Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T19:56:10.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic analysis of image acquisition, interpolation and scale-space smoothing

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Kalle Åström  and
Anders Heyden
Kalle Åström*
Affiliation:
Lund University
Anders Heyden*
Affiliation:
Lund University
*
∗ Postal address: Department of Mathematics, Lund University, Box 118, S-221 00 Lund, Sweden.
∗ Postal address: Department of Mathematics, Lund University, Box 118, S-221 00 Lund, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

In the high-level operations of computer vision it is taken for granted that image features have been reliably detected. This paper addresses the problem of feature extraction by scale-space methods. There has been a strong development in scale-space theory and its applications to low-level vision in the last couple of years. Scale-space theory for continuous signals is on a firm theoretical basis. However, discrete scale-space theory is known to be quite tricky, particularly for low levels of scale-space smoothing. The paper is based on two key ideas: to investigate the stochastic properties of scale-space representations and to investigate the interplay between discrete and continuous images. These investigations are then used to predict the stochastic properties of sub-pixel feature detectors.

The modeling of image acquisition, image interpolation and scale-space smoothing is discussed, with particular emphasis on the influence of random errors and the interplay between the discrete and continuous representations. In doing so, new results are given on the stochastic properties of discrete and continuous random fields. A new discrete scale-space theory is also developed. In practice this approach differs little from the traditional approach at coarser scales, but the new formulation is better suited for the stochastic analysis of sub-pixel feature detectors.

The interpolated images can then be analysed independently of the position and spacing of the underlying discretisation grid. This leads to simpler analysis of sub-pixel feature detectors. The analysis is illustrated for edge detection and correlation. The stochastic model is validated both by simulations and by the analysis of real images.

Keywords

Edge detectionsub-pixelerror analysisimage analysisfeature detection

MSC classification

Primary: 68U10: Image processing 60D05: Geometric probability and stochastic geometry 60G60: Random fields
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 4 , December 1999 , pp. 855 - 894
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adler, A. (1985). The Geometry of Random Fields. Wiley, New York.Google Scholar
Ahlfors, L. V. (1965). Complex Analysis. McGraw-Hill, New York.Google Scholar
Babaud, J., Witkin, A. P., Baudin, M. and Duda, R. O. (1986). Uniqueness of the Gaussian kernel for scale-space filtering. IEEE Trans. Pattern Analysis and Machine Intelligence 8, 26–33.Google Scholar
Bose, N. K. (1985). Digital Filters, Theory and Applications. North-Holland, Amsterdam.Google Scholar
Canny, F. (1986). A computational approach to edge detection. IEEE Trans. Pattern Analysis and Machine Intelligence 8, 676–698.Google ScholarPubMed
Cramér, H. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
Cressie, N. A. C. (1991). Statistics for Spatial Data. Wiley, New York.Google Scholar
De Michelli, E., Caprile, B., Ottonello, P. and Torre, V. (1989). Localization and noise in edge detection. IEEE Trans. Pattern Analysis and Machine Intelligence 10, 1106–1117.Google Scholar
Deriche, R. (1987). Using Canny's criteria to derive an optimal edge detector recursively implemented. Int. J. Computer Vision 1, 167–187.Google Scholar
Faugeras, O. (1993). Three-Dimensional Computer Vision. MIT Press, Cambridge, MA.Google Scholar
Hardy, G. H., Littlewood, J. E. and Pólya, G. (1959). Inequalities. 3rd edn., Cambridge University Press, Cambridge.Google Scholar
Hecht, E. (1987). Optics. Addison-Wesley, Reading, MA.Google Scholar
Hildreth, E. and Marr, D. (1980). Theory of edge detection. Proc. Roy. Soc. Lond. 207, 187–217.Google Scholar
Iijima, T. (1980). Basic theory on normalization of a pattern (in case of typical one-dimensional pattern). Bull. Elec. Laboratory 26, 368–388. (In Japanese.)Google Scholar
Koenderink, J. J. (1984). The structure of images. Biol. Cybernetics 50, 363–370.Google Scholar
Kotelnikov, V. A. (1933). On the transmission capacity of ‘ether’. In Proc. First All-union Conference on Questions of Communications.Google Scholar
Kundur, D., and Hatsinakos, D. (1996). Blind image deconvolution. IEEE Signal Processing Magazine, pp. 43–64.Google Scholar
Lindeberg, T. (1994). Scale-Space Theory in Computer Vision. Kluwer, Dordrecht.Google Scholar
Lindgren, G. and Rychlik., I. (1995). How reliable are contour curves—confidence sets for level contours. Bernoulli 1, 301–319.CrossRefGoogle Scholar
Marr, D. (1982). Vision. W. H. Freeman, New York.Google Scholar
Nalwa, V. S. and Binford, T. O. (1986). On detecting edges. IEEE Trans. Pattern Analysis and Machine Intelligence 8, 699–714.CrossRefGoogle ScholarPubMed
Nyquist, H. (1928). Certain topics in telegraph transmission theory. AIEE Trans. 47, 617–644.Google Scholar
Perona, P. and Malik, J. (1990). Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Analysis and Machine Intelligence 12, 629–639.Google Scholar
Pratt, B. (1978). Digital Image Processing. Wiley-Interscience, New Jersey.Google Scholar
Sapiro, G. and Tannenbaum, A. (1994). On affine invariant scale-space. Int. J. Computer Vision 11, 25–44. J. Func. Anal . 119, 79–120.Google Scholar
Shannon, C. E. (1949). Communication in the presence of noise. Proc. IRE 37.CrossRefGoogle Scholar
Torre, V. and Poggio, A. (1986). On edge detection. IEEE Trans. Pattern Analysis and Machine Intelligence 8, 147–163.Google Scholar
Witkin, A. P. (1983). Scale-space filtering. In Proc. Eighth International Joint Conference on Artificial Intelligence, Karlsruhe, West Germany, ed. Bunby, A.. William Kaufman, Los Altos, pp. 1019–1022.Google Scholar