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


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.


Weighted nonlocal Laplacian Low dimensional manifold model Nonlocal methods Point cloud 

Mathematics Subject Classification

65D05 65D25 41A05 


  1. 1.
    Aharon, M., Elad, M., Bruckstein, A.: K-svd—an algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans. Signal Process. 54, 4311–4322 (2006)CrossRefzbMATHGoogle Scholar
  2. 2.
    Arias, P., Caselles, V., Sapiro, G.: A variational framework for non-local image inpainting. In: Proceedings of the 7th International Conference on EMMCPVR (2009)Google Scholar
  3. 3.
    Buades, A., Coll, B., Morel, J.-M.: A non-local algorithm for image denoising. IEEE Comput. Soc. Conf. Comput. Vis. Pattern Recognit. (CVPR) 2, 60–65 (2005)zbMATHGoogle Scholar
  4. 4.
    Buades, A., Coll, B., Morel, J.-M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4, 490–530 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carlsson, G., Ishkhanov, T., de Silva, V., Zomorodian, A.: On the local behavior of spaces of natural images. Int. J. Comput. Vis. 76, 1–12 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chatterjee, P., Milanfar, P.: Patch-based near-optimal image denoising. IEEE Trans. Image Process. 21, 1635–1649 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Comaniciu, D., Meer, P.: Mean shift: a robust approach toward feature space analysis. IEEE Trans. Pattern Anal. Mach. Intell. 24, 603–619 (2002)CrossRefGoogle Scholar
  8. 8.
    Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3D transform-domain collaborative filtering. IEEE Trans. Image Process. 16, 2080–2095 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Facciolo, V.C.G., Arias, P., Sapiro, G.: Exemplar-based interpolation of sparsely sampled image. In: Proceedings of the 7th International Conference on EMMCPVR (2009)Google Scholar
  10. 10.
    Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7, 1005–1028 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kheradmand, A., Milanfar, P.: A general framework for regularized, similarity-based image restoration. IEEE Trans. Image Process. 23, 5136–5151 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lee, A.B., Pedersen, K.S., Mumford, D.: The nonlinear statistics of high-contrast patches in natural images. Int. J. Comput. Vis. 54, 83–103 (2003)CrossRefzbMATHGoogle Scholar
  13. 13.
    Li, Z., Shi, Z.: A convergent point integral method for isotropic elliptic equations on point cloud. SIAM Multiscale Model. Simul. 14, 874–905 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Li, Z., Shi, Z., Sun, J.: Point integral method for solving poisson-type equations on manifolds from point clouds with convergence guarantees. Commun. Comput. Phys. 22, 228–258 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Osher, S., Shi, Z., Zhu, W.: Low dimensional manifold model for image processing. SIAM J. Imaging Sci. (2017, to appear)Google Scholar
  16. 16.
    Perea, J.A., Carlsson, G.: A klein-bottle-based dictionary for texture representation. Int. J. Comput. Vis. 107, 75–97 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Peyré, G.: Image processing with non-local spectral bases. Multiscale Model. Simul. 7, 703–730 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Peyré, G.: Manifold models for signals and images. Comput. Vis. Image Underst. 113, 248–260 (2009)CrossRefGoogle Scholar
  19. 19.
    Peyré, G.: A review of adaptive image representations. IEEE J. Sel. Top. Signal Process. 5, 896–911 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shi, Z., Osher, S., Zhu, W.: Weighted nonlocal laplacian on interpolation from sparse data. J. Sci. Comput. (2017, to appear)Google Scholar
  21. 21.
    Shi, Z., Sun, J.: Convergence of the point integral method for poisson equation on point cloud. Res. Math. Sci. (2017, to appear)Google Scholar
  22. 22.
    Shi, Z., Sun, J., Tian, M.: Harmonic extension on point cloud. mathscidoc:1609.19003Google Scholar
  23. 23.
    Smith, S.M., Brady, J.M.: Susan—a new approach to low level image processing. Int. J. Comput. Vis. 23, 45–78 (1997)CrossRefGoogle Scholar
  24. 24.
    Spira, A., Kimmel, R., Sochen, N.: A short time Beltrami kernel for smoothing images and manifolds. IEEE Trans. Image Process. 16, 1628–1636 (2007)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Talebi, H., Milanfar, P.: Global image denoising. IEEE Trans. Image Process. 23, 755–768 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: Proceedings of the Sixth International Conference on Computer Vision, p. 839 (1998)Google Scholar
  27. 27.
    Wang, Y., Szlam, A., Lerman, G.: Robust locally linear analysis with applications to image denoising and blind inpainting. SIAM J. Imaging Sci. 6, 526–562 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Yin, R., Gao, T., Lu, Y.M., Daubechies, I.: A tale of two bases: local-nonlocal regularization on image patches with convolution framelets. arXiv:1606.01377 (2016)
  29. 29.
    Yu, G., Sapiro, G., Mallat, S.: Solving inverse problems with piecewise linear estimators: from gaussian mixture models to structured sparsity. IEEE Trans. Image Process. 21, 2481–2499Google Scholar
  30. 30.
    Yu, G., Sapiro, G., Mallat, S.: Image modeling and enhancement via structured sparse model selection. In: Proceedings of the 17th IEEE International Conference on Image Processing (ICIP) (2010)Google Scholar
  31. 31.
    Zhou, M., Chen, H., Paisley, J., Ren, L., Li, L., Xing, Z., Dunson, D., Sapiro, G., Carin, L.: Nonparametric bayesian dictionary learning for analysis of noisy and incomplete images. IEEE Trans. Image Process. 21, 130–144 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Zoran, D., Weiss, Y.: From learning models of natural image patches to whole image restoration. In: International Conference on Computer Vision (2011)Google Scholar

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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