# Using wavelet decomposition to determine the dimension of structures from projected images

## TL;DR

This paper introduces a new method using wavelet decomposition to estimate the fractal dimension of 3D structures from their 2D projected images, applied to astrophysical data.

## Contribution

A novel wavelet-based technique is introduced to infer 3D fractal dimensions from projected images without assumptions on embedding dimensions.

## Key findings

- Wavelet power spectra scale as Pj=2−jγ, where γ is the fractal dimension of the structure.
- The method was applied to Cassiopeia A, revealing different fractal dimensions for gas at different temperatures.
- Fractal dimensions ranged from γ=1.69 for CO emissions to γ=2.49 for X-ray emitting gas.

## Abstract

Turbulence is a fundamental process in a wide range of natural and laboratory systems. Experiments and observations often measure projected images of tracers in turbulent flows. Because the underlying structures are fractal, such observations are projections of fractals. This paper presents a general technique for measuring the fractal dimension of structures from their projected images. We apply this method to two-dimensional astronomical images and show that wavelet-based measurements can be used to infer the fractal dimension of a structure in 3D. Applying the technique to JWST infrared and X-ray images of the supernova remnant Cassiopeia A reveals markedly different fractal dimensions for gas at different temperatures, providing a tool for probing the instabilities that generate these complex structures.

Mesoscale structures in turbulent media can often be described as fractional dimensional across a wide range of scales. The goal of this paper is to determine the structure’s dimension from a projected image. Our method exploits the laws of scaling of wavelet power spectra under projection and does not carry any restrictions on the embedding and projected dimensions. We show that the wavelet power spectrum of a projected γ dimensional measure is Pj=2−jγ, where j is the wavelet scale. We contrast the wavelet method with the popular box-counting approach. For projected images, the use of box-counting at fixed thresholds often leads to erroneous results. We apply the method to James Webb Space Telescope (JWST) infrared and Chandra X-ray observations of the supernova remnant Cassiopeia A. We find that the emissions can be represented by projections of mesoscale substructures with fractal dimensions varying from γ=1.69±0.02 for the warm CO layer observed by JWST, up to γ=2.49±0.03 for the hot X-ray emitting gas layer in the supernova remnant.