2.1. EMU and RadioGalaxyNET

The Evolutionary Map of the Universe (EMU; Norris et al. Reference Norris2011; Hopkins et al. Reference Hopkins2025) is a wide-field radio continuum survey, using the ASKAP telescope. The primary goal of EMU is to make a deep ( $\sim\! 900$ MHz) radio continuum survey of the entire Southern sky at a resolution of $\sim\! 15''$ and sensitivity of $\sim\! 25\,\unicode{x03BC}$ Jy/beam. It is anticipated that EMU will detect and catalogue tens of millions sources, many of which will be extended and complex. A key motivation for this work is the analysis of large numbers of radio sources, for which we start by drawing from EMU data. In preparation for the full EMU survey (Norris et al. Reference Norris2011), the EMU Pilot Survey (EMU-PS; Norris et al. Reference Norris2021) was conducted to test the survey strategy and data processing pipeline. Phase 1 of the EMU-PS was limited to a contiguous region of $270\,\text{deg}^2$ within the Dark Energy Survey region (DES; Abbott et al. Reference Abbott2018), lying at a declination $\lt-30^\circ$ and Galactic latitude of $\gt+20^\circ$ , with a sensitivity of 25– $35\,\unicode{x03BC}$ Jy/beam at 944 MHz.

RadioGalaxyNET (Gupta et al. Reference Gupta, Hayder, Norris, Huynh and Petersson2024a) is developed from a curated dataset of DRAGNs classified by professional astronomers, designed for training machine learning models in radio morphology recognition. Because of the manual nature of identifying the original dataset, the DRAGNs used will tend to favour higher signal-to-noise systems, although this is not a fixed or quantitative threshold. It simply arises since very low surface brightness objects are harder to visually identify. The full RadioGalaxyNET dataset contains $2\,800$ EMU-PS $15' \times 15'$ cutouts in three channels (unprocessed radio, processed radio, and infrared), resulting in a total of $4\,155$ annotated sources, since some cutouts contain multiple sources. The annotations consist of radio morphological class information, bounding box details for associated components of each radio galaxy, radio galaxy segmentation masks, and the host galaxy positions from the infrared images. This work primarily analyses a subset of the sources from the RadioGalaxyNET data.

In order to ensure a source is sufficiently resolved and contains enough observable structure for our analysis, we define a minimum area threshold of the radio cutouts of 20 times the ASKAP synthesised beam size (1 beam $\approx 13''\times11''$ ; Norris et al. Reference Norris2021). The minimum area for each cutout is therefore $2\,860$ arcsec $^{2}$ or $\sim\!715$ pixels (1 pixel $\approx 2$ arcsec in the EMU-PS mosaic). Below this value the analysis of the radio lobes can become compromised by the limited number of beams (resolution elements). To produce the cutout images from the EMU-PS, we first extracted the coordinates of the bounding boxes from the RadioGalaxyNET data. These coordinates were transformed to match the orientation of the EMU-PS mosaic, from which we directly produce the cutouts of the individual radio galaxies. There are 595 sources from RadioGalaxyNET that met the minimum size requirements, with an average cutout size of $\sim\! 3\,600$ pixels. The cutouts were made on the ‘unsmooth’ EMU-PS mosaic, in order to maximise the resolution in different regions.

2.2. 3CRR and an atlas of DRAGNs

The Third Cambridge Catalogue of Radio Sources (3C; Edge et al. Reference Edge, Shakeshaft, McAdam, Baldwin and Archer1959; Archer et al. Reference Archer1959) is a radio catalogue originally observed at 159 MHz, then revised at 178 MHz (3CR; Bennett & Simth Reference Bennett and Simth1962; Bennett Reference Bennett1962). The 3CR catalogue contained 328 sources and was one of the first comprehensive lists of extragalactic radio sources. However, the original survey had relatively limited angular resolution, leading to source blending and other biases (Véron Reference Véron1977). To improve upon this, a further revision of 173 bright radio sources at 178 MHz was conducted (3CRR; Laing, Riley, & Longair Reference Laing, Riley and Longair1983). While the initial 3CR selection suffered from resolution limitations, 3CRR sources have since been observed in much greater detail and resolution. The 3CRR catalogue contains all extragalactic radio sources in the Northern Hemisphere with a 178 MHz flux density greater than $10.9$ Jy.

An Atlas of DRAGNs (Leahy, Bridle, & Strom Reference Leahy, Bridle and Strom2013) is a subset of the 3CRR catalogue, consisting of the nearest 85 DRAGNs. The Atlas of DRAGNs websiteFootnote ^a contains high-quality images of the DRAGNs compiled from different studies. The images in the atlas have high resolution, and sufficient sensitivity and spatial frequency coverage to clearly and accurately image the faintest regions of a radio source. Most images are at frequencies near $1.5$ GHz, although the frequencies range from $0.3$ to $8.4$ GHz in the atlas. The difference in resolution between EMU-PS and 3CRR sources from this atlas is highlighted in Figure 1. There is also a dedicated page providing supplementary information about each radio source, containing FR classification, redshift, radio power, and linear size. Given the well-documented nature of these sources, we use this atlas as a reference dataset in this work.

Figure 1. Comparison of two double lobed radio sources to highlight the resolution between datasets. As observed in J202644.4 $-$ 552227 (left) and 3C285 (right), both sources show similar structural features. Angular size of each image shown in the bottom right.

The cutouts of the 3CRR sources, as well as the corresponding supplementary information, were directly downloaded from the Atlas of DRAGNs site. We do not use the full sample of 85 sources in this work. Through manual inspection, it was found that two sources had corrupt FITS files, two sources were not centred in the image, two were extremely bent (see Appendix A), and seven contained excessive structural complexity too intricate to be processed reliably in an automatic way (see Table 1). Our reference dataset therefore consists of 72 3CRR sources (17 FRI and 55 FRII).

