###### Abstract

In our study, we show a multiwavelength view of ACT-CL J0019.6+0336 (which
hosts a radio halo), to investigate the cluster dynamics, morphology, and ICM.
We use a combination of XMM-Newton images, Dark Energy Survey (DES) imaging
and photometry, SDSS spectroscopic information, and 1.16 GHz MeerKAT data to
study the cluster properties. Various X-ray and optical morphology parameters
are calculated to investigate the level of disturbance. We find disturbances
in two X-ray parameters and the optical density map shows elongated and
axisymmetric structures with the main cluster component southeast of the
cluster centre and another component northwest of the cluster centre. We also
find a BCG offset of $\sim$950 km/s from the mean velocity of the cluster, and
a discrepancy between the SZ mass, X-ray mass, and dynamical mass ($M_{X,500}$
and $M_{SZ,500}$ lies $>3\sigma$ away from $M_{\rm{dyn},500}$), showing that
J0019 is a merging cluster and probably in a post-merging phase.

###### keywords:

diffuse radio emission; galaxy clusters; multiwavelength; radio halo; cluster
dynamical state; morphology; ICM; turbulence; bent tailed galaxies

9 4 97 Academic Editors: Francesca Loi and Tiziana Venturi 6 October 2021 2
November 2021 8 November 2021 https://doi.org/10.3390/ galaxies9040097 A
Multiwavelength Dynamical State Analysis of ACT-CL J0019.6+0336A
Multiwavelength Dynamical State Analysis of ACT-CL J0019.6+0336 Denisha S.
Pillay 1,2,*, David J. Turner 3, Matt Hilton 1,2, Kenda Knowles 4,5, Kabelo C.
Kesebonye 1,2, Kavilan Moodley 1,2, Tony Mroczkowski 6, Nadeem Oozeer 7,8,
Christoph Pfrommer 9, Sinenhlanhla P. Sikhosana 1,2 and Edward J. Wollack 10
Firstname Lastname, Firstname Lastname and Firstname Lastname Pillay, D.S.;
Turner, D.J.; Hilton, M.; Knowles, K.; Kesebonye, K.C.; Moodley, K.;
Mroczkowski, T.; Oozeer, N.; Pfrommer, C.; Sikhosana, S.P.; et al.
Correspondence<EMAIL_ADDRESS>
##  1 Introduction

Multiwavelength observations of galaxy clusters carry an abundance of
information, giving insight into the intracluster medium (ICM), its thermal
and non-thermal components, and the cluster dynamics. Galaxy clusters merge
with neighbouring clusters through the most energetic events in the Universe
since the big bang (Puchwein and Bartelmann, 2007). The merger disturbs the
cluster’s natural state through merger-driven turbulence and often leave
imprints on the ICM.

Turbulence in the ICM also affects individual sources in the cluster,
particularly radio galaxies (Blanton et al., 2015). There are differences in
radio emission from cluster galaxies and isolated galaxies that are generally
attributed to the interaction of the emission with the ICM. Most radio
emissions from galaxies have a compact radio source associated with the active
galactic nuclei (AGN) and extended regions of radio emission (radio lobes)
quite distant from the compact radio source. The morphology of extended radio
emission associated with individual galaxies (such as bent tail galaxies) is
strongly influenced by the environment in which the galaxy exists (Sakelliou
and Merrifield, 2000). The motion of galaxy clusters through the dense cluster
gas is widely recognized as the mechanism which produces bent radio source
tails and the dynamic pressure responsible for a variety of other observed
source shapes (Sakelliou and Merrifield, 2000).

A fraction of the gravitational energy released during violent merger events
is converted into magnetic-field amplification and results in the acceleration
of high energy particles in the ICM, giving rise to diffuse radio sources
(Feretti et al., 2012). Cluster-scale diffuse radio sources have steep
spectra, exhibit low surface brightness, and are broadly categorized into
three groups: radio halos, radio mini halos, and radio relics. Radio halos and
mini halos are centrally located sources, whereas relics are found as
filamentary, elongated shapes located at the cluster periphery (see (van
Weeren et al., 2019) for a recent review).

Diffuse radio emission and multiwavelength studies provide more knowledge
about the link between the underlying thermal and non-thermal processes
present in the ICM (Brunetti and Jones, 2014). Several open questions
regarding cluster diffuse emission exist. It is not understood why some
merging clusters host radio halos and relics while others do not show any
evidence of extended radio emission (Cassano et al., 2010). The correlation
between non-thermal diffuse emission and the dynamical properties of the host
clusters is unclear and the underlying formation mechanisms are not well
understood (Cassano et al., 2010). These open questions have larger
implications on topics and processes which include dark matter, AGN feedback
mechanisms, and the large-scale structure of the universe. The sensitivity of
new generation radio telescopes will enable studies of large and diverse
samples of clusters at higher redshift and lower mass.

Previous studies of radio diffuse emission were limited to high mass and low
redshift samples, and clusters with high redshifts were generally solitary
detections (Lindner et al., 2014; Knowles et al., 2016). Large sample studies,
such as Cuciti et al. (2021), are crucial to a complete understanding of the
evolutionary life of radio diffuse emission, formation theories of radio
sources, and analysis of radio source statistics. The MeerKAT Exploration of
Relics, Giant Halos and Extragalactic Radio Sources (MERGHERS; (Knowles et
al., 2016)) project aims to be a mass-selected sample of $\sim$200
Sunyaev–Zel’dovich (SZ) selected clusters. These clusters will cover a
suitable range of criteria to ensure an unbiased study. In preparation for
such a large project, a pilot sample of 13 clusters has been obtained using
the MeerKAT telescope (Knowles et al., 2021).

In this paper, we perform a multiwavelength follow-up of ACT-CL J0019.6+0336
(hereafter J0019), presented in Knowles et al. (2021), to further understand
the dynamics of the cluster. The paper is structured as follows. In Section 2,
we show existing multiwavelength data on J0019. Section 3 shows the morphology
analysis methods and results, followed by the discussion and conclusion in
Sections 4 and 5 respectively. In this paper, we adopt a $\Lambda$CDM flat
cosmology with $H_{0}$ = 70 kms-1 Mpc-1 , $\Omega_{\Lambda}$ = 0.7,
$\Omega_{m}$ = 0.3.

##  2 Observations and Data

J0019 lies at $z=0.266$ and was first detected in a catalogue of galaxy
clusters derived by Zwicky et al. (1965) (ZwCl 0017.0+0320) and was later
observed by the Atacama Cosmology Telescope (ACT; (Hilton et al., 2018)). In
this section, we describe the multi-wavelength data and MeerKAT observations
used for this study. The relevant cluster properties are given in Table 2.

[H] Published properties of J0019. The position, redshift, SZ mass, and radio
properties are from Knowles et al. (2021), with the mass being the weak
lensing-calibrated value. Integrated 0.5–2.0 keV X-ray luminosity is
calculated within $R_{500}$ = 1389.5 kpc (spherical radius that encloses an
average density equal to 500 times the critical density at the cluster
redshift, z = 0.266). R.A.J2000 (deg) 4.91085 Dec.J2000 (deg) 3.60879 redshift
0.266 $M_{\rm 500c,SZ}$ ($10^{14}M_{\odot}$ ) 10.2 $\pm$ 2.28
$L_{500,X}(10^{44}$ ergs/s) 4.94 $\pm$ 0.032 $S_{\rm 1.16GHz}$ (mJy) 9.16
$\pm$ 0.57 log$P_{1.4GHz}$ (W/Hz) 24.13 $\pm$ 0.06

###  2.1 Millimetre

Studies such as Basu (2012) and Cuciti et al. (2015), have shown that samples
selected via their SZ signal show a higher detection rate than X-ray selected
samples and SZ selected samples have the benefit of the flux limit translating
directly into a mass limit (Sommer and Basu, 2014). Various SZ telescopes now
provide large SZ-detected cluster samples in which galaxy clusters are
detected via the distortion of the cosmic microwave background (through
inverse Compton scattering of high-energy electrons). J0019 was observed in
both Planck and ACT surveys (Hilton et al., 2021; Planck Collaboration et al.,
2015). We make use of the ACT observations because they have higher
sensitivity and resolution than the Planck observations.

J0019 is part of the fifth release of the ACT cluster catalogue ((Abbott et
al., 2018), ACT DR5) which contains more than 4000 SZ selected optically
confirmed clusters (Hilton et al., 2021). J0019 was observed as part of the
larger SZ-selected sample from ACT that forms the targets for MERGHERS with a
signal-to-noise ratio, S/N = 25.23 (at the reference 2.4’ filter scale)
(Hilton et al., 2021). This shows that J0019 is a very massive and energetic
cluster.

###  2.2 Optical

We make use of available Dark Energy Survey (DES) DR1 photometry for J0019.
zClusterhttps://github.com/ACTCollaboration/zCluster (accessed on 1 June
2021). estimates the photometric redshift of a particular galaxy cluster using
multi-band optical and infrared photometry. The optical photometry is obtained
using the zCluster code and the photometric redshift of the cluster is
estimated by the weighted sum of individual galaxy probability distributions
in the direction of each galaxy (Hilton et al., 2018). These probability
distributions are found using a template fitting method as seen in Benitez
(2000), where a set of default galaxy spectral energy distribution (SED)
templates (Hilton et al., 2021) are fitted to the observed broadband SED of
each galaxy. Moreover, spectroscopic observations are available for 17
possible cluster members (within $R_{500}$) in the Sloan Digital Sky Survey
(SDSS) DR16 database (Ahumada et al., 2020).

