Computing spatial information from Fourier coefficient distributions

TL;DR

This paper introduces a novel method to quantify spatial information in images using Fourier coefficient distributions and Shannon entropy, enabling effective analysis of complex data like phase transitions.

Contribution

The paper presents a new Fourier-based framework for computing spatial information, applicable to images and arbitrary-dimensional data, with demonstrated effectiveness on physical systems.

Findings

Accurately detects phase transitions in a 2D Ising model.

Provides a robust measure of spatial information in various data types.

Utilizes Fourier coefficient distributions for information quantification.

Abstract

We present an approach to computing spatial information based on Fourier coefficient distributions. The Fourier transform (FT) of an image contains a complete description of the image, and the values of the FT coefficients are uniquely associated with that image. For an image where the distribution of pixels is uncorrelated, the FT coefficients are normally distributed and uncorrelated. Further, the probability distribution for the FT coefficients of such an image can readily be obtained by Parseval's theorem. We take advantage of these properties to compute the spatial information in an image by determining the probability of each coefficient (both real and imaginary parts) in the FT, then using the Shannon formalism to calculate information. By using the probability distribution obtained from Parseval's theorem, an effective distance from the completely uncorrelated or most uncertain case is obtained. The resulting quantity is an information computed in k-space (kSI). This approach provides a robust, facile and highly flexible framework for quantifying spatial information in images and other types of data (of arbitrary dimensions). The kSI metric is tested on a 2D Ising ferromagnet, and the temperature-dependent phase transition is accurately determined from the spatial information in configurations of the system.