Abstract
In this paper, we describe the use of graph-spectral techniques and their relationship to Riemannian geometry for the purposes of segmentation and grouping. We pose the problem of segmenting a set of tokens as that of partitioning the set of nodes in a graph whose edge weights are given by the geodesic distances between points in a manifold. To do this, we commence by explaining the relationship between the graph Laplacian, the incidence mapping of the graph and a Gram matrix of scalar products. This treatment permits the recovery of the embedding coordinates in a closed form and opens up the possibility of improving the segmentation results by modifying the metric of the space in which the manifold is defined. With the set of embedding coordinates at hand, we find the partition of the embedding space which maximises both, the inter-cluster distance and the intra-cluster affinity. The utility of the method for purposes of grouping is illustrated on a set of shape silhouettes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. In: Neural Information Processing Systems, vol. 14, pp. 634–640 (2002)
Borg, I., Groenen, P.: Modern Multidimensional Scaling, Theory and Applications. Springer Series in Statistics. Springer, Heidelberg (1997)
Chavel, I.: Riemannian Geometry: A Modern Introduction. Cambridge University Press, Cambridge (1995)
Fiedler, M.: A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czech Math. Journal (25), 619–633 (1975)
Gower, J.C.: Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53, 325–328 (1966)
Ham, J., Lee, D.D., Mika, S., Scholkopf, B.: A kernel view of the dimensionality reduction of manifolds. In: Proc. Int. Conf. Machine Learning, pp. 369–376 (2004)
Luo, B., Robles-Kelly, A., Torsello, A., Wilson, R.C., Hancock, E.R.: A probabilistic framework for graph clustering. In: EEE International Conference on Computer Vision and Pattern Recognition, vol. I, pp. 912–919 (2001)
Perona, P., Freeman, W.T.: Factorization approach to grouping. In: Burkhardt, H.-J., Neumann, B. (eds.) ECCV 1998. LNCS, vol. 1406, pp. 655–670. Springer, Heidelberg (1998)
Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000)
Sarkar, S., Boyer, K.L.: Quantitative measures of change based on feature organization: Eigenvalues and eigenvectors. Computer Vision and Image Understanding 71(1), 110–136 (1998)
Scott, G.L., Longuet-Higgins, H.C.: Feature grouping by relocalisation of eigenvectors of the proximity matrix. In: British Machine Vision Conference, pp. 103–108 (1990)
Shi, J., Malik, J.: Normalized cuts and image segmentations. In: Proc. of the IEEE Conf. on Comp. Vision and Pattern Recognition, pp. 731–737 (1997)
Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)
Torgerson, W.S.: Multidimensional scaling I: Theory and method. Psychometrika 17, 401–419 (1952)
Young, G., Householder, A.S.: Discussion of a set of points in terms of their mutual distances. Psychometrika 3, 19–22 (1938)
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
Robles-Kelly, A. (2005). Segmentation via Graph-Spectral Methods and Riemannian Geometry. In: Gagalowicz, A., Philips, W. (eds) Computer Analysis of Images and Patterns. CAIP 2005. Lecture Notes in Computer Science, vol 3691. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11556121_81
Download citation
DOI: https://doi.org/10.1007/11556121_81
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28969-2
Online ISBN: 978-3-540-32011-1
eBook Packages: Computer ScienceComputer Science (R0)