Abstract

Kriging is an estimation technique that has been proved useful in image processing since it behaves, under regular sampling, as a convolution. The uncertainty obtained with kriging has also been shown to behave as a convolution for the case of regular sampling. The convolution kernel for the uncertainty exclusively depends on the spatial correlation properties of the image. In this work we obtain, first, analytical expressions for the uncertainty of 1D images with noise using this convolution procedure. Then, we use this uncertainty to propose a new criterion for determining whether a 1D image with noise is correctly sampled.

© 2008 Optical Society of America

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References

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  1. W. K. Pratt, Digital Image Processing (Wiley, 1978).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. K. F. Cheung and R. J. Marks II, “Imaging sampling below the Nyquist density without aliasing,” J. Opt. Soc. Am. A 7, 92-105 (1990).
    [CrossRef]
  6. M. Pawlak and U. Stadmüller, “Recovering band-limited signals under noise,” IEEE Trans. Inf. Theor. 42, 1425-1438(1996).
    [CrossRef]
  7. M. Unser, “Sampling--50 years after Shannon,” Proc. IEEE 88, 569-587 (2000).
    [CrossRef]
  8. R. Christiensen, Linear Models for Multivariate, Time Series, and Spatial Data (Springer-Verlag, 1985).
  9. N. Cressie, Statistics for Spatial Data (Wiley, 1991).
  10. E. Bernabeu, I. Serroukh, and L. M. Sanchez-Brea, “A geometrical model for wire optical diffraction selected by experimental statistical analysis,” Opt. Eng. 38, 1319-1325(1999).
    [CrossRef]
  11. W. Y. V. Leung, P. J. Bones, and R. G. Lane, “Statistical interpolation of sampled images,” Opt. Eng. 40, 547-553 (2001).
    [CrossRef]
  12. D. Mainy, J. P. Nectoux, and D. Renard, “New developments in data processing of noisy images,” Mater. Charact. 36, 327-334 (1996).
    [CrossRef]
  13. L. M. Sanchez-Brea and E. Bernabeu, “Determination of the optimum sampling frequency of noisy images by spatial statistics,” Appl. Opt. 44, 3276-3283 (2005).
    [CrossRef] [PubMed]
  14. L. M. Sanchez-Brea and E. Bernabeu, “Uncertainty estimation by convolution using spatial statistics,” IEEE Trans. Image Process. 15, 3131-3137 (2006).
    [CrossRef]
  15. ISO, Guide to the Expression of the Uncertainty in Measurement (International Organization for Standardization, 1995).
  16. L. M. Sanchez-Brea and E. Bernabeu, “On the standard deviation in CCD cameras: a variogram-based technique for non-uniform images,” J. Electron. Imaging 11, 121-126(2002).
    [CrossRef]
  17. Mathematica 5, Wolfram Research, Inc., 100 Trade Center Drive Champaign, Ill., USA, pp. 61820-7237; http://www.wolfram.com.
  18. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113(2001).
  19. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901(2003).
    [CrossRef] [PubMed]
  20. J. M. Rico-García, L. M. Sanchez-Brea, and J. Alda, “Application of tomographic techniques to the spatial-response mapping of antenna-coupled detectors in the visible,” Appl. Opt. 47, 768-775 (2008).
    [CrossRef] [PubMed]

2008 (1)

2006 (1)

L. M. Sanchez-Brea and E. Bernabeu, “Uncertainty estimation by convolution using spatial statistics,” IEEE Trans. Image Process. 15, 3131-3137 (2006).
[CrossRef]

2005 (1)

2003 (2)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901(2003).
[CrossRef] [PubMed]

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

2002 (1)

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

2001 (2)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113(2001).

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

2000 (1)

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

1999 (1)

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

1996 (2)

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

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

1990 (1)

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]

Alda, J.

Bernabeu, E.

L. M. Sanchez-Brea and E. Bernabeu, “Uncertainty estimation by convolution using spatial statistics,” IEEE Trans. Image Process. 15, 3131-3137 (2006).
[CrossRef]

L. M. Sanchez-Brea and E. Bernabeu, “Determination of the optimum sampling frequency of noisy images by spatial statistics,” Appl. Opt. 44, 3276-3283 (2005).
[CrossRef] [PubMed]

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

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

Bones, P. J.

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

Cheung, K. F.

Christiensen, R.

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

Cressie, N.

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

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901(2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113(2001).

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113(2001).

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113(2001).

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, and R. G. Lane, “Statistical interpolation of sampled images,” Opt. Eng. 40, 547-553 (2001).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901(2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113(2001).

Leung, W. Y. V.

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

Mainy, D.

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

Pawlak, M.

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

Pratt, W. K.

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

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901(2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113(2001).

Renard, D.

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

Rico-García, J. M.

Sanchez-Brea, L. M.

J. M. Rico-García, L. M. Sanchez-Brea, and J. Alda, “Application of tomographic techniques to the spatial-response mapping of antenna-coupled detectors in the visible,” Appl. Opt. 47, 768-775 (2008).
[CrossRef] [PubMed]

L. M. Sanchez-Brea and E. Bernabeu, “Uncertainty estimation by convolution using spatial statistics,” IEEE Trans. Image Process. 15, 3131-3137 (2006).
[CrossRef]

L. M. Sanchez-Brea and E. Bernabeu, “Determination of the optimum sampling frequency of noisy images by spatial statistics,” Appl. Opt. 44, 3276-3283 (2005).
[CrossRef] [PubMed]

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

E. Bernabeu, I. Serroukh, and 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, and 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 and U. Stadmüller, “Recovering band-limited signals under noise,” IEEE Trans. Inf. Theor. 42, 1425-1438(1996).
[CrossRef]

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]

Appl. Opt. (2)

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light--theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109-113(2001).

