Quantifying Symmetry: Transformation Information for Planetary Nebulae and Supernova Remnants

TL;DR

The paper presents Transformation Information (TI), a KL-divergence-based, non-parametric framework to detect and quantify symmetry in astrophysical images by comparing an image to its rotated or reflected versions. TI minima over transformation parameters reveal symmetry axes, and a thresholded silhouette variant emphasizes global morphology for outlining features. Applied to planetary nebulae, TI recovers axes corresponding to bipolar and multipolar lobes, while in supernova remnants it identifies axes associated with protrusions and rims; a two-value variant highlights additional silhouette-based symmetries. Furthermore, a minima-prominence-to-width descriptor enables population-level separation between Type Ia and core-collapse SNRs, illustrating TI’s utility for classification and comparative studies. The work provides a reproducible, data-driven approach to symmetry identification and opens avenues for feature-aware refinements and uncertainty quantification in large-sample astrophysical morphology analyses.

Abstract

We present a quantitative symmetry-identification pipeline for astrophysical images based on Transformation Information (TI), an information measure of self-similarity under geometric transformations. TI is expressed as a Kullback-Leibler (cross-entropy) divergence between an image and its rotated or reflected counterpart on the overlapping domain. By scanning rotation angles and reflection axes, we obtain TI curves whose local minima identify symmetry operations. We validate the method on a wind-rose pattern and then apply it to planetary nebulae, where the recovered axes trace bipolar and multipolar lobes consistent with morphology-based classifications. Applying TI to supernova remnants yields estimate axes associated with protrusions, rims, and substructure. To emphasize global morphology, we introduce a thresholded two-level variant that compares binary silhouettes and can reveal outline-driven symmetries. Finally, we quantify symmetry using a minima prominence-to-width score and show that this compact descriptor separates Type Ia and core-collapse remnants into distinct populations for an X-ray sample. TI provides a non-parametric, reproducible framework for symmetry identification, classification and population studies.

Figures (9)

