【作者】 李桂清；

【导师】 李华；

【作者基本信息】 中国科学院计算技术研究所 ， 计算机应用技术， 2001， 博士

【摘要】 本文研究细分曲面造型方法其及在图形学中的应用. 细分方法近年来已成为图形学领域的一项重要研究内容. 但是,要进一步拓广细分方法的应用范围(尤其在CAD 领域),还有很多工作要做. 例如,高阶连续细分模式的构造、细分方法与解析方法的融合、细分曲面的光顺、细分曲面间的逻辑运算、数学工具的改善以及满足各种需要的新模式的发展等等. 本文的目的是研究一些行之有效的造型方法,以进一步提高细分曲面的造型能力. 所做的工作包括尖锐特征造型、基于细分曲面的参数曲面混合、3D 离散光顺曲线和开网格上离散光顺曲面的构造、B 样条曲面法向插值及三角网格简化. 已有的尖锐特征造型方法基本都是在原来细分模式的基础上通过修改几何规则来达到目的. 存在如下不足之处:首先,新规则的构造不直观,即不容易从明确的几何意义来指导新面具(Mask)的选取;其次,由于新规则的引进,需要重新对这些特殊的规则进行连续性分析. 本文提出的方法通过修改控制网格的拓扑结构来实现尖锐特征造型. 具体地, 通过边和顶点分裂操作把控制网格上需要生成尖锐特征的相关顶点和内部边组成的路径变成两条由边界顶点和边界边组成的路径. 由于细分曲面边界由边界顶点唯一确定而与内部顶点无关,具有相同顶点位置的边界路径生成的边界曲线是重合的,这就保证了曲面沿这些分裂而成的边界是连续但不光滑的. 利用细分曲面对参数曲面进行混合及对参数曲面围成的N 边洞作G~1和G~2连续填充是基于这样的考虑: 希望既保持参数曲面的优势又能借助细分曲面特点处理复杂的曲面造型问题. 本文对已有的细分模式引入一种新的边界生成方法(称为轮廓删除法)以增强细分方法在边界上的造型能力. 在此基础上,只要适当地构造混合或填充曲面的初始控制网格,就可以使基于轮廓删除的细分曲面与相应的参数曲面之间C~2连续地拼接在一起. 一般说来曲率分布越简单则曲面越光顺. Schneider和Kobbelt 的平面离散布状样条曲线和封闭控制网格上的离散布状样条曲面正是利用曲率逐段或逐片线性分布构造出来的. 本文试图把平面离散布状样条的概念推广到三维空间,并在此基础上定义开网格上的离散布状样条曲面作为封闭控制网格上的离散布状样条曲面的推广. 文中引入的离散Frenet 标架的概念是本文这一工作的基础. 所更多还原

【Abstract】 This thesis studies modeling and application of subdivision surfaces. Subdivision method has become an important tool in computer graphics recently. To extend application areas (especially in the area of CAD) more widely, however, many problems have to be solved. The problems include constructing subdivision schemes of higher continuity, fusing subdivision and analytic methods, fairing subdivision surfaces, intersecting subdivision surfaces, developing mathematical tool for analysis of subdivision surfaces and designing new subdivision schemes satisfying all kinds of requirements etc. The thesis investigates some effective modeling approaches for enhancing modeling ability of the subdivision methods. One of the contributions of this thesis is to propose an approach to generate sharp features for subdivision surfaces. Most of the known methods produce the sharp features by modifying weights of geometric rules of primitive subdivision schemes. It is not intuitive to select new weights satisfying conditions. Moreover, continuity analysis must be performed again for introduction of new rules. The approach described in this thesis is based on topology modification of control meshes. The main operation is to update the connectivity relation of edges and vertices associated with sharp features such that the internal edges and vertices become boundary ones. It is easy to guarantee that the surfaces are not smooth along the boundary on account of the property that the boundary curves of the subdivision surfaces are uniquely determined by the boundary polygons of the meshes. The thesis also employs subdivision surfaces to blend parametric surfaces and fill n -sided holes surrounded by parametric surfaces, which is based upon the consideration that while modeling complicated objects we can combine the advantages of both parametric surfaces and subdivision surfaces. An approach (named skirt-removed approach) is presented for generating subdivision surface boundary by removing boundary vertices, edges and faces of control meshes. On basis of the skirt-removed approach a unified framework is built to blend multiple patches of parametric spline surfaces and to fill n -sided hole G 1and G 2 continuously with subdivision surfaces by appropriately constructing primitive control meshes. Generally speaking, the simpler the curvature distribution, the fairer the surface. Planar discrete clothoid splines (PDCSs) and discrete clothoid spline (DCS) surfaces of Schneider and Kobbelt are just constructed by assuming that the curvature of the curves or surfaces is piecewise linearly distributed. The thesis extends the notion of PDCS to three-dimension (3D) space and defines the DCS surfaces on open meshes. Discrete Frenet frame introduced in the thesis is the key ingredient of the work. The curvature of the 3D DCS and the DCS surface on an open mesh obtained by our method is piecewise linear distributed in terms of the least squares. In addition, the DCS surface has 3D DCSs as its boundary. The thesis has also investigated the application of the subdivision method. Firstly, properties of the subdivision method are applied to parametric surface modeling and thus algorithms are designed to interpolate vertices and their normal vectors with quadratic and bicubic B-spline surfaces respectively. Secondly, application on triangle mesh simplification is also considered. In a kind of simplification algorithms based on triangle collapse, new vertices should be computed for substituting the collapsed triangles. Some approaches calculate new vertices by resampling from a quadratic surface fitting the vertices near the collapsed triangles. The fitting operation is finished through solving a linear system which is probably unstable and even hasn’t solution due to the ill-conditioned data. Other algorithms simply select the vertices or centers of the collapsed triangles as new vertices. The solution may result in large error because of linear sample. The thesis treats the primitive surfaces as butterfly subdivision surfaces with meshes required decimation as primitive control meshes and makes out new vertices using butterfly scheme. Therefore it overcomes the disadvantages appearing in traditional approaches.更多还原