What is the Fourier Transform of a Spatial Point Process?

TL;DR

This paper defines and analyzes the Fourier transform of spatial point processes, addressing spectral moments, tapering, and isotropic representation to enhance spectral analysis of spatial data.

Contribution

It provides a comprehensive framework for defining and implementing the Fourier transform of spatial point processes, including spectral moments and isotropic representation.

Findings

Derived spectral moments using Campbell's theorem

Established methods for tapering in spectral analysis

Outlined procedures for isotropic Fourier transform representation

Abstract

This paper determines how to define a discretely implemented Fourier transform when analysing an observed spatial point process. To develop this transform we answer four questions; first what is the natural definition of a Fourier transform, and what are its spectral moments, second we calculate fourth order moments of the Fourier transform using Campbell's theorem. Third we determine how to implement tapering, an important component for spectral analysis of other stochastic processes. Fourth we answer the question of how to produce an isotropic representation of the Fourier transform of the process. This determines the basic spectral properties of an observed spatial point process.