Abstract

In this paper, a method is proposed to measure the shapes of specular surfaces with one-shot-projection of structured laser patterns. By intercepting the reflection of the reflected laser pattern twice with two diffusive planes, the closed form solution is achieved for each reflected ray. The points on the specular surface are reconstructed by computing the intersections of the incident rays and the reflected rays. The proposed method can measure both static and dynamic specular shapes due to its one-shot-projection, which is beyond the capability of most of state of art methods that need multiple projections. To our knowledge, the proposed method is the only method so far that could yield the closed form solutions for the dynamic and specular surfaces.

© 2015 Optical Society of America

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References

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  1. Z. Z. Wang, “Robust measurement of the diffuse surface by phase shift profilometry,” J. Opt. 16(10), 105407 (2014).
    [Crossref]
  2. C. Je, S. W. Lee, and R. H. Park, “Color-stripe permutation pattern for rapid structured-light range imaging,” Opt. Commun. 285(9), 2320–2331 (2012).
    [Crossref]
  3. C. Je, K. H. Lee, and S. W. Lee, “Multi-projector color structured-light vision,” Signal Process. Image Commun. 28(9), 1046–1058 (2013).
    [Crossref]
  4. R. Furukawa, H. Kawasaki, R. Sagawa, and H. Masuyama, “Single color one-shot scan using modified Penrose tiling pattern,” IET Comput. Vis. 7(5), 293–301 (2013).
    [Crossref]
  5. L. Huang, C. S. Ng, and A. K. Asundi, “Dynamic three-dimensional sensing for specular surface with monoscopic fringe reflectometry,” Opt. Express 19(13), 12809–12814 (2011).
    [Crossref] [PubMed]
  6. H. S. Song and Y. M. Zhang, “Three-dimensional reconstruction of specular surface for a gas tungsten arc weld pool,” Meas. Sci. Technol. 18(12), 3751–3767 (2007).
    [Crossref]
  7. W. J. Zhang, Y. K. Liu, and Y. M. Zhang, “Real-time measurement of the weld pool surface in GTAW process,” 2013 IEEE International Conference on Instrumentation and Measurement Technology, 1640–1645 (2013).
    [Crossref]
  8. W. Jang, C. Je, Y. Seo, and S. W. Lee, “Structured-light stereo: Comparative analysis and integration of structured-light and active stereo for measuring dynamic shape,” Opt. Lasers Eng. 51(11), 1255–1264 (2013).
    [Crossref]
  9. G. Vogiatzis, C. Hernandez, P. H. S. Torr, and R. Cipolla, “Multi-view stereo via volumetric graphcuts and occlusion robust photo consistency,” IEEE T. Pattern Anal. 29(12), 2241–2246 (2007).
    [Crossref]
  10. K. N. Kutulakos and E. Steger, “A Theory of Refractive and Specular 3D Shape by Light-Path Triangulation,” Int. J. Comput. Vis. 76(1), 13–29 (2007).
    [Crossref]
  11. A. Sanderson, L. Weiss, and S. Nayar, “Structured highlight inspection of specular surfaces,” IEEE T-PAMI 10(1), 44–55 (1988).
    [Crossref]
  12. K. Ikeuchi, “Determining surface orientations of specular surfacesby using the photometric stereo method,” IEEE T. Pattern Anal. 3(6), 661–669 (1981).
    [Crossref]
  13. M. F. Tappen, “Recovering Shape from a Single Image of a Mirrored Surface from Curvature Constraints,”IEEE Conference on CVPR 2545–2552 (2011).
    [Crossref]
  14. M. Oren and S. K. Nayar, “A theory of specular surface geometry,” Int. J. Comput. Vis. 24(2), 105–124 (1997).
    [Crossref]
  15. J. Y. Zheng, Y. Fukagawa, and N. Abe, “3D Surface Estimation and Model Construction from Specular Motion,” IEEE T. Pattern Anal. 19 (5), 513–520 (1997).
    [Crossref]
  16. J. Y. Zheng and A. Murata, “Acquiring a Complete 3D Model from Specular Motion under the Illumination of Circular-Shaped Light Sources,” IEEE T. Pattern Anal. 22(8), 913–920 (2000).
    [Crossref]
  17. D. Miyazaki, M. Kagesawa, and K. Ikeuchi, “Transparent surface modeling from a pair of polarization images,” IEEE T. Pattern Anal. 26(1), 73–82 (2004).
    [Crossref] [PubMed]
  18. Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE T. Pattern Anal. 22(11), 1330–1334 (2000).
    [Crossref]
  19. R.E. Smith, J.M. Price and L.M. Howser, “A smoothing algorithm using cubic spline function,” NASA TN D-7397 (1974).
  20. C. D. Boor, “A practical guide to spline,” Appl. Math. Sci. 27, 1–348 (1978).
  21. Z. Z. Wang, “Monitoring of GMAW weld pool from the reflected laser lines for real time control,” IEEE Trans. Ind. Inform. 10(4), 2073–2083 (2014).
    [Crossref]
  22. Z. Z. Wang, X. Y. Huang, R. G. Yang, and Y. M. Zhang, “Measurement of mirror surfaces using specular reflection and analytical computation,” Mach. Vis. Appl. 24(2), 289–304 (2013).
    [Crossref]