###  2.3 Radio

J0019 has been observed by the MeerKAT telescope for $\sim$24 min at L-band
($\sim$0.9–1.7 GHz) ((Knowles et al., 2021), PI:Knowles).First, second, and
third-generation MeerKAT calibration algorithms have been applied to the radio
data to correct for direction independent and dependent effects. First and
second-generation calibration was applied to the field using the oxkat
v1.0https://github.com/IanHeywood/oxkat (accessed on 1 June 2020.) pipeline
(Heywood, 2020), a Python-based reduction pipeline for MeerKAT. Third-
generation calibration has been applied to the field using
DDFACEThttps://github.com/saopicc/DDFacet (accessed on 1 July 2020). and
Killmshttps://github.com/saopicc/killMS (accessed on 1 July 2020). to correct
for corrupting artifacts in the radio image. The
KATBEAMhttps://github.com/ska-sa/katbeam (accessed on 1 August 2020). package
is used to create a primary beam corrected final image for analysis. A large
scale filtered image (contours overlaid in Figure 1) is also created through
an image-plane filtering technique introduced by Rudnick (2004), in which
extended emission without compact sources is highlighted by filtering out
emission on a scale 1–3 times the synthesised beam. The left panel of Figure 1
shows the final full-resolution radio map of the J0019 cluster region with
large-scale filtered image contours overlaid. A radio halo ($\sim$810 kpc in
size) has been detected with a 1.16 GHz flux density of $9.16\pm 0.57$ mJy.
The power value has been estimated as $P_{\rm 1.4GHz}$ = $24.13\pm 0.06$ W/Hz
by using a fiducial spectral index, $\alpha=-1.3\pm 0.4$ and extrapolating the
MeerKAT flux densities to 1.4 GHz (adopting the spectral power law convention
of Sν $\propto$ $\nu^{\alpha}$).

Figure 1: Left: Full-resolution MeerKAT L-band radio image for J0019 with
large scale filtered image contours overlaid. Contours are at [$-$3, 3, 5, 10]
× $\sigma$ (52 $\upmu$Jy/beam) of the filtered image. The synthesized beam for
both the full-resolution (red ellipse, 7.9′′ $\times$ 7.2′′, p.a. 162∘) and
filtered map (black circle, 21′′) are indicated in the lower left of the
panel. The dashed red circle denotes $R_{500kpc}$ centred on the ACT SZ peak,
which is marked by a black cross. Full resolution central rms noise is 18.1
$\upmu$Jy/beam. Right: XMM-Newton combined PN, MOS1, and MOS2 images of J0019
with the large-scale radio contours overlaid. The black cross represents the
ACT SZ peak and the green Xs are centred on the peak radio emission of sources
found in the MeerKAT radio image. Regions of the image masked after removing
X-ray point sources are indicated by yellow, dashed, excluded circles. The
image is unbinned and has been smoothed by a Gaussian with a kernel radius of
6 pixels (1 px = 4.35′′).

2

###  2.4 X-ray

We use archival data from the XMM-Newton space telescope to provide an X-ray
view of J0019 and probe its morphology. The only XMM data available for J0019
was obs ID: 0693010301, a pointed, 47.9 ks observation of the cluster taken in
2012. Events from all three EPIC (PN, MOS1, and MOS2) The XMM-Newton
observations include European Photon Imaging Camera (EPIC) data from the two
MOS (Metal Oxide Semi-conductor) CCD arrays and the pn CCD array. cameras have
been used. The XMM Cluster Survey ((Romer et al., 2001), XCS) cleaned the data
and performed source detection with a custom version of WAVDETECT (Freeman et
al., 2002) (called the XCS Automated Pipeline Algorithm (XAPA)). XAPA is run
on merged EPIC (PN+MOS1+MOS2) images, with a pixel size of 4.35′′, that have
been generated using events within an energy range of 0.5–2.0 keV. Once
sources in an image have been located, XAPA classifies them as either point or
extended.

We use XMM: Generate and Analyse (XGAXMM: Generate and Analyse GitHub
(accessed on 1 May 2021).; (Turner et al., in prep)), a new open-source Python
module created for XCS, to generate soft-band (0.5–2.0 keV) count-rate maps.
XGA then uses the XCS source catalogue to produce a mask to remove irrelevant
sources. It identifies the XCS source that matches J0019, then masks all other
sources. The final processed X-ray image is shown in the right panel of Figure
1.

The unabsorbed X-ray luminosity in the 0.5–2.0 keV energy band was measured by
performing simultaneous fits to spectra generated within $R_{500}$Spherical
radius that encloses an average density equal to 500 times the critical
density at the cluster redshift, z = 0.266. of J0019, centred on the Knowles
et al. (2021) coordinates. We fit absorbed (with tbabs, Wilms et al. (2000))
plasma emission models (APEC, (Smith et al., 2001))to the three spectra, with
temperatures linked, metallicity fixed at 0.3 $Z_{\odot}$, redshift fixed at
0.266, and nH fixed at $0.029\times 10^{22}$ cm-2 (taken from the full-sky HI
survey by the HI4PI Collaboration et al. (2016)). The fit is performed using
XGA’s XSPEC (Arnaud, 1996) interface, SAS v17.0.0, and XSPEC v12.10.1. We
measure a temperature of $T=6.7\pm 0.1$ keV within $R_{500}$ and a bolometric
X-ray luminosity of $L_{X}=(1.86\pm 0.02)\times 10^{45}$ ergs/s. Comparing to
the Mx-Tx relation in Kotov and Vikhlinin (2006), we estimate a mass within
R500, M${}_{X}\sim$ (6.5$~{}\pm~{}1.3)\times 10^{14}M_{\odot}$.

##  3 Cluster Morphology

Galaxy clusters form via a hierarchical sequence of accretion and mergers of
smaller interacting substructures (Brunetti and Jones, 2014). Current
observations of radio diffuse emission favour theories of merger-driven radio
halo and radio relic formations (van Weeren et al., 2019). Therefore, it is
important to understand the dynamical state of J0019. Numerous studies, e.g.,
Böhringer et al. (2010); Wen and Han (2013) have analyzed the substructure of
clusters to determine their dynamical states using multiwavelength data. By
studying the form or shape of the cluster we can infer the dynamics. We use
available multiwavelength information to estimate a variety of X-ray and
optical-derived parameters.

###  3.1 Optical Morphology

zCluster retrieves and uses redshift probability distributions to measure a
galaxy cluster’s photometric redshift. Thereafter, we create a projected 2-D
density map of the galaxy cluster using the right ascension (in degrees),
declination (in degrees), photometric redshift of the cluster, and a catalogue
of redshift probability distributions of individual galaxies. We use the right
ascension and declination of each galaxy to position them onto a 2-D map as
done in Wen and Han (2013). The galaxy coordinates are converted into
Cartesian coordinates and centred at the input R.A. and Dec. (SZ peak), on a
projected 4 Mpc $\times$ 4 Mpc map. The map is made up of 20 $\times$ 20 bins,
each with 0.2 Mpc length. The positions of the galaxies in Cartesian
coordinates are obtained by finding the angular distance between the
coordinates of individual galaxies and the centre of the cluster.

The peak of the probability distribution gives the maximum likelihood of a
galaxy having a specific redshift (Hilton et al., 2018). We obtain a weighting
for our plot by integrating the probability distribution of each galaxy around
the photometric redshift (bounds of integration from z_phot - delta_z to
z_phot + delta_z). The zCluster code takes the above-calculated quantities and
produces a 2-D projected density map of the galaxy cluster. Thereafter, the
map is convolved with a Gaussian kernel (with smoothing length = 2 pixels) to
smooth the map.

A density uncertainty map (right panel of Figure 2) is created using Monte
Carlo simulations. We add Gaussian-distributed noise to the galaxy photometry
at each Monte Carlo step and recompute the photometric redshifts, probability
distributions, and density maps. After 1000 iterations, the 68th percentile of
the maps is taken as an estimate of the error map.

Figure 2: Left: Density map of J0019 obtained from zField using photometry
from the DES DR1 database and z = 0.266. The projected density map is 4 Mpc
$\times$ 4 Mpc with each pixel 0.2 Mpc $\times$ 0.2 Mpc in size. Black large-
scale radio contours are overlaid and the red cross indicates the SZ peak of
the cluster. The units of the colour scale are counts per pixel. Right:
Corresponding error map for the J0019 galaxy density map.

2

The density map for J0019 (left panel of Figure 2) is made using zField (a
subclass of zCluster) in which we have used photometric redshift, z = 0.266,
and a redshift range, delta_z = 0.2. The weighting for this density map is
obtained by integrating the probability distribution for each galaxy around z
= 0.266 (bounds of integration from 0.266 $-$ 0.2 to 0.266 + 0.2).

####  3.1.1 Centre Shift

The centre shift method used in this work is a simple method to determine the
dynamical state of a cluster by calculating the offset between the SZ peak and
the peak in the optical density map. The centre of the cluster generally hosts
the brightest cluster galaxy (BCG). In relaxed clusters, the centre of galaxy
clusters traced by the gas using X-ray, or the SZ effect, coincides (within a
few kpc) with the centre of the galaxy cluster traced optically. This can be
contrasted with the centres of mergers (determined using X-ray and SZ data),
which can be hundreds of kpc away from the centroid determined optically as
seen in Pandge et al. (2019).

We measure the distance between the coordinates of the maximum of the density
plot and the coordinates of the cluster SZ peak (Kolokotronis et al., 2001).
After finding the coordinates of the peak value of the density plot (left
panel of Figure 2), we use the distance Formula (1) to find the centre shift.

$CS=\sqrt{(x_{m}-x_{0})^{2}+(y_{m}-y_{0})^{2}}\times 0.2\,,$ (1)

where $x_{m}$ and $y_{m}$ are the x and y coordinates of the peak of the
density plot and $x_{0}$ and $y_{0}$ are the coordinates of the centre of the
plot (SZ peak). We multiply the distance formula by the size of 1 pixel in Mpc
(0.2), to obtain CS in the projected Mpc.

