Overcoming Confusion Noise with Hyperspectral Imaging from PRIMAger

The PRobe far-Infrared Mission for Astrophysics (PRIMA) concept aims to perform mapping with spectral coverage and sensitivities inaccessible to previous FIR space telescopes. PRIMA’s imaging instrument, PRIMAger, provides unique hyperspectral imaging simultaneously covering 25–235 µ m. We synthesise images representing a deep, 1500 hr deg − 2 PRIMAger survey, with realistic instrumental and confusion noise. We demonstrate that we can construct catalogues of galaxies with a high purity ( > 95 per cent) at a source density of 42k deg − 2 using PRIMAger data alone. Using the XID+ deblending tool we show that we measure fluxes with an accuracy better than 20 per cent to flux levels of 0.16, 0.80, 9.7 and 15 mJy at 47.4, 79.7, 172, 235 µ m respectively. These are a factor of ∼ 2 and ∼ 3 fainter than the classical confusion limits for 72–96 µ m and 126–235 µ m, respectively. At 1 . 5 ⩽ 𝑧 ⩽ 2, we detect and accurately measure fluxes in 8–10 of the 10 channels covering 47–235 µ m for sources with 2 ≲ log ( SFR ) ≲ 2 . 5, a 0.5 dex improvement on what might be expected from the classical confusion limit. Recognising that PRIMager will operate in a context where high quality data will be available at other wavelengths, we investigate the benefits of introducing additional prior information. We show that by introducing even weak prior flux information when employing a higher source density catalogue (more than one source per beam) we can obtain accurate fluxes an order of magnitude below the classical confusion limit for 96–235 µ m.


INTRODUCTION
The process of star formation is integral to understanding galaxy evolution (e.g.Madau & Dickinson 2014).However, a significant fraction of the UV emission from hot massive stars, which trace star formation, is obscured by dust and re-emitted at far-infrared (FIR) wavelengths (Cardelli et al. 1989;Calzetti et al. 2000;Burgarella et al. 2013a).
Previous studies have attempted to derive FIR-related properties of galaxies, and correct for dust attenuation, in order to determine physical properties without directly observing in the FIR, e.g. using the IRX- relation to determine IR luminosity (Meurer et al. 1999) or using energy-balancing SED fitting procedures to determine star-formation rates (SFR) and dust attenuation (Małek et al. ★ E-mail: j.donnellan@sussex.ac.uk 2018).However, there is no clear agreement within the literature that such approaches are universally applicable or accurate, e.g. with deviations from the IRX- relation being found (Narayanan et al. 2018) as well as discrepancies between SFRs and dust attenuation values obtained when fitting SEDs with and without IR photometry (Riccio et al. 2021;Pacifici et al. 2023).
Moreover, the total emission we receive from galaxies in the infrared, the cosmic infrared background (CIB, Puget et al. 1996), forms roughly half of the total extragalactic background light (e.g.Hauser & Dwek 2001;Dole et al. 2006).The discovery of this high CIB, along with wide-field FIR and sub-mm surveys, revealed a population of galaxies which are heavily enshrouded in dust and are known as dusty star-forming galaxies (see Casey et al. 2014, for a review).The most luminous DSFGs are thought to be the most intense stellar nurseries in the Universe, with incredibly high SFRs (Rowan-Robinson et al. 2018) and are therefore of crucial importance when it comes to understanding the cosmic star-formation history of the Universe (Long et al. 2022).Observations at short wavelengths can detect DSFGs, but they may be misidentified as unobscured galaxies at higher redshifts (e.g.Zavala et al. 2023).Statistical studies of populations of these DSFGs show clearly that their FIR luminosity is significant at most epochs and dominates the luminosity density of the universe at some (e.g.Gruppioni et al. 2013a;Burgarella et al. 2013b).
Observations across the FIR wavelength range are therefore required in order to better characterise and constrain the physical properties of galaxies.However, due to the opacity of the atmosphere for much of the FIR wavelength range, these observations must be conducted either from the stratosphere or space.Previous space-based FIR observatories have each significantly enhanced our understanding of the dusty Universe but have also been limited in their capabilities.The first spaced-based telescope to survey the full sky at IR wavelengths was the Infrared Astronomical Satellite (IRAS, Neugebauer et al. (1984)), but was only able to conduct shallow observations, detecting only the most luminous IR galaxies (LIRGS).The Infrared Space Observatory (ISO Kessler et al. 1996) provided spectroscopy at IR wavelengths (see Genzel & Cesarsky 2000, for a review) but was limited to observing the local Universe due to low sensitivity.Imaging and spectroscopy from the Spitzer Space Telescope (Werner et al. 2004) greatly advanced our understanding of obscured star-formation, with the Spitzer/MIPS 24µm emission widely used as a tracer of obscured luminosity and SFR (Reddy et al. 2008;Elbaz et al. 2011;Shivaei et al. 2017).Spitzer, however, was ultimately limited to only five years of cold mission and its spectroscopy was mainly limited to wavelengths below 35µm.The Herschel Space Observatory (Pilbratt et al. 2010) significantly extended the survey parameter space, discovering extreme DSFGs (e.g.Riechers et al. 2013) and was able to further constrain the CIB (Viero et al. 2015;Béthermin et al. 2017;Duivenvoorden et al. 2020) and the evolution of the IR luminosity function (Gruppioni et al. 2010(Gruppioni et al. , 2013b) ) but could only image in a small number of broad bands.Overall, however, these past missions were limited in imaging to a small number of broad-bands which were not able to capture all the features across the IR range.Previous IR spectroscopy suffered from limited sensitivity or wavelength coverage.
Imaging data from these previous observatories were also limited in depth due to what is known as confusion noise (particularly Spitzer/MIPS (Dole et al. 2004) and Herschel/SPIRE (Nguyen et al. 2010)).FIR space-based telescopes suffer from poor angular resolution due to the limited mirror sizes, which leads to the blending of sources when the telescope beam is large compared to their average separation.This gives rise to confusion noise, which increases with the observed wavelength for a given mirror size.
Identifying the need to improve upon our coverage of the FIR sky, NASA released an Announcement of Opportunity (AO) for an Astrophysics Probe Explorer limited to two themes as recommended by the National Academies' 2020 Decadal Review, one of which being a far infrared imaging or spectroscopy mission.In response to this call, The PRobe far-infrared Mission for Astrophysics (PRIMA) concept mission has been developed 1 .PRIMA is a 1.8m space-based telescope which will be cryogenically-cooled to 4.5 K.This FIR observatory has two planned instruments; the Far-Infrared Enhanced Survey Spectrometer (FIRESS) and the PRIMA Imaging Instrument (PRIM-Ager).FIRESS is a spectrometer covering the 24-235 µm wavelength range in 4 grating modules with spectral resolution  = /Δ ∼ 100.
1 https://prima.ipac.caltech.edu/A high resolution mode will allow it to reach  of thousands across full band.The PRIMAger instrument is composed of two bands.The first offers hyperspectral imaging with a  ∼ 10 providing 12 independent flux measurements from 25 to 80 µm while the second provides 4 broad band filters between 96 and 235 µm, all sensitive to polarization.Both instruments will operate with 100 mK cooled kinetic inductance detectors allowing for an incomparable improvement of sensitivity in FIR.As an observatory PRIMA will cover a wide range of science topics such as, but not limited to, origins of planetary atmospheres, evolution of galaxies and build-up of dust and metals through cosmic time (Moullet et al. 2023).
PRIMAger will be able to provide significantly improved sensitivity compared to previous FIR spaced-based imaging instruments, e.g. by over ∼2 orders of magnitude for point sources compared to Herschel/PACS (see Section 2.2 for more details on PRIMAger sensitivity capabilities).However, to realise the full benefit from this sensitivity, it will be essential to reduce the impact of confusion.
Various statistical methods have been developed to overcome the problems presented by confusion when estimating fluxes from FIR maps.One such tool is XID+, developed by Hurley et al. (2017).It is a deblending tool which uses a probabilistic Bayesian framework in which to include prior information on galaxy positions and fluxes and to obtain the full posterior probability distribution for fluxes.Positional priors can come from short-wavelength FIR maps, or from catalogues at other (e.g.near-IR) wavelengths.Hurley et al. (2017) found that XID+ performs better on flux accuracy and flux uncertainty accuracy for simulated SPIRE maps than previous prior-based source extraction tools, such as DESPHOT and LAMBDAR (e.g.XID+ at 10mJy had similar accuracy to LAMBDAR at 70mJy), and has been utilised in performing source extraction for the Herschel Extragalactic Legacy Project (HELP; Shirley et al. 2019Shirley et al. , 2021) ) and is now a tool used in the wider community (e.g.Shim et al. 2023).
This paper will demonstrate that by utilising the flux modelling capabilities of XID+, accurate flux measurements of galaxies can be obtained below the classical confusion limit from simulated PRIM-Ager maps.In Section 2, we outline how the simulated PRIMAger maps are generated and the confusion noise estimated.Blind source detection is performed on the maps in Section 3 to produce prior catalogues with positions to be used to de-blend the confused maps.In Section 4, we explore how prior information affects the flux modelling of XID+.In Section 5, we show how XID+ performs in terms of measured flux accuracy across the whole simulated dataset and compare to the classical confusion limits.We then discuss the implications of these results on which galaxies we are able to determine the physical properties of in Section 6 and make final conclusions in Section 7.

