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Using Wavelet Decomposition to Determine the Dimension of Structures from Projected Images

Svitlana Mayboroda, David N Spergel

TL;DR

The paper tackles the challenge of inferring the fractal dimension $\gamma$ of a $\gamma$-dimensional measure from projected images, where traditional box-counting and Fourier methods fail due to projection and aliasing. It develops a wavelet-based framework, showing that under a Copernican assumption the projected wavelet power spectrum satisfies $P_j \propto 2^{-j\gamma}$, enabling robust estimation of $\gamma$ from 2D projections regardless of the ambient dimension $N$ and projection dimension $D$. The authors derive the theory, demonstrate it on a synthetic Menger sponge, and apply it to Cas A using JWST and Chandra data, obtaining $\gamma \approx 1.7$ for warm CO and $\gamma \approx 2.5$ for hot X-ray gas, consistent with MHD instabilities shaping mesoscale structure. This approach provides a general, quantitative tool for comparing observations with theory in high-dimensional, projected data, with potential applications across data sciences and astrophysics.

Abstract

Mesoscale structures can often be described as fractional dimensional across a wide range of scales. We consider a $γ$ dimensional measure embedded in an $N$ dimensional space and discuss how to determine its dimension, both in $N$ dimensions and projected into $D$ dimensions. It is a highly non-trivial problem to decode the original geometry from lower dimensional projection of a high-dimensional measure. The projections are space-feeling, the popular box-counting techniques do not apply, and the Fourier methods are contaminated by aliasing effects. In the present paper we demonstrate that under the "Copernican hypothesis'' that we are not observing objects from a special direction, projection in a wavelet basis is remarkably simple: the wavelet power spectrum of a projected $γ$ dimensional measure is $P_j \propto 2^{-jγ}$. This holds regardless of the embedded dimension, $N$, and the projected dimension, $D$. This approach could have potentially broad applications in data sciences where a typically sparse matrix encodes lower dimensional information embedded in an extremely high dimensional field and often measured in projection to a low dimensional space. Here, we apply this method to JWST and Chandra observations of the nearby supernova Cas A. We find that the emissions can be represented by projections of mesoscale substructures with fractal dimensions varying from $γ= 1.7$ for the warm CO layer observed by JWST, up to $γ= 2.5$ for the hot X-ray emitting gas layer in the supernova remnant. The resulting power law indicates that the emission is coming from a fractal dimensional mesoscale structure likely produced by magneto-hydrodynamical instabilities in the expanding supernova shell.

Using Wavelet Decomposition to Determine the Dimension of Structures from Projected Images

TL;DR

The paper tackles the challenge of inferring the fractal dimension of a -dimensional measure from projected images, where traditional box-counting and Fourier methods fail due to projection and aliasing. It develops a wavelet-based framework, showing that under a Copernican assumption the projected wavelet power spectrum satisfies , enabling robust estimation of from 2D projections regardless of the ambient dimension and projection dimension . The authors derive the theory, demonstrate it on a synthetic Menger sponge, and apply it to Cas A using JWST and Chandra data, obtaining for warm CO and for hot X-ray gas, consistent with MHD instabilities shaping mesoscale structure. This approach provides a general, quantitative tool for comparing observations with theory in high-dimensional, projected data, with potential applications across data sciences and astrophysics.

Abstract

Mesoscale structures can often be described as fractional dimensional across a wide range of scales. We consider a dimensional measure embedded in an dimensional space and discuss how to determine its dimension, both in dimensions and projected into dimensions. It is a highly non-trivial problem to decode the original geometry from lower dimensional projection of a high-dimensional measure. The projections are space-feeling, the popular box-counting techniques do not apply, and the Fourier methods are contaminated by aliasing effects. In the present paper we demonstrate that under the "Copernican hypothesis'' that we are not observing objects from a special direction, projection in a wavelet basis is remarkably simple: the wavelet power spectrum of a projected dimensional measure is . This holds regardless of the embedded dimension, , and the projected dimension, . This approach could have potentially broad applications in data sciences where a typically sparse matrix encodes lower dimensional information embedded in an extremely high dimensional field and often measured in projection to a low dimensional space. Here, we apply this method to JWST and Chandra observations of the nearby supernova Cas A. We find that the emissions can be represented by projections of mesoscale substructures with fractal dimensions varying from for the warm CO layer observed by JWST, up to for the hot X-ray emitting gas layer in the supernova remnant. The resulting power law indicates that the emission is coming from a fractal dimensional mesoscale structure likely produced by magneto-hydrodynamical instabilities in the expanding supernova shell.

Paper Structure

This paper contains 6 sections, 13 equations, 4 figures.

Table of Contents

  1. 1. Introduction
  2. 2. Measuring Fractal Dimension
  3. 3. Measuring Fractal Dimension from Projected Images
  4. 4. Application: Cas A
  5. Conclusions
  6. Acknowledgments

Figures (4)

  • Figure 1: The upper left panel shows the projected density of a six level Menger sponge, a fractal of dimension $\gamma \sim 2.767$. The projection is along a line that is 30 degrees off of the principal axis. The upper right panel shows the wavelet amplitude versus scale and fits for two different ranges: a fit from 4-256 yields a slope of 2.773 and a fit of 8-128 yields a slope of 2.652. The lower left panel show the power spectrum of the Fourier transform of the image. Because of the finite size of the object, the FFT method yields 1.912 for the fractal dimension. When tested on Note that a Menger sponge embedded in a periodic box ($T^3$), it yields a value close to the correct value. The lower right panel shows the inferred dimension using a threshold method as a function of threshold. The value in the figure should be compared to $\gamma -1 = 1.767$.
  • Figure 2: This figure shows the box count versus box size for the projected Menger sponge. For box thresholds set at 0.0, 0.1, 0.2, 0.5, 0.8 and 0.9, the best fit lines (shown as dashed red) have slope 1.902, 1.894, 1.803, 1.521, 1.175 and 0.870. The correct answer should be $\sim 1.727$: while the box counting method can yield convincing power laws, it produces incorrect results when applied to projected density fields.
  • Figure 3: X-ray and Infrared images of Cas A. The first three images are Chandra X-ray images at 0.5-1.5 keV, 1.5-3.0 keV, and 4.0-6.0 keV. The lower right figure is derived Rho2024 by subtracting the F365W JWST NIRCAM image from F444W JWST NIRCAM image to remove the synchrotron emission. We display the image of the data provided by Tea Temim from the JWST Cas A survey team.
  • Figure 4: Wavelet Power Spectrum of 2D Image of Cas A for gas at different frequencies. Here, we use Ricker wavelets in the analysis. The upper left is for 0.5-1.5 keV X-ray emission. The upper right is for 1.5-3.0 keV X-ray emission and the lower left is for 4.0-6.0 keV. The lower right is for the cleaned JWST image. The break from the power law at the largest scales is due to the fractal description of the emitting surface failing at scales comparable to the radius of Cas A. Because of JWST's better angular resolution, the lower right spectrum has a larger dynamic range.