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Nonlocal PDEs on Graphs: From Tug-of-War Games to Unified Interpolation on Images and Point Clouds

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Abstract

In this paper, we introduce a new general class of partial difference operators on graphs, which interpolate between the nonlocal \(\infty \)-Laplacian, the Laplacian, and a family of discrete gradient operators. In this context, we investigate an associated Dirichlet problem for this general class of operators and prove the existence and uniqueness of respective solutions. We show that a certain partial difference equation based on this class of operators recovers many variants of a stochastic game known as ‘Tug-of-War’ and extends them to a nonlocal setting. Furthermore, we discuss a connection with certain nonlocal partial differential equations. Finally, we propose to use this class of operators as general framework to solve many interpolation problems in a unified manner as arising, e.g., in image and point cloud processing.

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References

  1. Alpaydin, E., Alimoglu, F.: Pen-based Recognition of Handwritten Digits Data Set. University of California, Irvine, Machine Learning Repository (1998)

  2. Alpaydin, E., Kaynak, C.: Optical Recognition of Handwritten Digits Data Set. UCI Machine Learning Repository (1998)

  3. Andreu, F., Mazón, J., Rossi, J., Toledo, J.: A nonlocal p-Laplacian evolution equation with Neumann boundary conditions. J. Math. Pures Appl. 90(2), 201–227 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Andreu-Vaillo, F., Mazón, J.M., Rossi, J.D., Toledo-Melero, J.J.: Nonlocal Diffusion Problems, vol. 165. American Mathematical Society, Rhode Island (2010)

  5. Arias, P., Facciolo, G., Caselles, V., Sapiro, G.: A variational framework for exemplar-based image inpainting. Int. J. Comput. Vis. 93(3), 319–347 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arya, S., Mount, D.M., Netanyahu, N.S., Silverman, R., Wu, A.Y.: An optimal algorithm for approximate nearest neighbor searching fixed dimensions. J. ACM 45(6), 891–923 (1998). doi:10.1145/293347.293348

    Article  MathSciNet  MATH  Google Scholar 

  7. Bai, X., Sapiro, G.: Geodesic matting: a framework for fast interactive image and video segmentation and matting. Int. J. Comput. Vis. 82(2), 113–132 (2009)

    Article  Google Scholar 

  8. Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, pp. 417–424. ACM Press/Addison-Wesley Publishing Co. (2000)

  9. Bertozzi, A.L., Flenner, A.: Diffuse interface models on graphs for classification of high dimensional data. Multisc. Model. Simul. 10(3), 1090–1118 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bertrand, G.: On topological watersheds. J. Math. Imaging Vis. 22(2–3), 217–230 (2005)

    Article  MathSciNet  Google Scholar 

  11. Bougleux, S., Elmoataz, A., Melkemi, M.: Local and nonlocal discrete regularization on weighted graphs for image and mesh processing. Int. J. Comput. Vis. 84(2), 220–236 (2009)

    Article  Google Scholar 

  12. Boykov, Y.Y., Jolly, M.P.: Interactive graph cuts for optimal boundary & region segmentation of objects in nd images. In: Eighth IEEE International Conference on Computer Vision, 2001. ICCV 2001. Proceedings, vol. 1, pp. 105–112. IEEE (2001)

  13. Bresson, X., Thomas, L., Uminsky, D., von Brecht, J.H.: Multiclass total variation clustering. In: Burges, C.J.C., Bottou, L., Ghahramani, Z., Weinberger, K.Q. (eds.) NIPS, pp. 1421–1429 (2013)

  14. Buades, A., Coll, B., Morel, J.M.: Nonlocal image and movie denoising. Int. J. Comput. Vis. 76(2), 123–139 (2008)

    Article  Google Scholar 

  15. Bühler, T., Hein, M.: Spectral clustering based on the graph p-laplacian. In: Proceedings of the 26th Annual International Conference on Machine Learning, pp. 81–88. ACM (2009)

  16. Chambolle, A., Lindgren, E., Monneau, R.: The Hölder infinite Laplacian and Hölder extensions. ESAIM Contr. Optim. CA. 18(3), 799–835 (2012)

  17. Chan, T.F., Kang, S.H., Shen, J.: Euler’s elastica and curvature-based inpainting. SIAM J. Appl. Math. 63(2), 564–592 (2003)

