Abstract

In many industrial inspection systems, it is required to have a high-precision three-dimensional measurement of an object under test. A popular technique is phase-measuring profilometry. In this paper, we develop some phase-shifting algorithms (PSAs). We propose a novel smoothness constraint in a regularization framework; we call this the R-PSA method and show how to obtain the desired phase measure with an iterative procedure. Both the simulation and experimental results verify the efficacy of our algorithm compared with current multiframe PSAs for interferometric measurements.

© 2012 Optical Society of America

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References

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  1. A. N. Ng, E. Y. Lam, R. Chung, K. S. Fung, and W. Leung, “Reference-free machine vision inspection of semiconductor die images,” Int. J. Image Graphics 9, 133–152 (2009).
    [CrossRef]
  2. G. Zeng, Y. Matsushita, L. Quan, and H.-Y. Shum, “Interactive shape from shading,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 343–350.
  3. M. Ishihara and H. Sasaki, “High-speed 3D shape measurement using a nonscanning multiple-beam confocal imaging system,” Proc. SPIE 3478, 68–75 (1998).
    [CrossRef]
  4. D. Scharstein and R. Szeliski, “A taxonomy and evaluation of dense two-frame stereo correspondence algorithms,” Int. J. Comput. Vis. 47, 7–42 (2002).
    [CrossRef]
  5. D. Caspi, N. Kiryati, and J. Shamir, “Range imaging with adaptive color structured light,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 470–480 (1998).
    [CrossRef]
  6. X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
    [CrossRef]
  7. V. Srinivasan, H. Liu, and M. Halioua, “Automated phase-measuring profilometry: a phase mapping approach,” Appl. Opt. 24, 185–188 (1985).
    [CrossRef]
  8. J. Cheng, R. Chung, E. Y. Lam, and K. S. Fung, “Bit-pairing codification for binary pattern projection system,” in International Conference on Pattern Recognition (IEEE, 2006), Vol. 2, pp. 263–266.
  9. J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, and Y. Xu, “Optimization of bit-pairing codification with learning for 3D reconstruction,” Int. J. Image Graphics 7, 445–462(2007).
    [CrossRef]
  10. J. Salvi, J. Pagés, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827–849 (2004).
    [CrossRef]
  11. M. Dong, R. Chung, E. Y. Lam, and K. S. Fung, “Height inspection of wafer bumps without explicit 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 33, 112–121 (2010).
    [CrossRef]
  12. J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, F. Wang, and W. Leung, “Structure-light based sensing using a single fixed fringe grating: fringe boundary detection and 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 31, 19–31(2008).
    [CrossRef]
  13. V. Srinivasan, H. Liu, and M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23, 3105–3108 (1984).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. P. J. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
    [CrossRef]
  17. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef]
  18. Y. Shu, R. Chung, Z. Tan, J. Cheng, E. Y. Lam, K. S. Fung, and F. Wang, “Projection optics design for tilted projection of fringe pattern,” Opt. Eng. 47, 053002 (2008).
    [CrossRef]
  19. S. Boyd and L. Vandenberghe, Convex Optimization(Cambridge University, 2004).
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    [CrossRef]
  21. Å. Björck, Numerical Methods for Least Squares Problems(Society for Industrial and Applied Mathematics, 1996).
  22. A. Blake and A. Zisserman, Visual Reconstruction (MIT, 1987).

2010 (1)

M. Dong, R. Chung, E. Y. Lam, and K. S. Fung, “Height inspection of wafer bumps without explicit 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 33, 112–121 (2010).
[CrossRef]

2009 (1)

A. N. Ng, E. Y. Lam, R. Chung, K. S. Fung, and W. Leung, “Reference-free machine vision inspection of semiconductor die images,” Int. J. Image Graphics 9, 133–152 (2009).
[CrossRef]

2008 (2)

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, F. Wang, and W. Leung, “Structure-light based sensing using a single fixed fringe grating: fringe boundary detection and 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 31, 19–31(2008).
[CrossRef]

Y. Shu, R. Chung, Z. Tan, J. Cheng, E. Y. Lam, K. S. Fung, and F. Wang, “Projection optics design for tilted projection of fringe pattern,” Opt. Eng. 47, 053002 (2008).
[CrossRef]

2007 (1)

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, and Y. Xu, “Optimization of bit-pairing codification with learning for 3D reconstruction,” Int. J. Image Graphics 7, 445–462(2007).
[CrossRef]

2004 (1)

