# Search for Magnetic Monopoles with ten years of the ANTARES neutrino
telescope

A. Albert  S. Alves  M. André  M. Anghinolfi  G. Anton  M. Ardid  S. Ardid
J.-J. Aubert  J. Aublin  B. Baret  S. Basa  B. Belhorma  M. Bendahman  F.
Benfenati  V. Bertin  S. Biagi  M. Bissinger  J. Boumaaza  M. Bouta  M.C.
Bouwhuis  H. Brânzaş  R. Bruijn  J. Brunner  J. Busto  B. Caiffi  D. Calvo  A.
Capone  L. Caramete  J. Carr  V. Carretero  S. Celli  M. Chabab  T. N. Chau
R. Cherkaoui El Moursli  T. Chiarusi  M. Circella  A. Coleiro  R. Coniglione
P. Coyle  A. Creusot  A. F. Díaz  G. de Wasseige  C. Distefano  I. Di Palma
A. Domi  C. Donzaud  D. Dornic  D. Drouhin  T. Eberl  T. van Eeden  D. van
Eijk  N. El Khayati  A. Enzenhöfer  P. Fermani  G. Ferrara  F. Filippini  L.
Fusco  J. García  Y. Gatelet  P. Gay  H. Glotin  R. Gozzini  R. Gracia Ruiz
K. Graf  C. Guidi  S. Hallmann  H. van Haren  A.J. Heijboer  Y. Hello  J.J.
Hernández-Rey  J. Hößl  J. Hofestädt  F. Huang  G. Illuminati  C. W. James  B.
Jisse-Jung  M. de Jong  P. de Jong  M. Kadler  O. Kalekin  U. Katz  N.R. Khan-
Chowdhury  A. Kouchner  I. Kreykenbohm  V. Kulikovskiy  R. Lahmann  R. Le
Breton  S. LeStum  D. Lefèvre  E. Leonora  G. Levi  M. Lincetto  D. Lopez-Coto
S. Loucatos  L. Maderer  J. Manczak  M. Marcelin  A. Margiotta  A. Marinelli
J.A. Martínez-Mora  B. Martino  K. Melis  P. Migliozzi  A. Moussa  R. Muller
L. Nauta  S. Navas  E. Nezri  B. Ó Fearraigh  A. Păun  G.E. Păvălaş  C.
Pellegrino  M. Perrin-Terrin  V. Pestel  P. Piattelli  C. Pieterse  C. Poirè
V. Popa  T. Pradier  N. Randazzo  D. Real  S. Reck  G. Riccobene  A. Romanov
A. Sánchez-Losa  F. Salesa Greus  D. F. E. Samtleben  M. Sanguineti  P.
Sapienza  J. Schnabel  J. Schumann  F. Schüssler  J. Seneca  M. Spurio  Th.
Stolarczyk  M. Taiuti  Y. Tayalati  S.J. Tingay  B. Vallage  V. Van Elewyck
F. Versari  S. Viola  D. Vivolo  J. Wilms  S. Zavatarelli  A. Zegarelli  J.D.
Zornoza  J. Zúñiga  Université de Strasbourg, CNRS, a UMR 7178, F-67000
Strasbourg, France  Université de Haute Alsace, F-68100 Mulhouse, France IFIC
- Instituto de Física Corpuscular (CSIC - Universitat de València) c/
Catedrático José Beltrán, 2 E-46980 Paterna, Valencia, Spain  Technical
University of Catalonia, Laboratory of Applied Bioacoustics, Rambla Exposició,
08800 Vilanova i la Geltrú, Barcelona, Spain  INFN - Sezione di Genova, Via
Dodecaneso 33, 16146 Genova, Italy  Friedrich-Alexander-Universität Erlangen-
Nürnberg, Erlangen Centre for Astroparticle Physics, Erwin-Rommel-Str. 1,
91058 Erlangen, Germany  Institut d’Investigació per a la Gestió Integrada de
les Zones Costaneres (IGIC) - Universitat Politècnica de València. C/ Paranimf
1, 46730 Gandia, Spain  Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille,
France  Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013
Paris, France  Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France
National Center for Energy Sciences and Nuclear Techniques, B.P.1382, R.
P.10001 Rabat, Morocco  University Mohammed V in Rabat, Faculty of Sciences, 4
av. Ibn Battouta, B.P. 1014, R.P. 10000 Rabat, Morocco  INFN - Sezione di
Bologna, Viale Berti-Pichat 6/2, 40127 Bologna, Italy  Dipartimento di Fisica
e Astronomia dell’Università, Viale Berti Pichat 6/2, 40127 Bologna, Italy
INFN - Laboratori Nazionali del Sud (LNS), Via S. Sofia 62, 95123 Catania,
Italy  University Mohammed I, Laboratory of Physics of Matter and Radiations,
B.P.717, Oujda 6000, Morocco  Nikhef, Science Park, Amsterdam, The Netherlands
Institute of Space Science, RO-077125 Bucharest, Măgurele, Romania
Universiteit van Amsterdam, Instituut voor Hoge-Energie Fysica, Science Park
105, 1098 XG Amsterdam, The Netherlands  INFN - Sezione di Roma, P.le Aldo
Moro 2, 00185 Roma, Italy  Dipartimento di Fisica dell’Università La Sapienza,
P.le Aldo Moro 2, 00185 Roma, Italy  LPHEA, Faculty of Science - Semlali, Cadi
Ayyad University, P.O.B. 2390, Marrakech, Morocco.  INFN - Sezione di Bari,
Via E. Orabona 4, 70126 Bari, Italy  Department of Computer Architecture and
Technology/CITIC, University of Granada, 18071 Granada, Spain  Université
Paris-Sud, 91405 Orsay Cedex, France  Laboratoire de Physique Corpusculaire,
Clermont Université, Université Blaise Pascal, CNRS/IN2P3, BP 10448, F-63000
Clermont-Ferrand, France  LIS, UMR Université de Toulon, Aix Marseille
Université, CNRS, 83041 Toulon, France  Dipartimento di Fisica
dell’Università, Via Dodecaneso 33, 16146 Genova, Italy  Royal Netherlands
Institute for Sea Research (NIOZ), Landsdiep 4, 1797 SZ ’t Horntje (Texel),
the Netherlands  Géoazur, UCA, CNRS, IRD, Observatoire de la Côte d’Azur,
Sophia Antipolis, France  International Centre for Radio Astronomy Research -
Curtin University, Bentley, WA 6102, Australia  Huygens-Kamerlingh Onnes
Laboratorium, Universiteit Leiden, The Netherlands  Institut für Theoretische
Physik und Astrophysik, Universität Würzburg, Emil-Fischer Str. 31, 97074
Würzburg, Germany  Dr. Remeis-Sternwarte and ECAP, Friedrich-Alexander-
Universität Erlangen-Nürnberg, Sternwartstr. 7, 96049 Bamberg, Germany
Mediterranean Institute of Oceanography (MIO), Aix-Marseille University,
13288, Marseille, Cedex 9, France; Université du Sud Toulon-Var, CNRS-INSU/IRD
UM 110, 83957, La Garde Cedex, France  INFN - Sezione di Catania, Via S. Sofia
64, 95123 Catania, Italy  Dpto. de Física Teórica y del Cosmos & C.A.F.P.E.,
University of Granada, 18071 Granada, Spain  IRFU, CEA, Université Paris-
Saclay, F-91191 Gif-sur-Yvette, France  INFN - Sezione di Napoli, Via Cintia
80126 Napoli, Italy  Museo Storico della Fisica e Centro Studi e Ricerche
Enrico Fermi, Piazza del Viminale 1, 00184, Roma  INFN - CNAF, Viale C. Berti
Pichat 6/2, 40127, Bologna  Institut Universitaire de France, 75005 Paris,
France  Dipartimento di Fisica dell’Università Federico II di Napoli, Via
Cintia 80126, Napoli, Italy

###### Abstract

This work presents a new search for magnetic monopoles using data taken with
the ANTARES neutrino telescope over a period of 10 years (January 2008 to
December 2017). Compared to previous ANTARES searches, this analysis uses a
run-by-run simulation strategy, with a larger exposure as well as a new
simulation of magnetic monopoles taking into account the Kasama, Yang and
Goldhaber model for their interaction cross-section with matter. No signal
compatible with the passage of relativistic magnetic monopoles is observed,
and upper limits on the flux of magnetic monopoles with $\beta$ = v/c $\geq$
0.55, are presented. For ultra-relativistic magnetic monopoles the flux limit
is $\sim$ 7$\times$$10^{-18}$ $\rm cm^{-2}\;s^{-1}\;sr^{-1}$.

###### keywords:

ANTARES telescope, Magnetic Monopoles, Neutrino

††journal: Journal of High Energy Astrophysics

##  1 Introduction

The existence of magnetic charges has been considered since a long time ago.
The introduction of hypothetical magnetic charges and magnetic currents can
restore the symmetry in Maxwell’s equations with respect to magnetic and
electric fields. When investigating the symmetry between electricity and
magnetism, Paul Dirac proved in 1931 [1], that the introduction of Magnetic
Monopoles (MMs) can also elegantly solve the problem of the quantization of
electric charge. In addition to this, Grand Unified Theories (GUTs) [2] also
predicted that MMs could be created shortly after the Big Bang.

