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• W42448260 abstract "The eigen-structures (eigenvalues and eigenfunctions) of the Laplace-Beltrami operator have been widely used in a broad range of application fields that include mesh smoothing, compression, editing, shape segmentation, matching, and parametrization, among others. While the Laplace operator is defined (mathematically) for a smooth domain, the underlying manifold is often approximated by a discrete mesh. Hence, the spectral structure of the manifold Laplacian is estimated from some discrete Laplace operator constructed from this mesh.Recently, several different discretizations have been proposed, each with its own advantages and limitations. Although the eigen-structures have been found to be useful in graphics, not much is known about their behavior when a surface is deformed or modified. The objective of my thesis is two-fold. One is to study, and to develop theory for, changes in the eigen-structures of the discrete Laplace operator as the underlying mesh is changed. The other is to explore applications for the spectral theory of shape perturbations in areas like shape matching and deformation.In particular, our work shows that the discrete Laplace is stable against noise and sampling. We also show that both the discrete and continuous Laplace change continuously as the underlying mesh or surface is deformed continuously, without introducing changes to the topology. Not only do these results help in providing a better theoretical understanding of the discrete Laplace operator, they also give us a solid base for developing applications. Indeed, we present two such applications: one that deals with shape matching and another that performs fast mesh deformations. Specifically, combining our theoretical results with concepts from persistent homology, we create a concise global shape signature that can be used for matching different shapes. Given our results regarding similarity of eigen-structures of similar shapes, our matching algorithm allows us to match even partially scanned or incomplete models, regardless of their pose or orientation. We also present a framework that uses eigenvectors to create an implicit skeleton of a shape and use it to deform the shape, producing smooth and natural looking deformations. By using the eigenvectors, we are able to reduce the problem size from the number of mesh vertices (hundreds to millions) to the number of eigenvectors used (tens to hundreds)." @default.
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• W42448260 date "2012-01-01" @default.
• W42448260 modified "2023-09-26" @default.
• W42448260 title "Discrete laplace operator: theory and applications" @default.
• W42448260 hasPublicationYear "2012" @default.
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