Abstract
A basic requirement of scale-space representations in general is that of scale causality, which states that local extrema in the image should not be enhanced when resolution is diminished. We consider a special class of nonlinear scale-spaces consistent with this constraint, which can be linearised by a suitable isomorphism in the grey-scale domain so as to reproduce the familiar Gaussian scale-space. We consider instances in which nonlinear representations may be the preferred choice, as well as instances in which they enter by necessity. We also establish their relation to morphological scale-space representations based on a quadratic structuring function.
Similar content being viewed by others
References
Bloch, I. and Maître, H. 1995. Fuzzy mathematical morphologies: A comparative study. Pattern Recognition, 28(9):1341–1387.
Dorst, L. and van den Boomgaard, R. 1994. Morphological signal processing and the slope transform. Signal Processing, 38:79–98.
Florack, L.M.J. 1997. Image Structure, volume 10 of Computational Imaging and Vision Series. Kluwer Academic Publishers: Dordrecht, The Netherlands.
Florack, L.M.J., Maas, R., and Niessen, W.J. 1999. Pseudo-linear scale-space theory. International Journal of Computer Vision, 31(2/3):247–259.
Hendee, W.R. and Wells, P.N.T. (Eds.). 1993. The Perception of Visual Information. Springer-Verlag: New York.
Iijima, T. 1962. Basic theory on normalization of a pattern (in case of typical one-dimensional pattern). Bulletin of Electrical Laboratory, 26:368–388. (in Japanese).
Kimmel, R. and Sochen, N.A. 1999. Geometric-variational approach for color image enhancement and segmentation. In M. Nielsen, P. Johansen, O. F. Olsen, and J. Weickert (Eds.). Scale-Space Theories in Computer Vision: Proceedings of the Second International Conference, Scale-Space'99, Corfu, Greece, volume 1682 of Lecture Notes in Computer Science, pp. 294–305, Springer-Verlag: Berlin.
Kimmel, R., Sochen, N.A., and Malladi, R. 1997. From high energy physics to low level vision. In B.M. ter Haar Romeny and M.A. Vierqever (Eds.). Scale-Space Theory in Computer Vision: Proceedings of the First International Conference, Scale-Space'97, lltrecht, The Netherlands, volume 1252 of Lecture Notes in Computer Science, Springer-Verlag: Berlin.
Koenderink, J.J. 1984. The structure of images. Biological Cybernetics, 50:363–370.
Koenderink, J.J. and van Doorn, A.J. 1999. The structure of locally orderless images. International Journal of Computer Vision, 31(2/3):159–168.
Lax, P. D. 1973. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM.
Lindeberg, T. 1994. Scale-Space Theory in Computer Vision. The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, Dordrecht, The Netherlands.
Marr, D.C. and Hildreth, E.C. 1980. Theory of edge detection. Proceedings of the Royal Society of London, B, 207:187–217.
Misner, C.W., Thorne, K.S., and Wheeler, J.A. 1973. Gravitation. Freeman: San Francisco.
Nielsen, M., Florack, L.M.J., and Deriche, R. 1997. Regularization, scale-space and edge detection filters. Journal of Mathematical Imaging and Vision, 7(4):291–307.
Olver, P.J. 1986. Applications of Lie Groups to Differential Equations, volume 107 of Graduate Texts in Mathematics. Springer-Verlag.
Otsu, N. 1981. Mathematical studies on feature extraction in pattern recognition. Ph.D. thesis, Electrotechnical Laboratory, Ibaraki, Japan (in Japanese).
Smoller, J. 1994. Shock Waves and Reaction-Diffusion Equations. Grundlehren der mathematischenWissenschaften. Springer-Verlag, New York.
Sporring, J., Nielsen, M., Florack, L.M.J., and Johansen, P. (Eds.). 1997. Gaussian Scale-Space Theory, volume 8 of Computational Imaging and Vision Series. Kluwer Academic Publishers: Dordrecht, The Netherlands.
ter HaarRomeny, B.M., Florack, L.M.J., Koenderink, J.J., and Viergever, M.A. (Eds.). 1997. Scale-Space Theory in Computer Vision: Proceedings of the First International Conference, Scale-Space'97, Utrecht, The Netherlands, volume 1252 of Lecture Notes in Computer Science. Springer-Verlag, Berlin.
van den Boomgaard, R. 1992a. Mathematical morphology: extensions towards computer vision. Ph.D. thesis, University of Amsterdam.
van den Boomgaard, R. 1992b. The morphological equivalent of the Gauss convolution. Nieuw Archief voorWiskunde, 10(3):219–236.
van den Boomgaard, R. and Dorst, L. 1997. The morphological equivalent of the Gaussian scale-space. In Gaussian Scale-Space Theory, Vol. 8 of Computed Imaging and Vision Series, Sporring, M. Nielsen, L.M.J. Florack and P. Johnsen (Eds.), Kluwer Academic Publishers: Dordrecht, The Netherlands, pp. 203–220.
van den Boomgaard, R. and Smeulders, A.W.M. 1994. The morphological structure of images, the differential equations of morphological scale-space. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(11):1101–1113.
Witkin, A.P. 1983. Scale-space filtering. In Proceedings of the International Joint Conference on Artificial Intelligence, Karlsruhe, Germany, pp. 1019–1022.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Florack, L. Non-Linear Scale-Spaces Isomorphic to the Linear Case with Applications to Scalar, Vector and Multispectral Images. International Journal of Computer Vision 42, 39–53 (2001). https://doi.org/10.1023/A:1011185417206
Issue Date:
DOI: https://doi.org/10.1023/A:1011185417206