####  3.1.2 Asymmetry Parameter

Another way to identify the morphology and dynamical state of a galaxy cluster
is to study and classify the asymmetry of the cluster. The morphology of
astronomical systems is closely related to their symmetry. If a cluster is
asymmetric, then it has been disturbed in some way and formed substructures
(Wen and Han, 2013). An efficient way to estimate the asymmetry of a cluster
is to flip the density map horizontally and vertically and then compare it to
the original image Conselice et al. (2000). The asymmetry parameter is
calculated using the following formula (Conselice et al., 2000):

$A^{2}=\frac{\sum(I_{0}-I_{\phi})^{2}}{\sum(2I_{0}^{2})}\,,$ (2)

where $I_{0}$ is the original image and $I_{\phi}$ is the image that has been
flipped horizontally and vertically.

###  3.2 X-ray Morphology

In this section, we show the calculations for the three main X-ray
morphological parameters to further study the cluster dynamics and compare our
results with those available in the literature. Following the work of Santos
et al. (2008), Poole et al. (2006), and Buote and Tsai (1995), we calculate
the power ratio, centroid shift value, and concentration parameter of J0019.

####  3.2.1 Concentration Parameter

Relaxed clusters generally host luminous cool cores in their centres whereas
merging clusters show core disturbance (Fabian et al., 1984). The
concentration parameter is defined as the ratio of the X-ray integrated fluxes
in the cluster core and the larger-scale region of the galaxy cluster (Santos
et al., 2008). The concentration parameter $c_{SB}$, following Santos et al.
(2008), is calculated in two circular regions; a core radius of 100 kpc and an
outer radius of 500 kpc using:

$c_{SB}=\frac{S<(100\rm~{}kpc)}{S<(500\rm~{}kpc)},$ (3)

where $S$ is the X-ray surface brightness centred on the X-ray peak, within a
particular radius (i.e., 100 kpc and 500 kpc). Cassano et al. (2010)
classifies a cluster as relaxed if $c_{SB}$ > 0.2 and disturbed if $c_{SB}$ <
0.2.

####  3.2.2 Centroid Shift

The centroid shift method is the most robust X-ray morphology estimator, as it
is most sensitive to the dynamical state while being least sensitive to image
quality (Poole et al., 2006). The centroid shift is defined as the distance
between the X-ray peak and the centre of mass, in units of aperture radius,
$R_{ap}$, and is calculated using the following equation:

$w=\left[\frac{1}{N-1}\sum(\Delta_{i}-\langle\Delta\rangle)^{2}\right]^{1/2}\times\frac{1}{R_{ap}}$
(4)

where $\Delta_{i}$ is the distance between the X-ray peak and the centroid of
the $i$th aperture and $\Delta$ is the mean of the sample of $\Delta_{i}$s.

This equation calculates the offset ($\Delta_{i}$) between the X-ray peak and
the centroid using a series of concentric circles. The centroid of the $i$th
aperture is the centre of mass within the $i$th aperture. A sample of
$\Delta_{i}$s are computed within a circular aperture of radius $R_{ap}$,
ranging from $R_{ap}$ = 500 kpc to 0.05 $\times R_{ap}$, decreasing in steps
of 5$\%$, as seen in Poole et al. (2006). Weißmann et al. (2013) classifies a
cluster as relaxed if $w$ < 0.01 and disturbed if $w$ > 0.01.

####  3.2.3 Power Ratios

The power ratio probes the underlying mass distribution to analyze the
substructure of a cluster. It is calculated using a multipole decomposition of
the potential of the two-dimensional projected mass distribution within a
certain aperture (Buote and Tsai, 1995). Instead of mass, it is applied to the
X-ray surface brightness. The third-order power ratio is one of the best
indicators of the dynamical state of a cluster (Rasia et al., 2013). The
general m-order power ratio (where _m_ > 0) is defined as:

$P_{m}=\frac{1}{2m^{2}R_{ap}^{2m}}(a_{m}^{2}+b_{m}^{2})\;\;\;{\rm and}$ (5)
$P_{0}=a_{0}{\rm ln}(R_{ap})^{2}$ (6)

where $a_{0}$ is the total intensity within aperture $R_{ap}$. $a_{m}$ and
$b_{m}$ are generic moments in polar coordinates, $R$ and $\phi$, given by:

$a_{m}(r)=\int_{R^{\prime}\lesssim
R_{ap}}S(x^{\prime})(R^{\prime})^{m}cos(m\phi^{\prime})d^{2}x^{\prime}$ (7)
$b_{m}(r)=\int_{R^{\prime}\lesssim
R_{ap}}S(x^{\prime})(R^{\prime})^{m}sin(m\phi^{\prime})d^{2}x^{\prime}$ (8)

The $P_{2}$ quadrupole power represents the cluster ellipticity, $P_{3}$
represents the bimodal distribution, and $P_{4}$ is similar to $P_{2}$, but
has power on much smaller scales. $P_{3}$ is most suitable to identify the
presence of substructures or asymmetries (Rasia et al., 2013). We make use of
the normalized hexapole moment, $P_{3}/P_{0}$, to provide a clear measure of
substructure calculated in an aperture of radius, $R_{ap}$ = 500 kpc, centred
on the X-ray cluster centroid as seen in Buote and Tsai (1995). Based on
previous studies Weißmann et al. (2013), a cluster is disturbed if
$P_{3}/P_{0}>10^{-7}$.

###  3.3 Results and Comparison with the Literature

To compare our optical results to the literature, the above methods were
applied to a sample of 98 galaxy clusters found in Wen and Han (2013), with
known cluster dynamical states. With the set of results obtained from this
sample, we can categorize clusters as relaxed or unrelaxed based on the
criteria shown in Figure 3.

We find that galaxy clusters with an asymmetry value, A${}^{2}>0.1$ or a
centre shift value, CS > 0.4 are all dynamically disturbed. Galaxy clusters
with centre shift values, CS < 0.4, and asymmetry values < 0.1 could be
classified as either relaxed or disturbed.

Applying the methods from Sections 3.1.1 and 3.1.2 on the density map of J0019
(3),we obtain a centre shift value, CS = 0.20 $\pm$ 0.16 Mpc, and an asymmetry
value, $A^{2}$ = 0.068 $\pm$ $4.9\times 10^{-4}$.

The statistical uncertainties on the above parameters are estimated by Monte
Carlo simulations in which we add Gaussian distributed noise to the galaxy
photometry. The density map is recreated at each Monte Carlo step and after
1000 iterations, the standard deviation of the sample is taken as an estimate
of the uncertainty.

Figure 3:  Plot of asymmetry parameter vs. centre shift for a sample of 98
clusters from Wen and Han (2013) with known cluster dynamical states. We find
that all clusters with an asymmetry value greater than $\sim$0.1 (denoted by
the vertical dashed line) are unrelaxed. We also find all clusters with a
centre shift value greater than $\sim$0.4 (denoted by the horizontal dashed
line) are unrelaxed. An asymmetry parameter less than $\sim$0.1 and a centre
shift value less than 0.4 show a mixture of relaxed and unrelaxed clusters. CS
is in Mpc. The red cross represents J0019.

The X-ray image has been smoothed with a Gaussian kernel of smoothing length 4
pixels for calculations. The following results were obtained:
$P_{3}/P_{0}=(8.93\pm 0.4)\times 10^{-8}$, $w=0.03\pm 0.01$, $c_{SB}=0.13\pm
0.002$. The analysis methods in Sections 3.1.1 and 3.1.2 have previously
(Weißmann et al., 2013) been applied to a sample of 80 galaxy clusters
observed with the XMM-Newton telescope and have classified a cluster as
disturbed when $P_{3}/P_{0}>10^{-7}$ and $w>0.01$. Another study (Cassano et
al., 2010) also characterized cluster substructures based on the power ratios,
centroid shift, and X-ray brightness concentration parameter estimated from
Chandra X-ray images. They define a cluster to be dynamically disturbed if its
morphological parameters satisfy the following conditions:
$P_{3}/P_{0}>1.2\times 10^{-7},c_{SB}<0.2$ and $w>0.012$. Two of the estimated
morphological parameters ($c_{SB}$ and $w$) satisfy the above conditions,
classifying J0019 as a dynamically disturbed cluster with $P_{3}/P_{0}$
falling $\sim 3\sigma$ away from the threshold value of $10^{-7}$.

The statistical uncertainties on the morphological parameters are estimated
from a set of 1000 Monte Carlo simulations in which Poisson/shot noise is
added to the observational XMM X-ray data. Uncertainties are estimated from
the standard deviation of a sample of 1000 simulated images with added noise.

To verify our calculations of the three morphological parameters, we use XMM-
Newton data on a known merging cluster, ACT-CL J0528.8$-$3927, and compare the
derived morphological parameters to the known literature values found in
Weißmann et al. (2013) and Lovisari et al. (2017). All estimated values
obtained by our code lie within $1\sigma$ of the literature values.

###  3.4 Optical Redshift Distribution

The redshift distribution of 17 spectroscopically confirmed SDSS DR16 (Ahumada
et al., 2020) cluster member galaxies are used to gauge any disturbance in the
cluster. This distribution is shown in Figure 4 where there is an indication
of bimodal structure in the histogram.

Figure 4: Histogram showing the redshift distribution for 17 spectroscopically
confirmed cluster members within $R_{500}$ from SDSS, DR16. Here, $v=0$ is
defined as the cluster systemic redshift of $z=0.266$. The bin width is 350
km/s. The vertical black dashed line shows the velocity of the BCG of the main
component.

The standard deviation of the redshift distribution corresponds to the
velocity dispersion, $\sigma_{v}$. Following Munari et al. (2013), we use the
calculated velocity dispersion and the galaxies-based scaling relation to
determine $M_{200}$ and $R_{200}$, using a value of $h=0.7$ in the Munari et
al. (2013) equation. For comparison with other results, we convert our M200
estimates into M500, following the appropriate relation given in Duffy et al.
(2008). The quantities M500 and R500 have been rescaled from M200 and R200
assuming a concentration parameter, $c=3$. We integrate a NFW profile Navarro
et al. (1997) and interpolate to determine M500 and R500. We estimate
uncertainties on all cluster properties by bootstrap resampling 1000 times.

