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.
The author gratefully acknowledges essential contributions by Hans Meine and Peer Stelldinger and many fruitful discussions with Bernd Neumann, Hans-Siegfried Stiehl, and Fred Hamprecht.
Chapter PDF
Similar content being viewed by others
Keywords
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
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)
Bracewell, R.N.: The Fourier Transform and its Applications. McGraw-Hill, New York (1978)
Canny, J.: A Computational Approach to Edge Detection. IEEE Trans. Pattern Analysis and Machine Intelligence 8(6), 679–698 (1986)
Deriche, R., Giraudon, G.: A computational approach for corner and vertex detection. Intl. Journal of Computer Vision 10(2), 101–124 (1993)
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)
Klette, R., Rosenfeld, A.: Digital Geometry. Elsevier, Amsterdam (2004)
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)
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)
Pavlidis, T.: Algorithms for Graphics and Image Processing. Computer Science Press, Rockville (1982)
Rohr, K.: Localization Properties of Direct Corner Detectors. Journal of Mathematical Imaging and Vision 4, 139–150 (1994)
Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, New York (1982)
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)
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)
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)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Köthe, U. (2008). What Can We Learn from Discrete Images about the Continuous World?. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds) Discrete Geometry for Computer Imagery. DGCI 2008. Lecture Notes in Computer Science, vol 4992. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79126-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-540-79126-3_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79125-6
Online ISBN: 978-3-540-79126-3
eBook Packages: Computer ScienceComputer Science (R0)
