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

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 P_j = 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.