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Pawas Ranjan

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SODA Conference 2013 Conference Paper

Weighted Graph Laplace Operator under Topological Noise

  • Tamal K. Dey
  • Pawas Ranjan
  • Yusu Wang 0001

Recently, various applications have motivated the study of spectral structures (eigenvalues and eigenfunctions) of the so-called Laplace-Beltrami operator of a manifold and their discrete versions. A popular choice for the discrete version is the so-called Gaussian weighted graph Laplacian which can be applied to point cloud data that samples a manifold. Naturally, the question of stability of the spectrum of this discrete Laplacian under the perturbation of the sampled manifold becomes important for its practical usage. Previous results showed that the spectra of both the manifold Laplacian and discrete Laplacian are stable when the perturbation is “nice” in the sense that it is restricted to a diffeomorphism with minor area distortion. However, this forbids, for example, small topological changes.

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SODA Conference 2010 Conference Paper

Convergence, Stability, and Discrete Approximation of Laplace Spectra

  • Tamal K. Dey
  • Pawas Ranjan
  • Yusu Wang 0001

Spectral methods have been widely used in a broad range of applications fields. One important object involved in such methods is the Laplace-Beltrami operator of a manifold. Indeed, a variety of work in graphics and geometric optimization uses the eigen-structures (i. e, the eigenvalues and eigenfunctions) of the Laplace operator. Applications include mesh smoothing, compression, editing, shape segmentation, matching, parameterization, and so on. While the Laplace operator is defined (mathematically) for a smooth domain, these applications often approximate a smooth manifold by a discrete mesh. The spectral structure of the manifold Laplacian is estimated from some discrete Laplace operator constructed from this mesh. In this paper, we study the important question of how well the spectrum computed from the discrete mesh approximates the true spectrum of the manifold Laplacian. We exploit a recent result on mesh Laplacian and provide the first convergence result to relate the spectrum constructed from a general mesh (approximating an m -manifold embedded in ℝ d ) with the true spectrum. We also study how stable these eigenvalues and their discrete approximations are when the underlying manifold is perturbed, and provide explicit bounds for the Laplacian spectra of two “close” manifolds, as well as a convergence result for their discrete approximations. Finally, we present various experimental results to demonstrate that these discrete spectra are both accurate and robust in practice.

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