The Vector Heat Method
The Vector Heat Method
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DOI：
10.1145/3243651
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发表时间：
2018-05
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通讯作者：
Nicholas Sharp;Yousuf Soliman;Keenan Crane
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作者：
Nicholas Sharp;Yousuf Soliman;Keenan Crane
本文描述了一种在曲面上高效计算切向量平行移动的方法，或者更一般地说，是在弯曲流形上计算任何向量值数据的平行移动的方法。更准确地说，它通过沿着最短测地线进行平行移动，将在任何区域上定义的向量场扩展到域的其余部分。这种基本操作使得能够实现快速、稳健的算法，用于外推水平集速度、求指数映射的逆、计算任意分布的几何中位数和卡彻尔/弗雷歇均值、构建重心 Voronoi图以及寻找顺序一致的地标。我们表明，不必通过显式追踪测地线来评估平行移动，而是可以通过涉及联络拉普拉斯算子的短时热流来计算它。因此，可以通过求解三个预分解的线性系统来实现移动，每个线性系统都类似于一个标准的泊松问题。为了实现该方法，我们只需要一个离散的联络拉普拉斯算子，我们针对各种几何数据结构（点云、多边形网格等）对其进行了描述。我们还研究了我们方法的数值行为，通过实验表明它在细化下收敛，并扩充了内蕴德劳内三角剖分的构建，以便它们可用于切向量场处理的情境中。
This article describes a method for efficiently computing parallel transport of tangent vectors on curved surfaces, or more generally, any vector-valued data on a curved manifold. More precisely, it extends a vector field defined over any region to the rest of the domain via parallel transport along shortest geodesics. This basic operation enables fast, robust algorithms for extrapolating level set velocities, inverting the exponential map, computing geometric medians and Karcher/Fréchet means of arbitrary distributions, constructing centroidal Voronoi diagrams, and finding consistently ordered landmarks. Rather than evaluate parallel transport by explicitly tracing geodesics, we show that it can be computed via a short-time heat flow involving the connection Laplacian. As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem. To implement the method, we need only a discrete connection Laplacian, which we describe for a variety of geometric data structures (point clouds, polygon meshes, etc.). We also study the numerical behavior of our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations so that they can be used in the context of tangent vector field processing.