# Observational Signatures of Circumbinary Discs II: Kinematic Signatures in
Velocity Residuals

Josh Calcino1, Brodie Norfolk2, Daniel J. Price2, Thomas Hilder2, Jessica
Speedie3, Christophe Pinte2,4, Himanshi Garg2, Richard Teague5, Cassandra
Hall6,7, Jochen Stadler8,4
1Department of Astronomy, Tsinghua University, 30 Shuangqing Rd, 100084
Beĳing, China
2School of Physics and Astronomy, Monash University, Vic 3800, Australia
3Department of Physics & Astronomy, University of Victoria, Victoria, BC, V8P
5C2, Canada
4Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France
5Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA
6Department of Physics and Astronomy, The University of Georgia, Athens, GA
30602, USA
7Center for Simulational Physics, The University of Georgia, Athens, GA 30602,
USA
8Laboratoire Lagrange, Université Côte d’Azur, CNRS, Observatoire de la Côte
d’Azur, 06304 Nice, France 4
Contact e-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Kinematic studies of protoplanetary discs are a valuable method for uncovering
hidden companions. In the first paper of this series, we presented five
morphological and kinematic criteria that aid in asserting the binary nature
of a protoplanetary disc. In this work we study the kinematic signatures of
circumbinary discs in the residuals of their velocity maps. We show that
Doppler-flips, spiral arms, eccentric gas motion, fast flows inside of the
cavity, and vortex-like kinematic signatures are commonly observed. Unlike in
the planetary mass companion case, Doppler-flips in circumbinary discs are not
necessarily centred on a companion, and can extend towards the cavity edge. We
then compare the kinematic signatures in our simulations with observations and
see similarities to the Doppler-flip signal in HD 100546 and the vortex-like
kinematic signatures in HD 142527. Our analysis also reveals kinematic
evidence for binarity in several protoplantary disks typically regarded as
circumstellar rather than circumbinary, including AB Aurigae and HD 100546.

###### keywords:

protoplanetary discs — circumstellar matter — methods: numerical —
hydrodynamics

††pubyear: 2023††pagerange: Observational Signatures of Circumbinary Discs II:
Kinematic Signatures in Velocity Residuals–Observational Signatures of
Circumbinary Discs II: Kinematic Signatures in Velocity Residuals

##  1 Introduction

A detailed analysis of the gas kinematics in protoplanetary discs has proven
to be an invaluable tool for understanding the origin of the substructures
often found in them (Pinte et al., 2018, 2019; Casassus & Pérez, 2019; Disk
Dynamics Collaboration et al., 2020; Teague et al., 2021; Pinte et al.,
2023b). Since the velocity field of the gas is a direct product of the forces
acting on it, the gas kinematics are a powerful probe of the phenomena
responsible for the observed substructures (e.g. Dong et al., 2018b; Francis &
van der Marel, 2020; Pinte et al., 2023a; Bae et al., 2023) Different physical
phenomena can produce similar substructures in the gas and dust distribution.
For example, rings and gaps seen in the mm dust continuum can be produced by
planets (e.g. Dipierro et al., 2015), volatile snow lines (e.g. Zhang et al.,
2015), the irradiation instability (Wu & Lithwick 2021; Kutra et al. 2023;
however see Melon Fuksman & Klahr 2022; Pavlyuchenkov et al. 2022), MHD driven
zonal flows and winds (e.g. Suriano et al., 2018), plus many more (for a
review, see Bae et al., 2023). The gas kinematics break this degeneracy as
different models often predict different kinematic signatures (Pinte et al.,
2023b).

Ref. $q$ $a$ (au) $e$ $i$ $\omega$ $M_{\textrm{disc}}$ (M⊙)
$H/R_{\textrm{ref}}$ $R_{\textrm{ref}}$ $R_{\textrm{in}}$ (au)
$R_{\textrm{out}}$ (au) $\alpha_{\textrm{SS}}$ $N_{\textrm{Orbits}}$ Planet
(P) $2.5\times 10^{-3}$ 80 0.0 0.0 0.0 0.010 0.066 100 10 400 $5\times
10^{-3}$ 60 Multiple Planets (MP) $[2.5,1.25]\times 10^{-3}$ [75.6, 130] 0.0
0.0 0.0 0.010 0.066 100 10 400 $5\times 10^{-3}$ 60 Eccentric Planet (EP)
$2.5\times 10^{-3}$ 80 0.4 0.0 0.0 0.010 0.066 100 100 400 $5\times 10^{-3}$
60 No Over-density (NOD) 0.25 40 0.0 0 0 0.010 0.066 100 63 400 $5\times
10^{-3}$ 1100 Over-density (OD) 0.2 30 0.0 0 0 0.005 0.05 45 45 120 $1.5\times
10^{-3}$ 500 Eccentric Companion (EC) 0.1 40 0.4 0 0 0.010 0.066 100 90 400
$5\times 10^{-3}$ 80 Heavy Inclined Companion (HIC) 0.25 40 0.5 30 0 0.010
0.066 100 90 400 $5\times 10^{-3}$ 20 Polar Companion (PC) 0.2 40 0.5 90 90
0.010 0.066 100 90 400 $5\times 10^{-3}$ 60

Table 1: A summary of the initial conditions of the models presented in Paper
I that are used in the present work. Circumbinary disc models are highlighted
in bold font. Note that model OD is taken from Calcino et al. (2019), but the
disc parameters have been rescaled.

Recent works have focused on understanding how forming planets produce
velocity perturbations in the gas (Perez et al., 2015b; Pérez et al., 2018;
Teague et al., 2018; Bollati et al., 2021), developing tools to aid in their
detection and characterisation (Izquierdo et al., 2021; Terry et al., 2021).
One way to examine these perturbations is by subtracting a best-fit Keplerian
disc model (e.g. Pérez et al., 2018; Teague, 2019). After doing-so, the
location of a planet may be revealed by a change in sign of velocity residual
across a narrow azimuthal section of the disk, known as a Doppler-flip (Pérez
et al., 2018; Pinte et al., 2023b). One such Doppler-flip was discovered in HD
100546 by Pérez et al. (2020) and attributed to a $\sim 10$ MJ planetary
companion. However the signature is also consistent with a more massive
companion inside the cavity (Norfolk et al., 2022). If channel maps are
inspected, a perturbation in the iso-velocity curve, known as a “velocity
kink”, may reveal the location of the planet (Perez et al., 2015b; Pinte et
al., 2018).

Although there are several convincing indrect detections of planets using the
gas kinematics (e.g. Pinte et al., 2018, 2019; Izquierdo et al., 2023),
observations also reveal that disc kinematics can be complex (Pinte et al.,
2020; Huang et al., 2021). Planets are not the only potential source of
perturbations in protoplanetary discs. However, not much attention has been
paid to the dynamical effects of other sources. The most well studied is
gravitational instability (GI) (Hall et al., 2020; Longarini et al., 2021;
Veronesi et al., 2021; Terry et al., 2021; Lodato et al., 2023), but the
conditions for GI are only present in a handful of the most well-studied discs
(Paneque-Carreño et al., 2021; Veronesi et al., 2021; Lodato et al., 2023).
Expanding our understanding of the dynamical effects that other disc
substructure producing mechanisms have on the velocity field of the disc can
identify or rule out each mechanism. For example, dust horseshoes could be
vortices induced either in planet-hosting (Zhu et al., 2015) or circumbinary
discs (Rabago et al., 2023), or they could be over-dense lumps in eccentric
discs that lack vortical motion (Ragusa et al., 2017; Calcino et al., 2019;
Ragusa et al., 2020). Continuum observations alone may not be enough to
distinguish between these scenarios. It has been found that the anti-cyclonic
motion inside of a vortex can be detected in the velocity residuals, and
appears as a sign change in the velocity residuals across the major axis of
the vortex (Huang et al., 2018; Robert et al., 2020). Boehler et al. (2021)
claimed a potential detection of such a signature in HD 142527. However it is
known that this is a binary star system (Biller et al., 2012; Nowak et al.,
2024), and it is unclear how the central binary imprints its kinematic
signatures on the cavity.

In Calcino et al. (2023, hereafter Paper I) we showcased some of the kinematic
features expected to arise in circumbinary discs and developed five criteria
to infer binarity. We found three morphological features commonly associated
with circumbinary discs: i) a gas depleted central cavity; ii) spiral arms
inside or outside of the cavity; and iii) velocity kinks close to the central
cavity in the channel maps. We also devised two kinematic criteria which
quantify the degree to which the projected velocity field is perturbed.
Together these morphological and kinematic criteria can be used to distinguish
circumbinary discs from circumstellar discs under the influence of other
effects. In this paper we expand on these results with a more thorough
analysis of the kinematic features seen in circumbinary discs. We study the
substructures that arise in the velocity residuals of simulated circumbinary
discs. We find striking similarities between our simulations and observations
of well studied transitional discs.

We structure the paper as follows: In Section 2 we briefly present our
hydrodynamical simulations and synthetic observations, and introduce the
observational data used in this paper. In Section 3 we study the kinematics
and residuals of simulated circumbinary discs, identifying fast flows inside
the cavity and Doppler-flips as features that can appear. In Section 4 we then
demonstrate that these features can explain two examples of kinematic
residuals (vortex-like kinematics and Doppler-flips) seen in HD 142527 and HD
100546, and show fast flows are seen inside the cavities of several transition
discs. We summarise our findings in Section 5.

##  2 Methods and Observational Data

###  2.1 SPH Simulations and Synthetic Observations

We used the smoothed particle hydrodynamics (SPH; Monaghan 1992) code phantom
(Price et al., 2018a). We refer the reader to Paper I for a detailed
description of the gas-only SPH models we use in the present work, however we
provide a brief description here. A summary of the model parameters is
provided in Table 1. Our gas discs are initialised with $5\times 10^{6}$ SPH
particles following a surface density profile $\Sigma(R)\propto R^{-p}$ for
$R_{\textrm{in}}<R<R_{\textrm{out}}$, where we set $p=1$ in all models. We
assume a locally isothermal equation of state with a temperature profile
$T(R)\propto R^{-2q_{T}}$, and $q_{T}=0.25$ in all models. We set the aspect
ratio of the disc $H/R_{\textrm{ref}}$ at $R_{\textrm{ref}}$, with specific
values listed in Table 1. The central stars and companions are modelled as
sink particles (Bate et al., 1995), where for every simulation we assume a
primary companion mass of 2 M⊙. The SPH artificial viscosity
$\alpha_{\textrm{AV}}$ is used to produce a Shakura & Sunyaev (1973) alpha
viscosity $\alpha_{\textrm{SS}}$ following Lodato & Price (2010), where the
value for each model is specified in Table 1. The simulations presented in
this work are chosen to reflect a range of binary configurations and disc
morphologies without being exhaustive. For example, circular binaries can pump
large eccentricities in circumbinary discs, while eccentric binaries can
produce comparatively less eccentric discs (Hirsh et al., 2020). Furthermore,
circumbinary discs that are co-planar, arbitrarily mis-aligned (Nowak et al.,
2024), or polar aligned (Kennedy et al., 2019), are both theoretically and
observationally supported configurations.

Our synthetic observations are the same as shown in Paper I. We used the Monte
Carlo radiative transfer code mcfost (Pinte et al., 2006; Pinte et al., 2009).
Since our simulations did not include dust, we assumed a power-law grain size
distribution $dn/ds\propto s^{-3.5}$ for $0.03~{}\mu$m $\leq~{}s~{}\leq~{}1$
mm with the grains well-coupled to the gas. We assumed that
$T_{\textrm{dust}}=T_{\textrm{gas}}$ and all molecules are at local
thermodynamical equilibrium. We only show synthetic 12CO (3-2) line
observations in the work and adopted a CO ratio of 12CO/H${}_{2}=~{}10^{-4}$.
The CO abundance is affected by photodissociation and freeze out ($T=20$ K)
(Pinte et al., 2018). Changing the CO line used in our analysis does not
significantly alter the kinematic signatures in the velocity residuals we
later study. The primary star in each simulation has an effective temperature
$T_{\textrm{eff}}=8000$ K and radius $R=1.8\ \textrm{R}_{\odot}$, giving a
blackbody luminosity $\sim 12\ \textrm{L}_{\odot}$. The temperature and radius
of the companions are computed from their mass listed in Table 1 using the
stellar and planetary tracks from Siess et al. (2000) and Allard et al.
(2001). A disc inclination of $i=30^{\circ}$, a disc position angle of
PA$=0^{\circ}$ and a distance of 100 pc were assumed.

