Abstract
In this paper, we describe the use of Riemannian geometry, and in particular the relationship between the Laplace-Beltrami operator and the graph Laplacian, for the purposes of embedding a graph onto a Riemannian manifold. Using the properties of Jacobi fields, we show how to compute an edge-weight matrix in which the elements reflect the sectional curvatures associated with the geodesic paths between nodes on the manifold. We use the resulting edge-weight matrix to embed the nodes of the graph onto a Riemannian manifold of constant sectional curvature. With the set of embedding coordinates at hand, the graph matching problem is cast as that of aligning pairs of manifolds subject to a geometric transformation. We illustrate the utility of the method on image matching using the COIL database.
National ICT Australia is funded by the Australian Government’s Backing Australia’s Ability initiative, in part through the Australian Research Council.
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© 2006 Springer-Verlag Berlin Heidelberg
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Robles-Kelly, A., Hancock, E.R. (2006). Point Pattern Matching Via Spectral Geometry. In: Yeung, DY., Kwok, J.T., Fred, A., Roli, F., de Ridder, D. (eds) Structural, Syntactic, and Statistical Pattern Recognition. SSPR /SPR 2006. Lecture Notes in Computer Science, vol 4109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11815921_50
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DOI: https://doi.org/10.1007/11815921_50
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37236-3
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