zhao, X. Xu, and S. Q. Xie, “Survey paper: computer-aided inspection planning-the state of the art,” Comput. Ind. 60, 453–466 (2009).

Appl. Opt. (4)

Comput. Aided Des. (1)

Y. Li and P. Gu, “Free-form surface inspection techniques state of the art review,” Comput. Aided Des. 36, 1395–1417 (2004).
[CrossRef]

Comput. Ind. (1)

F. Zhao, X. Xu, and S. Q. Xie, “Survey paper: computer-aided inspection planning-the state of the art,” Comput. Ind. 60, 453–466 (2009).

IEEE Trans. Pattern Anal. Mach. Intell. (1)

P. J. Besl and N. D. McKay, “A method for registration of 3D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992).
[CrossRef]

Int. J. Adv. Manuf. Tech. (1)

M.-W. Cho, H. Lee, G.-S. Yoon, and J. Choi, “A feature-based inspection planning system for coordinate measuring machines,” Int. J. Adv. Manuf. Tech. 26, 1078–1087 (2005).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Eng. (4)

D. Khodadad, E. Hällstig, and M. Sjödahl, “Dual-wavelength digital holographic shape measurement using speckle movements and phase gradients,” Opt. Eng. 52, 101912 (2013).
[CrossRef]

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

P. Bergström, S. Rosendahl, P. Gren, and M. Sjödahl, “Shape verification using dual-wavelength holographic interferometry,” Opt. Eng. 50, 101503 (2011).
[CrossRef]

M. Sjödahl, “Dynamic properties of multispectral speckles in digital holography and image correlation,” Opt. Eng. 52, 101908 (2013).
[CrossRef]

Opt. Express (1)

Opt. Laser. Eng. (1)

Y. Fu, G. Pedrini, B. M. Hennelly, R. M. Groves, and W. Osten, “Dual-wavelength image-plane digital holography for dynamic measurement,” Opt. Laser. Eng. 47, 552–557 (2009).
[CrossRef]

Proc. SPIE (3)

R. Klette and K. Schlüns, “Height data from gradient fields,” Proc. SPIE 2908, 204–215 (1996).

D. Khodadad, E. Hällstig, and M. Sjödahl, “Shape reconstruction using dual wavelength digital holography and speckle movements,” Proc. SPIE 8788, 878801 (2013).

M. Sjödahl, E. Hällstig, and D. Khodadad, “Multi-spectral speckles: theory and applications,” Proc. SPIE 8413, 841306 (2012).

Other (6)

K. J. Gåsvik, Optical Metrology (Wiley, 1995).

T. Kreis, Holographic Interferometry Principles and Methods (Akademie Verlag, 1996).

J.-D. Durou, J.-F. Aujol, and F. Courteille, “Integrating the normal field of a surface in the presence of discontinuities,” in Energy Minimization Methods in Computer Vision and Pattern Recognition, Lecture Notes in Computer Science (Springer-Verlag, 2009), pp. 261–273.

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?” in Computer Vision ECCV 2006, Lecture Notes in Computer Science (Springer-Verlag, 2006), pp. 578–591.

T. Wei and R. Klette, “Depth recovery from noisy gradient vector fields using regularization,” in Computer Analysis of Images and Patterns, Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 116–123.

T. Wei and R. Klette, “Height from gradient using surface curvature and area constraints,” in 3rd Indian Conference on Computer Vision, Graphics and Image Processing (Allied Publishers, 2002), pp. 1–6.

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Figures (13)

Fig. 1.
Fig. 1.

Outline of the data analysis.

Fig. 2.
Fig. 2.

Schematic of set-up. BS: beam splitter.

Fig. 3.
Fig. 3.

Measured object. The measured section at the outermost trace is marked with a square.

Fig. 4.
Fig. 4.

Fourier spectrum.

Fig. 5.
Fig. 5.

Vectors related to the generation of speckle displacements across plane D because of a wavelength shift.

Fig. 6.
Fig. 6.

Schematic representation of the refocusing procedure with quantities included in the derivation of the dynamic-image speckle properties.

Fig. 7.
Fig. 7.

(a) Slopes in u-direction p and (b) slopes in v-direction, q.

Fig. 8.
Fig. 8.

Schematic illustration of triangular mesh.

Fig. 9.
Fig. 9.

(a) Weights for slopes in u-direction and (b) weights for slopes in v-direction.

Fig. 10.
Fig. 10.

Part of the CAD model that describes the ideal shape of the measured object.

Fig. 11.
Fig. 11.

(a) Regularization parameters for function values, μh, (b) regularization parameters for u-slopes, μu, and (c) regularization parameters for v-slopes, μv.

Fig. 12.
Fig. 12.

Surface representation from the gradient field analysis using regularization.

Fig. 13.
Fig. 13.

(a) Initial deviations and (b) final deviations (mm). (c) Resulting surface representation using phase data and the CAD model.

Tables (1)

Tables Icon

Algorithm 1 The ICP algorithm

Equations (16)

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

Δϕ=4πh/Λ,
A(ΔL)=2MΔL(Δkk)tanθ,
A(ΔL0+z)=C+Dz,
C=2MΔL0(Δkk)tanθ,
D=2M(Δkk)tanθ.
wu[i,j]=αα+δu2,wv[i,j]=αα+δv2,
δu=|p[i,j+1]p[i,j]|+|p[i,j]p[i,j1]|,δv=|q[i+1,j]q[i,j]|+|q[i,j]q[i1,j]|,
J(h)==1mwu[](hu[]p[])2+wv[](hv[]q[])2,
μh[i,j]=β(8wu[i1,j1]wu[i,j1]wu[i1,j]wu[i,j]wv[i1,j1]wv[i,j1]wv[i1,j]wv[i,j]),
μu[i,j]=γ(2wu[i,j]wu[i1,j]),
μv[i,j]=γ(2wv[i,j]wv[i,j1]).
K(h)==1nμh[](h[]h˜[])2,
L(h)==1mμu[](hu[]h˜u[])2+=1mμv[](hv[]h˜v[])2,
minhJ(h)+K(h)+L(h),
minhAhb22,
minR,ti=1Nd(Rpi+t,X)2,

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