The CO cubes from mcfost were then processed to mimic the finite resolution
and sensitivity capabilities of ALMA using the Python package pymcfost. The
final cubes had a spectral resolution of 250 ms-1, and were convolved with a
0.15$\arcsec$ Gaussian beam. Artificial noise was added with a flux of
$F_{\textrm{Noise}}=2.5$ mJy. Integrated intensity and velocity maps were made
from the cubes using bettermoments (Teague & Foreman-Mackey, 2018) where a cut
of $5\times$ the RMS noise level was applied. We fit the velocity maps using
eddy (Teague, 2019). Since the disc flaring is relatively small in our
simulations, we fit Keplerian disc models that do not account for the disc
flaring. The Keplerian disc models assumed a single central massive body. We
did not attempt to fit more complication rotation profiles (e.g. Ragusa et
al., 2024).

###  2.2 Observational Data

Object | Line | ALMA Project ID | Beam ($\arcsec$) |  $\Delta v$ (km s-1) | Peak SNR${}^{(\textrm{a})}$ |  $r_{\textrm{cavity}}$ [″] | Cube Source${}^{(\textrm{b})}$
---|---|---|---|---|---|---|---
HD 163296 |  12CO (2-1) | 2018.1.01055.L | 0.15$\times$0.15 | 0.2 | 115 | - | P/PC
MWC 480 |  12CO (2-1) | 2018.1.01055.L | 0.15$\times$0.15 | 0.2 | 70 | - | P/PC
GM Aur |  12CO (2-1) | 2018.1.01055.L | 0.15$\times$0.15 | 0.2 | 34 | 0.23 | P/PC
IM Lupi |  12CO (2-1) | 2018.1.01055.L | 0.15$\times$0.15 | 0.2 | 42 | - | P/PC
AS 209 |  12CO (2-1) | 2018.1.01055.L | 0.15$\times$0.15 | 0.2 | 69 | - | P/PC
HD 97048 |  13CO (3-2) | 2016.1.00826.S | 0.11$\times$0.07 | 0.12 | 35 | 0.24 | P/PC
HD 169142 |  12CO (2-1) | 2015.1.00490.S | 0.073$\times$0.067 | 0.167 | 36 | 0.23 | P/PC
|  | 2016.1.00344.S |  |  |  |  |
TW Hya |  12CO (2-1) | 2013.1.00387.S | 0.19$\times$0.17 | 0.04 | 275 | - | P/PC
|  | 2018.A.00021.S |  |  |  |  |
GG Tau |  12CO (2-1) | 2018.1.00532.S | 0.34$\times$0.27 | 0.08 | 88 | 1.6 | A
HD 142527 |  13CO (2-1) | 2015.1.01353.S | 0.35$\times$0.31 | 0.084 | 39 | 1.18 | P/PC
HD 100546 |  12CO (2-1) | 2016.1.00344.S | 0.08$\times$0.06 | 0.5 | 29 | 0.33 | P/PC
CQ Tau |  12CO (2-1) | 2013.1.00498.S | 0.12$\times$0.10 | 0.5 | 30 | 0.36 | P/PC
|  | 2016.A.00026.S |  |  |  |  |
|  | 2017.1.01404.S |  |  |  |  |
AB Aurigae |  12CO (2-1) | 2021.1.00690.S | 0.24$\times$0.17 | 0.042 | 86 | 0.98 | P/PC
J1604 |  12CO (2-1) | 2018.1.01255.S | 0.18$\times$0.15 | 0.1 | 84 | 0.56 | P/PC
LkCa 15 |  12CO (3-2) | 2018.1.00350.S | 0.07$\times$0.05 | 0.874 | 50 | 0.43 | P/PC
CS Cha |  12CO (3-2) | 2017.1.00969.S | 0.1$\times$0.09 | 0.25 | 59 | 0.25 | P/PC
Table 2: Summary of the observations used in this work.
${}^{(\textrm{a})}$Peak SNR refers to the peak signal-to-noise ratio seen in
the channels of the cube. ${}^{(\textrm{b})}$P/PC: Reimaged data cube. Public
data or obtained via private communication, A: Archival data product. HD
163296, MWC 480, GM Aur, IM Lupi, AS 209 (Öberg et al., 2021), HD 97048 (Pinte
et al., 2019), HD 169142 (Garg et al., 2022), TW Hya (Teague et al., 2022), GG
Tau (Phuong et al., 2020), HD 142527 (Garg et al., 2021), HD 100546 (Law et
al., 2022), CQ Tau (Wölfer et al., 2021), AB Aurigae (Speedie et al. in
press), J1604 (Stadler et al., 2023), LkCa 15 (this work), CS Cha (Kurtovic et
al., 2022).

We utilised ALMA Band 6 and Band 7 archival $\rm{}^{12}CO$ and $\rm{}^{13}CO$
line data. The majority of our data was obtained either via private
communication as self-calibrated data products, or via the ALMA archive,
calibrated using the CASA pipeline for the appropriate ALMA cycle. When
selecting our sources we did not aim to obtain a complete or unbiased sample
of discs, as we did not attempt to constrain statistical properties of the
disc population. Our sample consists of discs which have been previously
studied in other works (e.g. Francis & van der Marel, 2020; Wölfer et al.,
2023). Our sampled is based off the transition disc sampled compiled by Wölfer
et al. (2023), excluding sources where the noise levels are too high, or the
beam too large, to resolve the kinematics inside the cavity. We also included
the MAPS sample (Öberg et al., 2021). In a future work we will provide a
detailed study of the exoALMA sample. The specific details of the observations
are summarised in Table 2. We imaged/re-imaged our data using the tclean task
in CASA, with the Multiscale Clean deconvolver (Cornwell, 2008) with Briggs
weighting, and applied JvM (Jorsater & van Moorsel, 1995) and primary beam
correction. We generated total intensity and velocity maps using bettermoments
(Teague & Foreman-Mackey, 2018). The total intensity maps were used to obtain
the disc radius. We used both the intensity weighted average velocity (Moment
1) and quadratic methods to produce the velocity maps in this paper. When
generating our images we applied noise cuts of $5\times$ the RMS noise for all
cubes.

##  3 Kinematic Signatures in Velocity Residuals

Figure 1: The surface density (top row) with the residuals from Keplerian
rotation generated from CO (bottom row) for all our circumbinary disc models.
Significant deviations from Keplerian rotation is observed in all models.
Spiral arms and Doppler flips are common inside and co-located with the
cavity. Since we used a position angle of PA$\ =0^{\circ}$ to produce our
synthetic observations, the disc major axis lies along the $x$-axis. The
velocity residuals have been deprojected using our fixed viewing inclination
angle of $i=30^{\circ}$. Figure 2: Binaries produce Doppler flips: Deprojected
residuals from Keplerian rotation obtained using eddy for all circumbinary
models listed in Table 1 for four different viewing angles (left to right
columns). The horizontal and vertical grid lines are along the major and minor
axis of the disc, respectively. The dash-dot circle shows the approximate
radius of the cavity, obtained from the peak in the azimuthally averaged gas
surface density. Doppler flips and spiral arms are ubiquitous in and around
the cavity. Several prominent Doppler flips are marked with arrows. Figure 3:
Deprojected residuals from Keplerian rotation obtained from the velocity map
constructed from 12CO (3-2) (top row), 13CO (3-2) (middle row), and C18O (3-2)
lines. Each column is the same simulation but seen from a different viewing
angle.  Figure 4: The deprojected residuals from Keplerian rotation for model
OD with $\phi=0^{\circ}$ with three different values of the inner radius for
the fitting with eddy. The residuals can change sign depending on the
background model that is subtracted. The large amplitude residuals in the
inner regions are not sensitive to the model subtracted, while the lower
amplitude residuals in the outer disc are.

###  3.1 Doppler-Flips

We define a Doppler-flip as a change in sign of velocity across a narrow
azimuthal section of the disk (Pinte et al., 2023b). Planet-induced Doppler-
flips are produced due to a change in sign of the non-Keplerian gas motion and
are centred on the planet (e.g. see Pérez et al., 2018; Bollati et al., 2021;
Calcino et al., 2022). Both azimuthal and radial velocities deviations are
negative in front of the planet and positive behind it. Since there are both
azimuthal and radial velocity perturbations producing the sign-flip, a planet
at any azimuthal position in the disc can produce an observable Doppler-flip.
However, since the radial velocity perturbation dominates (Rafikov, 2002;
Calcino et al., 2022), the Doppler-flip should be stronger along the disc
minor axis than along the major axis.

Figure 1 shows the integrated gas surface density from the circumbinary disc
simulations and the velocity residuals from their velocity maps, obtained from
12CO (3-2) line emission using the Moment 1 method. We identify similar
signatures to planet induced Doppler-flips in the velocity residuals of our
models. In contrast to the planet induced Doppler-flips, those seen in
circumbinary discs are not centred on the companion and can be a significant
distance from it. Supersonic radial and azimuthal velocity perturbations are
produced as the companion launches spiral arms and interacts with gas trying
to flow inside and fill the cavity. They can also have sudden changes in sign
as both the primary and secondary stars launch spiral arms into slower moving
material. These large perturbations can persist outside of the gas depleted
central cavity and be spatially co-located with the peak in the background gas
surface density, which is where we expect large dust grains to accumulate.

Figure 2 shows the velocity residuals for our circumbinary disc models at
different viewing angles (labelled $\phi$). Sudden changes in the sign of the
velocity residuals — Doppler-flips — are abundant inside and around the
cavity. Arrows in Figure 2 indicate several prominent Doppler-flips. We also
mark the approximate radial location of the gas pressure maximum around the
cavity with a dot-dash circle. The location of the Doppler-flips relative to
the cavity also suggests they could be co-located with other features
attributed to circumbinary discs, such as dust rings or dust asymmetries.
Whether or not a Doppler-flip is seen can depend on the disc model that is
subtracted. In Figure 4 we show the velocity residuals of model OD with three
different Keplerian models which were obtained by changing the minimum radius,
$r_{\textrm{min}}$, that is used in the fit with eddy. The disc model in eddy
assumes the gas is on a circular Keplerian orbit around a single star.
Although the high amplitude Doppler-flips (shown with an arrow in all three
columns) do not change sign, lower amplitude residuals, particularly in the
outer disc (bottom right of right-most panel), can change sign. In principle
this is not just an issue when fitting the velocity map of known circumbinary
discs, but also for single stars (e.g. Teague et al., 2022). By their very
definition velocity residuals are model dependent, and the appearance of a
Doppler-flip is therefore also model dependent. However high amplitude
Doppler-flips are less dependent on the Keplerian model that is subtracted, as
shown in Figure 4.

These results demonstrate that the appearance of Doppler-flips, particularly
low velocity (i.e. order few hundred meters per second) are sensitive to the
disc model that is subtracted. Since we assume a Keplerian background model,
large residuals should be expected. However what is notable is that the large
residuals can still be visible in proximity to the cavity edge and gas
pressure maxima. For example, in the HIC model (fourth row, first column of
Figure 2) a Doppler-flip on the order of several hundred meters per second is
seen just interior to the gas pressure maxima at a radial location exterior to
the binary orbit.

The Doppler-flips we observe in our simulations are also sensitive to the CO
isotopologue they are observed with, as shown in Figure 3. In Figure 9 of
Paper I, using the same simulation we use here, we showed how velocity kinks
seen in the channel maps also depend on the CO isotopologue. In that Figure,
the apparent amplitude of the kink decreases due to the more highly perturbed
material responsible for the kink becomes unobservable due to the lower
abundance. The same effect results in the Doppler-flips becoming less apparent
in the residual maps of Figure 3.

The ubiquity of Doppler-flip features in a circumbinary disc suggest that they
should not be used as an indicator of planet-disc interactions, particularly
when they are in proximity of a gas and dust depleted cavity in a transition
disc. Contrary to those generated by planets, Doppler-flips in CBDs are not
necessarily directly associated with a companion, but rather they are a result
of spiral structure exterior to the binary orbit. The structure of the
Doppler-flips we observe are reminiscent of the one in HD 100546 reported by
Casassus & Pérez (2019), which we comment more on in Section 4.2.

###  3.2 Spiral Arms: The Origin of Doppler-Flips

Figure 5: The surface density (top row) with deprojected residuals from
Keplerian rotation generated from CO (bottom row) of models over-density (OD)
and no over-density (NOD). The over-density produces a Doppler-flip signature
across it’s major axis. There is also additional spiral structure through the
disc with OD than in no OD model, which has a very similar cavity size and
shape as model OD. These additional spiral are arising due to the over-density
feature, and are apparent in the velocity residuals.

Circumbinary discs are rich with spiral structure. These spirals are visible
in the velocity residuals. Planet-induced Lindblad spirals appear mostly as
radial perturbations and the radial velocity changes sign across the spiral
(Bollati et al., 2021; Calcino et al., 2022). The spiral arms induced by
stellar mass companions produce larger perturbations in the velocity field of
the gas than their planetary counterparts, although they are still mostly
generated by Lindblad resonances. The spiral structure is most abundant near
and inside the cavity. Since this area is also typically associated with
(sub-)mm dust continuum emission, spiral structures may also be imprinted on
the dust continuum emission. Planet induced spiral wakes are visible in the mm
dust continuum emission if the dust and gas are well coupled (e.g. Speedie et
al., 2022; Verrios et al., 2022). Poblete et al. (2019) showed that marginally
decoupled dust grains can produce dusty clumps around the central cavity. It
is plausible that the binary induced spiral structure may be imprinted in the
mm dust continuum if the grains are sufficiently coupled to the gas.

In Paper I we showed that spiral structure in and around the central cavity of
a CBD produces substantial velocity perturbations away from Keplerian
rotation. These perturbations are manifested as large residuals after
subtracting the best fit Keplerian disc model, and occasionally produce the
Doppler-flips we discussed in the previous section. Given the causal link
between spiral arms and Doppler-flips, we should then expect that a Doppler-
flip observed on the edge of a cavity, produced by an inner binary, should be
accompanied by spiral structure.

Spiral structure not only originates directly from the binary interacting with
the disc through Lindblad resonances, but also from the over-density that can
develop. In the left most column of Figure 5 we show our model with an over-
density (model OD) in integrated surface density with the corresponding
velocity residuals below. The over-density is highlighted by white and black
dashed lines in the surface density and velocity residuals, respectively.
There is additional spiral structure through the disc that is not seen in the
no over-density (NOD) model, shown in the right column, which has a very
similar cavity size and shape as model OD. The formation of the over-density
is likely related to the growth of the eccentric cavity, where a disc
initialised with a smaller cavity leads to a more prominent over-density
developing (see Section 4.3 of Ragusa et al., 2020, for more details). Thus
the spirals are a direct result of the over-density, and not due to the
eccentric cavity. These spirals propagate far from the cavity, and are also
apparent in the velocity residuals. Since our model does not include gas self-
gravity, the spiral structure produced from the over-density is not due to the
gravitational potential of the over-density exciting spiral wakes (e.g. see
Zhu & Baruteau, 2016). We note that a similar trailing spiral is seen in the
CBD simulations of Rabago et al. (2023). The spiral from the over-density is
generated by hydrodynamical effects: the over-density is a localised region of
higher density and pressure which perturbs the surrounding disc material as it
orbits, resulting in the additional spiral structures. We show in Section 4.1
that the over-density might be responsible for the extended spiral structure
seen in HD 142527.

###  3.3 Eccentric Cavity

In three of our models the gas orbiting around the cavity has a non-negligible
eccentricity which appears in the velocity residuals. These are models over-
density (OD), no over-density (NOD), and eccentric companion (EC). Large red
and blueshifted residuals become more apparent when the eccentricity vector of
the disk is pointing perpendicular to the line of sight, which is the case
when $\phi=90^{\circ}$ and $\phi=270^{\circ}$ in Figure 2.

The residuals from eccentric discs arise due to the sub and super-Keplerian
motion that occur at apoapsis and periapsis, respectively. When fitting a
velocity map is it common to mask the central regions (e.g. Teague et al.,
2021, 2022). For a disc with a cavity and substantial deviations from
Keplerian rotation, these non-Keplerian motions are also often masked out
(e.g. Boehler et al., 2021; Wölfer et al., 2021; Norfolk et al., 2022). As we
show in Figure 4, the final residual map that is obtained depends on the model
that is subtracted, which in our eccentric disc models is sensitive to the
minimum radius assumed. This issue could be alleviated by fitting an eccentric
disc rotation profile to the observations (e.g. Kuo et al., 2022; Yang et al.,
2023). For a circumbinary discs, we expect that the disc eccentricity
decreases as a function of disc radius, and hence only the regions close to
the cavity may be highly eccentric.

Aside from the kinematics (Kuo et al., 2022), an eccentric cavity might also
be inferred from the spatially resolved distribution of gas (through line
emission observations) and dust (through scattered light or thermal emission
observations; Dong et al. 2018a; Yang et al. 2023). If the location of the
primary star can be determined, deprojecting the spatially resolved
observations can also lead to constraints on the disc eccentricity (Yang et
al., 2023).

###  3.4 Fast Flows Inside the Cavity

Figure 6: The rotation curve along the disc major axis of each model presented
in Table 1 (blue lines) with the expected Keplerian rotation of the
unperturbed disc (orange curve). The vertical red lines mark the location of
the edge of the cavity as seen with C18O (Figure 5 of Paper I). Minor
deviations from Keplerian rotation are seen in the Planet and Multiple Planets
models, mostly due to the assumptions used to generated the Keplerian rotation
curve. Modest deviations from Keplerian rotation are seen inside the cavity of
the Eccentric Planet model, while in most of the circumbinary disc models
several km/s deviations are seen. Since radial velocities are largely
perpendicular to the line of sight along the major axis, the deviations from
Keplerian rotation are mostly due to azimuthal perturbations. The black
horizontal line represents the size of the beam major axis.

As we showed in Paper I, and in Figure 1, both radial and azimuthal velocity
perturbations increase inside the cavity of a circumbinary disc. For the
inclined companion models, non-negligible vertical motion is also observed.
After subtracting a best-fit Keplerian rotation field, these fast flows are
obvious in the residuals (Figure 1) and span the entire width of the cavity.
In some instances, the residuals change sign across the major axis of the disc
(e.g. first column of model NOD in Figure 2), indicating mostly radial flows
(e.g. Teague et al., 2019) (e.g. model HIC), while in other instances a
superposition of azimuthal and radial velocities lead to no obvious sign
change across either the major or minor axes (e.g. multiple viewing angles of
models OD and NOD).

To further explore how these fast flows appear in observations, Figure 6 shows
the velocity along the major axis of the disc neglecting any disc flaring with
the blue curves, while the orange curves show the expected mid-plane Keplerian
rotation neglecting any disc flaring and gas pressure support. Along the major
axis the radial velocity component of a disc is negligible since the major
velocity component is perpendicular to the line of sight. For the planet
simulations (model P and MP), the deviations from the estimated Keplerian
rotation are small. In contrast, the eccentric planet model and the
circumbinary disc models show larger deviations. These perturbations span
several beams and hence are not localised. They also exceed estimates of the
local sound speed by a factor of several. Highly perturbed rotations curves
inside of a cavity can thus indicate that the disc is circumbinary. Similar
perturbations are seen in the rotation curve and residuals inside of the
cavity of GG Tau and AB Aurigae (Rivière-Marichalar et al. 2019; see also
Section 4.3).

###  3.5 Vortex-like Kinematic Signatures

Figure 7: The surface density (top row) moment 1 (top row) with deprojected
residuals from Keplerian rotation generated from CO for model OD at different
time snapshots. Significant deviations from Keplerian rotation is observed in
all models. Spiral arms and Doppler flips are common inside and co-located
with the cavity.

Figure 7 shows the surface density and velocity residuals of the over-density
model for four different snapshots which trace the orbital phase of the over-
density. The over-dense feature arises as the eccentric cavity grows around
the binary (Ragusa et al., 2020), and is distinct from a vortex as it lacks
vortical motion (Ragusa et al., 2017). Since it orbits around the cavity at
the Keplerian frequency, it is also not a “traffic-jam” of dust grains at the
apocenter of an eccentric disc (Ataiee et al., 2013). In Paper I we showed
that the over-dense feature orbiting a circumbinary disc also shows a change
in velocity across the major axis of the over-density. Since the residuals are
dominated by excess velocities arising from the eccentric gas motion around
the cavity, the signature from the over-dense feature can be difficult to
distinguish.

In left column of Figure 5 we see that this change in velocity is apparent in
the velocity residual. The over-density is outlined with a dashed line in the
integrated surface density and velocity residual plots. A change in sign
across the major axis of the over-density is apparent. The inner binary
clearly complicates any analysis of the kinematics. This fact demonstrates
that even with high spectral and spatial resolution observations,
disentangling the velocity signature of the over-dense feature, or ‘vortex’ as
in Boehler et al. (2021), is difficult.

##  4 Applications to Observed Kinematic Features

###  4.1 HD 142527

Figure 8: The moment 1 (left column), moment 1 residuals from a best-fit
Keplerian disc model (middle column), and the peak intensity (right column) of
our over-density model (top row) and HD 142527 (bottom row). The over-dense
feature can reproduce the spiral structure seen in HD 142527 in both peak
intensity and in the velocity residuals. The location of the over-dense
feature is similar in both the model and observations. A Doppler flip across
the over-density is reproduced in our model.

HD 142527 is a well studied binary system containing a 0.1-0.4 M⊙ companion
(Biller et al., 2012; Close et al., 2014; Lacour et al., 2016; Christiaens et
al., 2018). It also displays non-Keplerian gas flows (Casassus et al., 2015;
Garg et al., 2021) inside of its $\sim$140 au cavity (Casassus et al., 2013;
Avenhaus et al., 2017).

This binary was subject to the study by Price et al. (2018b), where it was
shown that many of the morphological features in this disc can be explained by
the observed companion, assuming a rather large binary semi-major axis of
$\sim 30-40$ au. Recently, Nowak et al. (2024) constrained the semi-major axis
of HD 142527B to $\approx 10$ au, which is too small to completely explain the
observed cavity size. Nevertheless, spiral arms have been observed in the
outer disk in CO integrated emission and peak intensity (Christiaens et al.,
2014; Garg et al., 2021). They are also seen after subtraction of a best-fit
Keplerian disk model by Boehler et al. (2021), along with kinematic signatures
hypothesised to originate from anti-cyclonic motion (which we refer to as
“vortical motion”) inside the dust trap (Casassus et al., 2013; Ohashi, 2008).
We have shown that such kinematic features can also be seen in a circumbinary
over-density, and may explain both this feature and the spiral arms.

