Abstract
This chapter introduces the notion of classes of shapes that have descriptive proximity to each other in planar digital 2D image object shape detection. A finite planar shape is planar region with a boundary (shape contour) and a nonempty interior (shape surface). The focus here is on the triangulation of image object shapes, resulting in maximal nerve complexes (MNCs) from which shape contours and shape interiors can be detected and described. An MNC is collection of filled triangles (called 2-simplexes) that have a vertex in common. Interesting MNCs are those collections of 2-simplexes that have a shape vertex in common. The basic approach is to decompose an planar region containing an image object shape into 2-simplexes in such a way that the filled triangles cover either part or all of a shape. After that, an unknown shape can be compared with a known shape by comparing the measurable areas of a collection of 2-simplexes covering both known and unknown shapes. Each known shape with a known triangulation belongs to a class of shapes that is used to classify unknown triangulated shapes. Unlike the conventional Delaunay triangulation of spatial regions, the proposed triangulation results in simplexes that are filled triangles, derived by the intersection of half spaces, where the edge of each half space contains a line segment connected between vertices called sites (generating points). A straightforward result of this approach to image geometry is a rich source of simple descriptions of plane shapes of image objects based on the detection of nerve complexes that are maximal nerve complexes or MNCs. The end result of this work is a proximal physical geometric approach to detecting and classifying image object shapes.
This research has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) discovery grants 194376, 185986 and Instituto Nazionale di Alta Matematica (INdAM) Francesco Severi, Gruppo Nazionale per le Strutture Algebriche, Geometriche e Loro Applicazioni grant 9 920160 000362, n.prot U 2016/000036.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Many thanks to the reviewer for pointing this out.
References
Adamaszek, M., Adams, H., Frick, F., Peterson, C., Previte-Johnson, C.: Nerve complexes on circular arcs, 1–17. arXiv:1410.4336v1 (2014)
Ahmad, M., Peters, J.: Delta complexes in digital images. Approximating image object shapes, 1–18. arXiv:1706.04549v1 (2017)
Alexandroff, P.: Über den algemeinen dimensionsbegriff und seine beziehungen zur elementaren geometrischen anschauung. Math. Ann. 98, 634 (1928)
Alexandroff, P.: Elementary concepts of topology. Dover Publications, Inc., New York (1965). 63 pp, translation of Einfachste Grundbegriffe der Topologie. Springer, Berlin (1932), translated by Alan E. Farley, Preface by D. Hilbert, MR0149463
Borsuk, K.: On the imbedding of systems of compacta in simplicial complexes. Fund. Math. 35, 217–234 (1948)
Borsuk, K.: Theory of shape. Monografie Matematyczne, Tom 59. (Mathematical Monographs, vol. 59) PWN—Polish Scientific Publishers (1975). MR0418088, Based on K. Borsuk, Theory of shape. Lecture Notes Series, No. 28, Matematisk Institut, Aarhus Universitet, Aarhus (1971), MR0293602
Borsuk, K., Dydak, J.: What is the theory of shape? Bull. Austral. Math. Soc. 22(2), 161–198 (1981). MR0598690
Bourbaki, N.: Elements of Mathematics. General Topology, Part 1. Hermann & Addison-Wesley, Paris (1966). I-vii, 437 pp
C̆ech, E.: Topological Spaces. John Wiley & Sons Ltd., London (1966). Fr seminar, Brno, 1936–1939; rev. ed. Z. Frolik, M. Katĕtov
Concilio, A.D., Guadagni, C., Peters, J., Ramanna, S.: Descriptive proximities I: properties and interplay between classical proximities and overlap, 1–12. arXiv:1609.06246 (2016)
Concilio, A.D., Guadagni, C., Peters, J., Ramanna, S.: Descriptive proximities. Properties and interplay between classical proximities and overlap, 1–12. arXiv:1609.06246v1 (2016)
Delaunay: Sur la sphère vide. Izvestia Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk (7), 793–800 (1934)
Edelsbrunner, H.: Modeling with simplical complexes. In: Proceedings of the Canadian Conference on Computational Geometry, pp. 36–44, Canada (1994)
Edelsbrunner, H., Harer, J.: Computational Topology. An Introduction. American Mathematical Society, Providence, R.I. (2010). Xii+110 pp., MR2572029
Efremovič, V.: The geometry of proximity I (in Russian). Mat. Sb. (N.S.) 31(73)(1), 189–200 (1952)
Fontelos, M., Lecaros, R., López-Rios, J., Ortega, J.: Stationary shapes for 2-d water-waves and hydraulic jumps. J. Math. Phys. 57(8), 081520, 22 pp (2016). MR3541857
Gratus, J., Porter, T.: Spatial representation: discrete vs. continuous computational models: a spatial view of information. Theor. Comput. Sci. 365(3), 206–215 (2016)