Simulated PRIMAger Maps
To test how well PRIMAger will recover fluxes of sources in the presence of confusion noise, we utilise simulated PRIMAger maps generated by Béthermin et al. (2024, submitted) (hereafter referred to as B24) using the Simulated Infrared Dusty Extragalactic Sky (SIDES) simulation (Béthermin et al. 2017).The SIDES simulation, map generation process, estimation of baseline confusion limits and simple blind detection of sources in the absence of instrumental noise are all described and presented in B24, however we summarise some of the relevant information here.
PRIMAger will be able to conduct hyperspectral imaging with linear variable filters in two bands, PHI1 and PHI2, between 25 and 80 µm with  ∼ 10.For simplicity, we represent each of these bands with 6 continuous channels spanning the wavelength range of the band (PHI1_1 to PHI1_6 for band PHI1 and PHI2_1 to PHI2_6 for band PHI2).PRIMAger will also be able to image with polarimetry via 4 broad band channels (PPI1-PPI4), centered at 96, 126, 172 and 235 µm with  ∼ 4 sensitive to 3 angles of polarisation (see B24 for further discussion of PRIMAger's polarimetry capabilities).Table 1 includes the central wavelengths and estimated beam full-width half-maxima (FWHM) for each of the 12 representative channels for bands PHI1 and PHI2 as well as the 4 polarimetry channels.
PRIMAger will be able to observe simultaneously with all bands, however, due to their relative position on the focal plane, all bands observe different parts of the sky and mapping is needed to cover a region of interest.SIDES is a simulation of the extragalactic sky in the far-infrared and the millimetre domain, starting from a dark-matter halo light cone with galaxy properties generated using a semi-empirical model.It is able to reproduce a large set of observed galaxy properties such as the source number counts at various angular resolutions, the redshift distributions and the large-scale anisotropies of the CIB.For this work, the latest version of the simulation presented in Béthermin et al. (2022) is used.
The output from the SIDES simulation is a lightcone catalogue of 1.4 deg × 1.4 deg, 0 <  < 10, corresponding to a comoving volume of 0.17 Gpc 3 containing 5.6M galaxies.The catalogue also contains the fluxes of each source which are obtained by integrating the spectral energy distribution of the SIDES galaxies over the representative PRIMAger channels.Simulated maps which contain confusion noise but no instrumental noise are generated for each of the 16 channels (hereafter these maps are referred to as 'noiseless maps', with simulated instrumental noise being added to produce 'noisy maps' as described in Section 2.2).The noiseless maps are generated by attributing the flux of the sources to the centre of the pixels at which they are located and then convolving the map with the relevant beam profile.The beam profiles are assumed to be Gaussian with FWHM values given in Table 1.Map pixel sizes are 0.8, 1.3 and 2.3 arcsec for bands PHI1, PHI2 and PPI, respectively.Cutouts of the same region from the simulated PRIMAger maps in 6 of the channels are shown in Figure 1, with the effect of confusion noise clearly demonstrated as you move to longer wavelengths, whereby sources become increasingly blended.The estimation of the confusion noise from the maps containing no instrumental noise is discussed below.

Estimating Classical Confusion Noise
Confusion noise arises due to surface brightness fluctuations in the maps arising from the astronomical sources themselves, convolved with the telescope beam.The lowest flux at which an individual point-like source can be identified above that fluctuating background is called the confusion limit (Condon 1974).In order to estimate the confusion limit of a simulated noiseless PRIMAger map, B24 applied a 5-clipping process.The standard deviation, , of all pixels is computed, 5 positive outliers are masked and the standard deviation of the unmasked pixels is recomputed.This process is iterated until the standard deviation converges, giving the 1 conf confusion noise.The classical confusion limit is then defined as 5 times this confusion noise, values of which are given in Table 1 for each of the PRIMAger channels.

PRIMAger Sensitivities and Simulated Noise
With significantly improved sensitivity compared to previous FIR space-based telescopes, PRIMAger will reach a point source sensitivity of 220 and 300 µJy in bands PHI1 and PHI2, respectively, at the 5 level with integration time 10 hr deg −2 .Likewise, it will reach point source sensitivities of 200, 300, 400 and 500 µJy in bands PPI1, PPI2, PPI3 and PPI4, respectively.For comparison, the deepest surveys with Herschel at 110, 160 and 250 µm (close to the PPI2, PPI3, and PPI4 bands) reached 5 sensitivities of 1100, 2100, 3800 µJy in (see table 5 in Oliver et al. 2012) In order to make realistic PRIMAger maps, we add instrumental 2 noise to the simulated maps.We assume a deep, 1500 hr deg −2 survey which is expected to give 5 point source sensitivities of 88, 108, 29, 45, 67 and 82 µJy in bands PHI1, PHI2, PPI1, PPI2, PPI3 and PPI4 respectively.
We add Gaussian noise to each pixel based on the nominal point source sensitivities (5 inst ) for the considered survey design for each PRIMAger channel in Table 1.We assume no spatially correlated noise and that the instrumental noise is constant across each respective map.Maps which contain both confusion noise and this added instrumental noise are referred to as 'noisy maps' for the remainder of the paper.

