3.1. Image data

The simulated images are described in Paper I, and we summarise the key points here for convenience. The simulated beam size is set to 15 $^{\prime\prime}$ FWHM at a 4 $^{\prime\prime}$ /pixel image-scale, with a $20\,\mu$ Jy beam $^{-1}$ (RMS) noise floor. The CMP and EXT images consist of 9 075 sources at 15 $^{\prime\prime}$ in size, and 9 974 sources varying from 15 $^{\prime\prime}$ to 45 $^{\prime\prime}$ in size (with axis-ratios varying between 0.4 and 1), respectively. The CMP sources have a maximum peak flux density of $1\,$ Jy and minimum of $50\,\mu$ Jy. The EXT is a combination of simulated compact and extended sources with maxima of $1\,$ Jy and $1\,$ mJy, respectively. For our CMP and EXT images, the ratios ( $\sigma_A$ ) of the total component area (i.e., $\pi\Sigma_{i=1}^Na_ib_i$ for N components with semi-major axes, $a_i$ , and semi-minor axes, $b_i$ ) to total image area (i.e., $2^\circ\times2^\circ$ ) are 0.031 and 0.055, respectively, and are thus still above the confusion limit.

Figure 1 shows a $2^\circ\times2^\circ$ cutout from the centre of an EMU Phase 1 Pilot tile that we use for this analysis. This region provides a good mixture of compact, extended, and complex sources including diffuse extended emission. The simulated images do not include diffuse emission.

3.2. Processing speeds

In the course of our analyses, we were able to use Hydra to assess and compare the processing speeds of the different SFs. Table 2 shows order of magnitude CPU times for one PRD calculation step, during the optimisation process (Paper I).

Table 2. SF order of magnitude PRD CPU times (rounded to one significant figure) for CMP, EXT, and EMU images. The processing was done on a 2GHz 16 core (single threaded) Intel Xeon Processor with 60G of RAM running Ubuntu 20.04.3 LTS.

The wide range in CPU times is due to the background and noise estimators being used. For Caesar and Selavy we are using robust statistics (Whiting & Humphreys Reference Whiting and Humphreys2012; Riggi et al. Reference Riggi2016, Paper I), which has time complexity (Knuth Reference Knuth2000) $\mathcal{O}(n\,\log\,n)$ . Aegean also has the same time complexity, however it uses an inter-quartile range (IQR) estimator (Hancock et al. Reference Hancock, Murphy, Gaensler, Hopkins and Curran2012). PyBDSF uses the typical $\mu/\sigma$ estimator (Mohan & Rafferty Reference Mohan and Rafferty2015), which goes as $\mathcal{O}(n)$ . ProFound uses $\sigma$ -clipping (Robotham et al. Reference Robotham, Davies, Driver, Koushan, Taranu, Casura and Liske2018), which is also of that order. The CPU times can be seen to vary, sometimes significantly, between images.

3.3. Typhon statistics

3.3.1. Optimisation run results

Hydra’s Typhon software module (Paper I) was used to optimise Aegean, Caesar, ProFound, PyBDSF, and Selavy’s RMS and island parameters to a 90% PRD cutoff (Table 3). Aegean, PyBDSF, and Selavy RMS box parameters were pre-optimised (Table 4) using bane (Hancock et al. Reference Hancock, Cathryn and Hurley-Walker2018) as part of a process to minimise the mean noise (i.e., $\mu$ -optimisation, Paper I). Here we present the EMU $\mathcal{D/S}$ -image results, with relevant CMP and EXT image results from Paper I incorporated as appropriate for comparison.

Table 3. Typhon run statistics for $2^\circ\times2^\circ$ CMP and EXT and EMU images, with SF, image depth ( $\mbox{D}= \mathcal{D},\,\mathcal{S}$ ), RMS parameter ( $n_{rms}$ [ $\sigma$ ]) island parameter, ( $n_{island}$ [ $\sigma$ ]), source numbers (N), residual RMS ( $\mu$ Jy beam $^{-1}$ ), and residual MADFM (median absolute deviation from the median) ( $\mu$ Jy beam $^{-1}$ ) columns. $^{\rm a}$ The CMP and EXT results are reproduced from Paper I.

^aThe MADFM estimators are normalised by 0.6744888 (Whiting Reference Whiting2012).

Table 4. Hydra $\mu$ -optimised Aegean, PyBDSF, and Selavy box_size and step_size input parameters, $^{\rm a}$ for CMP, EXT, and EMU $\mathcal{D}/\mathcal{S}$ -images. The CMP and EXT results have been incorporated from Paper I.

$\!\!^{\rm a}$ Selavy only accepts box_size.

Table 3 summarises the Typhon run statistics for our image data. As noted in Paper I, the RMS and island parameters are similar for CMP and EXT $\mathcal{D}$ -images for a given SF; however, they are slightly lower for the real $\mathcal{D}$ -image. As for the $\mathcal{S}$ -image, the simulated and real parameters are similar.

Table 4 summarises the EMU $\mathcal{D/S}$ -image optimised RMS box parameters for Aegean, PyBDSF, and Selavy. First, we note that $\mu_{\mathcal{S}}/\mu_{\mathcal{D}}\sim 5$ for all images. This is consistent with the factor of 5 noise increase for the $\mathcal{S}$ -image. Secondly, $\mu^{\small {\rm EMU}}_{\mathcal{D}}\sim35\,\mu$ Jy beam $^{-1}$ falls between $\mu^{\small \rm{CMP}}_{\mathcal{D}}\sim22\,\mu$ Jy beam $^{-1}$ and $\mu^{\small \rm{EXT}}_{\mathcal{D}}\sim68\,\mu$ Jy beam $^{-1}$ (and similarly for $\mathcal{S}$ -images), suggesting the simulated images are well suited for our analysis.

Figure 2 shows stacked measures of $\mathcal{D/S}$ source detections (N) for the $\mathcal{D/S}$ -images, where we have included the simulated data from Paper I. For the $\mathcal{S}$ -images, for a given SF, N is similar between the CMP and EMU sources, but slightly higher for EXT sources. With the exception of Selavy, N is similar between SFs. As for the simulated $\mathcal{D}$ -images, the relative proportion of N between SFs is roughly the same for CMP and EXT sources, except for Selavy. In the EMU image, there is a significant increase in the relative proportion of N by ProFound. This suggests a possible excess of spurious detections, an increase in real detections missed by other SFs, real detections being split into more components than by other SFs, or a combination of these.

Figure 2. SF CMP, EXT, EMU $\mathcal{D/S}$ -image detection stacked plots (from the N columns of Table 3).

In Paper I, we examined the ratios of deep-to-injected ( $\mathcal{D}\;:\;\mathcal{J}$ ) and shallow-to-deep ( $\mathcal{S}\;:\;\mathcal{D}$ ) source detections, or recovery rates. Table 5 shows the $\mathcal{S}\;:\;\mathcal{D}$ recovery rates for our image data. The $\mathcal{S}\;:\;\mathcal{D}$ recovery rates are roughly of the same order between the CMP and EMU images, indicating the real image has a large population of compact sources (c.f. Figure 1). The recovery rate is higher for the EXT image, by a factor around 2–3, likely a reflection of the different flux density distribution of sources in this image. This would seem to suggest that the CMP image more closely models our real sample. This is also evident when the real image (Figure 1) is compared with the simulated images in Paper I.

Table 5. $\mathcal{S}\;:\;\mathcal{D}$ recovery rates.

Table 6. $\mathcal{D/S}$ (D) residual (res.) $|\mbox{RMS}-\mbox{MADFM}|/\mbox{MADFM}$ (%), RMS ( $\mu$ Jy beam $^{-1}$ ), and MADFM ( $\mu$ Jy beam $^{-1}$ ) statistics extracted from Table 3.

Figure 3. Major-axis distributions for EMU sources (showing size distributions from both $\mathcal{D}$ and $\mathcal{S}$ image measurements together). NB: Size estimates between SFs are not necessarily directly comparable as they are estimated using different methods (Paper I).