Table 1. List of sources and reasons for omission from analysis.

3.1. Pre-processing

Many of the EMU-PS sources contain a very bright core. As the metric calculations are based on the brightest pixel, this can influence the results. We therefore implement a core-removal process. We first apply a minimum flux threshold of $150\,\unicode{x03BC}$ Jy, corresponding to roughly $5\sigma$ of the EMU-PS imaging ( $\sigma = 25$ – $30\,\unicode{x03BC}$ Jy/beam, Norris et al. Reference Norris2021). Pixels with intensities less than this threshold are are set to zero to remove noise for the purposes of core identification only. Following the work outlined in Hancock et al. (Reference Hancock, Murphy, Gaensler, Hopkins and Curran2012), to highlight potential edges and features in an image we generate a curvature map by convolving the thresholded image with the following Laplacian kernel:

(1) \begin{equation} L^2_{xy} = \begin{bmatrix} 1 & \quad 1 & \quad 1 \\ 1 & \quad -8 & \quad 1 \\ 1 & \quad 1 & \quad 1 \\\end{bmatrix}.\end{equation}

The resulting curvature map emphasises regions with sharp intensity transitions, aiding in the accurate localisation of a core. We then generate a binary map with labelled features, calculate their sizes, and retain the largest components. If an image contains two lobes and a core (i.e. number of components $\gt 2$ ), a circular mask is applied around the local maximum nearest to the infrared coordinates of the host galaxy, which is expected to be coincident with the core. This process is fully automatic and demonstrated in Figure 2. Through an initial visual inspection of the 592 EMU sources, we estimate that 186 (30%) sources contain a dominant core, i.e. the central component was brighter than the two lobes. We then applied the above core-removal process to the 592 images and, after another visual inspection, the core of 102 sources was successfully removed. However, this process is not perfect (see Appendix A). We found that the number of components detected was sometimes incorrect, leading sources that did not have a dominant core to accidentally have a lobe masked. Sources with a dominant core remaining after the attempted masking (84) or an inadvertently masked lobe (28) were subsequently dropped from our analysis, leading to a final EMU-PS dataset consisting of 480 sources.

Figure 2. Visualisation of the core masking process. From left to right is the original EMU-PS radio image, the curvature map, the labelled components, and the final image with the core removed.

3.2. Asymmetry

The asymmetry of a galaxy has been defined in a number of ways in optical astronomy (e.g. Schade et al. Reference Schade1995; Abraham et al. Reference Abraham1996a, Reference Abraham1996b; Conselice Reference Conselice1997; Brinchmann et al. Reference Brinchmann1998). These methods provide a framework for morphological classification that can also be applied to radio sources. We adopt the approach to asymmetry as described in Conselice et al. (Reference Conselice, Bershady and Jangren2000) and Conselice (Reference Conselice2003), which compares a source image (I), with itself rotated by an angle of $180^\circ$ ( $I_{Rot}$ ). We define the centre of rotation as the brightest pixel in a given lobe. We produce the difference image between the original and rotated image, and take the absolute value of the sum over all pixels, $\eta$ , and normalising by the total flux of the original lobe, $I_{Tot}$ , to produce a shape asymmetry metric ( $A_{S}$ ). The absolute value of the difference of two images will produce twice the flux for a perfectly asymmetric source (i.e. each bright pixel has no corresponding bright pixel after the image is rotated), and we therefore include the factor of $1/2$ to ensure $A_{S}$ is always between 0 and 1:

(2) \begin{equation} A_{S} = \frac{1}{2} \sum_{i,j}^{\eta} \frac{|I - I_{Rot}|}{I_{Tot}}. \end{equation}

We also include a metric to describe the flux distribution across a lobe, defining a ratio asymmetry ( $A_{R}$ ), which is the absolute value of the ratio of the original and rotated lobe, summed over all pixels, $\eta$ (Equation 3). For a completely uniform (or symmetric) source, each pixel ratio would be 1, and the sum of the ratio image will be number of pixels. We therefore normalise by the number of pixels, meaning a perfectly uniform source will produce a minimum $A_{R}$ value of 1:

(3) \begin{equation} A_{R} = \frac{1}{\eta} \sum_{i,j}^{\eta} \big| \frac{I}{I_{Rot}} \big|. \end{equation}

Examples of the images of $A_{S}$ and $A_{R}$ (before summing the pixels) are shown in Figure 3.

Figure 3. Visualisation of both asymmetry calculations. From left to right: original lobe image, its rotated counterpart, the residual image after taking the absolute value of the difference between the original and rotated image, and the residual image after taking the ratio of the original and rotated images.

We can define analogous calculations to $A_{S}$ and $A_{R}$ to compare how asymmetric the two lobes ( $I_{1}$ and $I_{2}$ ) of a radio galaxy are to each other. We denote these new parameters with the subscript p to represent the comparison of the pair of lobes in the DRAGN system. We can define two shape asymmetry metrics, the first being the total asymmetry ( $A_{S,p}$ , Equation 4) which is weighted by the relative flux in each lobe. This takes a similar form to the $A_S$ calculation, however we instead normalise by the sum of the total flux in both lobes. To compare only how symmetric shapes of the lobes are, without the relative flux weighting, we must normalise each lobe by the total flux, $I_{Tot}$ before computing the difference ( $A'_{S,p}$ , Equation 5). The ratio asymmetry between lobes, $A_{R,p}$ (Equation 6), is determined in the same way as $A_{R}$ , using the absolute value of the ratio of one lobe with the rotated other, normalised by the number of pixels, $\eta$ . As before, we can also normalise by the total flux before computing the ratio ( $A'_{R,p}$ , Equation 7). As with the single-lobe calculations, $A_{S,p}$ and $A'_{S,p}$ both have values between 0 and 1, and $A_{R,p}$ and $A'_{R,p}$ are always greater than 1.