To demonstrate this, we rescale our model OD such that the primary star has a
mass of 1.8 M⊙, consistent with the mass of HD 142527 (Fukagawa et al., 2006).
The length is rescaled by a factor of 3.8 from the original model presented in
Calcino et al. (2019) such that the total diameter of the cavity along the
major axis is $\sim 280$ au. When performing our radiative transfer
calculations we closely follow the prescription summarised in Section 2.1 with
a few differences. We set the inclination to $i=27^{\circ}$ and position angle
$\textrm{PA}=-20^{\circ}$ (Avenhaus et al., 2014; Perez et al., 2015a). The
temperature and radius of the central star are set to 6500 K and 2.8 R⊙
(Mendigutía et al., 2014). For the companion we assume a temperature of $3000$
K and radius of 0.9 R⊙ (Lacour et al., 2016). We also scale the total gas mass
of our simulation to be consistent with the gas surface density estimates from
Garg et al. (2021).

Figure 9: The velocity map produced with the quadratic method (left column)
and the residuals from a best fit rotation profile (right column) of our
eccentric disc model (top row) and HD 100546 (bottom row). Our eccentric disc
model reproduces the kink in the velocity map and the excess in velocity
residuals around the cavity of the observations. We produce the Doppler flip
signature with our eccentric companion model.

We obtained the ALMA 13CO data (2015.1.01353.S, PI Christiaens) used in Garg
et al. (2021) and Boehler et al. (2021). We show the velocity map, velocity
map residuals, and 13CO peak intensity of this data and our model in Figure 8.
Firstly, we see that our model recreates the observed spiral arms in the outer
region of the disc found by (Christiaens et al., 2014) and further studied by
(Garg et al., 2021). These spirals are associated with perturbations in the
velocity field of the disc (see Figures 1 and 4 in Paper I) and appear in our
velocity residuals. These are also seen in the data of HD 142527, as reported
by Boehler et al. (2021). Our model is also able to recreate the Doppler-flip
co-located with the over-density. Boehler et al. (2021) suggest this feature
is arising due to vortical motion inside the dust trap in HD 142527, however
our results show that such a feature can be create without any vortical
motion. Indeed, while analysing our model we found that Doppler-flips close to
the cavity edge are exceptionally common, with some co-located with the over-
density while others are not. This should not be too surprising given the
complex kinematics seen in our simulations in Figures 1 and 3 of Paper I.

Recently, Rabago et al. (2023) studied misaligned circumbinary discs using the
grid code Athena++, allowing them to have fine control over the disc
viscosity. Most previous studies employ smoothed particle hydrodynamics (e.g.
Price et al., 2018b), where it is more difficult to study circumbinary discs
with low ($\alpha<10^{-4}$) viscosity. The results from Rabago et al. (2023)
mostly agree with previous SPH results regarding disc breaking, warping, and
cavity size. Interestingly in their $\alpha=10^{-5}$ simulation a long-lived
vortex is formed in the circumbinary disc. The anti-cyclonic motion inside the
vortex is distinct from the mostly solid-body motion of the lump seen in SPH
simulations. However it is unclear whether it would be possible to distinguish
between these two scenarios using disc kinematics alone as the velocity
residuals of a circumbinary disc are already complicated. We note that the
simulations by Rabago et al. (2023) also cannot explain the observed cavity
size given the $\sim 10$ au semi-major axis of HD 142527B. A low disc
viscosity alone cannot explain the large cavity size. Another companion inside
the cavity, or including additional physics in the simulation, may resolve
this issue.

We summarise that the spiral arms in CO integrated emission and peak
intensity, along with their associated kinematic signatures, may directly
arise from the horseshoe feature in HD 142527. It is unclear whether the
kinematics of the horseshoe feature can be interpreted as a vortex in a
circumbinary disc, as they are also consistent with an over-dense feature
lacking vortical motion. In light of the recent results of Nowak et al.
(2024), it is not clear if there is a direct causal link between the dust
over-density and the inner binary in HD 142527.

###  4.2 HD 100546

The Herbig Ae/Be star HD 100546 is surrounded by a transition disc which
displays all of the morphological features we can attribute to a circumbinary
disc. Firstly, it displays a cavity of radius $\sim 20$ au in the $\sim$mm
dust continuum which extends out to $\sim 40$ au (Wright et al., 2015; Pineda
et al., 2019; Pérez et al., 2020; Norfolk et al., 2021), with a faint ring
located at 200 au (Walsh et al., 2014; Fedele et al., 2021). The most central
dust continuum ring also shows an asymmetry, which in the higher resolution
observations of Pérez et al. (2020) resemble tightly wound spiral arms. A
cavity is also present in the 12CO, implying it is heavily depleted in not
just large dust grains, but also gas (Pineda et al., 2019; Pérez et al.,
2020). Spiral arms are also observed in scattered light close to the cavity
and on larger scales at the edge of the protoplanetary disc (Sissa et al.,
2018; Fedele et al., 2021). Finally, velocity kinks are seen in the 12CO iso-
velocity curves outside of and on the edge of the cavity (Pérez et al., 2020),
along with a Doppler-flip on top of the $\sim$mm continuum ring (Casassus &
Pérez, 2019). Kinematic perturbations are also observed inside the cavity
around the $\sim$mm continuum emission associated with the central star (Pérez
et al., 2020). Casassus & Pérez (2019) and Pérez et al. (2020) attributed
these kinematic signatures to a 5-10 M${}_{\textrm{J}}$ planet, and suggested
the Doppler-flip on top of the continuum emission results from gas flows close
to the planet. This scenario is in tension with the position of the continuum
emission; such a large planet will carve a gap and produce a pressure maximum
beyond its orbital location. Dust grains will accumulate inside this pressure
maximum and thus we should reasonably expect a Doppler-flip signature to be
seen well inside the dust cavity, and not on top of the dust continuum
emission.

Figure 1 of Casassus & Pérez (2019) shows the deprojected continuum emission
with the Doppler-flip superimposed. From this perspective, the Doppler-flip
looks to be associated with the spiral structures in the continuum ring.
Several of our circumbinary disc models do show sharp changes in gas velocity
associated with spiral arms on the cavity edge, potentially explaining this
feature in HD 100546. To demonstrate this we obtained the CO J=2–1 from Pérez
et al. (2020), made publicly available from Law et al. (2022). We also
reproduced our eccentric companion model with a new simulation with a binary
semi-major of 10 au, and eccentricity of 0.4. We produced synthetic CO J=2-1
observations assuming stellar parameters similar to those in HD 100546
($T=9750$ K, M${}_{\star}=2$ M⊙ Vioque et al., 2018).

We produce velocity maps using the quadratic method (Teague et al., 2018) and
compute their residuals using eddy. These are shown in Figure 9 for our model
(top row) and the observations (bottom row). As indicated by the dashed black
line, which marks the $v_{\textrm{los}}=0$ km/s iso-velocity curve, the
velocity kink seen in the channel maps (see Pérez et al., 2020; Norfolk et
al., 2022) also appears in the velocity map. Aside from the Doppler-flip
reported by Pérez et al. (2020), there also appear to be other significant
velocity perturbations further inside the dust cavity. The Doppler-flip
signature along with the other perturbations inside the cavity are reproduced
in our eccentric companion model.

Recently Casassus et al. (2022) proposed that the Doppler-flip signal in HD
100546 could be due to a disc eruption event or inflow onto the disc. The
latter hypothesis seems rather unlikely, as it is not clear how inflowing
material could produce both blue and redshifted emission. The disc eruption
scenario also lacks a physical mechanism to explain the eruption, and whether
such a scenario could produce both blue and redshifted super-sonic
perturbations that span roughly half the azimuthal extent of the disc (and
appear inside of the cavity) is not demonstrated. The association of a
Doppler-flip with spiral structure seen in the continuum emission (Casassus et
al., 2022), as well as emission from the shock tracing molecule SO (Booth et
al., 2018, 2023), is expected with our scenario of a massive internal
companion.

The main criticism of our proposal is that such a companion probably should
have been detected in the sparse aperture masking observations with VLT/SPHERE
by Pérez et al. (2020), and more recently by Stolker et al. (2023). However as
noted in Norfolk et al. (2022), the observations by Pérez et al. (2020)
conducted in May 2018, and more recently throughout 2021 and 2022 by (Stolker
et al., 2023), were conducted when a suspected companion detected by Brittain
et al. (2014) was predicted to be behind the near-side of the disc (Brittain
et al., 2019). Interestingly, Stolker et al. (2023) find a bright spot on the
near-side of the disc that is in close proximity to where one would expect
this potential companion to be located. However they conclude this bright spot
is due to bright forward scattering on the near-side of the disc. Additional
sparse aperture masking observations tracking these bright spots may help to
constrain their orbital motion and determine if either are consistent with
predictions made by Brittain et al. (2014); Brittain et al. (2019).

###  4.3 Fast-Flows Inside the Cavities of Transition Discs

Figure 10: The rotation curve along the disc major axis of the first 8
observations listed in Table 2 (blue lines) with the expected Keplerian
rotation of the unperturbed disc (orange curve). For the HD 163296, GM Aur, IM
Lupi, and MWC 480, we have taken the elevated emission surface into account in
the Keplerian rotation model. The major axis of the beam is shown as the
horizontal black line. Most of the discs in this Figure shown few deviations
away from the expected Keplerian rotation. Where such deviations do exist,
they tend to be within one or two beam sizes from the centre of the disc,
where the rotation profile of the disc is unresolved. IM Lupi shows some
larger deviations, however this is likely due to uncertainties in modelling
the highly elevated emitting layer, and it’s observational effects. TW Hya
shows modest deviations away from the Keplerian model that appear several
times larger than the beam, but these may be due to the dust continuum
subtraction process.  Figure 11: As in Figure 10 but for the last 8 discs
listed in Table 2. CS Cha, GG Tau, and HD 142527 are confirmed
circumbinary(-triple) discs. While all three of these discs so show some
perturbations away from Keplerian, they are most obvious in GG Tau. The
optically thin 13CO and 12CO cavities inside of HD 142527 and CS Cha limit our
ability to spatially resolve their gas kinematics. Modest perturbations are
also seen inside the cavities of J1604 and LkCa 15.

We show the velocity close to the disc major axis of the first 8 discs of
Table 2 in Figure 10, and the last 8 discs of Table 2 in Figure 11. The
observed velocities are taken assuming either an elevated emission surface
(for GM Aur, HD 163296, IM Lupi, and MWC 480, where the emission surfaces are
taken from Law et al. (2021)) or pure Keplerian rotation (all other
observations). We fit each velocity map with eddy and use the resulting best-
fit Keplerian model 2D velocity map to obtain the velocities along the same
path close to the disc major axis as the observations. In both Figures we
scale the line-of-sight velocity by $\sin(i)$, where $i$ is the inclination of
the system. The observations are shown with the blue lines, while the models
are shown with the orange lines.

Starting with Figure 10, we notice that close to the central regions of the
disc there can be large deviations from the best-fit model. Smearing of the
high velocity components close to the central regions, along with
uncertainties in the height of the emitting layer, can cause these effects. We
note that these perturbations tend to be about one or two beam sizes in
extent. Larger-scale perturbations are seen in IM Lupi and TW Hya. In IM Lupi,
the deviations are likely originating due to the CO emitting layer and disc
inclination producing a complex CO emission line profile which not taken into
account in our model. TW Hya appears to have kinematic perturbations that are
several beam widths across which may be due to continuum subtraction process
(e.g. see Boehler et al., 2021).

Turning our attention to Figure 11, we note that the $x$-axis has been
rescaled to the size of the cavity for each of the discs, as listed in Table
2. Several discs presented are confirmed circumbinary(or -triple) discs. These
are GG Tau (Leinert et al., 1993), HD 142527 (Biller et al., 2012), and CS Cha
(Guenther et al., 2007; Nguyen et al., 2012). Starting with GG Tau, CO
emission is seen within the cavity region, allowing for the highly perturbed
kinematics to be traced (Phuong et al., 2020). Significant deviations from the
best-fit Keplerian rotation curve are seen, which are consistent with the
perturbations we see in our models in Figure 6. Deviations from Keplerian
rotation are also seen in the rotation profiles of CS Cha and HD 142527, but
since CO emission is missing inside of the cavity, larger perturbations are
not seen.

