Variational geometry processing

Abstract

In this thesis, two basic topics in geometry processing—curve/surface smoothing and reconstruction are discussed. The variational approaches are used to address these two closely related problems. The methods are based on defining suitable cost functionals to be minimized, and the cost is the combination of a fidelity term and a smoothness term. By utilizing different representations of interfaces, the energy functional assumes different forms and is minimized by disparate methodologies. In this thesis 3 different representations and 4 different minimization methods are discussed. First two chapters use Partial-Differential-Equation-based (PDE-based) methods to solve curve smoothing problems, in which level-set modelling and phase-field modelling are used respectively. Piece-wise constant functions are used to represent interfaces by discon- tinues of the functions in Chapters 4, 5, and 6, and combinatorial optimization tech- niques are applied for the minimization problems. In particular, Chapter 4 discusses a general energy functional framework suitable for geometry processing applications and the corresponding graph-cuts minimization. Under the same variational framework, Chapter 5 solves the minimization problem via centroidal-Voronoi-tessellation-based (CVT-based) methods and discusses multi-phase the problems in terms of clustering language. Chapter 6 dedicates to the surface reconstruction problem, in which the functional is minimized by multi-way graph-cuts on a Delaunay-based tetrahedral mesh so that the advantages of explicit and implicit methods for surface reconstruc- tion are well integrated. Numerous examples substantiate the effectiveness, efficiency and robustness of the proposed methods. In Chapter 7, a systematic comparison is conducted through various examples.