Abstract
In this paper, we show how the heat-kernel can be used to construct a scale-space for image smoothing and edge detection. We commence from an affinity weight matrix computed by exponentiating the difference in pixel grey-scale and distance. From the weight matrix, we compute the graph Laplacian. Information flow across this weighted graph-structure with time is captured by the heat-equation, and the solution, i.e. the heat kernel, is found by exponentiating the Laplacian eigen-system with time. Our scale-space is constructed by varying the time parameter of the heat-kernel. The larger the time the greater the the amount of information flow across the graph. The method has the effect of smoothing within regions, but does not blur region boundaries. Moreover, the boundaries do not move with time and this overcomes one of the problems with Gaussian scale-space. We illustrate the effectiveness of the method for image smoothing and edge detection.
Keywords
- Heat Kernel
- Coarse Scale
- Laplacian Matrix
- Discrete Image
- Degree Matrix
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.
Chapter PDF
References
Canny, J.: A computational approach to edge detection. IEEE Trans. Pattern Anal. and Machine Intell. 8(6), 679–698 (1986)
Chung, F.R.K., Yau, S.-T.: Discrete green’s functions. J. Combin. Theory Ser., 191–214 (2000)
Geman, S., Geman, D.: Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans. Pattern Anal. and Machine Intell. 6(6), 721–741 (1984)
Gidas, B.: A renormalization group approach to image processing problems. IEEE Trans. Pattern Anal. and Machine Intell. 11(2), 164–180 (1989)
Golub, G.H., Van Loan, C.F.: Matrix Computations. John Hopkins Press, Baltimore (1989)
Hummel, R.: The scale-space formulation of pyramid data structures. In: Uhr, L. (ed.) Parallel Computer Vision. Academic Press, London (1987)
Babaud, J., Witkin, A., Baudin, M., Duda, R.: Uniqueness of the gaussian kernel for scale-space filtering. IEEE Trans. Pattern Anal. and Machine Intell. 8(1), 26–33 (1986)
Koenderink, J.: The structure of images. Biological Cybernetics 50, 363–370 (1984)
Kondor, R., Lafferty, J.: Diffusion kernels on graphs and other discrete structures. In: 19th Intl. Conf. on Machine Learning (ICML) [ICM 2002] (2002)
Lozano, M., Escolano, F.: A significant improvement of softassign with diffusion kernels. In: Fred, A., Caelli, T.M., Duin, R.P.W., Campilho, A.C., de Ridder, D. (eds.) SSPR&SPR 2004. LNCS, vol. 3138, pp. 76–84. Springer, Heidelberg (2004)
Meila, M., Shi, J.: A random walks view of spectral segmentation. In: Proceedings of AI and Statistics, AISTATS (2001)
Perona, P., Malik, J.: Scale-space and edge dectection using anisotropic diffusion. IEEE Trans. Pattern Anal. and Machine Intell. 12(7), 629–639 (1990)
Witkin, A.P.: Scale-space filtering. In: 8th Int. Joint Conf. on Artificial Intelligence, Karlsruhe, Germany, pp. 1019–1021 (1983)
Scholkopf, M., Smola, A.: Learning with kernels. MIT Press, Cambridge (2001)
Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(8), 888–905 (2000)
Wilson, R., Spann, M.: Finite prolate spheroidal sequences and their application ii: Image feature description and segmentation. IEEE Trans. Pattern Anal. and Machine Intell. 10(2), 193–203 (1988)
Yuille, A., Poggio, T.: Scaling theorems for zero crossings. IEEE Trans. Pattern Anal. and Machine Intell. 8(1), 15–25 (1986)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zhang, F., Hancock, E.R. (2005). Image Scale-Space from the Heat Kernel. In: Sanfeliu, A., Cortés, M.L. (eds) Progress in Pattern Recognition, Image Analysis and Applications. CIARP 2005. Lecture Notes in Computer Science, vol 3773. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11578079_20
Download citation
DOI: https://doi.org/10.1007/11578079_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-29850-2
Online ISBN: 978-3-540-32242-9
eBook Packages: Computer ScienceComputer Science (R0)
Publish with us
-
Published in cooperation with
http://www.iapr.org/
