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On the signature of an image

Author

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  • Diehl, Joscha
  • Ebrahimi-Fard, Kurusch
  • Harang, Fabian N.
  • Tindel, Samy

Abstract

Over the past decade, the importance of the 1D signature which can be seen as a functional defined over 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.

Suggested Citation

  • Diehl, Joscha & Ebrahimi-Fard, Kurusch & Harang, Fabian N. & Tindel, Samy, 2025. "On the signature of an image," Stochastic Processes and their Applications, Elsevier, vol. 187(C).
    • Handle: RePEc:eee:spapps:v:187:y:2025:i:c:s0304414925001024
    DOI: 10.1016/j.spa.2025.104661
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    References listed on IDEAS

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    1. Yue Wu & Guy M Goodwin & Terry Lyons & Kate E A Saunders, 2022. "Identifying psychiatric diagnosis from missing mood data through the use of log-signature features," PLOS ONE, Public Library of Science, vol. 17(11), pages 1-18, November.
    2. Quer-Sardanyons, Lluís & Tindel, Samy, 2007. "The 1-d stochastic wave equation driven by a fractional Brownian sheet," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1448-1472, October.
    3. Harang, Fabian A. & Tindel, Samy, 2021. "Volterra equations driven by rough signals," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 34-78.
    Full references (including those not matched with items on IDEAS)

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