# Comprehensive study of Lorentz invariance violation in atmospheric and long-
baseline experiments

Deepak Raikwal<EMAIL_ADDRESS>Harish-Chandra Research Institute, A
CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj -
211019  Homi Bhabha National Institute, Anushakti Nagar, Mumbai 400094, India
Sandhya Choubey<EMAIL_ADDRESS>Department of Physics, School of Engineering
Sciences, KTH Royal Institute of Technology,
AlbaNova University Center, Roslagstullsbacken 21, SE–106 91 Stockholm, Sweden
The Oskar Klein Centre, AlbaNova University Center, Roslagstullsbacken 21,
SE–106 91 Stockholm, Sweden  Monojit Ghosh<EMAIL_ADDRESS>Center of Excellence
for Advanced Materials and Sensing Devices, Ruder Bošković Institute, 10000
Zagreb, Croatia

###### Abstract

In this paper, we have presented a comprehensive study of Lorentz Invariance
Violation (LIV) in the context of the atmospheric neutrino experiment ICAL and
the long-baseline experiments T2HK and DUNE. Our study consists of the full
parameter space of the LIV parameters, i.e., six CPT-violating LIV parameters
($a_{\alpha\beta}$) and six CPT-conserving LIV parameters ($c_{\alpha\beta}$).
In this study, our objective is to calculate the upper bound on all the LIV
parameters with respect to the individual experiments as well as their
combination. Our results show that DUNE gives the best sensitivity for the
parameters $a_{ee}$, $a_{e\mu}$, $a_{e\tau}$, and $a_{\mu\tau}$ in its 7 years
of running, whereas ICAL gives the best sensitivity on $a_{\mu\mu}$,
$a_{\mu\tau}$, $c_{ee}$, $c_{\mu\mu}$, $c_{\tau\tau}$, and $c_{\mu\tau}$ in
its 10 years of running. For $a_{\tau\tau}$, the sensitivities of DUNE and
ICAL are almost same. The combination of T2HK, DUNE, and ICAL gives the best
sensitivity for $a_{e\mu}$ and $a_{ee}$ with respect to all the existing
bounds in the literature. For the CPT-even diagonal parameters $c_{ee}$ and
$c_{\mu\mu}$, our work provides the first-ever bounds.

††preprint: APS/123-QED

##  I Introduction

The Standard Model (SM) has been highly successful in explaining physics at
low energies. However, at the Planck scale ($M_{P}\sim 10^{19}$ GeV), physics
behaves differently, deviating from the rules set by the SM. It is widely
accepted that the SM is a low-energy approximation of a more comprehensive
theory that encompasses physics at both low energies and the Planck scale. At
the Planck scale, various unusual physics scenarios can arise, such as Lorentz
invariance violation (LIV) Kostelecky and Samuel (1989) and CPT violation,
indicating a departure from the exact symmetries preserved by the SM.
Consequently, the laws governing physics at the Planck scale and low energies
can be described by an effective quantum field theory that incorporates the
SM. One example of such a theory is the Standard Model Extension (SME), which
permits arbitrary coordinate-independent Lorentz invariance violation Colladay
and Kostelecky (1998).

The SME extends the SM by including additional terms in its Lagrangian that
can violate Lorentz symmetry in different ways. These violations can be
realized through spontaneous Lorentz violation proposed by string and field
theories of gravity, non-commutative field theories, quantum gravity, and
other approaches. The SME encompasses various particle sectors, including
quark, gluon, electron, muon, and neutrino sectors, and experimental bounds on
Lorentz violation in these sectors have been studied and reported in the
literature. Although no experimental evidence of Lorentz violation has been
found so far, sensitivity limits have been established by experiments. These
limits can be further improved through combined analyses of different
experiments. While certain parameters related to Lorentz violation in the
neutrino sector have not been extensively studied, ongoing research aims to
explore and refine our understanding in this area.

The interference phenomenon of neutrino oscillation, in which active neutrinos
oscillate among their flavors, provides a unique opportunity to probe LIV
effects. In the presence of LIV, the neutrino oscillation Hamiltonian gets
modified, and therefore terms describing LIV appear in the neutrino
oscillation probabilities. The LIV parameters that appear in the neutrino
oscillation probabilities can be either CPT-violating or CPT-conserving. In
this paper, we present a comprehensive analysis of LIV in the context of
upcoming long-baseline neutrino experiments T2HK Abe  _et al._ (2018a) in
Japan and DUNE Abi  _et al._ (2020a) in the USA, and the upcoming atmospheric
neutrino experiment ICAL Ahmed  _et al._ (2017) at the INO facility in India.
In particular, we study the capability of these experiments to put an upper
bound on the parameters of LIV. Note that the study of LIV in the context of
DUNE and ICAL has been performed earlier. In what follows next, we mention the
studies that have been carried out in the past and the novel features in our
work, which we will show for the first time. In the context of DUNE, Ref.
Barenboim  _et al._ (2019) obtained the upper bounds on the CPT-violating LIV
parameters, and Ref. Kumar Agarwalla and Masud (2020) studied the effect of
LIV on the determination of the octant of the atmospheric mixing angle
$\theta_{23}$ and on the determination of the leptonic phase $\delta_{\rm
CP}$. A combined bound on the CPT-violating LIV parameters for DUNE and the
long-baseline experiment option at KM3NeT Adrian-Martinez  _et al._ (2016),
namely P2O Akindinov  _et al._ (2019), has been obtained in Ref. Fiza  _et
al._ (2022). For ICAL, Ref. Sahoo  _et al._ (2022) calculated bounds on the
three off-diagonal CPT-violating LIV parameters, using events calculated in
the energy window 1 GeV to 25 GeV, with the assumption that the phases
associated with the LIV parameters have less than 1$\%$ effect on the results.
To the best of our knowledge, the study of LIV in the context of T2HK has not
been performed in the past111When our paper was in the final stage of
preparation, two papers appeared in the arXiv which studied LIV in long-
baseline experiments. Ref. Sarker  _et al._ (2023) studied LIV in the context
of DUNE, and Ref. Agarwalla  _et al._ (2023) studied LIV in the context of
DUNE and T2HK. In our paper, we will not discuss the results of these two
papers.. Apart from T2HK, DUNE, and ICAL, LIV has been studied in the context
of Super-Kamiokande Abe  _et al._ (2015), IceCube Aartsen  _et al._ (2018);
Crivellin  _et al._ (2021), T2K, and NO$\nu$A Majhi  _et al._ (2020); Lin  _et
al._ (2022); Rahaman (2021). In the present work, we aim to perform a detailed
study of LIV in the context of T2HK, DUNE, and ICAL and compare our results
with the existing bounds on the LIV parameters. In our study, we will consider
the complete LIV parameter space, namely, six CPT conserving LIV parameters
and six CPT violating LIV parameters. In addition, for ICAL we calculate
sensitivities for an extended energy range of up to 100 GeV and consider the
full effect of the phases associated with the parameters of LIV. We will show
how our results for ICAL improve in comparison to the old results. This
improvement comes mainly from the higher energy events included in our
analysis. Furthermore, our results will also show that the phases associated
with the LIV parameters have a non-trivial effect on the final results in
ICAL. Apart from the individual sensitivities of each experiment, we will also
show the combined sensitivity of these three experiments. Therefore, for the
very first time, our work demonstrates: (i) a study of LIV in the context of
T2HK, (ii) a study of CPT conserving LIV parameters in the context of DUNE and
ICAL, (iii) an analysis of the diagonal CPT violating LIV parameters in the
context of ICAL, (iv) a study of LIV with an extended energy window up to 100
GeV with the full effect of the phases associated with the parameters of LIV
in ICAL, (v) combined sensitivity for long-baseline and atmospheric
experiments, and (vi) a detailed comparison of all the available bounds.

The paper is organized as follows. In the next section, we will briefly
discuss the theory of LIV and show how it changes the Hamiltonian of the
neutrino oscillations. In section III, we will discuss the experimental
configurations that we use in our analysis. In section IV, we study the effect
of LIV at the probability level, and in section V, we present the sensitivity
of the experiments in terms of their capability to put an upper bound on the
LIV parameters. After that, in section VI, we compare our results with the
results obtained in the previous works. Finally, in section VII, we will
summarize our results and then conclude.

##  II LIV and its effect on neutrino oscillations

Lorentz Invariance violating neutrinos and antineutrinos are effectively
described by the Lagrangian density,

$\mathcal{L}=\frac{1}{2}\bar{\psi}(i\partial-M-\hat{\mathcal{Q}})\psi+h.c.$
(1)

where, $\hat{\mathcal{Q}}$ is generic Lorentz invariance violating operator
and fields related to neutrino and antineutrino are introduced by fermionic
spinors $\psi$ and $\bar{\psi}$. The first term of the Eq. (1) is the kinetic
term, the second term is the mass term and the third term is the Lorentz
invariance violating term. The Lorentz invariance violating part of the
Lagrangian can be written for renormalizable Dirac coupling as Kostelecký and
Mewes (2012)

$\displaystyle\mathcal{L}_{\rm
LIV}=\frac{-1}{2}\left[a^{\mu}_{\alpha\beta}\bar{\psi_{\alpha}}\gamma_{\mu}\psi_{\beta}+b^{\mu}_{\alpha\beta}\bar{\psi_{\alpha}}\gamma_{5}\gamma_{\mu}\psi_{\beta}\right]$
(2)
$\displaystyle\frac{-1}{2}\left[-ic^{\mu\nu}_{\alpha\beta}\bar{\psi_{\alpha}}\gamma_{\mu}\partial_{\nu}\psi_{\beta}-id^{\mu\nu}_{\alpha\beta}\bar{\psi_{\alpha}}\gamma_{5}\gamma_{\mu}\partial_{\nu}\psi_{\beta}\right]+h.c$

The observable effects on the left handed neutrinos are controlled by

$(a_{L})^{\mu}_{\alpha\beta}=(a+b)^{\mu}_{\alpha\beta},\hfill(c_{L})^{\mu\nu}_{\alpha\beta}=(c+d)^{\mu\nu}_{\alpha\beta},$
(3)

which are constant Hermitian matrices in the flavor space that can modify the
standard vacuum Hamiltonian. The first term is CPT-odd LIV term and second
term is CPT-even LIV term. In this work we will focus on the isotropic
component of the Lorentz invariance violating terms and we will fix the
$(\mu,\nu)$ indices to zero. To simplify our notation, from now on, we will
denote the parameters $(a_{L})^{0}_{\alpha\beta}$ and
$(c_{L})^{00}_{\alpha\beta}$ as $a_{\alpha\beta}$ and $c_{\alpha\beta}$.
Explicitly, one can write the Lorentz–violating contribution to the full
oscillation Hamiltonian as,

$H=UMU^{\dagger}+V_{e}+H_{\rm LIV},$ (4)

where U is the PMNS(Pontecorvo–Maki–Nakagawa–Sakata) Giganti  _et al._ (2018)
mixing matrix having three mixing angles $\theta_{12}$, $\theta_{13}$
,$\theta_{23}$ and one CP phase $\delta_{\rm CP}$, $M$ is the neutrino mass
matrix given by

$M=\frac{1}{2E}\begin{pmatrix}0&0&0\\\ 0&\Delta m^{2}_{21}&0\\\ 0&0&\Delta
m^{2}_{31}\end{pmatrix},$ (5)

with $\Delta m^{2}_{21}=m_{2}^{2}-m_{1}^{2}$ and $\Delta
m^{2}_{31}=m_{3}^{2}-m_{1}^{2}$ where $m_{1}$, $m_{2}$ and $m_{3}$ are the
masses of the active neutrinos, $V_{e}$ is the matter potential

$V_{e}=\pm\sqrt{2}G_{F}\begin{pmatrix}N_{e}&0&0\\\ 0&0&0\\\
0&0&0\end{pmatrix},$ (6)

where $G_{F}$ is the Fermi constant and $N_{e}$ is the electron density. The
$+$ sign in $V_{e}$ is for neutrinos and $-$ sign is for antineutrinos. The
term $H_{\rm LIV}$ is given by

$H_{\rm LIV}=\pm\begin{pmatrix}a_{ee}&a_{e\mu}&a_{e\tau}\\\
a_{e\mu}^{\star}&a_{\mu\mu}&a_{\mu\tau}\\\
a_{e\tau}^{\star}&a_{\mu\tau}^{\star}&a_{\tau\tau}\end{pmatrix}-\frac{4}{3}E\begin{pmatrix}c_{ee}&c_{e\mu}&c_{e\tau}\\\
c_{e\mu}^{\star}&c_{\mu\mu}&c_{\mu\tau}\\\
c_{e\tau}^{\star}&c_{\mu\tau}^{\star}&c_{\tau\tau}\end{pmatrix}.$ (7)

Note that the diagonal parameters in Eq. 7 are real while the off-diagonal
parameters are complex with a phase $\phi^{a/c}_{\alpha\beta}$
($\alpha\neq\beta$) associated with them. In the CPT violating first term, the
$+$ sign in $V_{e}$ is for neutrinos and $-$ sign is for antineutrinos. The
second term being CPT conserving is the same for neutrinos and antineutrinos.

