Abstract

In optical metrology the final experimental result is normally an image acquired with a CCD camera. Owing to the sampling at the image, an interpolation is usually required. For determining the error in the measured parameters with that image, knowledge of the uncertainty at the interpolation is essential. We analyze how kriging, an estimator used in spatial statistics, can generate convolution kernels for filtering noise in regularly sampled images. The convolution kernel obtained with kriging explicitly depends on the spatial correlation and also on metrological conditions, such as the random fluctuations of the measured quantity, and the resolution of the measuring devices. Kriging, in addition, allows us to determine the uncertainty of the interpolation, and we have analyzed it in terms of the sampling frequency and the random fluctuations of the image, comparing it with Nyquist criterion. By use of kriging, it is possible to determine the optimum-required sampling frequency for a noisy image so that the uncertainty at interpolation is below a threshold value.

© 2005 Optical Society of America

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References

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  1. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).
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    [CrossRef]
  3. A. Stern, B. Javidi, “Sampling in the light of Wigner distribution,” J. Opt. Soc. Am. A 21, 360–366 (2004).
    [CrossRef]
  4. A. Stern, B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
    [CrossRef]
  5. C. E. Shannon, “Communication in presence of noise,” Proc. IRE 37, 20–21 (1949).
    [CrossRef]
  6. G. C. Holst, CCD Arrays, Cameras, and Displays (SPIE, Bellingham, Wash., 1996).
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    [CrossRef]
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    [CrossRef]
  9. M. Pawlak, U. Stadmüller, “Recovering band-limited signals under noise,” IEEE Trans. Inf. Theory 42, 1425–1438 (1996).
    [CrossRef]
  10. M. Unser, “Sampling—50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
    [CrossRef]
  11. P. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).
  12. ISO, Guide to the Expression of the Uncertainty in Measurement (International Organization for Standardization, Geneva, Switzerland, 1995).
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  14. N. Cressie, Statistics for Spatial Data (Wiley, New York, 1991).
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    [CrossRef]
  16. D. Mainy, J. P. Nectoux, D. Renard, “New developments in data processing of noisy images,” Mater. Charact. 36, 327–334 (1996).
    [CrossRef]
  17. W. Y. V. Leung, P. J. Bones, R. G. Lane, “Statistical interpolation of sampled images,” Opt. Eng. 40, 547–553 (2001).
    [CrossRef]
  18. T. D. Pham, M. Wagner, “Image enhancement by kriging and fuzzy sets,” Int. J. Pattern Recognit. 14, 1025–1038 (2000).
    [CrossRef]
  19. G. Y. Hu, R. F. O’Connell, “Analytical inversion of symmetric tridiagonal matrices,” J. Phys. A 29, 1511–1513 (1996).
    [CrossRef]
  20. G. S. Ammar, W. B. Gragg, “Superfast solution of real positive definite Toeplitz systems,” SIAM J. Matrix Anal. Appl. 9, 61–76 (1988).
    [CrossRef]
  21. W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, New York, 1992).
  22. J. P. Chilès, P. Delfiner, Geostatistics (Wiley, New York, 1999).
    [CrossRef]
  23. L. M. Sanchez-Brea, E. Bernabeu, “On the standard deviation in charge-coupled device cameras: a variogram-based technique for nonuniform images,” J. Electron. Imaging 11, 121–126 (2002).
    [CrossRef]
  24. G. Cloud, Optical Methods of Engineering Analysis (Cambridge U. Press, Cambridge, UK, 1998).

2004 (2)

A. Stern, B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

A. Stern, B. Javidi, “Sampling in the light of Wigner distribution,” J. Opt. Soc. Am. A 21, 360–366 (2004).
[CrossRef]

2003 (1)

H. P. Urbach, “Generalised sampling theorem for band-limited functions,” Math. Comput. Modell. 38, 133–140 (2003).
[CrossRef]

2002 (1)

L. M. Sanchez-Brea, E. Bernabeu, “On the standard deviation in charge-coupled device cameras: a variogram-based technique for nonuniform images,” J. Electron. Imaging 11, 121–126 (2002).
[CrossRef]

2001 (1)