(4) \begin{equation} A_{S,p} = \frac{1}{2} \sum_{i,j}^{\eta} \frac{|I_{1} - I_{2,Rot}|}{I_{1, Tot} + I_{2, Tot}}, \end{equation}

(5) \begin{equation} A'_{S,p} = \frac{1}{2} \sum_{i,j}^{\eta} \left|\frac{I_{1}}{I_{1,Tot}} - \frac{I_{2}}{I_{2,Tot}}\right|, \end{equation}

(6) \begin{equation} A_{R,p} = \frac{1}{\eta}\sum_{i,j}^{\eta} \left| \frac{I_{1}}{I_{2,rot}} \right|, \end{equation}

(7) \begin{equation} A'_{R,p} = \frac{1}{\eta}\sum_{i,j}^{\eta} \left| \frac{I_{1}/I_{1,Tot}}{I_{2,rot}/I_{2,Tot}} \right|. \end{equation}

3.3. Blurriness

We calculate how blurry (or diffuse) radio lobes are by measuring the rate at which the flux of a lobe decreases from the hotspot radially outwards. We first define eight cardinal directions, taking slices of the pixel intensities radially outwards from the brightest pixel (i.e. the hotspot). To calculate the rate at which the flux decreases, we use the scipy.optimize curve_fit function to fit a straight line to each of the eight slices and extract the gradient (Figure 4). While we acknowledge that the pixel intensity is not linear, this approach provides a robust first-order estimate of blurriness, as more complex models (e.g. exponential or power law fits) introduce additional parameters that may not be necessary for comparative classification. The gradient of pixel intensities (normalised by the peak brightness) in the whole slice alone is not enough to determine how blurred or sharp a lobe is. We therefore also measure the gradient of the three outermost pixels in each slice. We chose to use 3 pixels as this is the minimum number of points we can have to fit a gradient, and limit the number of pixels from within the lobe. We average over all eight directions for both the full slice and outermost pixels ( $\bar{g}_{\text{full}}$ and $\bar{g}_{\text{outer}}$ , respectively). To estimate the how blurred a lobe is, we then take the ratio of $\bar{g}_{\text{full}}$ and $\bar{g}_{\text{outer}}$ :

(8) \begin{equation} B = \frac{\bar{g}_{\text{full}}}{\bar{g}_{\text{outer}}}. \end{equation}

Small B values (low blurriness) are produced when the average edge gradient of flux is steeper than the average gradient of the whole lobe. This could be indicative of a relatively constant brightness distribution across the lobe with a sharp drop to the background, possibly produced by a lobe expanding into a dense environment, such as within a galaxy cluster. Conversely, higher B values (high blurriness) may indicate more diffuse lobe structure and potentially a less dense environment.

Figure 4. Visualisation of the blurriness calculation, highlighting the 8 cardinal directions and corresponding normalised intensity slices measured away from the brightest pixel of a lobe.

3.4. Concentration

The concentration metric describes how the flux is distributed within a lobe. Our approach to concentration is analogous to the approach described in Abraham et al. (Reference Abraham, Valdes, Yee and van den Bergh1994), with concentration defined as the ratio of flux between inner and outer isophotes of normalised radii. We define concentration, C, as the ratio of radii containing 80% and 20% of the flux in a lobe, centred on the brightest pixel:

(9) \begin{equation} C = \frac{r_{80}}{r_{20}}. \end{equation}

To calculate the radius of each isophote, we sum the number of pixels until 20% and 80% of the total lobe flux is achieved. This sum is treated as an area, $\alpha$ , and converted into a circular radius, r, using the relationship $r = \sqrt{\alpha/\pi}$ . Here, C has a minimum value of 2 for a completely uniform source. Larger values of C mean the flux is more concentrated within the lobe, and can be indicative of a hotspot. A comparison showing high and low concentration lobes is presented in Figure 5.

Figure 5. Visualisation of the concentration metric comparing lobes with high and low concentration (left and right, respectively). Radii corresponding to 20% and 80% of the lobe flux shown in red and orange, respectively.

3.5. Disorder

We include a metric to quantify how perturbed a radio galaxy lobe is. Perturbations in the lobe structure can be caused by differing jet power, or from dynamic forces within the host galaxy environment. We refer to this metric as ‘disorder’, D, and following the definition of complexity by Watson (Reference Watson2012), define it as the ratio of the perimeter of a lobe squared and its area multiplied by $4\pi$ :

(10) \begin{equation} \text{D} = \frac{\text{Perimeter}^2}{4\pi\text{Area}}. \end{equation}

To determine the perimeter, we measure the length of a fitted contour around pixels above the minimum threshold (10% of the peak brightness). To calculate the area of the lobe, we simply sum the number of non-NaN pixels above the minimum threshold. The D metric therefore has a minimum value of 1 for a perfectly circular source. Figure 6 highlights the difference between lobes with high and low disorder.

Figure 6. Visualisation of the disorder metric comparing lobes with high and low disorder (left and right, respectively). Perimeter calculated by the length of the red contour.

3.6. Elongation

We also measure the elongation of a lobe for a radio source, which can be related to the power of a radio jet as more powerful radio jets can produce more extended structures. We define it as the ratio of the major axis and the minor axis of a bounding box fitted to a radio lobe:

(11) \begin{equation} \text{E} = \frac{\text{Major axis}}{\text{Minor axis}}. \end{equation}

This metric will have a minimum value of 1 for a square or circularly shaped object. We perform a principle component analysis (PCA) on the pixels above the minimum threshold to orient and fit the bounding box, where we can then extract the lengths of the major and minor axes of the box. This is shown in Figure 7.