For simulated and real images, there is some variability in the residual RMS values, with Selavy being consistently high for $\mathcal{D}$ -images and consistently low for $\mathcal{S}$ -images. On the whole, however, the residual MADFMs are consistent between all SFs for a given image. Table 6 summarises these results. The $|\mbox{RMS}-\mbox{MADFM}|/\mbox{MADFM}$ percentage differences are a measure of the robustness of the SF residual models (which is also reflection of the Hydra optimisation schema, Paper I).

Figure 4. CMP and EXT $\mathcal{C_D}$ (a and g), $\mathcal{R_D}$ (b and h), $\mathcal{C_S}$ (c and i), $\mathcal{R_S}$ (d and j), $\mathcal{C_{DS}}$ (e and k), and $\mathcal{R_{DS}}$ (f and l) vs. S/N. These results are reproduced from Paper I, Figures 14, 15, and 16.

3.3.2. Source size distributions

Figure 3 shows the major-axis distribution for the EMU image data. Both the $\mathcal{D}$ and $\mathcal{S}$ source detections are combined, as the $\mathcal{S}$ data provides additional information. Although the underlying sources are in common between $\mathcal{D}$ and $\mathcal{S}$ , the higher noise level means that the SFs are operating on quantitatively different pixel values, and may well derive different size estimates. Also, the numbers of detections for $\mathcal{S}$ are relatively small (Figure 2), so incorporating them here is unlikely to bias the results. Recall that size estimates for different SFs use different methods and are not necessarily directly comparable.

In Figure 3, all but one of the distributions peak at the scale corresponding to the beam size, consistent with point (i.e., compact) source detections. Caesar shows a distribution systematically offset by a factor of about 1.5 toward larger sizes. The results for Aegean, PyBDSF, and Selavy are similar, which is reassuring given that they use similar approaches in fitting elliptical Gaussians to source components (see Paper I). ProFound shows an excess of very small sources. Some of these will be due to noise spikes, and others to faint sources close to the detection threshold (Table 3) where ProFound only has a few pixels available from which to calculate its flux-weighted sizes, leading to underestimates. Caesar’s excess at small sizes is less than that from ProFound, although were the Caesar results to be shifted left by the factor 1.5 noted above, it would be roughly comparable. The origin of small sizes reported by Caesar might be due to deblending issues, as it has a relatively lower source detection number (Figure 2). In general, all SFs with detections below the beam size are expected to be contaminated, to some degree, by noise spikes.

As for EXT images, they have a similar behaviour to the EMU case, except to a lesser degree (Paper I). In the CMP case, there are significantly fewer detections below the beam size, with some overestimates of source sizes in the case of ProFound and PyBDSF. The overestimates are most likely due to nearby noise peaks being incorporated into a source fit, artificially enlarging the size.

Figure 5. EMU-image $\mathcal{C_{DS}}$ vs. S/N (a) and $\mathcal{R_{DS}}$ (b) vs. S/N, with S/N expressed as $\mathcal{D}$ -signal/ $\mathcal{S}$ -noise and $\mathcal{S}$ -signal/ $\mathcal{S}$ -noise, respectively.

3.4. Completeness and reliability

In Paper I, we explored $\mathcal{C_D}$ (and $\mathcal{C_S}$ ) and $\mathcal{R_D}$ (and $\mathcal{R_S}$ ) for CMP and EXT sources; for ease of discussion and comparison here, we have duplicated the relevant results in Figure 4, along with the new measurements from the EMU data in Figure 5. Aegean was found to have the best $\mathcal{C}$ statistic followed PyBDSF, ProFound, Caesar, and Selavy. Selavy, followed by Caesar, tended to miss bright sources, and were less reliable ( $\mathcal{R}$ ) at high S/N. In general, the statistics for all SFs are poorer for the EXT case, but followed the same performance trends as the CMP case. Some of this can been attributed to confusion.

Figure 5 shows the newly measured $\mathcal{C_{DS}}$ and $\mathcal{R_{DS}}$ metrics for the EMU image. The results are similar to the CMP image case (Figure 4). Here, the detections appear fairly complete for all SFs, with high completeness ( $\mathcal{C_{DS}}\gtrsim 0.9$ ) above $\mbox{S/N}\gtrsim 10$ dropping to $\mathcal{C_{DS}}\sim 0.5$ by $\mbox{S/N}\sim 5$ . Reliability is also generally high ( $\mathcal{R_{DS}}\gtrsim 0.9$ ) for most SFs, although Selavy’s performance here is notably poorer ( $0.7\lesssim \mathcal{R_{DS}}\lesssim 0.9$ ) across almost the full range of S/N. All SFs drop in reliability below $\mbox{S/N}\sim 10-20$ .

Figure 6. Flux density ratios ( $S_{out}/S_{in}$ ) for CMP (left), simulated-extended (middle), and real (right) sources for Aegean (a, b, and c), Caesar (c, e, and f), ProFound (g, h, and i), PyBDSF (j, k, and l) and Selavy (m, n, and o) vs. S/N, expressed as $\mathcal{J}$ -signal/ $\mathcal{D}$ -noise and $\mathcal{D}$ -signal/ $\mathcal{S}$ -noise for simulated and real sources, respectively. The $1\sigma$ (solid) and $3\sigma$ (dashed) curves are RMS noise ( $\sigma$ ) deviations from the $\mbox{flux-ratio}=1$ lines (dotted). Also shown are the detection threshold ( $n_{rms}$ ; dot-dashed curves) and nominal $5\sigma$ threshold (dotted curves). These curves are annotated in (c).

Figure 7. False positives vs. S/N for CMP (a), EXT (b), and real (c) sources, with S/N expressed as $\mathcal{J}$ -signal/ $\mathcal{S}$ -noise and $\mathcal{D}$ -signal/ $\mathcal{S}$ -noise for simulated and real sources, respectively.

3.5. Flux density ratios and $\boldsymbol{n}\boldsymbol\sigma$ scatter

Figure 6 shows the flux ratios, $S_{out}/S_{in}$ , for each of the SFs. For the simulated images, $S_{in}$ (expressed as S/N) represents the $\mathcal{J}$ -signal over the $\mathcal{D}$ -noise. For the real images it represents the $\mathcal{D}$ -signal over the $\mathcal{S}$ -noise. $S_{out}$ represents the detected measurement. The horizontal (dotted) lines represent $S_{out}=S_{in}$ , and the solid and dashed curves about these lines are the $1\sigma$ and $3\sigma$ deviations from $S_{in}$ , respectively. The dot-dashed curves are the detection thresholds (i.e., the RMS parameters, $n_{rms}$ , in Table 3, corresponding to 90% PRD), and the dotted curves represent nominal $5\sigma$ thresholds (c.f. Hopkins et al. Reference Hopkins2015). The sharp vertical cut off below $\mbox{S/N} \sim 2$ for the CMP images is a consequence of having no artificial input sources fainter than that level.

The form of the flux ratio diagrams for the EMU image follow that of the simulated images, however its $\mathcal{DS}$ source numbers are lower due to relying on detections in the $\mathcal{S}$ -image. Similar $\mathcal{DS}$ flux ratio statistics can be measured for the simulated cases (not shown). We use $\mathcal{D}$ diagnostics for the simulated images, as they are the most robust, whereas the $\mathcal{S}$ and $\mathcal{DS}$ metrics provide no extra useful information.

For the simulated cases all SFs, with the exception of Selavy, are seen to detect sources down to the specified RMS threshold (dot-dashed line). Selavy was given a $\sim$ $3.2\sigma$ threshold (Table 3), but only appears to recover sources down to a $5\sigma$ level (Figure 6 (m) and (n)). For the real image case, though, Selavy does appear to recover sources down to the nominal S/N threshold.