SOURCE DETECTION
XID+ deblends maps using the positions of known sources (Section 4.1) and therefore requires a catalogue containing their prior positions.Usually, such a prior catalogue is obtained from shorter wavelength ancillary data from other telescopes, as was done by Pearson et al. (2018) who used optical data in the COSMOS field to deblend Herschel/SPIRE maps with XID+.However, to demonstrate that it is possible to detect sources and accurately measure their fluxes entirely from PRIMAger data in a self-contained way, a source detection process is run on the simulated PRIMAger maps themselves.This is possible due to the wide spectral coverage of PRIMAger, particularly in the PHI1 band (25-43 µm).This band is not limited by confusion and allows for sources to be detected at multiple wavelengths.Additionally, this band will capture PAH emission lines from star-forming sources around cosmic noon, enhancing their detection probability.
In order to explore the impact of different prior knowledge on the flux accuracy of XID+ (Section 4.2), we consider two different source detection methods.

Blind Detection Without Instrumental Noise
A blind source detection algorithm performed on all of the noiseless PRIMAger maps by B24, who produced a catalogue of 101,540 sources from the 1.96 deg 2 maps.The basic algorithm they employ searches for local maxima within a 5 × 5 pixel region.The threshold was set to be 5 times the measured confusion noise.

Blind Detection On Wiener Filtered Maps With Instrumental Noise
In far-IR and sub-millimeter blind surveys, it is common to perform blind detection by cross-correlating the signal with the PSF of the instrument.This method is expected to maximize the signal-to-noise ratio (S/N) on isolated point sources with white noise.This is appropriate for shallow surveys, dominated by instrumental noise.However, for the deep observations planned for PRIMA on extra-galactic deep fields, the spatially correlated confusion noise is non-negligible even in the shortest PHI1 bands and starts to dominate the noise in the data at PHI2 and longer wavelengths.PSF matching filter, in this case, no longer maximises the S/N of point sources, rather it increases the confusion noise and reduces the completeness of blind detection.
To maximise the S/N of blind source detection in data with substantial confusion noise, previous blind far-IR and millimetre surveys have introduced a Wiener filter as the general form of matching filter kernel that optimise point source blind detection on confusionlimited data (Chapin et al. 2011;Geach et al. 2017;Shirley et al. 2021).The philosophy of this method is a compromise between the uncorrelated white instrumental noise (which benefits from a wider kernel) and other spatially correlated confusion noise (which benefits from a narrower kernel and local background removal) to maximise the signal-to-noise ratio of point source blind detection.Our construction of Wiener filter follows the principles in Chapin et al. (2011) and the similar frameworks of constructing blind catalog in HELP project (Shirley et al. 2021).We refer the readers to those references for details but summarise the outcomes of the match-filtering as follows.
We consider the total noise in the simulated data from PRIMAger observations as two main components: a white noise component coming from the instrumental noise and a confusion noise component coming from other point sources in the map.In each band, we take the instrumental and confusion noise level expected for 1500hour PRIMAger deep survey over 1 deg 2 , create the Wiener filters following Chapin et al. (2011) and derive the corresponding matchfiltered map.A comparison between the effective PSF profile after applying Wiener filter and instrument PSF filter to the simulated PRIMA observation is illustrated in Fig. 2. The effective PSFs after Wiener filtering have primary peaks narrower than the instrument PSF, this reduces source blending and improve the completeness of blind source detection in confusion-dominated PHI2 and PPI1-PPI4 bands.The higher order ringings feature from Wiener filtering introduces additional fake sources around in blind detection, which we identify and remove later.However, the impact of ringing are limited to regions around very bright sources and the corresponding fake sources could be removed based our knowledge on their relative intensity compared to the nearest bright sources.
The blind source detection in the match-filtered maps is made using the find_peak method provided by photutils.A source is identified if the central pixel is the maximum among all pixels in a 5 × 5 pixel region.The maps are calibrated following Chapin et al. (2011) in mJy/Beam such that the point source flux could be estimated directly from the peak.
To remove false sources created by Wiener filtering we examined all sources in the simulation that are bright enough to produce ringing features above the total noise level of the map,  total , ( 2 total =  2 inst +  2 conf , where  inst and  conf are the instrumental noise and the confusion noise, respectively).We then predict the expected intensity of the corresponding ringing features.Sources with fluxes less than five times the expected ringing feature intensity Table 1.Properties and detection limits in the 12 representative PRIMAger channels for the two hyperspectral bands, PHI1 and PHI2, and the four polarimetry broad band channels, PPI1-PPI4.Beam FWHMs are estimated (column 3) for the baseline telescope aperture (1.8 m) and detector and pixel layout.The point source sensitivities are given (column 4) for a deep survey observed for ∼1500 hr deg −2 in the absence of confusion.The classical confusion limit as estimated by B24 is also quoted (column 5) for each channel.This is defined as 5 times the confusion noise which is obtained by estimating the variance in each of the maps via an iterative clipping process in the absence of instrumental noise.The depth of each Wiener-filtered catalogue of monochromatic, blind detections at the 95 per cent purity level are presented (column 6) with details discussed in Section 3.2.The depth reached by the two runs of XID+ with two different prior catalogues are also provided (columns 7 and 8) and are discussed in Section 5.The Wiener-filtered prior catalogue is self-consistently derived from Wiener-filtered catalogues extracted from the synthetic data, the Deep prior catalogue comes from the input model and represents a prior catalogue from other observatories with weak flux priors.XID+ depths are the limiting fluxes defined in equation 3. N.B. the flux accuracy tolerance in the purity analysis of the Wiener-filtered catalogue is different from than used in the definition of XID+ limiting flux.Thus, their values are not directly comparable.Data are quoted to 3 significant figures.