W. Y. V. Leung, P. J. Bones, R. G. Lane, “Statistical interpolation of sampled images,” Opt. Eng. 40, 547–553 (2001).
[CrossRef]

2000 (2)

T. D. Pham, M. Wagner, “Image enhancement by kriging and fuzzy sets,” Int. J. Pattern Recognit. 14, 1025–1038 (2000).
[CrossRef]

M. Unser, “Sampling—50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
[CrossRef]

1999 (1)

E. Bernabeu, I. Serroukh, L. M. Sanchez-Brea, “A geometrical model for wire optical diffraction selected by experimental statistical analysis,” Opt. Eng. 38, 1319–1325 (1999).
[CrossRef]

1996 (3)

D. Mainy, J. P. Nectoux, D. Renard, “New developments in data processing of noisy images,” Mater. Charact. 36, 327–334 (1996).
[CrossRef]

M. Pawlak, U. Stadmüller, “Recovering band-limited signals under noise,” IEEE Trans. Inf. Theory 42, 1425–1438 (1996).
[CrossRef]

G. Y. Hu, R. F. O’Connell, “Analytical inversion of symmetric tridiagonal matrices,” J. Phys. A 29, 1511–1513 (1996).
[CrossRef]

1990 (1)

1988 (1)

G. S. Ammar, W. B. Gragg, “Superfast solution of real positive definite Toeplitz systems,” SIAM J. Matrix Anal. Appl. 9, 61–76 (1988).
[CrossRef]

1977 (1)

A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications,” Proc. IEEE 65, 1565–1596 (1977).
[CrossRef]

1949 (1)

C. E. Shannon, “Communication in presence of noise,” Proc. IRE 37, 20–21 (1949).
[CrossRef]

Ammar, G. S.

G. S. Ammar, W. B. Gragg, “Superfast solution of real positive definite Toeplitz systems,” SIAM J. Matrix Anal. Appl. 9, 61–76 (1988).
[CrossRef]

Bernabeu, E.

L. M. Sanchez-Brea, E. Bernabeu, “On the standard deviation in charge-coupled device cameras: a variogram-based technique for nonuniform images,” J. Electron. Imaging 11, 121–126 (2002).
[CrossRef]

E. Bernabeu, I. Serroukh, L. M. Sanchez-Brea, “A geometrical model for wire optical diffraction selected by experimental statistical analysis,” Opt. Eng. 38, 1319–1325 (1999).
[CrossRef]

Bevington, P.

P. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).

Bones, P. J.

W. Y. V. Leung, P. J. Bones, R. G. Lane, “Statistical interpolation of sampled images,” Opt. Eng. 40, 547–553 (2001).
[CrossRef]

Cheung, K. F.

Chilès, J. P.

J. P. Chilès, P. Delfiner, Geostatistics (Wiley, New York, 1999).
[CrossRef]

Christiensen, R.

R. Christiensen, Linear Models for Multivariate, Time Series, and Spatial Data (Springer-Verlag, Berlin, 1985).

Cloud, G.

G. Cloud, Optical Methods of Engineering Analysis (Cambridge U. Press, Cambridge, UK, 1998).

Cressie, N.

N. Cressie, Statistics for Spatial Data (Wiley, New York, 1991).

Delfiner, P.

J. P. Chilès, P. Delfiner, Geostatistics (Wiley, New York, 1999).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, New York, 1992).

Gragg, W. B.

G. S. Ammar, W. B. Gragg, “Superfast solution of real positive definite Toeplitz systems,” SIAM J. Matrix Anal. Appl. 9, 61–76 (1988).
[CrossRef]

Holst, G. C.

G. C. Holst, CCD Arrays, Cameras, and Displays (SPIE, Bellingham, Wash., 1996).

Hu, G. Y.

G. Y. Hu, R. F. O’Connell, “Analytical inversion of symmetric tridiagonal matrices,” J. Phys. A 29, 1511–1513 (1996).
[CrossRef]

Javidi, B.

A. Stern, B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

A. Stern, B. Javidi, “Sampling in the light of Wigner distribution,” J. Opt. Soc. Am. A 21, 360–366 (2004).
[CrossRef]

Jerri, A. J.

