OSA's Digital Library

Applied Optics

Applied Optics


  • Vol. 44, Iss. 16 — Jun. 1, 2005
  • pp: 3276–3283

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

Luis Miguel Sanchez-Brea and Eusebio Bernabeu  »View Author Affiliations

Applied Optics, Vol. 44, Issue 16, pp. 3276-3283 (2005)

View Full Text Article

Enhanced HTML    Acrobat PDF (570 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(040.1520) Detectors : CCD, charge-coupled device
(100.2650) Image processing : Fringe analysis
(100.2960) Image processing : Image analysis
(110.4280) Imaging systems : Noise in imaging systems
(120.3940) Instrumentation, measurement, and metrology : Metrology

Original Manuscript: July 29, 2004
Revised Manuscript: November 29, 2004
Manuscript Accepted: November 29, 2004
Published: June 1, 2005

Luis Miguel Sanchez-Brea and Eusebio Bernabeu, "Determination of the optimum sampling frequency of noisy images by spatial statistics," Appl. Opt. 44, 3276-3283 (2005)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).
  2. H. P. Urbach, “Generalised sampling theorem for band-limited functions,” Math. Comput. Modell. 38, 133–140 (2003). [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).
  7. A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications,” Proc. IEEE 65, 1565–1596 (1977). [CrossRef]
  8. K. F. Cheung, R. J. Marks, “Imaging sampling below the Nyquist density without aliasing,” J. Opt. Soc. Am. A 7, 92–105 (1990). [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).
  13. R. Christiensen, Linear Models for Multivariate, Time Series, and Spatial Data (Springer-Verlag, Berlin, 1985).
  14. N. Cressie, Statistics for Spatial Data (Wiley, New York, 1991).
  15. 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]
  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).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited