Over the past decade, the importance of the 1D signature which can be seen as a functional defined along a path, has been pivotal in both path-wise stochastic calculus and the analysis of time series data. By considering an image as a two-parameter function that takes values in a $d$-dimensional space, we introduce an extension of the path signature to images. We address numerous challenges associated with this extension and demonstrate that the 2D signature satisfies a version of Chen's relation in addition to a shuffle-type product. Furthermore, we show that specific variations of the 2D signature can be recursively defined, thereby satisfying an integral-type equation. We analyze the properties of the proposed signature, such as continuity, invariance to stretching, translation and rotation of the underlying image. Additionally, we establish that the proposed 2D signature over an image satisfies a universal approximation property.

翻译：过去十年间，作为沿路径定义的泛函，一维签名在路径随机微积分和时间序列数据分析中展现出关键作用。通过将图像视为取值于 $d$ 维空间的双参数函数，我们提出了路径签名在图像上的扩展。我们解决了该扩展过程中面临的诸多挑战，并证明了二维签名满足陈氏关系的一个变体，以及一种洗牌型乘积。此外，我们展示了二维签名的特定变体可递推定义，从而满足积分型方程。我们分析了所提签名的性质，包括连续性、对图像拉伸、平移和旋转的不变性。同时，我们证明了所提二维签名在图像上具有普适逼近性质。