A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications,” Proc. IEEE 65, 1565–1596 (1977).
[CrossRef]

Lane, R. G.

W. Y. V. Leung, P. J. Bones, R. G. Lane, “Statistical interpolation of sampled images,” Opt. Eng. 40, 547–553 (2001).
[CrossRef]

Leung, W. Y. V.

W. Y. V. Leung, P. J. Bones, R. G. Lane, “Statistical interpolation of sampled images,” Opt. Eng. 40, 547–553 (2001).
[CrossRef]

Mainy, D.

D. Mainy, J. P. Nectoux, D. Renard, “New developments in data processing of noisy images,” Mater. Charact. 36, 327–334 (1996).
[CrossRef]

Marks, R. J.

Nectoux, J. P.

D. Mainy, J. P. Nectoux, D. Renard, “New developments in data processing of noisy images,” Mater. Charact. 36, 327–334 (1996).
[CrossRef]

O’Connell, R. F.

G. Y. Hu, R. F. O’Connell, “Analytical inversion of symmetric tridiagonal matrices,” J. Phys. A 29, 1511–1513 (1996).
[CrossRef]

Pawlak, M.

M. Pawlak, U. Stadmüller, “Recovering band-limited signals under noise,” IEEE Trans. Inf. Theory 42, 1425–1438 (1996).
[CrossRef]

Pham, T. D.

T. D. Pham, M. Wagner, “Image enhancement by kriging and fuzzy sets,” Int. J. Pattern Recognit. 14, 1025–1038 (2000).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

Press, W. H.

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, New York, 1992).

Renard, D.

D. Mainy, J. P. Nectoux, D. Renard, “New developments in data processing of noisy images,” Mater. Charact. 36, 327–334 (1996).
[CrossRef]

Sanchez-Brea, L. M.

L. M. Sanchez-Brea, E. Bernabeu, “On the standard deviation in charge-coupled device cameras: a variogram-based technique for nonuniform images,” J. Electron. Imaging 11, 121–126 (2002).
[CrossRef]

E. Bernabeu, I. Serroukh, L. M. Sanchez-Brea, “A geometrical model for wire optical diffraction selected by experimental statistical analysis,” Opt. Eng. 38, 1319–1325 (1999).
[CrossRef]

Serroukh, I.

E. Bernabeu, I. Serroukh, L. M. Sanchez-Brea, “A geometrical model for wire optical diffraction selected by experimental statistical analysis,” Opt. Eng. 38, 1319–1325 (1999).
[CrossRef]

Shannon, C. E.

C. E. Shannon, “Communication in presence of noise,” Proc. IRE 37, 20–21 (1949).
[CrossRef]

Stadmüller, U.

M. Pawlak, U. Stadmüller, “Recovering band-limited signals under noise,” IEEE Trans. Inf. Theory 42, 1425–1438 (1996).
[CrossRef]

Stern, A.

A. Stern, B. Javidi, “Sampling in the light of Wigner distribution,” J. Opt. Soc. Am. A 21, 360–366 (2004).
[CrossRef]

A. Stern, B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

Teukolski, S. A.

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, New York, 1992).

Unser, M.

M. Unser, “Sampling—50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
[CrossRef]

Urbach, H. P.

H. P. Urbach, “Generalised sampling theorem for band-limited functions,” Math. Comput. Modell. 38, 133–140 (2003).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, New York, 1992).

Wagner, M.

T. D. Pham, M. Wagner, “Image enhancement by kriging and fuzzy sets,” Int. J. Pattern Recognit. 14, 1025–1038 (2000).
[CrossRef]

IEEE Trans. Inf. Theory (1)

M. Pawlak, U. Stadmüller, “Recovering band-limited signals under noise,” IEEE Trans. Inf. Theory 42, 1425–1438 (1996).
[CrossRef]

Int. J. Pattern Recognit. (1)

T. D. Pham, M. Wagner, “Image enhancement by kriging and fuzzy sets,” Int. J. Pattern Recognit. 14, 1025–1038 (2000).
[CrossRef]

J. Electron. Imaging (1)

L. M. Sanchez-Brea, E. Bernabeu, “On the standard deviation in charge-coupled device cameras: a variogram-based technique for nonuniform images,” J. Electron. Imaging 11, 121–126 (2002).
[CrossRef]

