Transformations quasi-conformes de maillages volumiques et applications en infographie
Geometric modeling is important for both computer graphics and engineering. Our ability to represent geometric information sets the limits and the ease with which we manipulate 3D objects. One such representation is the volume mesh, that is composed of polyhedra assembled to approximate a desired shape. Some applications, such as texturing and remeshing, benefit from deforming the mesh to a more regular domain in order to perform some operations. We say that a deformation is \emph{quasi-conformal} if its distortion is bounded.  In this thesis, we propose algorithms for quasi-conformal deformations of volume meshes. We study these deformations because of their good local shape preservation properties and because they are still relatively unknown to the graphics community, as opposed to their 2D counterparts. This research attempts to generalize \gilles{some well-known surface deformation concepts to volumes}.  First, we present a linear approach to quasi-conformality. We develop a method that deforms a solid to a parameterization domain using a linear least squares method. This method is fast and simple to implement, but the result is an approximation to quasi-conformality because distortion is not bounded. Second, we solve this latter limitation with a nonlinear approach based on vertex positions. We develop a technique to deform the parameterization domain to a solid shape using a nonlinear least squares method. Nonlinearity lets us include constraints that guarantee the injectivity of the deformation. Moreover, deforming the parameterization domain instead of the shape itself lets us use more general domains. Third, we present a nonlinear approach based on dihedral angles. Our method defines the deformation of the volume mesh using its dihedral angles instead of its vertex positions. This change of variables permits a natural expression of the bounds of the deformation distortion. We present some applications of this new approach that include volume parameterization, shape interpolation, mesh optimization, and mesh compression of tetrahedral meshes.
Compression Deformation Parameterization Quasiconformality Computer graphics Fitting Compression Optimization