Discrete connections for geometry processing
Discrete connections for geometry processing
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用于几何处理的离散连接
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Keenan Crane
中科院分区：
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作者：
Keenan Crane
连接提供了一种比较在几何空间的不同点上定义的局部量的方法。这篇论文发展了一种离散的连接理论，自然地导致了用于几何处理的实用、高效的数值算法。我们的公式是由现实世界的应用程序驱动的，其中网格可能是噪声的或粗略离散的。此外，由于我们的离散框架与光滑理论非常相似，我们可以利用现有的大量知识来开发和解释网格处理算法。本文的主要贡献是提出了一种新的算法来计算离散曲面上的平凡连接，这些曲面除了给定指数的一组孤立奇点外，在任何地方都尽可能光滑。连接通过与每一对偶边相关联的角度来表示，即离散角度值1-表。这些角度由线性系统的解确定，并且在某种意义上是全局最优的，因为它们描述了在具有规定的奇点集的所有解中最接近Levi-Civita的平凡联系。与以前的方法相比，我们的算法非常简单，并且可以使用网格处理和线性代数中的标准操作来实现。该解可用于构造具有规定奇点和方向约束的旋转对称方向场，这在四边形网格重新划分和纹理合成等应用中是必不可少的。
Connections provide a way to compare local quantities defined at different points of a geometric space. This thesis develops a discrete theory of connections that naturally leads to practical, efficient numerical algorithms for geometry processing. Our formulation is motivated by real-world applications where meshes may be noisy or coarsely discretized. Further, because our discrete framework closely parallels the smooth theory, we can draw upon a huge wealth of existing knowledge to develop and interpret mesh processing algorithms. The main contribution of this thesis is a new algorithm for computing trivial connections on discrete surfaces that are as smooth as possible everywhere but on a set of isolated singularities of given index. A connection is represented via an angle associated with each dual edge, i.e., a discrete angle-valued 1-form. These angles are determined by the solution to a linear system, and are globally optimal in the sense that they describe the trivial connection closest to Levi-Civita among all solutions with the prescribed set of singularities. Relative to previous methods our algorithm is surprisingly simple, and can be implemented using standard operations from mesh processing and linear algebra. The solution can be used to construct rotationally symmetric direction fields with a prescribed set of singularities and directional constraints, which are essential in applications such as quadrilateral remeshing and texture synthesis.