Figure 7. Visualisation of the elongation metric comparing lobes with high and low elongation (left and right, respectively). The bounding box of each lobe is shown in black.

3.7. Angle between lobes

We include a metric that estimates how bent a radio source is, the angle between lobes (ABL). The ABL can serve as a proxy for environmental effects, as bent radio galaxies are predominantly found in dense and dynamic environments such as galaxy clusters. To calculate the ABL, we treat the brightest pixels in each lobe as points in space, $\mathbf{L}_1$ and $\mathbf{L}_2$ , where we can then determine vectors to the core of the radio galaxy, $\mathbf{C}$ (Equations 11a and 11b). For the location of the core, we prioritise using the infrared coordinates of the host galaxy when available, otherwise we simply use the geometric centre of original radio galaxy cutout. Once the vectors have been determined, we compute the dot product (Equation 12) and magnitude of the vectors (Equations 12a and 12b) to extract the angle between lobes (Equation 13), which is then converted into degrees. This calculation is illustrated in Figure 8. We will refer to ABL, $A_{S,p}$ , $A'_{S,p}$ , $A_{R,p}$ , and $A'_{R,p}$ as source metrics hereafter.

(11a) \begin{align} \mathbf{v}_1 = \mathbf{L}_1 - \mathbf{C} = (x_1 - x_c, y_1 - y_c),\end{align}

(11b) \begin{align} \mathbf{v}_2 = \mathbf{L}_2 - \mathbf{C} = (x_2 - x_c, y_2 - y_c), \end{align}

(12) \begin{align} \mathbf{v}_1 \cdot \mathbf{v}_2 = (x_1 - x_c)(x_2 - x_c) + (y_1 - y_c)(y_2 - y_c), \end{align}

(12a) \begin{align} |\mathbf{v}_1| = \sqrt{(x_1 - x_c)^2 + (y_1 - y_c)^2}, \end{align}

(12b) \begin{align} |\mathbf{v}_2| = \sqrt{(x_2 - x_c)^2 + (y_2 - y_c)^2}, \end{align}

(13) \begin{align} \cos(\theta) = \frac{\mathbf{v}_1 \cdot \mathbf{v}_2}{|\mathbf{v}_1| |\mathbf{v}_2|} \Rightarrow \theta = \arccos\left( \frac{\mathbf{v}_1 \cdot \mathbf{v}_2}{|\mathbf{v}_1| |\mathbf{v}_2|} \right). \end{align}

Figure 8. Example calculation of the ABL for a radio source from the EMU-PS. Positions of the brightest pixels in each lobe and the location of the host galaxy shown by the white dots and labelled $\mathbf{L}_1$ , $\mathbf{L}_2$ and $\mathbf{C}$ , respectively. Vectors $\mathbf{v}_1$ and $\mathbf{v}_2$ shown by the dashed lines. Angle $\theta$ indicated by the arc between both vectors.

3.8. Resolution dependence of metrics

It is well known that metrics used in quantifying shapes, such as those presented in this work, are likely to be sensitive to the resolution of the images analysed. The shape of unresolved or marginally resolved sources (i.e. components only one or two beams wide) will be determined by the shape of the telescope synthesised beam, which is usually an elliptical Gaussian (Boyce et al. Reference Boyce2023). To understand the effect resolution has on each of our metrics, we convolve high resolution images with 3 different 2D Gaussian kernels, each with increasing standard deviation ( $\sigma =$ 3, 6, and 9) to produce images with progressively worse resolution. The convolution is performed in real space using direct kernel application. As the $\sigma$ increases, we begin to lose more and more structure until the lobes become unresolved (Figure 9). We chose to convolve the 3CRR images for convenience, but the following analysis should be independent of the input images.

Figure 9. Effect of convolving radio image of 4C73.08 (top left) with 2D Gaussian kernels of increasing $\sigma$ in units of pixels.

We apply each metric to the convolved images and compare them with the original values in Figure 10. We can see that the ABL calculation (Figure 10k) is the most robust against resolution differences. The scattered points in this metric result from location of the brightest pixel changing, often moving closer to the centre of a lobe after convolving. We see that convolving tends to produce smaller values for shape asymmetry measures ( $A_S$ , $A_{S,p}$ and $A'_{S,p}$ ). This is an expected result, as the flux in the lobes becomes more evenly distributed and circular in shape, resulting in lower asymmetry. This appears in Figure 10c, g, and h, where we can see smaller asymmetry values with increasing $\sigma$ .

Figure 10. Comparison of the metric outputs of original 3CRR images and convolved versions. Blue, orange and green points correspond to images convolved with Gaussian kernels with $\sigma = 3$ , 6, and 9 respectively. Black 1:1 line indicates no deviation in the metrics after convolving. Panels (a)–(e) have two points per source, one for each lobe.

The other metrics show less intuitive trends. The process of convolving reduces the brightness of the peak pixel, spreading that flux over the surrounding pixels. For the metrics that use threshold masking (all apart from the ABL and asymmetry metrics), using 10% of a lower peak brightness means more pixels lie above the threshold. With B for example, more dim pixels around the edge of the lobe are retained. These pixels would produce a smaller average outer gradient making the overall ratio a larger number, moving the points above the $1:1$ line (Figure 10a). This effect would be increased as sigma gets larger, which is why the $\sigma=9$ points sit higher than the $\sigma=6$ and $\sigma=3$ points. For concentration, this may increase $r_{80}$ at a faster rate than the $r_{20}$ , making the overall concentration measurement higher.

