Geometric signals

Abstract

In signal processing, a signal is a function. Conceptually, replacing a function by its graph, and extending this approach to a more abstract setting, we define a signal as a submanifold M of a Riemannian manifold (with corners) that satisfies additional conditions. In particular, it is a relative cobordism between two manifolds with boundaries. We define energy as the integral of the distance function to the first of these boundary manifolds. Composition of signals is composition of cobordisms. A "time variable" can appear explicitly if it is explictly given (for example, if the manifold is of the form $\Sigma\times [0,1]$). Otherwise, there is no designated "time dimension", although the cobordism may implicitly indicate the presence of dynamics. We interpret a local deformation of the metric as noise. The assumptions on M allow to define a map $M\to M$ that we call a Fourier transform. We prove inequalities that illustrate the properties of energy of signals in this setting.