Signal processing with the Euler calculus

Abstract: It happens that many of the transforms traditionally used in signal processing have natural analogs under the Euler integral, popularized by Baryshnikov and Ghrist. The properties of these transforms are sensitive to topological (as well as certain geometric) features in the sensor field and allow signal processing to be performed on structured, integer valued data, such as might be gathered from ad hoc networks of inexpensive sensors. For instance, the analog of the Fourier transform computes a measure of width of support for indicator functions. There are some notable challenges in this theory, some of which are present in traditional transform theory (such as the presence of sidelobes), and some which are new (such as the nonlinearity of the transform when extended to real-valued data). These challenges and some mitigation strategies will be presented as well as a showcase of the transforms and their capabilities.