2014 (2)

Z. Z. Wang, “Robust measurement of the diffuse surface by phase shift profilometry,” J. Opt. 16(10), 105407 (2014).
[Crossref]

Z. Z. Wang, “Monitoring of GMAW weld pool from the reflected laser lines for real time control,” IEEE Trans. Ind. Inform. 10(4), 2073–2083 (2014).
[Crossref]

2013 (4)

Z. Z. Wang, X. Y. Huang, R. G. Yang, and Y. M. Zhang, “Measurement of mirror surfaces using specular reflection and analytical computation,” Mach. Vis. Appl. 24(2), 289–304 (2013).
[Crossref]

C. Je, K. H. Lee, and S. W. Lee, “Multi-projector color structured-light vision,” Signal Process. Image Commun. 28(9), 1046–1058 (2013).
[Crossref]

R. Furukawa, H. Kawasaki, R. Sagawa, and H. Masuyama, “Single color one-shot scan using modified Penrose tiling pattern,” IET Comput. Vis. 7(5), 293–301 (2013).
[Crossref]

W. Jang, C. Je, Y. Seo, and S. W. Lee, “Structured-light stereo: Comparative analysis and integration of structured-light and active stereo for measuring dynamic shape,” Opt. Lasers Eng. 51(11), 1255–1264 (2013).
[Crossref]

2012 (1)

C. Je, S. W. Lee, and R. H. Park, “Color-stripe permutation pattern for rapid structured-light range imaging,” Opt. Commun. 285(9), 2320–2331 (2012).
[Crossref]

2011 (1)

2007 (3)

H. S. Song and Y. M. Zhang, “Three-dimensional reconstruction of specular surface for a gas tungsten arc weld pool,” Meas. Sci. Technol. 18(12), 3751–3767 (2007).
[Crossref]

G. Vogiatzis, C. Hernandez, P. H. S. Torr, and R. Cipolla, “Multi-view stereo via volumetric graphcuts and occlusion robust photo consistency,” IEEE T. Pattern Anal. 29(12), 2241–2246 (2007).
[Crossref]

K. N. Kutulakos and E. Steger, “A Theory of Refractive and Specular 3D Shape by Light-Path Triangulation,” Int. J. Comput. Vis. 76(1), 13–29 (2007).
[Crossref]

2004 (1)

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, “Transparent surface modeling from a pair of polarization images,” IEEE T. Pattern Anal. 26(1), 73–82 (2004).
[Crossref] [PubMed]

2000 (2)

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE T. Pattern Anal. 22(11), 1330–1334 (2000).
[Crossref]

J. Y. Zheng and A. Murata, “Acquiring a Complete 3D Model from Specular Motion under the Illumination of Circular-Shaped Light Sources,” IEEE T. Pattern Anal. 22(8), 913–920 (2000).
[Crossref]

1997 (2)

M. Oren and S. K. Nayar, “A theory of specular surface geometry,” Int. J. Comput. Vis. 24(2), 105–124 (1997).
[Crossref]

J. Y. Zheng, Y. Fukagawa, and N. Abe, “3D Surface Estimation and Model Construction from Specular Motion,” IEEE T. Pattern Anal. 19 (5), 513–520 (1997).
[Crossref]

1988 (1)

A. Sanderson, L. Weiss, and S. Nayar, “Structured highlight inspection of specular surfaces,” IEEE T-PAMI 10(1), 44–55 (1988).
[Crossref]

1981 (1)

K. Ikeuchi, “Determining surface orientations of specular surfacesby using the photometric stereo method,” IEEE T. Pattern Anal. 3(6), 661–669 (1981).
[Crossref]

1978 (1)

C. D. Boor, “A practical guide to spline,” Appl. Math. Sci. 27, 1–348 (1978).

Abe, N.

