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On the signature of an image

Joscha Diehl, Kurusch Ebrahimi-Fard, Fabian Harang, Samy Tindel

TL;DR

This work extends the path signature to two-dimensional image-indexed fields by introducing three 2D siganture variants for $X:[0,T]^2\to\mathbb{R}^d$: the $\mathsf{id}$-signature $\mathbf{S}^{\mathsf{id}}(X)$, the full 2D signature $\mathbf{S}(X)$, and the symmetrized signature $\mathbf{S}^{\mathrm{Sym}}(X)$. It develops a rigorous plane calculus with simplexes, horizontal/vertical Chen relations, and a 2D shuffle algebra, showing that the full signature satisfies a 2D shuffle relation while the $\mathsf{id}$-signature satisfies a Chen-type relation; the symmetrized variant combines both algebraic properties. The paper proves invariances (translation, stretching, and 90-degree rotation equivariance) and continuity, and demonstrates universality: the full signature can approximate continuous functionals on fields, with a proof that the signature determines the field up to mixed partial equivalence. It also discusses homotopy invariance limitations and open problems, including extensions to rough paths and probabilistic analyses of random fields. Overall, this provides a principled, algebraically rich framework for extracting and analyzing image features via higher-dimensional signatures that generalize the classical path 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.

On the signature of an image

TL;DR

This work extends the path signature to two-dimensional image-indexed fields by introducing three 2D siganture variants for : the -signature , the full 2D signature , and the symmetrized signature . It develops a rigorous plane calculus with simplexes, horizontal/vertical Chen relations, and a 2D shuffle algebra, showing that the full signature satisfies a 2D shuffle relation while the -signature satisfies a Chen-type relation; the symmetrized variant combines both algebraic properties. The paper proves invariances (translation, stretching, and 90-degree rotation equivariance) and continuity, and demonstrates universality: the full signature can approximate continuous functionals on fields, with a proof that the signature determines the field up to mixed partial equivalence. It also discusses homotopy invariance limitations and open problems, including extensions to rough paths and probabilistic analyses of random fields. Overall, this provides a principled, algebraically rich framework for extracting and analyzing image features via higher-dimensional signatures that generalize the classical path 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 -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.
Paper Structure (25 sections, 22 theorems, 182 equations, 1 figure)

This paper contains 25 sections, 22 theorems, 182 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Linear ODEs and the 1D signature
  3. Motivation from real-valued differential equations
  4. The algebra of signatures
  5. Simplexes and integration
  6. Rectangles and simplexes
  7. Some properties of integral operators
  8. Increments in the plane
  9. Definition of the 2D signature
  10. A first notion of 2D signature: the $\mathsf{id}$-signature
  11. Chen's relation: Horizontal and vertical
  12. Challenges with the $\mathsf{id}$ signature
  13. Definition of the full 2D signature
  14. Invariance properties
  15. Translation invariance
  16. ...and 10 more sections

Key Result

Lemma 2.1

Let $x:[0,T]\to\mathbb{R}^{d}$ be a $\mathcal{C}^{1}$ signal and let $\{A^{i}\ | \, i=1,\ldots,d\} \subset \mathbb{R}^{e\times e}$ be a collection of constant matrices with real entries. We consider the $\mathbb{R}^{e}$-valued solution $y$ to the linear equation for $(s,t)\in\Delta^{2}_{0,T}$, with a given initial condition $y_s \in\mathbb{R}^{d}$. Recalling a0, we set and $A^\mathsf{e}$ is the

Figures (1)

  • Figure 3.1: Typical example of rectangle $[\mathbf{s},\mathbf{t}]$ in $[0,T]^{2}$.

Theorems & Definitions (73)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Definition 3.3
  • Remark 3.4
  • Lemma 3.5
  • Definition 3.7
  • Definition 3.8
  • Lemma 3.9
  • ...and 63 more