【作者】 熊赟晖；

【导师】 韩国强；

【作者基本信息】 华南理工大学 ， 计算机应用技术， 2010， 博士

【摘要】 多边形网格已经成为三维形状的一种标准表示。不过从点云重构的三角网格,其三角形的形状、顶点的邻接关系大多不规则,使得网格模型的压缩、形状计算、结构分析等操作变得困难,常常会导致数值算法不稳定,影响对模型的正确分析与处理。重新网格化是提高模型质量、构造多分辨率表示的一种重要途径。三角网格模型在计算机图形学和3D游戏、动画、虚拟现实等应用领域被广泛采用。四边形网格及4-3混合网格则在参数曲面拟合、有限元计算、纹理映射和编辑变形等方面具有优势。本文对这两类网格的重网格化进行研究,但侧重于四边形重新网格化方法及相关技术。本文的贡献表现在三个方面:四边形网格上拉普拉斯算子的离散化,基于流线的三角网格重新网格化和基于Morse理论的四边形重新网格化。1)提出了一种新的四边形网格上的拉普拉斯算子,称之为平均拉普拉斯算子,简记为MLBO。采用穷举法,将顶点p相邻空间四边形全部的三角剖分的可能形式全部找出来,将这些可能的三角剖分结果的面积平均作为顶点p附近区域的一个估计,然后利用三角网格上的拉普拉斯算子给出四边形网格上权值对称的拉普拉斯算子。2)提出一种基于场的重新三角网格化方法,先在原三角网格模型上建立拉普拉斯标量场,据此生成两组夹角为60°的流线.然后,从这两组流线构造以菱形面为主的网格并三角化得到三角基网格.最后,在基网格的基础上,再次使用流线技术对原始数据进行向上重采样,得到模型的多分辨率表示.一般地,基网格的三角形接近等边三角形,实验分析表示所生成的三角网格模型具有较高的质量。3)在Morse函数理论的基础上,提出了一种从三角网格中建立四边形网格多分辨率表示的方法.先由人工指定或从拉普拉斯矩阵的特征函数中提取临界点,计算带约束的拉普拉斯方程得到光滑的Morse函数.函数的临界点(极大、极小和鞍点)有规律的分布在模型表面,在三角网格表面梯度场的引导下,生成临界点间流线,得到临界点间的拓扑关系.通过临界点交换规则,同样是采用流线的方法,得到更精细的四边形网格。最终无需参数化而仅用流线方法就可建立多分辨率四边形网格。更多还原

【Abstract】 With the development of 3D scanning and other data acquisition techniques, polygonal meshes have become a de facto representation for 3D shapes. Nevertheless, in triangular meshes directly reconstructed from point clouds, face shape is generally bad and vertices are usually irregular. This makes it difficult to perform subsequent operations such as compression, shape computation, and structure analysis, and even leads to unstable numerical algorithms in some serious cases. Remeshing is an important way to improve mesh quality and to construct multi-resolution representation.Triangular meshes are widely used in 3D game, computer animation, and virtual reality. On the other hand, quadrilateral meshes are more favorable in applications such as parametric surface fitting, texturing, and FEA-based simulations. Both triangular remeshing and quadrilateral remeshing are studied in this paper but with emphasis on application of quadriangulation techniques.The constributions of this paper are as follows.1) A new discrete form is proposed for evaluation of Laplace–Beltrami operator over a quadrilateral mesh. It is derived by applying the discrete LBO for triangular meshes to all possible triangulations of the quadrilateral mesh, and therefore is named as mean Laplace–Beltrami operator (abbr. as MLBO). A symmetric MLBO is also explored.2) A triangular remeshing algorithm is constructed by tracing streamlines of scalar fields. It firstly establishes a Laplacian field on a given original mesh, and then generates two groups of streamlines with one group being a 60°rotational version of the other. Secondly, a rhombus-dominant mesh is constructed from the two groups of streamlines and then triangulated into a triangular base mesh. A multiresolution representation can be obtained by upsampling the original mesh through refining the base meshusing the streamline tracing again. Theoretically, triangles of the remeshing results should be close to be equilateral.. It has been approved by experiments.3) A quad mesh based multiresolution framework is established for given triangular meshes based on Morse theory. Firstly, a smooth Morse function is defined as the solution of a Laplacian equation with constraints in which critical points are either specified by user or exacted from an eigenfunction of the Laplacian matrix of the mesh.According to Morse theory, a coarse quad mesh can be produced by connecting the critical points of the function carefully. Finaly, a critical point exchange rule is designed to generate the structure of the finer quad mesh whose connectivity is also generated using stream line tracing. Parameterization is not required in the process as all resolution levels are created using the streamlines method.更多还原