Determination of the optimum sampling frequency of noisy images by spatial statistics.

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.

[1]  Noel A Cressie,et al.  Statistics for Spatial Data. , 1992 .

[2]  B. Javidi,et al.  Analysis of practical sampling and reconstruction from Fresnel fields , 2004 .

[3]  A. J. Jerri.  Correction to "The Shannon sampling theorem—Its various extensions and applications: A tutorial review" , 1979 .

[4]  C.E. Shannon,et al.  Communication in the Presence of Noise , 1949, Proceedings of the IRE.

[5]  Eusebio Bernabeu,et al.  Geometrical model for wire optical diffraction selected by experimental statistical analysis , 1999 .

[6]  R. Christensen.  Linear Models for Multivariate, Time Series, and Spatial Data , 1997 .

[7]  S. Standard.  GUIDE TO THE EXPRESSION OF UNCERTAINTY IN MEASUREMENT , 2006 .

[8]  A. J. Jerri.  The Shannon sampling theorem—Its various extensions and applications: A tutorial review , 1977, Proceedings of the IEEE.

[9]  Ulrich Stadtmüller,et al.  Recovering band-limited signals under noise , 1996, IEEE Trans. Inf. Theory.

[10]  R. Marks,et al.  Imaging sampling below the Nyquist density without aliasing , 1990 .

[11]  P. R. Bevington,et al.  Data Reduction and Error Analysis for the Physical Sciences , 1969 .

[12]  M. Unser.  Sampling-50 years after Shannon , 2000, Proceedings of the IEEE.

[13]  Michael Wagner,et al.  Image Enhancement by Kriging and Fuzzy Sets , 2000, Int. J. Pattern Recognit. Artif. Intell..

[14]  Gerald C. Holst,et al.  CCD arrays, cameras, and displays , 1996 .

[15]  D. Renard,et al.  New developments in data processing of noisy images , 1996 .

[16]  R. F. O’Connell,et al.  Analytical inversion of symmetric tridiagonal matrices , 1996 .

[17]  W. Press,et al.  Numerical Recipes - Example Book (FORTRAN) - Second Edition , 1992 .

[18]  Philip J. Bones,et al.  Statistical interpolation of sampled images , 2001 .

[19]  Bahram Javidi,et al.  Sampling in the light of Wigner distribution. , 2004, Journal of the Optical Society of America. A, Optics, image science, and vision.

[20]  Gary Cloud,et al.  Optical Methods of Engineering Analysis , 1996 .

[21]  H. P. Urbach.  Generalised sampling theorem for band-limited functions , 2003 .

[22]  W. Gragg,et al.  Superfast solution of real positive definite toeplitz systems , 1988 .

[23]  Eusebio Bernabeu,et al.  On the standard deviation in charge-coupled device cameras: A variogram-based technique for nonuniform images , 2002, J. Electronic Imaging.