Extracting cell complexes from 4-dimensional digital images

• Autores: Ana María Pacheco Martínez
• Directores de la Tesis: Pascal Lienhardt (dir. tes.), Pedro Real Jurado (dir. tes.), Jesús Nuñez Valdés (dir. tes.)
• Lectura: En la Universidad de Sevilla ( España ) en 2012
• Idioma: inglés
• Tribunal Calificador de la Tesis: Pedro Real Jurado (presid.), Ángel Ramón Francés Román (secret.), Pascal Lienhardt (voc.), Patrizio Frosini (voc.), Yukiko Kenmochi (voc.)
• Materias:
  • Matemáticas
    • Ciencia de los ordenadores
      • Informática
    • Geometría
• Enlaces
  • Tesis en acceso abierto en: Idus
• Resumen

  • A digital image can be defined as a set of n-xels on a grid made up by n-cubes.

    Segmentation consists in computing a partition of an image into regions. The n-xels having similar characteristics (color, intensity, etc.) are regrouped. Schematically, each n-xel is assigned a label, and each region of the image is made up by n-xels with the same label.

    The methods "type" Marching cubes and Kenmochi et al. construct complexes representing the topology of the region of interest of a 3-dimensional binary digital image. In the first method, the algorithm constructs a simplicial complex, whose 0-cells are points of the edges of the dual grid. In the second one, the authors construct a cell complex on a dual grid, i.e. the 0-cells of the complex are vertices of the dual grid. In order to construct the complex, Kenmochi et al. compute (up to rotations) the different configurations of white and black vertices of a cube, and then, they construct the convex hulls of the black points of these configurations. These convex hulls define the cells of the complex, up to rotations.

    The work developed in this thesis extends Kenmochi et al. method to dimension 4. The goal is to construct a cell complex from a binary digital image defined on a dual grid. First, we compute the different configurations of white and black vertices of a 4-cube, up to isometries, and then, we construct the convex hulls defined by these configurations. These convex hulls are constructed by deforming the original 4-cube, and we distinguish several basic construction operations (deformation, degeneracy of cells, etc.). Finally, we construct the cell complex corresponding to the dual image by assembling the cells so obtained.