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Generalization of the Weighted Nonlocal Laplacian in Low Dimensional Manifold Model

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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.

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Authors and Affiliations

  1. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Zuoqiang Shi

  2. Department of Mathematics, University of California, Los Angeles, CA, 90095, USA

    Stanley Osher & Wei Zhu

Authors
  1. Zuoqiang Shi
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  2. Stanley Osher
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  3. Wei Zhu
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Correspondence to Zuoqiang Shi.

Additional information

Research supported by DOE-SC0013838 and NSF DMS-1118971. Z. Shi was partially supported by NSFC Grants 11371220, 11671005.

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Shi, Z., Osher, S. & Zhu, W. Generalization of the Weighted Nonlocal Laplacian in Low Dimensional Manifold Model. J Sci Comput 75, 638–656 (2018). https://doi.org/10.1007/s10915-017-0549-x

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  • DOI: https://doi.org/10.1007/s10915-017-0549-x

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