Digital geometry fundaments: application to plane recognition

Triangulation, quadrangulation problems and more generally 3D object polyhedrization are an important subject of research. In digital geometry, a 3D object is seen as a set of voxels placed in a representation space only constituted of integers. The objective of the polyhedrization is to obtain a complete description of the object with faces, edges and vertices. The recognition of digital planes is a first step which is very important. We focus on digital naive planes that have been studied through their configurations of tricubes: of (n,m)-cubes and connected or not connected voxels set. The link between the normal equation of a plane and configuration of voxels set has been studied by the construction of the corresponding Farey net. We can find many references about the recognition of digital planes. Some algorithms were related to the construction of the convex hull of the studied voxels set. Other approaches use linear programming, mean square approximation or Fourier-Motzkin transform. The first algorithms entirely discrete recognized rectangular pieces of naive planes. Wwe describe an incremental algorithm to recognize any coplanar voxels set as a digital naive plane by using Farey nets. Then we propose a polyhedrization method able to give all the digital naive planes of the surface of the 3D object.