Abstract
Computational vision often needs to deal with derivatives ofdigital images. Such derivatives are not intrinsic properties ofdigital data; a paradigm is required to make them well-defined.Normally, a linear filtering is applied. This can be formulated interms of scale-space, functional minimization, or edge detectionfilters. The main emphasis of this paper is to connect these theoriesin order to gain insight in their similarities and differences. We donot want, in this paper, to take part in any discussion of how edgedetection must be performed, but will only link some of the current theories. We take regularization (or functional minimization) as astarting point, and show that it boils down to Gaussian scale-space ifwe require scale invariance and a semi-group constraint to besatisfied. This regularization implies the minimization of afunctional containing terms up to infinite order of differentiation.If the functional is truncated at second order, the Canny-Deriche filter arises. It is also shown that higher dimensional regularizationboils down to a rotated version of the one dimensional case, whenCartesian invariance is imposed and the image is vanishing at theborders. This means that the results from 1D regularization can beeasily generalized to higher dimensions. Finally we show how anefficient implementation of regularization of order n can be made byrecursive filtering using 2n multiplications and additions peroutput element without introducing any approximation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Schwartz, “Theorie des distributions,” Actualités scientifigues et industrielles, Publications de l’Institut de Mathématique de l’Université de Strasbourg, Vols. I, II, 1950–1951.
A.P. Witkin, “Scale-space filtering,” Proc. of IJCAI, Karlsruhe, Germany, 1983.
J.J. Koenderink, “The structure of images,” Biol. Cybern., Vol. 50, pp. 363–370, 1984.
L. Florack, “The syntactical structure of scalar images,” Ph.D. thesis, University of Utrecht, 1993.
A.N. Tikhonov and V.Y. Arseninn, Solution of Ill-posed Problems, V.H. Winston & Sons, John Wiley, 1977.
T. Poggio, H. Voorhees, and A. Yuille, “A regularized solution to edge detection,” A.I. Memo 883, M.I.T, May 1985.
J.F. Canny, “A computational approach to edge detection,” IEEE PAMI, Vol. 8, No.6, Nov. 1986.
R. Deriche, “Using Canny's criteria to derive a recursively implemented optimal edge detector,” Int. Journal of Computer Vision, Vol. 1, No.2, pp. 167–187, 1987.
M. Nielsen, L. Florack, and R. Deriche, “Regularization, scalespace, and edge detection filters,” Proc. of ECCV 96, Cambridge, UK, April 14–18, 1996, pp. 70–81.
J. Fourier, The Analytic Theory of Heat, New York: Dover Publications, 1995, Original Théori Analytique de la Chaleur, Paris, 1822.
T. Lindeberg, Scale-Space Theory in Computer Vision, Kluwer Academic Publishers: Netherlands, 1994.
T. Lindeberg, “On the axiomatic foundations of linear scalespace: Combining semi-group structure with causality vs. scale invariance,” Technical report, CVAP-159, Royal Institute of Technology, Stockholm, Sweden, 1994.
E.J. Pauwels, P. Fiddelaers, T. Moons, and L.J.V. Gool, “An extended class of scale-invariant and recursive scale space filters,” Katholieke Universiteit Leuven, Tech. Rep. KUL/ESAT/MI2/9316, 1994.
D. Hilbert, “Ueber die vollen invariantsystemen,” Math. Annalen, Vol. 42, pp. 313–373, 1893.
W.E.L. Grimson, From Images to Surfaces, MIT Press: Cambridge, Massachusetts, 1981.
R. Deriche, “Fast algorithms for low-level vision,” IEEE PAMI, Vol. 12, No.1, pp. 78–88, Jan. 1990.
R. Deriche, “Recursively implementing the Gaussian and its derivatives,” Proc. 2nd Int. Conference on Image Processing, Singapore, Sept. 7–11, 1992, pp. 263–267.
S. Castan, J. Zhao, and J. Shen, “Optimal filter for edge detection methods and results,” ECCV 90, Antibes, France, May 1990.
M. Unser, A. Aldroubi, and M. Eden, “Recursive regularization filters: Design, properties and applications,” IEEE PAMI, Vol. 13, No.3, March 1991.
M. Nielsen, L. Florack, and R. Deriche, “Regularization and scale-space,” Research Report 2352, INRIA, Sophia-Antipolis, France, 1994, URL: http://www.inria.fr/rapports/sophia/RR-2352.html.
P. Perona and J. Malik, “Scale space and edge detection using anisotropic diffusion,” IEEE PAMI, Vol. 12, pp. 429–439, 1990.
N. Nordström, “Biased anisotropic diffusion—A unified regularization and diffusion approach to edge detection,” Image and Visual Computing, No. 8, pp. 318–327, 1990.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nielsen, M., Florack, L. & Deriche, R. Regularization, Scale-Space, and Edge Detection Filters. Journal of Mathematical Imaging and Vision 7, 291–307 (1997). https://doi.org/10.1023/A:1008282127190
Issue Date:
DOI: https://doi.org/10.1023/A:1008282127190