Figure 7 is a complementary diagnostic to Figure 6, and shows the corresponding false-positive distributions. This emphasises in another way the SF characteristics we have seen in earlier $\mathcal{R}$ distributions (Section 3.4). Specifically, for CMP sources (i.e., comparing Figures 4b and 7a), Aegean picks up false detections predominantly close to the S/N threshold, with ProFound and PyBDSF having false detections peaking between about S/N $\sim 5-10$ , arising from blended sources and overestimated source sizes. The false sources seen by each SF correspond to their deficit in $\mathcal{R}$ (Figures 4 and 5). In particular, Aegean displays higher levels of completeness, but at the expense of reliability at low S/N, with the converse being true for Selavy and Caesar. In this analysis, Aegean shows arguably the best performance, as is evident from its false-positive distribution being limited to low S/N, generally reliable flux densities seen in the limited scattering beyond $S_{out}/S_{in}>3\sigma$ , and good completeness to low S/N. This is not surprising, though, as Aegean is well designed for identifying and fitting point sources, and the simulated data used here do not include anything else.

The false-positive distributions for the EXT case (cf., Figures 4h and 7b) have similar characteristics to the CMP case (Figures 4b and 7a), but are broadened somewhat. As for the real case (Figure 7c), the distribution for Selavy becomes more prominent, as reflected in its $\mathcal{R_{DS}}$ (Figure 5b).

Pleasingly, none of the SFs show any systematic overestimate or underestimate of the flux densities (Figure 6). To quantify the reliability of the flux density measurements, we focus on the scatter in the distribution of $S_{out}/S_{in}$ as a function of S/N. We first define the fraction of sources $r_{n\sigma}(\hat{S}_i)$ above a S/N limit, $\hat{S}_i$ , with $S_{out}/S_{in}$ lying between $1-n\sigma$ and $1+n\sigma$ :

(1) \begin{equation} r_{n\sigma}(\hat{S}_i) = \frac{\displaystyle\left|\left\{\!\frac{S_{out}}{S_{in}}\!\left|1\!\!-\!\!\frac{n}{\hat{S}_{in}}\!\le\!\frac{S_{out}}{S_{in}}\!\le1\!+\!\!\frac{n}{\hat{S}_{in}}\!\ni\!\hat{S}_{in}\!\ge\!\hat{S_i}\right.\!\right\}\right|}{\displaystyle\left|\left\{\!\frac{S_{out}}{S_{in}}\!\left|\hat{S}_{in}\!\ge\!\hat{S_i}\!\right.\right\}\right|}\,, \end{equation}

where S represents flux density, and $\hat{S}$ represents S/N.Footnote ^d Using this, we define the scatter in $S_{out}/S_{in}$ outside an $n\sigma$ range as

(2) \begin{equation} s_{n\sigma}(\hat{S}_i)=1-r_{n\sigma}(\hat{S}_i)\,. \end{equation}

Table 7 shows the compiled statistics for $3\sigma$ scatter.

For the CMP case, Aegean and Selavy tend to show the least scatter. The scatter for PyBDSF and Selavy is slightly higher than that reported by Hopkins et al. (Reference Hopkins2015), who saw scatters above a $5\sigma$ detection threshold of 4.9% for PyBDSF and 3.3% for Selavy, compared to the values of $\sim$ 6% seen here.

On the whole, the scatter is comparable between the CMP and EMU sources, with the highest scatter for EXT sources. Aegean tends to deliver the least scatter in the flux density estimates, with Caesar, PyBDSF, and Selavy showing similar results at the next level, and ProFound always with the highest scatter. The results in the EMU image appear to show the least scatter overall, but recall that these measurements result from comparing the detections in the $\mathcal{S}$ -image against those in the $\mathcal{D}$ -image for each SF against itself. These are not directly comparable with simulated image cases where the scatter compares measured against known input values. The implication here is that flux density estimates inferred from $\mathcal{D}/\mathcal{S}$ comparisons may not reflect the full extent of the true underlying uncertainties.

Table 7. $3\sigma$ scatter ( $s_{3\sigma}(\hat{S}_i)$ , Equation (2)) at $\hat{S}_i=3,\,5,\,10$ , for SF CMP, EXT, and EMU source flux density ratios (Figure 6). Also shown are averaged $3\sigma$ scatters.

3.6. Hydra viewer cutout case studies

Here we present a gallery of Hydra Viewer cutouts, for the purpose of analysing frequently encountered anomalies and artifacts utilising the metrics explored in the previous subsections. They have been sorted into broad categories: edge detections (sources near image edges), blending/deblending, faint sources, component size errors, bright sources, and diffuse emission. With the exception of diffuse emission (seen only in the real image), all such artefacts are found in the SF catalogues in all of the images, to varying degrees.

3.6.2. Blended sources

For the simulated images there are example cases of unresolved compact sources: e.g., Figure 9a. Here match_ids 6200, 6202, and 6203, with injected flux densities of 0.0529, 0.4032, and 0.6928 mJy, respectively, appear as a single compact unresolved source. All SFs detect it, with measured flux densities of 1.1118, 1.1113, 1.2481, 1.1296, and 1.1230 mJy, for Aegean, Caesar, ProFound, PyBDSF, and Selavy, respectively. These values are consistent with the total injected flux $\sim$ $1.1489\,$ mJy. So the total flux is accounted for. Regardless, this causes a degradation in the $\mathcal{C_D}$ ; in particular, the $\mbox{S/N}\sim20$ bin of Figure 4a.

Figure 8. Edge Detection Infographic: Edge detection example cutouts for (a) CMP $\mathcal{D}$ -image (from the $\mbox{S/N}\sim1\,600\pm110$ bin of $\mathcal{C_D}$ , Figure 4a), (b) CMP $\mathcal{S}$ -image (from the $\mbox{S/N}\sim42.6\pm2.9$ bin of $\mathcal{C_S}$ , Figure 4c), and (c) EXT $\mathcal{D}$ -image (from the $\mbox{S/N}\sim0.318\pm0.036$ bin of $\mathcal{C_D}$ , Figure 4g). In examples (a) and (b), Caesar and Selavy failed to detect the injected sources at match_ids 4 and 7 270, respectively. In example (c), there is an injected source at match_id 6607 ( $\mbox{S/N}\sim0.34$ ) which is detected by ProFound ( $\mbox{S/N}\sim4.82$ ) and PyBDSF ( $\mbox{S/N}\sim6.39$ ). The remaining detections, by both SFs, at match_ids 6608 and 6609 are spurious, due to noise fluctuations.

Figure 9. Blended Sources Infographic: CMP (a–b) and real (EMU; c–f) $\mathcal{D}$ -image examples of blended sources. In example (a) (from the $\mbox{S/N}\sim19.6\pm1.3$ bin of $\mathcal{C_D}$ , Figure 7a), only ProFound missed detection of the isolated source at match_id 6201. The remaining injected sources overlap to make up a single unresolved compact object, which is identified as match_id 6203 by all SFs. In example (b) (from the $\mbox{S/N}\sim0.318\pm0.036$ bin of $\mathcal{C_D}$ , Figure 7a), there are two injected sources at match_ids 6665 ( $\mbox{S/N}\sim0.070$ ) and 6666 ( $\mbox{S/N}\sim0.336$ ). Only match_id 6666 is detected by ProFound ( $\mbox{S/N}\sim7.468$ ) and PyBDSF ( $\mbox{S/N}\sim12.446$ ). Examples (c) and (d) show a $\mathcal{D}$ -image and $\mathcal{S}$ -image cutouts, respectively, of a compact object with a diffuse tail. Only Caesar is able to resolve the $\mathcal{D}$ -image (into two components; top), but not the $\mathcal{S}$ -image (bottom) as the diffuse emission is washed out. Examples (e) and (f) are the corresponding a $\mathcal{D}$ -residual and $\mathcal{S}$ -residual image cutouts in the previous example, respectively, for Caesar. All SFs make $\mathcal{D}$ and $\mathcal{S}$ detections at match_id 742. Caesar separately detects the bright peak at match_id 743 in the $\mathcal{D}$ -image, but with no corresponding $\mathcal{S}$ match. This contributes to a reduction in the inferred $\mathcal{C_{DS}}$ for Caesar at $\mbox{S/N}\sim3.4$ in Figure 8a.

