Abstract
The aim of the talk at GSI’2013 is to introduce the audience to the concept of interleaving between filtrations, coming from applied topology, and to emphasize its influence on emerging trends in point cloud data analysis.
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Biasotti, S., De Floriani, L., Falcidieno, B., Frosini, P., Giorgi, D., Landi, C., Papaleo, L., Spagnuolo, M.: Describing shapes by geometrical-topological properties of real functions. ACM Comput. Surv. 40(4), 12:1–12:87 (2008)
Boissonnat, J.-D., Guibas, L.J., Oudot, S.Y.: Manifold reconstruction in arbitrary dimensions using witness complexes. In: Proc. 23rd Sympos. on Comp. Geom., pp. 194–203 (2007)
Carlsson, G., de Silva, V.: Zigzag persistence. Foundations of Computational Mathematics 10(4), 367–405 (2010)
Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.Y.: Proximity of persistence modules and their diagrams. In: Proc. 25th Annu. Symposium on Computational Geometry, pp. 237–246 (2009)
Chazal, F., Cohen-Steiner, D., Lieutier, A.: A sampling theory for compact sets in euclidean spaces. Discrete Comput. Geom. 41(3) (2009)
Chazal, F., de Silva, V., Glisse, M., Oudot, S.Y.: The structure and stability of persistence modules. Research Report arXiv:1207.3674 [math.AT] (July 2012)
Chazal, F., de Silva, V., Oudot, S.Y.: Persistence stability for geometric complexes. Research Report arXiv:1207.3885 [math.AT] (July 2012)
Chazal, F., Guibas, L.J., Oudot, S.Y., Skraba, P.: Analysis of scalar fields over point cloud data. In: Proc. 19th ACM-SIAM Sympos. on Discrete Algorithms, pp. 1021–1030 (2009)
Chazal, F., Guibas, L.J., Oudot, S.Y., Skraba, P.: Persistence-based clustering in riemannian manifolds. In: Proc. 27th Annu. ACM Sympos. on Comput. Geom., pp. 97–106 (June 2011)
Chazal, F., Lieutier, A.: Stability and computation of topological invariants of solids in ℝn. Discrete Comput. Geom. 37(4), 601–617 (2007)
Chazal, F., Oudot, S.Y.: Towards persistence-based reconstruction in Euclidean spaces. In: Proc. 24th ACM Sympos. Comput. Geom., pp. 232–241 (2008)
Chazal, F., Cohen-Steiner, D., Guibas, L.J., Mémoli, F., Oudot, S.: Gromov-hausdorff stable signatures for shapes using persistence. Comput. Graph. Forum 28(5), 1393–1403 (2009)
Cheng, S.-W., Dey, T.K., Ramos, E.A.: Manifold reconstruction from point samples. In: Proc. 16th Sympos. Discrete Algorithms, pp. 1018–1027 (2005)
Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete & Computational Geometry 37(1), 103–120 (2007)
de Silva, V.: A weak characterisation of the Delaunay triangulation. Geometriae Dedicata 135(1), 39–64 (2008)
de Silva, V., Carlsson, G.: Topological estimation using witness complexes. In: Proc. Sympos. Point-Based Graphics, pp. 157–166 (2004)
de Silva, V., Ghrist, R.: Homological sensor networks. Notices of the American Mathematical Society 54(1), 10–17 (2007)
Dey, T.K., Fan, F., Wang, Y.: Computing topological persistence for simplicial maps. Research Report arXiv:1208.5018 [cs.CG] (August 2012)
Dionysus. By Dmitriy Morozov, http://www.mrzv.org/software/dionysus/
Edelsbrunner, H., Harer, J.: Persistent homology—a survey. In: Goodman, J.E., Pach, J., Pollack, R. (eds.) Surveys on Discrete and Computational Geometry. Twenty Years Later, pp. 257–282. Amer. Math. Soc. (2008)
Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Proc. 41st Annu. IEEE Sympos. Found. Comput. Sci., pp. 454–463 (2000)
Hudson, B., Miller, G.L., Oudot, S.Y., Sheehy, D.R.: Topological inference via meshing. In: Proc. 26th Annual Symposium on Computational Geometry, pp. 277–286 (2010)
Lee, A.B., Pederson, K.S., Mumford, D.: The nonlinear statistics of high-contrast patches in natural images. Internat. J. of Computer Vision 54(1-3), 83–103 (2003)
Oudot, S.Y., Sheehy, D.R.: Zigzag zoology: Rips zigzags for homology inference. In: Proc. 29th Annual Symposium on Computational Geometry (to appear, 2013)
Sheehy, D.R.: Linear-size approximations to the vietoris-rips filtration. In: Proc. 28th Symposium on Computational Geometry, pp. 239–248 (2012)
Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)
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Chazal, F., Oudot, S.Y. (2013). Interleaved Filtrations: Theory and Applications in Point Cloud Data Analysis. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2013. Lecture Notes in Computer Science, vol 8085. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40020-9_65
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