Magnetic monopoles are topologically stable particles and carry a magnetic
charge defined as a multiple integer of the Dirac charge :

$g_{D}=\frac{\hbar c}{2e}=\frac{e}{2\alpha}=68.5e$ (1)

where $e$ is the electon electric charge, $c$ is the velocity of light in
vacuum, $\hbar$ is the Planck constant and $\alpha\simeq 1/137$ is the fine
structure constant. While Dirac demonstrated the consistency of MMs with
quantum mechanics, G. ’t Hooft [3] and Polyakov [4] proved the necessity of
MMs in GUTs. This led to the conclusion that any unification model in which
the U(1) subgroup of electromagnetism is embedded in a semi-simple gauge
group, which is spontaneously broken by the Higgs mechanism, possesses
monopole-like solutions. The masses of MMs can range from $10^{8}$ to
$10^{17}$ ${\rm GeV/c^{2}}$. Larger MM masses are expected if gravity is
brought into the GUT picture, as well as in some supersymmetric models [5].
Moreover, MMs would be created after the Big Bang (during the phase transition
of symmetry breaking), and they would be accelerated by galactic magnetic
fields. The rarity of GUT MMs is also a motivation to the scenario of
inflation [6].

In this paper, a search based on an ANTARES data set of 2480 days collected
from 2008 to 2017 is presented. This analysis improves our previous results
[7], it is based on a new selection yielding a better separation of the
putative MM signal, in different velocity $v=\beta c$ ranges, from the
background induced by atmospheric muons and neutrinos. The optimization of the
selection uses the Model Rejection Factor method [8] which allows for the
calculation of the sensitivity (and consequently of the upper limit on the
flux) using the Feldman-Cousins [9] statistical method. Differently from our
previous publication in which the Mott model [10] of cross-section with matter
was adopted, in this work, the simulation of MM interaction with matter relies
on the Kasama, Yang and Goldhaber (KYG) model [11].

The paper is organized as follows: a brief description of the ANTARES
telescope and the expected signal from magnetic monopoles is given in section
2. The simulation of signal and background is described in section 3, while
the trigger logic and reconstruction method is summarized in section 4. The
MM-sensitive observables, the selection strategy and the optimization method
are presented in section 5. The result of the search, the upper limit
calculation and the comparison with other experiments are discussed in section
6.

##  2 The ANTARES neutrino telescope and the expected signal from MMs

The ANTARES (Astronomy with a Neutrino Telescope and Abyss environmental
RESearch) detector [12] is a Cherenkov neutrino telescope, anchored at 2475 m
below the surface of the Mediterranean Sea and 40 km offshore from Toulon
(France). The detector contains 12 detection lines of about 350 m each,
horizontally spaced 60 m to 75 m apart and covering a surface area of about
0.1 km2. Each line has 25 floors with 3 optical modules containing each a 10
inch photomultiplier tube (PMT). These PMTs (Hamamatsu R7081-20) are sensitive
to photons in the wavelength range $\lambda$ $\sim[300,600]$ nm. PMT signals
with an amplitude higher than the threshold of 0.3 photo-electrons are
captured in a time window of 40 ns, digitized and registered as hits [13],
[14]. After digitization the data is then sent under the form of packages of
104 ms, to an on-shores farm of computers for further data processing and
filtering. Finally, the data goes through a system of triggers, in order to
select signals that may correspond to the passage of relativistic particles.

The signal of MMs in a neutrino telescope is similar to that of high energy
muons. Tompkins [15] showed that, similarly to electric charge, magnetically
charged particles produce Cherenkov emission when their velocity is higher
than the Cherenkov threshold $\beta_{th}=1/n,$ where $n$ is the phase
refractive index of the medium. Below Cherenkov threshold, the interaction of
MMs with electrons in water produces knock-on electrons (also called delta-
rays) that, in turn, induce Cherenkov light. The total number of Cherenkov
photons having a wavelength between 300 and 600 nm ($N_{\gamma}$), per unit
path length of the monopole ($dx$) is calculated using the Berger formula
[16], and can be determined by:

$\frac{dN_{\gamma}}{dx}=\int_{T_{0}}^{T_{m}}\frac{d^{2}N_{e}}{dT_{e}dx}\left[\int_{T_{0}}^{T_{e}}\frac{dN_{\gamma}}{dx_{e}}\left(\frac{dE_{e}}{dx_{e}}\right)^{-1}dE_{e}\right]dT_{e},$
(2)

The electron can induce light if its kinetic energy $T_{e}$ is above the
threshold T0=0.25 MeV. $T_{m}$ is the classical upper limit on the energy that
can be transferred to an atomic electron in a single collision with a MM,
$E_{e}$ is the total energy of the delta-ray, $\frac{d^{2}N_{e}}{dT_{e}dx}$ is
the distribution of delta-rays produced by a MM and $dx_{e}$ is the unit
length travelled by an electron. The maximum energy transfer can be
approximated by:

$T_{max}=2\textit{m}_{e}c^{2}\beta^{2}\gamma^{2},$ (3)

where $\gamma$ is the Lorentz boost factor. This maximum energy is defined as:

$\begin{cases}{\textit{T${}_{m}$}}=0.69\times{\textit{T${}_{max}$}}$ \\- for
the Mott cross-section model, corrected by Ahlen
\cite[cite]{[\@@bibref{Number}{Ahlen:1976jw}{}{}]}$,\\\
{\textit{T${}_{m}$=}}{\textit{ T${}_{max}$}}$ \\- for the KYG model
\cite[cite]{[\@@bibref{Number}{Kazama:1976fm}{}{}]}.$\end{cases}\,.$ (4)

Figure 1: Number of Cherenkov photons emitted per cm in the sea water from a
magnetic monopole (red line), and from delta-rays produced along its path
according to the Mott model [10] (blue line), and to the KYG model [11]
(magenta line) as a function of the velocity of the monopole. The direct
Cherenkov emission from a single muon is also shown as a comparison reference
(green line).

In this study, the KYG model is utilized for MMs cross-section with matter,
whereas the previous analyses [7], [17] used the Mott model. The difference in
the photon yields by a MM (the number of expected delta-rays as a function of
$\beta$) derived with the two models is shown in Fig.1. The analysis is also
improved by a ”run-by-run” simulation strategy [18], that treats each data run
individually, allowing for an accurate reproduction of the data taking
conditions. This change is motivated by the fact that the KYG model derives
from a more accurate estimation of the cross-section of MMs with matter. Since
the limits set by the IceCube collaboration [19] also used the KYG model, this
choice allows for a more coherent comparison between the results obtained with
the two experiments.

##  3 Monte Carlo simulations

For this analysis, a dedicated Monte Carlo (MC) production that includes MMs
based on the KYG model for the signal has been used. The background consisting
of atmospheric muons and neutrinos. In order to take into account the
variation of environmental conditions in sea water, which affects data
acquisition and the optical module efficiencies, the production of simulated
files containing signal and background events is performed based on run-by-run
simulation. This approach addresses each run of data separately by considering
its real conditions of acquisition.

The simulation of up-going MMs is carried out using 10 equally spaced velocity
ranges in the interval of $\beta$ $\in$ $[0.5500,0.9950]$. It relies on a
package which is adapted on generators used to simulate the passage of muons
in the detector [20]. For each velocity range, a total of 500 MM tracks per
run are generated uniformly in the surface of a cylindrical volume around the
detector and according to the detector acceptance [18]. The detector’s
response to MM signals, together with the emission, propagation and detection
of direct and delta-ray Cherenkov light are then simulated, with the photon
wavelength ranging between 300 and 600 nm to match the ANTARES
photomultipliers sensitive range.

The background consists mainly of up-going muons resulting from the
interaction of atmospheric neutrinos and down-going atmospheric muons mis-
reconstructed as up-going tracks. The generator MUPAGE [21] is used to
simulate the atmospheric muons based on the parameterization of the direction
and energy distributions of under-water muons taking into account the muon
bundle multiplicity.

##  4 Trigger and Reconstruction

The events considered in this analysis must fulfill the conditions applied by
the ANTARES triggers [14], which are based on local coincidences defined as
the occurrence of either two hits on two separate optical modules of a single
storey within 20 ns, or one single hit of large amplitude (more than 3 photo-
electrons). For this analysis a software trigger is defined as a combination
of two local coincidences in adjacent or next-to-adjacent storeys within 100
ns or 200 ns, respectively.

In order to reconstruct the passage of a MM in the detector, a modified
version of the fast tracking algorithm [22] is used. The algorithm searches
for a straight line (i.e, a track-like event) compatible with the large
amplitude hit positions and times under the Cherenkov photon emission
hypothesis while allowing for a variable effective particle speed
$v_{reco}$=$\beta_{reco}$ c. One example of simulated MM with $\beta$ $\in$
[0.9505, 0.9950[ is shown in Fig.2. Two different approaches are followed in
this study, depending on the velocity of the simulated MMs:

  * 1.