From the mean redshifts of the components, we find a mean velocity at
$z=0.265$. The velocity dispersionCorrected for the cosmology in this paper.,
$\sigma_{v}$, is 810 $\pm~{}160$ km/s. We obtain a dynamical mass of M200 =
(4.1$~{}\pm~{}1.8)\times 10^{14}M_{\odot}$ and M500 = 2.5$~{}\pm~{}1.2\times
10^{14}M_{\odot}$.

##  4 Discussion

It is believed that diffuse radio sources are associated with dynamically
disturbed clusters. We can conclude that J0019 is undergoing a merger event
through the detection of the radio halo shown in (Knowles et al., 2021). A
radio halo in J0019 would require a significantly substantial merger to power
it. Previous studies (Weißmann et al., 2013) have classified a cluster as
disturbed when $P_{3}/P_{0}>10^{-7}$ and $w>0.01$. The authors of (Cassano et
al., 2010) also characterized cluster substructures based on the power ratios,
centroid shift, and X-ray brightness concentration parameter estimated from
Chandra X-ray images. They define a cluster to be dynamically disturbed if its
morphological parameters satisfy the following conditions:
$P_{3}/P_{0}>1.2\times 10^{-7}$, $c_{SB}<0.2$, and $w>0.012$. The
morphological estimators for J0019 (X-ray concentration and X-ray centre shift
values), classify J0019 as a merging cluster and the X-ray $P_{3}/P0$, optical
centre shift, and optical asymmetry values are inconclusive. X-ray morphology
parameters (particularly the power ratios) are largely insensitive to
substructures along the line of sight. Spectroscopy of more cluster members
are needed to gauge any disturbed morphology in this direction.

The optical density map shows a displacement (0.2 Mpc) between the ACT cluster
SZ peak and the peak in the density map. Moreover, the density map is fairly
elongated and axisymmetric. The density map shows two components, one along a
northwestern direction (top) and another along a southeastern direction
(bottom). This is inferred through the ‘tails’ or ‘trails’ seen in these
regions of the density map.

The radio halo in the left panel of Figure 1 roughly follows the thermal gas
in the X-ray image (right panel of Figure 1) as seen in the level edge present
at the bottom of the diffuse structure and its overall shape in both images.
The thermal gas and diffuse emission follow an elongated shape with an
extended bottom left feature which hosts the BCG. The X-ray image orientation
of the extended bottom left feature indicates that gas in the subcluster is
undergoing ram pressure stripping as it interacts with the main cluster
component.

Furthermore, there is a velocity offset3.4 of $\sim$950 km/s between the mean
velocity of the cluster and that of the BCG, further indicating that J0019 is
a disturbed cluster. There is also a discrepancy between the SZ mass
($M_{SZ,500}$), the X-ray estimated mass ($M_{X,500}$), and dynamical mass
estimated from the optical redshift distribution ($M_{\rm{dyn},500}$).
$M_{X,500}$ and $M_{SZ,500}$ lie $>3\sigma$ away from $M_{\rm{dyn},500}$,
suggesting that the two components, northwest and southeast of the cluster
centre, have already merged and J0019 is now observed in a post-merger phase.
The gas traced by X-ray and SZ emission is completely disturbed during a
previous collision, showing an enhanced X-ray emission and hence, a
T${}_{X}-M_{X}$ overestimation.

The angular resolution and short baselines of the MeerKAT telescope allow us
to investigate both low surface brightness diffuse emission as well as compact
radio sources. The radio contours overlaid on the optical image are shown in
Figure 5 where 10 compact radio sources are identified within a radius, R =
500 kpc. The compact source properties are listed in Table 4.

Only one of the radio sources is found in the Faint Images of the Radio Sky at
Twenty-Centimetres survey (FIRST (Becker et al., 1995)). This source is
labelled S5 and has an integrated flux density, S = 0.56 mJy. Source S3 in
Figure 5 is associated with the BCG of the main cluster component and S4 is
associated with the BCG of the infalling subcluster. This is inferred from the
optically visible BCGs. There are two BCGs that are spatially separated, as
seen in the DES image in Figure 5. These BCGs are also separated in velocity
space as shown in Figure 4. This provides support for the existence of two
distinct galaxy populations.

Moreover, an arched structure is found within the radio halo in projection.
The optical image (Figure 5) shows no optical counterpart at the centre of
this arched structure, S1, suggesting that it is embedded in the diffuse
structure. There are, however, two optical galaxies at either end of the bent
radio structure. Spectroscopy of all cluster members are needed to gauge if
these galaxies are related to the radio emission or not.

Figure 5: Optical $gri$-band DES image with MeerKAT contours overlaid.
Contours are without short baselines and start at [3, 4.5, 6, 10, 20, 40, 80]
$\times$ $\sigma$ rms where the rms is the full resolution central noise
($\sim$18.1 $\upmu$Jy/beam). The white dashed circle represents $R_{500kpc}$.
The radio image beam is shown as the white ellipse in the lower left corner
(7.9′′ $\times$ 7.2′′, p.a. 12∘) and the red cross marks the position of the
ACT SZ peak. Individual radio galaxies are labelled from S1 to S10.

[H] Properties of cluster region radio sources. Source labels are shown in
Figure 5. The R.A. and Dec. values are for the peak source emission in the
radio map. Flux densities for the sources are measured in the
$1.16~{}\rm{GHz}$ map. Spectroscopic redshifts, zSpec, are from SDSS DR16. a
C: compact; T: resolved with tailed emission. Source R.A. Dec. Type _a_
S1.16GHz zSpec Notes (deg) (deg) (mJy) S1 4.90205 3.59912 T 0.30 $\pm$ 0.03 No
optical counterpart found S2 4.91539 3.59455 C 0.67 $\pm$ 0.03 S3 4.91164
3.59829 C 0.04 $\pm$ 0.01 0.26904 Associated with BCG of the main component S4
4.90705 3.62907 C 1.70 $\pm$ 0.03 0.26415 Associated with BCG of infalling
subcluster S5 4.89830 3.63448 C 0.69 $\pm$ 0.02 Detected in FIRST(0.56 mJy) S6
4.91872 3.60286 C 0.10 $\pm$ 0.01 S7 4.91153 3.64177 T 0.66 $\pm$ 0.03 S8
4.92836 3.60541 C 0.15 $\pm$ 0.02 S9 4.92806 3.63719 C 0.11 $\pm$ 0.01 S10
4.90081 3.61702 C 0.05 $\pm$0.01

2

##  5 Conclusions

A low surface brightness radio halo was detected in J0019 (Knowles et al.,
2021) with the MeerKAT telescope at L-band. The halo flux density estimated by
Knowles et al. (2021) is $S_{1.16}$ = $9.16\pm 0.57$ mJy. The calculated
k-corrected 1.4 GHz radio power, $P_{1.4GHz}$, is (1.0 $\pm$ 0.3) × $10^{24}$
W/Hz and has been estimated by extrapolating the 610 MHz flux density to 1.4
GHz using a theoretically motivated spectral index of $\alpha$ = $-$1.3 $\pm$
0.4. In this study, we performed a dynamical state analysis of J0019 using
both XMM-Newton X-ray observations and optical DES photometry. The estimated
morphology parameters are as follows: $P_{3}/P_{0}=(8.93\pm 0.4)\times
10^{-8}$, $w=0.03\pm 0.01$, $c_{SB}=0.13\pm 0.002$, $A^{2}$ =
0.068$~{}\pm~{}4.9\times 10^{-4}$, and CS = 0.20 $\pm$ 0.16 Mpc.

Although the cluster is known to be involved in a merger event through the
detection of the radio halo, the X-ray $P_{3}/P_{0}$ value, optical centre
shift, and optical asymmetry values in the morphology analysis are
inconclusive. The X-ray centroid shift and concentration values indicate a
dynamically disturbed system.

The optical density map shows two cluster components: one along the
northwestern and another in the southeastern direction. The BCG of the main
component lies in the southern region and the BCG of the infalling cluster
lies in the northern region as seen in the DES gri-band image and optical
galaxy density map. Moreover, the density map shows that the cluster is
elongated and axisymmetric.

The bimodality detected in the velocity distribution and the offset of the BCG
with respect to the mean velocity of the cluster suggests that the cluster
merger has a component in the line-of-sight direction. However, the DES
density maps, the MeerKAT radio halo, and XMM X-ray surface brightness shape
shows an elongation in the north–south direction, which suggests that the
collision also presents a velocity component contained in the plane of the sky
and follows this north–south axis.

With an elongated shape in the X-ray map, the clear bimodality and
axisymmetric structures in the optical maps, two X-ray morphology parameters,
the BCG offset by $\sim$950 km/s from the mean velocity, and the discrepancy
between the SZ mass ($M_{SZ,500}$ = (10.2 $\pm~{}2.28)\times
10^{14}M_{\odot}$), X-ray mass ($M_{X,500}\sim(6.5\pm 1.3)\times
10^{14}M_{\odot}$), and dynamical mass ($M_{\rm{dyn},500}$ = (2.5
$\pm~{}1.2)\times 10^{14}M_{\odot}$), we clearly show that J0019 is a merging
cluster and probably in a post-merging phase.

We can aim to obtain spectroscopy for the full field to identify more cluster
members, infer cluster dynamics, and model how the merger event takes place in
future work. An additional aim could be to study both the spectral index map
of the diffuse structure and the polarized radio map. The energetics of the
system can also be studied in future work with simulations.