J. Y. Zheng, Y. Fukagawa, and N. Abe, “3D Surface Estimation and Model Construction from Specular Motion,” IEEE T. Pattern Anal. 19 (5), 513–520 (1997).
[Crossref]

Asundi, A. K.

Boor, C. D.

C. D. Boor, “A practical guide to spline,” Appl. Math. Sci. 27, 1–348 (1978).

Cipolla, R.

G. Vogiatzis, C. Hernandez, P. H. S. Torr, and R. Cipolla, “Multi-view stereo via volumetric graphcuts and occlusion robust photo consistency,” IEEE T. Pattern Anal. 29(12), 2241–2246 (2007).
[Crossref]

Fukagawa, Y.

J. Y. Zheng, Y. Fukagawa, and N. Abe, “3D Surface Estimation and Model Construction from Specular Motion,” IEEE T. Pattern Anal. 19 (5), 513–520 (1997).
[Crossref]

Furukawa, R.

R. Furukawa, H. Kawasaki, R. Sagawa, and H. Masuyama, “Single color one-shot scan using modified Penrose tiling pattern,” IET Comput. Vis. 7(5), 293–301 (2013).
[Crossref]

Hernandez, C.

G. Vogiatzis, C. Hernandez, P. H. S. Torr, and R. Cipolla, “Multi-view stereo via volumetric graphcuts and occlusion robust photo consistency,” IEEE T. Pattern Anal. 29(12), 2241–2246 (2007).
[Crossref]

Huang, L.

Huang, X. Y.

Z. Z. Wang, X. Y. Huang, R. G. Yang, and Y. M. Zhang, “Measurement of mirror surfaces using specular reflection and analytical computation,” Mach. Vis. Appl. 24(2), 289–304 (2013).
[Crossref]

Ikeuchi, K.

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, “Transparent surface modeling from a pair of polarization images,” IEEE T. Pattern Anal. 26(1), 73–82 (2004).
[Crossref] [PubMed]

K. Ikeuchi, “Determining surface orientations of specular surfacesby using the photometric stereo method,” IEEE T. Pattern Anal. 3(6), 661–669 (1981).
[Crossref]

Jang, W.

W. Jang, C. Je, Y. Seo, and S. W. Lee, “Structured-light stereo: Comparative analysis and integration of structured-light and active stereo for measuring dynamic shape,” Opt. Lasers Eng. 51(11), 1255–1264 (2013).
[Crossref]

Je, C.

W. Jang, C. Je, Y. Seo, and S. W. Lee, “Structured-light stereo: Comparative analysis and integration of structured-light and active stereo for measuring dynamic shape,” Opt. Lasers Eng. 51(11), 1255–1264 (2013).
[Crossref]

C. Je, K. H. Lee, and S. W. Lee, “Multi-projector color structured-light vision,” Signal Process. Image Commun. 28(9), 1046–1058 (2013).
[Crossref]

C. Je, S. W. Lee, and R. H. Park, “Color-stripe permutation pattern for rapid structured-light range imaging,” Opt. Commun. 285(9), 2320–2331 (2012).
[Crossref]

Kagesawa, M.

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, “Transparent surface modeling from a pair of polarization images,” IEEE T. Pattern Anal. 26(1), 73–82 (2004).
[Crossref] [PubMed]

Kawasaki, H.

R. Furukawa, H. Kawasaki, R. Sagawa, and H. Masuyama, “Single color one-shot scan using modified Penrose tiling pattern,” IET Comput. Vis. 7(5), 293–301 (2013).
[Crossref]

Kutulakos, K. N.

K. N. Kutulakos and E. Steger, “A Theory of Refractive and Specular 3D Shape by Light-Path Triangulation,” Int. J. Comput. Vis. 76(1), 13–29 (2007).
[Crossref]

Lee, K. H.

C. Je, K. H. Lee, and S. W. Lee, “Multi-projector color structured-light vision,” Signal Process. Image Commun. 28(9), 1046–1058 (2013).
[Crossref]

Lee, S. W.

