We introduce a rotation-invariant representation of planar shapes. In particular, this representation encodes shapes as vectors such that the Euclidean distance between them serves as a valid shape distance. For standardized, star-shaped objects, we can deterministically create a sketched vector of dimension $O(1/\varepsilon)$ in $O((1/\varepsilon) \log (1/\varepsilon))$ time that approximates this shape distance to within $\varepsilon$. Moreover, because the representation is a standard Euclidean vector, we can directly and efficiently perform various data analyses, such as nearest neighbor search and clustering, in shape space, inherently invariant to the rotation of the shapes. We demonstrate this through a series of simple experiments. The key technical contribution operates on functions over $\mathbb{S}^1$, which we use to encode standardized objects. The most general rotation-invariant representation of these functions works through a map to an infinite-dimensional function space, parameterized by an offset parameter. By analyzing special discretized cases of these functions, we show that the representation is strictly injective up to the desired rotation and a mirror-flip-type operation we call \emph{reverse of complement} (RoC). While RoC status can be controlled by how the function is defined, it is inherent to the representation and required to be handled in the analysis. Regardless, the vectorized representation is robust to small shape perturbations, and hence discretizing the angles leads to the efficient approximation and algorithm.

翻译：我们提出了一种平面形状的旋转不变表示方法。具体而言，该表示将形状编码为向量，使得它们之间的欧几里得距离可作为有效的形状距离。对于标准化的星形物体，我们可以在 $O((1/\varepsilon) \log (1/\varepsilon))$ 时间内确定性地创建维度为 $O(1/\varepsilon)$ 的草图向量，该向量以 $\varepsilon$ 的误差逼近此形状距离。此外，由于该表示是标准欧几里得向量，我们可以在形状空间中直接高效地进行各种数据分析（如最近邻搜索和聚类），且这些分析天然具备对形状旋转的不变性。我们通过一系列简单实验对此进行了验证。核心技术贡献在于对 $\mathbb{S}^1$ 上的函数进行操作，我们利用这些函数对标准化物体进行编码。这些函数的最通用旋转不变表示通过映射到由偏移参数参数化的无限维函数空间来实现。通过分析这些函数的特殊离散化情形，我们证明该表示在旋转和一种称为"补码反转"（reverse of complement, RoC）的镜像翻转操作下是严格单射的。虽然 RoC 状态可通过函数定义方式控制，但它是该表示固有的属性，需要在分析中加以处理。无论如何，该向量化表示对微小形状扰动具有鲁棒性，因此对角度进行离散化可实现高效的近似与算法。