Figure 10. Deblending Issues Infographic: CMP PyBDSF $\mathcal{D}$ -residual-image (a) with Selavy $\mathcal{S}$ -residual-image (b) (from clump_id 875), CMP $\mathcal{D}$ -image (c) with ProFound $\mathcal{D}$ -residual-image (d) (from clump_id 4672), and real (EMU) $\mathcal{D}$ -image (e) with Aegean $\mathcal{D}$ -residual-image (f) (from clump_id 875) cutout examples of deblending issues. In example (a), only PyBDSF makes a detection at match_id 1287. In example (b), only Selavy makes a detection at match_id 1287. In example (c–d), ProFound provides a single flux-weighted component that blends these two adjacent sources, and which is best matched to match_id 6669. This leads to the result that match_id 6668 is deemed undetected in the $\mathcal{C_D}$ statistics (Figure 4a). In example (e–f), Aegean overestimates the extent of the vary faint (diffuse) source at match_id 1698 (bottom image, middle): i.e., $(a,b,\theta)\sim(130^{\prime\prime},\,57.6^{\prime\prime},\,45.8^\circ)$ , with major, minor, and position-angle components, respectively. There is no corresponding $\mathcal{S}$ match. This leads to a degradation in the $\mathcal{C_{DS}}$ S/N $\sim$ 25 bin of Figure 5a.

Blending is also encountered with overlapping point and extended sources, with the former having slightly lower S/N. Figure 9b shows an example of a very low S/N detection being influenced by an even fainter adjacent source. Here, only ProFound and PyBDSF, which have the lowest detection thresholds (Table 3), make detections, and only in the $\mathcal{D}$ -image. Both SFs treat the two adjacent injected sources as a single source. This leads to a degradation in $\mathcal{C_D}$ , but not $\mathcal{R_S}$ (Figure 4a–b).

For simple cases of real sources with diffuse emission, such as a compact object with a diffuse tail, Aegean, ProFound, PyBDSF, and Selavy tend to characterise them with a single flux-weighted component position, whereas Caesar tends to resolve them in $\mathcal{D}$ -images, but not $\mathcal{S}$ -images where the diffuse emission is diminished: e.g., Figure 9c–f. Here Caesar decomposes this object into two sources in the $\mathcal{D}$ -image (e), but not in the $\mathcal{S}$ -image (f). The remaining SFs detect this as a single source in both the $\mathcal{D}$ and $\mathcal{S}$ images. The diffuse emission is washed out in the $\mathcal{S}$ -image, leading to a slight ( $\sim3$ %) drop in the overall S/N.

3.6.3. Deblending issues

Deblending issues can occur in systems having a relatively low S/N neighbour to a brighter object, causing a positional fit which is offset from the true centre of the bright object: e.g., Figure 10a. This leads to a degradation in $\mathcal{R_{D}}$ (and $\mathcal{R_{DS}}$ ). In such cases, PyBDSF tends to bias its position estimates towards flux-weighted centers. The other SFs tend to be less biased in such a fashion: e.g., Figure 10b. ProFound tends to extend its component footprint to include faint sources: e.g., Figure 10d.

Figure 11 shows an example of an injected source with a flux density of $\sim$ $291\,\mu$ Jy which is erroneously measured by Caesar as $\sim$ $0.664\,\mu$ Jy. This is likely to be a consequence of a poor deblending of the two adjacent sources, initially identified by Caesar as a single island. The position is accurate, so it is counted as a correct match, but because $\mathcal{R_D}$ (Figure 4b) is calculated using the measured S/N, this source contributes to $\mathcal{R_D}$ at the unreasonably low value of $\mbox{S/N}\sim0.033$ . For this source, Aegean, PyBDSF, and Selavy correctly estimate the flux density as $281.9^{+4.2}_{-8.4}\,\mu$ Jy, and it contributes to $\mathcal{R_D}$ for those SFs at $\mbox{S/N}\sim12$ . It is worth noting here that requiring flux-density matching in the reliability metric calculation would lead (for Caesar) to this source being deemed a false detection, although it has been detected at the correct location. ProFound does not separately detect match_id 6668 (Figure 10d) as it has blended it with match_id 6669. Comparing Figures 11 and 10d we can see the similar islands detected by ProFound and Caesar, although ProFound’s boundary is tighter, which provides a clear indication of why only a single match was identified.

Figure 10e–f shows an example of a compact system with two roughly opposing diffuse tails, along with a faint diffuse neighbour to the south. All SFs detect the core at match_id 1697, in both the $\mathcal{D}$ and $\mathcal{S}$ images: e.g., Aegean in Figure 10f, and PyBDSF in Figure 12. Caesar and Selavy further detect aspects of the diffuse emission, with Caesar fitting it as an extended halo (with its fit slightly biased towards the faint neighbour at match_id 1698) and Selavy as a tail (match_id 1699), but only in the $\mathcal{D}$ -image. As for the faint neighbouring source, only Aegean and PyBDSF detect it (match_id 1698), and only in the $\mathcal{D}$ -image (Figure 10f). Aegean overestimates its extent (leading to an overestimated $\mbox{S/N} \sim 25$ ), likely due to inclusion of emission from the diffuse tails in its fit. As there are no corresponding $\mathcal{S}$ detections for match_ids 1698 and 1699, they contribute to a degradation in $\mathcal{C_{DS}}$ . ProFound, on the other hand, includes both the bright source and the faint southern neighbour together in the $\mathcal{D}$ -image (with its fit slightly biased towards the faint source, S/N $\sim$ 154) but detects only the bright source core in the $\mathcal{S}$ -image (S/N $\sim$ 145). This is a further example of the impact on $\mathcal{C_{DS}}$ of extended or faint diffuse emission, characterised differently by different SFs in the $\mathcal{D}$ -image but absent in the $\mathcal{S}$ -image, resulting in mismatches between the reported sets of sources.

Figure 11. Caesar clump_id 4627 $\mathcal{D}$ $0.916^\prime\times0.916^\prime$ residual-image cutout for simulated-compact sources. Caesar underestimates the flux density for match_id 6668 ( $\sim$ $0.664\,\mu$ Jy, compared to $\sim$ $291\,\mu$ Jy for the injected source), resulting in an artifact in the calculated reliability, placing this source at an artificially low S/N. This information was extracted from the $\mathcal{R_D}$ $\mbox{S/N}\sim0.0331\pm0.0036$ bin (Figure 4b).

3.6.6. Simulated anomalies

Figure 15 shows an example of injected sources spanning $0.28<\mbox{S/N}<80$ in clump_id 50, with details in Table 8, for our EXT $\mathcal{D}$ -image. This is an informative clump, as it encapsulates a moderately rich environment in which to explore detection anomalies. The $\mathcal{D}$ -residual-image cutouts from the four SFs that make detections are shown in Figure 16. All SFs make detections in the $\mathcal{D}$ and $\mathcal{S}$ images, except for Selavy which only finds sources in the $\mathcal{S}$ -image. This may arise from Selavy’s underlying Duchamp-style thresholding (Whiting et al. Reference Whiting2002).

Figure 12. PyBDSF clump_id 1472 $\mathcal{D}$ (top) and $\mathcal{S}$ (bottom) $1.69^\prime\times1.69^\prime$ residual-image cutouts for real (EMU) sources. PyBDSF fits the core of the source shown in Figure 10a, while leaving out the diffuse emission (i.e., it over-subtracts), in both $\mathcal{D}$ and $\mathcal{S}$ images. This contributes to the $\mathcal{C_{DS}}$ S/N $\sim$ 152 bin (Figure 4a).

Figure 13. Noise Spike Detection Infographic: CMP $\mathcal{D}$ -image (a), EMU PyBDSF $\mathcal{D}$ -residual-image (b), and EMU $\mathcal{D}$ -image (/w ProFound residual-image inset) (c) cutout examples of noise spike detection. In example (a) (from the $\mbox{S/N}\sim12.4\pm1.4$ bin of $\mathcal{R_D}$ , Figure 4b), the detections within this clump are anomalous, as there is no injected source. In example (b), Aegean and PyBDSF detect a $\mathcal{D}$ source (match_id 1549) here, although visually this object is consistent with a noise spike. With no corresponding $\mathcal{S}$ detection, such results contribute to the inferred $\mathcal{C_D}$ at low S/N (Figure 5a). Example (c): At first glance, match_id 9535 appears to be part of a faint ring structure, something for which ProFound is uniquely suited. The size of the emission is on the order of common EMU’s beam size ( $18^{\prime\prime}$ ), making it consistent with either a noise spike or a faint compact source.

