Abstract

A framework with a combination of the radial basis functions (RBFs) method and the least-squares integration method is proposed to improve the integration process from gradient to shape. The principle of the framework is described, and the performance of the proposed method is investigated by simulation. Improvement in accuracy is verified by comparing the result with the usual RBFs-based subset-by-subset stitching method. The proposed method is accurate, automatic, easily implemented, and robust and even works with incomplete data.

© 2013 Optical Society of America

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  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]
  2. K. Harding, “Industrial metrology: engineering precision,” Nat. Photonics 2, 667–669 (2008).
    [CrossRef]
  3. M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).
    [CrossRef]
  4. T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
    [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, 12809–12814 (2011).
    [CrossRef]
  6. L. Huang, C. S. Ng, and A. K. Asundi, “Fast full-field out-of-plane deformation measurement using fringe reflectometry,” Opt. Lasers Eng. 50, 529–533 (2012).
    [CrossRef]
  7. S. Ettl, J. Kaminski, M. C. Knauer, and G. Häusler, “Shape reconstruction from gradient data,” Appl. Opt. 47, 2091–2097 (2008).
    [CrossRef]
  8. W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
    [CrossRef]
  9. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  10. L. Huang and A. Asundi, “Improvement of least-squares integration method with iterative compensations in fringe reflectometry,” Appl. Opt. 51, 7459–7465 (2012).
    [CrossRef]

2012 (2)

L. Huang, C. S. Ng, and A. K. Asundi, “Fast full-field out-of-plane deformation measurement using fringe reflectometry,” Opt. Lasers Eng. 50, 529–533 (2012).
[CrossRef]

L. Huang and A. Asundi, “Improvement of least-squares integration method with iterative compensations in fringe reflectometry,” Appl. Opt. 51, 7459–7465 (2012).
[CrossRef]

2011 (1)

2008 (2)

2004 (3)

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

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

T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (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]

1980 (1)

Asundi, A.

Asundi, A. K.

L. Huang, C. S. Ng, and A. K. Asundi, “Fast full-field out-of-plane deformation measurement using fringe reflectometry,” Opt. Lasers Eng. 50, 529–533 (2012).
[CrossRef]

L. Huang, C. S. Ng, and A. K. Asundi, “Dynamic three-dimensional sensing for specular surface with monoscopic fringe reflectometry,” Opt. Express 19, 12809–12814 (2011).
[CrossRef]

Bothe, T.

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

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

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]

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]

Ettl, S.

Harding, K.

K. Harding, “Industrial metrology: engineering precision,” Nat. Photonics 2, 667–669 (2008).
[CrossRef]

Häusler, G.

S. Ettl, J. Kaminski, M. C. Knauer, and G. Häusler, “Shape reconstruction from gradient data,” Appl. Opt. 47, 2091–2097 (2008).
[CrossRef]

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

Huang, L.

Jüptner, W. P. O.

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

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

Kaminski, J.

S. Ettl, J. Kaminski, M. C. Knauer, and G. Häusler, “Shape reconstruction from gradient data,” Appl. Opt. 47, 2091–2097 (2008).
[CrossRef]

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

Knauer, M. C.

S. Ettl, J. Kaminski, M. C. Knauer, and G. Häusler, “Shape reconstruction from gradient data,” Appl. Opt. 47, 2091–2097 (2008).
[CrossRef]

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

Li, W.

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

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

Ng, C. S.

L. Huang, C. S. Ng, and A. K. Asundi, “Fast full-field out-of-plane deformation measurement using fringe reflectometry,” Opt. Lasers Eng. 50, 529–533 (2012).
[CrossRef]

L. Huang, C. S. Ng, and A. K. Asundi, “Dynamic three-dimensional sensing for specular surface with monoscopic fringe reflectometry,” Opt. Express 19, 12809–12814 (2011).
[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]

Southwell, W. H.

von Kopylow, C.

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

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

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Nat. Photonics (1)

K. Harding, “Industrial metrology: engineering precision,” Nat. Photonics 2, 667–669 (2008).
[CrossRef]

Opt. Eng. (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]

Opt. Express (1)

Opt. Lasers Eng. (1)

L. Huang, C. S. Ng, and A. K. Asundi, “Fast full-field out-of-plane deformation measurement using fringe reflectometry,” Opt. Lasers Eng. 50, 529–533 (2012).
[CrossRef]

Proc. SPIE (3)

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

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

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Entire 3D shape result can be reconstructed from a gradient through the five-step framework.

Fig. 2.
Fig. 2.

Stitching problem can be considered an issue of integration with central differential values. (a) Subsets with overlaps. (b) The grid model of the equivalent integration problem.

Fig. 3.
Fig. 3.

True height distribution (a) is generated as the ground truth. The gradient data (b) dz/dx and (c) dz/dy are generated with additive normally distributed angular noise.

Fig. 4.
Fig. 4.

With a comparison of results, the proposed method shows its superiority apparently. (a) Reconstructed height with subset-by-subset stitching. (b) Height error of subset-by-subset stitching. (c) Histogram of (b). (d) Reconstructed height with stitching with least-squares integration. (e) Height error of stitching with least-squares integration. (f) Histogram of (e).

Fig. 5.
Fig. 5.

Height errors become larger along with the noise being severer, but the stitching with least-squares integration method is superior to the subset-by-subset stitching.

Fig. 6.
Fig. 6.

Feasibility of the proposed framework is verified with an experiment. (a) The sample photo. (b) The captured fringe patterns under a fringe reflectometric system. (c) Slope data. (d) The reconstructed 3D shape.

Fig. 7.
Fig. 7.

With the proposed method, the integration is able to be accomplished with incomplete datasets. (a) Slope in x direction. (b) Slope in y direction. (c) 3D shape. (d) Height error.

Equations (4)

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

{zm,n+1zm,n=pm,n×(n+1n)=pm,n,m=1,,M,n=1,,N1zm+1,nzm,n=qm,n×(m+1m)=qm,n,m=1,,M1,n=1,,N,
DZ=G,
D=[10010001001000010011100011000011],Z=[z1,1z2,1zM,N],G=[p1,1p1,2pM,N1q1,1q2,1qM1,N].
z=3(1x)2·ex2(y+1)210(x5x3y5)·ex2y213e(x+1)2y2,

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