  18. Coifman, R.R., Maggioni, M.: Diffusion wavelets. Appl. Comput. Harmon. Anal. 21(1), 53–94 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Couprie, C., Grady, L., Najman, L., Talbot, H.: Power watershed: a unifying graph-based optimization framework. IEEE Trans. Pattern Anal. Mach. Intell. 33(7), 1384–1399 (2011)

    Article  Google Scholar 

  20. Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: minimum spanning forests and the drop of water principle. IEEE Trans. Pattern Anal. Mach. Intell. 31(8), 1362–1374 (2009)

    Article  Google Scholar 

  21. Criminisi, A., Pérez, P., Toyama, K.: Region filling and object removal by exemplar-based image inpainting. IEEE Trans. Image Process. 13(9), 1200–1212 (2004)

    Article  Google Scholar 

  22. Desquesnes, X., Elmoataz, A., Lézoray, O.: Eikonal equation adaptation on weighted graphs: fast geometric diffusion process for local and non-local image and data processing. J. Math. Imaging Vis. 46(2), 238–257 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Drábek, P.: The p-Laplacian–Mascot of nonlinear analysis. Acta Math. Univ. Comen. 76(1), 85–98 (2007)

    MathSciNet  MATH  Google Scholar 

  24. Efros, A.A., Leung, T.K.: Texture synthesis by non-parametric sampling. In: The Proceedings of the Seventh IEEE International Conference on Computer Vision, 1999, vol. 2, pp. 1033–1038. IEEE (1999)

  25. Elmoataz, A., Desquesnes, X., Lakhdari, Z., Lézoray, O.: Nonlocal infinity Laplacian equation on graphs with applications in image processing and machine learning. Math. Comput. Simul. 102, 153–163 (2014)

    Article  MathSciNet  Google Scholar 

  26. Elmoataz, A., Desquesnes, X., Lézoray, O.: Non-local morphological PDEs and-Laplacian equation on graphs with applications in image processing and machine learning. IEEE J. Sel. Top. Signal Process. 6(7), 764–779 (2012)

    Article  MATH  Google Scholar 

  27. Elmoataz, A., Lezoray, O., Bougleux, S.: Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing. IEEE Trans. Image Process. 17(7), 1047–1060 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Elmoataz, A., Lézoray, O., Bougleux, S., Ta, V.T.: Unifying local and nonlocal processing with partial difference operators on weighted graphs. In: International Workshop on Local and Non-local Approximation in Image Processing, pp. 11–26 (2008)

  29. Elmoataz, A., Toutain, M., Tenbrinck, D.: On the \(p\)-Laplacian and \(\infty \)-Laplacian on graphs with applications in image and data processing. SIAM J. Imaging Sci. 8(4), 2412–2451 (2015)

  30. Falcão, A.X., Stolfi, J., de Alencar Lotufo, R.: Alencar Lotufo, R.: The image foresting transform: theory, algorithms, and applications. IEEE Trans. Pattern Anal. Mach. Intell. 26(1), 19–29 (2004)

    Article  Google Scholar 

  31. Ghoniem, M., Chahir, Y., Elmoataz, A.: Nonlocal video denoising, simplification and inpainting using discrete regularization on graphs. Signal Process. 90(8), 2445–2455 (2010). doi:10.1016/j.sigpro.2009.09.004

    Article  MATH  Google Scholar 

  32. Ghoniem, M., Elmoataz, A., Lezoray, O.: Discrete infinity harmonic functions: towards a unified interpolation framework on graphs. In: 2011 18th IEEE International Conference on Image Processing (ICIP), pp. 1361–1364. IEEE (2011)

  33. Gilboa, G., Osher, S.: Nonlocal linear image regularization and supervised segmentation. Multisc. Model. Simul. 6(2), 595–630 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multisc. Model. Simul. 7(3), 1005–1028 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Grady, L.: Random walks for image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 28(11), 1768–1783 (2006)

    Article  Google Scholar 

  36. Hein, M., Bühler, T.: An inverse power method for nonlinear eigenproblems with applications in 1-spectral clustering and sparse PCA. In: Advances in Neural Information Processing Systems, pp. 847–855 (2010)