C. Je, K. H. Lee, and S. W. Lee, “Multi-projector color structured-light vision,” Signal Process. Image Commun. 28(9), 1046–1058 (2013).
[Crossref]

W. Jang, C. Je, Y. Seo, and S. W. Lee, “Structured-light stereo: Comparative analysis and integration of structured-light and active stereo for measuring dynamic shape,” Opt. Lasers Eng. 51(11), 1255–1264 (2013).
[Crossref]

C. Je, S. W. Lee, and R. H. Park, “Color-stripe permutation pattern for rapid structured-light range imaging,” Opt. Commun. 285(9), 2320–2331 (2012).
[Crossref]

Liu, Y. K.

W. J. Zhang, Y. K. Liu, and Y. M. Zhang, “Real-time measurement of the weld pool surface in GTAW process,” 2013 IEEE International Conference on Instrumentation and Measurement Technology, 1640–1645 (2013).
[Crossref]

Masuyama, H.

R. Furukawa, H. Kawasaki, R. Sagawa, and H. Masuyama, “Single color one-shot scan using modified Penrose tiling pattern,” IET Comput. Vis. 7(5), 293–301 (2013).
[Crossref]

Miyazaki, D.

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, “Transparent surface modeling from a pair of polarization images,” IEEE T. Pattern Anal. 26(1), 73–82 (2004).
[Crossref] [PubMed]

Murata, A.

J. Y. Zheng and A. Murata, “Acquiring a Complete 3D Model from Specular Motion under the Illumination of Circular-Shaped Light Sources,” IEEE T. Pattern Anal. 22(8), 913–920 (2000).
[Crossref]

Nayar, S.

A. Sanderson, L. Weiss, and S. Nayar, “Structured highlight inspection of specular surfaces,” IEEE T-PAMI 10(1), 44–55 (1988).
[Crossref]

Nayar, S. K.

M. Oren and S. K. Nayar, “A theory of specular surface geometry,” Int. J. Comput. Vis. 24(2), 105–124 (1997).
[Crossref]

Ng, C. S.

Oren, M.

M. Oren and S. K. Nayar, “A theory of specular surface geometry,” Int. J. Comput. Vis. 24(2), 105–124 (1997).
[Crossref]

Park, R. H.

C. Je, S. W. Lee, and R. H. Park, “Color-stripe permutation pattern for rapid structured-light range imaging,” Opt. Commun. 285(9), 2320–2331 (2012).
[Crossref]

Sagawa, R.

R. Furukawa, H. Kawasaki, R. Sagawa, and H. Masuyama, “Single color one-shot scan using modified Penrose tiling pattern,” IET Comput. Vis. 7(5), 293–301 (2013).
[Crossref]

Sanderson, A.

A. Sanderson, L. Weiss, and S. Nayar, “Structured highlight inspection of specular surfaces,” IEEE T-PAMI 10(1), 44–55 (1988).
[Crossref]

Seo, Y.

W. Jang, C. Je, Y. Seo, and S. W. Lee, “Structured-light stereo: Comparative analysis and integration of structured-light and active stereo for measuring dynamic shape,” Opt. Lasers Eng. 51(11), 1255–1264 (2013).
[Crossref]

Song, H. S.

H. S. Song and Y. M. Zhang, “Three-dimensional reconstruction of specular surface for a gas tungsten arc weld pool,” Meas. Sci. Technol. 18(12), 3751–3767 (2007).
[Crossref]

Steger, E.

K. N. Kutulakos and E. Steger, “A Theory of Refractive and Specular 3D Shape by Light-Path Triangulation,” Int. J. Comput. Vis. 76(1), 13–29 (2007).
[Crossref]

Torr, P. H. S.

G. Vogiatzis, C. Hernandez, P. H. S. Torr, and R. Cipolla, “Multi-view stereo via volumetric graphcuts and occlusion robust photo consistency,” IEEE T. Pattern Anal. 29(12), 2241–2246 (2007).
[Crossref]

Vogiatzis, G.

G. Vogiatzis, C. Hernandez, P. H. S. Torr, and R. Cipolla, “Multi-view stereo via volumetric graphcuts and occlusion robust photo consistency,” IEEE T. Pattern Anal. 29(12), 2241–2246 (2007).
[Crossref]

Wang, Z. Z.