Fast MMs are simulated with $\beta$ in the range [0.8170, 0.9950[, split into
4 equal intervals and are reconstructed with $\beta_{reco}$=1. In this
velocity region, relativistic MMs are supposed to emit a significantly larger
amount of Cherenkov light in the detector compared to muons, which is crucial
to isolate the signal from the background. Referring to Fig.1, the light yield
for a single muon is less than two orders of magnitude with respect to the one
produced by a MM. However, during the 10 years of analyzed data, events with
muon bundles with multiplicity up to $\sim$100 are expected. In order to
distinghish signal from background in this velocity range, the number of
storeys with fired PMTs ($\mathrm{N_{sh}}$) is used in the reconstruction
algorithm [22]. In Fig.2, each point represents such a ”storey hit”, in which
the center of the storey represents the position coordinates, the time
considered is the time of the first hit and the charge is equal to the sum of
the hit charges. $\mathrm{N_{sh}}$ is roughly proportional to the amount of
light emitted by the particle. A large value of this quantity selects
candidate MMs, which produce much more light compared to other background
particles (muons and neutrinos) reaching the deep detector location.

  * 2.

Slower MMs are simulated with $\beta$ in the interval [0.5500, 0.8170[. The
events are generated into 6 equally spaced intervals in this velocity range.
MMs simulated with velocities within $\beta$ $\in$[0.5500, 0.8170[ are
searched for with the parameter $\beta_{reco}$ used as a free parameter in the
reconstruction algorithm. The corresponding output value in the range
$\beta_{reco}\in$[0.5500, 0.8170[ is used as primary cut to isolate the signal
from the background, as atmospheric muons and neutrinos are mostly
reconstructed as relativistic ($\beta_{reco}$ $\sim$1) particles.

Figure 2: Event display of a simulated magnetic monopole in the range
$\beta\in$[0.9950, 0.9950] passing through the ANTARES telescope after
traversing the Earth (up-going event). Each individual graph represents one
detector line, the octagonal arrangement corresponds approximately to their
layout on the sea floor and for each line the detected photons are given as
function of their arrival time (x-axis) and height above the sea floor
(y-axis), their amplitude is color coded as well. The MM signals (red
hyperbolae) are clearly distinguishable from background photons (black
crosses).

##  5 Event selection

Events of interest for this study are selected by imposing several cuts to
reduce the background stemming from atmospheric muons and neutrinos. The first
selection parameter corresponds to the zenith angle $\theta_{reco}$ of the
reconstructed track. In order to select up-going events, using the Earth as a
filter, the condition $\theta_{reco}>90^{\circ}$ is required. Only events
reconstructed with at least 2 lines of the detector are considered to improve
the quality of the event reconstruction [23].

Figure 3: Distribution of $\mathrm{N_{sh}}$ for atmospheric neutrinos (red
histogram), atmospheric neutrinos (blue histogram) with an uncertainty band of
35% (gray band). Data is represented in black points and MMs signal in shown
in green. The plot in the left corresponds to the interval [0.5945, 0.6390[ of
$\beta$ and has an additional cut (beside the initial cuts described in
section 6) on $\beta_{reco}$ in [0.5945, 0.6390[, while the plot on the right
corresponds equivalently to the $\beta$ range [0.8615, 0.9060[. Both plots
correspond to 10 years of analyzed data.

The $\mathrm{N_{sh}}$ parameter introduced earlier is chosen as an event
energy proxy in this study. A final discriminant variable used to isolate the
MM signal from the background combines the quality parameter of the track
reconstruction t$\chi{{}^{2}}$ with the brightness of the event given by
$\mathrm{N_{sh}}$ reduced by the number of free parameters, $N_{df}$, used by
the reconstruction method. This variable, denoted $\alpha$, is empirically
defined as:

$\alpha=\frac{t\chi^{2}}{1.3+\left(0.04\times\left(N_{sh}-N_{df}\right)\right)^{2}},$
(5)

$N_{df}$ is equal to 6 when $\beta_{reco}$ is included in the reconstruction,
which is the case for slow MMs ($\beta_{reco}$ $\in$ [0.5500, 0.8170[), and to
5 when $\beta_{reco}$ is fixed to 1, corresponding to almost relativistic MMs.

The selection of MM events against the background is carried out under a
blinded strategy to avoid any bias. The optimization of the selection
parameters, $\alpha$, $\mathrm{N_{sh}}$ and $\beta_{reco}$ is made in six bins
of $\beta$ in the range [0.5500, 0.8170[ and four bins in the range [0.8170,
0.9950[. In these four bins, $\beta_{reco}$ is fixed to 1.

Figure 4: Scatter-plot of the two variables $\alpha$ and $\mathrm{N_{sh}}$ for
the MMs signal simulated with $\beta$ in the range [0.5945, 0.6390[ (left
plot) with an additional cut (beside the initial cuts described in section 6)
$\beta_{reco}$ $\in$ [0.5945, 0.6390[, and $\beta$ $\in$ [0.8615, 0.9060[
(right plot). The background regions consisting of atmospheric muons in red
and atmospheric neutrinos in blue, are distinguishable. The black lines
indicate the optimized cuts. Both plots correspond to 10 years of analyzed
data.

Fig.4 shows the event distribution of $\alpha$ versus $\mathrm{N_{sh}}$ for
MMs simulated in the $\beta$ ranges [0.5945, 0.6390[ and [0.8615, 0.9060[. The
MMs signal can be distinguished from the background (superimposed on the
signal in the figure) by applying cuts on $\alpha$ and $\mathrm{N_{sh}}$.

To compensate for the lack of statistics in the $\mathrm{N_{sh}}$ distribution
for the simulated sample of atmospheric muons, an extrapolation is performed
in the signal region, by fitting the histogram with a Landau distribution as
seen in Fig.5. The total number of background events used to calculate the
sensitivity includes the contribution of this extrapolation.

Figure 5: Distribution of $\mathrm{N_{sh}}$ for simulated atmospheric muons.
The red line represents the extrapolation in the signal region using a Landau
function. The extrapolation is taken into account for the calculation of the
the sensitivity. The plot corresponds to 10 years of analyzed data.

To obtain the best sensitivity, the Model Rejection Factor is optimized for
each velocity interval by relying on $\alpha$ and $\mathrm{N_{sh}}$ cuts. The
selection efficiency for signal events in the ten intervals of $\beta$ after
applying the cuts on $\alpha$, $\mathrm{N_{sh}}$ and $\beta_{reco}$ (the
latter, applied in the six bins of lower velocities) ranges from 16% to 51%.

The $90\%$ C.L. sensitivity $S_{90\%}$ is calculated with the Feldman-Cousins
[9] formula, considering events which follow a Poisonnian distribution:

$\centering
S_{90\%}[\mathrm{cm}^{-2}\>\mathrm{s}^{-1}\>\mathrm{sr}^{-1}]=\frac{\bar{\mu}_{90}\left(n_{b}\right)}{A_{eff}\left[\mathrm{cm}^{2}\;\mathrm{sr}\right]\times
T[\mathrm{s}]}\>,\@add@centering$ (6)

where T is the duration of data taking, $n_{b}$ representing the number of
expected background events in the 90% C.L. interval ($\mu_{90}$),
$\bar{\mu}_{90}$ and $A_{eff}$ are defined as:

$\centering\bar{\mu}_{90}\left(n_{b}\right)=\sum_{n_{obs}=0}^{\infty}\mu_{90}\frac{n_{b}^{n_{obs}}}{n_{obs}!}e^{-n_{b}},\@add@centering$
(7)

$\centering A_{eff}=\frac{n_{MM}}{\Phi_{MM}},\@add@centering$ (8)

where $\ n_{MM}$ is the number of MMs remaining after cuts,
$\Phi_{MM}[\mathrm{cm}^{-2}\,\mathrm{sr}^{-1}]$ represents the flux of
generated MMs and $n_{obs}$ is the total number of observed events. The Model
Rejection Factor technique consists in varying the cuts until the minimum flux
of Rejection Factor (RF) is found, which coincides with the best sensitivity
(see Fig.6). After the optimization of the rejection factor RF, the
sensitivity at 90% C.L. is derived using the Feldman-Cousins method.

Figure 6: The model rejection factor MRF as a function of $\alpha$ and
$\mathrm{N_{sh}}$ cuts represented here for $\beta_{reco}$ $\in$ [0.7725,
0.8170[ as an example.

##  6 Results

The upper limits obtained corresponding to 10 years of data, as well as the
cuts and the number of events remaining in each $\beta$ region, are summarized
in Table 1.

After applying the cuts on the totality of the data taken, no event survived
the selection. Fig.7 shows the obtained ANTARES upper limit on the flux for
Magnetic Monopoles, taking into account the full period of 2480 days of data
taking, compared to the upper limits on the flux found by the other
experiments, and including upper limits of the previous MM analysis (1012
days) carried out with the ANTARES telescope.

