Skip to main content

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
Cancel
×
Home
  • Home
Article
    • Aa
    • Aa
  • Cited by 13
  • Cited by
    Crossref Citations
    Crossref logo
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Snášel, Václav Nowaková, Jana Xhafa, Fatos and Barolli, Leonard 2017. Geometrical and topological approaches to Big Data. Future Generation Computer Systems, Vol. 67, p. 286.

    Athreya, Siva Löhr, Wolfgang and Winter, Anita 2017. The gap between Gromov-vague and Gromov–Hausdorff-vague topology. Stochastic Processes and their Applications, Vol. 126, Issue. 9, p. 2527.

    Cámara, Pablo G. 2017. Topological methods for genomics: present and future directions. Current Opinion in Systems Biology,

    Carlsson, Gunnar and Kališnik Verovšek, Sara 2017. Symmetric and r-symmetric tropical polynomials and rational functions. Journal of Pure and Applied Algebra, Vol. 220, Issue. 11, p. 3610.

    Garland, Joshua Bradley, Elizabeth and Meiss, James D. 2017. Exploring the topology of dynamical reconstructions. Physica D: Nonlinear Phenomena, Vol. 334, p. 49.

    Khasawneh, Firas A. and Munch, Elizabeth 2017. Chatter detection in turning using persistent homology. Mechanical Systems and Signal Processing, Vol. 70-71, p. 527.

    Rieck, B. and Leitte, H. 2017. Exploring and Comparing Clusterings of Multivariate Data Sets Using Persistent Homology. Computer Graphics Forum, Vol. 35, Issue. 3, p. 81.

    Venkataraman, Vinay Ramamurthy, Karthikeyan Natesan and Turaga, Pavan 2017. 2016 IEEE International Conference on Image Processing (ICIP). p. 4150.

    Verovšek, Sara Kališnik and Mashaghi, Alireza 2017. Extended Topological Persistence and Contact Arrangements in Folded Linear Molecules. Frontiers in Applied Mathematics and Statistics, Vol. 2,

    Cortis, Andrea 2017. Unconventional Resources Technology Conference.

    Lesnick, Michael 2017. The Theory of the Interleaving Distance on Multidimensional Persistence Modules. Foundations of Computational Mathematics, Vol. 15, Issue. 3, p. 613.

    Rieck, B. and Leitte, H. 2017. Persistent Homology for the Evaluation of Dimensionality Reduction Schemes. Computer Graphics Forum, Vol. 34, Issue. 3, p. 431.

    Yogeshwaran, D. Subag, Eliran and Adler, Robert J. 2017. Random geometric complexes in the thermodynamic regime. Probability Theory and Related Fields,

    ×
  • Get access
    Add to cart $35.00
    Check if you have access via personal or institutional login
    Log in Register Recommend to librarian

Topological pattern recognition for point cloud data*

  • Gunnar Carlsson (a1)
Abstract

In this paper we discuss the adaptation of the methods of homology from algebraic topology to the problem of pattern recognition in point cloud data sets. The method is referred to as persistent homology, and has numerous applications to scientific problems. We discuss the definition and computation of homology in the standard setting of simplicial complexes and topological spaces, then show how one can obtain useful signatures, called barcodes, from finite metric spaces, thought of as sampled from a continuous object. We present several different cases where persistent homology is used, to illustrate the different ways in which the method can be applied.

Copyright
COPYRIGHT: © Cambridge University Press 2014 
Footnotes
Hide All
*

Colour online for monochrome figures available at journals.cambridge.org//anu.