We see for disorder and elongation that the convolved images produce fairly stable values, that then decrease gradually with increasing $\sigma$ . This again is an expected result as the shape of the lobe becomes more circular with convolving, therefore making both disorder and elongation calculations approach unity. For disorder, the increase in the number of pixels will increase both the perimeter and area, however the perimeter $^2$ dependence will cause the disorder calculation to become much larger. For elongation, we expect the value to become closer to unity after convolving. It is possible that the increase in the number of dim pixels included some of the jet structure that was previously too dim, therefore increasing the overall elongation value. We have similar expectations with $A_R$ , in that as we convolve the distribution of flux is smoothed, so $A_R$ should approach unity. As before with elongation, the inclusion of jet structure that was previously too dim, before convolving, can again increase the values of $A_R$ .

The above analysis shows that the metrics can be highly resolution dependent, as evidenced in Figure 10, with many metrics showing high scatter when comparing original and convolved values. Higher resolution will enable the metrics to be more robust, as there is more structural information available to be extracted.

4.1. Single-lobe metrics

Figure 11 shows pairwise combinations of the single-lobe calculations (ABCDE), with the normalised kernel density estimate (KDE) for each metric in the diagonals. For most of the metrics, the EMU and 3CRR sources occupy similar areas of parameter space, indicating the minimum image area of 20 ASKAP beams for the EMU sources allows them to be sufficiently resolved for the metric results to be comparable with the 3CRR data. While some metric combinations show slight correlations (e.g. B and C) or more complicated relationships (e.g. $A_S$ and $A_R$ ) we do see high scatter in each subplot, indicating that these metrics my be independent. We calculated the Spearman rank correlation coefficient ( $\rho$ , Siegel & Castellan Reference Siegel and Castellan1988; Conover Reference Conover1999) on each of the unique metric combinations from Figure 11. Values of $\rho \approx 0$ correspond to little to no correlation, with $\rho \approx \pm 1$ correspond to a strong positive or negative correlation, respectively. We found that most of the metric combinations had $-0.054 \lt \rho \lt 0.27$ , with the exception of $A_S$ - $A_R$ ( $\rho$ = $0.72$ ) and D-E ( $\rho = 0.57$ ). This indicates that the metrics are, in fact, largely independent of each other and therefore measure fundamentally different structural features of a radio galaxy lobe.

Since the FRI/II dichotomy has been a very productive scheme for uncovering the physics of radio galaxies, we will explore to what extent these metrics can reproduce this dichotomy. BothFR-types occupy similar regions in metric space for the majority of our metrics, with the exception $A_S$ , $A_R$ where we see stronger separation between FRI and FRII sources. FRI sources also tend to have a broad range of values, when compared to the FRII sources, particularly in the B, C, and E metrics. This reinforces the need for additional metrics to completely characterise the shape of radio lobes. We explore correlations between metrics and FR distributions in more detail in subsequent sections.

4.1.1. Asymmetry

Evident in both the scatter and KDE panels in Figure 11, the FRI and FRII sources are found in different areas of the asymmetry parameter space. We see that for both $A_S$ and $A_R$ that the FRI sources tend be less asymmetric than FRII sources. While the EMU sources share the same region of parameter space as the FRIs there is evidence of a bimodal distribution, highlighted in the KDE of the $A_R$ metric. There is a highly concentrated population at very low $A_R$ , and another more scattered population at moderate to high $A_R$ . The initial comparison of EMU and 3CRR sources indicates that the low $A_R$ values coincide with the FRIs and the moderate to high $A_R$ values with FRIIs. We explore the $A_S$ - $A_R$ parameter space further in Figure 12.

Figure 12. Enlarged version of the $A_S$ - $A_R$ sub-panel from Figure 11 showing only EMU data. In this figure different regions of this parameter space are highlighted by different colours. Panels below are lobe cutouts for each highlighted region.

The concentrated tail at low $A_R$ values (e.g. the blue and green regions) may be produced when the brightest pixel is near the edge of the lobe cutout, therefore limiting the size of the cutout and causing the overall flux distribution to appear more uniform. This will result in lower values for the $A_R$ metric, forming the tail with $\log_{10}(A_R)$ values close to 0. We can relate this to the abundance of FRI sources present in the tail. FRI sources are often referred to as core-brightened, with the peak brightness occurring in the jets near the core. The code splits an image of a radio galaxy in half under the assumption that there will be a radio lobe in each half. As the FRI hotspots tend to be closer to the core, the brightest pixel for an FRI source would therefore be near the border of the lobe cutout, limiting the number of pixels in the asymmetry calculations. The tail is therefore likely an artefact of both the flux distribution of the sources and of how we isolate a lobe in our algorithm. For objects found in the more scattered section of this parameter space (e.g. the red and orange regions), the asymmetry metric appears to be working as expected, with the scatter resulting from actual variance in lobe structure in different sources.

In the purple region, there are two measurements with $\log_{10}(A_S)\gt0$ , which should not occur for a normally distributed source. We can see in the lobe cutouts for this region of parameter space that both contain two separate circular sources, with one near the centre and the other offset. From visual inspection, the central sources appear to have low rotational asymmetry, and will not contribute much to the asymmetry value. For the off-centre sources however, the rotation by $180^{\circ}$ will produce two bright spots after taking the absolute value, corresponding to a high $A_S$ value. We should also note that both of these sources have relatively low overall brightness, and, we can see some of the noise fluctuations in surrounding pixels.