Figure 14. $3^\prime\times3^\prime$ image cutout of clump_id 2071 of the EXT $\mathcal{D}$ -image. All SFs detected the injected source ( $\mbox{S/N}\sim226$ ) at match_id 4142, except for Caesar. Only Selavy detects the adjacent fainter source, match_id 4141, where $(\mbox{S/N})_{injected}\sim19$ . This information was extracted from the $\mathcal{C_D}$ $\mbox{S/N}\sim220\pm25$ bin of Figure 4g.

Figure 15. $3^\prime\times3^\prime$ cutout of clump_id 50 of the EXT $\mathcal{D}$ -image. The orange ellipses indicate injected sources, summarised in Table 8. The associated SF detections are shown in Figure 16. These sources span $0.28<\mbox{S/N}<80$ in $\mathcal{C_{D}}$ (Figure 4h).

For simulated images, based on the flux ratios in Figure 6, Selavy appears to produce a catalogue with an effective detection threshold of $5\sigma$ , regardless of its RMS parameter setting. Recalling that the EXT image contains extended-component elements with a maximum peak flux density of $1\,$ mJy, as opposed to $1\,$ Jy for its compact-component elements (Section 3.1), this suggests the issue may be related to the extended low surface brightness emission around such objects. There is a roughly 10% degradation in $\mathcal{C_{D}}$ , comparing the results for CMP sources alone (Figure 4a) to the EXT sources (Figure 4g). While this degradation approximates the fraction of injected EXT sources, not all the missed sources are extended. It does seem likely, though, that it is the EXT sources that contribute disproportionately to the shortfall in $\mathcal{C_{D}}$ .

Comparing $\mathcal{C_{S}}$ (Figure 4i) and $\mathcal{C_{D}}$ (Figure 4g) for EXT sources, the distributions are consistent with the addition of the 5 $\sigma$ noise level (shifting the S/N axis by 5 units), for most SFs (e.g., the value of $\mathcal{C_S}$ at $\mbox{S/N}=10$ is similar to the value of $\mathcal{C_D}$ at $\mbox{S/N}=5$ ). Selavy shows a different behaviour in that its performance in the $\mathcal{S}$ -image appears to be better than in the $\mathcal{D}$ -image. At the lowest S/N end this is related to faint low surface brightness objects lying below the detection threshold in the $\mathcal{S}$ -image, and not contributing to the $\mathcal{C_S}$ and $\mathcal{C_D}$ distributions. At moderate and high S/N, though, the reason for the difference is less clear. It may be that faint extended emission around bright objects misleads Selavy in the $\mathcal{D}$ -image, by being pushed below the noise level in the $\mathcal{S}$ -image it makes the bright central region of those sources more easily detectable.

It is clear from the details in Table 8 that the sources not explicitly detected by any SF are at low S/N. Their detection is also confounded by the overlap with the brightest source (match_id 114). Aegean and ProFound have handled this by extending the size of the detection associated with match_id 114, influenced by the additional low S/N emission. This leads to over-estimation by Aegean of the source flux density and mischaracterisation of its orientation (evidenced by the oversubtraction in its residual). ProFound simply associates all the flux with the single object, which leads to a flux density overestimate compared to the input flux density of match_id 114. Caesar searches for blobs nested inside large islands, allowing a more accurate characterisation of such overlapping sources. These residual statistics are demonstrated in Table 3. Here the residual statistic values are significantly lower for ProFound and Caesar than the other SFs.

3.6.7. Oversized components

For CMP sources, PyBDSF sees the largest number of false detections within the $\mathcal{R_D}$ $\mbox{S/N}\sim12$ bin of Figure 4b (a false/true ratio of 86/522). Figure 17a shows an example of one of these, illustrating a major failure mode for PyBDSF. It tends to overestimate source sizes due to inclusion of nearby noise spikes. At low or modest S/N even small noise fluctuations can lead it to expand its island in fitting to the source. Figure 17b shows another example, here with an excessively large size ( $\sim$ $67^{\prime\prime}$ ), associated with the dip in the $\mathcal{R_S}$ $\mbox{S/N}\sim12$ bin of Figure 4d. Figure 17c shows an example where a real injected source is the centre of a 10-component PyBDSF clump, spanning $\sim$ $156^{\prime\prime}$ .

Table 8. Summary of injected EXT-sources in Figure 15. The Detected column indicates if at least one SF has detected an injected source.

Turning to $\mathcal{R_{D}}$ (Figure 4b) and $\mathcal{R_{S}}$ (Figure 4d), at the lowest S/N levels ( $\mbox{S/N}\lesssim 3$ ) Caesar appears to be reporting large numbers of false sources, leading to poor reliability. All of these artifacts are of the same nature as shown in Figure 17d. They spatially coincide with the injected sources, but overestimate the flux density due to fitting an extreme ellipse with artificially large major axis. A catastrophic example of this effect is found in the $\mathcal{C_S}$ $\mbox{S/N}\sim 260$ bin of Figure 4g for EXT sources, shown in Figure 17 (e–f).

3.7. Diffuse emission case study

We now consider an example of a source with a combination of complex structures. The system is introduced in Figure 18, with bright compact features labelled (a) through (e), and diffuse extended emission as ( $\chi$ ), ( $\epsilon$ ), and ( $\lambda$ ). We distinguish the main complex, broadly referred to herein as a bent-tail source, although the details of its nature are likely much more complex, from the bright adjacent component. This main complex is given the label B2293 (not marked in the figure, with the prefix ‘B’ for ‘bent-tail’). The bright adjacent component (labelled C2294, with the prefix ‘C’ for ‘compact’) is identified here as a separate clump. While the emission in C2294 seems highly likely to be related to the emission from B2293, our approach to associating detections into clumps relies on their component footprints touching or overlapping. The compact nature of C2294 leads the SFs to characterise it with ellipses that do not overlap with any ellipse in B2293, and these are consequently marked as independent clumps. The source at match_id 2667, indicated in the figure, does end up being associated with clump B2293, although it is not physically associated with the complex system (B2293+C2294). Details are given in Table 9, where we have chosen to list the MADFM as a representative statistic to characterise residuals (e.g., Riggi et al. Reference Riggi2019), although the other statistics are still computed, and give similar results. This complex of emission provides a good illustration of how the different SFs perform with multiple overlapping, extended and diffuse structures.

C2294 corresponds to match_id 2668 of clump_id 2294. The clump information for this object is given in the lower partition of Table 9. We have not included the cutout images for this system (whose SF components are greyed out in Figure 19e–h), as, in general, we are focusing our attention on the complex source (B2293). The following discussion arises from consideration of the detections and residuals shown in Figures 18 and 19, and Table 9, with explicit reference where clarity is required.

The interpretation of $(\varepsilon)$ and $(\lambda)$ as purely diffuse emission is subjective, based only on their appearance in this radio image, without consideration of possible optical or infrared counterparts. Similarly, the component labelled $(\chi)$ appears subjectively to be a bright extension from (a). The annotations (a) through (e) are likewise subjectively identified as compact components. This choice is adequate for the current discussion given that this is the same information being used by the SFs.

Aegean cleanly identifies C2294 (match_id 2668) and the component at (a) (match_id 2660) in the $\mathcal{D}$ and $\mathcal{S}$ images. While it identifies several additional components (match_ids 2661, 2662, 2663, 2664, 2665, 2666) to characterise the complex of compact and diffuse emission, these poorly model the emission itself as evidenced in the residuals, for both $\mathcal{D}$ and $\mathcal{S}$ images. It is clear from the annotations, too, that the components identified in the $\mathcal{S}$ -image (match_ids 2661, 2663, 2664 only) are markedly different from their counterparts in the $\mathcal{D}$ -image. The component sizes tend to overestimate the extent of the true emission and are often poorly aligned with the structure. Such mismatches, as discussed above, lead to a reduction in $\mathcal{C_{DS}}$ and $\mathcal{R_{DS}}$ . For these complex structures, Aegean generally identifies source positions biased towards flux-weighted centers, with fitted component sizes compensating for the total flux within. This affects the position and size accuracy of the region it is trying to characterise.