Conceptualization, implementing methodology, interpretation of results,
analysis, draft writing and editing, D.S.P.; resources and draft writing,
D.J.T.; methodology, K.K., K.C.K., M.H. and D.S.P.; writing—review and
editing, D.S.P., K.K., M.H., K.M., T.M., N.O., C.P., S.P.S. and E.J.W.;
supervision, K.M., K.K. and M.H. All authors have read and agreed to the
published version of the manuscript.

The MeerKAT telescope is operated by the South African Radio Astronomy
Observatory (SARAO), which is a facility of the National Research Foundation,
an agency of the Department of Science and Innovation. DSP acknowledges
funding from the South African Radio Astronomy Observatory and the National
Research Foundation (NRF Grant Number: 96741). K.K. is supported by the New
Scientific Frontiers grant of the South African Radio Astronomy Observatory.
DT acknowledges support from the UK Science and Technology Facilities Council
via grant ST/P006760/1. CP acknowledges funding from the European Research
Council under ERC-CoG grant CRAGSMAN-646955.

Not applicable.

Not applicable.

The radio data underlying this article will be shared on reasonable request to
the corresponding author. XMM Newton observations and optical DES images and
photometry are publicly available.

###### Acknowledgements.

The authors thank the anonymous referees whose comments have greatly improved
the manuscript, and B. Partridge for useful comments on the original version.
This research made use of the Astropy, 13 (Astropy Collaboration et al. 2013,
2018), NumPy (Harris et al. 2020), and SciPy (Virtanen et al. 2020) Python
packages. Astropy is a community-developed core Python package for Astronomy.
The Common Astronomy Software Applications (CASA) package is developed by an
international consortium of scientists based at the National Radio
Astronomical Observatory (NRAO), the European Southern Observatory (ESO), the
National Astronomical Observatory of Japan (NAOJ), the Academia Sinica
Institute of Astronomy and Astrophysics (ASIAA), the CSIRO division for
Astronomy and Space Science (CASS), and the Netherlands Institute for Radio
Astronomy (ASTRON) under the guidance of NRAO. The authors declare no conflict
of interest. [custom] References

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# Human-Aware 3D Scene Generation with Spatially-constrained Diffusion Models

Xiaolin, Hong1,3, Hongwei, Yi211footnotemark: 1, Fazhi, He1, Qiong, Cao3

###### Abstract

Generating 3D scenes from human motion sequences supports numerous
applications, including virtual reality and architectural design. However,
previous auto-regression-based human-aware 3D scene generation methods have
struggled to accurately capture the joint distribution of multiple objects and
input humans, often resulting in overlapping object generation in the same
space. To address this limitation, we explore the potential of diffusion
models that simultaneously consider all input humans and the floor plan to
generate plausible 3D scenes. Our approach not only satisfies all input human
interactions but also adheres to spatial constraints with the floor plan.
Furthermore, we introduce two spatial collision guidance mechanisms: human-
object collision avoidance and object-room boundary constraints. These
mechanisms help avoid generating scenes that conflict with human motions while
respecting layout constraints. To enhance the diversity and accuracy of human-
guided scene generation, we have developed an automated pipeline that improves
the variety and plausibility of human-object interactions in the existing 3D
FRONT HUMAN dataset. Extensive experiments on both synthetic and real-world
datasets demonstrate that our framework can generate more natural and
plausible 3D scenes with precise human-scene interactions, while significantly
reducing human-object collisions compared to previous state-of-the-art
methods. Our code and data will be made publicly available upon publication of
this work.

Figure 1:  Our method generates more plausible 3D scenes given input human
motions and floor plans. It excels in two key aspects: (1) avoiding collisions
between humans and objects, as well as between objects, a significant
improvement over MIME (Yi et al. 2023), and (2) providing better support for
human-object interactions compared to DiffuScene (Tang et al. 2023).

## Introduction

Creating diverse and realistic 3D environments inhabited by humans is
essential for numerous applications, such as virtual reality (VR), interior
design and training embodied artificial intelligence (AI) agents (Zhang et al.
2019). For example, VR game designers aim to construct immersive environments
that enable participants to engage seamlessly and naturally with objects.
These demands have driven researchers to explore diverse scene generation
methods, propelling the rapid advancement of 3D scene synthesis (Luo et al.
2020; Paschalidou et al. 2021; Liu et al. 2023; Gao et al. 2023). Despite
recent progress, there remain significant challenges in generating visually
plausible scenes that are populated with appropriate furniture and adhere to
various human motions.

There has been a recent surge in research dedicated to the problem of human-
aware scene generation (Ye et al. 2022; Yi et al. 2023). A common approach in
these studies is to apply autoregressive models to sequentially place objects
based on input humans and already generated objects. However, these methods
often yield implausible scenes with object-object collisions. This limitation
primarily arises from the inherent inability of autoregressive models to
capture the joint distribution of multiple objects and multiple humans fully.
Consequently, exploring a generative method that can effectively model and
capture these complex distributions is crucial for generating realistic 3D
scenes.

More recently, diffusion-based approaches for scene synthesis (Tang et al.
2023; Yang et al. 2024) have shown an impressive ability to simplify the
approximation of the joint distribution of objects, thereby generating the
entire scenes at once and thus enhancing the realism of generated scenes.
Meanwhile, many studies in image generation (Saharia et al. 2022; Zhang, Rao,
and Agrawala 2023; Kawar et al. 2023) and human motion synthesis (Huang et al.
2023; Karunratanakul et al. 2023) have demonstrated that diffusion models can
effectively incorporate inference guidance to meet user-defined goals. Despite
these advancements, there is still no standard solution for generating
plausible 3D scenes that both support various human interactions and adhere to
spatial constraints, such as avoiding motion collisions and respecting room
boundaries.

To tackle these challenges, we propose SHADE, a Spatially-constrained Human-
Aware Diffusion based 3D Environments synthesis. As shown in Figure 1, our
method can generate plausible scene layouts that avoid collisions with humans
and between objects, while supporting various human activities such as sitting
and lying. Our key insight lies in innovative harnessing diffusion models to
simultaneously input all humans and the floor plan to generate holistic object
configurations. Specifically, we input the contact bounding boxes and free
space mask extracted from input human motions and the floor plan, following
(Yi et al. 2023). We then learn a diffusion model to capture the joint
distribution of objects, enabling the simultaneous generation of plausible
object placements and understanding the relationships between their
attributes. To further enhance the plausibility of generated scenes, we design
two spatial collision guidance functions: 1) motions collision avoidance that
calculates the 2D distances between objects and moving humans to prevent their
implausible penetrations, while 2) boundary constraint that penalizes the
distance by which objects extend beyond the floor plan, ensuring that object
placement respects room boundaries. During inference, we combine two guidance
functions with an object-object collision function (Yang et al. 2024). This
allows our diffusion model to generate collision-free scenes that respect
human movements, room boundaries, and prevent object overlap.

Beyond model design, we enhance human-aware scene generation by addressing key
data challenges in datasets like 3D FRONT HUMAN (Yi et al. 2023).
Specifically, we tackle two main issues: 1) incorrect penetrations in human-
object interactions hinder accurate spatial relationship modeling, and 2)
limited interaction diversity restricts model generalization. Our approach
includes automated techniques for adjusting translations to correct
penetrations and augmenting categories and orientations to diversify human-
object interactions. We use our calibrated dataset to train SHADE and evaluate
it on synthetic and real datasets. Our results, both quantitative and
qualitative, show that our method generates more plausible 3D scenes with
realistic human-scene interactions and effectively reduces human-object
collisions. These findings underscore the superiority of SHADE compared to
previous state-of-the-art methods.

Our main contributions are: (i) We propose a human-aware diffusion-based model
for generating realistic 3D scene layouts with plausible human-object
interactions in a single step. (ii) We design two spatial collision inference
strategies, motion collision avoidance and boundary constraint guidance, to
enhance scene fidelity by preventing collisions with human motions and
respecting room layouts. (iii) We devise an automated calibration pipeline to
enhance accuracy and diversity in human-object interactions within the
original dataset, thereby improving the generative quality and diversity of
human-aware scene synthesis.

## Related Works

Human-agnostic Scene Synthesis. The objective of human-agnostic scene
generative methods is to generate plausible and diverse scene layouts without
taking into account human activities (Wang et al. 2018; Zhou, While, and
Kalogerakis 2019; Ritchie, Wang, and Lin 2019; Wang, Yeshwanth, and Nießner
2020; Zhang et al. 2020a, b; Paschalidou et al. 2021; Tang et al. 2023; Liu et
al. 2023; Patil et al. 2023). Early works explore procedural modeling with
grammars (Müller et al. 2006) to place objects in the scenes, heavily relying
on manually designed rules and producing scenes with limited diversity.
Subsequently, graph-based approaches (Zhou, While, and Kalogerakis 2019; Li et
al. 2019; Luo et al. 2020; Gao et al. 2023) have been extensively developed to
represent 3D scenes as scene graphs and capture the underlying structure of
indoor scenes using graph neural networks. Unlike these studies, recent
autoregressive generation methods represent 3D scenes as object sequences and
learn autoregressive models to sequentially predict the next object
conditioned on already generated objects. The representative methods include
SceneFormer (Wang, Yeshwanth, and Nießner 2020), ATISS (Paschalidou et al.
2021), and CLIP-Layout (Liu et al. 2023). However, these autoregressive
methods fail to accurately capture the distribution of the entire object
sequence, which often results in generating overlapping objects in the same
place. More recently, some diffusion-based approaches (Tang et al. 2023; Yang
et al. 2024) have been proposed to ease the approximation of the joint
distribution of multiple objects, benefiting the generation of high-fidelity
scenes. Inspired by these works, we develop a spatially-constrained diffusion
method, SHADE. Unlike previous diffusion-based methods (Tang et al. 2023; Yang
et al. 2024), SHADE learns a diffusion model to capture the joint distribution
of objects conditioned on both human motions and the floor plan. Furthermore,
our SHADE integrates multiple spatial constraints as inference guidance to
improve scene plausibility.

