On length measures of planar closed curves and the comparison of convex shapes
On length measures of planar closed curves and the comparison of convex shapes
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DOI：
10.1007/s10455-021-09795-0
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发表时间：
2020-10
影响因子：
0.7
通讯作者：
N. Charon;T. Pierron
中科院分区：
文献类型：
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作者：
N. Charon;T. Pierron
在本文中，我们重新审视的概念，长度措施相关联的平面封闭曲线。这是一个特殊的情况下，面积措施的超曲面，介绍了早期领域的凸几何。曲线的长度测度是圆上的一个测度，直观地表示曲线上切向量指向某一方向的部分的长度。虽然平面闭曲线不以其长度测度为特征，但基本的Minkowski-Fenchel-Fenchen定理指出长度测度完全表征凸曲线模平移，使其成为研究凸对象几何性质的特别有用的工具。目前的工作，这最初是出于形状分析的问题，介绍了长度措施的一般类Lipschitz浸入和定向平面闭曲线，并推导出这类曲线的长度测量映射的一些基本性质。然后，我们专注于凸形状的情况下，并提出了几个新的结果。首先，我们证明了与Minkowski-Fenchel-Escheren定理给出的长度测度相关联的唯一凸曲线的等周特征，即它最大化共享相同长度测度的所有曲线之间的有符号面积。其次，我们解决了凸平面曲线之间的距离与相关的测地线路径的构造问题。为此目的，我们引入和研究一个新的距离空间的长度措施，对应于一个受约束的变体的Wasserstein度量的最佳运输，从这里我们可以诱导凸曲线之间的距离。我们还提出了一个原始对偶算法来数值计算这些距离和测地线，并显示了一些简单的模拟来说明这种方法。
In this paper, we revisit the notion of length measures associated to planar closed curves. These are a special case of area measures of hypersurfaces which were introduced early on in the field of convex geometry. The length measure of a curve is a measure on the circlethat intuitively represents the length of the portion of curve which tangent vector points in a certain direction. While a planar closed curve is not characterized by its length measure, the fundamental Minkowski–Fenchel–Jessen theorem states that length measures fully characterize convex curves modulo translations, making it a particularly useful tool in the study of geometric properties of convex objects. The present work, that was initially motivated by problems in shape analysis, introduces length measures for the general class of Lipschitz immersed and oriented planar closed curves, and derives some of the basic properties of the length measure map on this class of curves. We then focus specifically on the case of convex shapes and present several new results. First, we prove an isoperimetric characterization of the unique convex curve associated to some length measure given by the Minkowski–Fenchel–Jessen theorem, namely that it maximizes the signed area among all the curves sharing the same length measure. Second, we address the problem of constructing a distance with associated geodesic paths between convex planar curves. For that purpose, we introduce and study a new distance on the space of length measures that corresponds to a constrained variant of the Wasserstein metric of optimal transport, from which we can induce a distance between convex curves. We also propose a primal-dual algorithm to numerically compute those distances and geodesics, and show a few simple simulations to illustrate the approach.