Figure 16. $2.51^\prime\times2.51^\prime$ cutout, without (top) and with (bottom) annotations, of clump_id 50 of the EXT $\mathcal{D}$ -image. The corresponding injected sources are shown in Figure 15. Aegean (green), Caesar (cyan), ProFound (red), and PyBDSF (gold) make $\mathcal{D}$ and $\mathcal{S}$ detections at match_ids 110 and 114 only, except for Selavy which only finds them in the $\mathcal{S}$ -image (not shown).

Figure 17. Oversized Components Infographic: CMP PyBDSF $\mathcal{D}$ ((a) and (c)) and $\mathcal{S}$ (b) residual-image, Caesar $\mathcal{D}$ (d) residual-image, and EXT $\mathcal{D}$ -image (e) and Caesar $\mathcal{D}$ -residual-image (f) cutout examples of oversized components. Example (a) (from the $\mbox{S/N}\sim12.4\pm1.4$ bin of $\mathcal{R_D}$ , Figure 4b) is a spurious detection as there is no injected source. In example (b) (from the $\mbox{S/N}\sim12.4\pm1.4$ bin of $\mathcal{R_D}$ , Figure 4b), the detection by PyBDSF at match_id 1225 is spurious, as there is no injected source at this location. In example (c), the large-footprint detection by PyBDSF at match_id 2796 demonstrates one of its most frequent failure modes. Its flux density is 1.84 mJy, as compared to 0.0798 mJy for the injected source. In example (d) (from the $\mbox{S/N}\sim20.7\pm1.4$ bin of $\mathcal{C_{DS}}$ , Figure 4a), Caesar’s Gaussian fit at match_id 5806 extends well beyond its island. Its flux density estimate is 2.18 mJy compared to 0.14 mJy for the injected source. (For clump_ids 5804 and 5805 the injected flux densities are 0.07 mJy and 0.12 mJy, respectively.) In examples (e–f) (from the $\mbox{S/N}\sim259\pm39$ bin of $\mathcal{C_S}$ , Figure 4a), Caesar overestimates the flux density at match_id 2639, with $18\,$ mJy, compared to $0.074\,$ mJy for the injected source. Its respective semi-major and semi-minor axes are $920^{\prime\prime}$ and $11^{\prime\prime}$ , compared to $20^{\prime\prime}$ and $10^{\prime\prime}$ for the injected source.

Caesar robustly identifies C2294 (match_id 2668) and (a) (match_id 2660) in the $\mathcal{D}$ and $\mathcal{S}$ images. In the $\mathcal{D}$ -image, $(\chi)$ is incorporated into its estimate of the source (a). In the $\mathcal{S}$ -image, it resolves the two separately into (a) and $(\chi)$ (match_id 2667). Recall that Caesar is designed to be sensitive to diffuse emission (Riggi et al. Reference Riggi2016), which is evident in the extent of its residual-image footprint (representing the parent-blob, Figure 19). It correctly characterises the extent and overall shape of the emission. The individual components (child-blobs) are determined in its second iteration. Refinement of the parameters for that step may lead to improvements in the way the various components are characterised. This step was not deemed to be practical for the current implementation of Hydra, and it is also unclear whether fine-tuning for such an individual source would have unintended effects for rest of the image. This is an aspect that can be explored in future developments of the tool. Having acknowledged this point, in this implementation Caesar does poorly in terms of characterising most of the components, with the exception of detecting $(\varepsilon)$ in the $\mathcal{D}$ -image (match_id 2662 with S/N $\sim$ $0.2\,$ mJy). It attempts to fit everything by a single component (match_id 2661), both in the $\mathcal{D}$ ( $a\sim166^{\prime\prime}$ , $b\sim78.1^{\prime\prime}$ , and $\theta\sim8.09^\circ$ with $S\sim73.9\,$ mJy) and $\mathcal{S}$ ( $a\sim248^{\prime\prime}$ , $b\sim90.2^{\prime\prime}$ , and $\theta\sim20.2^\circ$ with $S\sim90.3\,$ mJy) images. Here a, b and $\theta$ are the fitted major and minor axes and position angle respectively. These correspond to S/N bins 152 and 181, respectively. The mismatch between these components in the $\mathcal{D}$ and $\mathcal{S}$ images contribute to a degradation in $\mathcal{C_{DS}}$ (Figure 5a).

ProFound does a good job of characterising the features of B2293+C2294 in the $\mathcal{D}$ -image, but less so in the $\mathcal{S}$ -image. As with Caesar, ProFound accurately characterises the extent and overall shape of the emission in general, as seen in the residual images for both $\mathcal{D}$ and $\mathcal{S}$ . It tends to merge brighter peaks with adjacent diffuse emission. The combinations $(a)+(\chi)$ , $(e)+(\varepsilon)+(d)+(\lambda)$ , (b), and (c) are identified in match_ids 2660, 2663, 2664, and 2665, respectively. This is consistent with ProFound’s design, identifying diffuse islands and characterising them through such flux-weighted components. In the $\mathcal{S}$ -image, ProFound does not do as well because much of the diffuse emission is washed out. This becomes clear by comparing the $\mathcal{D}$ and $\mathcal{S}$ residual images in Figure 19 through the reduction in its island size (this is also apparent with Caesar, but to a lesser degree). In the $\mathcal{S}$ -image ProFound breaks B2293 into two main components, everything along the chain from $(a)-(\lambda)$ in match_id 2660, and $(b)+(c)$ in match_id 2664. This again results in a mismatch between the components in the $\mathcal{D}$ and $\mathcal{S}$ images.

PyBDSF does not perform well for extended sources with diffuse emission. It tends to merge things together and attempt to fit complex regions with a single Gaussian. In the $\mathcal{D}$ -image, it merges the B2294+C2294 system into one component (match_id 2660), whereas in the $\mathcal{S}$ -image they are separated into two components, B2263 (match_id 2662) and C2294 (match_id 2668). Clearly it is representing the system largely as a single island in the $\mathcal{D}$ -image, due to the presence of the extended diffuse emission. In the $\mathcal{S}$ -image it is resolved into two, as the diffuse emission linking B2294 and C2294 is masked by the higher noise level. With our settings (Paper I), PyBDSF uses the number of peaks found within the island as its initial guess for the number of Gaussians to fit, iteratively reducing them until a good fit is achieved (Mohan & Rafferty Reference Mohan and Rafferty2015). Here, presumably, it is the diffuse emission that is affecting the fit quality, as can be seen in the residual images. This is also consistent with the failure modes seen in our simulated point source image, (Figure 17a–c). Figure 17c may be a good analogy, where PyBDSF fits a single component to a large region encompassing multiple injected sources presumably influenced also by peaks in the background noise, mimicking the diffuse emission in this case.

Selavy accurately characterises C2294 (match_id 2668) in the $\mathcal{D}$ and $\mathcal{S}$ images, but does poorly with B2294. B2294 is characterised by a single component in the $\mathcal{D}$ -image, at match_id 2660, and by three components in the $\mathcal{S}$ -image, at match_ids 2660, 2661, and 2664. It appropriately characterises $(a)+(\chi)$ in the $\mathcal{D}$ and (a) in the $\mathcal{S}$ , at match_id 2660. This is not unexpected, given the diffuse emission is reduced in the latter. Perhaps surprisingly, though, it misses the remainder of the emission beyond $(\chi)$ in the $\mathcal{D}$ -image. In the $\mathcal{S}$ -image it detects two components here, linking the emission spanning $(\chi)-(\lambda)$ (match_id 2661) and $(b)+(c)$ (match_id 2664). Selavy’s approach also includes a stage of rejecting poorly fit components (Whiting & Humphreys Reference Whiting and Humphreys2012), and that may be the issue with the lack of components reported for the emission in the $\mathcal{D}$ -image. This may have been slightly more successful in the $\mathcal{S}$ -image (due to a reduction in the diffuse emission). This contributes to a reduction in $\mathcal{R_{DS}}$ (Figure 5b), as the $\mathcal{S}$ detection at match_id 2661 has no $\mathcal{D}$ counterpart, seen in the degradation in the S/N $\sim$ 125 bin. This effect occurs elsewhere at similarly high S/N from other sources.