Channel
Central  The näive filter is optimal for flux estimation of isolated point sources, including non-local data with reduced weight to enhance the signal to instrumental noise ratio in comparison to a central, purely local, estimate.The Wiener filter has a much narrow central beam which provides some reduction in instrumental noise but balanced against adding in confusion noise from non-local data.The Wiener filter also has a negative feature (at around 7 ′′ in this example) which provides a local background subtraction, actively reducing confusion noise.
are considered as contaminated and removed from the blind detection catalog.We note that although this conservative cut could also remove some real faint sources this will be reflected in our completeness and flux accuracy estimates and further optimisation could improve our results.
Before constructing the prior list, it is critical to define a cut on the depth for blindly detected source catalogs to avoid significant contamination from false detections, while maintaining high completeness.Significant contamination could be a problem not because of the false objects themselves but also, through the XID+ modelling, reduce the flux accuracy for real sources.
Far-IR and submillimeter surveys usually set a flux cut based on the purity derived from statistical analysis.Purity is defined as the fraction of detected source above certain flux limit that have corresponding counterparts in the simulated input catalogue close enough in positions and fluxes.In our analysis, we consider that a correct counterpart to a blindly detected source satisfies the following criteria (similar to B24): (i) the positional offset between blindly detected source and the counterpart,  off , satisfies  off ⩽  FWHM /2, where  FWHM is the FWHM of the instrument PSF (ii) the observed flux (S obs ) of the blindly detected source and the true flux (S True ) of the counterpart satisfies  True /2 ⩽  obs ⩽ 2 True (iii) the counterpart is the brightest source that satisfies criteria (i) and (ii).
We crossmatch the blind detected source catalog with the simulation input catalogues using those criteria.For each band, we perform crossmatching on blindly detected sources with S/N>2.5.The purity of blindly detected sources above different flux limits are then de-rived accordingly.We choose a cut in observed flux corresponding to the 95 per cent purity and the flux threshold for each band are listed in Table .1.The resulting blindly detected single-band catalog reaches completeness of ∼ 83 per cent at PHI1 bands, ∼ 67 per cent at PHI2 bands and ∼ 75 per cent at PPI1-PPI4 bands on sources with S True >  total .These catalogs are further cross-matched from the shortest to the longest wavelength to obtain a unique list of priors from Wiener filtering.

XID+: A Probabilistic De-Blender
XID+ 3 , developed by Hurley et al. (2017), is a prior-based source photometry tool which is able to simultaniously estimate the fluxes of a collection of sources with known positions.The basic model of XID+ assumes that the input data () are maps with  1 ×  2 =  pixels, where the maps are formed from  known sources, with flux densities   and a background term accounting for unknown sources.The point response function (PRF, ) quantifies the contribution each source makes to each pixel in the map and is assumed to be a Gaussian.The map can therefore be described as follows: where the two independent noise terms represent the instrumental noise and the residual confusion noise which is modelled as Gaussian fluctuations about , a global background.XID+ undertakes an MCMC sampling from this probabilistic model to obtain the full posterior.Originally, Hurley et al. (2017) utilised the Bayesian inference tool, Stan, to perform the MCMC sampling.However, here we implement the Numpyro backend which is built into XID+ as it is faster.
The original XID+ applied a flat, uniform prior on the source fluxes (from zero flux to the highest pixel value in the map).However, later works (Pearson et al. 2017(Pearson et al. , 2018;;Wang et al. 2021) demonstrated that by applying more informative flux priors, e.g. from SED-fitting of ancillary photometry, provided improvements in flux accuracy and allowed fainter fluxes to be reliably measured.
We would expect the choice of prior information provided to affect the modelling accuracy.In the basic XID+ model described above, the possible prior information to include are (a) the positions of previously detected sources (i.e. the density of sources) and (b) the prior probability distributions of their fluxes.The following section investigates the impact of varying these two prior information dimensions on the flux modelling accuracy of XID+.

Impact of Prior Knowledge
In order to investigate the impact of the inclusion of prior knowledge on the modelling accuracy of XID+, we consider (a) varying the density of sources included in the prior source position catalogue as well as (b) varying the prior flux distribution.One would expect the flux modelling accuracy to improve as the density of the prior source position catalogue increases, as the more faint sources are included in the modelling, the fewer sources remain to contribute confusion.However, without any prior flux knowledge, there would be an upper limit, and even a reversal, to the gain in modelling 3 https://github.com/H-E-L-P/XID_plusA summary of these catalogues is presented in Table 2.Note that when considering application to real data (i) could, in principle, be generated from PRIMager map data if sufficiently deep that instrumental noise was negligible (ii) could be generated from the PRIMager survey data we are considering here (iii) would require catalogues generated from data from other telescopes.
The redshift distributions of the sources in each of the above catalogues is shown in Figure 3.A secondary peak of sources in the deep catalogue is present at  ≳ 6 due to the 3.3µm PAH emission line moving into the PHI1_1 channel which is used for the selection of sources for this particular catalogue.Figure 4 shows where the sources from each catalogue lie in the SFR-stellar mass plane for a single redshift bin (1.5 <  ⩽ 2.0) compared to values for the starforming main-sequence (MS) from the literature (Speagle et al. 2014;Pearson et al. 2018;Leslie et al. 2020;Leja et al. 2022).The Deep catalogue contains significantly more low-mass galaxies as well as a larger population of galaxies just below the MS, moving towards the quiescent region, across all masses.Conversely, the Wiener-filtered and B24 catalogues have a higher percentage of their total sources above the MS.
For each of the catalogues, XID+ is run on a sample of the data covering ∼ 0.12 deg 2 with uninformative, flat flux priors (i.e. with uniform flux priors on all sources ranging from zero to the highest pixel value in the respective map.) for the PPI1 channel.This channel is chosen as it is confusion-dominated (i.e. the instrumental noise is negligible compared to the confusion noise) but remains key for many of the PRIMA science goals.Additionally, XID+ was run with Gaussian flux priors centred on the sources' true flux, with standard deviations of 2.0, 1.0, 0.5, 0.3 times the true flux of the sources (i.e. with increasing prior flux knowledge).In a real survey, these flux prior constraints would likely come from predicted fluxes obtained from SED-fitting procedures utilising ancillary photometry (Section 6.3 discusses this further).The method used for measuring the performance of the flux modelling from XID+ for the above runs as well as the subsequent results are described in the following section.

Limiting Flux Statistic
In order to quantify the flux accuracy of XID+ for the varying prior knowledge parameters, we define the following statistics to describe the 'limiting flux' reached in each of the PRIMAger maps.Firstly, we quantify the deviation of the extracted fluxes,  obs from the true fluxes,  true , within bins of true flux using the median absolute deviation (scaled to a Gaussian),  MAD : where Δ =  obs −  true .We then define the limiting flux,  limiting , as the flux at which  MAD equals 0.2: This corresponds to the true flux at which the median deviation of the observed fluxes from the true values equals 20% of the true flux.The choice of this statistic and whether it is a reasonable measure of the flux down to which source fluxes can be accurately recovered is considered in Appendix A.
The prior flux knowledge is quantified as the true flux,  true , over the dispersion in the Gaussian flux prior,  prior .It is worth noting that the  true / prior ∼ 3.3 flux prior is only considered in order to investigate the upper bound of this parameter space, as if we were able to constrain the flux a priori this accurately then new data would add little! Figure 5 shows the limiting flux in the PPI1 channel as a function of the prior flux knowledge for all three prior catalogues.It highlights how increasing the prior flux knowledge for the shallower prior source catalogues (the B24 and the Wiener-filtered catalogues) provides negligible gains.Therefore, it is better to only apply a flat flux prior for prior source catalogues of source densities < 1. Increasing the prior flux knowledge at these source densities returns little gain but will likely introduce more assumptions into the modelling, depending on how the prior flux information is obtained.For deeper, higher source density prior catalogues, however, even weak information from flux priors can lead to substantial gains in the XID+ flux modelling accuracy and limiting flux.

