Abstract
This chapter introduces a number of basic types of shape classes commonly found in CW complexes. These shape classes are useful in clustering and separating subcomplexes in triangulated finite, bounded surface regions such as those found in visual scenes. Spatial shape classes derived from spatial proximities 
are examples of what Leader [1] called clusters. A Leader cluster is a collection of near sets, derived from a given member A of a proximity space X, by finding all subsets E of X that are near A. Each spatial shape class is a Leader cluster. Four types of spatial shape classes are considered in this chapter.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Many thanks to Talia Fernos [11] for posting this LG& TBQ (conference in Geometry, Topology, and Dynamics) announcement.
- 2.
Many thanks to Alessandro Granata for his permission to use his paintings in this study of
-based optical vortex nerve classes. Also, many thanks to M. Z. Ahmad for supplying the Matlab script used to find optical vortex nerves on triangulated digital images.
-based optical vortex nerve classes. Also, many thanks to M. Z. Ahmad for supplying the Matlab script used to find optical vortex nerves on triangulated digital images.References
Leader, S.: On clusters in proximity spaces. Fundam. Math. 47, 205–213 (1959)
Gellert, W., Küstner, H., Hellwich, M., H. Kästner, E.: The VNR Concise Encyclopedia of Mathematics, 760 p (56 plates). Van Nostrand Reinhold Co., New York, London (1977). ISBN: 0-442-22646-2, MR0644488; see Mathematics at a glance, A compendium. Translated from the German under the editorship of Hirsch, K.A. and with the collaboration of Pretzel, O., Primrose, E.J.F., Reuter, G.E.H., Stefan, A., Tropper, A.M., Walker, A., MR0371551
Peters, J.: Local near sets: pattern discovery in proximity spaces. Math. Comp. Sci. 7(1), 87–106 (2013). https://doi.org/10.1007/s11786-013-0143-z, MR3043920, ZBL06156991
Concilio, A.D., Guadagni, C., Peters, J., Ramanna, S.: Descriptive proximities. properties and interplay between classical proximities and overlap. Math. Comput. Sci. 12(1), 91–106 (2018). MR3767897, Zbl 06972895
Dareau, A., Levy, E., Aguilera, M., Bouganne, R., Akkermans, E., Gerbier, F., Beugnon, J.: Revealing the topology of quasicrystals with a diffraction experiment. Phys. Rev. Lett., arXiv 1607(00901v2), 1–7 (2017). https://doi.org/10.1103/PhysRevLett.119.215304
Fermi, M.: Why topology for machine learning and knowledge extraction. Mach. Learn. Knowl. Extr. 1(6), 1–6 (2018). https://doi.org/10.3390/make1010006
Fermi, M.: Persistent topology for natural data analysis - a survey. arXiv 1706(00411v2), 1–18 (2017)
Baikov, V., Gilmanov, R., Taimanov, I., Yakovlev, A.: Topological characteristics of oil and gass reservoirs and their applications. In: A.H. et. al. (ed.) Integrative Machine Learning, LNAI 10344, 182–193 pp. Springer, Berlin (2017)
Pellikka, M., Suuriniemi, S., Kettunen, L.: Homology in electromagnetic boundary value problems. Bound. Value Probl. 2010(381953), 1–18 (2010). https://doi.org/10.1155/2010/381953
Peters, J.: Computational proximity. Excursions in the topology of digital images. Intell. Syst. Ref. Libr. 102, Xxviii + 433 (2016). https://doi.org/10.1007/978-3-319-30262-1, MR3727129 and Zbl 1382.68008
Fernos, T.: LGTBQ: a conference in geometry, topology, and dynamics on the work of LGTBQ+ mathematicians, 10–14 June 2019, at the University of Michigan. Technical report, Deparment of Mathematics, University of Wisconsin (2018). http://www.math.wisc.edu/~kent/LG&TBQ.html
Ghrist, R.: Barcodes: the persistent topology of data. Bull. Amer. Math. Soc. (N.S.) 45(1), 61–75 (2008). MR2358377
Ghrist, R.: Elementary Applied Topology, Vi+269 pp. University of Pennsylvania, Philadelphia (2014). ISBN 978-1-5028-8085-7
Nye, J.: Events in fields of optical vortices: rings and reconnection. J. Opt. 18, 1–11 (2016). https://doi.org/10.1088/2040-8978/18/10/105602
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Peters, J.F. (2020). Leader Clusters and Shape Classes. In: Computational Geometry, Topology and Physics of Digital Images with Applications. Intelligent Systems Reference Library, vol 162. Springer, Cham. https://doi.org/10.1007/978-3-030-22192-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-22192-8_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-22191-1
Online ISBN: 978-3-030-22192-8
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)