Z. Z. Wang, “Robust measurement of the diffuse surface by phase shift profilometry,” J. Opt. 16(10), 105407 (2014).
[Crossref]

Z. Z. Wang, “Monitoring of GMAW weld pool from the reflected laser lines for real time control,” IEEE Trans. Ind. Inform. 10(4), 2073–2083 (2014).
[Crossref]

Z. Z. Wang, X. Y. Huang, R. G. Yang, and Y. M. Zhang, “Measurement of mirror surfaces using specular reflection and analytical computation,” Mach. Vis. Appl. 24(2), 289–304 (2013).
[Crossref]

Weiss, L.

A. Sanderson, L. Weiss, and S. Nayar, “Structured highlight inspection of specular surfaces,” IEEE T-PAMI 10(1), 44–55 (1988).
[Crossref]

Yang, R. G.

Z. Z. Wang, X. Y. Huang, R. G. Yang, and Y. M. Zhang, “Measurement of mirror surfaces using specular reflection and analytical computation,” Mach. Vis. Appl. 24(2), 289–304 (2013).
[Crossref]

Zhang, W. J.

W. J. Zhang, Y. K. Liu, and Y. M. Zhang, “Real-time measurement of the weld pool surface in GTAW process,” 2013 IEEE International Conference on Instrumentation and Measurement Technology, 1640–1645 (2013).
[Crossref]

Zhang, Y. M.

Z. Z. Wang, X. Y. Huang, R. G. Yang, and Y. M. Zhang, “Measurement of mirror surfaces using specular reflection and analytical computation,” Mach. Vis. Appl. 24(2), 289–304 (2013).
[Crossref]

H. S. Song and Y. M. Zhang, “Three-dimensional reconstruction of specular surface for a gas tungsten arc weld pool,” Meas. Sci. Technol. 18(12), 3751–3767 (2007).
[Crossref]

W. J. Zhang, Y. K. Liu, and Y. M. Zhang, “Real-time measurement of the weld pool surface in GTAW process,” 2013 IEEE International Conference on Instrumentation and Measurement Technology, 1640–1645 (2013).
[Crossref]

Zhang, Z. Y.

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE T. Pattern Anal. 22(11), 1330–1334 (2000).
[Crossref]

Zheng, J. Y.

J. Y. Zheng and A. Murata, “Acquiring a Complete 3D Model from Specular Motion under the Illumination of Circular-Shaped Light Sources,” IEEE T. Pattern Anal. 22(8), 913–920 (2000).
[Crossref]

J. Y. Zheng, Y. Fukagawa, and N. Abe, “3D Surface Estimation and Model Construction from Specular Motion,” IEEE T. Pattern Anal. 19 (5), 513–520 (1997).
[Crossref]

Appl. Math. Sci. (1)

C. D. Boor, “A practical guide to spline,” Appl. Math. Sci. 27, 1–348 (1978).

IEEE T-PAMI (1)

A. Sanderson, L. Weiss, and S. Nayar, “Structured highlight inspection of specular surfaces,” IEEE T-PAMI 10(1), 44–55 (1988).
[Crossref]

IEEE T. Pattern Anal. (6)

K. Ikeuchi, “Determining surface orientations of specular surfacesby using the photometric stereo method,” IEEE T. Pattern Anal. 3(6), 661–669 (1981).
[Crossref]

G. Vogiatzis, C. Hernandez, P. H. S. Torr, and R. Cipolla, “Multi-view stereo via volumetric graphcuts and occlusion robust photo consistency,” IEEE T. Pattern Anal. 29(12), 2241–2246 (2007).
[Crossref]

J. Y. Zheng, Y. Fukagawa, and N. Abe, “3D Surface Estimation and Model Construction from Specular Motion,” IEEE T. Pattern Anal. 19 (5), 513–520 (1997).
[Crossref]

J. Y. Zheng and A. Murata, “Acquiring a Complete 3D Model from Specular Motion under the Illumination of Circular-Shaped Light Sources,” IEEE T. Pattern Anal. 22(8), 913–920 (2000).
[Crossref]

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, “Transparent surface modeling from a pair of polarization images,” IEEE T. Pattern Anal. 26(1), 73–82 (2004).
[Crossref] [PubMed]

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE T. Pattern Anal. 22(11), 1330–1334 (2000).
[Crossref]

IEEE Trans. Ind. Inform. (1)

Z. Z. Wang, “Monitoring of GMAW weld pool from the reflected laser lines for real time control,” IEEE Trans. Ind. Inform. 10(4), 2073–2083 (2014).
[Crossref]

