On the signature of an image

TL;DR

This paper extends the concept of the path signature from one-dimensional paths to two-dimensional images, analyzing its mathematical properties and invariance, and demonstrating its potential for image analysis.

Contribution

The paper introduces a novel 2D signature for images, extending path signatures, and explores its properties, invariances, and approximation capabilities.

Findings

2D signature satisfies Chen's relation and shuffle product

Signature is invariant to image transformations like translation and rotation

Proven universal approximation property for the 2D signature

Abstract

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.