J. Salvi, J. Pagés, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827–849 (2004).
[CrossRef]

2002 (1)

D. Scharstein and R. Szeliski, “A taxonomy and evaluation of dense two-frame stereo correspondence algorithms,” Int. J. Comput. Vis. 47, 7–42 (2002).
[CrossRef]

2001 (1)

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
[CrossRef]

1998 (2)

D. Caspi, N. Kiryati, and J. Shamir, “Range imaging with adaptive color structured light,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 470–480 (1998).
[CrossRef]

M. Ishihara and H. Sasaki, “High-speed 3D shape measurement using a nonscanning multiple-beam confocal imaging system,” Proc. SPIE 3478, 68–75 (1998).
[CrossRef]

1995 (1)

1990 (1)

1988 (1)

1987 (1)

1985 (1)

1984 (1)

1982 (1)

C. C. Paige and M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Softw. 8, 43–71 (1982).
[CrossRef]

Ai, C.

Batlle, J.

J. Salvi, J. Pagés, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827–849 (2004).
[CrossRef]

Björck, Å.

Å. Björck, Numerical Methods for Least Squares Problems(Society for Industrial and Applied Mathematics, 1996).

Blake, A.

A. Blake and A. Zisserman, Visual Reconstruction (MIT, 1987).

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimization(Cambridge University, 2004).

Brophy, C. P.

Caspi, D.

D. Caspi, N. Kiryati, and J. Shamir, “Range imaging with adaptive color structured light,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 470–480 (1998).
[CrossRef]

Chen, W.

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
[CrossRef]

Cheng, J.

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, F. Wang, and W. Leung, “Structure-light based sensing using a single fixed fringe grating: fringe boundary detection and 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 31, 19–31(2008).
[CrossRef]

Y. Shu, R. Chung, Z. Tan, J. Cheng, E. Y. Lam, K. S. Fung, and F. Wang, “Projection optics design for tilted projection of fringe pattern,” Opt. Eng. 47, 053002 (2008).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, and Y. Xu, “Optimization of bit-pairing codification with learning for 3D reconstruction,” Int. J. Image Graphics 7, 445–462(2007).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, and K. S. Fung, “Bit-pairing codification for binary pattern projection system,” in International Conference on Pattern Recognition (IEEE, 2006), Vol. 2, pp. 263–266.

Chung, R.

M. Dong, R. Chung, E. Y. Lam, and K. S. Fung, “Height inspection of wafer bumps without explicit 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 33, 112–121 (2010).
[CrossRef]

A. N. Ng, E. Y. Lam, R. Chung, K. S. Fung, and W. Leung, “Reference-free machine vision inspection of semiconductor die images,” Int. J. Image Graphics 9, 133–152 (2009).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, F. Wang, and W. Leung, “Structure-light based sensing using a single fixed fringe grating: fringe boundary detection and 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 31, 19–31(2008).
[CrossRef]

Y. Shu, R. Chung, Z. Tan, J. Cheng, E. Y. Lam, K. S. Fung, and F. Wang, “Projection optics design for tilted projection of fringe pattern,” Opt. Eng. 47, 053002 (2008).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, and Y. Xu, “Optimization of bit-pairing codification with learning for 3D reconstruction,” Int. J. Image Graphics 7, 445–462(2007).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, and K. S. Fung, “Bit-pairing codification for binary pattern projection system,” in International Conference on Pattern Recognition (IEEE, 2006), Vol. 2, pp. 263–266.

de Groot, P. J.

Dong, M.

M. Dong, R. Chung, E. Y. Lam, and K. S. Fung, “Height inspection of wafer bumps without explicit 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 33, 112–121 (2010).
[CrossRef]

Eiju, T.

Fung, K. S.

M. Dong, R. Chung, E. Y. Lam, and K. S. Fung, “Height inspection of wafer bumps without explicit 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 33, 112–121 (2010).
[CrossRef]

A. N. Ng, E. Y. Lam, R. Chung, K. S. Fung, and W. Leung, “Reference-free machine vision inspection of semiconductor die images,” Int. J. Image Graphics 9, 133–152 (2009).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, F. Wang, and W. Leung, “Structure-light based sensing using a single fixed fringe grating: fringe boundary detection and 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 31, 19–31(2008).
[CrossRef]

Y. Shu, R. Chung, Z. Tan, J. Cheng, E. Y. Lam, K. S. Fung, and F. Wang, “Projection optics design for tilted projection of fringe pattern,” Opt. Eng. 47, 053002 (2008).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, and Y. Xu, “Optimization of bit-pairing codification with learning for 3D reconstruction,” Int. J. Image Graphics 7, 445–462(2007).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, and K. S. Fung, “Bit-pairing codification for binary pattern projection system,” in International Conference on Pattern Recognition (IEEE, 2006), Vol. 2, pp. 263–266.

