Abstract

The local model fitting (LMF) method is one of the useful single-shot surface profiling algorithms. The measurement principle of the LMF method relies on the assumption that the target surface is locally flat. Based on this assumption, the height of the surface at each pixel is estimated from pixel values in its vicinity. Therefore, we can estimate flat areas of the target surface precisely, whereas the measurement accuracy could be degraded in areas where the assumption is violated, because of a curved surface or sharp steps. In this paper, we propose to overcome this problem by weighting the contribution of the pixels according to the degree of satisfaction of the locally flat assumption. However, since we have no information on the surface profile beforehand, we iteratively estimate it and use this estimation result to determine the weights. This algorithm is named the iteratively-reweighted LMF (IRLMF) method. Experimental results show that the proposed algorithm works excellently.

© 2010 Optical Society of America

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References

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  1. J. H. Brunning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wave front measuring interferometer for testing optical surface and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef]
  2. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160(1982).
    [CrossRef]
  3. J. Kato, I. Yamaguchi, T. Nakamura, and S. Kuwashima, “Video-rate fringe analyzer based on phase-shifting electronic moiré patterns,” Appl. Opt. 36, 8403–8412 (1997).
    [CrossRef]
  4. D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
    [CrossRef]
  5. N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
    [CrossRef]
  6. M. Sugiyama, H. Ogawa, K. Kitagawa, and K. Suzuki, “Single-shot surface profiling by local model fitting,” Appl. Opt. 45, 7999–8005 (2006).
    [CrossRef]
  7. M. Takeda and T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
    [CrossRef]
  8. See http://www.cable-net.ne.jp/corp/torayins/SP-500.html for details.

2006 (1)

2005 (1)

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

1997 (1)

1996 (1)

M. Takeda and T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

1991 (1)

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

1982 (1)

1974 (1)

Abe, T.

M. Takeda and T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

Banyard, J. E.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Brangaccio, D. J.

Brock, N.

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Brunning, J. H.

Gallagher, J. E.

Hayes, J.

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Herriott, D. R.

Ina, H.

Kato, J.

Kimbrough, B.

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Kitagawa, K.

Kobayashi, S.

Kuwashima, S.

Millerd, J.

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Nakamura, T.

Nassar, N. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

North-Morris, M.

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Novak, M.

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Ogawa, H.

Rosenfeld, D. P.

Sugiyama, M.

Suzuki, K.

Takeda, M.

M. Takeda and T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160(1982).
[CrossRef]

Virdee, M. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

White, A. D.

Williams, D. C.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Wyant, J. C.

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Yamaguchi, I.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

M. Takeda and T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

Opt. Laser Technol. (1)

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147–150 (1991).
[CrossRef]

Proc. SPIE (1)

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak, and J. C. Wyant, “Dynamic interferometry,” Proc. SPIE 5875, 58750F (2005).
[CrossRef]

Other (1)

See http://www.cable-net.ne.jp/corp/torayins/SP-500.html for details.

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Figures (7)

Fig. 1
Fig. 1

Simulation results for an artificial bump.

Fig. 2
Fig. 2

Simulation results for an artificial sphere.

Fig. 3
Fig. 3

LMF with ideal window shape and WLMF with ideal weights for bump.

Fig. 4
Fig. 4

LMF with ideal window shape and WLMF with ideal weights for sphere.

Fig. 5
Fig. 5

Weights at the first iteration.

Fig. 6
Fig. 6

Actual measurement results of a real bump.

Fig. 7
Fig. 7

Weights at the first iteration (real bump).

Equations (14)

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g ( x , y ) a ( x , y ) + b ( x , y ) cos ( ϕ ( x , y ) + 2 π f x x + 2 π f y y ) ,
g ( x , y ) = a ( x , y ) + ξ c ( x , y ) φ c ( x , y ) + ξ s ( x , y ) φ s ( x , y ) ,
ξ c ( x , y ) : = b ( x , y ) cos ϕ ( x , y ) , φ c ( x , y ) : = cos ( 2 π f x x + 2 π f y y ) , ξ s ( x , y ) : = b ( x , y ) sin ϕ ( x , y ) , φ s ( x , y ) : = sin ( 2 π f x x + 2 π f y y ) .
g ¯ ( x , y ) : = a + ξ c φ c ( x , y ) + ξ s φ s ( x , y ) ,
ξ c : = b cos ϕ , ξ s : = b sin ϕ .
( a ^ , ξ ^ c , ξ ^ s ) : = arg min ( a , ξ c , ξ s ) i = 1 n ( g i g ¯ ( x i , y i ) ) 2 .
( a ^ , ξ ^ c , ξ ^ s ) = ( A A ) 1 A g ,
A = 1 φ c ( x 1 , y 1 ) φ s ( x 1 , y 1 ) 1 φ c ( x 2 , y 2 ) φ s ( x 2 , y 2 ) 1 φ c ( x n , y n ) φ s ( x n , y n ) , g = g 1 g 2 g n .
ϕ ^ : = arctan ( ξ ^ s / ξ ^ c ) + 2 m π ,
( a ˜ , ξ ˜ c , ξ ˜ s ) argmin ( a , ξ c , ξ s ) [ i = 1 n w i ( g i g ¯ ( x i , y i ) ) 2 ] .
( a ˜ , ξ ˜ c , ξ ˜ s ) = ( A WA ) 1 A Wg ,
W = diag ( w 1 , w 2 , , w n ) .
w i = c ( ( ϕ ( x 0 , y 0 ) ϕ ( x i , y i ) ) 2 + c ) 1 ,
w i = c ( ( ϕ ^ ( x 0 , y 0 ) ϕ ^ ( x i , y i ) ) 2 + c ) 1 ,

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