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

J. Phys. A (1)

G. Y. Hu, R. F. O’Connell, “Analytical inversion of symmetric tridiagonal matrices,” J. Phys. A 29, 1511–1513 (1996).
[CrossRef]

Mater. Charact. (1)

D. Mainy, J. P. Nectoux, D. Renard, “New developments in data processing of noisy images,” Mater. Charact. 36, 327–334 (1996).
[CrossRef]

Math. Comput. Modell. (1)

H. P. Urbach, “Generalised sampling theorem for band-limited functions,” Math. Comput. Modell. 38, 133–140 (2003).
[CrossRef]

Opt. Eng. (3)

E. Bernabeu, I. Serroukh, L. M. Sanchez-Brea, “A geometrical model for wire optical diffraction selected by experimental statistical analysis,” Opt. Eng. 38, 1319–1325 (1999).
[CrossRef]

W. Y. V. Leung, P. J. Bones, R. G. Lane, “Statistical interpolation of sampled images,” Opt. Eng. 40, 547–553 (2001).
[CrossRef]

A. Stern, B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

Proc. IEEE (2)

A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications,” Proc. IEEE 65, 1565–1596 (1977).
[CrossRef]

M. Unser, “Sampling—50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
[CrossRef]

Proc. IRE (1)

C. E. Shannon, “Communication in presence of noise,” Proc. IRE 37, 20–21 (1949).
[CrossRef]

SIAM J. Matrix Anal. Appl. (1)

G. S. Ammar, W. B. Gragg, “Superfast solution of real positive definite Toeplitz systems,” SIAM J. Matrix Anal. Appl. 9, 61–76 (1988).
[CrossRef]

Other (9)

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, New York, 1992).

J. P. Chilès, P. Delfiner, Geostatistics (Wiley, New York, 1999).
[CrossRef]

P. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).

ISO, Guide to the Expression of the Uncertainty in Measurement (International Organization for Standardization, Geneva, Switzerland, 1995).

R. Christiensen, Linear Models for Multivariate, Time Series, and Spatial Data (Springer-Verlag, Berlin, 1985).

N. Cressie, Statistics for Spatial Data (Wiley, New York, 1991).

G. C. Holst, CCD Arrays, Cameras, and Displays (SPIE, Bellingham, Wash., 1996).

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

G. Cloud, Optical Methods of Engineering Analysis (Cambridge U. Press, Cambridge, UK, 1998).

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Figures (10)

Fig. 1
Fig. 1

(a) Estimation by kriging for a quantity with spatial dependence f(x) = sin(2πx) when it is sampled with a frequency υ = 2.5. Solid curve, kriging estimation; dashed curve, noiseless simulation; dotted curve, error bands; circles, simulated measurements; I, resolution of the measuring devices. Here s = 0.001, I0 = 0.001. (b) Estimated (thick curve) and real (thin curve) uncertainties. We can see that there is a strong fluctuation in uncertainty estimation.

Fig. 2
Fig. 2

(a) Estimation by kriging for a quantity with spatial dependence f(x) = sin(2πx) when it is sampled with a frequency υ = 30. Solid curve, kriging estimation; dashed curve, noiseless simulation; dotted curve, error bands; circles, simulated measurements; I, resolution of the measuring devices. The parameters for the measuring process are s = 0.001, I0 = 0.001. (b) Estimated (thick curve) and real (thin curve) uncertainties. Now the estimated uncertainty is much lower and does not present fluctuations.

Fig. 3
Fig. 3

(a) Estimation by kriging of a quantity with spatial dependence f(x) = sin(2πx) when it is sampled with a frequency υ = 30. Solid curve, kriging estimation; dashed curve, noiseless simulation; dotted curve, error bands; circles, simulated measurements; I, resolution of the measuring devices. Here s = 0.3, I0 = 0.1. (b) Estimated (thick curve) and real (thin curve) uncertainties. Owing to the sampling, uncertainty decreases considerably (approximately 0.15), but it is higher than resolution I0.

