Abstract
This paper describes a new approach for regularising triangulated graphs. We commence by embedding the graph onto a manifold using the heat-kernel embedding. Under the embedding, each first-order cycle of the graph becomes a triangle. Our aim is to use curvature information associated with the edges of the graph to effect regularisation. Using the difference in Euclidean and geodesic distances between nodes under the embedding, we compute sectional curvatures associated with the edges of the graph. Using the Gauss Bonnet Theorem we compute the Gaussian curvature associated with each node from the sectional curvatures and through the angular excess of the geodesic triangles. Using the curvature information we perform regularisation with the advantage of not requiring the solution of a partial differential equation. We experiment with the resulting regularization process, and explore its effect on both graph matching and graph clustering.
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References
Bertalmio, M., Cheng, L.T., Osher, S., Sapiro, G.: Variational problems and partial differential equations on implicit surfaces. Journal of Computational Physics 174, 759–780 (2001)
Bougleux, S., Elmoataz, A.: Image smoothing and segmentation by graph regularization. LNCS, vol. 3656, pp. 745–752. Springer, Heidelberg (2005)
Boykov, Y., Huttenlocher, D.: A new bayesian framework for object recognition. In: Proceeding of IEEE Computer Society Conference on CVPR, vol. 2, pp. 517–523 (1999)
Chan, T., Osher, S., Shen, J.: The digital tv filter and nonlinear denoising. IEEE Trans. Image Process 10(2), 231–241 (2001)
Chan, T., Shen, J.: Variational restoration of non-flat image features: Models and algorithms. SIAM J. Appl. Math. 61, 1338–1361 (2000)
Cheng, L., Burchard, P., Merriman, B., Osher, S.: Motion of curves constrained on surfaces using a level set approach. Technical report, UCLA CAM Technical Report (00-32) (September 2000)
Chung, F.R.: Spectral graph theory. In: Proc. CBMS Regional Conf. Ser. Math., vol. 92, pp. 1–212 (1997)
Cox, T., Cox, M.: Multidimensional Scaling. Chapman-Hall, Boca Raton (1994)
Dubuisson, M., Jain, A.: A modified hausdorff distance for object matching, pp. 566–568 (1994)
ElGhawalby, H., Hancock, E.R.: Measuring graph similarity using spectral geometry. In: Campilho, A., Kamel, M.S. (eds.) ICIAR 2008. LNCS, vol. 5112, pp. 517–526. Springer, Heidelberg (2008)
Gauss, C.F.: Allgemeine Flächentheorie (Translated from Latin). W. Engelmann (1900)
Kimmel, R., Malladi, R., Sochen, N.: Images as embedding maps and minimal surfaces: Movies, color, texture, and volumetric medical images. International Journal of Computer Vision 39(2), 111–129 (2000)
Lim, B.P., Montenegro, J.F., Santos, N.L.: Eigenvalues estimates for the p-laplace operator on manifolds. arXiv:0808.2028v1 [math.DG], August 14 (2008)
Lopez-Perez, L., Deriche, R., Sochen, N.: The beltrami flow over triangulated manifolds. In: Sonka, M., Kakadiaris, I.A., Kybic, J. (eds.) CVAMIA/MMBIA 2004. LNCS, vol. 3117, pp. 135–144. Springer, Heidelberg (2004)
Luo, B., Wilson, R.C., Hancock, E.R.: Spectral embedding of graphs. Pattern Recogintion 36, 2213–2230 (2003)
Memoli, F., Sapiro, G., Osher, S.: Solving variational problems and partial differential equations, mapping into general target manifolds. Technical report, UCLA CAM Technical Report (02-04) (January 2002)
Osher, S., Shen, J.: Digitized pde method for data restoration. In: Anastassiou, E.G.A. (ed.) Analytical-Computational methods in Applied Mathematics, pp. 751–771. Chapman & Hall/CRC, New York (2000)
Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2001)
Sochen, N., Deriche, R., Lopez-Perez, L.: The beltrami flow over implicit manifolds. In: ICCV (2003)
Sochen, N., Deriche, R., Lopez-Perez, L.: Variational beltrami flows over manifolds. In: IEEE ICIP 2003, Barcelone (2003)
Sochen, N., Deriche, R., Lopez-Perez, L.: Variational beltrami flows over manifolds. Technical report, INRIA Resarch Report 4897 (June 2003)
Sochen, N., Kimmel, R.: Stereographic orientation diffusion. In: Proceedings of the 4th Int. Conf. on Scale-Space, Vancouver, Canada (October 2001)
Sochen, N., Kimmel, R., Malladi, R.: From high energy physics to low level vision. Report, LBNL, UC Berkeley, LBNL 39243, August, Presented in ONR workshop, UCLA, September 5 (1996)
Sochen, N., Kimmel, R., Malladi, R.: A general framework for low level vision. IEEE Trans. on Image Processing 7, 310–318 (1998)
Sochen, N., Zeevi, Y.: Representation of colored images by manifolds embedded in higher dimensional non-euclidean space. In: Proc. IEEE ICIP 1998, Chicago (1998)
Xiao, B., Hancock, E.R.: Heat kernel, riemannian manifolds and graph embedding. In: Fred, A., Caelli, T.M., Duin, R.P.W., Campilho, A.C., de Ridder, D. (eds.) SSPR&SPR 2004. LNCS, vol. 3138, pp. 198–206. Springer, Heidelberg (2004)
Young, G., Householder, A.S.: Disscussion of a set of points in terms of their mutual distances. Psychometrika 3, 19–22 (1938)
Zhou, D., Schlkopf, B.: Regularization on discrete spaces. In: Kropatsch, W.G., Sablatnig, R., Hanbury, A. (eds.) DAGM 2005. LNCS, vol. 3663, pp. 361–368. Springer, Heidelberg (2005)
Zhou, D., Schlkopf, B.: In: Chapelle, O., Schlkopf, B., Zien, A. (eds.) Semi-supervised learning, pp. 221–232 (2006)
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ElGhawalby, H., Hancock, E.R. (2009). Graph Regularisation Using Gaussian Curvature. In: Torsello, A., Escolano, F., Brun, L. (eds) Graph-Based Representations in Pattern Recognition. GbRPR 2009. Lecture Notes in Computer Science, vol 5534. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02124-4_24
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DOI: https://doi.org/10.1007/978-3-642-02124-4_24
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