  37. Johnson, A., Hebert, M.: Using spin images for efficient object recognition in cluttered 3d scenes. IEEE Trans. Pattern Anal. 21(5), 433–449 (1999)

    Article  Google Scholar 

  38. Kawohl, B.: Variations on the p-laplacian. In: Bonheure, D., Takac, P., et al. (eds.) Nonlinear Elliptic Partial Differential Equations. Contemporary Mathematics, vol. 540, pp. 35–46 (2011)

  39. Kawulok, M., Smolka, B.: Texture-adaptive image colorization framework. EURASIP J. Adv. Signal Process. 2011, 99 (2011)

    Article  Google Scholar 

  40. Kervrann, C.: An adaptive window approach for image smoothing and structures preserving. In: Pajdla, T., Matas, J. (eds.) ECCV (3), Lecture Notes in Computer Science, vol. 3023, pp. 132–144. Springer (2004)

  41. Keysers, D., Dahmen, J., Theiner, T., Ney, H.: Experiments with an extended tangent distance. In: ICPR, pp. 2038–2042 (2000)

  42. LeCun, Y., Cortes, C.: MNIST handwritten digit database. http://yann.lecun.com/exdb/mnist/ (2010)

  43. Lee, Y.S., Chung, S.Y.: Extinction and positivity of solutions of the p-Laplacian evolution equation on networks. J. Math. Anal. Appl. 386(2), 581–592 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. Leifman, G., Tal, A.: Mesh colorization. Comput. Graph. Forum 31(2), 421–430 (2012). http://dblp.uni-trier.de/db/journals/cgf/cgf31.html#LeifmanT12

  45. Leifman, G., Tal, A.: Pattern-driven colorization of 3d surfaces. In: CVPR, pp. 241–248 (2013)

  46. Levin, A., Lischinski, D., Weiss, Y.: Colorization using optimization. ACM Trans. Graph. 23(3), 689–694 (2004). doi:10.1145/1015706.1015780

    Article  Google Scholar 

  47. Lindqvist, P.: Notes on the p-Laplace equation. Univ. (2006)

  48. Lozes, F., Elmoataz, A., Lézoray, O.: Nonlocal processing of 3d colored point clouds. In: 2012 21st International Conference on Pattern Recognition (ICPR), pp. 1968–1971 (2012)

  49. Lozes, F., Elmoataz, A., Lezoray, O.: Partial difference operators on weighted graphs for image processing on surfaces and point clouds. IEEE Trans. Image Process. 23(9), 3896–3909 (2014). doi:10.1109/TIP.2014.2336548

    Article  MathSciNet  Google Scholar 

  50. Manfredi, J.J., Parviainen, M., Rossi, J.D.: Dynamic programming principle for tug-of-war games with noise. ESAIM Control Optim. Calc. Var. 18(01), 81–90 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  51. Manfredi, J.J., Parviainen, M., Rossi, J.D.: On the definition and properties of p-harmonious functions. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze-Serie V 11(2), 215 (2012)

    MathSciNet  MATH  Google Scholar 

  52. Markle, W.: The development and application of colorization. SMPTE J. 93(7), 632–635 (1984)

    Article  Google Scholar 

  53. Meyer, F.: Topographic distance and watershed lines. Signal Process. 38(1), 113–125 (1994)

    Article  MATH  Google Scholar 

  54. Mitra, N.J., Nguyen, A., Guibas, L.: Estimating surface normals in noisy point cloud data. Int. J. Comput. Geom. Appl. 14(04–05), 261–276 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  55. Neuberger, J.M.: Nonlinear elliptic partial difference equations on graphs. Exp. Math. 15(1), 91–107 (2006)

  56. Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces, vol. 153. Springer Science & Business Media (2006)

  57. Park, J.H., Chung, S.Y.: Positive solutions for discrete boundary value problems involving the p-Laplacian with potential terms. Comput. Math. Appl. 61(1), 17–29 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  58. Peres, Y., Pete, G., Somersille, S.: Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones. Calc. Var. Partial Differ. Equ. 38(3–4), 541–564 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  59. Peres, Y., Schramm, O., Sheffield, S., Wilson, D.: Tug-of-war and the infinity Laplacian. J. Am. Math. Soc. 22(1), 167–210 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  60. Peres, Y., Sheffield, S., et al.: Tug-of-war with noise: a game-theoretic view of the \( p \)-laplacian. Duke Math. J. 145(1), 91–120 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  61. Rusu, R.B., Blodow, N., Marton, Z.C., Beetz, M.: Aligning point cloud views using persistent feature histograms. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, 2008. IROS 2008, pp. 3384–3391. IEEE (2008)