Choice of Prior Source Catalogue
To investigate the flux modelling performance of XID+ across the full simulated PRIMAger dataset, we will continue with both the Wiener-filtered and the Deep prior source catalogues.The former will be used as the benchmark as it is generated from the more realistic maps which include instrumental noise, providing a robust and conservative estimate of a realistic blind source detection process.No prior flux information will be used with this catalogue as to avoid introducing assumptions for little gain.This run will provide the most conservative limiting flux results.
The Deep prior source catalogue is not generated from PRIM-Ager's capabilities or from the maps themselves.However, a catalogue of such source density and depth is possible to obtain from wide-field surveys conducted by higher resolution observatories, such as The Nancy Grace Roman Space Telescope.It is important to understand how much can be gained from utilising such rich ancillary datasets.Additionally for this run, we include prior flux information for each source as a Gaussian distribution centered on the true flux,  true , with a spread of  prior =  true .This is to test the flux modelling performance in the more informative prior knowledge regime.
The blind detection catalogue produced by B24 represents what is possible to achieve in the limit of no instrumental noise (i.e. for very deep surveys).However, due to it being produced from the noiseless simulated PRIMAger maps, rather than the more realistic maps with added simulated noise which XID+ will be run on, it will not be considered further.

XID+ Photometry with Wiener-filtered Prior Catalogue
Proceeding with the Wiener-channeled prior catalogue, we ran XID+ was run on each of the 16 noisy PRIMAger maps independently with flat flux priors.Note that the maps used as data input to XID+ are not filtered in any way, they are simply the simulated maps containing both confusion and instrumental noise (as described in Section 2.2).The output from XID+ is the full posterior distribution for the flux in the channel corresponding to the map of each source in the prior catalogue, including the correlation between sources.The measured flux of a given source for a particular channel is quoted as the median of its marginalised posterior flux distribution.
Figure 6 shows the scaled MAD,  MAD (defined by equation( 2)), The limiting fluxes reached by XID+ in each of the 16 noisy PRIM-Ager maps are shown in Figure 7 by the dashed blue line.These are compared to the classical confusion limits for each map as calculated by B24 (solid orange line, also given in Table 1).For all six PHI1 maps, which are limited by the instrumental noise rather than the confusion noise, XID+ is able to accurately measure source fluxes down to within a factor of 1.35 of the 5 instrumental noise.For the remaining, redder maps which are confusion-dominated (bands PHI2 and PPI), XID+ reaches a limiting flux below the classical confusion limit in each channel.As the bottom panel of Figure 7 shows, the gain in depth relative to the classical confusion limit steadily increases through the representative channels of band PHI2.Starting at the PHI2_1 channel, accurate fluxes are recovered down to the confusion limit of this channel.By PHI2_6, fluxes which are a factor of ∼ 2 below the respective confusion limit are accurately recovered.For the 4 PPI channels (96-235µm), this is improved to a factor of ∼ 3.
These results are also compared to two galaxy SED models from Kirkpatrick et al. (2015).One is a star-forming galaxy template at  = 2 and a luminosity of  = 10 12.3  ⊙ with no AGN emission contributing to the total IR luminosity (f AGN = 0), shown by the dotted grey line in Figure 7.The other SED template, however, has f AGN = 0.44 and is shown by the solid grey line.Distinguishing between these two types of objects is important to the extragalactic science case for PRIMA.Being able to do so enables the study of De-blending with positional and weak flux priors consistently attains 5 depths more than an order of magnitude fainter than the classical confusion limit at  > 100 m.This figure shows that using XID+, SEDs from typical galaxies at  = 2 can be measured to  = 126 m using only positional priors (derived from the Wiener-filtered map), and out to the longest PRIMAger PPI channel ( = 235 m) with the addition of a weak intensity prior.Top: Limiting flux density as a function of wavelength covering the 12 representative channels of the two LVF PRIMAger bands, PHI1 (25-43µm) and PHI2 (47-80µm), and the 4 PPI channels (96-235µm).Blue dashed line shows the limiting flux density, as defined in Sec.4.2.1, reached by XID+ with flat flux priors and the Wiener-filtered detection prior catalogue.The dash-dotted pink line shows the results from XID+ with the Deep prior catalogue and flux priors with  s,prior =  true .The orange solid line shows the classical confusion limits from B24 and the red triangles show the 5 baseline point source sensitives in each of the channels.Also plotted are two model SEDs of galaxies from Kirkpatrick et al. (2015) at z=2 with luminosity L = 10 12.3  ⊙ , corresponding to the knee of the FIR luminosity function at this redshift (Magnelli et al. 2014), and fraction of luminosity from AGN emission, f AGN , of 0 and 0.4, shown by the dotted grey line and the solid gray line respectively.Bottom: Limiting flux density reached by XID+ relative to the 5 confusion limits for the 10 reddest channels which are confusion-dominated.
the impact that AGN have on galaxy evolution.The limiting flux results from XID+ show that accurate fluxes can be obtained for both objects up to ⩽ 100µm, spanning a range where there is significant distinction between these two SEDs.

XID+ Photometry with Deep Prior Catalogue
Figure 7 also shows the results from the XID+ run with the Deep prior source catalogue with flux prior knowledge of  prior / true = 1 (pink dash-dotted line).Utilising these more informative priors allows for significantly deeper limiting fluxes to be reached, particularly for the PPI1-PPI3 channels where the limiting flux is more than an order of magnitude below the classical confusion limits.Additionally, for the PHI1 band and blue PHI2 channels, the limiting flux is pushed down to the instrumental noise of the simulated survey.
Comparing again against the two model SED templates with differing f AGN , these deeper limiting fluxes allow for these two objects to be accurately observed in the two reddest PRIMAger channels (PPI3 and PPI4).

Alternative methods
In this paper we have focused on quantifying how much fainter than the näive, classical confusion we can probe with PRIMAger using modern, but relatively well-established techniques.However, it is important to note that the hyperspectral capabilities of PRIMAger will lend themselves well to more sophisticated techniques which are likely to do better.The rich spectral information available in PRIMAger (including the continuous linear variable filters in PH1 and PH2) can augment the spatial information.In this paper we have concentrated on using the high resolution at short wavelengths to provide positional priors at longer wavelengths.However, this does not exploit the fact that different types of galaxies and galaxies at different redhshifts have different spectral signatures.Simultaneously modelling the spatial and spectral information, which is possible in the XID+ framework, would improve these results.Even a relatively simple stepwise approach of stepping through the channels one-byone and using the short wavelengths to inform the flux priors of the longer wavelengths would yield benefits (e.g.Wang et al. 2024, sub.and see Section 6.3) .Furthermore, sophisticated tools are being developed rapidly in the context of AI and machine learning and we note in Appendix B that impressive deconvolution results are not limited to this prior-based deblending technique, but are a general property of the hyperspectral imaging dataset.
PRIMager will also be working alongside spectral imaging capabilities from the FIRESS instrument.Tools like CIGALE (Boquien et al. 2019) can model simultaneously PRIMAger photometry and FIRESS spectral data allowing us to consistently model dust and gas.
It is also worth noting that our investigation has been restricted to deep surveys, future work is needed to assess wide surveys.As shown in B24 the wide fields surveys will also be affected by confusion, albeit to a less extent and being confusion limited at longer wavelengths than the deep surveys.For the wide surveys the reduced sensitivity at shorter wavelengths will have an impact on the prior catalogues that can be self-consistently constructed from PRIMA data, and hence the deblending performance.This can also be addressed by using multi-band techniques in the detection process, e.g.generalising the Wiener filtering to multi-bands.

