What Can We Learn from Discrete Images about the Continuous World?

  • Ullrich Köthe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4992)

Abstract

Image analysis attempts to perceive properties of the continuous real world by means of digital algorithms. Since discretization discards an infinite amount of information, it is difficult to predict if and when digital methods will produce reliable results. This paper reviews theories which establish explicit connections between the continuous and digital domains (such as Shannon’s sampling theorem and a recent geometric sampling theorem) and describes some of their consequences for image analysis. Although many problems are still open, we can already conclude that adherence to these theories leads to significantly more stable and accurate algorithms.

Keywords

Point Spread Function Homotopy Type Sampling Theorem Discrete Image Noise Standard Deviation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Bouma, H., Vilanova, A., van Vliet, L.J., Gerritsen, F.A.: Correction for the Dislocation of Curved Surfaces Caused by the PSF in 2D and 3D CT Images. IEEE Trans. Pattern Analysis and Machine Intelligence 27(9), 1501–1507 (2005)CrossRefGoogle Scholar
  2. 2.
    Bracewell, R.N.: The Fourier Transform and its Applications. McGraw-Hill, New York (1978)MATHGoogle Scholar
  3. 3.
    Canny, J.: A Computational Approach to Edge Detection. IEEE Trans. Pattern Analysis and Machine Intelligence 8(6), 679–698 (1986)CrossRefGoogle Scholar
  4. 4.
    Deriche, R., Giraudon, G.: A computational approach for corner and vertex detection. Intl. Journal of Computer Vision 10(2), 101–124 (1993)CrossRefGoogle Scholar
  5. 5.
    Förstner, W.: Image Preprocessing for Feature Extraction in Digital Intensity, Color and Range Images. In: Proc. Summer School on Data Analysis and the Statistical Foundations of Geomatics. Lecture Notes in Earth Science, Springer, Berlin (1999)Google Scholar
  6. 6.
    Klette, R., Rosenfeld, A.: Digital Geometry. Elsevier, Amsterdam (2004)MATHGoogle Scholar
  7. 7.
    Köthe, U.: Edge and Junction Detection with an Improved Structure Tensor. In: Michaelis, B., Krell, G. (eds.) DAGM 2003. LNCS, vol. 2781, pp. 25–32. Springer, Heidelberg (2003)Google Scholar
  8. 8.
    Köthe, U., Stelldinger, P., Meine, H.: Provably Correct Edgel Linking and Subpixel Boundary Reconstruction. In: Franke, K., Müller, K.-R., Nickolay, B., Schäfer, R. (eds.) DAGM 2006. LNCS, vol. 4174, pp. 81–90. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Pavlidis, T.: Algorithms for Graphics and Image Processing. Computer Science Press, Rockville (1982)Google Scholar
  10. 10.
    Rohr, K.: Localization Properties of Direct Corner Detectors. Journal of Mathematical Imaging and Vision 4, 139–150 (1994)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, New York (1982)MATHGoogle Scholar
  12. 12.
    Stelldinger, P., Köthe, U., Meine, H.: Topologically Correct Image Segmentation Using Alpha Shapes. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 542–554. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Unser, M., Aldroubi, A., Eden, M.: B-Spline Signal Processing. IEEE Trans. Signal Processing 41(2), 821–833 (part I), 834–848 (part II) (1993)Google Scholar
  14. 14.
    Weickert, J., Scharr, H.: A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance. J. Visual Comm. Image Repr. 13(1/2), 103–118 (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ullrich Köthe
    • 1
  1. 1.Multi-Dimensional Image Processing GroupUniversity of HeidelbergGermany

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