Human-aware Scene Synthesis. This branch of scene synthesis focuses on
producing plausible scenes in which the input human motions can naturally take
place (Qi et al. 2018; Nie et al. 2022; Ye et al. 2022; Yi et al. 2023). To
this end, Pose2Room (Nie et al. 2022) proposes a pose-conditioned generative
model to predict object configurations from human pose trajectories. However,
Pose2Room can only predict contact objects rather than an entire scene. In
contrast, SUMMON (Ye et al. 2022) learns a ContactFormer to generate
affordable objects that contact with humans and employs an autoregressive
model to complete the scene. Similar to (Paschalidou et al. 2021), MIME (Yi et
al. 2023) learn an autoregressive model to sequentially predict object
placements based on the contact information and free space extracted from
human motions and floor plan. Unlike these approaches, SHADE is a diffusion-
based, non-autoregressive method that inherently explores the relationships
between all object attributes to generate more plausible furniture layouts.
Moreover, our method incorporates spatial-constrained guidance into the
generation process, thereby enhancing scene plausibility.

## Method

Figure 2:  Overview of our method. SHADE learns a diffusion model to gradually
clean the noisy scene $\mathbf{x}_{T}$ by simultaneously considering the
contact bounding boxes, free-space mask, floor plan, and time step. During
inference, SHADE applies three spatial collision guidance functions to ensure
the generation of plausible scenes that avoid conflicts with human motions,
room boundaries, as well as prevent object overlap.

The overview of SHADE is presented in Fig. 2. Specifically, our method firstly
inputs the contact bounding boxes and free space mask extracted from given
human motions and the floor plan, and extracts their embeddings with
respective encoders. Conditioned on these embeddings, we learn a scene
diffusion model to capture the joint distribution of multiple objects. This
enables us to generate plausible object configurations by exploring the
relationships between their attributes. During inference, we apply three
spatial collision guidance functions to guide the diffusion model in
generating plausible 3D scenes that avoid conflicts with human motions and
adhere to layout constraints. In the following, we first describe the problem
formulation of our task and then detail our human-aware diffusion model and
inference guidance.

### Problem Formulation

Given input human motions $\mathcal{H}$ and an empty floor plan $\mathcal{F}$,
our goal is to generate a 3D scene $\mathcal{S}$ that can support various
human interactions and movements. Following (Paschalidou et al. 2021; Yi et
al. 2023; Tang et al. 2023), the scene $\mathcal{S}$ is represented as an
unordered set of $N$ objects, denoted as $\mathbf{x}=\\{o_{i}\\}^{N}_{i=1}$.
Each object $o_{i}=\\{k_{i},\ell_{i},s_{i},r_{i}\\}$ consists of a semantic
label $k_{i}\in\mathbb{R}^{K}$ out of $K$ categories, location
$\ell_{i}\in\mathbb{R}^{3}$, size $s_{i}\in\mathbb{R}^{3}$ and orientation
$r_{i}\in\mathbb{R}^{1}$. Based on human-object interaction, these objects
$\mathcal{O}$ can be categorized into two kinds: 1) contact objects
$\mathcal{Q}=\\{q\\}^{L}_{i=1}$, which are in contact with humans, and 2) non-
contact objects $\bar{\mathcal{Q}}=\\{\bar{q}\\}^{N-L}_{i=1}$ without any
interaction with humans. To condition the 3D scene generation with human
motions, we extract both contact humans and free-space humans from the input
motions, following (Yi et al. 2023). Specifically, contact humans indicate the
location and category of contact objects, represented as the collection of
contact boxes $\mathcal{C}=\\{c_{i}\\}^{L}_{i=1}$. Free-space humans define
the walkable area of a room, specifying regions where objects cannot be
placed. This information is represented as a binary free-space mask
$\mathcal{FS}$ by projecting all foot contact points on the floor plan.
Formally, conditioned on the floor plan $\mathcal{F}$, the free-space mask
$\mathcal{FS}$ and all contact humans $\mathcal{C}$, we learn a generative
model to predict the contact objects $\mathcal{Q}$ and the non-contact objects
$\bar{\mathcal{Q}}$, such that they can support all human interactions while
adhering to the constraints imposed by human motions $\mathcal{H}$ and the
floor plan $\mathcal{F}$.

### Human-aware Scene Synthesis

The diffusion model provides a robust framework for scene generation by
learning the joint distribution of multiple objects, which is essential for
creating plausible scenes without overlapping objects. In the following
sections, we detail how we incorporate the diffusion model with all input
humans and the floor plan to generate realistic 3D scenes.

Diffusion and Generative Process. Denote $\mathbf{x}_{0}\sim
q(\mathbf{x}_{0})$ as a clean scene sampled from the training data, we
gradually add Gaussian noise to $\mathbf{x_{0}}$ with a forward diffusion
process $q(\mathbf{x}_{t+1}|\mathbf{x}_{t})$ of length $T$. After $T$
diffusion steps, it approximates Gaussian noise
$\mathbf{x}_{T}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$. To generate scenes
conditioned on human motions and floor plan, our diffusion model learns a
reverse denoising process $p_{\theta}(\mathbf{x}_{t-1}|\mathbf{x}_{t},C)$ to
convert $\mathbf{x}_{T}$ back to $\mathbf{x}_{0}$:

$p_{\theta}(\mathbf{x}_{t-1}|\mathbf{x}_{t},C)=\mathcal{N}(\mathbf{x}_{t-1};\mu_{\theta}(\mathbf{x}_{t},t,C),(1-\alpha_{t})\mathbf{I}),$
(1)

where $\theta$ are the model parameters, $\alpha_{t}$ depends on a pre-defined
variance schedule, $C$ denotes the conditioning signals. As illustrated in
Fig. 2, we select $C=\\{\mathcal{C},\mathcal{F},\mathcal{FS}\\}$ as the
conditioning signals, enabling the diffusion model to generate 3D scenes by
leveraging the information from all contact humans $\mathcal{C}$, the floor
plan $\mathcal{F}$, and the free-space mask $\mathcal{FS}$. According to
$\epsilon-$prediction (Ho, Jain, and Abbeel 2020a),
$\mu_{\theta}(\mathbf{x}_{t},t,C)$ can be re-parameterized as:

$\mu_{\theta}(\mathbf{x}_{t},t,C)=\frac{1}{\sqrt{\alpha_{t}}}\left(\mathbf{x}_{t}-\frac{1-\alpha_{t}}{\sqrt{1-\bar{\alpha}_{t}}}{\bm{\epsilon}_{\theta}(\mathbf{x}_{t},t,C)}\right),$
(2)

where $\bar{\alpha}_{t}=\prod_{i=1}^{t}\alpha_{i}$,
$\bm{\epsilon}_{\theta}(\mathbf{x}_{t},t,C)$ is a denoising network that
predicts the noise applied to $\mathbf{x}_{0}$ from a noisy scene
$\mathbf{x}_{t}$ and the conditions $C$. Finally, we can reconstruct a clean
scene $\mathbf{x_{0}}$ as follows:

$p_{\theta}(\mathbf{x}_{0}|C)=p(\mathbf{x}_{T})\prod_{t=1}^{T}p_{\theta}(\mathbf{x}_{t-1}|\mathbf{x}_{t},C),$
(3)

where $p_{\theta}(\mathbf{x}_{0}|C)$ denotes the probability of scene
$\mathbf{x}_{0}$ conditioned on $C$. Mathematically, we can maximize the
conditional probability $p_{\theta}(\mathbf{x}_{0}|C)$ by training the
denoising network $\bm{\epsilon}_{\theta}(\mathbf{x}_{t},t,C)$ with a
simplified objective (Ho, Jain, and Abbeel 2020a):

$\displaystyle\mathcal{L}_{sim}$
$\displaystyle=\mathbb{E}_{\bm{\epsilon},t,\mathbf{x}_{0}}\left[\left\|\bm{\epsilon}-\bm{\epsilon}_{\theta}(\mathbf{x}_{t},t,C)\right\|^{2}\right]$
(4)
$\displaystyle=\mathbb{E}_{\bm{\epsilon},t,\mathbf{x}_{0}}\left[\left\|\bm{\epsilon}-\bm{\epsilon}_{\theta}(\sqrt{\bar{\alpha}_{t}}\mathbf{x}_{0}+\sqrt{1-\bar{\alpha}_{t}}\bm{\epsilon},t,C)\right\|^{2}\right].$

During scene generation, our diffusion model starts from a Gaussian noise
$\mathbf{x}_{T}$ and gradually predicts a cleaner scene with Eq. 1 and 2. In
each denoising step, the diffusion model generates all objects within the
scene simultaneously, which inherently captures the relationships between all
object attributes to enhance scene plausibility. After $T$ steps, we generate
a plausible scene $\mathbf{x}_{0}$ that can afford various human interactions,
such as touching, sitting, and lying.

### Inference Guidance for Spatial Constraints

In human-aware scene generation, the generated scenes must avoid colliding
with human motions and respect room boundary constraints. To enhance these, we
incorporate inference guidance into the generation process. Specifically, we
introduce two spatial collision guidance functions to improve further the
plausibility of 3D scenes generated by the diffusion model: 1) the motion
collision avoidance function pushes objects back when they collide with human
motions, and 2) the boundary constraint function repositions objects that are
outside the floor plan.