Properties of Galaxies Accessible to PRIMAger
Having quantified how accurately we can measure fluxes and hence the flux limits at which we can accurately determine fluxes (i.e. to "detect" galaxies) it is important to consider the implications for studies of galaxy properties.It is thus instructive to consider the detectability of galaxies in physical parameter space.

Redshift, SFR Plane
We firstly consider the detectability of the significant star formation (as traced by the FIR luminosity density) as a function of SFR and redshift.The underlying FIR luminosity density of the SIDES simulation, , as a function of SFR and redshift is indicated by the grey-scale contours in Figure 8.For each channel we translate from a limiting flux from XID+ to a limiting SFR as follows.We select all sources from the SIDES simulation whose true fluxes are within 10 percent of the limiting flux.The limiting SFR is defined as the median SFR of these sources.
These are shown in Figure 8 for the limiting fluxes from the two XID+ runs with the Wiener-filtered and Deep prior catalogues by the blue dashed and pink dash-dotted lines, respectively, for two of the channels.The region of the z-SFR plane above these lines are where sources have fluxes in the given channel which can be accurately measured for the given method.
The limiting boundary due to the confusion limit is also estimated in the same way (solid orange lines).As can be seen in the right panel of Figure 8, the confusion limit in the PPI2 channel prohibits sources which form the peak of the luminosity density from being reliably recovered.Utilising XID+ allows for this peak to begin to be probed even with the low source density prior catalogue and no flux prior information.With the more extensive prior catalogue with additional prior flux information, the full peak of the luminosity density can be explored.

Stellar mass, SFR Plane
We can consider that a source is recovered if it has a true flux in at least one channel for the confusion-dominated maps (PHI2_1-PPI4; 47-235µm) which is above the corresponding limiting flux in that channel from XID+.Revisiting the stellar mass-SFR plane for a single redshift bin of 1.5 <  ⩽ 2.0 for the sources in the two prior catalogues used for the two XID+ runs (originally shown by the left and right panels in Figure 4), we can identify which of these sources are recovered.The right-hand panels in Figure 9 show the recovered sources meeting the above criteria from the two XID+ runs relative to the star-forming main-sequence (MS) from the literature.The left-hand panels show the sources which are recovered above the classical confusion limits determined by B24.For each M * -SFR bin, the average number of channels in which the sources within that bin are recovered is also calculated and shown by the colour-scale.Additionally, an average SFR value for all sources which are detected in 2, 8 and 10 of the PHI2_1-PPI4 channels are also shown.The latter two ensure that at least two detections are made in the 96-235µm channels and therefore robustly recovering a given galaxy in the FIR regime.
For the Wiener-filtered prior catalogue (top panels), XID+ recovers a comparable number of sources for this prior catalogue and redshift bin as those recovered above the classical confusion limits.This is due to the majority of the sources being detected in the shortest wavelength channel considered for the selection, PHI2_1 (47µm), where the limiting flux from this run of XID+ is comparable to the classical confusion limit.However, XID+ is able to recover these sources in more channels.As such, it is able to accurately sample the FIR regime of galaxy SEDs (by detecting the galaxy and measuring its flux to an accuracy of better than 20 per cent in 8-10 channels in PHI2 and PPI bands) down to log 10 (SFR) ∼2-2.5  ⊙ yr −1 .This provides an improvement of 0.5 dex compared to what can be recovered above the classical confusion limits.Also shown is the SFR of the knee of the FIR luminosity function at  = 1.75 from Magnelli et al. (2014), which has log( * ) = 12.16  ⊙ and a corresponding log(SFR) = 2.33  ⊙ yr −1 , using the conversion from Kennicutt & Evans (2012).XID+ is able to recover this knee of the FIR luminosity function in at least 8 of the 10 channels covering 47-235 µm.
For the Deep prior catalogue run of XID+ utilising weak prior flux information, this is improved further, detecting sources in 8-10 channels down to log 10 (SFR) ∼1.6-2.1, which is ∼ an order of magnitude below what is reached for sources above the classical confusion limits.Moreover, this run of XID+ is able to recover ∼3.6x more sources than those above the classical confusion limits for this prior source catalogue and redshift bin.

Obtaining Prior Flux Information
For the runs of XID+ which have included prior flux knowledge, we have employed a toy model to represent the constraining power of the prior flux knowledge.In reality, these flux priors would need to be obtained via some modelling of the SED of the source.When PRIMA is launched there will be a wealth of deep ancillary photometry available from contempoary missions and ground-based facilities.SED-fitting of this data could be performed to estimate the flux of the source in the particular channel map which requires de-blending.The choice of SED modelling procedure as well as the type of ancillary data available to fit (e.g.radio and MIR photometry vs.only UV/optical) will inevitably impact the accuracy of the modelling (Pacifici et al. 2023;Thorne et al. 2023).Moreover, even if the flux prior information is not constraining for specific sources and is only representative of typical galaxy populations, modelling these will still reduce the confusion noise.Exploiting prior flux information would still allow atypical galaxies to be detected by looking for cases where the posterior significantly departs from the prior distribution or where the model does not fit the data well, through posterior predictive checking.
In addition to utilising ancillary photometry, another approach which only requires data from PRIMAger is possible due to the probe's extensive spectral coverage and resolution.Source detection and photometry can be performed on the shorter-wavelength band PHI1 maps (i.e. < 40µm), which are not confusion-dominated.Probabilistic SED fitting using an SED library (e.g. from CIGALE) can then be performed on these extracted fluxes to estimate, for example, the fluxes of the sources in the next 3 channels (in ascending wavelength order), providing the flux priors to be used to de-blend the corresponding maps.This step-wise method can be repeated so that the reddest and most confused maps will have prior flux information which is determined from the shorter wavelength maps.Applying this step-wise method and determining whether it can provide prior flux knowledge which is sufficiently informative is beyond the scope of this paper, but we outline it here to be tested in the context of PRIMAger in future work.