IEEE Trans. Image Process. (1)

L. M. Sanchez-Brea and E. Bernabeu, “Uncertainty estimation by convolution using spatial statistics,” IEEE Trans. Image Process. 15, 3131-3137 (2006).
[CrossRef]

IEEE Trans. Inf. Theor. (1)

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

J. Electron. Imaging (1)

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

J. Opt. Soc. Am. A (1)

Mater. Charact. (1)

D. Mainy, J. P. Nectoux, and 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. (2)

E. Bernabeu, I. Serroukh, and 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, and R. G. Lane, “Statistical interpolation of sampled images,” Opt. Eng. 40, 547-553 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901(2003).
[CrossRef] [PubMed]

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]

Other (5)

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

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

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

Mathematica 5, Wolfram Research, Inc., 100 Trade Center Drive Champaign, Ill., USA, pp. 61820-7237; http://www.wolfram.com.

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

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Figures (5)

Fig. 1
Fig. 1

For the case of 1D sinusoidal image with amplitude A and noise s and for device resolution I = 0.001 A , several sampling frequencies are represented: (a) N EQ ( x ) using Eq. (10), when s / A = 0.25 ; (b)  σ ( x ) using Eq. (11), when s / A = 0.25 ; (c)  σ ( x ) using Eq. (11), when s / A = 0.025 . In all the figures, the black lines correspond to the technique proposed in this work, and gray lines are obtained with standard kriging.

Fig. 2
Fig. 2

(a)  N EQ ( x ) function and (b) uncertainty for a sinusoidal image with Δ x / p = 0.5 (low sampling limit). The solid lines correspond to the results by convolution using Eqs. (10, 11), respectively, and the dashed line corresponds to the results obtained with ordinary kriging technique. For this case, we have used s / A = 0.25 and I 0 / A = 0.001 .

Fig. 3
Fig. 3

(a)  N EQ ( x ) function for a sinusoidal image with Δ x / p = 0.1 (high sampling limit). The solid line corresponds to the results by convolution using Eq. (10) (fluctuating line), and the dashed line corresponds to the results obtained with ordinary kriging technique. For this case, we have used s / A = 0.25 , and I 0 / A = 0.001 . (b) Uncertainty using Eq. (11) for the case of (a).

Fig. 4
Fig. 4

(a) Maximum uncertainty σ max ( ν , s ) obtained using Eq. (16) and (b) contours of constant maximum uncertainty.

Fig. 5
Fig. 5

Maximum uncertainty σ max ( ν , s ) obtained using Eq. (16) for s = ( 0.1 , 0.2 , , 1 ) (solid lines) and uncertainty for the Nyquist criterion (dashed line).

Equations (20)

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σ 2 ( x ) = I 2 + s 2 N EQ ( x ) ,
N EQ ( x ) = D M ( x ) * ( x ) ,
DM ( x ) = N EQ ( x ) | N = 1 = γ ( 0 ) 2 γ ( | x | ) γ ( 0 ) ,
2 γ ( h ) = 1 | N ( h ) | N ( h ) ( Z i Z j ) 2 .
γ ( h ) = s 2 + A 2 sin 2 ( π h / p ) .
γ ( h ) = s 2 + ( h p / ( π A ) ) 2 .
DM ( h ) = 1 1 + 2 π 2 ( A h s p ) 2 ,
N EQ ( x ) = k = 1 1 + 2 π 2 ( A / s p ) 2 ( x x 0 k Δ x ) 2 ,
k = 1 a k 2 + b k + c = π d { cot [ π 2 ( b d 2 ) ] cot [ π 2 ( b + d 2 ) ] } ,
N EQ ( x ) = 1 2 2 s p A Δ x sinh ( 2 s p A Δ x ) cosh ( 2 s p A Δ x ) cos [ 2 π ( x x 0 Δ x ) ] ,
σ 2 ( x ) = I 2 + 2 A s Δ x p cosh ( 2 s p A Δ x ) cos ( 2 π x x 0 Δ x ) sinh ( 2 s p A Δ x ) = I 2 + 2 s 2 cosh ( 2 ν N SNR ) cos [ 2 π ν ( x x 0 ) ] ν N SNR sinh ( 2 ν N SNR ) ,
N EQ ( x ) 1 1 + 2 ( SNR / ν N ) 2 sin 2 [ π ( x x 0 ) / Δ x ] .
σ 2 ( x ) = I 2 + s 2 + 2 ( A ν N ) 2 sin 2 [ π ν ( x x 0 ) ] ,
N EQ ( x ) 1 2 ν N SNR .
σ 2 ( x ) = I 2 + 2 A s ν N ,
σ max 2 = I 2 + 2 s 2 SNR ν N coth ( ν N 2 SNR ) .
σ max 2 = I 2 + s 2 + 2 ( A v N ) 2 ,
v T 2 A σ max 2 I 2 s 2 v p .
σ max 2 = I 2 + 2 A s v N ,
v T = 2 A s σ max 2 I 2 v p .

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