$\beta$ Interval | $\beta_{reco}$ | $\beta$ cut | $\alpha$ cut | $\mathrm{N_{sh}}$ cut | Expected | Observed | Flux upper limit
---|---|---|---|---|---|---|---
background | events | $\left[\mathrm{cm}^{-2}\;\mathrm{s}^{-1}\;\mathrm{sr}^{-1}\right]$
[0.5500, 0.5945[ | Fitted | [0.5500, 0.5945[ |  $<$ 12.4 | $\geq 41$ | 5 $\times 10^{-5}$ | 0 | 8.4 $\times 10^{-18}$
[0.5945, 0.6390[ | Fitted | [0.5945, 0.6390[ |  $<$ 10.8 | $\geq 45$ | 2 $\times 10^{-5}$ | 0 | 1.0 $\times 10^{-17}$
[0.6390, 0.6835[ | Fitted | [0.6390, 0.6835[ |  $<$ 8.8 | $\geq 51$ | 3 $\times 10^{-4}$ | 0 | 6.5 $\times 10^{-18}$
[0.6835, 0.7280[ | Fitted | [0.6835, 0.7280[ |  $<$ 5.2 | $\geq 68$ | 2 $\times 10^{-4}$ | 0 | 6.7 $\times 10^{-18}$
[0.7280, 0.7725[ | Fitted | [0.7280, 0.7725[ |  $<$ 3.6 | $\geq 85$ | 5 $\times 10^{-4}$ | 0 | 7.0 $\times 10^{-18}$
[0.7725, 0.8170[ | Fitted | [0.7725, 0.8170[ |  $<$ 2.6 | $\geq 86$ | 8 $\times 10^{-4}$ | 0 | 3.7 $\times 10^{-18}$
[0.8170, 0.8615[ | 1 | - |  $<$ 0.6 | $\geq 102$ | 0.29 | 0 | 2.8 $\times 10^{-18}$
[0.8615, 0.9060[ | 1 | - |  $<$ 0.3 | $\geq 105$ | 0.18 | 0 | 1.2 $\times 10^{-18}$
[0.9060, 0.9505[ | 1 | - |  $<$ 0.3 | $\geq 105$ | 0.18 | 0 | 8.8 $\times 10^{-19}$
[0.9505, 0.9950] | 1 | - |  $<$ 0.3 | $\geq 105$ | 0.18 | 0 | 7.3 $\times 10^{-19}$

Table 1: The optimized cuts, the number of background events remaining after
cuts, the number of observed events remaining after the cuts and the upper
limit on the flux obtained in each $\beta$ range, for the full analyzed data
sample corresponding to 10 years live time.

Above the Cherenkov threshold ($\beta$=0.76), where the emission of Cherenkov
light is direct and the impact of the cross-section models can be neglected,
the improvement in the upper limit on the flux for MMs compared to the results
found in the previous MMs search (ANTARES limit [7] 1012 days) is mainly due
to the increase in the statistics, that is data taking time. Below the
Cherenkov threshold a much substantial improvement is observed in the
sensitivity, which is also a consequence of the choice of the KYG model, that
can be considered more promising for the lower velocities (see Fig.1), in view
of the improved description of MM cross-section and the increase in the light
yield. In this region, the additional cut on $\beta_{reco}$ eliminated the
majority of the background, this additional cut explains the significant drop
in the number of remaining background events observed in Table 1 for
$\beta<0.8170$. The background in the range of $\beta\geq 0.8170$ is dominated
by atmospheric muons, while in the lower velocity range the background is
dominated by atmospheric neutrinos.

Figure 7: ANTARES 90% C.L upper limit on the flux for MMs corresponding to 10
years of analyzed data (2480 days, red line) compared to other experiments:
ANTARES previous upper limit on the flux (brown line [7]), IceCube (blue [19]
and green lines [24]), MACRO (cyan line [25]) and Baikal (magenta line [26]),
as well as the theoretical Parker bound (black line [27]).

##  7 Conclusion

In this work, the results of a new analysis searching for magnetic monopoles
with velocities ranging in the interval of $\beta$ $\in$ [0.5500,0.9950[
crossing the neutrino telescope is presented, using an optimized simulation
strategy of MMs based on the KYG model, and a larger exposure. A Model
Rejection Factor method, relying on the cuts on observable parameters, as the
number of hits in the detector and the quality of the reconstructed event, has
been employed to optimize the sensitivity for each of the 10 intervals of
$\beta$ considered. After the analysis of the full data sample (2480 days), no
event survived the selection and upper limits on the flux are set for each of
the 10 intervals. The choice of the KYG model for the MM cross-section with
matter led to the improvement in the upper limit on the flux for low
velocities with respect to ANTARES previous result, as well as the extra cut
that is applied in this region, which allowed a better background rejection.
The upper limit on the flux obtained in this analysis is between $7.3\times
10^{-19}$ and $1.0\times 10^{-17}$ ${\rm cm^{-2}\;s^{-1}\;sr^{-1}}$, and holds
for MMs with mass $\mathrm{M}\gtrsim 10^{11}\;\mathrm{GeV/c^{2}}$, due to the
requirement to cross the Earth diameter [28], which can be considered a
competitive result, in particular in view of the modest size of ANTARES with
respect to IceCube.

## Acknowledgements

The authors acknowledge the financial support of the funding agencies: Centre
National de la Recherche Scientifique (CNRS), Commissariat à l’énergie
atomique et aux énergies alternatives (CEA), Commission Européenne (FEDER fund
and Marie Curie Program), Institut Universitaire de France (IUF), LabEx
UnivEarthS (ANR-10-LABX-0023 and ANR-18-IDEX-0001), Région Île-de-France (DIM-
ACAV), Région Alsace (contrat CPER), Région Provence-Alpes-Côte d’Azur,
Département du Var and Ville de La Seyne-sur-Mer, France; Bundesministerium
für Bildung und Forschung (BMBF), Germany; Istituto Nazionale di Fisica
Nucleare (INFN), Italy; Nederlandse organisatie voor Wetenschappelijk
Onderzoek (NWO), the Netherlands;Executive Unit for Financing Higher
Education, Research, Development and Innovation (UEFISCDI), Romania;
Ministerio de Ciencia, Innovación, Investigación y Universidades (MCIU):
Programa Estatal de Generación de Conocimiento (refs. PGC2018-096663-B-C41,
-A-C42, -B-C43, -B-C44) (MCIU/FEDER), Generalitat Valenciana: Prometeo
(PROMETEO /2020/019), Grisolía (refs. GRISOLIA /2018/119,
/2021/192) and GenT (refs. CIDEGENT/2018/034, /2019/043, /2020/049, 021/023)
programs, Junta de Andalucía (ref. A-FQM-053-UGR18), La Caixa Foundation (ref.
LCF/BQ/IN17/11620019), EU: MSC program (ref. 101025085), Spain; Ministry of
Higher Education, Scientific Research and Innovation, Morocco, and the Arab
Fund for Economic and Social Development, Kuwait. We also acknowledge the
technical support of Ifremer, AIM and Foselev Marine for the sea operation and
the CC-IN2P3 for the computing facilities.

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s\models_{{\cal LQF}}t$ implies that $(r\land s)\lor\neg
e_{d}(s)\models_{{\cal LQF}}\neg e_{d}(s)\lor t$. Therefore, $r\lor\neg
e_{d}(s)\models_{{\cal LQF}}\neg e_{d}(s)\lor t$ because for each valuation
$v$, $v(e_{d}(s))$ is a central element and $v(s\lor\neg e_{d}(s))=1$. Since
$r\models_{{\cal LQF}}r\lor\neg e_{d}(s)$ then $r\models_{{\cal LQF}}\neg
e_{d}(s)\lor t$. Thus, $T\models_{{\cal LQF}}\neg e_{d}(s)\lor t$ and, by
Theorem 7.6, $T\vdash\neg e_{d}(s)\lor t$.

On the other hand, let us suppose that $T\vdash\neg e_{d}(s)\lor t$. Then,
there exist $t_{1}\ldots t_{n}\in T$ such that, by defining
$r=t_{1}\land\ldots\land t_{n}$, $r\models_{{\cal LQF}}\neg e_{d}(s)\lor t$.
Therefore, we also have that $r\land e_{d}(s)\models_{{\cal
LQF}}e_{d}(s)\land(\neg e_{d}(s)\lor t)$ and, consequently, $r\land
e_{d}(s)\models_{{\cal LQF}}e_{d}(s)\land t$ because for each valuation $v$,
$v(e_{d}(s)\land(\neg e_{d}(s)\lor t))=v(e_{d}(s)\land t)$. Since
$e_{d}(s)\land t\models_{{\cal LQF}}t$ we have that

$r\land e_{d}(s)\models_{{\cal LQF}}t.$ (39)

Let $v$ be a valuation such that $v(T)=1$ and $v(s)=1$. Then, $v(r)=1$ and
$v(e_{d}(s))=e_{d}(v(s))=e_{d}(1)=1$. Thus, by Eq.(39), $v(t)=1$ proving that
$T\cup\\{s\\}\models_{{\cal LQF}}t$. Hence, by Theorem 7.6,
$T\cup\\{s\\}\vdash t$.

$\Box$

## Acknowledgments

This work is supported by i) MIUR, project PRIN 2017: Theory and applications
of resource sensitive logics, CUP: 20173WKCM5.