Halioua, M.

Hariharan, P.

Ishihara, M.

M. Ishihara and H. Sasaki, “High-speed 3D shape measurement using a nonscanning multiple-beam confocal imaging system,” Proc. SPIE 3478, 68–75 (1998).
[CrossRef]

Kiryati, N.

D. Caspi, N. Kiryati, and J. Shamir, “Range imaging with adaptive color structured light,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 470–480 (1998).
[CrossRef]

Lam, E. Y.

M. Dong, R. Chung, E. Y. Lam, and K. S. Fung, “Height inspection of wafer bumps without explicit 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 33, 112–121 (2010).
[CrossRef]

A. N. Ng, E. Y. Lam, R. Chung, K. S. Fung, and W. Leung, “Reference-free machine vision inspection of semiconductor die images,” Int. J. Image Graphics 9, 133–152 (2009).
[CrossRef]

Y. Shu, R. Chung, Z. Tan, J. Cheng, E. Y. Lam, K. S. Fung, and F. Wang, “Projection optics design for tilted projection of fringe pattern,” Opt. Eng. 47, 053002 (2008).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, F. Wang, and W. Leung, “Structure-light based sensing using a single fixed fringe grating: fringe boundary detection and 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 31, 19–31(2008).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, and Y. Xu, “Optimization of bit-pairing codification with learning for 3D reconstruction,” Int. J. Image Graphics 7, 445–462(2007).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, and K. S. Fung, “Bit-pairing codification for binary pattern projection system,” in International Conference on Pattern Recognition (IEEE, 2006), Vol. 2, pp. 263–266.

Leung, W.

A. N. Ng, E. Y. Lam, R. Chung, K. S. Fung, and W. Leung, “Reference-free machine vision inspection of semiconductor die images,” Int. J. Image Graphics 9, 133–152 (2009).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, F. Wang, and W. Leung, “Structure-light based sensing using a single fixed fringe grating: fringe boundary detection and 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 31, 19–31(2008).
[CrossRef]

Liu, H.

Matsushita, Y.

G. Zeng, Y. Matsushita, L. Quan, and H.-Y. Shum, “Interactive shape from shading,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 343–350.

Ng, A. N.

A. N. Ng, E. Y. Lam, R. Chung, K. S. Fung, and W. Leung, “Reference-free machine vision inspection of semiconductor die images,” Int. J. Image Graphics 9, 133–152 (2009).
[CrossRef]

Oreb, B. F.

Pagés, J.

J. Salvi, J. Pagés, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827–849 (2004).
[CrossRef]

Paige, C. C.

C. C. Paige and M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Softw. 8, 43–71 (1982).
[CrossRef]

Quan, L.

G. Zeng, Y. Matsushita, L. Quan, and H.-Y. Shum, “Interactive shape from shading,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 343–350.

Salvi, J.

J. Salvi, J. Pagés, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827–849 (2004).
[CrossRef]

Sasaki, H.

M. Ishihara and H. Sasaki, “High-speed 3D shape measurement using a nonscanning multiple-beam confocal imaging system,” Proc. SPIE 3478, 68–75 (1998).
[CrossRef]

Saunders, M. A.

C. C. Paige and M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Softw. 8, 43–71 (1982).
[CrossRef]

Scharstein, D.

D. Scharstein and R. Szeliski, “A taxonomy and evaluation of dense two-frame stereo correspondence algorithms,” Int. J. Comput. Vis. 47, 7–42 (2002).
[CrossRef]

Shamir, J.

D. Caspi, N. Kiryati, and J. Shamir, “Range imaging with adaptive color structured light,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 470–480 (1998).
[CrossRef]

Shu, Y.

Y. Shu, R. Chung, Z. Tan, J. Cheng, E. Y. Lam, K. S. Fung, and F. Wang, “Projection optics design for tilted projection of fringe pattern,” Opt. Eng. 47, 053002 (2008).
[CrossRef]

Shum, H.-Y.

G. Zeng, Y. Matsushita, L. Quan, and H.-Y. Shum, “Interactive shape from shading,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 343–350.

Srinivasan, V.

Su, X.

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
[CrossRef]

Szeliski, R.