IET Comput. Vis. (1)

R. Furukawa, H. Kawasaki, R. Sagawa, and H. Masuyama, “Single color one-shot scan using modified Penrose tiling pattern,” IET Comput. Vis. 7(5), 293–301 (2013).
[Crossref]

Int. J. Comput. Vis. (2)

K. N. Kutulakos and E. Steger, “A Theory of Refractive and Specular 3D Shape by Light-Path Triangulation,” Int. J. Comput. Vis. 76(1), 13–29 (2007).
[Crossref]

M. Oren and S. K. Nayar, “A theory of specular surface geometry,” Int. J. Comput. Vis. 24(2), 105–124 (1997).
[Crossref]

J. Opt. (1)

Z. Z. Wang, “Robust measurement of the diffuse surface by phase shift profilometry,” J. Opt. 16(10), 105407 (2014).
[Crossref]

Mach. Vis. Appl. (1)

Z. Z. Wang, X. Y. Huang, R. G. Yang, and Y. M. Zhang, “Measurement of mirror surfaces using specular reflection and analytical computation,” Mach. Vis. Appl. 24(2), 289–304 (2013).
[Crossref]

Meas. Sci. Technol. (1)

H. S. Song and Y. M. Zhang, “Three-dimensional reconstruction of specular surface for a gas tungsten arc weld pool,” Meas. Sci. Technol. 18(12), 3751–3767 (2007).
[Crossref]

Opt. Commun. (1)

C. Je, S. W. Lee, and R. H. Park, “Color-stripe permutation pattern for rapid structured-light range imaging,” Opt. Commun. 285(9), 2320–2331 (2012).
[Crossref]

Opt. Express (1)

Opt. Lasers Eng. (1)

W. Jang, C. Je, Y. Seo, and S. W. Lee, “Structured-light stereo: Comparative analysis and integration of structured-light and active stereo for measuring dynamic shape,” Opt. Lasers Eng. 51(11), 1255–1264 (2013).
[Crossref]

Signal Process. Image Commun. (1)

C. Je, K. H. Lee, and S. W. Lee, “Multi-projector color structured-light vision,” Signal Process. Image Commun. 28(9), 1046–1058 (2013).
[Crossref]

Other (3)

R.E. Smith, J.M. Price and L.M. Howser, “A smoothing algorithm using cubic spline function,” NASA TN D-7397 (1974).

M. F. Tappen, “Recovering Shape from a Single Image of a Mirrored Surface from Curvature Constraints,”IEEE Conference on CVPR 2545–2552 (2011).
[Crossref]

W. J. Zhang, Y. K. Liu, and Y. M. Zhang, “Real-time measurement of the weld pool surface in GTAW process,” 2013 IEEE International Conference on Instrumentation and Measurement Technology, 1640–1645 (2013).
[Crossref]

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Figures (17)

Fig. 1
Fig. 1 Principle of the Proposed Method
Fig. 2
Fig. 2 Virtual Camera
Fig. 3
Fig. 3 Demonstration of incident ray determination
Fig. 4
Fig. 4 Illustration of practical system
Fig. 5
Fig. 5 Established system.
Fig. 6
Fig. 6 Captured laser points by different cameras.
Fig. 7
Fig. 7 Reconstructed convex mirror 1 (a) result by the system without re-calibration and smoothing factor α=0.9 ; (b) result by the re-calibrated system and smoothing factor α=0.1 ; (c) result by the re-calibrated system and smoothing factor α=0.5 ; (b) result by the re-calibrated system and smoothing factor α=0.9 ; (The scale is mm)
Fig. 8
Fig. 8 Reconstructed convex mirror 2 (a) result by the system without re-calibration and smoothing factor α=0.9 ; (b) result by the re-calibrated system and smoothing factor α=0.1 ; (c) result by the re-calibrated system and smoothing factor α=0.5 ; (b) result by the re-calibrated system and smoothing factor α=0.9 ; (The scale is mm)
Fig. 9
Fig. 9 Reconstructed convex mirror 3 (a) result by the system without re-calibration and smoothing factor α=0.9 ; (b) result by the re-calibrated system and smoothing factor α=0.1 ; (c) result by the re-calibrated system and smoothing factor α=0.5 ; (b) result by the re-calibrated system and smoothing factor α=0.9 ; (The scale is mm)
Fig. 10
Fig. 10 Reconstructed dynamic specular surface (a) result by the system without re-calibration and smoothing factor α=0.9 ; (b) result by the re-calibrated system and smoothing factor α=0.1 ; (c) result by the re-calibrated system and smoothing factor α=0.5 ; (b) result by the re-calibrated system and smoothing factor α=0.9 ; (The scale is mm)
Fig. 11
Fig. 11 Reconstruction of a flat mirror (the scale is mm)
Fig. 12
Fig. 12 Simulation results of reconstructing a flatmirror and a convex mirror in MATLAB
Fig. 13
Fig. 13 Visual comparison of reconstruction result a real weld pool by the proposed method with those by the state of art methods [6] [7](a) result by the proposed method; (b) result by [7]; (c) result by [6](the scale is mm)
Fig. 14
Fig. 14 Reconstructed weld pool at time   T 2 =16.7ms (the scale is mm)
Fig. 15
Fig. 15 Reconstructed weld pool at time   T 2 =33.4ms (the scale is mm)
Fig. 16
Fig. 16 Reconstructed weld pool at time   T 2 =50.1ms (the scale is mm)
Fig. 17
Fig. 17 Reconstructed weld pool at time   T 2 =66.8ms (the scale is mm)