Two other discs stand out as having large perturbations inside of their
cavities: AB Aur and HD 100546. AB Aur has a claimed detection of a companion
by Currie et al. (2022), and their astrometric fitting suggests it is highly
eccentric and inclined with respect to the disc. Such a body could lead to
significant perturbations inside the cavity, larger than those suggested by
our planet-hosting disc models in Paper I. However it is not clear whether
this detection is a feature on the disc surface or truly an embedded
protoplanet (Zhou et al., 2023; Biddle et al., 2024, however see Currie 2024).
The outer disc appears to be highly perturbed, and potentially gravitationally
unstable (Speedie et al. in press), which may be responsible for the
perturbations in this region. But the central cavity is optically thin in 13CO
(Rivière-Marichalar et al., 2020), possibly ruling out GI as the primary
source of perturbations in this region.

HD 100546 has had several claimed detections of protoplanets, both inside
(Currie et al., 2015) and outside (Quanz et al., 2013) the central cavity.
Figure 6 shows that planetary mass companions do not produce large amplitude
perturbations inside of the cavities they generate. The highly perturbed
kinematics inside of the cavity are suggestive of a more massive companion
generating the central cavity and the other observed morphological features
seen in this disc.

Modest perturbations are also seen in both J1604 and LkCa 15. For J1604, it is
not entirely clear whether these perturbations are rotational or out of the
plane of the disc (Stadler et al., 2023). It is feasible that either a
planetary mass companion on a non-circular orbit, or a more massive companion,
can produce such perturbations (see Figure 7). In LkCa 15, the relatively low
signal-to-noise, sparse spectral resolution, and relatively large disc
flaring, make interpreting the perturbed kinematics difficult. However the
channel maps, shown in Figure 12, reveal that there is a large velocity kink
in proximity to the cavity. In Paper I, we showed that such a signature is
produced by a stellar mass companion inside the cavity. Deeper and higher
spectral resolution observations will help to resolve the nature of these
perturbations

Figure 12: The CO (3-2) emission around LkCa 15. The white ellipses mark the
inner and outer extent of the two bright continuum emission rings (Facchini et
al., 2020). Kinks in the iso-velocity curves are seen inside and in proximity
to the continuum emission, and are marked with the white arrows.

##  5 Summary

Circumbinary discs generate diverse kinematic and morphological features, some
of which are observed in transitional discs that are not yet known to be
circumbinary. By using simulations and synthetic observations, we have shown
that Doppler-flips, spiral arms, and vortex-like features can appear in the
velocity residuals and potentially be misinterpreted as evidence for planet-
disc interactions. Two of these features have previously been reported in
transition discs. Firstly, vortex-like kinematics were reported by Boehler et
al. (2021) spatially co-located with the dust trap in HD 142527. We show in
Section 4.1 that the vortex-like kinematics can be reproduced in a
circumbinary disc model that does not contain a vortex. The complexity of the
kinematics of circumbinary discs mean that it will be difficult to be
confidently attribute velocity perturbations inside of the dust trap to
vortical motion. Secondly, a Doppler-flip in the disc around HD 100546 was
reported by Pérez et al. (2020) and attributed to a multi-Jupiter mass planet.
Recently, Casassus et al. (2022) attribute the Doppler-flip to a mass eruption
event from a $\sim 10$ ME planet. In Section 4.2 we show that an eccentric
internal companion can reproduce the observed Doppler-flip, as well as other
significant velocity perturbations around the central cavity.

Additionally, we analyse the rotation curves of 16 protoplanetary and
transitional discs in the literature. We show that two transition discs, AB
Aur and HD 100546, have highly perturbed rotation curves inside of their
cavities, which may be consistent with our proposition that such perturbations
are generated by massive (i.e. above planetary mass) companions. Other
transition discs, such as J1604 and LkCa 15 have peculiar kinematic signatures
inside of their cavities which appear consistent with companion-disc
interactions.

The gas kinematics of transition discs can reveal what physical mechanism is
responsible for their origin. A sizeable and high fidelity sample, such as
that soon to be released by the exoALMA111https://www.exoalma.com/ survey,
will allow for a more detailed study on the statistical properties of the
kinematic perturbations seen inside the cavities of transition discs.

## Acknowledgements

We thank the anonymous referee for their useful comments which helped improved
the clarity of the manuscript. This work is supported by the National Natural
Science Foundation of China under grant No. 12233004 and 12250610189. Je.S. is
supported by the Natural Sciences and Engineering Research Council of Canada
(NSERC). DJP and CP are grateful for Australian Research Council funding via
DP180104235, DP220103767 and DP240103290. Jo.S. has received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (PROTOPLANETS, grant agreement No.
101002188). This paper makes use ALMA data with project IDs listed in Table 2.
ALMA is a partnership of ESO (representing its member states), NSF (USA) and
NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ.

## Data Availability Statement

The simulation dump files and selected radiative transfer models will be made
available at 10.5281/zenodo.10706132. Observational data from MAPS can be
obtained from https://alma-maps.info/data.html. Additional observational data
can be found in the citations in the caption of Table 2, or by contacting the
authors. phantom is publicly available at
https://github.com/danieljprice/phantom, while mcfost is available at
https://github.com/cpinte/mcfost.