##  III Experimental configuration

In this section, we discuss the experimental configurations of T2HK, DUNE, and
ICAL that we use in our analysis. As these experiments have different
combinations of energy, baselines, and matter effects, the sensitivity of one
experiment can be complementary to another. We simulate T2HK and DUNE using
GLoBES Huber  _et al._ (2005, 2007), and for ICAL, we use the software from
the INO collaboration. In order to implement LIV in GLoBES, we have written an
independent probability engine and then included it in GLoBES. We also use the
same probability engine for ICAL.

###  III.1 ICAL detector at the INO facility

####  III.1.1 Detector

The India-based neutrino observatory (INO) will house a 50 kton magnetized
iron calorimeter (ICAL) detector to study oscillations of the atmospheric
neutrinos. The resistive plate chambers (RPCs) Ahmed  _et al._ (2017) will be
the active detector elements in ICAL, while iron will be the target for
atmospheric neutrinos. ICAL will be optimized to be sensitive primarily to
atmospheric $\nu_{\mu}$ and $\bar{\nu}_{\mu}$. The detector’s structure, with
horizontal layers of iron interspersed with RPCs, allows it to have nearly
complete coverage to the direction of incoming neutrinos, except for those
that produce nearly horizontally traveling $\mu^{\pm}$. As a result, it is
sensitive to a wide range of neutrino path lengths. ICAL will be sensitive to
muon energy, muon angle, and total hadron energy Lakshmi  _et al._ (2014)
deposited. RPCs with fast response times can distinguish between upward and
downward moving $\mu^{\pm}$. ICAL will be magnetized to about 1.5 T and hence
will be able to distinguish between $\nu_{\mu}$ and $\bar{\nu}_{\mu}$. Our
analysis uses a muon energy range of $E_{\mu}$ = 1 GeV to 100 GeV. The ICAL
detector simulations use the CERN GEANT4 Agostinelli  _et al._ (2003)-based
package developed by the ICAL collaboration. In our analysis, we will consider
a run-time of 10 years.

####  III.1.2 Flux

When primary cosmic ray protons and nuclei interact in the atmosphere, they
produce secondary ”atmospheric” cosmic rays made of hadrons and their decay
products. The spectrum of these secondaries peaks in the GeV range but extends
to high energy with a power-law spectrum. Neutrinos are the final component of
secondary cosmic radiation because they interact weakly, and they include
$\nu_{e}$, $\bar{\nu}_{e}$, $\nu_{\mu}$, and $\bar{\nu}_{\mu}$. The neutrino
flux varies with location and also has seasonal variations, as described in
Ref Honda  _et al._ (2015). We used the atmospheric flux generated for the
Theni site by Honda et al. Honda  _et al._ (2015). The flux was generated for
both solar maximum and solar minimum with rock on top of the detector. We use
the average of both fluxes for our simulations.

####  III.1.3 Event Generator

The GENIE event generator uses Monte Carlo simulations to generate neutrino
nucleon interactions. It supports various neutrino flux formats, including
HAKKM, BGLRS, and FLUKA for atmospheric neutrino flux, as well as any flux
distribution in the form of a polynomial equation. It can produce cross-
section files for any material or neutrino with energies ranging from a few
MeV to several hundred GeV, and it also includes the cross-section files
generated and distributed by the Fermi collaboration with GENIE MC. In GENIE,
it’s possible to specify the detector geometry using GEANT4. To adapt GENIE
for ICAL, the ICAL collaboration has made the necessary modifications to the
software AJMI and MAJUMDER (2017).

####  III.1.4 Binning Scheme

We use data binned in three variables: the energy of the observed muon
$E^{obs}_{\mu^{\pm}}$, the direction of the observed muon
$\cos(\theta^{obs}_{\mu^{\pm}})$, and the energy of the observed hadron
$E^{obs}_{had}$. We calculate the hadron energy as $E^{true}_{had}$ =
$E_{\nu}$ \- $E_{\mu}$. At the generator level, we obtain the true muon energy
and angle, which may differ from the observed quantities due to the detector
response. To incorporate neutrino oscillations, we apply the re-weighting
algorithm Barger  _et al._ (1980) and then implement the detector response to
the events. To account for detector efficiency, we use muon and hadron lookup
Chatterjee  _et al._ (2014); Devi  _et al._ (2013) tables provided by the INO
collaboration. We assume a Gaussian distribution for the muon energy and angle
to account for detector resolution. We use the Vavilov distribution function
for the hadron energy Devi  _et al._ (2013). We bin the data in the scheme
presented in Table-1.

Observable | Range | Bin width | No. of bins
---|---|---|---
$E^{obs}_{\mu}$(GeV)(15 bins) | [1,11] | 1 | 10
| [11,21] | 5 | 2
| [21,25] | 4 | 1
| [25,50] | 25 | 1
| [50,100] | 50 | 1
$\cos\theta^{obs}_{\mu}$ (15 bins) | [-1.0,0.0] | 0.1 | 10
| [0.0,1] | 0.2 | 5
$E^{obs}_{had}$ (GeV) (4bins) | [0,2] | 1 | 2
| [2,4] | 2 | 1
| [4,15] | 11 | 1

Table 1: The number of bins and bin widths of the three observables $E_{\mu}$,
$\cos\theta^{obs}_{\mu}$ and $E_{had}^{obs}$ are given in the table which we
use in our analysis for ICAL.

####  III.1.5 Systematic uncertainties

For the atmospheric neutrino studies, the dominant uncertainties come from
uncertainties in the atmospheric neutrino flux calculations. For the fluxes we
take the following systematic uncertainties into account: an overall flux
normalization error of 20$\%$, correlated tilt error of 5$\%$ and correlated
zenith angle error of 5$\%$. In addition, we take a cross-section error of 10%
to account for the uncertainty in the neutrino-nucleon cross-sections.
Finally, we should have systematic uncertainties coming from the ICAL detector
configuration. The ICAL collaboration takes this to be a consolidated 5$\%$
systematic uncertainty, uncorrelated amongst the bins. These errors are put
separately for neutrinos and antineutrinos. The magnitude of the systematic
errors are taken to be the same for neutrino and antineutrino events.

###  III.2 DUNE

DUNE (Deep Underground Neutrino Experiment) is a versatile neutrino experiment
with multiple scientific objectives. It is suitable for long baseline studies,
atmospheric neutrino studies, and indirect dark matter detection studies. To
simulate DUNE for the long-baseline case, we use the most recent DUNE
configuration files provided by the collaboration Abi  _et al._ (2021). The
neutrino source is located at Fermilab and has a beam power of 1.2 MW with a
total exposure of (1.1 - 1.9) x 1021 POT/year. The beam power will be upgraded
to 2.4 MW after 7 years of running. We assume a total run-time of 7 years,
with 3.5 years in neutrino mode and the remaining 3.5 years in antineutrino
mode. The on-axis 40 kt liquid argon far detector (FD) is housed at the
Homestake Mine in South Dakota over a 1300 km baseline. For both the
appearance and disappearance channels, we consider neutrino and antineutrino
energies ranging from 0 to 20 GeV. The reconstructed energy is divided into a
total of 71 bins, with 64 bins having widths of 0.125 GeV in the energy range
of 0 to 8 GeV, and 7 bins with variable widths beyond 8 GeV. We include the
”wrong-sign” components in the beam when calculating events, whether signal or
background, for both $\nu_{e}/\bar{\nu}_{e}$ and $\nu_{\mu}/\bar{\nu}_{\mu}$
candidate events. The flux uncertainty for DUNE is approximately 8$\%$ and is
dominated by hadron production uncertainties. The flux uncertainty for the
near detector/far detector (ND/FD) is approximately 0.5$\%$ at the peak but
rises to approximately 2$\%$ in the falling edge and is dominated by focusing
effects.

###  III.3 T2HK

For T2HK, we have utilized the configuration described in Ref. Abe  _et al._
(2018b). The experiment consists of two water-Cerenkov detector tanks, each
with a fiducial volume of 187 kt, located at Kamioka, which is 295 km away
from the neutrino source at J-PARC. The beam power is 1.3 MW, and the total
exposure is $27\times 10^{21}$ protons on target, which corresponds to ten
years of operation. The total run-time is divided into five years in neutrino
mode and five years in antineutrino mode. For systematic errors, we considered
an overall normalization error of 4.71$\%$ (4.13$\%$) for the appearance
(disappearance) channel in neutrino mode and 4.47$\%$ (4.15$\%$) for the
appearance (disappearance) channel in antineutrino mode. Both the signal and
the background have the same systematic error. In T2HK, we have used 98
equispaced bins in the energy range of 0.2 - 10 GeV.

##  IV Discussion at the probability level

In this section, we will analyze the impact of LIV on the probability levels
for all three experiments. For our analysis, we have considered one parameter
at a time and studied the sensitivity of each individual parameter. Therefore,
in our study, we have six independent CPT-conserving parameters and six
independent CPT-violating parameters. Note that if we consider all the
parameters at the same time instead of one at a time, the number of
independent parameters in each CPT-conserving case and CPT-violating case
would be five. This is because, when we consider all the parameters at the
same time, we can subtract a matrix proportional to identity without changing
the oscillation probabilities. Therefore, only two of the diagonal parameters
are left to be independent. The values of the LIV parameters used in the
probability plots are mentioned in the panel titles/legends. The values of the
oscillation parameters used in these figures are given in Table 2. For
$\delta_{\rm CP}$, we consider the value $0^{\circ}$ for ICAL and $0^{\circ}$
and $-90^{\circ}$ for T2HK and DUNE. Our choices of $\delta_{\rm CP}$ are
motivated by the current best-fit value of this parameter as obtained by the
currently running experiment T2K Bronner (2022) and NO$\nu$A Hartnell (2022).
All the figures are given for normal ordering of the neutrino masses, i.e.,
$\Delta m^{2}_{31}>0$.

###  IV.1 ICAL

To illustrate the effect of LIV on ICAL, we calculate the disappearance
channel probability ($\nu_{\mu}\rightarrow\nu_{\mu}$) with and without LIV,
and present the difference between them as oscillograms in the $\cos\theta$ \-
$E$ plane, where $E$ and $\theta$ here refer to the neutrino energy and
neutrino zenith angle. For ICAL the relevant energy region of the oscillograms
is 1 - 100 GeV. In Fig. 1 (2), we present the probability oscillogram for
$a_{\alpha\beta}$ ($c_{\alpha\beta}$), respectively. The first and second rows
are for the diagonal LIV parameters, while the third and fourth rows are for
the off-diagonal LIV parameters. The first and third rows correspond to
neutrinos, while the second and fourth rows correspond to antineutrinos. The
value of $\phi^{a/c}_{\alpha\beta}$ for the off-diagonal LIV parameters is
taken as zero in these figures.

Figure 1: Probability oscillogram for $\Delta P_{\nu_{\mu}\nu_{\mu}}$ and
$\Delta P_{\bar{\nu}_{\mu}\bar{\nu}_{\mu}}$ channels for $a_{ee}$,
$a_{\mu\mu}$, $a_{\tau\tau}$, $a_{e\mu}$, $a_{e\tau}$ and $a_{\mu\tau}$. For
ICAL, the relevant energy region of the oscillograms is 1-100 GeV. The red
line shows the region relevant for DUNE and the black line shows the region
relevant for T2HK.

Figure 2: Probability oscillograms for $\Delta P_{\nu_{\mu}\nu_{\mu}}$ and
$\Delta P_{\bar{\nu}_{\mu}\bar{\nu}_{\mu}}$ channels for $c_{ee}$,
$c_{\mu\mu}$, $c_{\tau\tau}$, $c_{e\mu}$, $c_{e\tau}$ and $c_{\mu\tau}$. For
ICAL, the relevant energy region of the oscillograms is 1-100 GeV. The red
line shows the region relevant for DUNE and the black line shows the region
relevant for T2HK.

Observations from these oscillograms are as follows. For the diagonal CPT-
violating LIV parameters, we observe that the effect of LIV is greater in
$a_{\mu\mu}$ and $a_{\tau\tau}$ than in $a_{ee}$. In $a_{ee}$, the effect is
more pronounced in neutrinos than in antineutrinos. For neutrinos, $a_{ee}$
can be probed in the energy range of 1 GeV to 20 GeV, while for antineutrinos,
it can be probed below 1 GeV. For $a_{\mu\mu}$ and $a_{\tau\tau}$, the effect
of LIV is similar, with some differences along the $\cos\theta$ axis. The
effect of LIV on these two parameters is more significant in antineutrinos
than in neutrinos.

Regarding the off-diagonal CPT-violating LIV parameters, we see that the
effect of LIV on $a_{e\mu}$ and $a_{e\tau}$ is similar, with some differences
along the $\cos\theta$ axis. The effect of LIV on these two parameters is more
prominent in neutrinos than in antineutrinos. Among the three off-diagonal LIV
parameters, $a_{\mu\tau}$ is more sensitive to LIV. For this parameter, the
effect of LIV is opposite in neutrinos and antineutrinos, with the blue and
yellow colors interchanged in the panels. Thus, we expect to have strong
bounds on $a_{\mu\mu}$, $a_{\tau\tau}$, and $a_{\mu\tau}$ in ICAL as far as
CPT-violating LIV parameters are concerned, compared to the other three
parameters. This is consistent with our analytical understanding of these
parameters. It has been shown that in the disappearance channel probabilities,
the parameters $a_{\mu\mu}$, $a_{\tau\tau}$, and $a_{\mu\tau}$ contribute at
the leading order Sahoo  _et al._ (2022).

