Parametrization Of The Closed Triangular Surfaces For Surface Fitting

Surface parametrization is one of important problems in computer graphics and computer aided geometric design.It plays increasingly important roles in many fields of CG/CAGD.such as texture mapping,scattered data fitting,surface approximation and remeshing,morphing,and so on.Polygonal meshes,especially triangular meshes,are the most popular representation of surfaces,because of the flexibility and the high accuracy.All the surfaces computed in the thesis are triangular meshes.The parametrization of triangular meshes can be described as follows:Given a manifold triangular mesh which is composed by a set of spatial points and a suitable manifold parameter field,to seek for a one-to-one mapping for points from the parameter field to the triangular mesh,which causes that the mesh on the parameter field and the given mesh are topological isomorphism.With guaranteeing the triangles on the parameter field not overlap,the smallest distortion of some geometry measure between the two mesh is tried.Because of the surfaces'complexity and the applications' variety,there is not the best parametrization method.For a certain application,some parametrization methods might behave better than the others.The choice of different parametrizations depends heavily on the application details.In this thesis,we proposed the parametrization methods of closed surfaces for surface fitting.For genus zero closed surfaces,the parameter domain is sphere,so the cutting of the surface meshes in the parametrization process is avoided.This kind of parametrization, named as spherical parametrization,becomes the popular topic.We proposed a method to parameterize a genus zero mesh so that a surface fitting algorithm with the PHT-splines can generate good results.First,by generalizing Eck's planar parametrization,the model of spherical parametrization based on minimizing the discrete harmonic energy is given,but the computation is much more complicated. Then,two stable methods,the hierarchical method and the Lagrange-Newton method,are introduced.The hierarchical method computes the mesh vertices layer by layer,so the computing time is reduced greatly.The Lagrange-Newton method improved the details in the result.Through some examples,we can see that this spherical parametrization behaves better for surface fitting.If the number of the vertices on the given mesh is large scale,we use the mesh simplification before Lagrange-Newton parametrization,and through reverted the parametrization can be obtained quickly.For genus-one closed surfaces,the parametrization is toroidal,whose parameter domain is torus.We define the discrete harmonic energy and build the model of toroidal parametrization.The scale and the nonlinear degree of the model are much high,so the initial value is important to decide the solution.We cut the given genus one mesh using a closed circle which cannot shrink to a point in the mesh. The cut surface is mapped into the cylinder,and then is mapped to the torus by the formula.This result is a parametrization,but the triangles do not have good shapes.Setting the current result as an initial value,and find the optimum solution of the model using the Lagrange-Newton method.After the two steps,the toroidal parametrization can be obtained effectively.We propose a closed surface parametrization method based on a new theory of curves and surfaces.Curves and surfaces are the fundamental research objects in computer graphics and computer aided geometric design.We proposed a new concept:curves and surfaces are defined in the form of parametric mapping based on implicit domain.This expression combines the advantages of parametric and implicit expression.We define the implied parametric curves and surfaces,compute geometric invariants,and achieve the transformation between the implied parametric curves/surfaces and parametric/implicit curves/surfaces.Using the implied parametric theory,we map a given genus zero mesh surface to sphere,and obtain spherical parametrization.