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# Self-Aligned Concave Curve: Illumination Enhancement for Unsupervised
Adaptation

Wenjing Wang  Peking UniversityBeijingChina<EMAIL_ADDRESS>,  Zhengbo Xu
Peking UniversityBeijingChina<EMAIL_ADDRESS>,  Haofeng Huang  Peking
UniversityBeijingChina<EMAIL_ADDRESS>and  Jiaying Liu  Peking
UniversityBeijingChina<EMAIL_ADDRESS>
(2022)

###### Abstract.

Low light conditions not only degrade human visual experience, but also reduce
the performance of downstream machine analytics. Although many works have been
designed for low-light enhancement or domain adaptive machine analytics, the
former considers less on high-level vision, while the latter neglects the
potential of image-level signal adjustment. How to restore underexposed
images/videos from the perspective of machine vision has long been overlooked.
In this paper, we are the first to propose a learnable illumination
enhancement model for high-level vision. Inspired by real camera response
functions, we assume that the illumination enhancement function should be a
concave curve, and propose to satisfy this concavity through discrete
integral. With the intention of adapting illumination from the perspective of
machine vision without task-specific annotated data, we design an asymmetric
cross-domain self-supervised training strategy. Our model architecture and
training designs mutually benefit each other, forming a powerful unsupervised
normal-to-low light adaptation framework. Comprehensive experiments
demonstrate that our method surpasses existing low-light enhancement and
adaptation methods and shows superior generalization on various low-light
vision tasks, including classification, detection, action recognition, and
optical flow estimation. All of our data, code, and results will be available
online upon publication of the paper.

Low-light, domain adaptation, low-level vision, high-level vision

††journalyear: 2022††copyright: acmcopyright††conference: Proceedings of the
30th ACM International Conference on Multimedia; October 10–14, 2022; Lisboa,
Portugal††booktitle: Proceedings of the 30th ACM International Conference on
Multimedia (MM ’22), October 10–14, 2022, Lisboa, Portugal††price: 15.00††doi:
10.1145/3503161.3547991††isbn: 978-1-4503-9203-7/22/10††ccs: Computing
methodologies Image processing††ccs: Computing methodologies Computer vision
problems

##  1\. Introduction

Insufficient lighting is a common kind of image degradation, caused by dark
environments, corrupted equipment, or improper shooting settings. It may
degrade the visual quality of images, causing low visibility, lost details,
and aesthetic distortion. Besides, with the rise of machine learning, visual
analytics has been playing an increasingly critical role in many applications.
Low light can also pose threats to machine analytics, presenting challenges to
high-level vision tasks in the low-light condition, such as nighttime
autonomous driving and surveillance video analysis.

The research of restoring low-light images has attracted wide attention. From
early manually designed algorithms (Jobson et al., 1997) to recent data-driven
deep models (Liu et al., 2021a), a great number of works have effectively
improved human visual quality of low-light images. However, most existing low-
light enhancement methods do not take machine vision into account. Some
methods introduce noise removal (Wei et al., 2018) or detail reconstruction
(Xu et al., 2020b), which enhances visual quality but might mislead high-level
analytics models. Some other methods take semantic perception into
consideration (Liu et al., 2020), but they still focus on human vision and
have unsatisfactory performance on downstream high-level vision tasks.

In recent years, to promote the development of machine analytics in low-light
scenarios, many datasets have been built (Yang and Ye Yuan, 2020; Loh and
Chan, 2019), giving birth to a series of machine vision models tailored for
underlit environments. With a large amount of training data, many methods
improve normal light frameworks by incorporating the properties of
insufficient lighting conditions (Cui et al., 2021). Some works further
explore the task of adapting normal light models to low-light without task-
specific annotation, which is more universal and application-driven (Wu et
al., 2021; Wang et al., 2021a; Lengyel et al., 2021). However, existing
normal-to-low light adaptation algorithms either rely on multiple sources (Wu
et al., 2021), adopt troublesome multi-stage and multi-level processes (Wang
et al., 2021a), or fail in darker cases (Lengyel et al., 2021). Moreover, most
adaptive methods concentrate on the high-dimensional features of machine
analytics models, but neglect the characteristics of input images themselves.

Different from existing normal-to-low light adaptation methods, we make full
use of the potential of illumination adjustment. We first build an
illumination enhancement model which can maximize the model’s abilities while
being easy to learn. Inspired by camera response functions, we design a new
model paradigm: deep concave curve, which can determine the new pixel value in
the enhanced result with a high degree of freedom. To effectively satisfy
concavity, we propose to first predict a non-positive second derivative, then
apply discrete integral implemented by convolutions. To train this model
towards unsupervised adaptation, we design asymmetric self-supervised
alignment. On the normal light side, we learn decision heads with a self-
supervised pretext task. Then on the low-light side, we fix the decision heads
and let our model improve the pretext task performance through enhancing the
input image. In this way, even without annotated data, our model can learn how
to make the machine analytics model better perceive the enhanced low-light
image. To make full use of image information and provide good guidance for
illumination enhancement, we propose a new rotated jigsaw permutation task.
Experiments show that our model architecture and training design are
compatible with each other. On one hand, our self-learned strategy can better
restore illumination compared with other feature adaptation strategies; on the
other hand, our deep concave curve can best maximize the potential of self-
learned illumination alignment.

The proposed illumination enhancement model, self-aligned concave curve
(SACC), can serve as a powerful tool for unsupervised low-light adaptation.
Although SACC does not require normal or low-light annotations and does not
even adjust the downstream model, it achieves superior performance on a
variety of low-light vision tasks. To further deal with noises and semantic
domain gaps, we propose to adapt downstream analytics models by pseudo
labeling. Finally, we build an adaptation framework SACC+, which is concise
and easy to implement but can outperform existing low-light enhancement and
adaptation methods by a large margin.

In summary, our contributions are threefold:

  * •

We are the first to propose a learnable pure illumination enhancement model
for high-level vision. Inspired by camera response functions, we design a deep
concave curve. Through discrete integral, the concavity constraint can be
satisfied through the model architecture itself.

  * •

Towards unsupervised normal-to-low light adaptation, we design an asymmetric
cross-domain self-supervised training strategy. Guided by the rotated jigsaw
permutation pretext task, our curve can adjust illumination from the
perspective of machine vision.

  * •

To verify the effectiveness of our method, we explore various high-level
vision tasks, including classification, detection, action recognition, and
optical flow estimation. Experiments demonstrate our superiority over both
low-light enhancement and adaptation state-of-the-art methods.

##  2\. Related Works

Low-light Enhancement.

Early methods manually design illumination models and enhancement strategies.
In the Retinex theory (Land, 1977), images are decomposed into reflectance
(albedo) and shading (illumination). On this basis, Retinex-based methods (Guo
et al., 2017; Fu et al., 2016) first decompose images and then either
separately or simultaneously process the two components. Histogram
equalization and its variants (Pizer et al., 1990) instead redistribute the
intensities on the histogram.

Recent methods are mainly based on deep learning. Some mod-

els mimic the Retinex decomposition process (Wei et al., 2018; Zhang et al.,
2019). RUAS (Liu et al., 2021a) unrolls the optimization process of Retinex-
inspired models and searches desired network architectures. EnlightenGAN
(Jiang et al., 2021b) introduces adversarial learning. Zero-DCE (Guo et al.,
2020) designs a curve-based low-light enhancement model and learns in a zero-
reference way. Some methods also target RAW images (Chen et al., 2018), videos
(Chen et al., 2019; Jiang and Zheng, 2019), and introduce extra light sources
(Xiong et al., 2021; Xia et al., 2021). Interested readers may refer to (Liu
et al., 2021c) and (Li et al., 2021b) for comprehensive surveys.

Existing low-light enhancement methods disregard downstream machine learning
tasks. In comparison, our model targets high-level vision and greatly benefits
machine vision in low-light scenarios.

High-Level Vision in Low-light scenarios.

With an increasing demand for autonomous driving and surveillance analysis,
low-light high-level vision has attracted ever-higher attention in recent
years. For dark object detection, Sasagawa et al. (Sasagawa and Nagahara,
2020) merged pretrained models in different domains with glue layers and
generative knowledge distillation. MAET (Cui et al., 2021) learns through
jointly decoding degrading transformations and detection predictions. HLA-Face
(Wang et al., 2021a) adopts a joint pixel-level and feature-level adaptation
framework. For nighttime semantic segmentation, DANNet (Wu et al., 2021)
employs adversarial training to adapt models in one stage without additional
day-night image transferring. For general tasks, CIConv (Lengyel et al., 2021)
designed a color invariant representation. Some works also focus on tasks of
image retrieval (Jenícek and Chum, 2019), depth estimation (Wang et al.,
2021b), and matching (Song et al., 2021) in low-light conditions.

Despite all these progress on high-level vision in low-light scenarios, many
methods rely on low-light annotations, which are neither robust nor flexible
enough. Existing unsupervised adaptation methods focus on feature migration
and ignore the importance of pixel-level adjustment. Based on illumination
enhancement, we propose a new method for low-light adaptation that outperforms
existing methods by a wide margin.

##  3\. Deep Concave Curve

In this section, we introduce the motivation and detailed architecture of our
illumination enhancement model.

###  3.1. From CRF to Concave Curve

Digital photographic cameras use camera response functions (CRFs) when mapping
irradiance to intensities. Although scene illumination changes linearly on the
irradiance level, to fit the logarithmic perception of human vision, cameras
employ non-linear CRFs, making illumination adjustment complicated on the
intensity level. To utilize the linearity of irradiance, some low-light
enhancement methods (Huang et al., 2020, 2022) transform intensities to
irradiance, adjust the irradiance, and then map irradiance back to
intensities. However, back-and-forth irradiance $\leftrightarrow$ intensity
mapping is inconvenient and difficult to introduce high-level machine vision
guidance.