Now, let us discuss the case for CPT-conserving parameters. From Eq. 7, we see
that there is a factor $\frac{4}{3}E$ that appears with the CPT conserving LIV
parameters. Therefore, we expect that ICAL will be sensitive to these
parameters for higher values of $E$. For the diagonal parameters, the behavior
of $c_{\alpha\alpha}$ is similar to that of $a_{\alpha\alpha}$. From the
panels, we see that the effect of LIV is weak in $c_{ee}$ compared to the
other two parameters. For $c_{ee}$, the effect of LIV is visible in the energy
range of 2 GeV to 20 GeV in neutrinos and 0.1 GeV to 10 GeV for antineutrinos.
For $c_{\mu\mu}$ and $c_{\tau\tau}$, the effect is similar for neutrinos and
antineutrinos, with some small differences along the $\cos\theta$ axis. For
the off-diagonal parameter $c_{e\mu}$, the effect of LIV is mostly the same in
neutrinos and antineutrinos. For this parameter, the effect of LIV is visible
in the energy range of 1-10 GeV, which has the most matter effect for
neutrinos but is absent for antineutrinos. For $c_{e\tau}$, the effect of LIV
is more prominent in neutrinos than in antineutrinos. The parameter
$c_{\mu\tau}$ has a similar effect in neutrinos and antineutrinos. From the
above discussion, we understand that ICAL will have weaker bounds on $c_{ee}$
as compared to the other CPT-conserving LIV parameters.

###  IV.2 DUNE

In Fig. 3 and Figure 4, we present the appearance channel
($\nu_{\mu}\rightarrow\nu_{e}$) probabilities as a function of energy ($E$)
for DUNE baseline, considering $a_{\alpha\beta}$ and $c_{\alpha\beta}$,
respectively. The first and second rows in both figures correspond to diagonal
LIV parameters, while the third and fourth rows correspond to off-diagonal LIV
parameters. In the first and third rows, we set the value of $\delta_{\rm CP}$
to $0^{\circ}$, whereas in the second and fourth rows, it is set to
$-90^{\circ}$. In each panel, the black curve represents standard oscillations
in the three-flavor scenario. For diagonal LIV parameters, we present curves
for both positive and negative values, and for non-diagonal LIV parameters, we
show curves for four values of $\phi^{a/c}_{\alpha\beta}$: $0^{\circ}$,
$90^{\circ}$, $180^{\circ}$, and $270^{\circ}$. These figures are only for
neutrinos. To understand the effect of LIV in the disappearance channel, we
refer to Fig.1 and Fig.2. The red line indicates the relevant regions in the
oscillograms for DUNE.

Figure 3: The $\nu_{e}$ appearance probability plots for $a_{ee}$,
$a_{\mu\mu}$, $a_{\tau\tau}$ (first/second row is for $\delta_{\rm
CP}=0^{\circ}/-90^{\circ}$) and $a_{e\mu}$, $a_{e\tau}$, $a_{\mu\tau}$
(third/fourth row is for $\delta_{\rm CP}=0^{\circ}/-90^{\circ}$) for DUNE
setup. In all panels the probabilities for standard three generation
oscillations are shown by the solid black curves.

Figure 4: The $\nu_{e}$ appearance probability plots for $c_{ee}$,
$c_{\mu\mu}$, $c_{\tau\tau}$ (first/second row is for $\delta_{\rm
CP}=0^{\circ}/-90^{\circ}$) and $c_{e\mu}$, $c_{e\tau}$, $c_{\mu\tau}$
(third/fourth row is for $\delta_{\rm CP}=0^{\circ}/-90^{\circ}$) for DUNE
setup. In all panels the probabilities for standard three generation
oscillations are shown by the solid black curves.

From these figures, we can deduce the following physics points. From the
analytical calculation of the appearance channel probability it has been shown
that the LIV parameters $a_{e\mu}$, $a_{e\tau}$ and $a_{ee}$ dominate at the
leading order Kumar Agarwalla and Masud (2020). Therefore at DUNE, we expect
to obtain good sensitivity for these parameters. Amongst $a_{\alpha\alpha}$,
we observe that DUNE is more sensitive to the parameters $a_{ee}$ and
$a_{\tau\tau}$ as compared to $a_{\mu\mu}$. The effect of LIV for the
parameter $a_{ee}$ is opposite to that of $a_{\tau\tau}$. For positive values
of $a_{ee}$, the probability is higher than the probability in the standard
three-flavour scenario, and for a negative value of $a_{ee}$, the probability
is lower than the probability in the standard three-flavour scenario. This
behaviour is opposite for $a_{\tau\tau}$. The effect of LIV on $a_{\mu\mu}$ is
negligible. Regarding $a_{\alpha\beta}$ with $\alpha\neq\beta$, the effect of
LIV in DUNE is greater in $a_{e\mu}$ and $a_{e\tau}$ compared to
$a_{\mu\tau}$. For a given value of $a_{e\mu}$, the difference with the
standard oscillation probability is greater for $\phi^{a}_{e\mu}=0^{\circ}$
and $180^{\circ}$ at $\delta_{\rm CP}=0^{\circ}$, and it is greater for
$\phi^{a}_{e\mu}=90^{\circ}$ and $270^{\circ}$ at $\delta_{\rm
CP}=-90^{\circ}$. For $a_{e\tau}$, the behavior is opposite to that of
$a_{e\mu}$, i.e., for $\delta_{\rm CP}=0^{\circ}$, the separation of the
probability from the standard oscillation probability in the presence of
$a_{e\tau}$ is greater at $\phi^{a}_{e\tau}=90^{\circ}$ and $270^{\circ}$, and
for $\delta_{\rm CP}=-90^{\circ}$, the separation is greater for
$\phi^{a}_{e\tau}=0^{\circ}$ and $180^{\circ}$. The effect of $a_{\mu\tau}$
for the appearance channel is seen to be negligible for DUNE.

In general, the appearance channel in DUNE is less sensitive to CPT-conserving
LIV parameters as compared to CP-violating LIV parameters. The behavior of
$c_{\alpha\alpha}$ is seen to be similar to that of $a_{\alpha\alpha}$. There
is more sensitivity to parameters $c_{ee}$ and $c_{\tau\tau}$ as compared to
$c_{\mu\mu}$. The effect of the parameter $c_{ee}$ is opposite to that of
$c_{\tau\tau}$. For a positive values of $c_{ee}$, the probability is lower
compared to the probability in the standard three-flavor scenarios, while for
negative values of $c_{ee}$, the probability is higher than the probability in
the standard three-flavor scenarios. The trend is opposite for $c_{\tau\tau}$.
The effect of $c_{\mu\mu}$ is negligible.

The behavior of $c_{\alpha\beta}$ with $\alpha\neq\beta$ is similar to that of
$a_{\alpha\beta}$. The effect is more pronounced for $c_{e\mu}$ and
$c_{e\tau}$ as compared to $c_{\mu\tau}$. For a given value of $c_{e\mu}$, the
difference with the standard oscillation probability is higher for
$\phi^{c}_{e\mu}=0^{\circ}$ and $180^{\circ}$ when $\delta_{\rm
CP}=0^{\circ}$, and it is higher for $\phi^{c}_{e\mu}=90^{\circ}$ and
$270^{\circ}$ when $\delta_{\rm CP}=-90^{\circ}$. For $c_{e\tau}$ the
situation is opposite to what we see for $c_{e\mu}$. The effect of
$c_{\mu\tau}$ is negligible.

Upon examining the previous discussion it becomes clear that the appearance
channel of DUNE is weakly sensitive to the parameters $a_{\mu\mu}$,
$a_{\mu\tau}$, $c_{\mu\mu}$, and $c_{\mu\tau}$. However, the disappearance
channel exhibits better sensitivity to these parameters. Thus, by combining
information from both channels, it will be possible to achieve some
sensitivity for these four parameters.

###  IV.3 T2HK

Fig. 5 depicts the same information for the appearance channel as shown in
Fig. 3, but for the T2HK baseline. For the disappearance channel in T2HK we
refer to the black line in Fig. 1. From the figures we note that the effect of
LIV parameters in T2HK has features similar to what we saw in the case of
DUNE. However, for T2HK the separation between the probabilities corresponding
to LIV and standard three-flavor oscillation is considerably less than that in
DUNE. Thus, we expect to have the same behavior as in DUNE, but with
significantly less sensitivity.

Figure 5: The $\nu_{e}$ appearance probability plots for $a_{ee}$,
$a_{\mu\mu}$, $a_{\tau\tau}$ (first/second row is for $\delta_{\rm
CP}=0^{\circ}/-90^{\circ}$) and $a_{e\mu}$, $a_{e\tau}$, $a_{\mu\tau}$
(third/fourth row is for $\delta_{\rm CP}=0^{\circ}/-90^{\circ}$) for T2HK
setup. In all panels the probabilities for standard three generation
oscillations are shown by the solid black curves.

We do not present any figures for CP conserving LIV parameters for T2HK
because we have explicitly checked that for the T2HK baseline and energy, the
probability curves corresponding to standard oscillation and the probability
curves for different values of $c_{\alpha\beta}$ are almost inseparable.

##  V Results

In this section we will present the capability of T2HK, DUNE and ICAL to limit
the LIV parameters. We will estimate the sensitivity by calculating a
$\chi^{2}$ function defined as Moon Moon Devi and Dighe (2014):

$\chi^{2}=\sum_{i}2\left[(T_{i}-D_{i})-D_{i}ln(\frac{T_{i}}{D_{i}})\right]+\sum_{j}\xi^{2}_{j}$

(8)

where the sum is over the energy bins for T2HK and DUNE whereas in ICAL the
sum is over $E_{\mu}^{obs}$, $\theta_{\mu}^{obs}$ and $E_{\rm had}^{obs}$
bins. The number of events in each bin for theory (data) is given by $T_{i}$
$(D_{i})$. To implement the systematic errors, the theory events are varied
as:

$T_{i}=T^{0}_{i}\left(1+\sum_{j}\pi^{j}_{i}\xi_{j}\right)$ (9)

where $T^{0}_{i}$ is the corresponding number of events in theory without
systematic errors. The pull parameters $\xi_{j}$ correspond to different
sources of systematic uncertainties.

The oscillation parameters that we use in our calculations are specified in
Table 2. In our analysis, we use $\Delta m^{2}_{\rm eff}$ given by Nunokawa
_et al._ (2005); Raut (2013)

(10)

We perform minimisation of the $\chi^{2}$ over $\theta_{23}$ in the range
$40^{\circ}$ to $51^{\circ}$, $|\Delta m^{2}_{\rm eff}|$ within its current
3$\sigma$ range, and the mass ordering. The phase $\delta_{\rm CP}$ is varied
over their full range for DUNE and T2HK. For ICAL, we have set $\delta_{\rm
CP}=0^{\circ}$ both in theory and data since the $\chi^{2}$ for ICAL depends
weakly on $\delta_{\rm CP}$. For T2HK and DUNE, we consider two values of
$\delta_{\rm CP}$(true), namely $0^{\circ}$ and $-90^{\circ}$. Our analysis
keeps $\Delta m^{2}_{21}$, $\theta_{13}$, and $\theta_{12}$ fixed. We present
our results for true normal ordering of the neutrino masses. For true inverted
ordering the plots are qualitatively similar, so we do not present them here
for brevity.

$\Delta m^{2}_{21}$(eV${}^{2})$ |  $\Delta m^{2}_{\rm eff}$(eV${}^{2})$ | $\sin^{2}\theta_{12}$ | $\sin^{2}\theta_{23}$ | $\sin^{2}2\theta_{13}$ | $\delta_{\rm CP}$
---|---|---|---|---|---
7.42$\times 10^{-5}$ | 2.49$\times 10^{-3}$ | 0.33 | 0.5 | 0.0875 | $0^{\circ}/-90^{\circ}$
fix | 2.38-2.62$\times 10^{-3}$ | fix | 0.41-0.6 | fix | $0-360^{\circ}$

Table 2:  Oscillation parameters used in analysis

###  V.1 Sensitivity for $a_{\alpha\alpha}$

Let us begin our discussion with the diagonal CPT-violating LIV parameters. In
Fig. 6, we present the bounds on the parameters in the $\chi^{2}$ vs
$a_{\alpha\alpha}$ plane. The left column is for $\delta_{\rm CP}=0^{\circ}$,
and the right column is for $\delta_{\rm CP}=-90^{\circ}$. In each column,
different panels correspond to different $a_{\alpha\alpha}$ parameters. In
each panel, we present the individual sensitivities of T2HK, DUNE, ICAL, and
the combined sensitivity of all these three experiments.