We conducted simulations to investigate the features observed in Figure 12, particularly what produces the concentrated tail and what causes sources to have $\log_{10}(A_S)\gt0$ . To simulate a perfectly symmetric lobe (red points), with the only asymmetry coming from noise fluctuations we first populate a 1D array with values ranging between $1\times 10^{-5}$ and 1, referred to as array x. We then generate a second 1D array, y, which is identical to array x except for the addition of random noise, normally distributed around zero with an amplitude of $0.1$ :

\begin{equation*} y = x + \text{noise},\end{equation*}

where we then calculate $A_S$ and $A_R$ by substituting x for lobe 1 and y for lobe 2 into Equations (2) and (3), respectively.

To simulate an asymmetric lobe (blue points), we generated 15 different $25 \times 25$ Gaussian profiles, with ranging intensities, $\sigma$ , skew, and rotation (Figure 13). We loop over each of the profiles, systematically adding noise until we have the same number of data points as our EMU-PS data (480), repeating the asymmetry calculations. The noise added in this loop is normally distributed around zero, with amplitude ranging from $0.015$ to $0.9$ .

Figure 13. Collection of the 15 different 2D Gaussian profiles used in the simulations.

Finally for the noise-only simulation (green points), we randomly populate a $15 \times 15$ array with Gaussian noise matching the noise distribution of the EMU-PS, i.e. centred on zero with $\sigma = 23\,\unicode{x03BC}$ Jy/beam (Norris et al. Reference Norris2021).

The asymmetry calculations on the simulated data (Figure 14) revealed that the noise-only points explicitly exist at $\log_{10}(A_S)\gt0$ . This is likely due to normalising by the total flux. The sum of an array populated by Gaussian noise distributed around zero approaches 0 as we increase the number of points in the array. This means that for the noise-only points (i.e. sources with poor signal-to-noise), we normalise by a very small number, leading to $\log_{10}(A_S)$ values greater than 0. In this work we wanted to investigate how the metrics perform on a variety of different sources, and we do not include any minimum signal to noise cutoff.

Figure 14. Enlarged version of the $A_S$ - $A_R$ sub-panel from Figure 11 with only EMU data, comparing how the asymmetry parameters characterise different simulated data. Grey, blue, red, and green correspond to data points from the EMU-PS, simulated asymmetric lobes, simulated symmetric lobes, and Gaussian distributed noise, respectively.

Figure 15. Enlarged version of the B-C sub-panel from Figure 11. EMU sources are represented by grey dots, and 3CRR sources colour-coded by their FR classification: FRI sources are shown with blue triangles and FRII sources with green crosses.

While the initial asymmetry results (both $A_S$ and $A_R$ ) taken at face value suggest that FRII sources tend to be more asymmetric than FRIs, they need to be interpreted with caution. The way $A_S$ and $A_R$ are defined, using the brightest pixel as a centre of rotation, the tendency is for FRI sources to have fewer pixels, and a more symmetrical structure within the cutout for each lobe. As discussed earlier, a limited number of pixels (i.e. if the bright spots of a lobe are near the edge of a lobe cutout) can produce a low asymmetry value. Conversely, FRII lobes tend to have hotspots, and hence a brightest pixel, positioned such that more of the actual lobe structure is included in the measurement, leading to higher values of asymmetry. This is not to say that these measures are wrong, but it is important to interpret them given how they are defined, reflecting the degree of asymmetry only within the region sampled.

4.1.3. Disorder and elongation

We explore connections between $A_S$ -D and C-D in Figures 16 and 17, respectively. We find that $A_S$ and D are very weakly correlated ( $\rho = 0.16$ and $0.077$ for EMU and 3CRR, respectively), and that there are no sources with both high disorder and low asymmetry. Similarly, we find that C and D are weakly correlated ( $\rho = 0.27$ for both EMU and 3CRR), and there are no sources with both high disorder and low concentration. Highly disordered structures tend to coincide with more asymmetric or diffuse features, making it less likely for a source to simultaneously have low asymmetry or high concentration. From the KDE distributions in Figure 11, while the FRI sources appear to have slightly larger values of D than FRII sources, this distinction is not clear. This could be due to the lower jet power and brightness of the FRI sources than those FRII sources. The lobe structure may therefore be more sensitive to noise fluctuations, increasing the overall disorder of a lobe. The lack of FR separation in C and D suggest that either these metrics may not be sufficiently sensitive to hotspots and jet power, or that other factors, such as variations in lobe brightness distribution and environmental effects, play a dominant role in determining concentration and disorder.

Figure 19. Pairwise comparisons of the source metrics (ABL, $A_{S,p}$ , $A'_{S,p}$ , $A_{R,p}$ , $A'_{R,p}$ ) calculated on radio galaxies from the EMU-PS and 3CRR surveys. The scatter plots show the relationships between these metrics, with normalised KDEs along the diagonal displaying the distribution of each. EMU sources are represented by grey dots, and 3CRR sources colour-coded by their FR classification: FRI sources are shown with blue triangles and FRII sources with green crosses.

Figure 16. Enlarged version of the $A_S$ -D sub-panel from Figure 11. EMU sources are represented by grey dots, and 3CRR sources colour-coded by their FR classification: FRI sources are shown with blue triangles and FRII sources with green crosses.

Figure 17. Enlarged version of the C-D sub-panel from Figure 11. EMU sources are represented by grey dots, and 3CRR sources colour-coded by their FR classification: FRI sources are shown with blue triangles and FRII sources with green crosses.

Figure 18. Enlarged version of the D-E sub-panel from Figure 11. EMU sources are represented by grey dots, and 3CRR sources colour-coded by their FR classification: FRI sources are shown with blue triangles and FRII sources with green crosses.