Motion Collision Avoidance. Without directly calculating the mesh
interpenetration between moving humans and objects, we estimate motion
collision scores using the predicted bounding boxes of objects
$\\{\hat{o}_{i}\\}^{N}_{i=1}$ in the scene and free-space mask $\mathcal{FS}$
induced by human movements. Specifically, we project the free-space mask
$\mathcal{FS}$ onto a bird’s-eye view to produce a 2D human motion map, from
which we calculate the 2D transform distance map $\mathcal{M}$. Using this
representation, we measure the motion collision scores by:

$\mathcal{J}_{m}=\sum_{i=1}^{N}\text{SDF}(\hat{o}_{i},\mathcal{M}),$ (5)

where $\text{SDF}(\hat{o}_{i},\mathcal{M})$ queries the 2D distance value
between each object and human motion map. A positive distance indicates a
collision between the object and human motions.

Boundary Constraint. Besides satisfying motion collision constraints, the
generated objects must also meet boundary constraints, meaning all objects
should be located within the given floor plan. To this end, we introduce an
additional boundary constraint function to penalize objects that exceed the
boundaries. Similar to motion collision avoidance, we extract the out-of-
bounds region from the 2D projection of the floor plan $\mathcal{F}$ and
calculate its 2D transform distance map $\mathcal{B}$. The distance that
objects extend beyond the boundaries is then measured by calculating their 2D
distance from $\mathcal{B}$:

$\mathcal{J}_{b}=\sum_{i=1}^{N}\text{SDF}(\hat{o}_{i},\mathcal{B}).$ (6)

Overall Inference Guidance. To generate a more plausible scene that respects
spatial constraints from human motions and room layout, we integrate motion
collision avoidance $\mathcal{J}_{m}$ and boundary constraint
$\mathcal{J}_{b}$ into the inference guidance of our diffusion model. Inspired
by (Yang et al. 2024), we also employ the object collision avoidance
$\mathcal{J}_{o}$ to penalize the collision between objects. Finally, our
overall inference guidance can be formulated as
$\mathcal{J}=\mathcal{J}_{m}+\mathcal{J}_{b}+\mathcal{J}_{o}$, where
$\mathcal{J}_{o}=\sum_{i,j,i\neq j}\mathbf{IoU}({o}_{i},o_{j})$ measures the
objects collision score by calculating the 3D IoU of each object pair.

During generative process, the gradient of $\mathcal{J}$ perturbs the
predicted $\hat{\mathbf{x}}_{0}$ at each denoising step as
$\tilde{\mathbf{x}}_{0}=\hat{\mathbf{x}}_{0}-\gamma\nabla\mathcal{J}(\hat{\mathbf{x}}_{0}),$
where $\gamma$ controls the guidance strength. Then, the predicted mean
$\mathbf{\mu}_{\theta}$ can be computed with the updated clean scene
$\tilde{\mathbf{x}}_{0}$ as (Ho et al. 2022; Rempe et al. 2023). This approach
effectively pushes the generated scenes towards the desired spatial
constraints, ensuring they can avoid colliding with human motions and room
boundaries, and prevent object collision.

## Dataset Calibration of 3D FRONT HUMAN

In this section, we introduce the calibration pipeline designed to tackle two
major data challenges in the 3D FRONT HUMAN dataset (Yi et al. 2023). Our
pipeline begins with translation modification to correct penetrations in
human-object interactions, thereby improving spatial accuracy. Next, it
employs category and orientation augmentation to increase the diversity of
interactions, which enhances model generalization. These techniques are
explained in greater detail below.

Translation Modification. To address incorrect penetrations in human-object
interactions, we adjust the translation parameters of humans to avoid
implausible contact with objects. Specifically, we modify these parameters
within a range of $[-2m,2m]$ centered on the contact object. We then evaluate
the plausibility of the interactions using two indicators: the 3D IoU score
(Zhou et al. 2019) between the bounding boxes of the human and object, and the
human-scene interpenetration error $\mathbf{{E}_{pen}}$ (Shen et al. 2023):

$\mathbf{{E}_{pen}}=\sum_{i=1}^{V}\mathds{1}_{x<0}\left[sdf\left(v_{i},\mathcal{S}\right)\right]\cdot\left|sdf\left(v_{i},\mathcal{S}\right)\right|,$
(7)

where $V$ is the number of human mesh vertices,
$sdf\left(v_{i},\mathcal{S}\right)$ is the signed distance from vertex $v_{i}$
to scene $\mathcal{S}$, and $\mathds{1}_{x<0}[\cdot]$ is an indicator function
that returns 1 when the condition is met, and 0 otherwise. The translation
modification is considered complete when $\mathbf{{E}_{pen}}$ falls below the
safe bound $\sigma_{1}$ and the 3D IoU is higher than $\sigma_{2}$.
Empirically, we set $\sigma_{1}=20$, $\sigma_{2}=90\%$ for sitting and lying
humans, and $\sigma_{2}=50\%$ for touching humans.

Category Augmentation. After refining the implausible human-object
interpenetration via translation modification, we perform category
augmentation to enhance the diversity of human-object interactions. The
motivation behind category augmentation is that people interact with objects
in various ways in real human activities, such as lying on a bed or sitting on
it. To capture these behaviors, we augment the dataset by randomly replacing
half of the contact humans with alternate modes of interaction, thereby
fabricating a refreshed set of human-object interactions. For instance, we
might transform a scene from “lying on the bed” to “sitting on the bed”, as
shown in Fig. 3 b.

Orientation Augmentation. Considering that people might interact with objects
from various angles, we also introduce orientation augmentation to enrich the
dataset. To achieve this, we firstly add random noise to orientation
parameters of contact humans. Next, we adjust the translation parameters of
contact humans until their interactions with objects meet the same criterion
proposed in the translation modification. This ensures that contact humans can
interact with objects in a varied and realistic manner.

Figure 3: Comparison between 3D-FRONT HUMAN (Yi et al. 2023) and our
calibrated dataset. We correct human-object penetrations through translation
modification to improve spatial accuracy. Additionally, we apply category and
orientation augmentation to enhance the diversity in interactions.

To verify the effectiveness of our calibrated pipeline, we report
$\mathbf{E_{pen}}$ and 3D IoU score for 3D-FRONT HUMAN and our calibrated
dataset in Tab. 1. It shows that the calibration pipeline effectively reduces
human-scene interpenetration errors and improves human-object interactions of
3D-FRONT HUMAN across all room types. The qualitative comparisons presented in
Figure 3 also demonstrate that our calibrated dataset provides more plausible
and diverse human-object interactions.

Table 1: Quantitative comparisons between the 3D-FRONT HUMAN (Yi et al. 2023)
and our calibrated dataset on human-scene interaction metrics.

Room Type | Dataset |  $\mathbf{E_{pen}}$ $\downarrow$ | 3D IoU $\uparrow$
---|---|---|---
Bedroom | 3D-FRONT HUMAN (Yi et al. 2023) | 226.97 | 0.75
Our calibrated dataset | 16.01 | 0.94
Living | 3D-FRONT HUMAN (Yi et al. 2023) | 125.49 | 0.77
Our calibrated dataset | 19.50 | 0.94
Dining | 3D-FRONT HUMAN (Yi et al. 2023) | 133.34 | 0.78
Our calibrated dataset | 20.72 | 0.95

## Experiments

Implementations. Following the default settings in (Ho, Jain, and Abbeel
2020b), the forward process variances are set to constants increasing linearly
from $\beta_{1}$=$10^{-4}$ to $\beta_{T}$=$0.02$. Unlike previous studies
(Tang et al. 2023; Yang et al. 2024), we reduce the number of diffusion steps
to $T$=100 from 1000, thereby substantially speeding up the sampling process.
During training, we sample the diffusion step $t$ from a uniform distribution
at each iteration and use the Adam (Kingma and Ba 2014) optimizer with a
learning rate of $1\times 10^{-4}$ and no weight decay. Moreover, we apply
random global rotation augmentation between [0, 360] degrees on the entire
scene, including the floor plane, all objects, all contact humans, and the
free space. Finally, we train SHADE on a Nvidia Tesla A100 GPU for 625K
iterations with a batch size of 128. During inference, we use the DDPM sampler
(Ho, Jain, and Abbeel 2020b) to obtain the object properties and apply
inference guidance in the last five steps.

Network Architecture. As depicted in Fig. 2, we implement SHADE with a
transformer-based architecture composed of 4 DiT blocks with adaLN-Zero
(Peebles and Xie 2022). Each block has a latent dimension of 512, and each
attention layer consists of 8 heads. As input, we extract the layout embedding
from the 3D point clouds of the floor plan $\mathcal{F}$ and the free-space
mask $\mathcal{FS}$ using a PointNet (Qi et al. 2017). Meanwhile, all objects
and contact humans are encoded by a fully connected network (Tang et al.
2023), and the time step $t$ is encoded using a positional embedding (Tancik
et al. 2020). Each object embedding is then treated as an input token and fed
into the denoising network. To ensure the scene generation is aware of
different contact humans and room layout, we sum the embedding vectors of each
contact human, floor plan, and $t$ as conditions for each contact object, from
which we regress the transformation parameters of each adaLN-Zero block. These
parameters are then applied to normalize the embedding vector of the
corresponding contact object. This strategy effectively guides the diffusion
model to generate plausible contact objects for different contact humans. Note
that the generation of non-contact objects only condition on the sum of the
embedding vectors of layout and $t$. Additionally, the self-attention (Vaswani
et al. 2017) layer in each adaLN-Zero block models the spatial relationships
between all objects within the scene, thereby enhancing scene plausibility.

Dataset. We conduct experiments on the calibrated 3D FRONT HUMAN dataset,
which contains a total of 5,689 bedrooms, 2,987 living rooms, and 2,549 dining
rooms. We use 21 object categories for the bedrooms, and 24 for the living
rooms and dining rooms. Following (Yi et al. 2023), for each kind of room, we
split the data into 80% for training, 10% for validation, and 10% for testing.
We train and validate our model on the training and validation sets
respectively, and evaluate it on the test set for each room type. During
inference, we use the 3D-FUTURE dataset (Fu et al. 2021) for object retrieval,
as suggested in (Paschalidou et al. 2021).

