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International Workshop on Graph-Based Representations in Pattern Recognition

GbRPR 2017: Graph-Based Representations in Pattern Recognition pp 13–22Cite as

Shape Simplification Through Graph Sparsification

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Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10310))

Abstract

In this paper, we draw on Spielman and Srivastava’s method for graph sparsification in order to simplify shape representations. The underlying principle of graph sparsification is to retain only the edges which are key to the preservation of desired properties. In this regard, sparsification by edge resistance allows us to preserve (to some extent) links between protrusions and the remainder of the shape (e.g. parts of a shape) while removing in-part edges. Applying this idea to alpha shapes (abstract representations which have a huge number of edges) opens up a way of introducing a hierarchy of the edge strength, thus being relevant for shape analysis and interpretation.

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Notes

  1. 1.

    In this paper, the parameter \(\epsilon \) controls the number of samples needed by the process, whereas the weight for choosing the edges is given by effective resistances.

References

  1. Achlioptas, D.: Database-friendly random projections. In: Buneman, P. (ed.) Proceedings of the Twentieth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, PODS, 21–23 May 2001, Santa Barbara, California, USA. ACM (2001)

    Google Scholar 

  2. Batson, J.D., Spielman, D.A., Srivastava, N.: Twice-ramanujan sparsifiers. SIAM J. Comput. 41(6), 1704–1721 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Batson, J.D., Spielman, D.A., Srivastava, N., Teng, S.: Spectral sparsification of graphs: theory and algorithms. Commun. ACM 56(8), 87–94 (2013)

    Article  Google Scholar 

  4. Benczúr, A.A., Karger, D.R.: Approximating s-t minimum cuts in Õ(\({n}^{\text{2}}\)) time. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, STOC, Philadelphia, Pennsylvania, USA, 22–24 May 1996, pp. 47–55 (1996)

    Google Scholar 

  5. Berger, M., Tagliasacchi, A., Seversky, L.M., Alliez, P., Guennebaud, G., Levine, J.A., Sharf, A., Silva, C.T.: A survey of surface reconstruction from point clouds. Comput. Graph. Forum 36, 301–329 (2016)

    Article  Google Scholar 

  6. Edelsbrunner, H.: Alpha shapes - survey. In: Tessellations in the Sciences: Virtues, Techniques and Applications of Geometric Tilings. Springer Verlag (2011)

    Google Scholar 

  7. Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  8. Edelsbrunner, H., Kirkpatrick, D., Seidel, R.: On the shape of a set of points in the plane. IEEE Trans. Inf. Theory 29(4), 551–559 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28, 511–533 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Edelsbrunner, H., Mücke, E.P.: Three-dimensional alpha shapes. ACM Trans. Graph. 13(1), 43–72 (1994)

    Article  MATH  Google Scholar 

  11. Naylor, B., Bajaj, C., Edelsbrunner, H., Kaufman, A., Rossignac, J.: Computational representations of geometry. In: SIGGRAPH 1996 Course Notes (1996)

    Google Scholar 

  12. Paoluzzi, A., Bernardini, F., Cattani, C., Ferrucci, V.: Dimension-independent modeling with simplicial complexes. ACM Trans. Graph. 12(1), 56–102 (1993)

    Article  MATH  Google Scholar 

  13. Qiu, H., Hancock, E.R.: Clustering and embedding using commute times. IEEE Trans. Pattern Anal. Mach. Intell. 29(11), 1873–1890 (2007)

    Article  Google Scholar 

  14. Rudelson, M., Vershynin, R.: Sampling from large an approach through geometric functional analysis. J. ACM 54(4), 21 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Spielman, D.A., Srivastava, N.: Graph sparsification by effective resistances. SIAM J. Comput. 40(6), 1913–1926 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Spielman, D.A., Teng, S.: Spectral sparsification of graphs. SIAM J. Comput. 40(4), 981–1025 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

F. Escolano and M. Curado are funded by Project TIN2015-69077-P of the Spanish Government.

Author information

Authors and Affiliations

  1. Department of Computer Science and AI, University of Alicante, 03690, Alicante, Spain

    Francisco Escolano & Manuel Curado

  2. CNR-IMATI, Via de Marini, 6 (Torre di Francia), 16149, Genova, Italy

    Silvia Biasotti

  3. Department of Computer Science, University of York, York, YO10 5DD, UK

    Edwin R. Hancock

Authors
  1. Francisco Escolano
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  2. Manuel Curado
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  3. Silvia Biasotti
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  4. Edwin R. Hancock
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Corresponding author

Correspondence to Francisco Escolano .

Editor information

Editors and Affiliations

  1. Università degli Studi di Salerno, Fisciano, Italy

    Pasquale Foggia

  2. Chinese Academy of Sciences, Beijing, China

    Cheng-Lin Liu

  3. Università degli Studi di Salerno, Fisciano, Italy

    Mario Vento

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Escolano, F., Curado, M., Biasotti, S., Hancock, E.R. (2017). Shape Simplification Through Graph Sparsification. In: Foggia, P., Liu, CL., Vento, M. (eds) Graph-Based Representations in Pattern Recognition. GbRPR 2017. Lecture Notes in Computer Science(), vol 10310. Springer, Cham. https://doi.org/10.1007/978-3-319-58961-9_2

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  • DOI: https://doi.org/10.1007/978-3-319-58961-9_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-58960-2

  • Online ISBN: 978-3-319-58961-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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