## References

  * Allard et al. (2001) Allard F., Hauschildt P. H., Alexander D. R., Tamanai A., Schweitzer A., 2001, ApJ, 556, 357
  * Ataiee et al. (2013) Ataiee S., Pinilla P., Zsom A., Dullemond C. P., Dominik C., Ghanbari J., 2013, A&A, 553, L3
  * Avenhaus et al. (2014) Avenhaus H., Quanz S. P., Schmid H. M., Meyer M. R., Garufi A., Wolf S., Dominik C., 2014, ApJ, 781, 87
  * Avenhaus et al. (2017) Avenhaus H., et al., 2017, AJ, 154, 33
  * Bae et al. (2023) Bae J., Isella A., Zhu Z., Martin R., Okuzumi S., Suriano S., 2023, in Inutsuka S., Aikawa Y., Muto T., Tomida K., Tamura M., eds, Astronomical Society of the Pacific Conference Series Vol. 534, Protostars and Planets VII. p. 423 (arXiv:2210.13314), doi:10.48550/arXiv.2210.13314
  * Bate et al. (1995) Bate M. R., Bonnell I. A., Price N. M., 1995, MNRAS, 277, 362
  * Biddle et al. (2024) Biddle L. I., Bowler B. P., Zhou Y., Franson K., Zhang Z., 2024, arXiv e-prints, p. arXiv:2402.12601
  * Biller et al. (2012) Biller B., et al., 2012, ApJ, 753, L38
  * Boehler et al. (2021) Boehler Y., et al., 2021, A&A, 650, A59
  * Bollati et al. (2021) Bollati F., Lodato G., Price D. J., Pinte C., 2021, MNRAS, 504, 5444
  * Booth et al. (2018) Booth A. S., Walsh C., Kama M., Loomis R. A., Maud L. T., Juhász A., 2018, A&A, 611, A16
  * Booth et al. (2023) Booth A. S., Ilee J. D., Walsh C., Kama M., Keyte L., van Dishoeck E. F., Nomura H., 2023, A&A, 669, A53
  * Brittain et al. (2014) Brittain S. D., Carr J. S., Najita J. R., Quanz S. P., Meyer M. R., 2014, ApJ, 791, 136
  * Brittain et al. (2019) Brittain S. D., Najita J. R., Carr J. S., 2019, ApJ, 883, 37
  * Calcino et al. (2019) Calcino J., Price D. J., Pinte C., van der Marel N., Ragusa E., Dipierro G., Cuello N., Christiaens V., 2019, MNRAS, 490, 2579
  * Calcino et al. (2022) Calcino J., Hilder T., Price D. J., Pinte C., Bollati F., Lodato G., Norfolk B. J., 2022, ApJ, 929, L25
  * Calcino et al. (2023) Calcino J., Price D. J., Pinte C., Garg H., Norfolk B. J., Christiaens V., Li H., Teague R., 2023, arXiv e-prints, p. arXiv:2306.07909
  * Casassus & Pérez (2019) Casassus S., Pérez S., 2019, ApJ, 883, L41
  * Casassus et al. (2013) Casassus S., et al., 2013, Nature, 493, 191
  * Casassus et al. (2015) Casassus S., et al., 2015, ApJ, 812, 126
  * Casassus et al. (2022) Casassus S., Cárcamo M., Hales A., Weber P., Dent B., 2022, ApJ, 933, L4
  * Christiaens et al. (2014) Christiaens V., Casassus S., Perez S., van der Plas G., Ménard F., 2014, ApJ, 785, L12
  * Christiaens et al. (2018) Christiaens V., et al., 2018, A&A, 617, A37
  * Close et al. (2014) Close L. M., et al., 2014, ApJ, 781, L30
  * Cornwell (2008) Cornwell T. J., 2008, IEEE Journal of Selected Topics in Signal Processing, 2, 793
  * Currie (2024) Currie T., 2024, Research Notes of the American Astronomical Society, 8, 146
  * Currie et al. (2015) Currie T., Cloutier R., Brittain S., Grady C., Burrows A., Muto T., Kenyon S. J., Kuchner M. J., 2015, ApJ, 814, L27
  * Currie et al. (2022) Currie T., et al., 2022, Nature Astronomy, 6, 751
  * Dipierro et al. (2015) Dipierro G., Price D., Laibe G., Hirsh K., Cerioli A., Lodato G., 2015, MNRAS, 453, L73
  * Disk Dynamics Collaboration et al. (2020) Disk Dynamics Collaboration et al., 2020, arXiv e-prints, p. arXiv:2009.04345
  * Dong et al. (2018a) Dong R., et al., 2018a, ApJ, 860, 124
  * Dong et al. (2018b) Dong R., Najita J. R., Brittain S., 2018b, ApJ, 862, 103
  * Facchini et al. (2020) Facchini S., et al., 2020, A&A, 639, A121
  * Fedele et al. (2021) Fedele D., Toci C., Maud L., Lodato G., 2021, A&A, 651, A90
  * Francis & van der Marel (2020) Francis L., van der Marel N., 2020, ApJ, 892, 111
  * Fukagawa et al. (2006) Fukagawa M., Tamura M., Itoh Y., Kudo T., Imaeda Y., Oasa Y., Hayashi S. S., Hayashi M., 2006, ApJ, 636, L153
  * Garg et al. (2021) Garg H., et al., 2021, MNRAS, 504, 782
  * Garg et al. (2022) Garg H., et al., 2022, MNRAS, 517, 5942
  * Guenther et al. (2007) Guenther E. W., Esposito M., Mundt R., Covino E., Alcalá J. M., Cusano F., Stecklum B., 2007, A&A, 467, 1147
  * Hall et al. (2020) Hall C., et al., 2020, arXiv e-prints, p. arXiv:2007.15686
  * Hirsh et al. (2020) Hirsh K., Price D. J., Gonzalez J.-F., Ubeira-Gabellini M. G., Ragusa E., 2020, MNRAS, 498, 2936
  * Huang et al. (2018) Huang P., Isella A., Li H., Li S., Ji J., 2018, ApJ, 867, 3
  * Huang et al. (2021) Huang J., et al., 2021, ApJS, 257, 19
  * Izquierdo et al. (2021) Izquierdo A. F., Testi L., Facchini S., Rosotti G. P., van Dishoeck E. F., 2021, A&A, 650, A179
  * Izquierdo et al. (2023) Izquierdo A. F., Testi L., Facchini S., Rosotti G. P., van Dishoeck E. F., Wölfer L., Paneque-Carreño T., 2023, A&A, 674, A113
  * Jorsater & van Moorsel (1995) Jorsater S., van Moorsel G. A., 1995, AJ, 110, 2037
  * Kennedy et al. (2019) Kennedy G. M., et al., 2019, Nature Astronomy, 3, 230
  * Kuo et al. (2022) Kuo I. H. G., Yen H.-W., Gu P.-G., Chang T.-E., 2022, ApJ, 938, 50
  * Kurtovic et al. (2022) Kurtovic N. T., et al., 2022, A&A, 664, A151
  * Kutra et al. (2023) Kutra T., Wu Y., Lithwick Y., 2023, arXiv e-prints, p. arXiv:2309.16065
  * Lacour et al. (2016) Lacour S., et al., 2016, A&A, 590, A90
  * Law et al. (2021) Law C. J., et al., 2021, ApJS, 257, 4
  * Law et al. (2022) Law C. J., et al., 2022, arXiv e-prints, p. arXiv:2205.01776
  * Leinert et al. (1993) Leinert C., Zinnecker H., Weitzel N., Christou J., Ridgway S. T., Jameson R., Haas M., Lenzen R., 1993, A&A, 278, 129
  * Lodato & Price (2010) Lodato G., Price D. J., 2010, MNRAS, 405, 1212
  * Lodato et al. (2023) Lodato G., et al., 2023, MNRAS, 518, 4481
  * Longarini et al. (2021) Longarini C., Lodato G., Toci C., Veronesi B., Hall C., Dong R., Patrick Terry J., 2021, ApJ, 920, L41
  * Melon Fuksman & Klahr (2022) Melon Fuksman J. D., Klahr H., 2022, ApJ, 936, 16
  * Mendigutía et al. (2014) Mendigutía I., Fairlamb J., Montesinos B., Oudmaijer R. D., Najita J. R., Brittain S. D., van den Ancker M. E., 2014, ApJ, 790, 21
  * Monaghan (1992) Monaghan J. J., 1992, ARA&A, 30, 543
  * Nguyen et al. (2012) Nguyen D. C., Brandeker A., van Kerkwijk M. H., Jayawardhana R., 2012, ApJ, 745, 119
  * Norfolk et al. (2021) Norfolk B. J., et al., 2021, MNRAS, 502, 5779
  * Norfolk et al. (2022) Norfolk B. J., et al., 2022, ApJ, 936, L4
  * Nowak et al. (2024) Nowak M., Rowther S., Lacour S., Meru F., Nealon R., Price D. J., 2024, arXiv e-prints, p. arXiv:2402.03595
  * Öberg et al. (2021) Öberg K. I., et al., 2021, ApJS, 257, 1
  * Ohashi (2008) Ohashi N., 2008, Ap&SS, 313, 101
  * Paneque-Carreño et al. (2021) Paneque-Carreño T., et al., 2021, ApJ, 914, 88
  * Pavlyuchenkov et al. (2022) Pavlyuchenkov Y. N., Maksimova L. A., Akimkin V. V., 2022, Astronomy Reports, 66, 800
  * Perez et al. (2015a) Perez S., et al., 2015a, ApJ, 798, 85
  * Perez et al. (2015b) Perez S., Dunhill A., Casassus S., Roman P., Szulágyi J., Flores C., Marino S., Montesinos M., 2015b, ApJ, 811, L5
  * Pérez et al. (2018) Pérez S., Casassus S., Benítez-Llambay P., 2018, MNRAS, 480, L12
  * Pérez et al. (2020) Pérez S., et al., 2020, ApJ, 889, L24
  * Phuong et al. (2020) Phuong N. T., et al., 2020, A&A, 635, L9
  * Pineda et al. (2019) Pineda J. E., et al., 2019, ApJ, 871, 48
  * Pinte et al. (2006) Pinte C., Ménard F., Duchêne G., Bastien P., 2006, A&A, 459, 797
  * Pinte et al. (2009) Pinte C., Harries T. J., Min M., Watson A. M., Dullemond C. P., Woitke P., Ménard F., Durán-Rojas M. C., 2009, A&A, 498, 967
  * Pinte et al. (2018) Pinte C., et al., 2018, ApJ, 860, L13
  * Pinte et al. (2019) Pinte C., et al., 2019, Nature Astronomy, 3, 1109
  * Pinte et al. (2020) Pinte C., et al., 2020, ApJ, 890, L9
  * Pinte et al. (2023a) Pinte C., et al., 2023a, MNRAS,
  * Pinte et al. (2023b) Pinte C., Teague R., Flaherty K., Hall C., Facchini S., Casassus S., 2023b, in Inutsuka S., Aikawa Y., Muto T., Tomida K., Tamura M., eds, Astronomical Society of the Pacific Conference Series Vol. 534, Protostars and Planets VII. p. 645 (arXiv:2203.09528), doi:10.48550/arXiv.2203.09528
  * Poblete et al. (2019) Poblete P. P., Cuello N., Cuadra J., 2019, MNRAS, 489, 2204
  * Price et al. (2018a) Price D. J., et al., 2018a, Publ. Astron. Soc. Australia, 35, e031
  * Price et al. (2018b) Price D. J., et al., 2018b, MNRAS, 477, 1270
  * Quanz et al. (2013) Quanz S. P., Avenhaus H., Buenzli E., Garufi A., Schmid H. M., Wolf S., 2013, ApJ, 766, L2
  * Rabago et al. (2023) Rabago I., Zhu Z., Martin R. G., Lubow S. H., 2023, MNRAS, 520, 2138
  * Rafikov (2002) Rafikov R. R., 2002, ApJ, 569, 997
  * Ragusa et al. (2017) Ragusa E., Dipierro G., Lodato G., Laibe G., Price D. J., 2017, MNRAS, 464, 1449
  * Ragusa et al. (2020) Ragusa E., Alexander R., Calcino J., Hirsh K., Price D. J., 2020, MNRAS, 499, 3362
  * Ragusa et al. (2024) Ragusa E., Lynch E., Laibe G., Longarini C., Ceppi S., 2024, A&A, 686, A264
  * Rivière-Marichalar et al. (2019) Rivière-Marichalar P., Fuente A., Baruteau C., Neri R., Treviño-Morales S. P., Carmona A., Agúndez M., Bachiller R., 2019, ApJ, 879, L14
  * Rivière-Marichalar et al. (2020) Rivière-Marichalar P., et al., 2020, A&A, 642, A32
  * Robert et al. (2020) Robert C. M. T., Méheut H., Ménard F., 2020, A&A, 641, A128
  * Shakura & Sunyaev (1973) Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337
  * Siess et al. (2000) Siess L., Dufour E., Forestini M., 2000, A&A, 358, 593
  * Sissa et al. (2018) Sissa E., et al., 2018, A&A, 619, A160
  * Speedie et al. (2022) Speedie J., Booth R. A., Dong R., 2022, ApJ, 930, 40
  * Stadler et al. (2023) Stadler J., et al., 2023, A&A, 670, L1
  * Stolker et al. (2023) Stolker T., et al., 2023, arXiv e-prints, p. arXiv:2312.11989
  * Suriano et al. (2018) Suriano S. S., Li Z.-Y., Krasnopolsky R., Shang H., 2018, MNRAS, 477, 1239
  * Teague (2019) Teague R., 2019, The Journal of Open Source Software, 4, 1220
  * Teague & Foreman-Mackey (2018) Teague R., Foreman-Mackey D., 2018, bettermoments: A robust method to measure line centroids, doi:10.5281/zenodo.1419754, https://doi.org/10.5281/zenodo.1419754
  * Teague et al. (2018) Teague R., Bae J., Bergin E. A., Birnstiel T., Foreman-Mackey D., 2018, ApJ, 860, L12
  * Teague et al. (2019) Teague R., Bae J., Bergin E. A., 2019, Nature, 574, 378
  * Teague et al. (2021) Teague R., et al., 2021, ApJS, 257, 18
  * Teague et al. (2022) Teague R., et al., 2022, ApJ, 936, 163
  * Terry et al. (2021) Terry J. P., Hall C., Longarini C., Lodato G., Toci C., Veronesi B., Paneque-Carreño T., Pinte C., 2021, arXiv e-prints, p. arXiv:2108.11282
  * Veronesi et al. (2021) Veronesi B., Paneque-Carreño T., Lodato G., Testi L., Pérez L. M., Bertin G., Hall C., 2021, ApJ, 914, L27
  * Verrios et al. (2022) Verrios H. J., Price D. J., Pinte C., Hilder T., Calcino J., 2022, ApJ, 934, L11
  * Vioque et al. (2018) Vioque M., Oudmaijer R. D., Baines D., Mendigutía I., Pérez-Martínez R., 2018, A&A, 620, A128
  * Walsh et al. (2014) Walsh C., et al., 2014, ApJ, 791, L6
  * Wölfer et al. (2021) Wölfer L., et al., 2021, A&A, 648, A19
  * Wölfer et al. (2023) Wölfer L., et al., 2023, A&A, 670, A154
  * Wright et al. (2015) Wright C. M., et al., 2015, MNRAS, 453, 414
  * Wu & Lithwick (2021) Wu Y., Lithwick Y., 2021, ApJ, 923, 123
  * Yang et al. (2023) Yang H., Fernandez-Lopez M., Li Z.-Y., Stephens I. W., Looney L. W., Lin Z.-Y. D., Harrison R., 2023, arXiv e-prints, p. arXiv:2304.02937
  * Zhang et al. (2015) Zhang K., Blake G. A., Bergin E. A., 2015, ApJ, 806, L7
  * Zhou et al. (2023) Zhou Y., et al., 2023, AJ, 166, 220
  * Zhu & Baruteau (2016) Zhu Z., Baruteau C., 2016, MNRAS, 458, 3918
  * Zhu et al. (2015) Zhu Z., Dong R., Stone J. M., Rafikov R. R., 2015, ApJ, 813, 88


# Object-conditioned Bag of Instances for
Few-Shot Personalized Instance Recognition

###### Abstract

Nowadays, users demand for increased personalization of vision systems to
localize and identify personal instances of objects (_e.g_., my dog rather
than dog) from a few-shot dataset only. Despite outstanding results of deep
networks on classical label-abundant benchmarks (_e.g_., those of the latest
YOLOv8 model for standard object detection), they struggle to maintain within-
class variability to represent different instances rather than object
categories only. We construct an Object-conditioned Bag of Instances (OBoI)
based on multi-order statistics of extracted features, where generic object
detection models are extended to search and identify personal instances from
the OBoI’s metric space, without need for backpropagation. By relying on
multi-order statistics, OBoI achieves consistent superior accuracy in
distinguishing different instances. In the results, we achieve $77.1\%$
personal object recognition accuracy in case of $18$ personal instances,
showing about $12\%$ relative gain over the state of the art.

Index Terms—  Personalization, Object Recognition, Instances, Few-Shot
Learning

##  1 Introduction

Smart devices are starting to be ubiquitous in everyday life [1] and their
users are demanding for instance-level personalized detection of vision
systems mounted on such devices [2, 3]. For example, vacuum cleaners can now
monitor the behavior of users’ specific pets, and stay away from those
specific pets that are mostly scared by the robot’s noise [4]. Nonetheless,
users do not provide many labeled examples, being a time-consuming operation.
Therefore, we introduce a new task of few-shot instance-level personalization
of object detection models to detect and recognize personal instances of
objects (_e.g_., dog1 and dog2 rather than just dog). The limited availability
of the data distinguishes our task from previous instance-level
personalization attempts [5, 6]. To the best of our knowledge, previous works
assume large availability of labelled data and finetune (FT) the models
through computationally expensive updates. However, FT-based methods
inevitably fail when few-shot samples are provided [7, 8, 9, 10].