Guadagni, C.: Bornological convergences on local proximity spaces and \(\omega _{\mu }\)-metric spaces. Ph.D. thesis, Università degli Studi di Salerno, Salerno, Italy (2015). Supervisor: A. Di Concilio, 79 pp
Kokkinos, I., Yuille, A.: Learning an alphabet of shape and appearance for multi-class object detection. Int. J. Comput. Vis. 93(2), 201–225 (2011). https://doi.org/10.1007/s11263-010-0398-7
Leray, J.: L’anneau d’homologie d’une reprësentation. Les Comptes rendus de l’Académie des sciences 222, 1366–1368 (1946)
Lodato, M.: On topologically induced generalized proximity relations, Ph.D. thesis. Rutgers University (1962). Supervisor: S. Leader
Maggi, F., Mihaila, C.: On the shape of capillarity droplets in a container. Calc. Var. Partial Differ. Equ. 55(5), 55:122 (2016). MR3551302
Naimpally, S., Peters, J.: Topology with Applications. Topological Spaces via Near and Far. World Scientific, Singapore (2013). Xv + 277 pp, American Mathematical Society. MR3075111
Opelt, A., Pinz, A., Zisserman, A.: Learning an alphabet of shape and appearance for multi-class object detection. Int. J. Comput. Vis. 80(1), 16–44 (2008). https://doi.org/10.1007/s11263-008-0139-3
Peters, J.: Near sets. General theory about nearness of sets. Applied. Math. Sci. 1(53), 2609–2629 (2007)
Peters, J.: Near sets. Special theory about nearness of objects. Fundamenta Informaticae 75, 407–433 (2007). MR2293708
Peters, J.: Near sets: an introduction. Math. Comput. Sci. 7(1), 3–9 (2013). http://doi.org/10.1007/s11786-013-0149-6. MR3043914
Peters, J.: Topology of Digital Images - Visual Pattern Discovery in Proximity Spaces, vol. 63. Intelligent Systems Reference Library. Springer (2014). Xv + 411 pp. Zentralblatt MATH Zbl 1295, 68010
Peters, J.: Visibility in proximal Delaunay meshes and strongly near Wallman proximity. Adv. Math. Sci. J. 4(1), 41–47 (2015)
Peters, J.: Computational Proximity. Excursions in the Topology of Digital Images. Intelligent Systems Reference Library 102. Springer (2016). Viii + 445 pp. http://doi.org/10.1007/978-3-319-30262-1
Peters, J.: Proximal relator spaces. Filomat 30(2), 469–472 (2016). MR3497927
Peters, J.: Two forms of proximal physical geometry. axioms, sewing regions together, classes of regions, duality, and parallel fibre bundles, 1–26. arXiv:1608.06208 (2016). To appear in Adv. Math. Sci. J. 5 (2016)
Peters, J.: Foundations of Computer Vision. Computational Geometry, Visual Image Structures and Object Shape Detection. Intelligent Systems Reference Library 124. Springer International Publishing, Switzerland (2017). I-xvii, 432 pp. http://doi.org/10.1007/978-3-319-52483-2
Peters, J.: Computational Topology of Digital Images. Visual Structures and Shapes. Intelligent Systems Reference Library. Springer International Publishing, Switzerland (2018)
Peters, J., Guadagni, C.: Strongly near proximity and hyperspace topology, 1–6. arXiv:1502.05913 (2015)
Peters, J., Ramanna, S.: Maximal nucleus clusters in Pawlak paintings. Nerves as approximating tools in visual arts. In: Proceedings of Federated Conference on Computer Science and Information Systems 8 (ISSN 2300-5963), 199–2022 (2016). http://doi.org/10.15439/2016F004
Peters, J., Tozzi, A., Ramanna, S.: Brain tissue tessellation shows absence of canonical microcircuits. Neurosci. Lett. 626, 99–105 (2016)
Rényi, A.: On measures of entropy and information. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 547–547. University of California Press, Berkeley (2011). Math. Sci. Net. Review MR0132570
de Verdière, E., Ginot, G., Goaoc, X.: Multinerves and helly numbers of acylic families. In: Proceedings of 28th Annual Symposium on Computational Geometry, pp. 209–218 (2012)
Voronoï, G.: Nouvelles applications des paramètres continus à la théorie des formes quadratiques. premier mémoir. J. für die reine und angewandte math. 133, 97–178 (1907). JFM 38.0261.01
Wallman, H.: Lattices and topological spaces. Ann. Math. 39(1), 112–126 (1938)
Ziegler, G.: Lectures on polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995). X+370 pp. ISBN: 0-387-94365-X, MR1311028
Ziegler, G.: Lectures on Polytopes. Springer, Berlin (2007). http://doi.org/10.1007/978-1-4613-8431-1
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Peters, J.F., Ramanna, S. (2018). Shape Descriptions and Classes of Shapes. A Proximal Physical Geometry Approach. In: Stańczyk, U., Zielosko, B., Jain, L. (eds) Advances in Feature Selection for Data and Pattern Recognition. Intelligent Systems Reference Library, vol 138. Springer, Cham. https://doi.org/10.1007/978-3-319-67588-6_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-67588-6_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-67587-9
Online ISBN: 978-3-319-67588-6
eBook Packages: EngineeringEngineering (R0)