• Figure 1: Our implementation of the transformation information (TI) symmetry identifier, demonstrated on a wind-rose synthetic image. Right panels depict the output of our processing pipeline. Upper right is the rotational TI measure. Dashed red vertical lines indicate identified valleys, denoting angles $\theta_{symm}^{rot}$ for which rotating the image by $\theta_{rot}$ results in a local TI minima, corresponding to a rotational symmetry. The bottom right panel is the reflection TI measure, with dashed blue lines indicating $\theta_{ref}$, for which a reflection about a line at $\theta_{ref}$ produces a local minima in the TI measure. Left is our annotated image, where we denote $\theta_{rot}$/$\theta_{ref}$ (dashed blue/red lines) on the image. The symmetry axes corresponding to $\theta_{ref}$, set at $\theta^{ref}_{symm}=\theta_{ref}+90^\circ$ are denoted as solid yellow lines, and serve to guide the eye for symmetric features along this axis. Rotational symmetries are denoted with a curved arrow, where $\theta_{symm}^{rot}$ is denoted near the arrow head. Dashed red lines indicate $\theta_{symm}^{rot}$ angles, with an additional line at $\theta=0$ for reference. Two panels at the center demonstrate the subtraction between the transformed image and the original, for two examples: $\theta_{rot}=270^\circ$ (top) and $\theta_{ref}=135^\circ$ (bottom), scaled with the color bar between the two panels. For our wind-rose example, all reflection symmetries matching each opposite spike pair and rotational symmetries corresponding to rotations from one spike to the next are consistent with base truth assumptions.
• Figure 2: Our implementation of the TI symmetry identifier for the PN M1-37. Dashed-red/blue lines depict axis along which the transformations reveal the PN is symmetrical, and in solid lines the corresponding symmetry lines themselves. For the reflection symmetry the identified axes at $\theta_{symm}\approx77^\circ, 94^\circ$ degrees (solid-yellow) correspond to the (split, see lower half) vertical structure, but missing the axes at $\approx125^\circ$ and $\approx55^\circ$. An equatorial bright symmetric indent is also tagged at $\theta_{symm}=3^\circ\approx0^\circ$. Overall, the north-south symmetry is captured by a reflection axis $\theta_{ref}=4^\circ\approx0^\circ$, and a rotation axis at $\theta_{symm}^{rot}=179^\circ\approx180^\circ$. Rotation TI also recovered the missing $\theta_{symm}=55^\circ+180^\circ$.
• Figure 3: Four more analyzed PNe, where we denote the identified symmetry axis identified by the TI measure with yellow lines. We label the panels with their designation at the top left, and in an inset at the top right we add the original nebulae images, as taken from the Bruce Balick planetary nebulae catalog. We present the full TI profiles for these PNe in Appendix \ref{['appendix:TIplots']}. (a) MyCn 18: Two clear axis are identified, corresponding to the hourglass lobes (north-south, $\theta_{symm}=86^\circ\approx90^\circ$) and the secondary axes in the equatorial plane ($\theta_{symm}=175^\circ\approx180^\circ$). A rotational symmetry with $\theta_{symm}^{rot}=179^\circ\approx180^\circ$ corresponds to the general $180^\circ$ symmetry. (b) NGC 6543: four reflection symmetries and one rotational symmetry. The general elongated structure is identified by $\theta_{symm}=58^\circ$. Large features corresponding to the complicated ringed structure are matched by both $\theta_{symm}=22^\circ, 109^\circ, 143^\circ$ and the rotational symmetry of $\theta_{symm}^{rot}=156^\circ$. The many concentric circular shells (e.g., Guerrero2020_NGC6543rings) did not interfere with the TI measure as these are approximately uniformly distributed along all angles. (c) NGC 5307: The main north-south elongated structure featuring two opposite bright regions is identified by the $\theta_{symm}=89^\circ, 94^\circ$ axes. The bright knots opposite at $\theta_{feature}\approx150^\circ$ triggered an identified axes at $\theta_{symm}=153^\circ$ and potentially with the two north-south knots, $\theta_{symm}=174^\circ$. A rotational symmetry axis at $\theta_{symm}^{rot}=172^\circ\approx180^\circ$ is likely a combination of the bright knots and the general polar shape. (d) NGC 7293: Four reflection axes are detected, roughly $45^\circ$ from one another. $\theta_{symm}=34^\circ$ appears to coincide with the general elongated shape, and the other axes at $\theta_{symm}=79^\circ,124^\circ,171^\circ$ with bright rims that are equidistant and located opposite from the center. A rotational symmetry axis $\theta_{symm}^{rot}=179^\circ$ corresponds to a bright bipolar shape.
• Figure 4: Application of the TI measure on a soft X-ray image of the Vela SNR (eROSITA DR1 data, see details and references in SokerShishkin2025_Vela). To separate the Vela SNR from two other SNRs in the same patch of sky, we excise the Puppis A SNR from the image replacing a circle around its' center with the average of its surrounding, and use an image of the lower energy spectrum where the Vela Jr. SNR is less visible. We chose the center location to be the deduced center between opposite clumps (blue star in Figure 1 of SokerShishkin2025_Vela). Several projectiles (clumps) and their bow-shock wake are accompanied by bright X-ray emission. We denote previously identified clumps with their associated designated letters as overlay (post-TI analysis). Two symmetry axes along the horizontal image direction $\theta_{symm}=-24^\circ,35^\circ$ correspond to the presence of several projectiles and the slightly extended structure in this general direction. A symmetry axis at $\theta_{symm}=94^\circ$ correlates to a previously identified north-south axis Mayer_etal_2023_VelaSokerShishkin2025_Vela, and emphasizes the elongated shape extending between features D and E. Rotational symmetries at $\theta_{symm}^{rot}=139^\circ,155^\circ$ appear to correspond to aligning projectile track features in the outer structure of the remnant. We illustrate this with curved concentric arrows between E and A,B.
• Figure 5: As Figure \ref{['fig:PNe']}, but for four SNRs. (a) N132D: A reflection symmetry axis at $\theta_{symm}=138^\circ$ corresponds to the extended structure (image top-left to bottom-right). Two additional symmetry axes at $\theta_{symm}=15^\circ,65^\circ$ match different parts of the bright arc at the remnant bottom, or folding along the image top-left ($\theta_{ref}=155^\circ$). Two rotation symmetries correlate with moving outer protrusions (indicated with red arrows). (b) Simeis 147: Several symmetry axes are identified, two coincide with the extended shape (at $\theta_{symm}=70^\circ,94^\circ$) and three ($\theta_{symm}=27^\circ,116^\circ,160^\circ$) correlate with folded-over bright zones along the outer shape. A rotational symmetry with $\theta_{symm}^{rot}=265^\circ=-95^\circ\approx90^\circ$ seems to match bright features or protrusions in the outer shape, and $\theta_{symm}^{rot}=190^\circ\approx180^\circ$ match the overall extended shape. (c) N63A: Two groups of reflection symmetries are identified; $\theta_{symm}=131^\circ$, corresponding to folding over the elongated axis (top left to bottom right in the image). Notable features coinciding over this axis are the two protruding features on both ends of the remnant. $\theta_{symm}=49^\circ$, similarly folds over the narrow direction, reflecting around the large tail at the image bottom right and matching the two smaller protrusions at the image top left (Ear 1 and Ear 2, see Karagoz_etal2023_N63A for their notations). A rotational symmetry of $\theta_{symm}^{rot}=185^\circ\approx180^\circ$ further cements a symmetrical opposing morphological structure. (d) G321.3-3.9: Of the four reflective symmetry axes, two axes $\theta_{symm}=114^\circ,133^\circ$ coincide with two structures: a protrusion on the image top left and a rimmed bubble at the image bottom right. A rotational symmetry of $\theta_{symm}^{rot}=12^\circ$ matches both the width of said bubble and the distance to a structure adjacent to the top protrusion. Bright rim-like features on the remnant narrow section (horizontal, at $\theta_{axis}\approx20^\circ$) are matched with both $\theta_{symm}=17^\circ$ and $\theta_{symm}^{rot}=150^\circ$.