Numerical and variational aspects of mesh parameterization and editing [article]

Rhaleb Zayer, Universität Des Saarlandes, Universität Des Saarlandes
A surface parameterization is a smooth one-to-one mapping between the surface and a parametric domain. Typically, surfaces with disk topology are mapped onto the plane and genus-zero surfaces onto the sphere. As any attempt to flatten a non-trivial surface onto the plane will inevitably induce a certain amount of distortion, the main concern of research on this topic is to minimize parametric distortion. This thesis aims at presenting a balanced blend of mathematical rigor and engineering
more » ... d engineering intuition to address the challenges raised by the mesh parameterization problem. We study the numerical aspects of mesh parameterization in the light of parallel developments in both mathematics and engineering. Furthermore, we introduce the concept of quasi-harmonic maps for reducing distortion in the fixed boundary case and extend it to both the free boundary and the spherical case. Thinking of parameterization in a more general sense as the construction of one or several scalar fields on a surface, we explore the potential of this construction for mesh deformation and surface matching. We propose an "on-surface parameterization" for guiding the deformation process and performing surface matching. A direct harmonic interpolation in the quaternion domain is also shown to give promising results for deformation transfer. many persons. The first and foremost debt of gratitude is owed to Hans-Peter Seidel. His trust, vision, and commitment are the ground on which this inquiry could gradually develop. I also thank the members of my thesis committee, Bruno Lévy and Denis Zorin, for many valuable suggestions and for their interest in reviewing the work in this thesis. During the time I have spent at Max Planck Institute, I was fortunate to work closely with several talented people who left their fingerprints on the development of the ideas in this work. Unfortunately I can only mention few of their names. Special thanks to Christian Rössl whose experience and knowledge helped much in the shaping of this work. Common projects together with Ioannis Ivrissimtziss, Holger Theisel, Hitoshi Yamauchi, Stefan Gumhold, Zachi Karni, and Christian Theobalt opened wider possibilities for me and gave me the chance to learn from their broad expertise. The special environment at MPI had a great impact on the progress of this work. Credit and a big round of thanks go to all the AG4 members thanks to whom the academic and social life could not have been more pleasant. Keeping to the tradition, the people who keep you from becoming insane are mentioned last. I am most grateful to my family, and to Birgitta, Adel, and Trish for their tremendous understanding and ongoing inspiration and support.
doi:10.22028/d291-25917 fatcat:2om3ybbdvrfs3frmj7rhwpcszq