Figure 6: $\chi^{2}$ as a function of LIV parameters $a_{ee}$, $a_{\mu\mu}$
and $a_{\tau\tau}$ for true $\delta_{\rm CP}$ = $0^{\circ}$ (left column) and
$-90^{\circ}$ (right column) in ICAL, DUNE, T2HK and combined.

From the discussion on the probabilities, we understood that ICAL has weaker
sensitivity on $a_{ee}$ as compared to DUNE. On the other hand the sensitivity
to $a_{\mu\mu}$ is poor in the appearance channel for DUNE, making it more
difficult to measure this parameter in DUNE as compared to $a_{ee}$. For
$a_{\tau\tau}$ both DUNE and ICAL are sensitive. T2HK was seen to have the
weakest dependence on LIV. These features are clearly visible in Fig. 6. From
the panels we see that DUNE has the best sensitivity for $a_{ee}$, with better
sensitivity expected for $\delta_{\rm CP}=-90^{\circ}$ as compared to
$\delta_{\rm CP}=0^{\circ}$. For $a_{\mu\mu}$ and $a_{\tau\tau}$, both ICAL
and DUNE have comparable sensitivities. Also, for these parameters we expect
similar sensitivity for both choices of $\delta_{\rm CP}$. For all the cases,
T2HK has the weakest sensitivity. As expected, when we combine all the three
experiments, we obtain the best possible sensitivity for all the parameters.
In Table 3 we have listed the 95$\%$ C.L. sensitivity limit of these
parameters for both $\delta_{\rm CP}=0^{\circ}$ and $-90^{\circ}$.

###  V.2 Sensitivity for $a_{\alpha\beta}$

Let us now discuss the sensitivities for the non-diagonal CPT violating LIV
parameters. In Fig. 7 we present contour plots at 95$\%$ C.L. (2 dof) in the
$|a_{\alpha\beta}|$ vs $\phi^{a}_{\alpha\beta}$ plane. The left column is for
$\delta_{\rm CP}=0^{\circ}$ and the right column is for $\delta_{\rm
CP}=-90^{\circ}$. In each column, different panels corresponds to different
$a_{\alpha\beta}$ parameters. In each panel, we have presented the individual
sensitivities of T2HK, DUNE, ICAL and the combined sensitivity of all these
three experiments.

Figure 7: 95$\%$ C.L. (2 dof) contour plots between $|a_{\alpha\beta}|$ and
$\phi^{a}_{\alpha\beta}$. Left column is for $\delta_{\rm CP}=0^{\circ}$ and
right column is for $\delta_{\rm CP}=-90^{\circ}$.

From the discussion on the probability level, it is evident that ICAL has good
sensitivity for $a_{\mu\tau}$ while long-baseline experiments are expected to
have good sensitivity for parameters $a_{e\mu}$ and $a_{e\tau}$. Furthermore,
it has been observed that between the two long-baseline experiments, T2HK has
weaker sensitivity than DUNE. This is consistent with the features seen in
Fig.7. For $a_{\mu\tau}$, ICAL has the best sensitivity, while DUNE is also
has reasonably sensitive. The upper bound for this parameter is obtained at
$\phi^{a}_{\mu\tau}=90^{\circ}$ for all the three experiments and both values
of $\delta_{\rm CP}$. For $a_{e\mu}$, T2HK has weaker sensitivity than DUNE
but better than ICAL. In ICAL, the upper bounds for $a_{e\mu}$ and $a_{e\tau}$
correspond to $\phi^{a}_{e\mu}/\phi^{a}_{e\tau}=120^{\circ}$. As long-baseline
experiments and ICAL depend differently on $\phi^{a}_{\alpha\beta}$ for
$a_{e\mu}$ and $a_{e\tau}$, combining all three experiments is expected to
provide the best possible sensitivity. Table 3 lists the 95$\%$ C.L.(1 dof)
sensitivity limit for these parameters for $\delta_{\rm CP}=0^{\circ}$ and
$-90^{\circ}$. To obtain the limits quoted in this table, we have minimised
the $\chi^{2}$ over $\phi^{a}_{\alpha\beta}$.

LIV parameters | ICAL | DUNE | T2HK | Combined
---|---|---|---|---
$a_{ee}$ | -6/4.5 | -2.84/2.56(-2.7/2.5) | -26/21.6(-8.5/8.8) | -1.95/2.0(-1.9/2.0)
$a_{\mu\mu}$ | -2.2/2.2 | -2.9/2.5(-2.7/2.4) | -12/13(-12/13.4) | -1.3/1.3(-1.3/1.4)
$a_{\tau\tau}$ | -2.3/2.2 | -2.9/2.5(-2.7/2.4) | -7.7/8.0(-8.4/7.5) | -1.5/1.5(-1.4/1.4)
$a_{e\mu}$ | 3.3 | 0.64(0.56) | 2.2(2.0) | 0.45(0.4)
$a_{e\tau}$ | 4.4 | 1.1(0.9) | 5.8(2.3) | 0.6(0.63)
$a_{\mu\tau}$ | 1.0 | 1.95(2.0) | 6.8(6.9) | 0.95(0.96)

Table 3: 95$\%$ C.L. (1 dof) bounds for ICAL, DUNE, T2HK and their combination
in the units of $10^{-23}$ GeV. For $a_{\alpha\alpha}$, we have given two
values. One corresponds to $+$ve values of the parameters and the other for
$-$ve values. For DUNE, T2HK and Combined, we have given two different limits,
one is for true $\delta_{\rm CP}=0^{\circ}$ (outside parentheses) and other is
for true $\delta_{\rm CP}=-90^{\circ}$ (inside parentheses).

###  V.3 Sensitivity for $c_{\alpha\alpha}$

In this subsection, we will discuss the sensitivity of the different
experiments to the diagonal CPT conserving LIV parameters. Fig. 8 presents the
same information as Fig. 6 but for $c_{\alpha\alpha}$. We have not provided
any curves for T2HK in these panels, as its sensitivity to CPT conserving LIV
parameters is very weak.

Figure 8: $\chi^{2}$ as a function of LIV parameters $c_{ee}$, $c_{\mu\mu}$
and $c_{\tau\tau}$ for true $\delta_{\rm CP}$ = $0^{\circ}$ (left column) and
$-90^{\circ}$ (right column) in ICAL, DUNE and combined.

The probability discussion had shown that ICAL has better sensitivity to
diagonal CPT-conserving LIV parameters than DUNE, except for $c_{ee}$. We had
also seen that for long-baseline experiments the behavior of
$c_{\alpha\alpha}$ with respect to the appearance channel probabilities is
similar to that of $a_{\alpha\alpha}$, but the changes in the probability due
to $c_{\alpha\alpha}$ are somewhat lower than those due to $a_{\alpha\alpha}$.
From Fig. 8 DUNE gives the best sensitivity for $c_{ee}$, while ICAL is better
for the two. Combining the two experiments improves the sensitivity for
$c_{ee}$, but for the other two parameters adding DUNE provides only a slight
improvement. Sensitivities for all three parameters are similar for both
values of $\delta_{\rm CP}$. Table 4 lists the 95$\%$ C.L. (1 dof) sensitivity
limit for these parameters for both choices of $\delta_{\rm CP}=0^{\circ}$ and
$-90^{\circ}$.

###  V.4 Sensitivity for $c_{\alpha\beta}$

In this subsection, we will discuss the sensitivity for the off-diagonal CPT
conserving LIV parameters. Fig. 9 is the same as Fig. 7 but for
$c_{\alpha\beta}$ (when $\alpha\neq\beta$). Again we do not present the curves
for T2HK.

Figure 9: 95$\%$ C.L. (2 dof) contour plots between $|c_{\alpha\beta}|$ and
$\phi^{c}_{\alpha\beta}$. Left column is for $\delta_{\rm CP}=0^{\circ}$ and
right column is for $\delta_{\rm CP}=-90^{\circ}$.

For these set of parameters, we should get sizable amount of sensitivity from
ICAL. Regarding the long-baseline experiments, as understood from the
probabilities, the behavior of $c_{\alpha\beta}$ is similar as that of
$a_{\alpha\beta}$ with a lower sensitivity. For $c_{e\mu}$, DUNE gives better
sensitivity than ICAL except $\delta_{\rm CP}=-90^{\circ}$ and
$\phi^{c}_{e\mu}=180^{\circ}$. For $c_{e\tau}$, the sensitivity of DUNE is
better than ICAL in some of the regions depending on the values of
$\delta_{\rm CP}$ and $\phi^{c}_{e\tau}$. For DUNE, the upper bound
corresponds to $\phi^{c}_{e\mu}=50^{\circ}$ for $\delta_{\rm CP}=0^{\circ}$
and $\phi^{c}_{e\mu}=90^{\circ}$ for $\delta_{\rm CP}=-90^{\circ}$ in
$c_{e\mu}$. For $c_{e\tau}$, the upper bound comes at
$\phi^{c}_{e\tau}=10^{\circ}$ for $\delta_{\rm CP}=0^{\circ}$ and
$\phi^{c}_{e\tau}=90^{\circ}$ for $\delta_{\rm CP}=-90^{\circ}$ in DUNE. For
ICAL, the upper bound comes at $\phi^{c}=90^{\circ}$ for both $c_{e\mu}$ and
$c_{e\tau}$. For these two parameters, the sensitivity improves when DUNE and
ICAL are combined and in this case the upper bound corresponds to around
$\phi^{c}_{e\mu}=90^{\circ}$. For $c_{\mu\tau}$, the sensitivity of DUNE is
weak as compared to ICAL. Therefore when combined the improvement in the
sensitivity is very marginal. In this case the upper bound corresponds to
$\phi^{c}_{\mu\tau}=90^{\circ}$. For this parameter, the sensitivity is
similar for both the values of $\delta_{\rm CP}$. In Table 4 we have listed
the 95$\%$ C.L.(1 dof) sensitivity limit of these parameters for both values
of $\delta_{\rm CP}$. To obtain the limits quoted in this table, we have
minimised the $\chi^{2}$ over $\phi^{c}_{\alpha\beta}$.

LIV parameters | ICAL | DUNE | Combined
---|---|---|---
$c_{ee}$ | -62/80 | -86/124(-90/140) | -41/80(-43/65)
$c_{\mu\mu}$ | -9.4/7.3 | -136/103(187/140) | -7.8/6(-7.5/7.6)
$c_{\tau\tau}$ | -8/8.5 | -155/112(-153/113) | -6.1/6.5(-6.9/6.3)
$c_{e\mu}$ | 7.8 | 8.2(8.9) | 5.5(5.2)
$c_{e\tau}$ | 16.8 | 34(32.02) | 11(13.8)
$c_{\mu\tau}$ | 1.6 | 11.0(11.0) | 1.6(1.6)

Table 4: 95$\%$ C.L.(1 dof) bounds for ICAL, DUNE and their combination in the
units of $10^{-25}$. For $c_{\alpha\alpha}$ parameters, we have given two
values. One is corresponding to $+$ve value and another is for $-$ve value.
For DUNE, T2HK and their combination, we have given two different limits, one
is for $\delta_{\rm CP}=0^{\circ}$(outside parenthesis) and other is for
$\delta_{\rm CP}=-90^{\circ}$(inside parenthesis).

##  VI Comparison with previous results

In this section, we will compare the results obtained in our study with
previous results. It is important to note that in our work Lorentz invariance
violation is isotropic in nature and this manifests as an alteration of the
neutrino oscillation probabilities. However, Lorentz invariance violation can
be also anisotropic and its effect can be studied by looking at the sidereal
variations in the event rates in an experiment. All the bounds that we provide
in this section are for isotropic LIV. For bounds on the anisotropic LIV, we
refer to Ref. Kostelecky and Russell (2011). In Table 5, we have compiled all
existing bounds on LIV parameters, along with our bounds obtained from the
combined analysis of ICAL, T2HK, and DUNE. For $a_{\alpha\beta}$, we present
the bounds obtained from ICAL+T2HK+DUNE, while for $c_{\alpha\beta}$, we
present the bounds obtained from ICAL+DUNE. Individual bounds from these
experiments can be found in Tables 3 and 4. Additionally, we have presented a
bar chart in Fig. 10 to show the comparison between our bounds and the current
best available bounds on LIV parameters.

