平方根速度框架下$R^N$中PL曲线的精确匹配
Precise Matching of PL Curves in $R^N$ in the Square Root Velocity Framework
摘要：
Srivastava等人提出的平方根速度函数（SRVF）已被证明是比较$R^N$模参数化中绝对连续曲线的一种有效方法。基于这种方法已经发表了几篇计算论文。在本文中，我们仔细建立的SRVF方法的理论基础。特别是，我们分析的商建设的一组绝对连续的曲线模组（或在某些情况下，半群）的reparametrizations，证明了一个重要的定理结构的闭轨道需要在这个商建设。我们观察到，在绝对连续曲线的空间中，分段线性曲线的集合相对于SRVF度量是稠密的。最后，给定两条分段线性曲线，我们建立了一个精确的算法来产生这两条曲线之间的最佳匹配。这也导致精确确定对应于这些曲线的商空间中的点之间的测地线。在过去，这种测地线只使用动态规划的方法来近似。我们展示了这种算法的例子。
英文摘要：
The square root velocity function (SRVF), introduced by Srivastava et al, has proved to be an effective way to compare absolutely continuous curves in $R^N$ modulo reparametrization. Several computational papers have been published based on this method. In this paper, we carefully establish the theoretical foundations of the SRVF method. In particular, we analyze the quotient construction of the set of absolutely continuous curves modulo the group (or in some cases, semigroup) of reparametrizations, proving an important theorem about the structure of the closed orbits required in this quotient construction. We observe that the set of piecewise linear curves is dense in the space of absolutely continuous curves with respect to the SRVF metric. Finally, given two piecewise linear curves, we establish a precise algorithm for producing the optimal matching between these curves. This also results in a precise determination of the geodesic between the points in the quotient space corresponding to these curves. In the past, this geodesic has only been approximated using the method of Dynamic Programming. We show examples resulting from this algorithm.
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