Over the past decade, the path signature has emerged a powerful tool in data science, particularly for time series analysis and pathwise stochastic calculus, due to its efficient computability via dynamic programming based on Chen’s identity. We define signature for images by extending the concept of path signature — a particular 1-parameter object derived from iterated integrals along paths — to surfaces by introducing a second parameter. This extension involves iterated surface integrals, which adds significant complexity. Our research identified two distinct approaches to defining surface signature. The first, recursively defined via integral-type equations, satisfies a Chen-type relation and a shuffle-type product, while exhibiting continuity, as well as invariance to stretching, translation, and rotation. It also possesses a universal approximation property. Notably, this approach has already inspired follow-up work, demonstrating its practical value in image analysis. The second approach, on the other hand, is based on the non-commutative Stokes’ theorem and draws inspiration from higher gauge theory, an advanced topic in modern mathematical physics. It employs crossed modules of Lie groups and Lie algebras to capture surface features extending beyond line integrals along boundary loops. Further developments include a Magnus-type expression for the logarithm of the surface signature and a sewing lemma in the crossed module framework, paving the way for new directions in image analysis.