Ph.D Thesis | |

Ph.D Student | Ben-Chen Mirela |
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Subject | Discrete Geometric Algorithms for Mesh Processing |

Department | Department of Computer Science |

Supervisor | PROF. Chaim Craig Gotsman |

Full Thesis text |

Geometric modeling deals with representing real objects in a virtual world. A popular geometric representation is the polygonal model. For many applications the need arises to improve the model while preserving its intrinsic geometric property, that it still describes the same "real" object. In this research, we investigate a few such applications - conformal mappings of 3D models to the plane, shape retrieval, space deformation and animation transfer. Although the applications are quite different, they use similar geometric concepts and mathematical machinery, the most noticeable being the notion of conformality as a change to the shape which preserves its essence.

We first consider the problem of planar mesh parameterization. We show how a simple discretization of a classical equation for conformal maps on continuous surfaces can be applied to generate high quality planar mappings in an efficient manner. In addition, we show how the conformal factor, a function on the mesh, which is related to the local scaling the surface should undergo in order to be flattened to the plane, can be used a shape signature, for shape matching and retrieval.

Later, we address the problem of 3D shape editing and deformation, useful in animation applications. Deformation tasks are extremely time consuming, so the challenge there is "say less, do more", the user should specify as little as possible, and the deformation method should deduce the rest. Here, again, conformal and harmonic maps play an important role, as they allow the user to modify the global shape, while preserving the small details. We introduce a novel space deformation method based on harmonic maps, and show in addition how to use the same framework in order to transfer animations from one shape to another.