D. Scharstein and R. Szeliski, “A taxonomy and evaluation of dense two-frame stereo correspondence algorithms,” Int. J. Comput. Vis. 47, 7–42 (2002).
[CrossRef]

Tan, Z.

Y. Shu, R. Chung, Z. Tan, J. Cheng, E. Y. Lam, K. S. Fung, and F. Wang, “Projection optics design for tilted projection of fringe pattern,” Opt. Eng. 47, 053002 (2008).
[CrossRef]

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization(Cambridge University, 2004).

Wang, F.

Y. Shu, R. Chung, Z. Tan, J. Cheng, E. Y. Lam, K. S. Fung, and F. Wang, “Projection optics design for tilted projection of fringe pattern,” Opt. Eng. 47, 053002 (2008).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, F. Wang, and W. Leung, “Structure-light based sensing using a single fixed fringe grating: fringe boundary detection and 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 31, 19–31(2008).
[CrossRef]

Wyant, J. C.

Xu, Y.

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, and Y. Xu, “Optimization of bit-pairing codification with learning for 3D reconstruction,” Int. J. Image Graphics 7, 445–462(2007).
[CrossRef]

Zeng, G.

G. Zeng, Y. Matsushita, L. Quan, and H.-Y. Shum, “Interactive shape from shading,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 343–350.

Zisserman, A.

A. Blake and A. Zisserman, Visual Reconstruction (MIT, 1987).

ACM Trans. Math. Softw. (1)

C. C. Paige and M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Softw. 8, 43–71 (1982).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Electron. Packag. Manuf. (2)

M. Dong, R. Chung, E. Y. Lam, and K. S. Fung, “Height inspection of wafer bumps without explicit 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 33, 112–121 (2010).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, F. Wang, and W. Leung, “Structure-light based sensing using a single fixed fringe grating: fringe boundary detection and 3D reconstruction,” IEEE Trans. Electron. Packag. Manuf. 31, 19–31(2008).
[CrossRef]

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

D. Caspi, N. Kiryati, and J. Shamir, “Range imaging with adaptive color structured light,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 470–480 (1998).
[CrossRef]

Int. J. Comput. Vis. (1)

D. Scharstein and R. Szeliski, “A taxonomy and evaluation of dense two-frame stereo correspondence algorithms,” Int. J. Comput. Vis. 47, 7–42 (2002).
[CrossRef]

Int. J. Image Graphics (2)

A. N. Ng, E. Y. Lam, R. Chung, K. S. Fung, and W. Leung, “Reference-free machine vision inspection of semiconductor die images,” Int. J. Image Graphics 9, 133–152 (2009).
[CrossRef]

J. Cheng, R. Chung, E. Y. Lam, K. S. Fung, and Y. Xu, “Optimization of bit-pairing codification with learning for 3D reconstruction,” Int. J. Image Graphics 7, 445–462(2007).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

Y. Shu, R. Chung, Z. Tan, J. Cheng, E. Y. Lam, K. S. Fung, and F. Wang, “Projection optics design for tilted projection of fringe pattern,” Opt. Eng. 47, 053002 (2008).
[CrossRef]

Opt. Lasers Eng. (1)

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
[CrossRef]

Pattern Recogn. (1)

J. Salvi, J. Pagés, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827–849 (2004).
[CrossRef]

Proc. SPIE (1)

M. Ishihara and H. Sasaki, “High-speed 3D shape measurement using a nonscanning multiple-beam confocal imaging system,” Proc. SPIE 3478, 68–75 (1998).
[CrossRef]

Other (5)

G. Zeng, Y. Matsushita, L. Quan, and H.-Y. Shum, “Interactive shape from shading,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 343–350.

J. Cheng, R. Chung, E. Y. Lam, and K. S. Fung, “Bit-pairing codification for binary pattern projection system,” in International Conference on Pattern Recognition (IEEE, 2006), Vol. 2, pp. 263–266.

S. Boyd and L. Vandenberghe, Convex Optimization(Cambridge University, 2004).

Å. Björck, Numerical Methods for Least Squares Problems(Society for Industrial and Applied Mathematics, 1996).

A. Blake and A. Zisserman, Visual Reconstruction (MIT, 1987).

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Figures (8)

Fig. 1.
Fig. 1.

Setup for the three-dimensional phase reconstruction system.

Fig. 2.
Fig. 2.

Reconstruction results measured in radians from two sets of four-frame phase shifts. “Shift 1” is {0,π/2,π,3π/2} (displayed in blue), while “Shift 2” is {0,π/8,13π/8,15π/8} (displayed in red).