Tables (2)

Tables Icon

Table 1 Comparison of the original and searched coefficients for plane3

Tables Icon

Table 2 Comparison of the original and searched coefficients for plane4

Equations (56)

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

x l i x 0 i a i = y l i y 0 i b i = z l i z 0 i c i = t i ;    l=1  6
d 2 = | ( P 1 P 0 )×( P 0 C ) | 2 | P 1 P 0 | 2   
[ V 1 i , V 2 i , V 3 i ] T [ x,y,z ] T = [ b 1 i , b 2 i , b 3 i ] T  
V 1 i = [ ( z 1 i z 0 i ) 2 + ( y 1 i y 0 i ) 2 ( x 0 i x 1 i )( y 1 i y 0 i ) ( x 0 i x 1 i )( z 1 i z 0 i ) ] T   
V 2 i = [ ( x 0 i x 1 i )( y 1 i y 0 i ) ( z 1 i z 0 i ) 2 + ( x 1 i x 0 i ) 2 ( y 0 i y 1 i )( z 1 i z 0 i ) ] T  
V 3 i = [ ( x 0 i x 1 i )( z 1 i z 0 i ) ( y 0 i y 1 i )( z 1 i z 0 i ) ( x 1 i x 0 i ) 2 + ( y 1 i y 0 i ) 2 ] T
b 1 i =[ ( x 1 i z 0 i x 0 i z 1 i )( z 1 i z 0 i )+( x 1 i y 0 i x 0 i y 1 i )( y 1 i y 0 i ) ]
b 2 i =[ ( x 1 i y 0 i x 0 i y 1 i )( x 0 i x 1 i )+( y 1 i z 0 i y 0 i z 1 i )( z 1 i z 0 i ) ]
b 3 i =[ ( y 1 i z 0 i y 0 i z 1 i )( y 0 i y 1 i )+( x 1 i z 0 i x 0 i z 1 i )( x 0 i x 1 i ) ]   
[ V 1 , V i ,, V N ] T [ x,y,z ] T = [ b 1 , b i , b N ] T
  π 1 = [0,0,1,0] T
π= ( P T ) 1 π 1
P= [ 1 0 0 1 0 C x 0 C y 0 0 0 0 1      f 0     1 ] 1 [ r 0 r 1 r 3 r 4 r 2 t x r 5 t y r 6 r 7 0 0 r 8 t z 0 1 ]
r1= [ r 0 , r 3 , r 6 ] T = K 1 h 1 | | K 1 h 1 | |
r2= [ r 1 , r 4 , r 7 ] T = K 1 h 2 | | K 1 h 1 | |
T= [ T X , T Y , T Z ] T = K 1 h 3 | | K 1 h 1 | |
r3= [ r 2 , r 5 , r 8 ] T =r1×r2
K=[ f 0 C x 0 f C y 0 0 1 ]
H ^ = argmin H ( i=1 N ( x i x i ' )   2 + ( y i y i ' ) 2 )
x i = H 11 X i + H 12 Y i + H 13 H 31 X i + H 32 Y i + H 33 ;     i=1N
y i = H 21 X i + H 22 Y i + H 23 H 31 X i + H 32 Y i + H 33 ;     i=1N
H=[ H 11 H 12 H 13 H 21 H 22 H 23 H 31 H 32 H 33 ]
Z i = π( 1 ) X i +π( 2 ) Y i +π( 4 ) π( 3 )
X i X 0 i A i = Y i Y 0 i B i = Z i Z 0 i C i = T i
d i = ( X i x i ) 2 + ( Y i y i ) 2 + ( Z i z i ) 2