4.1. Detection confidence

Given Hydra uses multiple SFs, it leads to the possibility of exploring cross-comparison metrics. For example, Figure 21 shows detection confidence charts, indicating coincident detections between SFs. The bar on the far left shows the number of sources detected in common between all 5 SFs. The next bar shows the numbers of sources from each SF detected in common by 4 SFs, and so on. This metric uses a ‘majority rules’ process to help determine whether a detection is more likely to be real. If a source is only picked up by a single SF, chances are that detection is spurious (e.g., Figure 13c). The more SFs that agree on a source, the more likely it is to be real. This could be used to constrain metrics, such as completeness, reliability, flux ratios, etc. This could be particularly useful in refining such statistics for real images, where the true underlying source population is not known a priori.

Table 9. Cluster table information for clump_ids 2293 (upper partition) and 2294 (lower partition). The cutouts shown in Figure 19 are for the upper partition. The source component footprint parameters, a, b, and $\theta$ , correspond to the major axis, minor axis, and position angle, respectively. The S/N is calculated using the noise in the $\mathcal{S}$ -image, estimated using Bane. The MADFMs are computed within the source component footprints and normalised with respect to their areas for the residual images. The values for ProFound are its flux-weighted estimates (Paper I).

Figure 19. $4.73^\prime\times4.73^\prime$ $\mathcal{D}$ and $\mathcal{S}$ real-image (a–b), and Aegean (c–d), Caesar (e–f), ProFound (g–h), PyBDSF (i–j), and Caesar (k–l) residual-image cutouts for clump_id 2293 of the B2293+C2294 bent-tail system. Only the Caesar and ProFound cutouts are annotated with match_id information (Table 9) for clarity.

Figure 20. Diffuse Emission Case Study Summary: Summary of bent-tail system (B2293+C2294) diffuse emission case study (see Section 3.7, Figures 18 and 19, and Table 9).

One should be cautious, however, not to take this as ipso facto true. For example, a detection by all SFs could be a bright artifact or noise spike that mimics a source. At the other extreme one could end up excluding real sources only detected by finders well-adapted to recognising them (e.g., Figure 22). The middle ground may indicate the nature of detection, such as compact and/or extended sources with or without diffuse emission, depending on a SF’s strengths (Paper I).

The number of possible cross-source-SF diagnostics that could be developed is substantial. Consequently we have only briefly touched on this subject here, leaving further details for future investigation.

4.2. Processing residual images

The existence of initially undetected sources in the residual images suggests that running a second iteration of the SFs on such residual images may potentially improve on the completeness of the delivered catalogues. This may not be practical for Caesar or ProFound, however, as they compute residuals by subtracting out all of the flux within an island; although, one could create residual images from their island components. Aegean, PyBDSF, and Selavy are perhaps more suitable, as they are Gaussian-fit based. There is, however, a challenge that arises from over-subtraction, where the initial source list includes overestimated sources (e.g., Figure 10a). Here, the residuals reflect a poor initial fit, rather than true sources remaining to be detected in such a second pass.

Figure 21. Real image $\mathcal{D}$ (top) and $\mathcal{S}$ (bottom) detection confidence charts. The stacked bars indicate agreement between SF detections. From left to right, 5 SFs agree, 4 SFs agree, etc.

Figure 22. Example of a diffuse VLASS source, uniquely detected by ProFound in a comparison study with PyBDSF (Boyce Reference Boyce2020). The source was found in a $3^{\prime}\times3^{\prime}$ region centered at J112328+064341. The cutout was extracted from a QL image tile, available at NRAO (https://science.nrao.edu), and then processed using ProFound in R Studio.

The PyBDSF residual $\mathcal{D}$ -image in Figure 16, which emphasises the undetected sources at match_ids 109, 111, 112, and 113, suggests that a second detection pass on such residual images may have merit. How best to implement such an approach remains a challenge, as it may introduce new false detections if residuals such as seen in Figure 16 with Aegean dominate, and likewise for more complex objects, containing diffuse emission.

In order to explore this further, let us consider the residual statistics of our case-study system (i.e., clump_id 2293 in Table 9) and the full EMU image (Table 3), summarised in Table 10. Three potential approaches for reprocessing residual images are as follows:

1. 1. process all $2^\circ\times2^\circ$ residual images,

2. 2. process the $2^\circ\times2^\circ$ residual image with the best MADFM,

3. 3. process an aggregate clump-based residual-image consisting of the best MADFM’s on a clump-by-clump basis.

Approach 1 is ruled out as being computationally impractical. To explore the other options, we consider the MADFMs for CMP and EXT sources in Table 3, and real sources in Table 10.

Given that ProFound and Caesar assign all flux in an island to a source, their residuals are zero by default at the location of sources, leading to their MADFM statistics understandably being the lowest. In order to compare them fairly to the other SFs, residual images created instead from their inferred Gaussian components would be necessary. The analysis of the complex system B2293+C2294 (Figure 18) suggests that ProFound would likely produce residual MADFMs similar to the other SFs, and potentially perform better in the case of complex extended sources with diffuse emission. Caesar would probably not fare as well, given that Hydra is not implementing its full capability for decomposing complex structures. For simplicity in the remainder of this discussion we focus on the other three SFs as potential starting places for image reprocessing.

Considering approach 2, we see that Aegean and PyBDSF have identical MADFMs in the full image. It would be easy to choose one at random, or invoke the other residual image metrics as a further discriminator. Regardless, the next challenge will be understanding and accounting for the artifacts associated with the residuals of the chosen SF. Discriminating in a second SF run between a truly overlooked component and a residual peak left by a poor first-pass subtraction seems on its face to be intractable. One approach may be not to treat each iterative pass as independent and concatenate the output catalogues, but instead to first combine the components together where they overlap in order to better reconstruct the true underlying flux distribution for each complex.

Table 10. Residual image statistics for the full $2^{\circ}\times 2^{\circ}$ EMU pilot sample image and for clump_id 2293 drawn from it (extracted from Tables 3 and 9, respectively). The MADFMs are normalised by cutout area, and are in units of mJy arcmins $^{-2}$ beam $^{-1}$ ). N is the component count.

Approach 3 is perhaps more unconventional, as it draws on the aggregate of results from the different SFs. By using the cutouts with the best MADFM in each case, though, it is likely to contain fewer artifacts than the monolithic approach 2 above. Figure 23 shows a distribution of MADFMs for Aegean, PyBDSF, and Selavy. Here, we only include a MADFM if it is the lowest for each clump. In other words, each clump is represented only once, and the best MADFM for it contributes to the distribution for that SF. At low MADFM values each SF appears to contribute roughly equally to the set of best residual cutouts. The corresponding aggregate residual image would be roughly a homogeneous mixture of results from the three SFs. For increasing MADFM, Selavy contributes fewer of the best residuals, followed by PyBDSF, with Aegean contributing the most at the higher values of MADFM. For our sample complex system, the aggregate would include 8 components within clump_id 2293 (which contains B2293) arising from Aegean (Table 10) and one component within clump_id 2294 (which only contains C2294) from Selavy (Table 9 bottom partition). This approach may have some merit, but clearly also adds a layer of complexity that may prove more challenging than desired.

An alternative approach would be to first identify compact components only and produce a residual image by subtracting those, presuming that all finders perform (reasonably) well on such compact emission. This would leave a residual image consisting only of emission with more complex structure. This would then need to be processed using a SF that has demonstrated good performance with complex structures, established with another run of Hydra. This may provide the opportunity to better characterise extended emission in the vicinity of bright compact emission. If this model of multi-pass source-finding is promising, it could be further implemented iteratively, perhaps with S/N limits imposed on the initial pass(es) to first characterise and remove the bright sources, and progress to the faint population in later iterations. Either of the approaches 2 or 3 could be considered in these scenarios. The results from each pass in such approaches would then have to be incorporated with the existing cluster catalogue and appropriately tagged. While this iterative approach was one of the motivations for developing Hydra, it is clear that a considerable amount of further analysis would be required to demonstrate its value.