Fig. 3.
Fig. 3.

Reconstruction results measured in radians from the same PSA with and without blurring effect (displayed in magenta and blue, respectively).

Fig. 4.
Fig. 4.

Typical semiconductor sample called an LGA. (a) Image of an LGA sample; (b) golden pad in LGA.

Fig. 5.
Fig. 5.

Simulation experiments for comparing the improvement on accuracy between the PSA and the R-PSA. The reconstruction results are measured in micrometers. (a) Synthetic background intensity along the profile. (b)  Synthetic fringe contrast along the profile. (c) Cross section from the substrate to the golden pad. (d) Comparison of the results from PSA and R-PSA with four images.

Fig. 6.
Fig. 6.

Error ranges at different noise levels for PSA and R-PSA (units in micrometers). Results from (a) PSA3, (b) R-PSA3, (c) PSA4, (d) R-PSA4, (e) PSA5, (f) R-PSA5.

Fig. 7.
Fig. 7.

Surface reconstruction of a golden pad in an LGA sample. (a) Image of a golden pad. (b) Projected fringe image. (c) Fringe contrast image. (d) Profile from PSA3. (e) Profile after smoothing. (f) Profile from R-PSA3.

Fig. 8.
Fig. 8.

Surface reconstruction of a DPAK sample. (a) Image of a DPAK sample. (b) Profile from PSA4. (c) Profile after smoothing. (d) Profile from R-PSA4. (e) Comparison of the cross section on the column at x=13.

Tables (1)

Tables Icon

Table 1. Comparison of the Standard Deviation of the Reconstruction Error

Equations (26)

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

ϕh(x,y)=2π(tanα+tanβ)Ph(x,y),
ϕ(x,y)=ϕr(x,y)+ϕh(x,y).
I1(x,y)=B(x,y)+F(x,y)cosϕ(x,y)+N1(x,y),
Ik(x,y)=B(x,y)+F(x,y)cos[ϕ(x,y)+(k1)π/2]+Nk(x,y),
tanϕ(x,y)=I4(x,y)I2(x,y)I1(x,y)I3(x,y),
ϕh(x,y)=arctan{I4(x,y)I2(x,y)I1(x,y)I3(x,y)}ϕr(x,y).
Ek(x,y)=Ik(x,y){B(x,y)+F(x,y)cos[ϕ(x,y)+sk]}.
E(x,y)=k=1nEk2(x,y).
Ek(x,y)=Ik(x,y){B(x,y)+Fc(x,y)·(cossk)+Fs(x,y)·(sinsk)}.
M0=[1coss1sins11coss2sins21cossnsinsn],d0=[I1(x0,y0)I2(x0,y0)In(x0,y0)],andv0=[B(x0,y0)Fc(x0,y0)Fs(x0,y0)],
E(x0,y0)=M0v0d022
M0=[1cos0sin01cosπ2sinπ21cosπsinπ1cos3π2sin3π2]=[110101110101].
v0=(M0TM0)1M0Td0,
B(x0,y0)=14[I1(x0,y0)+I2(x0,y0)+I3(x0,y0)+I4(x0,y0)],
cosϕ(x0,y0)=I1(x0,y0)I3(x0,y0)2F(x0,y0),
sinϕ(x0,y0)=I4(x0,y0)I2(x0,y0)2F(x0,y0).
E(x,y)=k=1nEk2(x,y)+λ1R1(x,y)+λ2R2(x,y),
R1(x,y)=[F(x+1,y)F(x,y)]2,
R2(x,y)=[F(x,y+1)F(x,y)]2,
ET=x,yE(x,y)=x,y,kEk2(x,y)+λ1x,yR1(x,y)+λ2x,yR2(x,y).
R1(x,y)[Fc(x+1,y)Fc(x,y)]2+[Fs(x+1,y)Fs(x,y)]2,R2(x,y)[Fc(x,y+1)Fc(x,y)]2+[Fs(x,y+1)Fs(x,y)]2.
Dxv22=x,yR1(x,y)andDyv22=x,yR2(x,y).
v=(MTM+λ1DxTDx+λ2DyTDy)1MTd.
minimizex,y,k(Ik(x,y){B(x,y)+F(x,y)cos[ϕ(x,y)+sk]})2such thatcos2ϕ(x,y)+sin2ϕ(x,y)=1.
λ1(x,y)=C1C2+R1(x,y),
λ2(x,y)=C1C2+R2(x,y).

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