In our work, we utilize the latest YOLOv8 [11] efficient detection model, and
we enable personalized instance recognition via backpropagation-free
Prototypes-based Few-Shot Learners (PFSLs), such as [12, 13]. In short, PFSLs
learn a metric space in which classification is performed by computing
distances to prototypical representations of each class.

In this context, we extend PFSLs to support object-class conditioned search,
and we call these approaches Object-conditioned Bag of Instances (OBoI), since
they contain instance-level prototypes. Our approach enriches any OBoI method
by augmenting localized encoder embeddings (EEs) of the input object via
multi-order statistics to construct a richer metric space, where instance-
specific patterns are separable. We compute augmented EEs (AEEs) via a
reduction module similar to recent pooling schemes [14, 15, 16, 17] to
characterize the distribution of the specific instances from the few-shot
labelled data. A concurrent work [14] applies ensemble learning on multi-order
features learned separately; however, their focus is neither personalized
instance recognition nor object detection, and they require gradient-based
training. A backpropagation-free approach, instead, could be especially useful
where dynamic compilers are not available for the target hardware. Our OBoIs
with AEEs significantly increase model personalization, alleviating neural
collapse [18, 19], _i.e_., a state at which within-class variability of hidden
layer outputs is completely lost due to the object-level optimization
objective. Our main novelties are:

  1. 1.

We propose a novel task of few-shot personalization of object detectors to
recognize instances of objects;

  2. 2.

We extend PFSLs via object-level conditioning (OBoIs);

  3. 3.

We further design a multi-order feature space where personal instances can be
separated via a backpropagation-free metric learning on few-shot labelled user
data only;

  4. 4.

OBoIs provide superior results on both same and other domain data (11-22% and
7-18% relative gains respectively).

Fig. 1: For each object (_e.g_., ball), multiple instances (_e.g_., ball1,
ball2) are acquired across different sequences.

##  2 Few-Shot Personalized Instance Recognition

In our setup, we aim at personalizing generic object detection models to
recognize objects by a set of instance-level labels.

We are given a generic object detection model $M_{o}$ (_e.g_., YOLOv8 [11])
which has been trained on a labelled object-level dataset
$\mathcal{T}_{o}=\\{x_{o,k},y_{o,k}\\}_{k=1}^{N_{o}}$, whose labels
$y_{o,k}\in\mathcal{C}_{o}$ belong to an object-level class set
$\mathcal{C}_{o}$ (_e.g_.,
$\mathcal{C}_{o}=\\{\mathrm{ball},\mathrm{bottle}\\}$). We target
personalization of the classification ability of $M_{o}$ to recognize a set of
personal classes given few-shot labelled samples
$\mathcal{T}_{i}=\\{x_{i,k},y_{i,k}\\}_{k=1}^{N_{i}}$, where ${N_{o}\gg
N_{i}}$ and labels $y_{i,k}\in\mathcal{C}_{i}$ belong to an instance-level
class set ${\mathcal{C}_{i}}$ (_e.g_.,
${\mathcal{C}_{i}=\\{\mathrm{ball}_{\mathrm{1}},\mathrm{ball}_{\mathrm{2}},\mathrm{bottle}_{\mathrm{1}}\\}}$).
We remark that, in this work, we focus on the detector’s classification part
and we do not update the localization part. In other words, we assume that
there exists an instance-to-object function $f(\cdot)$ mapping each class
$c\in\mathcal{C}_{i}$ to a label in $\mathcal{C}_{o}$; _i.e_.,
$f:\mathcal{C}_{i}\to\mathcal{C}_{o}$ (_e.g_.,
$f(\mathrm{ball}_{\mathrm{1}})=\mathrm{ball}$), therefore there is no need to
update the detector’s regression part.

In the analyses, we implement the model $M_{o}$ by YOLOv8, pre-train the model
on MSCOCO [20] and then on $\mathcal{T}_{o}$, which contains samples from
Open-Images-V7 (OIV7) [21] of the same object-level classes as in the personal
dataset $\mathcal{T}_{i}$; _i.e_., such that $f(\cdot)$ exists. We used the
default learning parameters [11] for pre-training. We design several setups
whereby we assign a few samples to the training set and the remaining ones to
the testing ($80\%$) and validation ($20\%$) sets.

Datasets. We use CORe50 [6] or iCubWorld Transformations (iCWT) [22] as the
personalized recognition datasets. CORe50: we consider a subset of 45 personal
instances (_i.e_., $|\mathcal{C}_{i}|=45$) belonging to 9 object-level classes
(_i.e_., $|\mathcal{C}_{o}|=9$), acquired over 11 variable-background
sequences, _i.e_., different domains (see Fig. 1). iCWT: we consider a subset
of 9 object-level classes with 10 personal instances each acquired under 5
sequences with diverse affine transformations of the items. On both datasets,
we restrict the personalization stage to the frames being correctly labelled
by YOLOv8n, maintaining a balanced number of samples per instance and per
sequence.

Metrics. We compute the instance recognition accuracy averaged within each
object-level class ($\mathrm{Acc}_{o}$, %) and averaged over all instances
($\mathrm{Acc}_{i}$, %) on the test set. We define the relative gain between
two methods obtaining $\mathrm{Acc}_{i,1}$ and $\mathrm{Acc}_{i,2}$
($\mathrm{Acc}_{i,2}>\mathrm{Acc}_{i,1}$) as: $\Delta\triangleq
100\cdot(\mathrm{Acc}_{i,2}-\mathrm{Acc}_{i,1})/(\mathrm{Acc}_{i,1})$.

Fig. 2: A generic object detector (_e.g_., YOLOv8) is adapted to detect
personal instances via our backpropagation-free Object-conditioned Bag of
Instances (OBoI) approach with augmented embeddings by multi-order statistics.

Table 1: $\mathrm{Acc}_{i}$ of OBoIs on 1SAS (same domain). | # Instances per object class
---|---
2 | 3 | 4 | 5
Random | 50.00 | 33.33 | 25.00 | 20.00
FT | 54.12 | 34.43 | 24.11 | 21.41
SimpleShot | 68.84 | 52.69 | 42.44 | 38.46
\+ AEE (ours) | 76.34 | 62.04 | 51.79 | 46.30
$\Delta$ | +10.9 | +17.7 | +22.0 | +20.4
ProtoNet | 68.84 | 54.88 | 47.97 | 43.98
\+ AEE (ours) | 77.08 | 64.57 | 55.81 | 53.39
$\Delta$ | +12.0 | +17.7 | +16.3 | +21.4

##  3 Object-conditioned Bag of Instances

We propose a lightweight module that can be integrated into any object
detection network. Our solution is based on three key components: (i) an
object detection network, with (ii) a multi-order statistical augmentation of
embeddings for (iii) instance-level recognition via an OBoI. Next, we outline
how we construct our OBoI to personalize object detectors pre-trained on
$\mathcal{T}_{o}$ on the server side (see Fig. 2).

In our case, $\mathcal{T}_{o}$ is made of subsets of the generic OIV7 dataset.
The model $M_{o}$ is then adapted to recognize personal user-specific object
instances. Without loss of generality, we assume that $M_{o}=D_{o}\circ E_{o}$
can be decomposed into an encoder $E_{o}$ and a detection head $D_{o}$. Given
a sample from $\mathcal{T}_{i}$, $(x_{i,k},y_{i,k})\in\mathcal{T}_{i}$, where
${x_{i,k}\in\mathbb{R}^{H\times W\times 3}}$ is an RGB input image with size
$H\times W$ and $y_{i,k}$ the associated personal instance label, we pass
$x_{i,k}$ through the model obtaining the object-level predicted label
($\hat{y}_{o,k}$) and bounding box.

We rescale the predicted coordinates by $H/H^{\prime}=W/W^{\prime}$ to match
the low-resolution spatial dimensions $H^{\prime}\times W^{\prime}$ of the
$D$-dimensional features ${e_{i,k}\triangleq
E_{o}(x_{i,k})\in\mathbb{R}^{H^{\prime}\times W^{\prime}\times D}}$ (for the
sake of simplicity, we assume that only one object is present in each image,
but the same rationale applies in presence of multiple objects seamlessly). We
then build a binary mask $S_{i,k}$ to discard regions outside the bounding
box, and we apply it via the Hadamard product obtaining
${e^{\prime}_{i,k}=S_{i,k}\odot e_{i,k}}$ which is then passed through a
reduction operation. In order to characterize the instance-level distribution
from the few input samples, we extract and concatenate the first $R$
statistical moments to form
$v_{i,k}\triangleq\mathrm{concat}(m_{1},\dots,m_{R})$ being
$\mathrm{concat}(\cdot)$ the concatenation operation and $m_{n}$ the $n$-th
order central statistical moment [23] computed over the features in
$e^{\prime}_{i,k}$ corresponding to non-zero entries in $S_{i,k}$.

Finally, we use the vector $v_{i,k}$ as the input to some PFSL. In our case,
we use PFSLs to identify instance-level classes on a metric space spanned by
multi-order statistics. We refer to [12, 13] for more details. Additionally,
we condition the search for representations of personal objects only within
the instances whose object category matches $\hat{y}_{o,k}$, thus simplifying
the search of the correct nearest instance-level prototype.

The overall pipeline can be thought of as an Object-conditioned Bag of
Instances (OBoI) since generic category-level output is converted to specific
personal-level output via conditional nearest prototype selection. Our setup
and method are fully compatible with the key requirement of continually
learning new instances over time [6, 5, 16, 24, 25, 26]; whenever a user
presents a new instance to be recognized, we can include new instance-level
prototypes in the OBoI at any time with no accuracy degradation with respect
to the case where all instances are available from the beginning of the
adaptation process.

##  4 Experimental Results

Most of the evaluation focuses on the personal instance-level accuracy, since
our modules do not influence in any way the object-level detection accuracy
and bounding box regression. For the sake of clarity, we consider that each
input sample contains one single instance. Nonetheless, our method can handle
multiple instances in input samples by running the instance-level prototype
search for each input object independently. Provided that the general object
detection results are accurate, our results would not change.

Unless otherwise stated, we report all results on YOLOv8n, being the most
suitable for deployed applications, and in the case of 2 instances per each
object-level category.

Same domain. The first scenario we design considers 1-Shot from All Sequences
(1SAS), therefore the same domain is seen during few-shot training (one sample
per each sequence) and testing (all remaining samples from all sequences).
Table 1 reports $\mathrm{Acc}_{i}$ in multiple setups having a variable number
of instances per object-level class. First, we observe that gradient-based
fine-tuning methods (_e.g_., FT) are not effective and obtain comparable
results to a random classifier (lower bound). OBoI via PFSL methods such as
SimpleShot [13] and ProtoNet [12] show large gains compared to FT by learning
a metric space from the extracted features. In both cases, the augmentation of
embeddings via our multi-order statistics boost the recognition accuracy
significantly, especially in presence of multiple instances per object.
Remarkably, we can personalize YOLOv8n to achieve $77.08\%$ $\mathrm{Acc}_{i}$
when detecting $18$ personal instances via just a few samples and via a
backpropagation-free approach, assuming that a correct object-level
classification and bounding box regression are output from the detection head.
Fig. 3(a) reports $\mathrm{Acc}_{i}$ and $\mathrm{Acc}_{o}$ of OBoIs via
ProtoNet in the case of 2 instances per object. We consider three
configurations for ProtoNet: at the logits level (_i.e_., the output of the
last layer of the detector’s head), at the encoder embedding level (_i.e_.,
the output of the detector’s encoder) or via our multi-statistics augmented
encoder embeddings (_i.e_., with features augmented via multi-order
statistics). We observe that our proposed solution consistently improves or
achieves comparable results on every object class.