( d i ) 2 = ( A i T i + X 0 i a i t i x 0 i ) 2 + ( B i T i + Y 0 i b i t i y 0 i ) 2 + ( C i T i + Z 0 i c i t i z 0 i ) 2
T i = ρ 1 t i + ρ 2
ρ 1 = a i A i + b i B i + c i C i A i A i + B i B i + C i C i
ρ 2 = A i x 0 i + B i y 0 i + C i z 0 i A i X 0 i B i Y 0 i C i Z 0 i A i A i + B i B i + C i C i
t i = μ 1 + μ 2 + μ 3 σ
σ= ( A i ρ 1 a i ) 2 + ( B i ρ 1 b i ) 2 + ( C i ρ 1 c i ) 2
μ 1 =( A i ρ 1 a i )( A i ρ 2 + X 0 i x 0 i )
μ 2 =( B i ρ 1 b i )( B i ρ 2 + Y 0 i y 0 i )
μ 3 =( C i ρ 1 c i )( C i ρ 2 + Z 0 i z 0 i )
T i = ρ 1 μ 1 + μ 2 + μ 3 σ + ρ 2
[ x i y i z i ]= μ 1 + μ 2 + μ 3 σ [ a i b i c i ]+[ x 0 i y 0 i z 0 i ]
[ X i Y i Z i ]=( ρ 1 μ 1 + μ 2 + μ 3 σ + ρ 2 )[ A i B i C i ]+[ X 0 i Y 0 i Z 0 i ]
[ X r i Y r i Z r i ]= 1 2 [ X i Y i Z i ]+ 1 2 [ x i y i z i ]
P f ( 1: N t )= argmin f ( α j=1 N t | P r ( [ v( i ) ] )f( v( i ) ) | 2 +( 1α ) | d 2 f( t ) d t 2 | 2 dt )
v(i)=1+(i1)/M    i=1: N t ;  N t =M×N
x x 0 a = y y 0 b = z2 c =t
x x 0 +1 a = y y 0 b = z2 c =t
X x 0 a = Y y 0 b = Z2 c =T
X x 0 +1 a = Y y 0 b = Z2 c =T
X 2 = x 0 a( x 0 π 4 ( 1 )+ y 0 π 4 ( 2 )+ π 4 ( 4 ) ) a π 4 ( 1 )+b π 4 ( 2 )c π 4 ( 3 )
Y 2 = y 0 b( x 0 π 4 ( 1 )+ y 0 π 4 ( 2 )+ π 4 ( 4 ) ) a π 4 ( 1 )+b π 4 ( 2 )c π 4 ( 3 )
Z 2 =2+ c( x 0 π 4 ( 1 )+ y 0 π 4 ( 2 )+ π 4 ( 4 ) ) a π 4 ( 1 )+b π 4 ( 2 )c π 4 ( 3 )
Z 2 = π 4 (1) X 2 + π 4 (2) Y 2 + π 4 (4) π 4 (3)
Z 2 e = π 4 e (1) X 2 + π 4 e (2) Y 2 + π 4 e (4) π 4 e (3)
X x 0 a = Y y 0 b = Z2 c e =T
X x 0 +1 a = Y y 0 b = Z2 c e' =T
c e = π 3 e (1) X 2 + π 3 e (2) Y 2 + π 3 e (4) π 3 e (3) π 4 e ( 1 ) X 4 + π 4 e ( 2 ) Y 4 + π 4 e ( 4 ) π 4 e ( 3 )
c e' = π 3 e (1) X 2 ' + π 3 e (2) Y 2 ' + π 3 e (4) π 3 e (3) π 4 e ( 1 ) X 4 ' + π 4 e ( 2 ) Y 4 ' + π 4 e ( 4 ) π 4 e ( 3 )
Z d r =T| c e c e ' |
Z d = max i=1..N Z i r min i=1N Z i r
  θ ^ = argmin θ ( Z d )

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