Figure 4: Qualitative comparison on the test split in calibrated 3D FRONT HUMAN. Compared with existing state-of-the-art methods, our method generates more plausible scenes that avoid conflict with free-space humans and room boundaries, and present fewer overlapping objects.  Table 2: Quantitative comparison on the test split of the Calibrated 3D FRONT HUMAN dataset. We compare SHADE with MIME and DiffuScene on human-scene interaction score 3D IoU, scene plausibility metrics $\mathbf{Col_{mot}}$, $\mathbf{R_{out}}$, $\mathbf{Col_{obj}}$, and standard perceptual quality scores FID, CKL. The best scores are highlighted in bold, and the second best scores are underlined. Room Type | Method | 3D IoU $\uparrow$ |  $\mathbf{Col_{mot}}$ $\downarrow$ |  $\mathbf{R_{out}}$ $\downarrow$ |  $\mathbf{Col_{obj}}$ $\downarrow$ | FID $\downarrow$ | CKL $\downarrow$
---|---|---|---|---|---|---|---
Bedroom | MIME (Yi et al. 2023) | 0.905 | 0.068 | 0.003 | 0.020 | 37.32 | 0.021
DiffuScene (Tang et al. 2023) | 0.550 | 0.169 | 0.005 | 0.049 | 42.83 | 0.026
SHADE (Ours) | 0.915 | 0.051 | 0.006 | 0.016 | 41.59 | 0.023
Living | MIME (Yi et al. 2023) | 0.899 | 0.041 | 0.002 | 0.060 | 34.25 | 0.013
DiffuScene (Tang et al. 2023) | 0.301 | 0.184 | 0.003 | 0.117 | 38.15 | 0.031
SHADE (Ours) | 0.913 | 0.029 | 0.002 | 0.023 | 35.71 | 0.008
Dining | MIME (Yi et al. 2023) | 0.924 | 0.031 | 0.002 | 0.067 | 34.61 | 0.015
DiffuScene (Tang et al. 2023) | 0.280 | 0.153 | 0.003 | 0.115 | 41.35 | 0.040
SHADE (Ours) | 0.903 | 0.023 | 0.003 | 0.050 | 39.99 | 0.012
Table 3: Quantitative comparison on the PROXD qualitative dataset (Hassan et
al. 2019).

Method | 3D IoU $\uparrow$ |  $\mathbf{Col_{mot}}$ $\downarrow$ |  $\mathbf{R_{out}}$ $\downarrow$ |  $\mathbf{Col_{obj}}$ $\downarrow$
---|---|---|---|---
MIME (Yi et al. 2023) | 0.883 | 0.224 | 0.002 | 0.144
DiffuScene (Tang et al. 2023) | 0.046 | 0.202 | 0.003 | 0.096
SHADE (Ours) | 0.881 | 0.171 | 0.003 | 0.070

Baselines. We compare our method with MIME (Yi et al. 2023) and DiffuScene
(Tang et al. 2023) using their official implementations. MIME is a
transformer-based autoregressive method for human-aware scene generation,
while DiffuScene is a diffusion-based method that learns 3D scene
distributions without conditioning on the floor plan. For a more fair
comparison, we adapt DiffuScene on the 2D-floor plan with the free-space mask
to enhance its perception of human motions. All methods are trained on our
calibrated 3D FRONT HUMAN dataset.

Evaluation Metrics. Following (Yi et al. 2023), we evaluate all methods on the
plausibility of human-object interaction and the realism of the generated
scenes. As suggested in (Yi et al. 2023), we use the 3D IoU score to measure
the plausibility of human-object interaction, by calculating the intersection
ratio between input contact bounding boxes and generated objects.
Additionally, we employ $\mathbf{Col_{mot}}$ (Yi et al. 2023) to evaluate
collision between generated objects and free-space humans, and
$\mathbf{Col_{obj}}$ (Yang et al. 2024) to represent the collision rate
between objects in the generated scene. To measure the violation of the room
layout, we compute the collision rate between generated objects and the areas
outside the floor plan, denoted as $\mathbf{R_{out}}$. Furthermore, following
(Paschalidou et al. 2021), we calculate the Fréchet inception distance (FID)
score and the category KL divergence (CKL) between generated scenes and real
scenes to measure the realism of generated scenes. All these evaluation
experiments are conducted on the test split of the calibrated 3D FRONT HUMAN
dataset.

Figure 5: Ablation on spatial collision guidance functions. The top row shows
the scenes generated without guidance (w/o), with red boxes indicating
constraint violations, while the bottom row shows scenes generated with
guidance (w), where green boxes highlight the improvements achieved by the
proposed guidance.  Table 4: Ablation study on different spatial collision
guidance functions. The best scores are highlighted in bold, and the second
best scores are underlined.

Motion Collision | Room Boundary | Object Collision |  $\mathbf{Col_{mot}}$($\downarrow$) |  $\mathbf{R_{out}}$($\downarrow$) |  $\mathbf{Col_{obj}}$($\downarrow$)
---|---|---|---|---|---
✗ | ✗ | ✗ | 0.082 | 0.003 | 0.020
✓ | ✗ | ✗ | 0.034 | 0.008 | 0.023
✗ | ✓ | ✗ | 0.100 | 0.002 | 0.021
✗ | ✗ | ✓ | 0.084 | 0.004 | 0.008
✓ | ✓ | ✓ | 0.051 | 0.006 | 0.016

Figure 6: Ablation study on varying numbers of free-space and contact humans.
(a) As the number of free-space humans increases, SHADE generates fewer
objects in the scene. (b) Providing more contact humans as input, SHADE
generates more occupied objects to interact with them.

### Human-aware Scene Synthesis

Figure 4 exhibits the generation ability of our method and baselines for
different room types. Since DiffuScene only considers floor plan and free-
space information, it fails to generate appropriate objects to interact with
contact humans. Additionally, DiffuScene also generates objects colliding with
free-space humans. Both MIME and our method can generate reasonable objects to
support various contact humans, such as placing a bed under a sitting human or
a sofa under a lying human. However, MIME still generates objects in free-
space humans or outside the floor plan (see the second column of Fig. 4). In
contrast, our method generates more plausible scenes that can avoid colliding
with free-space humans and room boundaries, as well as prevent object overlap.

These observations are validated by the quantitative comparisons of various
evaluation metrics presented in Table 2. Our method outperforms MIME in 3D IoU
scores for both the bedroom and living room and performs comparably in the
dining room. Furthermore, our method significantly reduces motion collisions
($\mathbf{Col_{mot}}$) and object collisions ($\mathbf{Col_{obj}}$) across all
room types when compared to the baselines. Although MIME achieves better FID
scores, our method consistently shows better CKL scores, indicating better
alignment with the categorical distribution of real data. Overall, these
results demonstrate the effectiveness of our framework.

Following MIME (Yi et al. 2023), we test SHADE on a real dataset of human
motion to evaluate the generalization of our method. We use 5 input motions
from the PROXD (Hassan et al. 2019) dataset and generate 10 scenes for each
motion sequence. The quantitative results of our method and the baselines are
reported in Table 3. Note that all methods are not fine-tuned on the PROXD
dataset. For human-object interaction plausibility, our method achieves a 3D
IoU score of 0.881, outperforming DiffuScene and being comparable to MIME. In
terms of scene plausibility, our method surpasses the other methods on the
motion collision metric $\mathbf{Col_{mot}}$ and the object collision metric
$\mathbf{Col_{obj}}$, indicating that our method can generate more realistic
scenes with fewer human-object and object-object collisions. Figure 1 presents
the qualitative visualization of all models’ generations.

### Ablation Studies

Inference Guidance. We investigate the impact of each spatial collision
guidance function on the bedroom and present the results in Tab. 4. Compared
to scene generation without any guidance function, incorporating the motion
collision avoidance function reduces the $\mathbf{Col_{mot}}$ metric to 0.034,
demonstrating its effectiveness. Similar conclusions can be drawn from the
results of the other two spatial collision guidance functions. It is
noteworthy that these collision-based guidance functions can negatively affect
each other. For example, while the room boundary constraint function improves
the $\mathbf{R_{out}}$ metric, it can degrade the other metrics. This is
reasonable because room boundary guidance pushes objects within the floor
plan, potentially leading lead to increased collisions with free-space humans
($\mathbf{Col_{mot}}$) and other objects ($\mathbf{Col_{obj}}$). To balance
these effects, we integrate all spatial collision functions, thereby achieving
better overall performance in scene plausibility. Fig. 5 provides a
qualitative visualization of the effect of each spatial collision guidance
function. The improvements shown in the second row of Fig. 5 confirm that our
collision-based guidance functions can significantly enhance the plausibility
of 3D scenes.

Number of Input Humans. In Fig. 6, we further investigate the impact of input
humans on scene generation by varying the number of free-space humans and
contact humans provided as input. Our qualitative results indicate that as the
density of free-space humans increases, our method generates fewer objects in
the scenes. Additionally, when given more contact humans, we produce more
occupied objects to interact with them. These findings demonstrate the
flexible scene generation capabilities of our method, influenced by the
varying numbers of input humans.

## Conclusions

We introduced SHADE, a spatially-constrained diffusion model for human-aware
3D scene synthesis. SHADE learns a scene diffusion model that simultaneously
considers all input humans and floor maps to generate plausible furniture
layouts. During scene generation, SHADE integrates the motion collision
avoidance, boundary constraint and object collision avoidance as guidance to
further enhance the scene plausibility. Additionally, we devise an automated
calibration pipeline to improve the spatial accuracy and diversity of human-
object interactions in existing human-aware 3D scene dataset, thereby
enhancing the generation ability of SHADE. The quantitative and qualitative
results showcase promising improvements in synthetic and real-world HSI
datasets, demonstrating the effectiveness of our framework and calibration
pipeline.

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