The Euler-Poincaré Formula Through Contact Surfaces of Voxelized Objects

 

H. Sánchez-Cruz*¹, H. Sossa-Azuela², U-D. Braumann^3,4, E. Bribiesca⁵

 

¹ Centro de Ciencias Básicas Universidad Autónoma de Aguascalientes. Aguascalientes, Ags. México, * hsanchez@correo.uaa.mx

² Centro de Investigación en Computación Instituto Politécnico Nacional. México, D. F., México.

³ lnterdisziplinäres Zentrum für Bioinformatik Universität Leipzig, Leipzig, Germany.

⁴ Institut für Medizinische Informatik, Statistik und Epidmiologie, Universität Leipzig, Leipzig, Germany.

⁵ Instituto de Investigaciones en Matemáticas Aplicadas y Sistemas. UNAM. D.F., México.

 

ABSTRACT

Two new versions of the Euler-Poincaré formula are proposed considering two new defined cuboids: the tetra-voxel and the octo-voxel, without losing information on the number of vertices and edges. The well-known relationship between contact and enclosing surface concepts, as well as the relationships between vertices, edges and enclosing surfaces, allowed us to compute an innovative algorithm for obtaining alternative versions of the Euler-Poincaré formula. This is a very important topological descriptor of 3D binary images. We considered not only topological but geometric aspects. Our method was compared to other proposals, obtaining that our proposed contact surface-based method offers more advantages.

Keywords: Euler number, Euler characteristic, Euler-Poincaré, contact surfaces, tetra-voxels, octo-voxels, edges, vertices.

 

RESUMEN

Se proponen dos nuevas versiones de la fórmula Euler-Poncaré. Para ello se consideran dos definiciones de cuboides: los tetra-voxeles y los octo-voxeles, de forma que no haya pérdida de información en el número de vértices y aristas. La conocida relación entre superficie envolvente y superficie de contacto, así como sus relaciones con los vértices y aristas, nos permitió implementar un nuevo algoritmo para obtener versiones alternativas de la fórmula Euler-Poincaré, la cual es un descriptor topológico muy importante para imágenes binarias 3D. No solamente consideramos los aspectos geométricos sino también topológicos. El método propuesto fue comparado con otros, y obtuvimos que el nuestro, basado en la superficie de contacto, ofrece mayores ventajas.

 

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