CONCLUSIONS
In this work, we have shown that confusion mitigation methods utilising positional priors successfully demonstrated on Herschel datasets will allow PRIMAger to reliably extract fluxes well below the classical confusion limits.
We have tested these mitigation methods on mock data that simulate a 1500 hr deg −2 depth hyperspectal imaging survey with PRIMager, from 25-235µm, using a sky, observatory and instrument model that provides maps with realistic confusion and "instrumental" noise.
We have demonstrated that we can produce catalogues of galaxies with high purity from the PRIMager images alone (i.e.blindly) using a Wiener-filter optimised to suppress both forms of noise.Specifically, we have produced catalogues with 95 per cent purity reaching 55-117 µJy in S obs in the six PHI1 bands, where the majority of sources are first detected.This blind catalog also reaches a completeness of ∼ 83 per cent on sources with S True >  total in PHI1 bands, with a source density 42k deg −2 (or ∼ 0.5 sources per beam in PPI1 band).
We have then shown that we are able to accurately recover the fluxes of these high purity PRIMager sources from 25-235µm with no prior flux information using the Bayesian probabilistic deblending code XID+.We demonstrated that flux accuracy within 20 per cent of the true flux values are obtained below the confusion limits for all the confusion-dominated maps.A gain of a factor of ∼2 below the classical confusion limits (as estimated by B24) is achieved between 72-96 µm, as shown in Figures 6 and 7.This increases to a factor of ∼3 for 126-235 µm (the reddest channels in the PHI2 band).This allows PRIMAger to recover SEDs out to  = 126 µm for sources at the knee of the infrared luminosity function for  = 2, as shown in Figure 7.
We have also shown that even greater improvements are possible with the introduction of additional prior information, e.g.arising from the detection in, and spectral energy distribution modelling of, other wavelengths with data from other contemporary observatories.We have investigated the impact of increasing the source density of the prior position catalogue alongside varying prior flux knowledge on the flux modelling accuracy of XID+.De-blending of sources at high densities (> 1 source per beam), or equivalently lower fluxes, benefits significantly from adding prior flux information to XID+.We show that with weak prior flux information (a Gaussian prior with dispersion equal to the flux) accurate fluxes for sources are recovered at  < 80µm down to the instrumental noise level of the survey.This same catalog and flux prior results in recovering fluxes about an order of magnitude below the classical confusion limit at 96-172µm, and a factor of 6 below the classical confusion confusion at 235µm.
We have also shown that de-blending with XID+ allows a survey such as the one described for PRIMAger to detect and measure accurately source fluxes for galaxies which contribute to the bulk of the IR luminosity density.Additionally, we have demonstrated that XID+ is able to sample the FIR regime of galaxy SEDs with accurate flux measurements in 8-10 of the 10 channels covering 47-235µm for sources with log(SFR) ∼ 2-2.5 at 1.5 ⩽  ⩽ 2.0.This improves upon what can be achieved above the classical confusion limits by 0.5 dex, as shown in the top panel of Figure 9.Most importantly, these observations are self-contained as the prior source catalogues can be obtained from the shorter wavelength PRIMAger maps, where confusion noise is not dominant, and are subsequently used to deblend the longer wavelength maps and accurately measure source fluxes.
We have therefore demonstrated that imaging data from PRIM-Ager will not be limited by näive, classical confusion noise if deblending with XID+ is employed.Accurate flux measurements below the confusion limits are therefore currently achievable using data from PRIMAger in a self-contained way.
Further improvement can also be achieved both by utilising ancillary data to provide additional prior source positions and prior flux information, and also, with PRIMAger data along by using shorter wavelength data to provide improved priors by utilising XID+ in a step-wise process as described in Section 6.3.

CONTRIBUTIONS
The contributions of the authors using the Contributor Roles Taxonomy (CRediT) were as follows.A1.For the PPI maps, the instrumental noise is negligible compared to the confusion noise and therefore the clipped-variance of the noisy map is comparable to that of the noiseless map.For the PHI2 maps, the clipped variance of the noisy map is greater than the confusion limits, as expected because the instrumental noise is non-negligible.This direct measurement of the total noise in the map is also greater than simply adding the two noise components in quadrature as confusion noise is non-Gaussian.
Secondly, we estimate the MAD of the pixel values as a more robust estimator of the dispersion of the pixel values in the noisy maps (scaled by a factor of 1.4862, being the ratio between MAD and  for a Gaussian distribution).This is shown in Figure A1 as the dash-dotted orange line.It is clear that this statistic naturally returns a lower estimate of the noise than the clipped variance.This is partially because the MAD statistic ignores both the positive and negative tails, and while the clipping only removes the positive tail, but more importantly because the MAD statistic removes much more of the tails.
We now turn to consider the dispersion metrics at the locations of sources.To investigate this, we conduct a näive photometry to measure the fluxes of the Wiener-filtered catalogue sources in the noisy maps.This involves simply reading the value of the map at the position on the source (as the maps are in units of mJy/beam).The MAD-based limiting flux statistic is then applied to these näive photometry measurements.The results are shown by the blue dotted line in Figure A1 and are consistent with the MAD measure of the total noise in the maps.Both, however, are systematically lower than the clipped variance measure of noise (∼ 10-20 per cent lower), implying that some of the gains from XID+ compared to the classical confusion limits are due to this choice of the statistic.Despite this, the results from XID+ remain below all of the measures of the total noise for all confusion-dominated maps.

APPENDIX B: ROBUSTNESS OF DECONVOLUTION TECHNIQUES
The impressive deconvolution achieved from the hyperspectral imaging is not unique to the XID+ algorithms.In this appendix we demonstrate this through an alternative method for prior catalogue generation, using a machine learning model trained to super-resolve the full range of PRIMA bands into a single output.We use a denoising autoencoder adapted from that developed for Herschel SPIRE 500 m imaging by Lauritsen et al. (2021).Unlike XID+, it does not assume the positions of a set of point sources extracted at shorter wavelengths as a Bayesian prior, but rather predicts the properties of the image from the training set of shorter wavelength data.The model was trained using cut-outs of the simulated hyperspectral PRIMAger imaging from this paper, with the target resolution for the longest wavelength data being that of the shortest wavelength imaging, i.e. a resolution improvement of a factor of approximately five. Figure B1 shows the results of this deconvolution in a segment of the simulated image; also shown is the PRIMA catalogue generated target image for the same region of sky.The model has never been exposed to this target image.A comprehensive analysis of the statistical properties of this alternative PRIMA deconvolution is deferred to a later paper, though it is clear that the deconvolution capacity is a general property of PRIMA's hyperspectral imaging, and not simply specific to prior-based deblending algorithms.

Figure 1 .
Figure 1.Simulated PRIMAger maps including both "instrumental noise" and source confusion, illustrating the transition from instrumental dominated at short wavelengths to confusion dominated at longer wavelengths.The sources are drawn from the SIDES simulations.The instrumental noise synthesises observations of 1500 hr deg −2 and is discussed in Section 2.2.Cutouts are 4 ′ × 4 ′ in representative channels with  = 10 in bands PHI1 and PHI2 and  = 4 in band PPI.

Figure 2 .
Figure2.The effective Point Spread Function (PSF) profile in a PPI1 image after applying: a Wiener filter (solid line); or a näive PSF filter (dashed line).The näive filter is optimal for flux estimation of isolated point sources, including non-local data with reduced weight to enhance the signal to instrumental noise ratio in comparison to a central, purely local, estimate.The Wiener filter has a much narrow central beam which provides some reduction in instrumental noise but balanced against adding in confusion noise from non-local data.The Wiener filter also has a negative feature (at around 7 ′′ in this example) which provides a local background subtraction, actively reducing confusion noise.