  62. Schnabel, R., Wahl, R., Klein, R.: Efficient ransac for point-cloud shape detection. In: Computer Graphics Forum, vol. 26, pp. 214–226. Wiley Online Library (2007)

  63. Schönlieb, C.B., Bertozzi, A.: Unconditionally stable schemes for higher order inpainting. Commun. Math. Sci. 9(2), 413–457 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  64. Sethian, J.A.: Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. In: Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (1999)

  65. Sinop, A.K., Grady, L.: A seeded image segmentation framework unifying graph cuts and random walker which yields a new algorithm. In: IEEE 11th International Conference on Computer Vision, 2007. ICCV 2007, pp. 1–8. IEEE (2007)

  66. Sýkora, D., Buriánek, J., Žára, J.: Unsupervised colorization of black-and-white cartoons. In: NPAR, pp. 121–128 (2004)

  67. Ta, V.T., Elmoataz, A., Lézoray, O.: Adaptation of eikonal equation over weighted graph. In: Scale Space and Variational Methods in Computer Vision, pp. 187–199. Springer, Berlin, Heidelberg (2009)

  68. Ta, V.T., Elmoataz, A., Lézoray, O.: Nonlocal PDEs-based morphology on weighted graphs for image and data processing. IEEE Trans. Image Process. 20(6), 1504–1516 (2011)

    Article  MathSciNet  Google Scholar 

  69. Toutain, M., Elmoataz, A., Lozes, F., Mansouri, A.: Non-local discrete \(\infty \)-Poisson and Hamilton Jacobi equations—from stochastic game to generalized distances on images, meshes, and point clouds. J. Math. Imaging Vis. 55(2), 229–241 (2016). doi:10.1007/s10851-015-0592-x

    Article  MathSciNet  MATH  Google Scholar 

  70. Vincent, L., Soille, P.: Watersheds in digital spaces: an efficient algorithm based on immersion simulations. IEEE Trans. Pattern Anal. Mach. Intell. 13(6), 583–598 (1991)

    Article  Google Scholar 

  71. Von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)

    Article  MathSciNet  Google Scholar 

  72. Yatziv, L., Sapiro, G.: Fast image and video colorization using chrominance blending. IEEE Trans. Image Process. 15(5), 1120–1129 (2006)

    Article  Google Scholar 

  73. Zelnik-manor, L., Perona, P.: Self-tuning spectral clustering. In: Advances in Neural Information Processing Systems, vol. 17, pp. 1601–1608. MIT Press (2005)

  74. Zhou, D., Schölkopf, B.: Discrete regularization. In: Semi-Supervised Learning, pp. 221–232. MIT Press (2006)

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Acknowledgments

AE is supported by the ANR GRAPHSIP, and MT is supported by a European FEDER Grant (PLANUCA Project).

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Authors and Affiliations

  1. Laboratoire LIGIM, Université de Paris-Est Marne-La-Valée, Cité Descartes, Bâtiment Copernic - 5, bd Descartes, Champs sur Marne, 77454, Marne-la-Vallée Cedex 2, France

    Abderrahim Elmoataz

  2. Université de Caen Normandie, GREYC Laboratory, 6 Boulevard Maréchal Juin, 14050, Caen Cedex, France

    Abderrahim Elmoataz & François Lozes

  3. Datexim SAS, 51 Avenue de la Côte de Nacre, 14000, Caen Cedex, France

    Matthieu Toutain

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  1. Abderrahim Elmoataz
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Correspondence to Abderrahim Elmoataz.

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Elmoataz, A., Lozes, F. & Toutain, M. Nonlocal PDEs on Graphs: From Tug-of-War Games to Unified Interpolation on Images and Point Clouds. J Math Imaging Vis 57, 381–401 (2017). https://doi.org/10.1007/s10851-016-0683-3

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