Experiments |  Details |  95$\%$ C.L.(1 dof) $a_{\alpha\beta}$ in GeV $(c_{\alpha\beta})$ |  Ref.
---|---|---|---
SK (Atmospheric) |  SK atmospheric data |  $e\mu$ = 1.8$\times 10^{-23}$ (8$\times 10^{-27}$) |  Abe  _et al._ (2015)
|  |  $e\tau$ = 2.8$\times 10^{-23}$ (9.3$\times 10^{-25}$) |
|  |  $\mu\tau$ = 5.1$\times 10^{-24}$ (4.4$\times 10^{-27}$) |
IceCube |  IceCube data analyzed for $\mu\tau$ LIV |  $\mu\tau$ = 2.0$\times 10^{-24}$ (2.7$\times 10^{-28}$) $90\%$ C.L. |  Aartsen  _et al._ (2018)
|  parameters for 3, 4, 5, 6 and 7 dim. operators |  $\tau\tau$ = 2.0$\times 10^{-26}$ (2.0$\times 10^{-31}$) $90\%$ C.L. |
ICAL |  Atmospheric neutrino simulated |  $e\mu$ = 1.34$\times 10^{-23}$ (N.A) |  Sahoo  _et al._ (2022)
|  between 1-25 GeV range for $a_{\alpha\beta}$ |  $e\tau$ = 1.58$\times 10^{-23}$ (N.A) |
|  |  $\mu\tau$ = 2.2$\times 10^{-24}$ (N.A) |
DUNE |  Long-baseline neutrino simulated |  $e\mu$ = 7$\times 10^{-24}$ (N.A) |  Barenboim  _et al._ (2019)
|  between 0.2-10 GeV range for $a_{\alpha\alpha}$ |  $e\tau$ = 1.0$\times 10^{-23}$ (N.A) |
|  and $a_{\alpha\beta}$ |  $\mu\tau$ = 1.7$\times 10^{-23}$ (N.A) |
|  |  $ee$ = (-2.5, 3.2)$\times 10^{-23}$ (N.A) |
|  |  $\mu\mu$ = (-3.7, 2.8)$\times 10^{-23}$ (N.A) |
T2K+NOVA |  NOVA and T2K long-baseline |  $e\mu$ = 3.6$\times 10^{-23}$ (N.A) |  Majhi  _et al._ (2020)
|  experiments simulation for LIV |  $e\tau$ = 1.08$\times 10^{-22}$ (N.A) |
|  sensitivity for $a_{\alpha\alpha}$ and $a_{\alpha\beta}$ |  $\mu\tau$ = 8$\times 10^{-23}$ (N.A) |
|  |  $ee$ = (-5.5, 3.4)$\times 10^{-22}$ (N.A) |
|  |  $\mu\mu$ = (-1.07, 1.18)$\times 10^{-22}$ (N.A) |
|  |  $\tau\tau$ = (-1.12, 0.9)$\times 10^{-22}$ (N.A) |
DUNE+P2O |  Long-baseline DUNE and P2O data |  $e\mu$ = 4.7$\times 10^{-24}$ (N.A) |  Fiza  _et al._ (2022)
|  simulated for LIV sensitivity for $a_{\alpha\alpha}$ |  $e\tau$ = 6$\times 10^{-24}$ (N.A) |
|  and $a_{\alpha\beta}$ |  $\mu\tau$ = 1.3$\times 10^{-23}$ (N.A) |
|  |  $ee$ = (-2.6, 3.3)$\times 10^{-23}$ (N.A) |
|  |  $\mu\mu$ = (-1.5, 1.6)$\times 10^{-23}$ (N.A) |
ICAL+T2HK+DUNE |  Combining atmospheric and |  $e\mu$ = 4$\times 10^{-24}$ (5.1$\times 10^{-25}$) |  This
($a_{\alpha\beta}$) |  long-baseline experiments with $a_{\alpha\alpha}$, |  $e\tau$ = 6$\times 10^{-24}$ (1.1$\times 10^{-24}$) |  work
|  $a_{\alpha\beta}$, $c_{\alpha\alpha}~{}{\rm and}~{}c_{\alpha\beta}$. ICAL with extended |  $\mu\tau$ = 9.5$\times 10^{-24}$ (1.6$\times 10^{-25}$) |
ICAL+DUNE |  range 1-100 GeV. DUNE with recent |  $ee$ = (-2.1, 2.1)$\times 10^{-23}$ ((-4.2, 6.5)$\times 10^{-24}$) |
($c_{\alpha\beta}$) |  TDR included. Exploring $\phi_{\alpha\beta}$ impact |  $\mu\mu$ = (-1.8, 1.9)$\times 10^{-23}$ ((-7.4, 6)$\times 10^{-25}$) |
|  on experiments and combined analysis. |  $\tau\tau$ = (-1.5, 1.5)$\times 10^{-23}$ ((-6.7, 6.4)$\times 10^{-25}$) |

Table 5: 95$\%$ C.L. (1 dof) limit of the LIV parameters from available
literature, except for IceCube. For IceCube, the bounds are available only at
90% C.L.

Figure 10: In the left panel, we have shown our limit for CPT odd parameters
(red bars) and current best bounds (green bars) which are listed in Table 5.
In right panel, we have shown results for CPT even parameters. Note that the
current bound for $c_{\tau\tau}$ is at 90% C.L whereas all the other bounds
are at 95% C.L.

It is worth noting that the study of LIV parameters in the context of DUNE and
ICAL has been previously conducted. However, our analysis presents some new
features, which we would like to highlight.

  * •

Ref.Sahoo  _et al._ (2022) previously studied CPT odd LIV parameters for off-
diagonal elements ($a_{\alpha\beta}$, where $\alpha\neq\beta$) for ICAL,
assuming that complex phases have a negligible effect on the sensitivity. They
used the Nuance MC generator for event generation and limited the muon energy
range from 1 to 25 GeV. However, in our analysis, we considered the impact of
nontrivial $\phi_{\alpha\beta}$ on the LIV parameters and used the GENIE MC
generator, which employs updated cross-section values. Moreover, we
implemented an extended binning scheme, which expanded the muon energy range
from 25 GeV to 100 GeV. As a result, our sensitivity bounds are more stringent
than those obtained in Ref.Sahoo  _et al._ (2022).

  * •

In Ref.Barenboim  _et al._ (2019), the authors studied CPT odd LIV parameters
in DUNE, while Ref.Fiza  _et al._ (2022) combined P2O with DUNE. Both studies
used the conceptual design report of DUNE (CDR) Acciarri  _et al._ (2016) for
detector configuration, efficiency, resolution, and systematic uncertainties.
In our analysis, we used the recently published technical design report of
DUNE (TDR) Abi  _et al._ (2020b) to build our detector setup and consider new
efficiency, resolution, and systematic uncertainties. As a result, our
sensitivity is improved compared to Ref. Barenboim  _et al._ (2019).

Now let us compare the bounds for the individual parameters as obtained from
different experiments.

  * •

The CPT odd diagonal parameters $a_{\alpha\alpha}$ have been studied in the
context of DUNE Barenboim  _et al._ (2019), T2K+NO$\nu$A Majhi  _et al._
(2020), and DUNE+P2O Fiza  _et al._ (2022). In Refs.Fiza  _et al._ (2022) and
Barenboim  _et al._ (2019), only two independent diagonal parameters,
$a_{ee}-a_{\tau\tau}$ and $a_{\mu\mu}-a_{\tau\tau}$, have been considered. In
Ref.Majhi  _et al._ (2020), all three diagonal parameters, $a_{ee}$,
$a_{\mu\mu}$, and $a_{\tau\tau}$, have been taken as independent parameters.
In our work, we have also considered all three diagonal parameters as
independent parameters. From Table 5, we can see that our results provide the
best sensitivity for $a_{ee}$.

  * •

The CPT-odd off-diagonal parameters $a_{\alpha\beta}$ (for $\alpha\neq\beta$)
have been studied in all the experiments listed in Table 5. Among them, the
combination of ICAL, T2HK, and DUNE provides the best sensitivity for the
parameter $a_{e\mu}$.

  * •

We have studied the CPT-odd diagonal parameter $c_{\alpha\alpha}$ in
atmospheric neutrinos using the ICAL detector setup and in the long baseline
experiment DUNE. For the parameter $c_{\tau\tau}$, IceCube Aartsen  _et al._
(2018) provides a stronger bound than our results222Note that the bound for
IceCube, listed in Table 5, is given at 90$\%$ C.L., whereas our results are
given at 95$\%$ C.L.. The other two CPT-even diagonal parameters $c_{ee}$ and
$c_{\mu\mu}$ have not been studied in any literature. Our work provides the
first-ever bounds on these parameters.

  * •

The CPT-even off-diagonal parameters $c_{\alpha\beta}$ (for $\alpha\neq\beta$)
have been studied in the context of SK Abe  _et al._ (2015) and IceCube
Aartsen  _et al._ (2018). In our work, we have simulated results for DUNE and
ICAL detector setups. Our results are comparable to SK for $c_{e\tau}$ only.
For $c_{e\mu}$, SK has given the best limit, and for $c_{\mu\tau}$, IceCube
has provided the strongest limit.

##  VII Summary and conclusion

In this paper, we presented a comprehensive study of Lorentz invariance
violation (LIV) in the context of atmospheric neutrino experiment ICAL and
long-baseline experiments T2HK and DUNE. We considered the full parameter
space of LIV parameters, which includes six CPT-violating LIV parameters
($a_{\alpha\beta}$) and six CPT-conserving LIV parameters ($c_{\alpha\beta}$).
Our objective was to calculate the upper bound on all LIV parameters for
individual experiments and their combination. Note that while some LIV
parameters have been studied in the context of ICAL and DUNE before, the
specifications used in our study differ from previous ones. For DUNE, we used
the configuration from the latest technical design report, while for ICAL, we
considered an extended energy region of 1 GeV to 100 GeV. Furthermore, we
studied the effect of phases associated with LIV parameters in ICAL, which was
not considered in earlier studies. As a result, our results are better as
compared to previous ones.

The sensitivity of ICAL to LIV parameters mainly comes from the disappearance
channel, whereas for DUNE and T2HK, the sensitivity can come from both the
appearance and the disappearance channels. At the probability level, we showed
that ICAL is mainly sensitive to $a_{\mu\mu}$, $a_{\tau\tau}$, and
$a_{\mu\tau}$, amongst the CPT-violating LIV parameters. For CPT-conserving
LIV parameters, ICAL is sensitive to all parameters except $c_{ee}$. For DUNE,
we showed that the appearance channel is not very sensitive to the parameters
$a_{\mu\mu}$, $a_{\mu\tau}$, $c_{\mu\mu}$, and $c_{\mu\tau}$, while being
reasonably sensitive to the others. Sensitivity to some of these could still
come from DUNE’s disappearance channel. For T2HK, the sensitivity to the CPT
odd parameters is weak, while the sensitivity to the CPT even parameters is
negligible.

At the $\chi^{2}$ level, we found that the results are consistent with the
conclusions that we derived from the oscillation probabilities. Among the
three experiments, DUNE gives the best sensitivity on $a_{ee}$, $a_{e\mu}$,
$a_{e\tau}$, and $a_{\mu\tau}$, whereas ICAL gives the best sensitivity on
$a_{\mu\mu}$, $c_{ee}$, $c_{\mu\mu}$, $c_{\tau\tau}$, $c_{e\mu}$, $c_{e\tau}$,
and $c_{\mu\tau}$. For $a_{\tau\tau}$, the sensitivity of DUNE and ICAL is
similar. When comparing the existing bounds on the LIV parameters from
different experiments with the combination of T2HK, DUNE, and ICAL (i.e.,
bounds from ICAL+T2HK+DUNE on $a_{\alpha\beta}$ and bounds from ICAL+DUNE on
$c_{\alpha\beta}$), we find that for the parameters $a_{ee}$ ($a_{\mu\mu}$),
our results are better (comparable) with DUNE+P2O. For the parameter
$a_{e\mu}$, the strongest bounds are from our results. For $c_{e\mu}$, SK has
given the best limit, and for $c_{\mu\tau}$, IceCube has given the strongest
limit. The bounds from SK on $c_{e\tau}$ are similar to what we obtained from
the combination of DUNE and ICAL. For the diagonal CPT-conserving isotropic
LIV parameters, our work provides the first ever bounds on $c_{ee}$ and
$c_{\mu\mu}$.

###### Acknowledgements.

This work is performed by the members of the INO-ICAL collaboration. We thank
the members of the INO-ICAL collaboration for their valuable comments and
constructive inputs. The HRI cluster computing facility
(http://cluster.hri.res.in) is gratefully acknowledged. This work has been in
part funded by Ministry of Science and Education of Republic of Croatia grant
No. KK.01.1.1.01.0001. This project has received funding/support from the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska -Curie grant agreement No 860881-HIDDeN.