We propose to simplify the above complex pipeline into one single intensity-
level adjustment, which is denoted by $g$. We first analyze what form $g$
should take. Ignoring spatial variations like lens fall-off (Asada et al.,
1996), vignetting, and signal-dependent noise, CRF can be assumed to be the
same for each pixel in an image. Accordingly, we set $g$ to be spatially
shared. Second, to follow the numerical range of pixels and preserve order,
$g$ should pass $(0,0)$, $(1,1)$, and increase monotonically. Additionally,
although pixels are discrete, we want $g$ to appear roughly continuous, i.e.,
like a curve. Despite the above constraints, the solution space of $g$ is
still too large, making optimization difficult.

DoRF (Grossberg and Nayar, 2004) is a dataset of real camera CRFs. As shown in
Fig. 1, we find that most CRFs are concave. Statistically, 89.5% of the
discrete second derivatives are negative. This is because human perception of
brightness is more sensitive to relative differences between darker tones than
differences between lighter tones. Accordingly, enhancing illumination on
irradiance results in concave transformation on intensities. Therefore, we
assume that $g$ should be concave.

With the concavity constraint narrowing the solution space, in Sec. 4 we will
develop a learning strategy, through which an appropriate $g$ can be
automatically found.

Figure 1. Left: real camera CRFs from the DoRF (Grossberg and Nayar, 2004)
dataset. Right: the heat map of second-order derivatives in DoRF. Each column
represents a CRF. Figure 2. The framework of our model. Our model first
predicts a non-negative minus second derivative $-\nabla^{2}c$, and then
integrates and normalizes it into a concave curve $g$, which controls the
illumination of the enhanced result. The model is trained in an asymmetric
self-supervised way. Based on a pretrained and fixed-weight feature extractor,
we first train a pretext head on normal light images, and then train our model
with the fixed-weight pretext head on low-light images.

###  3.2. Form and Implementation

Based on the above analysis, our illumination enhancement adjustment function
should be monotonically increasing and concave. The model is called: deep
concave curve.

Now we introduce the detailed design of our model. We use a neural network $C$
to predict $g$ from the input low-light image $I_{L}$. Our $g$ is a mapping
from original pixel values to new pixel values. If the color space has $P$
numeric values (e.g., 256 for 8-bit images), then $g$ will be an
$\mathbb{R}^{P}$ vector. For original pixels of value $(p-1)/P$ on the input
image $I_{L}$, where $p\in\\{1,2,..,P\\}$, their new pixel values would be the
$p$-th element of $g$. For ease of understanding, we provide an example in the
supplementary.

Intuitively, the conditions of monotonic increasing and concavity can be
fulfilled by penalty terms on the discrete first and second derivatives.
However, this naïve strategy will introduce new losses, adding the burden of
balancing multiple learning objectives.

We propose to satisfy these two constraints through the model architecture
itself. Instead of directly estimating $g$, we first predict a minus second
derivative $-\nabla^{2}c=C(I_{L})$. By setting the last layer of network $C$
to ReLU, we ensure that $-\nabla^{2}c\geq 0$. Then, we integrate
$-\nabla^{2}c$ into $c$. Finally, to fit the range of $[0,1]$, we normalize
$c$ into $g=\text{Norm}(c)$. Since $-\nabla^{2}c\geq 0$, $g$ is guaranteed to
be concave.

The discrete integral from $-\nabla^{2}c$ to $\nabla c$ can be implemented as
subsequence summation. Denote matrix-vector multiplication $\nabla
c=A\times(-\nabla^{2}c)$, we set $A=[a_{ij}]$ as an upper triangular matrix:

(1) $\displaystyle a_{ij}=\left\\{\begin{aligned} 1,&&i\leq j,\\\
0,&&i>j.\end{aligned}\right.\vspace{-0.5mm}$

Since $-\nabla^{2}c\geq 0$ and $a_{ij}\geq 0$, we can derive that $\nabla
c\geq 0$, thus $g$ is guaranteed to increase monotonically. Similarly, denote
$c=B\times(\nabla c)$, we set $B=[b_{ij}]$ as a strictly lower triangular
matrix:

(2) $\displaystyle\vspace{-0.5mm}b_{ij}=\left\\{\begin{aligned} 0,&&i\leq
j,\\\ 1,&&i>j.\end{aligned}\right.\vspace{-0.5mm}$

For simplicity, we merge the two integral into one step
$c=D\times(-\nabla^{2}c)$, where $D=B\times A$ is pre-calculated. Since the
first element of $g$ is known to be zero, we only need to predict the other
$P-1$ values, i.e., $-\nabla^{2}c\in\mathbb{R}^{P-1}$ and
$c\in\mathbb{R}^{P}$. Finally, for normalization, we divide the vector by its
maximum value: $g=\text{Norm}(c)=c/||c||_{\infty}$.

###  3.3. Network Architecture

We predict an independent $g$ for each color channel. For RGB images, we have
$g_{R}$, $g_{G}$, $g_{B}$. Since different color channels share the same
summation matrix, we put $g_{R}$, $g_{G}$, and $g_{B}$ side-by-side to form a
1D tensor, and implement integral as a 1D convolution with 256 kernels of size
1. The convolution weight is fixed to $D=B\times A$.

The architecture is illustrated in the red part of Fig. 2. To increase
receptive field and acceleration, we first down-sample the input image to
$16\times 16$ resolution. Curve predictor $C$ consists of a shallow U-net
(Ronneberger et al., 2015), two convolutions, and one fully connected layer.
For 8-bit RGB, the output dimension is 765. Then, we reshape the output into
255$\times$3, and feed it to the integral 1D convolution. Finally, we apply
$g$ on the image $I_{L}$, which is efficiently implemented with the
index_select function in PyTorch (Paszke et al., 2019).

##  4\. Asymmetric Self-Supervised Alignment

In the previous section, we have proposed the deep concave curve for
illumination enhancement. In this section, we will introduce how to train it
towards machine vision.

###  4.1. How to Guide Illumination Enhancement

Our goal is to align the brightness of low-light images to normal light and
improve the performance of downstream tasks in low-light conditions. The main
idea is to employ high-level vision models as guidance. We first pretrain
downstream models on normal light images, then fix their backbones and let our
deep concave curve bridge the feature-level gap between low and normal light.

The challenge is that our deep concave curve can only adjust illumination,
namely signal restoration, but low and normal light also differs in semantics.
To narrow feature-level gaps, existing methods adopt discrepancy metrics (Sun
and Saenko, 2016) and adversarial learning (Ganin and Lempitsky, 2015).
However, discrepancy metrics and adversarial discriminators bring extra
semantic supervision, which can mislead our model and make training hard to
converge.

To disentangle signal restoration and semantics, we adopt cross-domain self-
supervised pretext tasks. The framework is shown in Fig. 2. We first train
normal light self-supervised learning with a fixed-weight feature backbone and
a trainable multilayer perceptrons (MLP) head. Although enhanced low-light
data and normal light data may contain different objects and scenes, they all
follow the distribution of natural images. Therefore, the normal light
pretrained model can also accomplish pretext tasks on enhanced low-light
images. Using this property, we fix both the backbone and the head, and train
our deep concave curve by the self-supervised loss on enhanced low-light
images. Since semantics are taken care of by the MLP head, our deep concave
curve will not be disturbed by it. Through training to improve pretext task
performance, our model learns how to make the feature extractor better
perceive the enhanced low-light image, therefore implicitly benefiting
downstream tasks in low-light conditions.

###  4.2. Pretext Task Design

Now we need to find a pretext task that is able to provide sufficient
supervision for illumination adaptation. Contrastive learning (He et al.,
2020) is one of the most popular self-supervised learning methods nowadays.
However, it adopts signal-related augmentations such as color jittering, which
may destroy the model’s perception of illumination. Moreover, contrastive
learning maximizes the feature distance between different samples, which may
mislead the enhancement model into generating diverse but abnormal results.
Rotation prediction (Gidaris et al., 2018) and jigsaw puzzling (Noroozi and
Favaro, 2016) do not interfere with the enhancement process, but they are not
powerful enough, providing limited supervision for illumination adaptation.

To effectively guide the enhancement model, we propose rotated jigsaw puzzles.
We first rotate the input image by random angles, then apply 3$\times$3 jigsaw
shuffling and ask the network to recognize the permutation. Compared with
traditional jigsaw puzzling, rotation augmentation can increase the difficulty
of the pretext task, which forces the MLP heads to better understand
semantics, thus introduces more supervision for adaptation.

The learning objective is as follows:

(3) $\displaystyle\mathcal{L}_{L}$
$\displaystyle=\mathcal{L}_{c}(\hat{p}^{r-jig}_{L},p^{r-jig}_{L}),$
$\displaystyle\mathcal{L}_{N}$
$\displaystyle=\mathcal{L}_{c}(\hat{p}^{r-jig}_{N},p^{r-jig}_{N}),$

where $\mathcal{L}_{c}$ is the cross-entropy loss, $\hat{p}^{r-jig}$ is the
predicted jigsaw permutation of the randomly rotated and jigsaw shuffled
image, $p^{r-jig}$ is the ground truth permutation, $L$ stands for low light
and $N$ stands for normal light.

##  5\. Framework Design and Analysis

###  5.1. Unsupervised Low-light Adaptation

With the above designs, we propose a new unsupervised low-light adaptation
framework: self-aligned concave curve (SACC). Given a pretrained normal-light
downstream model, we use its feature backbone to train our deep concave curve
with Eq. (3). While testing, we first enhance the input image and then apply
the downstream model. Different from existing low-light adaptation methods,
SACC is blind to both normal and low-light annotations and does not even need
to adjust the downstream model. Although only performing illumination
enhancement, SACC achieves superior performance by taking full advantage of
image data and machine features.

