Abstract

Least-squares integration is one of the most effective and widely used methods for shape reconstruction from gradient data, which result from gradient measurement techniques. However, its reconstruction accuracy is limited due to the imperfection of the Southwell grid model, which is commonly applied in the least-squares integration method. An operation with iterative compensations is therefore proposed, especially for the traditional least-squares integration method, to improve its integration accuracy. Simulation and experiment are carried out to verify the feasibility and superiority of the proposed operation. This compensatory operation with iterations is suggested, and its good performance on integration accuracy improvement is shown.

© 2012 Optical Society of America

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References

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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. 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]
  3. 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]
  4. B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).
  5. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
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  8. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
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  9. 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]
  10. R. Legarda-Saenz and A. Espinosa-Romero, “Wavefront reconstruction using multiple directional derivatives and Fourier transform,” Opt. Eng. 50, 040501 (2011).
    [CrossRef]
  11. J. Koskulics, S. Englehardt, S. Long, Y. Hu, and K. Stamnes, “Method of surface topography retrieval by direct solution of sparse weighted seminormal equations,” Opt. Express 20, 1714–1726 (2012).
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2012 (1)

2011 (1)

R. Legarda-Saenz and A. Espinosa-Romero, “Wavefront reconstruction using multiple directional derivatives and Fourier transform,” Opt. Eng. 50, 040501 (2011).
[CrossRef]

2004 (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]

2001 (1)

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).

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)

1977 (3)

Bothe, T.

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]

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]

Englehardt, S.

Espinosa-Romero, A.

R. Legarda-Saenz and A. Espinosa-Romero, “Wavefront reconstruction using multiple directional derivatives and Fourier transform,” Opt. Eng. 50, 040501 (2011).
[CrossRef]

Fried, D. L.

Häusler, G.

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]

Hu, Y.

Hudgin, R. H.

Jüptner, W. P. O.

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]

Kaminski, J.

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.

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]

Koskulics, J.

Legarda-Saenz, R.

R. Legarda-Saenz and A. Espinosa-Romero, “Wavefront reconstruction using multiple directional derivatives and Fourier transform,” Opt. Eng. 50, 040501 (2011).
[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]

Long, S.

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).

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.

Stamnes, K.

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]

J. Opt. Soc. Am. (4)

J. Refractive Surg. (1)

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).

Opt. Eng. (2)

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

R. Legarda-Saenz and A. Espinosa-Romero, “Wavefront reconstruction using multiple directional derivatives and Fourier transform,” Opt. Eng. 50, 040501 (2011).
[CrossRef]

Opt. Express (1)

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 (6)

Fig. 1.
Fig. 1.

In Southwell grid model, the points for height estimation are at the same locations of those whose gradient data have been measured out.

Fig. 2.
Fig. 2.

True shape is designed as the ground truth, and its corresponding gradient data are analytically calculated in simulation to test the performance of the iterative compensation. (a) The true shape, and the corresponding true slopes in x-direction p=dz/dx (b) and in y-direction q=dz/dy (c).

Fig. 3.
Fig. 3.

Integration results with the traditional method and the proposed iterative compensation are compared, showing the validity of the iterations. (a) Height errors with no compensation with the traditional method, (b) after the first compensation, (c) after the second compensation, (d) after the third compensation, (e) standard deviations of height error before and after compensations, (f) one-line profile of the errors with the traditional method and proposed compensation approach, and (g) the reconstructed result after the third compensation.

Fig. 4.
Fig. 4.

Integrate gradient data with noise. (a) True shape to be reconstructed, (b) slope p=dz/dx, and (c) slope p=dz/dy.

Fig. 5.
Fig. 5.

A comparison is carried out between the traditional method and the proposed iterative compensation with the existence of noise on gradient data. (a) Height error before compensation, (b) height error after iterative compensation, and (c) profiles of height error in one line (y=0).

Fig. 6.
Fig. 6.

A comparison of the traditional and proposed methods is also carried out with experimental data from the fringe reflection technique. (a) Fringe patterns from fringe reflection technique, (b) slope in x-direction, (c) slope in y-direction, (d) integrated mirror surface by using proposed method, and (e) the shape difference between proposed method and the traditional least squares integration method.

Equations (8)

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

{pi,j+1+pi,j2=zi,j+1zi,jxi,j+1xi,j,i=1,,M,j=1,,N1qi+1,j+qi,j2=zi+1,jzi,jyi+1,jyi,j,i=1,,M1,j=1,,N,
{zi,j+1zi,j=12(pi,j+1+pi,j)(xi,j+1xi,j),i=1,,M,j=1,,N1zi+1,jzi,j=12(qi+1,j+qi,j)(yi+1,jyi,j),i=1,,M1,j=1,,N,
DZ=G,
D=[10010001001000010011100011000011][(M1)N+M(N1)]×MN,
Z=[z1,1z2,1zM,N]MN×1,
G=12[(p1,2+p1,1)(x1,2x1,1)(p1,3+p1,2)(x1,3x1,2)(pM,N+pM,N1)(xM,NxM,N1)(q2,1+q1,1)(y2,1y1,1)(q3,1+q2,1)(y3,1y2,1)(qM,N+qM1,N)(yM,NyM1,N)][(M1)N+M(N1)]×1.
z=0.3cos(0.4x2+2x)cos(0.4y2+2y)+0.7cos[(x3+y2)/4π],
z=3(1x)2·ex2(y+1)210(x5x3y5)·ex2y213e(x+1)2y2,

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