Precise Matching of PL Curves in $R^N$ in the Square Root Velocity Framework
Precise Matching of PL Curves in $R^N$ in the Square Root Velocity Framework
复制标题
平方根速度框架中 $R^N$ 中 PL 曲线的精确匹配
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
E. Klassen
中科院分区：
文献类型：
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作者：
S. Lahiri;Daniel Robinson;E. Klassen
由Srivastava等人引入的平方根速度函数（SRVF）已被证明是比较R^N模重参数化中绝对连续曲线的有效方法。基于这种方法已经发表了几篇计算论文。在本文中，我们认真地建立了SRVF方法的理论基础。特别地，我们分析了模于重参数化群（或某些情况下半群）的绝对连续曲线集的商构造，证明了这种商构造所需的闭轨道结构的一个重要定理。我们观察到分段线性曲线集相对于SRVF度量在绝对连续曲线空间中是密集的。最后，给出了两条分段线性曲线，建立了曲线间最优匹配的精确算法。这也可以精确地确定与这些曲线相对应的商空间中点之间的测地线。在过去，这个测地线只能用动态规划的方法来近似。我们将展示由该算法产生的示例。
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.