【作者】 何军；

【导师】 张彩明；

【作者基本信息】 山东大学 ， 计算机软件与理论， 2009， 博士

【摘要】 随着三维扫描和相关技术的进步,三维数字几何模型已成为一种新兴的数字媒体,在三维游戏,计算机动画,电影特效,工业造型设计,计算机仿真以及数字文化遗产保护等方面取得了日益广泛的应用。针对这类数据进行高效的处理,也成为计算机图形学的一个研究热点,即数字几何处理(DGP:Disital GeometryProcessing)。数字几何处理研究的问题主要包括三维数字几何模型的建模、处理和应用等方面。近十多年来,无论是学术界还是工业界,数字几何处理都引起了人们的关注,技术也得到了很大的发展,但是,随着相应研究的不断深入和应用范围的愈加广泛,数字几何处理研究中仍然充满着挑战。本文主要讨论曲面形式的三维模型表示。曲面一般以连续形式或者离散形式表示。连续形式主要包括样条曲面、隐式曲面和细分曲面,而离散形式主要是网格和点云。基于网格形式表示的数字几何,本文主要围绕网格上的样条曲面拟合和变形技术进行了深入探讨,创新点主要包括以下三个方面:(1)提出一种在不规则四边形网格上构造有理双三次样条曲面的方法对任意拓扑四边形网格上,通过均匀双三次B样条基函数的分解和子基函数的分类,将B样条曲面方法推广到任意四边形网格,从而在任意网格上构造样条曲面。给定一个任意四边形控制网格,首先对每个控制点构造一个基函数:然后所有控制点加权组合形成整体曲面。构造的曲面是分片双三次有理参数多项式曲面。新方法可看成是B样条曲面构造方法的扩展,如果控制网格是规则四边形网格,那么构造得到的曲面与均匀双三次B样条曲面是一致的。(2)提出使用法向叉乘目标函数项以及带约束的特征保持尺寸调整方法模型复用中,对已有模型调整尺寸以适合应用的需求是非常常见的问题。对此,本文提出新的特征保持的模型尺寸调整方法。新方法通过对每条边的缩放来驱动模型尺寸的调整,然后以缩放前后三角形法向的叉乘为目标函数项进行优化,几何意义上,该法向叉乘项表示极小化每个三角形的法向变化:Willmore能量意义上,该项可以近似地极小化缩放带来的能量变化。另外,对需要精确保持某些特征区域的模型,采用带约束的尺寸调整方法,通过引入拉格朗日乘数来求解满足约束条件的优化问题,从而能够精确保持模型的重要特征,目前已有的三维模型尺寸调整方法还不能做到这一点。(3)提出一种特征敏感的模型变形方法特征对于形状表达和模型的理解与分析具有特别重要的作用。因此,变形过程中,应该很好的保持特征细节不变。基于特征敏感度量,新方法更加注重保持网格模型的特征细节区域。具体是通过在对偶网格中使用特征敏感度量的思想,直接将法向量考虑进来,导出特征敏感的拉普拉斯算子。与已有离散拉普拉斯算子相比,该算子对特征细节更加敏感,用于变形时能够更好的保持模型的特征细节。另外,在变形中通过将对偶网格中的1-邻域四面体作为基本变形单元,并且引入四面体约束,通过极小化该四面体底面三角形的扭曲和相应高的变化来尽量保持四面体的形状不变。从而,新方法能够在变形过程中很好的保持模型的特征细节。同时,新方法的时间复杂性与已有线性拉普拉斯变形方法接近。更多还原

【Abstract】 With the fast development of 3D scanning and related techniques, 3D models are now widely used as an emerging type of media in 3D games, computer animation, industrial model design, computer simulation, digital cultural heritage protection and so on. Techniques for processing 3D geometry data have also become a hot research topic in computer graphics, which is called Digital Geometry Processing (DGP). The research contents of DGP cover modeling, processing and application of 3D digital geometric models. No matter in industry or academic world, much attention has been paid on DGP, and techniques have been greatly developed. However, as the related research continues to develop and the applied range is being enlarged, there are still many challenges in DGP.In this thesis we focus on 3D representation based on surfaces. 3D surfaces can be classified into two categories: continuous form surfaces and discrete form surfaces. Continuous form surfaces include spline surfaces, implicit surfaces, subdivision surfaces and so on. And discrete form surfaces include mesh and point cloud. Based on mesh representation, this thesis presents several novel techniques for spline surface fitting and deformation for 3D models. The main contributions are as follows:(1) We propose a new method for constructing rational bi-cubic spline surfaces on irregular meshes.For spline surface fitting on arbitrary meshes, we extend B-spline method to irregular meshes through the decomposition and classification of uniform bi-cubic B-spline basis function. Given a quad mesh of control points, a basis function is constructed for each control point. Then the surface is defined by the weighted combination of all the control points using their associated basis functions. This surface is a piecewise bi-cubic rational parametric polynomial surface. It is an extension to uniform B-spline surfaces in the sense that its definition is an analogy of the B-spline surface, and it produces a uniform bi-cubic B-spline surface if the control mesh is a regular quad mesh. (2) We present to use the cross product of normals as the object function term and propose a new method to resize models with constraints.To reuse existing mesh models, resizing is often necessary to satisfy engineering requirements. We present a new resizing method in this thesis. The resizing is driven by scaling each edge of the mesh. And an objective function component, which is expressed as the cross product of normal vectors of triangles before and after scaling, is devised to optimize the resizing model. Its geometric meaning is to minimizing the variation of normal vector of every triangle. In the sense of Willmore energy, the cross product item can approximately minimize the variation of the Willmore energy before and after resizing. For some important featured regions of the model, whose shape should be accurately preserved, a constrained resizing method is also presented. Lagrange multiplier method is used to preserve the shape of these regions. However, existing resizing methods have not considered the accurate preserving of these important featured regions.(3) We present a feature sensitive deformation method.Features are crucial for accurate representing geometry, as well as understanding and analysis of models. Therefore, the feature details should be well-preserved during deformation. Based on feature sensitive metric, we pay more attention on the feature regions of the mesh models. Firstly, taking unit normal vectors into account, we derive a FS Laplacian operator, which is more sensitive to featured regions of mesh models than existing operators. Secondly, we use the 1-ring tetrahedron in the dual mesh as the basis deformation cell. To preserve the shape of the tetrahedron, we introduce linear tetrahedron constraints minimizing both the distortion of the base triangle and the change of the corresponding height. These ensure that geometric details are accurately preserved during deformation. The time complexity of our new method is similar to that of existing linear Laplacian methods.更多还原