Besides insufficient light, there are other low-light factors, such as color
bias and noise. For the former, SACC can handle color bias. First, SACC has
independent curves for each color channel. Second, downstream feature
extractors are usually trained on high-quality normal-light images without
color bias, thus encouraging our model to restore balanced color. For the
latter, real camera noise is quite hard to remove, but improving the model’s
robustness to noise is much easier. On this basis, we adopt pseudo labeling to
adapt the downstream model. We predict labels of the enhanced low-light data,
remove predictions of low confidence, and then fine-tune the downstream model
on normal light data with labels and low-light data with pseudo labels. Our
strategy can not only implicitly handle noises but also narrow other kinds of
domain gaps, such as scene differences. This advanced version is called SACC+.

###  5.2. Analysis: Why Self-Supervised Alignment?

Table 1. Effects of different learning strategies for guiding our illumination enhancement model. Category | Strategy | mAP (%)
---|---|---
Baseline | - | 16.09
Discrepancy Metrics |  CMD (Zellinger et al., 2017) | 28.90
MMD (Borgwardt et al., 2006) | 33.76
Adversarial Learning |  LSGAN (Mao et al., 2017) | 38.05
|  MoCo (He et al., 2020) | 38.46
Self-Supervised |  Rotation (Gidaris et al., 2018) | 39.59
Learning |  Jigsaw (Noroozi and Favaro, 2016) | 41.01
| Rotated Jigsaw (proposed) | 41.31

In the following, we verify our self-supervised alignment strategy through
experiments. The analysis is based on DARK FACE (Yang and Ye Yuan, 2020) and
WIDER FACE (Yang and Ye Yuan, 2020). The goal is to improve the performance of
WIDER-FACE-pretrained DSFD (Li et al., 2019) on DARK FACE by enhancing input
images. We do not use multiscale testing for simplicity.

Adapting from DARK FACE to WIDER FACE is especially hard since WIDER FACE is
collected from the Internet while DARK FACE is captured at nighttime streets.
Due to this huge scene gap, discrepancy metrics do not perform well as shown
in Table 1. Adversarial learning leads to a better result, but the training is
very unstable. In experiments, the balance between the low-light enhancement
model and the discriminator often crashes. In comparison, self-supervised
learning is more effective and easier to train. The contrastive learning
method MoCo (He et al., 2020) has been proved powerful for training networks
from scratch. But for adapting illumination, MoCo is less effective than
rotation (Gidaris et al., 2018) and jigsaw (Noroozi and Favaro, 2016).
Finally, our rotated jigsaw strategy achieves the best result, which is in
line with our motivation to increase the difficulty of the pretext task.

###  5.3. Analysis: Why Deep Concave Curve?

Table 2. Effects of different low-light enhancement backbones under our asymmetric self-supervised strategy. Category | Method | mAP (%)
---|---|---
Baseline | - | 16.09
Other |  EnlightenGAN (Jiang et al., 2021b) | 18.42
Zero-DCE (Guo et al., 2020) | 25.36
Gamma Correction $x^{\gamma}$ | 38.25
Ours | No Constraint | 12.44
$\nabla g\geq$ 0 | 34.40
$\nabla g\geq$ 0 and $\nabla^{2}g\leq$ 0 (proposed) | 41.31
$\nabla g\geq$ 0, $\nabla^{2}g\leq$ 0, and $\nabla^{3}g\geq$ 0 | 40.74
Table 3. Comparison with other low-light enhancement networks under different training strategies. Method | Version | mAP (%)
---|---|---
Baseline | - | 16.09
| Original | 28.85
EnlightenGAN (Jiang et al., 2021b) | Retrained | 20.08
|  Original + Eq. (3) | 32.88
| Original | 38.32
Zero-DCE (Guo et al., 2020) | Retrained | 32.53
|  Original + Eq. (3) | 40.54
SACC (Ours) | - | 41.31

Next, we raise another question: under the proposed training strategy, is deep
concave curve superior to other network architectures?

We first explore two representative models: EnlightenGAN (Jiang et al., 2021b)
and Zero-DCE (Guo et al., 2020). EnlightenGAN adopts a widely-used U-net
(Ronneberger et al., 2015) like image-to-image architecture. Zero-DCE adjusts
images by element-wise second-order curves. We retrain these two networks with
Eq. (3). Other experimental settings are the same as Sec. 5.2. As shown in
Table 2, their mAP scores are less than 30. In Fig. 3(b) and (c), we can see
that there are weird line artifacts. As far as we are concerned, these lines
are trivial solutions for the rotated jigsaw permutation problem. In other
words, EnlightenGAN and Zero-DCE get overfitted to Eq. (3), therefore cannot
generalize to the downstream high-level vision task. In comparison, our deep
concave curve obtains better visual results and mAP scores. This is because
our spatially sharing, monotonically increasing and concavity constraints act
as regularization and prevent the model from carving cheating symbols or hints
on images.

Naïve gamma correction $x^{\gamma}$ ($0$$<$$\gamma$$<$$1$) has been commonly
used for low-light enhancement, but the adaptation performance is not as good
as ours as shown in Table 2. This is because, for each color channel,
$x^{\gamma}$ only has one learnable parameter $\gamma$. In contrast, our curve
has 255 parameters. Compared with $x^{\gamma}$, our curve has a higher degree
of freedom and thus is more powerful.

We further explore the necessity of each constraint in Table 2 and Fig. 4.
Without any constraint, the enhanced results are far from natural images. With
only $\nabla g\geq$ 0, the curve does not look continuous and leads to low
contrast and dull color as shown in Fig. 4(b). With $\nabla g\geq$ 0 and
$\nabla^{2}g\leq$ 0, the model achieves the best result. We also try
$\nabla^{3}g\geq$ 0, but the mAP score gets slightly worse and the image gets
partially over-exposed. The reason might be that too strong constraints limit
the expression ability of the model. Also, integrating too many times makes
the weights of the integral matrix extremely large, resulting in small
gradients and increasing the difficulty of neural network training.

Figure 3. Effects of asymmetric self-supervised learning with different low-
light enhancement model backbones. Figure 4. Low-light enhancement results
(left) and curve shapes (right) under different curve form constraints.

Finally, we discuss: if the key lies in preventing overfitting to Eq. (3),
will other ‘regularizations’ also work? We explore the constraints of
adversarial learning in EnlightenGAN and a group of spatial, exposure, color,
and smoothness non-reference loss functions in Zero-DCE. As shown in Table 3,
when we retrain EnlightenGAN and Zero-DCE on DARK FACE and WIDER FACE, their
performances get degraded, because their learning strategies are specially
designed for their original datasets. Then, we try to keep their loss
functions on their original datasets, but add Eq. (3) on DARK FACE and WIDER
FACE. In Table 3, we can see that the mAP scores are higher, demonstrating the
effectiveness of Eq. (3) on other models. This also indicates that adversarial
learning and non-reference loss functions are effective ‘regularizations’ as
well. But even when using Eq. (3), EnlightenGAN and Zero-DCE still perform
worse than our deep concave curve. This is because adversarial and non-
reference learning introduces extra conflicts against high-level vision, while
our deep concavity is a purer regularization, and thus can maximize the
potential of Eq. (3).

###  5.4. Analysis: Denoising or SACC+?

Table 4. Effects of denoising enhanced low-light images and the proposed advanced version SACC+. Method | Version | mAP (%)
---|---|---
Baseline | SACC | 41.31
Denoising |  SACC + BM3D (Dabov et al., 2007) | 24.51
SACC + Neighbor2Neighbor (Huang et al., 2021) | 40.54
Proposed | SACC+ | 45.51

We further discuss the necessity of denoising. We evaluate two representative
methods: the widely-used BM3D (Dabov et al., 2007) and the learning-based
Neighbor2Neighbor (Huang et al., 2021). These two methods can remove noise
from the perspective of human vision. However, denoising also blurs details,
which impedes machine vision. As a result, the accuracy of face detection
degrades as shown in Table 4. In comparison, fine-tuning with low-light pseudo
labels greatly improves the performance, supporting our design of SACC+.

Figure 5. Example classification result. The original image is wrongly
categorized as “Chair” with a confidence of 0.98, after enhancement, it can be
correctly recognized as “Cat”.

##  6\. Experiments

In this section, we provide implementation details, benchmarking results, and
more experimental analysis and applications.

###  6.1. Implementation Details

Our deep concave curve can be applied to various models and vision tasks. In
this paper, we show the results of ResNet (He et al., 2016), DSFD (Li et al.,
2019) (VGG (Simonyan and Zisserman, 2015) backbone), I3D (Carreira and
Zisserman, 2017) (3D-CNN), and PWC-Net (Sun et al., 2018) (feature pyramid
backbone), involving four different visual tasks.

For SACC, we first pretrain the pretext task head on normal light images with
the feature extractor fixed. Then, we fix the pretext task and train our deep
concave curve on low light images. For SACC+, we first fix the pretrained SACC
and then fine-tune the downstream model on normal-light and SACC-enhanced low-
light data. Due to page limit, detailed experimental settings for each task

are provided in the supplementary material.

