# On the Computation of the Number of Bubbles and Tunnels of a 3-D Binary Object

### Humberto Sossa

#### Abstract

Two formulations and two general procedures useful to compute the number of bubbles and tunnels of 3-D binary objects are introduced in this paper. The first formulation is useful to compute the number of bubbles or voids of the object, while the second one is only useful to compute the number of tunnels or holes of the object. The first procedure allows obtaining, in two steps, the number of bubbles and tunnels of a 3-D object. Finally, the second procedure permits computing, in three steps, the number of bubbles and tunnels of several 3-D objects from an image containing them. Correctness of the functioning of the two formulations is established theoretically. Results with a set of images are provided to demonstrate the utility and validity of the second proposed procedures.

#### References

- Uchiyama, T. et al., 1999. Three-Dimensional Microstructural Analysis of Human Trabecular Bone in Relation to Its Mechanical Properties. Bone 25(4):487- 491.
- Vogel, HJ. and Roth, K., 2001. Quantitative morphology and network representation of soil pore structure. Advances in Water Resources 24:233-242.
- Velichko, A. et al., 2008. Unambiguous classification of complex microstructures by their three-dimensional parameters applied to graphite in cast iron. Acta Materialia 56:1981-1990.
- Roque, WL. et al., 2009. The Euler-Poincaré characteristic applied to identify low bone density from vertebral tomographic images. Rev Bras Reumatol 49(2):140-52.
- Lin, X. et al., 2008. A New Approach to Compute the Euler Number of 3D Image. In 3rd IEEE Conference on Industrial Electronics and Applications, pp. 1543- 1546.
- Lee, CN. et al., 1991. Winding and Euler numbers for 2D and 3D digital images, CVGIP: Graph. Models Image Process, 53(6):522-537.
- Lee, CN. et al. 1993. Holes and Genus of 2D and 3D digital images, CVGIP: Graph. Models Image Process, 55(1):20-47.
- Gray, SB., 1970. Local Properties of Binary Images in Two and Three Dimensions, Boston: Information International Inc.
- Park, CM. and Rosenfeld, A., 1971, Connectivity and Genus in Three Dimension, TR-156, Computer Vision Laboratory, Computer Science Center, University of Maryland, College Park, MD.
- Kong, TY. and Rosenfeld, A., 1987. Digital Topology: Introduction and Survey, Computer vision Graphic and Image Processing, 48:357-393.
- Lee, CN. and Rosenfeld, A., 1987. Computing the Euler Number of a 3D Image, In Proceedings of the IEEE First International Conference on Computer Vision, pp. 567-571.
- Bonnassie, A. et al., 2001. Shape description of threedimensional images based on medial axis. In Proceedings of the 2001 International Conference on Image Processing, pp. 931-934.
- Toriwaki, J. and Yonekura, T., 2002. Euler Number and Connectivity Indexes of a Three Dimensional Digital Picture. Forma, 17:183-209.
- Schladitz, K. et al., 2006. Measuring Intrinsic Volumes in Digital 3D Images. A. Kuba, L.G. Nyúl, and K. Palágyi (Eds.): In DGCI 2006, LNCS 4245, pp. 247-258.
- Saha, PK. and Chaudhuri, BB., 1995. A new approach to computing the Euler Characteristic. Pattern Recognition 28(12):1955-1963.
- Gonzalez, R. and Woods, R., 2002. Digital Image Processing. Prentice Hall.

#### Paper Citation

#### in Harvard Style

Sossa H. (2016). **On the Computation of the Number of Bubbles and Tunnels of a 3-D Binary Object** . In *Proceedings of the 5th International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,* ISBN 978-989-758-173-1, pages 17-23. DOI: 10.5220/0005629800170023

#### in Bibtex Style

@conference{icpram16,

author={Humberto Sossa},

title={On the Computation of the Number of Bubbles and Tunnels of a 3-D Binary Object},

booktitle={Proceedings of the 5th International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,},

year={2016},

pages={17-23},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005629800170023},

isbn={978-989-758-173-1},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 5th International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,

TI - On the Computation of the Number of Bubbles and Tunnels of a 3-D Binary Object

SN - 978-989-758-173-1

AU - Sossa H.

PY - 2016

SP - 17

EP - 23

DO - 10.5220/0005629800170023