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Journal of Scientific Computing

, Volume 75, Issue 2, pp 638–656 | Cite as

Generalization of the Weighted Nonlocal Laplacian in Low Dimensional Manifold Model

  • Zuoqiang Shi
  • Stanley Osher
  • Wei Zhu
Article

Abstract

In this paper we use the idea of the weighted nonlocal Laplacian (Shi et al. in J Sci Comput, 2017) to deal with the constraints in the low dimensional manifold model (Osher et al. in SIAM J Imaging Sci, 2017). In the original LDMM, the constraints are enforced by the point integral method. The point integral method provides a correct way to deal with the constraints, however it is not very efficient due to the fact that the symmetry of the original Laplace–Beltrami operator is destroyed. WNLL provides another way to enforce the constraints in LDMM. In WNLL, the discretized system is symmetric and sparse and hence it can be solved very fast. Our experimental results show that the computational cost is reduced significantly with the help of WNLL. Moreover, the results in image inpainting and denoising are also better than the original LDMM and competitive with state-of-the-art methods.

Keywords

Weighted nonlocal Laplacian Low dimensional manifold model Nonlocal methods Point cloud 

Mathematics Subject Classification

65D05 65D25 41A05 

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina
  2. 2.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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