*Forma,* Vol. 17 (No. 3), pp. 183-209, 2002

*Review*

## Euler Number and Connectivity Indexes of a Three Dimensional Digital Picture

Junichiro Toriwaki^{1}* and Tatsuhiro Yonekura^{2}

^{1}Department of Information Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

^{2}Department of Computer and Information Sciences, School of Engineering, Ibaraki University, 4-12-1, Nakanarusawa-cho, Hitachi, Ibaraki 316-8511, Japan

*E-mail address: toriwaki@nuie.nagoya-u.ac.jp

(Received July 25, 2002; Accepted September 4, 2002)

**Keywords: **
Digital Picture Processing, Euler Number, Connectivity, Digital Topology, Three Dimensional Picture

**Abstract. **
Fundamental properties of topological structure of a 3D digitized picture are presented including the concept of neighborhood and connectivity among volume cells (voxels) of 3D digitized binary pictures defined on a cubic grid, the concept of simplicial decomposition of a 3D digitized object, and two algorithms for calculating the Euler number (genus). First we define four types of connectivity. Second we present two algorithms to calculate the Euler number of a 3D figure. Thirdly we introduce new local features called the connectivity number (CN) and the connectivity index to study topological properties of a 3D object in a 3D digitized picture. The CN at a 1-voxel (=a voxel with the value 1) x is defined as the change in the Euler number of the object caused by changing the value of x into zero, that is, caused by deleting x. Finally, by using them, we prove a necessary and sufficient condition that a 1-voxel is deletable. A 1-voxel x is said to be deletable if deletion of x causes no decrease and no increase in the numbers of connected components, holes and cavities in a given 3D picture.

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