Fig. 4
Fig. 4

(a) Σ(ν, s ¯, Ī) parameter for the 1D signal f(x) = sin(2πx) when it is sampled with different frequencies and random fluctuations. (b) Contour.

Fig. 5
Fig. 5

Plot showing how Σ decreases with the sampling frequency for two values of the random fluctuations: s ¯ = 0.5 (thin curve) and s ¯ = 0 (thick curve). When s ¯ is small, Σ decreases strongly between υ = 2 and 5. (arb. u.)−1. When s ¯ = 0.5, Σ decreases more gradually. Dashed lines represent the uncertainty for Nyquist criterion.

Fig. 6
Fig. 6

(a) Σ(ν, s ¯, Ī) parameter for the 2D image f(x) = sin(2πx)sin(2πy) when it is sampled with different frequencies and random fluctuations. (b) Contour.

Fig. 7
Fig. 7

Plot showing how Σ decreases with sampling frequency for two values of the random fluctuations: s = 0.5 (thin curve) and s = 0 (thick curve). When s is small, Σ decreases strongly between υ = 2 and 5 (arb. u.)−1. When s = 0.5, Σ decreases more gradually. Dashed lines represent the uncertainty for the Nyquist criterion.

Fig. 8
Fig. 8

Σ(ν, s ¯, Ī) parameter for several values of s ¯ and Ī. Solid curves are estimated with Eq. (17), and dots are obtained with the general expression for kriging [Eq. (16)]. The function used to obtain this result is f(x) = sin(2πx), with a sampling frequency ν = 50.

Fig. 9
Fig. 9

(a) Two-dimensional image obtained with a CCD camera consisting of the fringe pattern of a 350-µm defect obtained with the shadow moiré technique. It can be observed that the image presents noise. (b) Same 2D image of (a) after it was processed with kriging. (c) Profile of (a) and (b) showing the experimental data (circles), interpolation (solid curve), and the error bars (dashed curve).

Fig. 10
Fig. 10

Profile of the convolution kernel Λ for the interpolation of Fig. 9(a) obtained with Eq. (8). Because we have considered the quantity to be isotropic, the convolution kernel presents revolution symmetry.

Equations (18)

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σ ( Q ¯ ) = [ I 2 + s 2 ( Q ) N ] 1 / 2 ,
2 γ ( h ) = E [ ( Z i Z j ) 2 ] = 1 N ( h ) i = 1 N ( h ) ( Z i Z j ) 2 ,
m ( x ) = E [ Z ( x ) ] = l = 0 p β l f l ( x ) ,
Z ( x ) = m ( x ) + e ( x ) ,
Z ( x ) = i = 1 N λ i ( x ) Z i ,
σ 2 ( x ) = E { [ Z ( x ) Z 0 ( x ) ] 2 }
i = 0 N λ i ( x ) f l ( x ) = 1 , l = 0 , , p .
λ T ( x ) = [ γ + FH g ] T Γ 1 ,
λ ( x ) = [ λ 1 ( x ) , , λ N ( x ) ] , γ = [ γ ( x x 1 ) , , γ ( x x N ) ] T , f = [ f 0 ( x ) , , f p ( x ) ] T , [ Γ ] i , j = γ ( x i x j ) , [ F ] l , j = f l ( x j ) , H = ( F T Γ 1 F ) 1 , g = ( f F T Γ 1 γ ) .
λ i ( x ) = Λ ( x ) * δ ( x x i ) ,
Z ( x ) = Λ ( x ) * Π ( x ) ,
σ 2 ( x ) = γ T Γ 1 γ g T H g .
γ γ ¯ = γ γ ( 0 ) 2 ,
Γ i , j Γ ¯ i , j = Γ i , j γ ( 0 ) δ i , j I i I j ,
σ 1 2 ( x ) = 2 γ ¯ 1 / Γ ¯ i , j 1 = 2 γ ( x ) γ ( 0 ) + I 0 2 ,
Z i = f ( x i ) + p 1 + p 2 ,
Σ ( ν , s ¯ , Ī ) = max [ | σ ¯ ( x ) | ] ν , s , I ,
Σ ( ν , s ¯ , Ī ) = [ ( ν , s ¯ , Ī = 0 ) 2 + Ī 2 ] 1 / 2 .

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