## References

  * Kostelecky and Samuel (1989) V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989).
  * Colladay and Kostelecky (1998) D. Colladay and V. A. Kostelecky, Phys. Rev. D 58, 116002 (1998), arXiv:hep-ph/9809521 .
  * Abe  _et al._ (2018a) K. Abe _et al._ (Hyper-Kamiokande), PTEP 2018, 063C01 (2018a), arXiv:1611.06118 [hep-ex] .
  * Abi  _et al._ (2020a) B. Abi _et al._ (DUNE), (2020a), arXiv:2002.03005 [hep-ex] .
  * Ahmed  _et al._ (2017) S. Ahmed _et al._ (ICAL), Pramana 88, 79 (2017), arXiv:1505.07380 [physics.ins-det] .
  * Barenboim  _et al._ (2019) G. Barenboim, M. Masud, C. A. Ternes, and M. Tórtola, Phys. Lett. B 788, 308 (2019), arXiv:1805.11094 [hep-ph] .
  * Kumar Agarwalla and Masud (2020) S. Kumar Agarwalla and M. Masud, Eur. Phys. J. C 80, 716 (2020), arXiv:1912.13306 [hep-ph] .
  * Adrian-Martinez  _et al._ (2016) S. Adrian-Martinez _et al._ (KM3NeT), J. Phys. G 43, 084001 (2016), arXiv:1601.07459 [astro-ph.IM] .
  * Akindinov  _et al._ (2019) A. V. Akindinov _et al._ , Eur. Phys. J. C 79, 758 (2019), arXiv:1902.06083 [physics.ins-det] .
  * Fiza  _et al._ (2022) N. Fiza, N. R. Khan Chowdhury, and M. Masud, (2022), arXiv:2206.14018 [hep-ph] .
  * Sahoo  _et al._ (2022) S. Sahoo, A. Kumar, and S. K. Agarwalla, JHEP 03, 050 (2022), arXiv:2110.13207 [hep-ph] .
  * Sarker  _et al._ (2023) A. Sarker, A. Medhi, and M. M. Devi, (2023), arXiv:2302.10456 [hep-ph] .
  * Agarwalla  _et al._ (2023) S. K. Agarwalla, S. Das, S. Sahoo, and P. Swain, (2023), arXiv:2302.12005 [hep-ph] .
  * Abe  _et al._ (2015) K. Abe _et al._ (Super-Kamiokande), Phys. Rev. D 91, 052003 (2015), arXiv:1410.4267 [hep-ex] .
  * Aartsen  _et al._ (2018) M. G. Aartsen _et al._ (IceCube), Nature Phys. 14, 961 (2018), arXiv:1709.03434 [hep-ex] .
  * Crivellin  _et al._ (2021) A. Crivellin, F. Kirk, and M. Schreck, JHEP 04, 082 (2021), arXiv:2009.01247 [hep-ph] .
  * Majhi  _et al._ (2020) R. Majhi, S. Chembra, and R. Mohanta, Eur. Phys. J. C 80, 364 (2020), arXiv:1907.09145 [hep-ph] .
  * Lin  _et al._ (2022) H.-X. Lin, J. Tang, S. Vihonen, and P. Pasquini, Phys. Rev. D 105, 096029 (2022), arXiv:2111.14336 [hep-ph] .
  * Rahaman (2021) U. Rahaman, Eur. Phys. J. C 81, 792 (2021), arXiv:2103.04576 [hep-ph] .
  * Kostelecký and Mewes (2012) V. A. Kostelecký and M. Mewes, Physical Review D 85 (2012), 10.1103/physrevd.85.096005.
  * Giganti  _et al._ (2018) C. Giganti, S. Lavignac, and M. Zito, Progress in Particle and Nuclear Physics 98, 1 (2018).
  * Huber  _et al._ (2005) P. Huber, M. Lindner, and W. Winter, Comput. Phys. Commun. 167, 195 (2005), arXiv:hep-ph/0407333 .
  * Huber  _et al._ (2007) P. Huber, J. Kopp, M. Lindner, M. Rolinec, and W. Winter, Comput. Phys. Commun. 177, 432 (2007), arXiv:hep-ph/0701187 .
  * Lakshmi  _et al._ (2014) S. M. Lakshmi, A. Ghosh, M. M. Devi, D. Kaur, S. Choubey, A. Dighe, D. Indumathi, M. V. N. Murthy, and M. Naimuddin, Journal of Instrumentation 9, T09003 (2014).
  * Agostinelli  _et al._ (2003) S. Agostinelli, J. Allison, and G. collaboration, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 506, 250 (2003).
  * Honda  _et al._ (2015) M. Honda, M. S. Athar, T. Kajita, K. Kasahara, and S. Midorikawa, Phys. Rev. D 92, 023004 (2015).
  * AJMI and MAJUMDER (2017) A. AJMI and G. MAJUMDER, Pramana 88, 55 (2017).
  * Barger  _et al._ (1980) V. Barger, K. Whisnant, S. Pakvasa, and R. J. N. Phillips, Phys. Rev. D 22, 2718 (1980).
  * Chatterjee  _et al._ (2014) A. Chatterjee, K. K. Meghna, R. Kanishka, T. Thakore, V. Bhatnagar, R. Gandhi, D. Indumathi, N. K. Mondal, and N. Sinha, Journal of Instrumentation 9, P07001 (2014).
  * Devi  _et al._ (2013) M. M. Devi, A. Ghosh, D. Kaur, L. S. Mohan, S. Choubey, A. Dighe, D. Indumathi, S. Kumar, M. V. N. Murthy, and M. Naimuddin, JINST 8, P11003 (2013), arXiv:1304.5115 [physics.ins-det] .
  * Abi  _et al._ (2021) B. Abi _et al._ (DUNE), (2021), arXiv:2103.04797 [hep-ex] .
  * Abe  _et al._ (2018b) K. Abe _et al._ (Hyper-Kamiokande), PTEP 2018, 063C01 (2018b), arXiv:1611.06118 [hep-ex] .
  * Bronner (2022) C. Bronner,  (2022), 10.5281/ZENODO.6683821.
  * Hartnell (2022) J. Hartnell,  (2022), 10.5281/ZENODO.6683827.
  * Moon Moon Devi and Dighe (2014) S. K. A. Moon Moon Devi, Tarak Thakore and A. Dighe, JHEP (2014), 10.1007/JHEP10(2014)189, arXiv:1406.3689 [hep-ph] .
  * Nunokawa  _et al._ (2005) H. Nunokawa, S. J. Parke, and R. Zukanovich Funchal, Phys. Rev. D 72, 013009 (2005), arXiv:hep-ph/0503283 .
  * Raut (2013) S. K. Raut, Mod. Phys. Lett. A 28, 1350093 (2013), arXiv:1209.5658 [hep-ph] .
  * Kostelecky and Russell (2011) V. A. Kostelecky and N. Russell, Rev. Mod. Phys. 83, 11 (2011), arXiv:0801.0287 [hep-ph] .
  * Acciarri  _et al._ (2016) R. Acciarri _et al._ (DUNE), (2016), arXiv:1601.02984 [physics.ins-det] .
  * Abi  _et al._ (2020b) B. Abi _et al._ (DUNE), JINST 15, T08010 (2020b), arXiv:2002.03010 [physics.ins-det] .


# Motion Segmentation from a Moving Monocular Camera

Yuxiang Huang1 and John Zelek1 1Yuxiang Huang and 1John Zelek are with Systems
Design Engineering Department, University of Waterloo, Waterloo, N2L 3G1,
Canada. {yuxiang.huang<EMAIL_ADDRESS>
###### Abstract

Identifying and segmenting moving objects from a moving monocular camera is
difficult when there is unknown camera motion, different types of object
motions and complex scene structures. To tackle these challenges, we take
advantage of two popular branches of monocular motion segmentation approaches:
point trajectory based and optical flow based methods, by synergistically
fusing these two highly complementary motion cues at object level. By doing
this, we are able to model various complex object motions in different scene
structures at once, which has not been achieved by existing methods. We first
obtain object-specific point trajectories and optical flow mask for each
common object in the video, by leveraging the recent foundational models in
object recognition, segmentation and tracking. We then construct two robust
affinity matrices representing the pairwise object motion affinities
throughout the whole video using epipolar geometry and the motion information
provided by optical flow. Finally, co-regularized multi-view spectral
clustering is used to fuse the two affinity matrices and obtain the final
clustering. Our method shows state-of-the-art performance on the KT3DMoSeg
dataset, which contains complex motions and scene structures. Being able to
identify moving objects allows us to remove them for map building when using
visual SLAM or SFM.

##  I INTRODUCTION

The objective of motion segmentation is to divide a video frame into regions
segmented by common motions. Motion segmentation from a moving camera is
particularly important as it has various applications in areas like autonomous
navigation, robotics and SLAM. In a dynamic scene, the video camera is moving
at an unknown velocity with respect to the environment. Such scenarios pose
many challenges to motion segmentation methods such as motion degeneracy,
motion parallax and motion on the epipolar plane [1]. Existing monocular
motion segmentation methods often fail when facing these challenges [2, 3, 4],
or need specific assumptions on the scene structure, object classes or motion
types [5, 6, 7].

In order to overcome these limitations and achieve high quality instance
motion segmentation results regardless of scene structures and motion types,
we draw our inspiration from two branches of well studied motion segmentation
approaches: optical flow based methods and point trajectory based methods.
These two types motion cues are not only complementary in nature (long-term vs
instantaneous motion), but they can also be used to derive highly
complementary geometric and motion models for different motion types and scene
structures: points trajectory based methods, when analyzed using epipolar
geometry, will fail if the motion is mainly on the epipolar plane, but it is
robust to depth variations, perspective effects and motion parallax; on the
other hand, optical flow based methods will fail on these challenges, but it
is robust to motions on the epipolar plane. We propose to combine these two
complimentary motion cues at object level to obtain a robust and comprehensive
motion representation of the scene. By using the state-of-the-art methods for
object recognition, detection, segmentation and tracking, we can easily obtain
an objectiveness prior (i.e., an initial grouping) of all motion data. This
approach enables us to analyze motions of each individual object, which is
crucial for both robots and humans to build their situational awareness and
scene understanding capabilities [8, 9].

To build our motion segmentation framework, we first leverage state-of-the-art
deep learning models [10, 11, 12, 13] to recognize, detect, segment and track
any common objects throughout the video. Then, for every object in the video,
we obtain (1) a set of sparse point trajectories for each object and (2) a
dense optical flow mask for each object on each frame. By using object-
specific sparse point trajectories and optical flow masks, we are able to
derive object-specific geometric models (i.e. fundamental matrices based on
epipolar geometry) and instantaneous motion models respectively on every frame
pair where the object is visible. By fusing these two highly complementary
models, we are theoretically able to model the vast majority of motions even
in complex scenes. Our experiments show significant improvements on motion
segmentation results in challenging scenarios, highlighting the approach’s
potential in real-world applications. In summary, the key contributions of our
paper are as follows:

  1. 1.

We combine the well-studied fundamental matrix motion model and the optical
flow based instantaneous motion model using multi-view spectral clustering to
model multiple complex motions in challenging scenes.

  2. 2.

We show how to model different motions at object level by incorporating per-
frame objectiveness prior obtained from recent computer vision foundational
models.

  3. 3.

We achieve state-of-the-art result on the challenging KT3DMoSeg dataset [4] in
terms of both producing high-quality point trajectory clustering and pixel-
level masks for individual moving instances.

##  II RELATED WORK

Monocular motion segmentation can be broadly categorized into three groups:
(1) Intensity based methods [14, 15, 16, 17, 18], (2) sparse correspondence
based methods [19, 20, 21, 22, 23, 24, 25, 4, 26] and (3) deep learning
methods [6, 27, 28, 7, 29, 30, 31, 5, 32].

###  II-A Intensity Based Methods

Intensity based methods can be further categorized into indirect and direct
methods. Indirect methods [15, 16, 17] rely on pixel-wise correspondences as
input, and produce a pixel-wise segmentation mask indicating different motion
groups. Such input is usually optical flow, which assumes the brightness or
color intensity of every point in the scene remains the same throughout the
whole sequence. In contrast, direct methods [14] do not require optical flow –
they combine the two processes of optimizing for the brightness consistency
constraint and estimating the motion models together and directly take a pair
of images as input. Most recent works on intensity based methods use optical
flow based indirect methods, possibly due to the fast advance in optical flow
estimation [33, 34]. Such methods usually adopt an iterative optimization
approach to estimate the motion model and motion region simultaneously.

Intensity-based methods tend to perform well on scenes without strong depth
variations or motion parallax as the motion flow vectors projected to a 2D
image from the 3D space are determined by both the depth and the screw motion
of objects [35]. Therefore, if the scene contains strong depth variation (e.g.
road scenes), intensity-based methods will fail to distinguish if a part of
the image is moving independently or is just at a different depth from its
surroundings.

###  II-B Sparse Correspondence Based Methods

Sparse correspondence based methods can be further categorized into two-frame
and multi-frame methods. Two frame methods [19, 20] usually recover motion
parameters by solving an iterative energy minimization problem of finding a
certain number of geometric models (e.g., fundamental matrices) on a set of
matched feature points, to minimize an energy function that evaluates the
quality of the overall clustering of correspondences. Multi-frame methods
often operate on manually corrected trajectory points obtained from a dense
optical flow tracker. Such methods usually perform clustering on affinity
matrices constructed using the results of geometric model fitting [25, 4, 26]
or pairwise affinities derived from spatio-temporal motion cues and appearance
cues [22, 24].

Point trajectories, when analyzed using epipolar geometry, are robust to depth
variations in the scene, but are prone to motions on the epipolar plane. [26]
uses trifocal tensor as a more robust model to analyze point trajectories.
Trifocal tensor is more robust to noise and motions on the epipolar plane, but
it is harder to optimize and prone to failure when the three cameras are close
to being colinear [1], which can often happen on road scenes. [4, 2] produce
promising results by combining multiple geometric models, but still fails to
produce a coherent and consistent segmentation on some scenes. Methods using
spatio-temporal information and appearance cues [22, 24] suffer from similar
issues. In addition, although they tend to perform a bit better than geometric
methods on motions with less rigidity [36], they perform worse on scenes with
motion parallax or strong camera motions.