Figure 3 .
Figure3.Redshift distribution for the three prior source catalogues considered in Section 4.2: The Deep catalogue (pink) of ∼590,000 sources which all have a flux greater than 1µJy in the PHI1_1 channel, the catalogue of blindly detected sources in the noiseless maps (green) from B24 of ∼102,000 sources and the catalogue of blindly detected sources in the Wiener-filtered, noisy maps (blue) with ∼83,000 sources.

Figure 4 .
Figure 4. Location of sources within the three catalogues considered in Section 4.2 on the stellar mass-SFR plane for a single redshift bin of 1.5 <  ⩽ 2.0.For reference we indicate a range of reported locations for the star-forming "main sequence" in the literature (Speagle et al. 2014; Pearson et al. 2018; Leslie et al. 2020; Leja et al. 2022).The number of sources in each M * -SFR bin is shown by the colour scale for the Deep, B24 and Wiener-filtered catalogues in the left, middle and right panels, respectively.

Figure 5 .
Figure 5.The limiting flux, as defined in Section 4.2.1, reached by XID+ in the PPI1 channel as a function of the prior flux knowledge.Prior knowledge is defined as the true flux of the sources,  true divided by the spread on the Gaussian flux prior,  prior (i.e. as the prior flux knowledge increases, the spread on the flux prior decreases).Results are shown for a sample of the data (totalling ∼0.12 deg 2 ) from the three prior source catalogues described in Section 4.2: Weiner-filtered catalogue (green line with cross markers); blind detected catalogue from B24 (blue line with diamond markers); and the Deep catalogue (pink line with triangle markers).For the Deep catalogue beyond  true / prior > 1 (not plotted, but indicated by the dashed line) the limiting flux is ∼ 1µJy, i.e. flux of the faintest source, indicating that the modelling is performing as well as possible.Source densities for each of the catalogues are indicated in the legend.The orange dash-dotted line shows the classical confusion limit for the PPI1 channel estimated by B24.

Figure 6 .
Figure 6.XID+ flux accuracy as a function of true flux for Wiener-filtered prior in the 10 reddest PRIMAger channels (coloured solid lines).Flux accuracy is quantified as the scaled Median Absolute Deviation (MAD),  MAD as defined in equation 2, of the ratios of measured source fluxes from XID+.The horizontal black dashed line shows the 'limiting flux' threshold at  MAD = 0.2, which represents measured flux accuracy of 20 per cent (5).The true flux at which the coloured solid lines intercept with this threshold is taken to be the 'limiting flux' for the given channel, as defined in equation 3.

Figure 7 .
Figure7.Limiting flux density as a function of wavelength from 25-235µm for XID+ deblending.De-blending with positional and weak flux priors consistently attains 5 depths more than an order of magnitude fainter than the classical confusion limit at  > 100 m.This figure shows that using XID+, SEDs from typical galaxies at  = 2 can be measured to  = 126 m using only positional priors (derived from the Wiener-filtered map), and out to the longest PRIMAger PPI channel ( = 235 m) with the addition of a weak intensity prior.Top: Limiting flux density as a function of wavelength covering the 12 representative channels of the two LVF PRIMAger bands, PHI1 (25-43µm) and PHI2 (47-80µm), and the 4 PPI channels (96-235µm).Blue dashed line shows the limiting flux density, as defined in Sec.4.2.1, reached by XID+ with flat flux priors and the Wiener-filtered detection prior catalogue.The dash-dotted pink line shows the results from XID+ with the Deep prior catalogue and flux priors with  s,prior =  true .The orange solid line shows the classical confusion limits from B24 and the red triangles show the 5 baseline point source sensitives in each of the channels.Also plotted are two model SEDs of galaxies fromKirkpatrick et al. (2015) at z=2 with luminosity L = 10 12.3  ⊙ , corresponding to the knee of the FIR luminosity function at this redshift(Magnelli et al. 2014), and fraction of luminosity from AGN emission, f AGN , of 0 and 0.4, shown by the dotted grey line and the solid gray line respectively.Bottom: Limiting flux density reached by XID+ relative to the 5 confusion limits for the 10 reddest channels which are confusion-dominated.

Figure 8 .
Figure 8. Regions of the redshift-SFR plane accessible to PRIMAger at the given limiting fluxes reached by XID+ in the PHI2_3 (left) and PPI2 (right) channels.Sources which lie within above the lines are those whose fluxes can be accurately measured in the given channels using XID+.Additionally, the FIR luminosity density as a function of redshift and SFR is shown by the grey-scale contours.The classical confusion limits from B24 in each channel are shown by the solid orange lines.The limiting fluxes from XID+ with the Wiener-filtered prior catalogue and the Deep prior catalogue with flux priors of  s,prior =  true are shown by the dashed blue lines and the dash-dotted pink lines, respectively.

Figure 9 .
Figure9.Locations of galaxies recoverable by XID+ in the stellar mass-SFR plane for a single redshift bin of 1.5 <  ⩽ 2.0, colour-coded by the number of bands in which they could be detected.A galaxy is considered detectable if it has a true flux above the limiting flux in at least one channel for the confusiondominated maps (PHI2_1-PPI4; 47-235µm).The limiting fluxes are taken as the classical confusion limits from B24 for the top left and bottom left panels and as the limiting flux results from XID+ runs with the Wiener-filtered and Deep prior catalogues for the top right and bottom right panels, respectively.Colour-scale shows the average number of channels which the sources within each M * -SFR bin are detected in.Overplotted are the average SFR values for all sources which are detected in 2, 8 and 10 of the PHI2_1-PPI4 channels, shown by the dotted, dash-dotted and dashed grey horizontal lines, respectively.The orange dashed horizontal line shows the SFR of the knee of the FIR luminosity function at  = 1.75 fromMagnelli et al. (2014).The star-forming main-sequence curves from the literature are indicated by the shaded region.This shows that XID+ can recover multi-band photometry into the main sequence with galaxies detected self-consistently by PRIMA (top right) and substantially spanning the main-sequence with deeper (external) prior catalogues (bottom right).

Figure B1 .
Figure B1.This figure shows 424 × 424 arcsec 2 postage stamps of the simulated 235µm PRIMA SIDES image (left) compared to the autoencoder predicted image (right), showing a resolution increase of around a factor of 5. Also shown is the catalogue generated 'target' image (centre), which has never been seen by the predicting model.The flux scale is in Jy/beam.A comparable deconvolution product can be created from the XID+ prior-based deblending.

Table 2 .
Number of sources in each of the three prior source catalogues over 1.96 deg 2 explored in Section 4.2.B24 uses a basic peak detection on noisefree (confusion only) maps; the Wener-filtered catalogue is an extraction from a simulation of the deep, 1500 hr deg −2 , survey; the Deep catalogue comes from the simulated input catalogue and represents a prior catalogue from other facilities.The source density is given as number of sources per band PPI1 beam.The corresponding flux depth from the simulation input catalogue at this source density is also provided.