Table 2: $\mathrm{Acc}_{i}$ of OBoIs on 1S1S (other domain). | # Instances per object class
---|---
2 | 3 | 4 | 5
Random | 50.00 | 33.33 | 25.00 | 20.00
FT | 48.12 | 35.98 | 21.74 | 22.53
SimpleShot | 60.62 | 43.95 | 34.31 | 29.84
\+ AEE (ours) | 65.46 | 49.62 | 40.52 | 35.33
$\Delta$ | +8.0 | +12.9 | +18.1 | +18.4
ProtoNet | 60.33 | 46.52 | 37.13 | 32.13
\+ AEE (ours) | 64.37 | 51.39 | 40.70 | 36.77
$\Delta$ | +6.7 | +10.5 | +9.6 | +14.4

(a) 1SAS (same domain).

(b) 1S1S (other domain).

Fig. 3: $\mathrm{Acc}_{i}$ and per-object $\mathrm{Acc}_{o}$ for OBoIs via
ProtoNet. EE: encoder embeddings. AEE: augmented EE.

Other domain. We designed a more realistic, yet challenging, setup considering
1-Shot from 1 Sequence (1S1S) during training and all remaining samples during
testing. The model experiences different domains at training time (one sample
from the first sequence only) and at testing time (all remaining samples from
all sequences). Table 2 summarizes the main results. Reducing the training
samples further decreased the accuracy of FT, compared to the 1SAS setup. Also
SimpleShot and ProtoNet show lower accuracy, due to the fewer training samples
and the domain gap of the 1S1S setup. Nonetheless, they exhibit large gains
over FT. Our method shows a significant improvement of the performance in
every case, even in the presence of a domain shift, and especially in case of
multiple instances per object category. We argue that the gain attained by our
approach is lower than the previous one due to the difficulty in reliably
matching multi-order statistics between a single input sample from a single
domain and several target samples from several domains. Fig. 3(b) reports
$\mathrm{Acc}_{o}$; similarly to the previous case, we confirm that our
solution obtains robust results across most of the classes.

Variable training shots are studied in Fig. 4, where we observe that OBoIs
with our AEE improve personal recognition accuracy regardless of the number of
available training samples (_i.e_., shots) for both ProtoNet and SimpleShot.

Fig. 4: $\mathrm{Acc}_{i}$ at variable shots. Samples are drawn randomly from each sequence. PN: ProtoNet, SS: SimpleShot. Table 3: Ablation on YOLOv8 models. General object detection results are computed on the subset from OIV7. Personal instance recognition is evaluated on the subset from CORe50; we report a single score that is the $\mathrm{mean}_{p}(\mathrm{Acc}_{i,p})$ where $p$ indicates the setup with $p$ instances per object ($p\in\\{2,3,4,5\\}$). YOLOv8: | n | s | m | l | x
---|---|---|---|---|---
| Size [MB] | 5.9 | 21.4 | 77.0 | 83.6 | 130.4
Detection | Precision | 72.0 | 73.2 | 70.4 | 72.4 | 77.2
Recall | 72.5 | 70.3 | 74.4 | 75.5 | 77.3
mAP50 | 77.7 | 77.6 | 79.3 | 78.2 | 78.7
mAP50-95 | 60.9 | 61.7 | 62.3 | 62.7 | 63.1
1S1S | ProtoNet | 44.03 | 43.56 | 48.44 | 55.38 | 58.77
\+ AEE (ours) | 48.31 | 48.47 | 56.30 | 56.88 | 60.84
$\Delta$ | +9.72 | +11.27 | +16.23 | +2.72 | +3.52
1SAS | ProtoNet | 53.92 | 53.86 | 61.66 | 64.16 | 66.60
\+ AEE (ours) | 62.71 | 66.67 | 75.29 | 75.21 | 74.65
$\Delta$ | +16.31 | +23.79 | +22.10 | +17.23 | +12.09

Other YOLOv8 sizes are evaluated in Table 3 on both general object detection
and personalized instance recognition task. Larger YOLOv8 models can improve
detection performance, and this correlates with the personal instance
recognition accuracy. The improvement of larger YOLOv8

Table 4: $\mathrm{Acc}_{i}$ on iCWT. PN: ProtoNet. | # Instances per object class
---|---
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
1SAS | PN | 82.3 | 66.5 | 62.6 | 58.8 | 46.9 | 46.0 | 43.6 | 41.5 | 41.5
\+ AEE | 85.8 | 75.8 | 71.9 | 67.2 | 57.7 | 55.8 | 52.4 | 49.5 | 47.8
$\Delta$ | +4.4 | +13.9 | +14.9 | +14.3 | +22.9 | +21.5 | +20.2 | +19.1 | +15.2
1S1S | PN | 71.7 | 52.9 | 51.7 | 47.2 | 39.5 | 38.7 | 36.1 | 33.4 | 31.1
\+ AEE | 79.6 | 60.1 | 56.5 | 49.8 | 41.1 | 40.6 | 37.5 | 34.2 | 33.0
$\Delta$ | +11.1 | +13.5 | +9.4 | +5.5 | +4.1 | +5.0 | +3.8 | +2.4 | +6.1

models comes at a cost of a significantly larger model size and slower FPS:
YOLOv8x improves personal recognition by about $25\%$ compared to YOLOv8n,
while having about 22$\times$ larger size and $3.6\times$ slower inference.
The final choice depends on the hardware specifications of target devices.

Computational inference time of our AEE on top of the OBoI with ProtoNet
increases by as little as $0.8\%$ making our method lightweight with a nearly
negligible impact.

Additional ablation studies to evaluate our design choices are reported here
on ProtoNet. Removing the object-level conditioning lowered $\mathrm{Acc}_{i}$
in the 1SAS setup from $77.1\%$ of our approach (ProtoNet + AEE) to $70.9\%$,
showing a relative gain of about $3\%$ compared to the baseline ($68.8\%$).
This is due to the larger search space in metric learning. Removing the mask
$S_{i,k}$ leads to more background noise regions to flow into prototype
computation, and it decreases accuracy by $2.6\%$ in the 1SAS setup, and by
even more ($5.1\%$) in the 1S1S setup, since background varies across
different sequences.

Another personal dataset (iCWT) is shown in Table 4 against the highest
baseline ProtoNet. Our approach exhibits robust gains across all setups
ranging from 18 to 90 instances.

##  5 Conclusion

In this paper, we introduced few-shot instance-level personalization of object
detectors. We proposed a new method (OBoI) to personalize detection models to
recognize user specific instances of object categories. OBoI is a
backpropagation-free metric learning approach on a multi-order statistics
feature space. We believe that this setup and our method could pave the way to
personal instance-level detection and could stimulate future research and
applications.

## References

  * [1] Ovidiu Vermesan and Joël Bacquet,  Internet of Things–The Call of the Edge: Everything Intelligent Everywhere,  CRC Press, 2022.
  * [2] Junfeng He, Khoi Pham, Nachiappan Valliappan, Pingmei Xu, Chase Roberts, Dmitry Lagun, and Vidhya Navalpakkam,  “On-device few-shot personalization for real-time gaze estimation,”  in ICCVW, 2019.
  * [3] Nidhi Arora, Daniel Ensslen, Lars Fiedler, Wei Wei Liu, Kelsey Robinson, Eli Stein, and Gustavo Schüler,  “The value of getting personalization right - or wrong - is multiplying,”  McKinsey & Company, pp. 1–12, 2021.
  * [4] “An Intelligent At-Home Helper – How the Bespoke Jet Bot™ AI+ Takes Care of Your Pet When You’re Away,” https://tinyurl.com/srjs6zcn.
  * [5] Raffaello Camoriano, Giulia Pasquale, Carlo Ciliberto, Lorenzo Natale, Lorenzo Rosasco, and Giorgio Metta,  “Incremental robot learning of new objects with fixed update time,”  in ICRA. IEEE, 2017, pp. 3207–3214.
  * [6] Vincenzo Lomonaco and Davide Maltoni,  “CORe50: a new dataset and benchmark for continuous object recognition,”  in CoRL, 2017, pp. 17–26.
  * [7] Yu-Xiong Wang, Liangke Gui, and Martial Hebert,  “Few-shot hash learning for image retrieval,”  in ICCVW, 2017, pp. 1228–1237.
  * [8] Deunsol Jung, Dahyun Kang, Suha Kwak, and Minsu Cho,  “Few-shot metric learning: Online adaptation of embedding for retrieval,”  in ACCV, 2022.
  * [9] Jiancai Zhu, Jiabao Zhao, Jiayi Zhou, Liang He, Jing Yang, and Zhi Zhang,  “Uncertainty-aware few-shot class-incremental learning,”  in ICASSP, 2023, pp. 1–5.
  * [10] Aymane Abdali, Vincent Gripon, Lucas Drumetz, and Bartosz Boguslawski,  “Active learning for efficient few-shot classification,”  in ICASSP, 2023, pp. 1–5.
  * [11] “Yolov8 by ultralytics,” https://github.com/ultralytics/,  Accessed: 2023-08-20.
  * [12] Jake Snell, Kevin Swersky, and Richard Zemel,  “Prototypical networks for few-shot learning,”  NeurIPS, 2017.
  * [13] Yan Wang, Wei-Lun Chao, Kilian Q Weinberger, and Laurens Van Der Maaten,  “Simpleshot: Revisiting nearest-neighbor classification for few-shot learning,”  arXiv:1911.04623, 2019.
  * [14] Sai Yang, Fan Liu, Delong Chen, and Jun Zhou,  “Few-shot classification via ensemble learning with multi-order statistics,”  IJCAI, 2023.
  * [15] Umberto Michieli, Pablo Peso Parada, and Mete Ozay,  “Online continual learning in keyword spotting for low-resource devices via pooling high-order temporal statistics,”  INTERSPEECH, 2023.
  * [16] Umberto Michieli and Mete Ozay,  “Online continual learning for robust indoor object recognition,”  IROS, 2023.
  * [17] Umberto Michieli and Mete Ozay,  “HOP to the Next Tasks and Domains for Continual Learning in NLP,”  in AAAI, 2024.
  * [18] Vignesh Kothapalli, Ebrahim Rasromani, and Vasudev Awatramani,  “Neural collapse: A review on modelling principles and generalization,”  TMLR, 2023.
  * [19] Vardan Papyan, XY Han, and David L Donoho,  “Prevalence of neural collapse during the terminal phase of deep learning training,”  PNAS, vol. 117, no. 40, pp. 24652–24663, 2020.
  * [20] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick,  “Microsoft coco: Common objects in context,”  in ECCV. Springer, 2014, pp. 740–755.
  * [21] Alina Kuznetsova, Hassan Rom, Neil Alldrin, Jasper Uijlings, Ivan Krasin, Jordi Pont-Tuset, Shahab Kamali, Stefan Popov, Matteo Malloci, Alexander Kolesnikov, et al.,  “The open images dataset v4: Unified image classification, object detection, and visual relationship detection at scale,”  IJCV, vol. 128, no. 7, pp. 1956–1981, 2020.
  * [22] Giulia Pasquale, Carlo Ciliberto, Francesca Odone, Lorenzo Rosasco, and Lorenzo Natale,  “Are we done with object recognition? the icub robot’s perspective,”  Robotics and Autonomous Systems, vol. 112, pp. 260–281, 2019.
  * [23] Athanasios Papoulis and S Unnikrishna Pillai,  Probability, random variables and stochastic processes,  2002\.
  * [24] Umberto Michieli and Pietro Zanuttigh,  “Continual semantic segmentation via repulsion-attraction of sparse and disentangled latent representations,”  in CVPR, 2021, pp. 1114–1124.
  * [25] Umberto Michieli and Pietro Zanuttigh,  “Incremental learning techniques for semantic segmentation,”  in ICCVW, 2019.
  * [26] Timothée Lesort, Vincenzo Lomonaco, Andrei Stoian, Davide Maltoni, David Filliat, and Natalia Díaz-Rodríguez,  “Continual learning for robotics: Definition, framework, learning strategies, opportunities and challenges,”  Information Fusion, 2020.