###  II-C Deep Learning Based Methods

Deep learning based methods usually takes a pair or a sequence of input frames
as input and directly produces a either a binary segmentation mask of moving
vs static objects [6, 27, 28], or a multi-label segmentation mask showing
different objects of different motions [7, 29, 30, 31, 5, 32]. Most deep
learning methods use CNNs amd their network architecture can be broadly
summarized to have the following main components: (1) a module to extract the
appearance information from consecutive frames, (2) a module to extract motion
information from the same frames, (3) a module to fuse the appearance and
motion information, and (4) a decoder to generate the final segmentation.
These methods are usually fully-supervised and require a large amount of
training data. Besides, these methods are often only able to perform binary
motion detection (moving vs. static) [7, 31, 28], or they are only able to
detect instance motions for specific scenes and limited number of object types
they are trained on [30, 5]

##  III METHODOLOGY

Figure 1: Motion Segmentation Pipeline

###  III-A Object Recognition, Detection, Segmentation and Tracking

Figure 1 shows a diagram of our motion segmentation pipeline. In order to
identify all motions in a video sequence at object level, we first identify
every common object in the video and track their movements throughout the
video. We achieve this by using the most recent foundational models in object
recognition (Recognize Anything Model)[12], detection (Grounding DINO model)
[11] and segmentation (Segment Anything Model) [10], and a state-of-the-art
object tracker (DeAOT) [13]. We adapt our preprocessing pipeline from Segment
and Track Anything (SAMTrack) [37], which is an object segmentation and
tracking framework made of the Grounding DINO model, Segment Anything Model
(SAM) and the DeAOT tracker. SAMTrack allows the user to segment and track any
specific objects in the video with a text prompt. To make our system fully
automatic and universal to all motions in most scenes, we avoid using the
user-defined text prompt by adding RAM at the beginning of our pipeline to
automatically recognize any common objects in the video. In summary, our whole
preprocessing pipeline consists of the following main steps: 1) Use RAM to
recognize any common objects in the first frame of the video; 2) Feed the
output of RAM as a text prompt to the Grounding DINO model to obtain object
bounding boxes; 3) Feed these bounding boxes to SAM to obtain an instance
segmentation mask of the first frame. Non-max suppression was used to remove
objects with an IoU score $>$ 0.5 and instances whose bounding boxes are
larger than half the image are also removed; 4) Use the DeAOT tracker to track
each object’s mask throughout the entire video. In order to detect new objects
entering the scene in the middle of the video, we run step 1) every l frames
to check if there are new objects. If so, we then run steps 2) to 4) to
segment and track the new objects together with the existing objects in the
frame. The number l is video-specific, for example, more dynamic videos with
more objects entering and leaving the scene would benefit from a smaller l.

###  III-B Obtaining Point Trajectories and Optical Flow Masks

Once we have the instance segmentation mask for every frame of the video, we
then need to obtain a set sparse of point trajectories and a dense optical
flow mask on every object at every frame where it’s visible. Essentially, we
aim to obtain instance point trajectories and instance optical flows. We use
one of the state-of-the-art methods [38] to obtain optical flow for all
frames, then label each flow vector with the object label of the pixel.
Ideally, point trajectories need to be sampled from each object and
automatically tracked at each frame using a point tracker (e.g., [39, 40]),
however, for benchmarking purposes, we use the manually corrected point
trajectories provided by the KT3DMoSeg dataset. We assign every point
trajectory with an initial object label using the per-frame instance
segmentation masks. If a trajectory does not belong to any object, we label it
as the background. Due to inaccuracy in instance segmentation and tracking, a
point trajectory can be identified to be on different objects or background in
different frames. In such case, we assign its object label to be the most
frequent label it is identified as. In the future, we will use a point tracker
as well as robust point sampling and occlusion handling techniques to
automatically generate point trajectories for all detected objects.

###  III-C Model Fitting

After obtaining instance point trajectories and optical flow masks for each
frame, we compute the motion models of each object to model its motion
throughout the video. To compute the epipolar geometry based motion models
using point trajectories, we compute a fundamental matrix of each object
between every f frames by solving ${p^{\prime}TFp=0}$ using the eight-point
algorithm. If a degenerate case is encountered for the fundamental matrix, we
do not use it. For the optical flow based motion model, we use a full
quadratic motion model with 12 parameters to model the instantaneous object
screw motion:

$\begin{split}f(x,y)=&(a+bx+cy+dx^{2}+exy+fy^{2},\\\
&g+hx+iy+jx^{2}+kxy+ly^{2})\end{split}$ (1)

where (x, y) is the 2D coordinates of the pixels relative to the image center.
Since we already have the instance optical flow field, we can obtain the
following equation:

$\begin{split}f(x,y)=(u,v)\end{split}$ (2)

where $(u,v)$ is the optical flow vector of the pixel. We fit the function 1
above to the the optical flow vectors of every object and solve for the 12
parameters representing the object motion model by optimizing the mean squared
error. We use this specific motion model as it’s a simplified version of the
classic Longuet-Higgins and Pruzdny model equations [41], which model’s the
instantaneous screw motion of rigid objects at arbitrary depth. Since it’s not
possible to solve for the depth of each pixel, this motion model assumes the
objects’ depths are only slightly different. It was shown to perform well on
scenes with limited motion parallax [32], nevertheless, it often fails when
there is strong motion parallax and depth variations.

###  III-D Constructing Affinity Matrices

After all fundamental matrices and optical flow motion models are computed,
each object will have a fundamental matrix between every f frames and an
optical flow motion model between every two frames. By fitting every object’s
trajectory points and optical flow vectors to every other object’s fundamental
matrix and optical flow motion model on the same frame pair, we can obtain the
residuals of every object to all other objects’ motion models respectively. We
use Sampson distance as the residual for fundamental matrix [1] and mean
squared error for optical flow motion model. Assuming there are k objects in
the scene, for the i-th object at the m-th frame pair, we obtain the following
residual vectors under the fundamental matrix and optical flow motion models:

$\begin{split}{\boldsymbol{r}_{o}}_{i}^{m}=[{r_{o}}_{i,1}^{m},{r_{o}}_{i,2}^{m},...,{r_{o}}_{i,k}^{m}],\\\
{\boldsymbol{r}_{f}}_{i}^{m}=[{r_{f}}_{i,1}^{m},{r_{f}}_{i,2}^{m},...,{r_{f}}_{i,k}^{m}]\end{split}$

where ${r_{o}}_{i,k}^{m}$ is the mean residual for fitting the parametric
motion model of object i on the optical flow vectors of object k between
frames m and m \+ 1, and ${r_{f}}_{i,k}^{m}$ is the mean Sampson error for
fitting the fundamental matrix of object i on the trajectory points of object
k between frames m and m + f. We construct two affinity matrices encapsulating
the pairwise motion affinities between each pair of objects using the ordered
residual kernal (ORK) [42]. More specifically, for each object, we sort its
residual vectors in ascending order and define a threshold to select the
smallest t-th residual as inliers. We define
${\boldsymbol{c}_{i}}=\\{0,1\\}^{K}$ as the inlier mask to denote if an object
i is an inlier for each of the K motion models, and the pairwise motion
affinity between objects i and j can be computed as
${\boldsymbol{a}_{ij}=\boldsymbol{c}_{i}^{\intercal}\boldsymbol{c}_{j}}$,
which denotes the co-occurrence between two objects as an inlier of all motion
models. ORK is robust to outliers and makes the affinity matrix more adaptive
to different scenes by reducing the need to set scene specific inlier
thresholds.

###  III-E Co-Regularized Multi-view Spectral Clustering

After constructing the affinity matrices, we use the epsilon-neighborhood
scheme [25] to sparsify the affinity matrices. We adapt co-regularized multi-
view spectral clustering [43] to fuse the two affinity matrices together to
obtain the final clustering of object motions and trajectory points. Co-
regularized multi-view spectral clustering uses an regularization term to
encourage consensus between different views and is shown to perform well on
fusing multiple geometric models for a consistent representation of data [4].

##  IV PRELIMINARY RESULTS & CONCLUSIONS

Figure 2: Trajectory clustering results and the generated segmentation masks
by [4] (top) and our method (bottom)

We tested our method on the KT3DMoSeg dataset since it is the only existing
dataset involving challenging scenes and multiple complex motions. Since
KT3DMoSeg uses pre-defined point trajectories, it could occur that an object
segmented by our preprocessing pipeline has fewer than 7 point trajectories.
In this case, it’s not possible to compute fundamental matrices for the
object, but we can still obtain its residual vectors by fitting its trajectory
points (if there is any) and the optical flow vectors on the motion models of
other objects to compute its pairwise motion affinity scores. Figure 2 shows a
qualitative comparison between the segmentation masks generated by our method
and a baseline we created using a state-of-the-art method [4] whose code is
publicly available. To establish the baseline, we prompt SAM using the
clustered trajectories of [4] (top left) to produce the segmentation mask.
Even though [4] achieves very low clustering error rate on this sequence, the
quality of the generated segmentation mask is still worse than ours since a
few wrongly labeled trajectory points can easily mislead SAM. It’s also not
able to recognize which motion is background. Figure 3 shows qualitative
ablation study and comparison to state-of-the-art methods. Table I shows
quantitative results comparing to existing methods. Our method outperforms the
state-of-the-art both qualitatively and quantitatively. We also compare the
total running time (measured on an Intel 13900K CPU and an NVIDIA RTX 4090
GPU) of our method with [4] for reference. Our method is more than twice as
slow since it’s not yet optimized.

Figure 3: Qualitative ablation study and comparison with state-of-the-art methods on the Seq38_Clip02 sequence. From top to bottom: fundamental matrix only, optical flow only, fundamental matrix + optical flow, MVC [4], CMFO [2] TABLE I: Quantitative results in terms of average classification error (%) and total running time (s). Lower is better Methods | Avg. Error Rate (%) | Running Time (s)
---|---|---
LSA [44] | 38.30 | -
GPCA [45] | 34.60 | -
ALC [46] | 24.31 | -
SSC [23] | 33.88 | -
LRR [47] | 33.67 | -
MVC [4] | 10.99 | 1409.6
CMFO [2] | 6.73 | -
Ours | 5.78 | 3230.1