We find D and E are moderately correlated ( $\rho = 0.57$ and $0.51$ for EMU and 3CRR sources, respectively). The comparison between the D and E metrics in Figure 18 highlights a general logarithmic envelope, with no sources exhibiting both high elongation and low disorder. Highly elongated jets or lobes will be more susceptible to environmental interactions, turbulence, and instabilities, all of which can contribute to increased disorder. There is also no clear separation between FR types in this parameter space. This suggests that disorder and elongation alone are not primary distinguishing factors in the classical FR classification. Instead, these metrics may be more reflective of environmental influences and source dynamics rather than intrinsic differences between FR types.

4.2. Source metrics

As before with the single-lobe calculations, Figure 19 shows the pairwise combinations of the source metrics (ABL, $A_{S,p}$ , $A'_{S,p}$ , $A_{R,p}$ , and $A'_{R,p}$ ), again with the normalised KDE along the diagonal. We can again see that the EMU and 3CRR sources occupy the same areas of parameter space, with differentiation between FRI and FRII sources in each of the asymmetry metrics. FRIIs tend to be more asymmetric than the FRIs. This is likely due to the different structure around the hotspots in each of the radio lobes produced from the higher power jets than in FRI sources. Further, the brightest parts of a FRI source tend to be near the core, where the jets are still highly collimated. This, in conjunction with the limited number of resolution elements for some sources (see Section 4.1.1), results in lower asymmetry for the FRI sources. The features observed when comparing the shape and ratio asymmetries for the source asymmetry calculations are notably similar to those seen in the single-lobe asymmetry results. We expect both lobes of a radio source to have similar shape and structure when produced from jets of the same power and in the same environment. For the small number of cases where the environment is not consistent for both lobes (e.g. if the galaxy is on the edge of a cluster), than we expect to see higher asymmetry between lobes. We do however see some variation when comparing the flux-weighted ( $A_S$ and $A_R$ ) and normalised versions ( $A'_S$ and $A'_R$ ) of the asymmetry metrics. Looking at the KDEs for each, we can see that the flux-weighted calculations show a wider spread of values than the normalised ones. This is likely a direct result of the difference in brightness between the two lobes. This difference could be from jets with differing power or a projection effect with the emission of the closer lobe being brighter. This analysis supports the findings of Rodman et al. (Reference Rodman2019), who studied the relationship between length asymmetries of radio lobe and the environment properties of the host AGN. They found that the FRI type sources tended to have smaller asymmetries than FRII sources, and that there was a stronger anti-correlation between length ratio and ambient galaxy density than with lobe luminosity and environment. However, we acknowledge that this study had a limited sample size (6 FRI, 16 FRII, 1 hybrid), with limited positive results for FRIs. Our findings are also consistent with Yates-Jones, Shabala, & Krause (Reference Yates-Jones, Shabala and Krause2021).

4.2.1. Angle between lobes

Figure 20 shows that the majority of EMU and FRII sources show little to no bending. There is a distinct separation between FR types at an ABL of roughly $160^\circ$ , with FRIs tending to be more bent than FRIIs. Specifically, we find that $53.21\%$ of EMU sources and $76.36\%$ of 3CRR FRII sources have an ABL $\gt150^\circ$ , with only $29.41\%$ of FRI sources with an ABL $\gt150^\circ$ . This is to be expected as FRIs have lower power and are often found in higher density environments (like galaxy clusters) than FRIIs. In addition to the lower jet power, FRIs also tend to have slower jet speed. The jets are then more likely to be slowed down through entrainment (e.g. Laing & Bridle Reference Laing and Bridle2014), therefore making them easier to bend (Banfield et al. Reference Banfield2015) than FRII sources.

Figure 20. Enlarged version of the ABL KDE sub-panel from Figure 11. EMU sources are represented in grey, and 3CRR sources colour-coded by their FR classification: FRI sources are shown in blue and FRII sources in green.

4.3. Data clustering in parameter space

To confirm that the metrics presented in this work are connected with physically meaningful aspects of radio galaxy morphology, we used Gaussian Mixture Modelling (GMM; VanderPlas Reference VanderPlas2016), via the sklearn.mixture GaussianMixture package. GMM models the data as a probabilistic mixture of Gaussian distributions, where each cluster is characterised by a mean and covariance structure rather than a strict threshold in any single metric (see e.g. Bishop Reference Bishop2006). This allows the clustering process to account for both individual contributions and correlations between metrics, meaning that the separation of clusters emerges from their joint distribution rather than from any one dominant parameter. In this way, GMM can identify if there are multiple populations within a dataset. Before applying GMM, we normalised all metrics to a standard scale spanning a range of $-1$ to 1 with a mean of 0, to ensure that each feature contributes equally to the clustering process.

In order to compare against FRI and FRII populations, we specified the GMM to produce two clusters, treating the ABCDE and source metrics as features. We did GMM clustering with the single-lobe metrics (ABCDE) and source metrics separately. This was because the source measurements contain half as many data points as the single-lobe measurements (i.e. one measurement per source rather than one per lobe). Doubling the source metrics to match the number of single-lobe measurements would artificially inflate their influence in the clustering process, making subsequent interpretation more ambiguous.

Apart from a mean vector and a covariance matrix, GMM also returns a mixture coefficient (weight) that reflects the prior probability of a data point belonging to that component. These mixture weights are not associated with individual features but instead represent the relative prevalence of each Gaussian component in the data. To explore how each feature (or metric in this case) contributes to the separation between clusters, we examined the differences between means of each feature across the two clusters. This provides a simple estimate of which features vary most between clusters, offering insight into their relative contribution to the clustering structure. Although this is not a formal measure of feature importance, it allows us to visualise which metrics show the strongest differences between clusters. This is explored for the single-lobe metrics in Figure 21. We can see that the metrics $A_S$ , $A_R$ , B, and D have the highest influence in creating the clusters, with C and E showing little to no contribution. For the source metrics (Figure 22), metrics $A_{S,p}$ , $A'_{S,p}$ and ABL, have the highest influence in creating the clusters.

