论文标题
在平面封闭曲线的长度测量和凸形的比较上
On length measures of planar closed curves and the comparison of convex shapes
论文作者
论文摘要
在本文中，我们重新审视了与平面封闭曲线相关的长度度量的概念。这些是在凸线几何学领域早期引入的高层曲面面积测量的特殊情况。曲线的长度度量是圆圈$ \ mathbb {s}^1 $的度量，直觉代表曲线部分的长度，该曲线部分指向沿特定方向指向的曲线部分。尽管平面封闭曲线的特征在于其长度度量，但基本的Minkowski-Fenchel-Jessen定理指出，长度度量完全表征了凸曲线模型翻译，这使其成为研究凸的几何特性的特别有用工具。目前的工作最初是由形状分析中的问题引起的，它引入了Lipschitz浸入和定向的平面封闭曲线的长度度量，并衍生了该类别曲线上长度测量图的一些基本属性。然后，我们专门针对凸形的情况，并提出几个新结果。首先，我们证明了与Minkowski-fenchel-Jessen定理给出的一定长度度量相关的唯一凸曲线的等等表征，即它在所有相同长度度量的曲线之间最大化了签名区域。其次，我们解决了构建距离之间的距离的问题，该距离与凸面平面曲线之间的相关地球路径。为此，我们介绍并研究了长度测量空间的新距离，该距离与最佳运输的瓦斯汀度量的约束变体相对应，我们可以从中诱导凸曲线之间的距离。我们还提出了一种原始的二算法，以数值计算这些距离和地球学，并显示一些简单的模拟以说明方法。
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 circle $\mathbb{S}^1$ that 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.
