Theory And Application Of Splines Over Hierarchical T-meshes

Geometric modeling is the kernel of Computer-Aided Geometric Design(CAGD) and Computer Graphics(CG),the main contents of which are representation,design, display,analysis for curves,surfaces,solids and so on.Free-form curves and surfaces modeling is the most important technology in geometric modeling.At present,interpolation, fitting and approximation have formed a theoretical system of geometric modeling,whose principal parts are the Non-uniform Rational B-spline(NURBS) and implicit algebraic surfaces.The method of NURBS has become a standard in Computer-Aided Design/Manufacture(CAD/CAM) because of its uniform mathematical model.Although NURBS has advantages such as parametric representation and exact expression of conical surface,its topology must be restricted on tensor-product meshes,which is a serious drawback.This means that typically,a large number of NURBS control points serve no purpose other than satisfying topological constraints. In order to overcome these problems,Sederberg[Sederberg 2003,Sederberg 2004]introduced T-spline which allows T-junctions in the control nets.T-spline is a generalization of NURBS surface which is capable of significantly reducing the number of superfluous control points.Deng et al.invented the spline over T-mesh in[Deng 2006],which is single polynomial in each cell of the T-mesh.This thesis focuses on the splines over hierarchical T-meshes.First,we discuss the dimension of the spline space over hierarchical T-meshes S(m,n,m-1,n-1,(?)) for m = n = 2 and give a lower bound of dimension for higher degree spline spaces.The most important technique in the dimension analysis is the linear space embedding by an operator of mixed partial derivative,which embeds the space S(m,n,m - 1,n - 1,(?)) into the space S(m - 1,n - 1,m - 2,n - 2,(?)).Then,we define a new type of spline - submesh spline,which is defined in terms of some tensor-product B-splines.We provide an effective algorithm to locate the valid tensor-product submeshes and define the submesh functions over them. The submesh functions do not form a basis for the spline space,but they have some good properties such as nonnegativity,local support and forming a partition of unity.The local refinement algorithm and the surface fitting algorithm for open mesh models based on submesh splines are given as well.Our method needs less control points compared with PHT-splines in[Deng 2008].Besides,based on the implicit splines over 3D hierarchical T-meshes and the moving parabolic approximation(MPA) method in[Yang 2007b],we develop a selfadaptive implicit surface reconstruction method for unorganized point clouds.In order to reconstruction the geometric details,we subdivide the T-meshes level by level adaptively according to the approximation error.For every new basis vertex, we estimate the geometric information by MPA algorithm and compute the control coefficients for the basis functions of this basis vertex.Then we can get the implicit surfaces quickly without solving linear equations.In the end,we propose a type of C~1-continuous bicubic(denoted as(3,3,1,1)) hierarchical B-splines from the general hierarchical B-splines.Then we present a transformation algorithm between(3,3,1,1) hierarchical B-splines and(3,3,1,1) polynomial splines over hierarchical T-meshes.A surface expressed in terms of hierarchical B-splines can be exactly converted into a polynomial splines over hierarchical T-meshes,and vice versa.However,the conversion from T-spline in[Sederberg 2003]to hierarchical B-splines is approximation.