Euler-Bessel and Euler-Fourier Transforms

TL;DR

This paper introduces topological Euler-based Bessel and Fourier transforms, establishing their properties and applying them to target reconstruction and shape discrimination with sensor data.

Contribution

It defines the topological Bessel transform, relates it to the Fourier transform via logarithmic blowup, and develops a Morse index formula for these transforms.

Findings

Established the topological Bessel transform and its properties.

Derived a relationship between Bessel and Fourier transforms using logarithmic blowup.

Applied the transforms to target localization and shape discrimination with sensor data.

Abstract

We consider a topological integral transform of Bessel (concentric isospectral sets) type and Fourier (hyperplane isospectral sets) type, using the Euler characteristic as a measure. These transforms convert constructible $\zed$-valued functions to continuous $\real$-valued functions over a vector space. Core contributions include: the definition of the topological Bessel transform; a relationship in terms of the logarithmic blowup of the topological Fourier transform; and a novel Morse index formula for the transforms. We then apply the theory to problems of target reconstruction from enumerative sensor data, including localization and shape discrimination. This last application utilizes an extension of spatially variant apodization (SVA) to mitigate sidelobe phenomena.