Figure 23. Distribution of clumps with the lowest MADFM wrt Aegean, PyBDSF, and Selavy.

5.1. Source finder comparison and performance

Hydra was used to explore the performance of SFs through $\mathcal{C}$ and $\mathcal{R}$ statistics, and their size and flux-density measurements, as well as through targeted case studies (Section 3.6). The most significant differences appeared when encountering real sources with diffuse emission. Other frequently encountered anomalies were with sources at image edges, blending/deblending, component size errors, and poorly characterised bright sources.

In terms of characterising sources, all SFs seem to handle compact objects well. With the exception of Selavy, this generally applies to extended objects as well. In the case of compact objects with diffuse emission, Aegean, ProFound, and PyBDSF tend to characterise them as point sources, whereas Caesar deblends them into cores with diffuse halos, and Selavy as cores with jet-like structures. This is all consistent with how the SFs are designed (Paper I). For more complex sources with diffuse emission, Aegean performs best at characterising their complexity through a combination of elliptical Gaussian components, with PyBDSF and Selavy overestimating component sizes, missing regions of flux, or fitting components misled by adjacent flux peaks. Caesar and ProFound characterise extended islands well, but their corresponding components are not always robust representations of the whole.

In moving from simulated to real images, we find the major distinction comes when considering diffuse sources. ProFound performs the best when it comes to complex sources with diffuse emission, characterising them as islands of flux, but without resolving them further (by design, Robotham et al. Reference Robotham, Davies, Driver, Koushan, Taranu, Casura and Liske2018). Caesar in general performs similarly, although it is not implemented optimally in the current version of Hydra. Caesar can implement different RMS and island parameters for the parent and child segments in order to improve the deblending of complex structures (Riggi et al. Reference Riggi2016, Reference Riggi2019). These are currently defined by Hydra to have the same settings, though, due to its present implementation requiring a single pair of RMS and island parameters for each SF (Paper I). In short, given the SCORPIO survey analysis by Riggi et al. (Reference Riggi2016), Caesar is expected to outperform ProFound in terms of its deblending approach, although it may still not characterise diffuse emission to the same degree (e.g., compare Caesar and ProFound footprints in Figure 19e–f and g–h, respectively). PyBDSF and Selavy have similar performance issues. They both tend to merge together sources embedded in regions of diffuse emission, leading to different characterisations in the $\mathcal{D}$ and $\mathcal{S}$ images depending on the degree of diffuse emission present, with a corresponding impact on $\mathcal{C_{DS}}$ and $\mathcal{R_{DS}}$ . Aegean is similar, but performs slightly better in $\mathcal{C_{DS}}$ and $\mathcal{R_{DS}}$ perhaps due to its approach of using a curvature map to tie its Gaussian fits to peaks of negative curvature, which may reduce the degree to which the fitting differs in the $\mathcal{D}$ and $\mathcal{S}$ images.

Figure 3 shows that all but one of the SFs have source size distributions peaking at the scale corresponding to the beam size, consistent with point source detections. Caesar’s distribution is systematically offset by a factor of about 1.5 toward larger sizes, consistent with its performance seen in the examples so far, where its attempts to capture diffuse emission can lead to overestimates of source sizes. The distribution for ProFound has fewer very extended sources, suggesting the largest sources identified by Aegean, PyBDSF, and Selavy are likely to be poor fits with overestimated sizes. ProFound also shows an excess of very small sources, likely to be noise spikes or faint sources where the true extent of the flux distribution is masked by the noise. This is a consequence of the size metric given by ProFound which is linked to the size of the island containing only those pixels lying above the detection threshold, in contrast to the fitted Gaussian sizes from the other SFs. It is natural that these sizes will be smaller than the beam size for faint objects. Caesar shows similar results to ProFound at the small scales albeit offset to larger sizes by the factor 1.5 already noted. These outcomes are in line with the specific examples described above for the B2293+C2294 complex structure. Comparing Figure 3 with the distributions for EXT sources in Paper I, the distributions show largely the same characteristics, in particular with ProFound fitting fewer extreme extended sources, and Caesar performing similarly at those large sizes to the remaining SFs. The main difference in the real image appears to be the excess of artificially small source sizes fit by ProFound and Caesar, and Caesar’s systematic overestimate of sizes.

5.2. Processing of residual images

Hydra produces an aggregated clump catalogue consisting of one row per clump ID, from the cluster catalogue of the SFs with best residual RMS, MADFM, and $\Sigma I^2$ metrics, normalised by clump cutout area. Hydra also has summary information of residual RMS, MADFM, and $\Sigma I^2$ statistics for the entire $\mathcal{D}$ and $\mathcal{S}$ images, and independent $\mathcal{D}$ and $\mathcal{S}$ catalogues for each SF. There are cases where taking a catalogue generated from the SF that provides the ‘best’ fit on a source-by-source basis (a heterogeneous catalogue) may be advantageous, and Hydra provides this option. This is not possible with individual SFs alone. The more traditional homogeneous catalogue, separated by SF, is also available.

This leads to the concept of creating residual images from either heterogeneous or homogeneous catalogues. In either case, the residual image would be created based on either of the residual RMS, MADFM, and $\Sigma I^2$ metrics, or some other metric. The simplest concept would be to create the residual $\mathcal{D}$ -images by subtracting out only compact sources (i.e., clumps consisting only of single components from any of the SFs). Running Hydra again on the residual may deliver better results for extended emission in the vicinity of bright compact emission. This approach is promising, as it is very similar to the operations of Caesar, which reprocesses its blobs after subtracting their compact components (Riggi et al. Reference Riggi2016). This is also planned for future versions of Hydra.

5.3. Final thoughts

The past two decades have seen an explosion in technologies, providing radio telescope facilities with the ability to perform deep large sky-coverage surveys (Lacy et al. Reference Lacy2020; Norris et al. Reference Norris2011, Reference Norris2021), or transient surveys with modest sky coverage at high cadence (Banyer et al. Reference Banyer, Murphy, Ballester, Egret and Lorente2012; Murphy et al. Reference Murphy2013). This has led to researching and developing SF technologies capable of handling such data volumes and rates (Hopkins et al. Reference Hopkins2015; Riggi et al. Reference Riggi2016, Reference Riggi2019; Robotham et al. Reference Robotham, Davies, Driver, Koushan, Taranu, Casura and Liske2018; Hale et al. Reference Hale, Robotham, Davies, Jarvis, Driver and Heywood2019; Bonaldi et al. Reference Bonaldi2021). These case studies involved the fine tuning of parameters by the SF experts in order to get the best performance. The advent of large scale surveys such as EMU (Norris et al. Reference Norris2021), aiming to detect up to 40 million sources, makes such fine-tuning difficult at best.

We developed Hydra to automate the process of SF optimisation and to provide data products and diagnostics to allow for comparison studies between different SFs. Hydra is designed to be extensible and user friendly. Each SF is containerised in modules with standardised interfaces, allowing for optimisation through RMS and island parameters, which are common to the traditional class of SFs. The parameters are optimised by constraining the false detection rate (e.g., Williams et al. Reference Williams2016; Hale et al. Reference Hale, Robotham, Davies, Jarvis, Driver and Heywood2019). New modules can be added to Hydra by following a standardised set of rules.

Future improvements to Hydra include adding the island catalogues where provided by SFs, improvements to optimisation schemes through using parameters more finely tuned for different SFs, development of completeness and reliability metrics for handling complex sources with diffuse emission, and methods of flux recovery through processing residual images with compact components removed. Catalogue post-processing is also an option. Once the false-detection fraction has been established for a given SF, a user-specified limit to false detections can be translated to an effective S/N threshold and only sources detected above that threshold will be provided to the user. In such a scenario the full set of detected sources would still be retained in Hydra’s tar archives for subsequent exploration as needed, but only those above the requested threshold will be presented in the Hydra viewer tools. Implementation of multi-processor options where SFs support that is also planned. Hydra is being explored for integration into the EMU and VLASS pipelines. The heterogeneous nature of Hydra data, arising from multiple SFs, will add versatility to future radio survey data processing.