Euler integral transforms and applications

Michael Robinson (Univ. Pennsylvania)

Abstract: The old idea of using the combinatorial Euler characteristic as a valuation to define an integration theory found application to sensor networks in a recent paper of Baryshnikov and Ghrist. They showed that a dense network of sensors, each of which produces an integer count of nearby targets could be integrated to yield a total count of the targets within the sensor field even if the target support regions overlap. The resulting algorithm is reminiscent of signal processing techniques, though it uses integer-valued data points. Seeing as a primary tool of signal processing is the integral transform, a question is "are there integral transforms in this theory?" It happens that many of the transforms traditionally used in harmonic analysis have natural analogs under the Euler integral. 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.