Abstract
Given a 3D binary digital image I, we define and compute an edge-weighted tree, called Homological Region Tree (or Hom-Tree, for short). It coincides, as unweighted graph, with the classical Region Adjacency Tree of black 6-connected components (CCs) and white 26-connected components of I. In addition, we define the weight of an edge (R, S) as the number of tunnels that the CCs R and S “share”. The Hom-Tree structure is still an isotopic invariant of I. Thus, it provides information about how the different homology groups interact between them, while preserving the duality of black and white CCs.
An experimentation with a set of synthetic images showing different shapes and different complexity of connected component nesting is performed for numerically validating the method.
Work supported by the Spanish research projects TOP4COG:MTM2016-81030-P (AEI/FEDER, UE), COFNET (AEI/FEDER, UE) and the VPPI of University of Seville.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ansaldi, S., De Floriani, L., Falcidieno, B.: Geometric modeling of solid objects by using a face adjacency graph representation. In: ACM SIGGRAPH Computer Graphics, vol. 19, no. 3, pp. 131–139. ACM, July 1985
Bertrand, G.: Simple points, topological numbers and geodesic neighborhoods in cubic grids. Pattern Recogn. Lett. 15, 1003–1011 (1994)
Cardoze, D.E., Miller, G.L., Phillips, T.: Representing topological structures using cell-chains. In: Kim, M.-S., Shimada, K. (eds.) GMP 2006. LNCS, vol. 4077, pp. 248–266. Springer, Heidelberg (2006). https://doi.org/10.1007/11802914_18
Costanza, E., Robinson, J.: A region adjacency tree approach to the detection and design of fiducials. In: Video, Vision and Graphics, pp. 63–99 (2003)
Cucchiara, R., Grana, C., Prati, A., Seidenari, S., Pellacani, G.: Building the topological tree by recursive FCM color clustering. In: Object Recognition Supported by User Interaction for Service Robots, vol. 1, pp. 759–762. IEEE, August 2002
Cohn, A., Bennett, B., Gooday, J., Gotts, N.: Qualitative spacial representation and reasoning with the region connection calculus. GeoInformatica 1(3), 275–316 (1997)
Delgado-Friedrichs, O., Robins, V., Sheppard, A.: Skeletonization and partitioning of digital images using discrete Morse theory. IEEE Trans. Pattern Anal. Mach. Intell. 37(3), 654–666 (2015)
Díaz-del-Río, F., Real, P., Onchis, D.: Labeling color 2D digital images in theoretical near logarithmic time. In: Felsberg, M., Heyden, A., Krüger, N. (eds.) CAIP 2017. LNCS, vol. 10425, pp. 391–402. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-64698-5_33
Díaz-del-Río, F., Real, P., Onchis, D.M.: A parallel homological spanning forest framework for 2D topological image analysis. Pattern Recogn. Lett. 83, 49–58 (2016)
Forman, R.: Morse theory for cell complexes. Adv. Math. 134, 90–145 (1998)
Klette, R., Rosenfeld, A.: Digital Geometry Geometric: Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)
Klette, G.: Skeletons in digital image processing. CITR, The University of Auckland, New Zealand (2002)
Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Elsevier, Amsterdam (2004)
Kong, T.Y., Rosenfeld, A.: Topological Algorithms for Digital Image Processing, vol. 19. Elsevier, Amsterdam (1996)
Kong, T.Y., Roscoe, A.W.: A theory of binary digital pictures. Comput. Vis. Graph. Image Process. 32(2), 221–243 (1985)
Kovalevsky, V.: Algorithms in digital geometry based on cellular topology. In: Klette, R., Žunić, J. (eds.) IWCIA 2004. LNCS, vol. 3322, pp. 366–393. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30503-3_27
Lienhardt, P.: Topological models for boundary representation: a comparison with n-dimensional generalized maps. Comput. Aided Des. 23(1), 59–82 (1991)
Molina-Abril, H., Real, P., Nakamura, A., Klette, R.: Connectivity calculus of fractal polyhedrons. Pattern Recogn. 48(4), 1150–1160 (2015)
Molina-Abril, H., Real, P.: Homological spanning forest framework for 2D image analysis. Ann. Math. Artif. Intell. 64(4), 385–409 (2012)
Pavlidis, T.: Algorithms for Graphics and Image Processing. Springer, Heidelberg (1997)
Real, P., Molina-Abril, H., Díaz-del-Río, F., Blanco-Trejo, S., Onchis, D.: Enhanced parallel generation of tree structures for the recognition of 3D images. In: Carrasco-Ochoa, J., Martínez-Trinidad, J., Olvera-López, J., Salas, J. (eds.) MCPR 2019. LNCS, vol. 11524, pp. 292–301. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-21077-9_27
Real, P., Diaz-del-Rio, F., Onchis, D.: Toward parallel computation of dense homotopy skeletons for nD digital objects. In: Brimkov, V.E., Barneva, R.P. (eds.) IWCIA 2017. LNCS, vol. 10256, pp. 142–155. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59108-7_12
Rosenfeld, A.: Adjacency in digital pictures. Inf. Control 26(1), 24–33 (1974)
Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, Cambridge (1982)
Stell, J., Worboys, M.: Relations between adjacency trees. Theoret. Comput. Sci. 412(34), 4452–4468 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Real, P., Molina-Abril, H., Díaz-del-Río, F., Blanco-Trejo, S. (2019). Homological Region Adjacency Tree for a 3D Binary Digital Image via HSF Model. In: Vento, M., Percannella, G. (eds) Computer Analysis of Images and Patterns. CAIP 2019. Lecture Notes in Computer Science(), vol 11678. Springer, Cham. https://doi.org/10.1007/978-3-030-29888-3_30
Download citation
DOI: https://doi.org/10.1007/978-3-030-29888-3_30
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-29887-6
Online ISBN: 978-3-030-29888-3
eBook Packages: Computer ScienceComputer Science (R0)