## References

  * [1] R. I. Hartley and A. Zisserman, _Multiple View Geometry in Computer Vision_ , 2nd ed. Cambridge University Press, ISBN: 0521540518, 2004.
  * [2] Z. Xi, J. Liu, B. Luo, and Q. Qin, “Multi-Motion Segmentation: Combining Geometric Model-Fitting and Optical Flow for RGB Sensors,” _IEEE Sensors Journal_ , vol. 22, no. 7, pp. 6952–6963, Apr. 2022, conference Name: IEEE Sensors Journal.
  * [3] F. Xu, G. Gu, K. Ren, and W. Qian, “Motion Segmentation by New Three-View Constraint from a Moving Camera,” _Mathematical Problems in Engineering_ , vol. 2015, pp. 1–14, 2015. [Online]. Available: http://www.hindawi.com/journals/mpe/2015/546580/
  * [4] X. Xu, L. F. Cheong, and Z. Li, “Motion Segmentation by Exploiting Complementary Geometric Models,” in _2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition_. Salt Lake City, UT, USA: IEEE, June 2018, pp. 2859–2867. [Online]. Available: https://ieeexplore.ieee.org/document/8578400/
  * [5] E. Mohamed, M. Ewaisha, M. Siam, H. Rashed, S. Yogamani, W. Hamdy, M. El-Dakdouky, and A. El-Sallab, “Monocular Instance Motion Segmentation for Autonomous Driving: KITTI InstanceMotSeg Dataset and Multi-Task Baseline,” in _2021 IEEE Intelligent Vehicles Symposium (IV)_. Nagoya, Japan: IEEE Press, July 2021, pp. 114–121. [Online]. Available: https://doi.org/10.1109/IV48863.2021.9575445
  * [6] J. Vertens, A. Valada, and W. Burgard, “SMSnet: Semantic motion segmentation using deep convolutional neural networks,” in _2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , Sept. 2017, pp. 582–589, iSSN: 2153-0866.
  * [7] M. Ramzy, H. Rashed, A. E. Sallab, and S. Yogamani, “RST-MODNet: Real-time Spatio-temporal Moving Object Detection for Autonomous Driving,” Dec. 2019, arXiv:1912.00438 [cs, stat] version: 1. [Online]. Available: http://arxiv.org/abs/1912.00438
  * [8] Y. Jiang, Q. Xu, K. Ma, Z. Yang, X. Cao, and Q. Huang, “What to Select: Pursuing Consistent Motion Segmentation from Multiple Geometric Models,” _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 35, no. 2, pp. 1708–1716, May 2021, number: 2\. [Online]. Available: https://ojs.aaai.org/index.php/AAAI/article/view/16264
  * [9] H. Bavle, J. L. Sanchez-Lopez, C. Cimarelli, A. Tourani, and H. Voos, “From SLAM to Situational Awareness: Challenges and Survey,” _Sensors_ , vol. 23, no. 10, p. 4849, Jan. 2023, number: 10 Publisher: Multidisciplinary Digital Publishing Institute. [Online]. Available: https://www.mdpi.com/1424-8220/23/10/4849
  * [10] F. Rajič, L. Ke, Y.-W. Tai, C.-K. Tang, M. Danelljan, and F. Yu, “Segment Anything Meets Point Tracking,” July 2023, arXiv:2307.01197 [cs]. [Online]. Available: http://arxiv.org/abs/2307.01197
  * [11] S. Liu, Z. Zeng, T. Ren, F. Li, H. Zhang, J. Yang, C. Li, J. Yang, H. Su, J. Zhu, and L. Zhang, “Grounding DINO: Marrying DINO with Grounded Pre-Training for Open-Set Object Detection,” Mar. 2023, arXiv:2303.05499 [cs]. [Online]. Available: http://arxiv.org/abs/2303.05499
  * [12] Y. Zhang, X. Huang, J. Ma, Z. Li, Z. Luo, Y. Xie, Y. Qin, T. Luo, Y. Li, S. Liu, Y. Guo, and L. Zhang, “Recognize Anything: A Strong Image Tagging Model,” June 2023, arXiv:2306.03514 [cs]. [Online]. Available: http://arxiv.org/abs/2306.03514
  * [13] Z. Yang and Y. Yang, “Decoupling Features in Hierarchical Propagation for Video Object Segmentation,” _Advances in Neural Information Processing Systems_ , vol. 35, pp. 36 324–36 336, Dec. 2022. [Online]. Available: https://proceedings.neurips.cc/paper˙files/paper/2022/hash/eb890c36af87e4ca82e8ef7bcba6a284-Abstract-Conference.html
  * [14] S. Negahdaripour and B. K. P. Horn, “Direct Passive Navigation,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. PAMI-9, no. 1, pp. 168–176, Jan. 1987\. [Online]. Available: https://ieeexplore.ieee.org/document/4767884
  * [15] H. Sekkati and A. Mitiche, “A variational method for the recovery of dense 3D structure from motion,” _Robotics and Autonomous Systems_ , vol. 55, no. 7, pp. 597–607, July 2007. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0921889006001941
  * [16] A. Wedel, D. Cremers, T. Pock, and H. Bischof, “Structure- and motion-adaptive regularization for high accuracy optic flow,” in _2009 IEEE 12th International Conference on Computer Vision_ , Sept. 2009, pp. 1663–1668, iSSN: 2380-7504.
  * [17] P. Bideau and E. Learned-Miller, “It’s Moving! A Probabilistic Model for Causal Motion Segmentation in Moving Camera Videos,” in _Computer Vision – ECCV 2016_ , B. Leibe, J. Matas, N. Sebe, and M. Welling, Eds. Cham: Springer International Publishing, 2016, vol. 9912, pp. 433–449, series Title: Lecture Notes in Computer Science. [Online]. Available: http://link.springer.com/10.1007/978-3-319-46484-8˙26
  * [18] P. Bideau, A. RoyChowdhury, R. R. Menon, and E. Learned-Miller, “The Best of Both Worlds: Combining CNNs and Geometric Constraints for Hierarchical Motion Segmentation,” in _2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition_. Salt Lake City, UT, USA: IEEE, June 2018, pp. 508–517. [Online]. Available: https://ieeexplore.ieee.org/document/8578158/
  * [19] A. Delong, A. Osokin, H. N. Isack, and Y. Boykov, “Fast approximate energy minimization with label costs,” in _2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition_ , June 2010, pp. 2173–2180, iSSN: 1063-6919.
  * [20] H. Isack and Y. Boykov, “Energy-Based Geometric Multi-model Fitting,” _International Journal of Computer Vision_ , vol. 97, no. 2, pp. 123–147, Apr. 2012. [Online]. Available: http://link.springer.com/10.1007/s11263-011-0474-7
  * [21] D. Hutchison, T. Kanade, J. Kittler, J. M. Kleinberg, F. Mattern, J. C. Mitchell, M. Naor, O. Nierstrasz, C. Pandu Rangan, B. Steffen, M. Sudan, D. Terzopoulos, D. Tygar, M. Y. Vardi, G. Weikum, T. Brox, and J. Malik, “Object Segmentation by Long Term Analysis of Point Trajectories,” in _Computer Vision – ECCV 2010_ , K. Daniilidis, P. Maragos, and N. Paragios, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010, vol. 6315, pp. 282–295, series Title: Lecture Notes in Computer Science. [Online]. Available: http://link.springer.com/10.1007/978-3-642-15555-0˙21
  * [22] T. Brox and J. Malik, “Object segmentation by long term analysis of point trajectories,” in _Proceedings of the 11th European conference on Computer vision: Part V_ , ser. ECCV’10. Berlin, Heidelberg: Springer-Verlag, Sept. 2010, pp. 282–295.
  * [23] E. Elhamifar and R. Vidal, “Sparse Subspace Clustering: Algorithm, Theory, and Applications,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 35, no. 11, pp. 2765–2781, Nov. 2013. [Online]. Available: http://ieeexplore.ieee.org/document/6482137/
  * [24] P. Ochs, J. Malik, and T. Brox, “Segmentation of Moving Objects by Long Term Video Analysis,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 36, no. 6, pp. 1187–1200, June 2014, conference Name: IEEE Transactions on Pattern Analysis and Machine Intelligence.
  * [25] T. Lai, H. Wang, Y. Yan, T.-J. Chin, and W.-L. Zhao, “Motion Segmentation Via a Sparsity Constraint,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 18, no. 4, pp. 973–983, Apr. 2017, conference Name: IEEE Transactions on Intelligent Transportation Systems.
  * [26] F. Arrigoni, L. Magri, and T. Pajdla, “On the Usage of the Trifocal Tensor in Motion Segmentation,” in _Computer Vision – ECCV 2020_ , A. Vedaldi, H. Bischof, T. Brox, and J.-M. Frahm, Eds. Cham: Springer International Publishing, 2020, vol. 12365, pp. 514–530, series Title: Lecture Notes in Computer Science. [Online]. Available: https://link.springer.com/10.1007/978-3-030-58565-5˙31
  * [27] M. Siam, H. Mahgoub, M. Zahran, S. Yogamani, M. Jagersand, and A. El-Sallab, “MODNet: Motion and Appearance based Moving Object Detection Network for Autonomous Driving,” in _2018 21st International Conference on Intelligent Transportation Systems (ITSC)_ , Nov. 2018, pp. 2859–2864, iSSN: 2153-0017.
  * [28] M. Bosch, “Deep Learning for Robust Motion Segmentation with Non-Static Cameras,” Feb. 2021, arXiv:2102.10929 [cs]. [Online]. Available: http://arxiv.org/abs/2102.10929
  * [29] A. Dave, P. Tokmakov, and D. Ramanan, “Towards Segmenting Anything That Moves,” in _2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW)_. Seoul, Korea (South): IEEE, Oct. 2019, pp. 1493–1502. [Online]. Available: https://ieeexplore.ieee.org/document/9022103/
  * [30] Z. Cao, A. Kar, C. Hane, and J. Malik, “Learning Independent Object Motion From Unlabelled Stereoscopic Videos,” in _2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_. Long Beach, CA, USA: IEEE, June 2019, pp. 5587–5596. [Online]. Available: https://ieeexplore.ieee.org/document/8953391/
  * [31] M. Faisal, I. Akhter, M. Ali, and R. Hartley, “EpO-Net: Exploiting Geometric Constraints on Dense Trajectories for Motion Saliency,” in _2020 IEEE Winter Conference on Applications of Computer Vision (WACV)_. Snowmass Village, CO, USA: IEEE, Mar. 2020, pp. 1873–1882. [Online]. Available: https://ieeexplore.ieee.org/document/9093589/
  * [32] E. Meunier, A. Badoual, and P. Bouthemy, “EM-Driven Unsupervised Learning for Efficient Motion Segmentation,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 45, no. 4, pp. 4462–4473, Apr. 2023, conference Name: IEEE Transactions on Pattern Analysis and Machine Intelligence.
  * [33] Z. Teed and J. Deng, “RAFT: Recurrent All-Pairs Field Transforms for Optical Flow,” arXiv, Tech. Rep. arXiv:2003.12039, Aug. 2020, arXiv:2003.12039 [cs] type: article. [Online]. Available: http://arxiv.org/abs/2003.12039
  * [34] D. Sun, C. Herrmann, F. Reda, M. Rubinstein, D. Fleet, and W. T. Freeman, “Disentangling Architecture and Training for Optical Flow,” Sept. 2022, arXiv:2203.10712 [cs]. [Online]. Available: http://arxiv.org/abs/2203.10712
  * [35] A. Mitiche and J. Aggarwal, _Computer Vision Analysis of Image Motion by Variational Methods_ , ser. Springer Topics in Signal Processing. Cham: Springer International Publishing, 2014, vol. 10. [Online]. Available: http://link.springer.com/10.1007/978-3-319-00711-3
  * [36] J. Lezama, K. Alahari, J. Sivic, and I. Laptev, “Track to the future: Spatio-temporal video segmentation with long-range motion cues,” in _CVPR 2011_ , June 2011, pp. 3369–3376, iSSN: 1063-6919.
  * [37] Y. Cheng, L. Li, Y. Xu, X. Li, Z. Yang, W. Wang, and Y. Yang, “Segment and Track Anything,” May 2023, arXiv:2305.06558 [cs]. [Online]. Available: http://arxiv.org/abs/2305.06558
  * [38] D. Sun, C. Herrmann, F. Reda, M. Rubinstein, D. J. Fleet, and W. T. Freeman, “Disentangling Architecture and Training for Optical Flow,” in _Computer Vision – ECCV 2022_ , ser. Lecture Notes in Computer Science, S. Avidan, G. Brostow, M. Cissé, G. M. Farinella, and T. Hassner, Eds. Cham: Springer Nature Switzerland, 2022, pp. 165–182.
  * [39] A. W. Harley, Z. Fang, and K. Fragkiadaki, “Particle Video Revisited: Tracking Through Occlusions Using Point Trajectories,” in _Computer Vision – ECCV 2022_ , ser. Lecture Notes in Computer Science, S. Avidan, G. Brostow, M. Cissé, G. M. Farinella, and T. Hassner, Eds. Cham: Springer Nature Switzerland, 2022, pp. 59–75.
  * [40] C. Doersch, Y. Yang, M. Vecerik, D. Gokay, A. Gupta, Y. Aytar, J. Carreira, and A. Zisserman, “TAPIR: Tracking Any Point with per-frame Initialization and temporal Refinement,” June 2023, arXiv:2306.08637 [cs]. [Online]. Available: http://arxiv.org/abs/2306.08637
  * [41] H. C. Longuet-Higgins and K. Prazdny, “The Interpretation of a Moving Retinal Image,” _Proceedings of the Royal Society of London. Series B, Biological Sciences_ , vol. 208, no. 1173, pp. 385–397, 1980, publisher: The Royal Society. [Online]. Available: https://www.jstor.org/stable/35316
  * [42] T.-j. Chin, H. Wang, and D. Suter, “The Ordered Residual Kernel for Robust Motion Subspace Clustering,” in _Advances in Neural Information Processing Systems_ , vol. 22. Curran Associates, Inc., 2009. [Online]. Available: https://proceedings.neurips.cc/paper˙files/paper/2009/hash/b337e84de8752b27eda3a12363109e80-Abstract.html
  * [43] A. Kumar, P. Rai, and H. Daume, “Co-regularized Multi-view Spectral Clustering,” in _Advances in Neural Information Processing Systems_ , vol. 24. Curran Associates, Inc., 2011. [Online]. Available: https://papers.nips.cc/paper˙files/paper/2011/hash/31839b036f63806cba3f47b93af8ccb5-Abstract.html
  * [44] J. Yan and M. Pollefeys, “A General Framework for Motion Segmentation: Independent, Articulated, Rigid, Non-rigid, Degenerate and Non-degenerate,” in _Computer Vision – ECCV 2006_ , ser. Lecture Notes in Computer Science, A. Leonardis, H. Bischof, and A. Pinz, Eds. Berlin, Heidelberg: Springer, 2006, pp. 94–106.
  * [45] R. Vidal, R. Tron, and R. Hartley, “Multiframe Motion Segmentation with Missing Data Using PowerFactorization and GPCA,” _International Journal of Computer Vision_ , vol. 79, no. 1, pp. 85–105, Aug. 2008. [Online]. Available: https://doi.org/10.1007/s11263-007-0099-z
  * [46] S. Rao, R. Tron, R. Vidal, and Y. Ma, “Motion Segmentation in the Presence of Outlying, Incomplete, or Corrupted Trajectories,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 32, no. 10, pp. 1832–1845, Oct. 2010, conference Name: IEEE Transactions on Pattern Analysis and Machine Intelligence.
  * [47] G. Liu, Z. Lin, S. Yan, J. Sun, Y. Yu, and Y. Ma, “Robust Recovery of Subspace Structures by Low-Rank Representation,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 35, no. 1, pp. 171–184, Jan. 2013. [Online]. Available: http://ieeexplore.ieee.org/document/6180173/