Figure 21. Comparison of the relative contributions of single-lobe metrics in distinguishing between GMM clusters.

Figure 22. Comparison of the relative contributions of source metrics in distinguishing between GMM clusters.

Figure 23 compares the distributions of the GMM clusters with the FR distributions of the single-lobe metrics from Figure 11. The GMM clusters match closely with the FR distributions (cluster 1 with FRI, cluster 2 with FRII) in our metric space. The cluster and FR distributions are particularly close in the $A_R$ , and D metrics, reflecting the relative importance of these metrics in the clustering process. The clusters made with the source metrics also align closely with FR distributions (Figure 24), particularly in $A_{R,p}$ and $A'_{R,p}$ . This suggests that while the GMM relies more heavily on $A_{S,p}$ and $A'_{S,p}$ to separate the data probabilistically, the actual FR classification differences manifest more strongly in the $A_{R,p}$ and $A'_{R,p}$ distributions. The results highlighted in Figures 23 and 24 confirm that both the ABCDE and source metrics are physically motivated and are not arbitrary shape analysis calculations.

Figure 23. Comparison of the distributions of the GMM clusters with the FR distributions of the single-lobe metrics from Figure 11. Blue and green lines correspond to the distributions of FRI and II sources respectively. Red and purple dashed lines correspond to GMM clusters 1 and 2 respectively. Note how for $A_R$ and D, the distributions are quite different for FRIs and FRIIs, and are well-matched to the Cluster 1 and 2 distributions, respectively.

To assess the performance of the clustering in distinguishing between FR morphologies, we compute completeness and reliability for each class. We define completeness as the fraction of sources of a given FR type that are correctly assigned to the corresponding cluster. Specifically for FRI sources, completeness is defined as the number of FRI sources in the FRI-predominant cluster divided by the total number of FRI sources. Reliability quantifies how pure a given cluster is with respect to an FR type. We define it as the fraction of sources in the FRI-predominant cluster that are actually classified as FRI. The same definitions apply to FRII sources. A high completeness value indicates that most FRI (or FRII) sources are correctly assigned, while a high reliability value suggests that the cluster is largely free from contamination by the other FR type. These values, for both single-lobe and source metrics, are compiled in Table 5. On average, the clustering is more successful at grouping the FRII sources together, likely due to larger number of FRIIs in our sample.

Figure 24. Comparison of the distributions of the GMM clusters with the FR distributions of the source metrics from Figure 11. Blue and green lines correspond to the distributions of FRI and II sources respectively. Red and purple dashed lines correspond to GMM clusters 1 and 2 respectively. Again, note how for $A'_{S,p}$ , $A'_{R,p}$ and ABL, the difference in distributions between FRIs and FRIIs, that are well approximated by the distributions of Clusters 1 and 2, respectively.

To further quantify the similarity between distributions of the FR types with their respective clusters, we perform the Kolmogorov–Smirnov (KS, Kolmogorov Reference Kolmogorov1933; Smirnov Reference Smirnov1948) test. Specifically we use the two-sample KS test, which takes the null hypothesis that two distributions come from the same population. If the p-value of the KS test is small ( $p\lt0.05$ ), we can conclude that the two distributions were sampled from different underlying populations. We used the KS test to compare FRI with cluster 1 and FRII with cluster 2, compiling the resulting p-values in Table 6. We can see that for all metrics $p\gt\gt0.05$ , consistent with the null hypothesis, that both are drawn from the same population. This further reinforces that the sources in cluster 1 are consistent with the properties of FRIs, and those in cluster 2 with FRIIs.

Table 5. Completeness and reliability of GMM clusters mapped to FR classifications, using the single-lobe (ABCDE) and source metrics (ABL, $A_{S,p}$ , $A'_{S,p}$ , $A_{R,p}$ , $A'_{R,p}$ ).

Table 6. Kolmogorov–Smirnov test results comparing FRI sources with Cluster 1 and FRII sources with Cluster 2 for both the single-lobe metrics (ABCDE), and source metrics.

While we do not expect a perfect 1:1 correlation between the GMM clusters and the FR classifications, the observed correspondence between them in this simple comparison strongly reinforces the value of quantifying radio galaxy structure through the ABCDE and source metrics. The fact that these metrics can probabilistically separate sources into distinct clusters that correspond to known FR types (to some degree) demonstrates the utility of these metrics as a flexible tool for classifying and quantifying radio galaxy morphology.

4.4. Computational efficiency

To estimate how computationally efficient our algorithm is, we timed how long it took to compute both the ABCDE and source metrics for all images in the EMU and 3CRR datasets. These computations were performed on an Apple MacBook Pro with an M4 chip and 16 GB of RAM, running in a Python notebook. We repeated this process 10 times for both datasets, then took the average. We obtained an average computation time of $10.56$  s for the 480 EMU images, averaging $0.022$ s per image. For the 72 3CRR images, we found an average computation time of $5.02$  s, averaging $0.070$ s per image. The average number of pixels in the EMU images was $3\,596.88$ , whereas for the higher-resolution 3CRR images, the average number of pixels was $115\,729.85$ . The difference in computation time between EMU and 3CRR is therefore likely dependent on the number of pixels in the image. This demonstrates that the algorithm performs better than linear, as a roughly $30\times$ increase in the number of pixels resulted in only a $\sim\! 3\times$ increase in computation time on average. This efficiency highlights the algorithm’s scalability, making it feasible for large-scale surveys like EMU. Further optimisations, such as parallelising computations or leveraging GPU acceleration, may improve performance even further.