Footnotes
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

J. Bardeen , J. Bond , N. Kaiser and A. Szalay (1986), ‘The statistics of peaks of Gaussian random fields’, Astrophys. J. 304, 15–61.
C. Bennett , R. Hill , G. Hinshaw , M. Nolta , N. Odegard , L. Page , D. Spergel , J. Weiland , E. Wright , M. Halpern , N. Jarosik , A. Kogut , M. Limon , S. Meyer , G. Tucker and E. Wollack (2003), ‘First-year Wilkinson Microwave Anisotropy Probe (WMAP)* observations: Foreground emission’, Astrophys. J. Supplement Series 148, 97.
P. Buneman (1974), ‘A note on the metric properties of trees’, J. Combin. Theory B 17, 48–50.
D. Burago , Y. Burago and S. Ivanov (2001), A Course in Metric Geometry, Vol. 33 of Graduate Studies in Mathematics, AMS.
G. Carlsson (2009), ‘Topology and data’, Bull. Amer. Math. Soc. 46, 255–308.
G. Carlsson and V. de Silva (2010), ‘Zigzag persistence’, Found. Comput. Math. 10, 367–405.
G. Carlsson and A. Zomorodian (2009), ‘The theory of multidimensional persistence’, Discrete Comput. Geom. 42, 71–93.
G. Carlsson , T. Ishkhanov , V. de Silva and A. Zomorodian (2008), ‘On the local behavior of spaces of natural images’, Internat. J. Computer Vision 76, 1–12.
G. Carlsson , V. de Silva and D. Morozov (2009), Zigzag persistent homology and real-valued functions. In Proc. 25th Annual Symposium on Computational Geometry, ACM, pp. 247–256.
G. Carlsson , A. Zomorodian , A. Collins and L. Guibas (2005), ‘Persistence barcodes for shapes’, Internat. J. Shape Modeling 11, 149.
J. Chan , G. Carlsson and R. Rabadan (2013), ‘Topology of viral evolution’, Proc. Nat. Acad. Sci. 110, 18566–18571.
F. Chazal , D. Cohen-Steiner and Q. Merigot (2011), ‘Geometric inference for probability measures’, Found. Comput. Math. 11, 733–751.
D. Cohen-Steiner , H. Edelsbrunner and J. Harer (2007), ‘Stability of persistence diagrams’, Discrete Comput. Geom. 37, 103–120.
D. Cohen-Steiner , H. Edelsbrunner , J. Harer and Y. Mileyko (2010), ‘Lipschitz functions have Lp-stable persistence’, Found. Comput. Math. 10, 127–139.
D. Donoho (1999), ‘Wedgelets: Nearly minimax estimation of edges’, Ann. Statist. 27, 859–897.
W. Doolittle (1999), ‘Phylogenetic classification and the universal tree’, Science 284 (5423), 2124–2128.
H. Edelsbrunner , D. Letscher and A. Zomorodian (2002), ‘Topological persistence and simplification’, Discrete Comput. Geom. 28, 511–533
S. Eilenberg (1944), ‘Singular homology theory’, Ann. of Math. 45, 407–447.
P. Gabriel (1972), ‘Unzerlegbare Darstellungen I’, Manuscr. Math. 6, 71–103.
A. Greven , P. Pfafelhuber and A. Winter (2009), ‘Convergence in distribution of random metric measure spaces’, Probab. Theory Rel. Fields 145, 285–322.
J. Gott , M. Dickinson and A. Melott (1986), ‘The sponge-like topology of large-scale structure in the universe’, Astrophys. J. 306, 341–357.
A. Hamilton , J. Gott and D. Weinberg (1986), ‘The topology of the large-scale structure of the universe’, Astrophys. J. 309, 1–12.
J. van Hateren and A. van der Schaaf (1998), ‘Independent component filters of natural images compared with simple cells in primary visual cortex’, Proc. R. Soc. Lond. B 265, 359–366.
J. Irwin and B. Schoichet (2005), ‘ZINC: A free database of commercially available compounds for virtual screening’, J. Chem. Inf. Model. 45, 177–182.
H. Karcher (1977), ‘Riemannian center of mass and mollifier smoothing’, Comm. Pure Appl. Math. 30, 509–541.
A. Lee , K. Pedersen and D. Mumford (2003), ‘The non-linear statistics of high contrast patches in natural images’, Internat. J. Computer Vision 54, 83–103.
Y. Mileyko , S. Mukherjee and J. Harer (2011), ‘Probability measures on the space of persistence diagrams’, Inverse Problems 27, 1–22.
J. Milnor (1963), Morse Theory, Princeton University Press.
M. Nicolau , A. Levine and G. Carlsson (2011), ‘Topology based data analysis identiies a subgroup of breast cancers with a unique mutational proile and excellent survival. Proc. Nat. Acad. Sci. 108, 7265–7270.
P. Niyogi , S. Smale and S. Weinberger (2008), ‘Finding the homology of submanifolds with high conidence from random samples’, Discrete Comput. Geom. 39, 419–441.
C. Park , P. Pranav , P. Chingangram , R. van de Weygaert , B. Jones , G. Vegter , I. Kim , J. Hidding and W. Helwing (2013), ‘Betti numbers of Gaussian fields’, J. Korean Astron. Soc. 46, 125–131.
J. Perea and G. Carlsson (2014), ‘A Klein bottle-based dictionary for texture representation’, Internat. J. Computer Vision 107, 75–97.
G. Segal (1968), ‘Classifying spaces and spectral sequences’, Inst. Hautes Études Sci. Publ. Math. 34, 105–112.
T. Sousbie , C. Pichon and H. Kawahara (2011), ‘The persistent cosmic web and its filamentary structure II: Illustrations’, Mon. Not. R. Astron. Soc. 414, 384–403.
R. van de Weygaert , G. Vegter , H. Edelsbrunner , B. Jones , P. Pranav , C. Park , W. Hellwing , B. Eldering , N. Kruithof , E. Bos , J. Hidding , J. Feldbrugge , E. ten Have , M. van Engelen , M. Caroli and M. Teillaud (2011), Alpha, Betti, and the Megaparsec Universe: On the topology of the cosmic web. In Transactions on Computational Science XIV ( M.L. Gavrilova et al., eds), Vol. 6970 of Lecture Notes in Computer Science, Springer, pp. 60–101.
A. Zomorodian (2010), The tidy set: A minimal simplicial set for computing homology of clique complexes. In Proc. 2010 Annual Symposium on Computational Geometry, ACM, pp. 257–266.
A. Zomorodian and G. Carlsson (2005), ‘Computing persistent homology’, Discrete Comput. Geom. 33, 247–274.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
  • URL: /core/journals/acta-numerica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

CAPTCHA *

Skip to the audio challenge
×

Metrics

Full text views

Total number of HTML views: 2
Total number of PDF views: 210 *
Loading metrics...

Abstract views

Total abstract views: 1062 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd September 2017. This data will be updated every 24 hours.

Cancel
Confirm
×