Abstract
Contours or closed planar curves are common in many domains. For example, they appear as object boundaries in computer vision, isolines in meteorology, and the orbits of rotating machinery. In many cases when learning from contour data, planar rotations of the input will result in correspondingly rotated outputs. It is therefore desirable that deep learning models be rotationally equivariant. In addition, contours are typically represented as an ordered sequence of edge points, where the choice of starting point is arbitrary. It is therefore also desirable for deep learning methods to be equivariant under cyclic shifts. We present RotaTouille, a deep learning framework for learning from contour data that achieves both rotation and cyclic shift equivariance through complex-valued circular convolution. We further introduce and characterize equivariant non-linearities, coarsening layers, and global pooling layers to obtain invariant representations for downstream tasks. Finally, we demonstrate the effectiveness of RotaTouille through experiments in shape classification, reconstruction, and contour regression.
Abstract (translated)
轮廓或封闭的平面曲线在许多领域都很常见。例如，在计算机视觉中，它们作为物体边界出现；在气象学中，它们表现为等值线；在旋转机械中，则是其轨道。在很多情况下，从轮廓数据学习时，输入的平面旋转会导致相应的输出旋转。因此，希望深度学习模型具有旋转等变性（即对旋转保持不变的关系）。此外，轮廓通常被表示为一系列边缘点的有序序列，并且起点的选择是任意的。因此，也希望深度学习方法在循环位移下保持等变性。 我们提出了RotaTouille，这是一种基于复杂值圆卷积实现同时具有旋转和循环位移等变性的深度学习框架，以从轮廓数据中进行学习。此外，我们引入并描述了等变非线性、细化层以及全局池化层，以便为下游任务获取不变表示。最后，通过形状分类、重构及轮廓回归的实验，展示了RotaTouille的有效性。
URL
https://arxiv.org/abs/2508.16359