Table 5. Adaptive low-light classification comparison results. $\dagger$ denotes that the low-light enhancement model is retrained. Category | Method | Top-1 (%)
---|---|---
Baseline |  ResNet-18 (He et al., 2016) | 55.04
Fully Supervised |  Finetuned ResNet-18 (He et al., 2016) | 71.52
|  RetinexNet (Wei et al., 2018) | 44.72
|  Zero-DCE$\dagger$ (Guo et al., 2020) | 50.80
|  EnlightenGAN (Jiang et al., 2021b) | 57.76
|  Zero-DCE++ (Li et al., 2021a) | 58.56
|  MF (Fu et al., 2016) | 59.20
Low-Light |  KinD (Zhang et al., 2019) | 59.28
Enhancement |  LIME (Guo et al., 2017) | 59.44
|  Zero-DCE (Guo et al., 2020) | 59.44
|  RUAS$\dagger$ (Liu et al., 2021a) | 59.84
|  EnlightenGAN$\dagger$ (Jiang et al., 2021b) | 60.24
|  LLFlow (Wang et al., 2022) | 60.72
|  SACC (Ours) | 61.44
|  CMD (Zellinger et al., 2017) | 55.92
Unsupervised |  AdaBN (Li et al., 2017) | 59.68
Domain |  DANN (Ganin and Lempitsky, 2015) | 59.76
Adaptation |  CIConv (Lengyel et al., 2021) | 60.96
|  SACC+ (Ours) | 63.92
Figure 6. Subjective comparison results for dark face detection. $\dagger$
denotes that the low-light enhancement model is retrained. The color of the
bounding boxes represents the confidence of recognition, with yellow
indicating higher confidence.

###  6.2. Low-light Object Classification

We first evaluate our approach by one of the most fundamental vision tasks:
image classification. CODaN (Lengyel et al., 2021) is a 10-class dataset for
low-light adaptation. It contains 13k daytime and 2.5k nighttime images. We
use its official daytime setting and split 1.25k nighttime images for
training/validation and the others for testing. The task is to adapt ResNet-18
(He et al., 2016) from daytime to nighttime. Our method is compared with nine
low-light enhancement and four unsupervised domain adaptation methods.
Enhancement methods only adjust input images, while adaptation methods also
adjust the classifier. They all cannot obtain low-light labels. We
additionally show the result of fully supervised learning as a reference of
performance upper bound, where the baseline is trained with low-light labels.

The results are shown in Table 5. Low-light enhancement methods restore input
low-light images from the perspective of human visual quality and neglect
machine vision, thus resulting in limited performance on CODaN. Unsupervised
domain adaptation methods adapt the classifier to nighttime. CMD (Zellinger et
al., 2017), AdaBN (Li et al., 2017), and DANN (Ganin and Lempitsky, 2015) are
designed for normal light, but the day-night domain gap is more complex.
Accordingly, their accuracies are all below 60%. CIConv (Lengyel et al., 2021)
uses a color invariant layer. Benefitting from its robust feature, CIConv is
superior to other low-light enhancement and unsupervised domain adaptation
methods. But compared with our methods, CIConv is less effective. Through
adjusting illumination from the perspective of machine vision, our SACC best
adapts the daytime classifier to nighttime. With the help of pseudo label
fine-tuning to adjust the downstream classifier, the accuracy of our SACC+ is
even higher. Fig. 5 shows an example result. $-\nabla^{2}c$ is very discrete,
but after integral $g$ appears like a continuous curve. As the input image
suffers from red color bias, our model brightens more on the green channel,
which adjusts the color balance.

Table 6. Adaptive dark face detection comparison results. $\dagger$ denotes that the low-light enhancement model is retrained. Category | Method | mAP (%)
---|---|---
Baseline |  DSFD (Li et al., 2019) | 16.09
Fully Supervised |  Finetuned DSFD (Li et al., 2019) | 45.99
|  RetinexNet (Wei et al., 2018) | 12.04
|  KinD (Zhang et al., 2019) | 15.84
|  EnlightenGAN$\dagger$ (Jiang et al., 2021b) | 20.77
|  EnlightenGAN (Jiang et al., 2021b) | 31.31
|  Zero-DCE $\dagger$ (Guo et al., 2020) | 37.30
Low-Light |  LLFlow (Wang et al., 2022) | 37.41
Enhancement |  RUAS$\dagger$ (Liu et al., 2021a) | 38.36
|  LIME (Guo et al., 2017) | 40.71
|  Zero-DCE++ (Li et al., 2021a) | 40.90
|  Zero-DCE (Guo et al., 2020) | 41.27
|  MF (Fu et al., 2016) | 41.43
|  SACC (Ours) | 44.57
|  CIConv (Lengyel et al., 2021) |  04.40
Unsupervised |  OSHOT (D’Innocente et al., 2020) | 25.38
Domain |  Progressive DA (Hsu et al., 2020) | 28.47
Adaptation |  Pseudo Labeling (Inoue et al., 2018) | 35.07
HLA-Face (Wang et al., 2021a) | 44.44
|  SACC+ (Ours) | 46.91

###  6.3. Dark Face Detection

We further conduct a benchmark on unsupervised dark face detection. WIDER FACE
(Yang et al., 2016) contains 32k images of various events and scenes. DARK
FACE (Yang and Ye Yuan, 2020) contains 10k nighttime street scene images. We
use their official splittings. The baseline is DSFD (Li et al., 2019).

Results are shown in Table 6. Although RetinexNet (Wei et al., 2018) and KinD
(Zhang et al., 2019) consider detail reconstruction and noise reduction, they
introduce extra artifacts for machine vision and thus degrade the performance
of face detection. Other low-light enhancement methods can benefit DSFD, but
their mAP scores are lower than ours. These methods may make enhanced images
look unnatural in machine vision, which can mislead the face detector as shown
in Fig. 6.

Figure 7. Optical flow estimation results of the same scene under different
illumination levels.

Unsupervised domain adaptation methods OSHOT (D’Innocente et al., 2020),
Progressive DA (Hsu et al., 2020), and Pseudo Labeling (Inoue et al., 2018)
cannot effectively handle the normal-low light gap. Although CIConv (Lengyel
et al., 2021) is effective on CODaN, due to the much lower illumination of
nighttime street scenes, the color invariant layer of CIConv fails on DARK
FACE. As visualized in Fig. 6(i), the representation of CIConv suffers from
severe noise. HLA-Face (Wang et al., 2021a) is the state-of-the-art for WIDER-
DARK unsupervised adaptation. It adopts a complex low-level and high-level
joint adaptation strategy. In comparison, our SACC only adjusts illumination
but still outperforms HLA-Face. Our advanced version SACC+ even outperforms
the fully supervised upper bound, demonstrating the great potential of our
method.

Now we analyze the generalization of DSFD-trained SACC to other face
detectors: PyramidBox (Tang et al., 2018) and MogFace (Liu et al., 2021b). As
shown in Table 7, our performance is consistently good even for unseen
downstream models, indicating that high-level vision knowledge is universal
among different machine models.

Table 7. More results on DARK FACE (Yang and Ye Yuan, 2020). Red denotes the best and blue denotes the second best performance. | Detection mAP (%) | NR-IQA
---|---|---
| (Tang et al., 2018) | (Li et al., 2019) | (Liu et al., 2021b) | AVG | (Mittal et al., 2013) | (Liu et al., 2014)
Original | 13.99 | 16.09 | 16.36 | 15.48 |  8.5 | 24.1
RetinexNet (Wei et al., 2018) | 11.42 | 12.04 | 14.56 | 12.67 |  7.7 | 21.9
KinD (Zhang et al., 2019) | 15.61 | 15.84 | 21.27 | 17.57 |  9.8 | 42.2
EnlightenGAN$\dagger$ (Jiang et al., 2021b) | 19.54 | 20.77 | 24.02 | 21.44 |  9.0 | 22.6
EnlightenGAN (Jiang et al., 2021b) | 28.45 | 31.31 | 35.79 | 31.85 |  9.7 | 14.1
Zero-DCE$\dagger$ (Guo et al., 2020) | 33.41 | 37.30 | 37.75 | 36.15 |  6.7 | 18.4
LLFlow (Wang et al., 2022) | 32.84 | 37.41 | 41.08 | 37.11 |  8.4 | 10.3
RUAS$\dagger$ (Liu et al., 2021a) | 32.77 | 38.36 | 40.71 | 37.28 |  6.2 |  4.4
LIME (Guo et al., 2017) | 35.69 | 40.71 | 42.82 | 39.74 |  6.5 |  5.7
Zero-DCE++ (Li et al., 2021a) | 35.56 | 40.90 | 43.45 | 39.97 |  6.4 |  6.7
Zero-DCE (Guo et al., 2020) | 35.95 | 41.27 | 43.62 | 40.28 |  6.4 |  7.9
MF (Fu et al., 2016) | 37.49 | 41.43 | 43.87 | 40.93 |  6.5 |  8.2
SACC (Ours) | 39.20 | 44.57 | 46.45 | 43.41 |  6.3 |  3.2

###  6.4. Low-light Action Recognition

Table 8. Video action recognition comparison results. Original |  StableLLVE (Zhang et al., 2021) |  SMOID (Jiang and Zheng, 2019) |  SACC (Ours)
---|---|---|---
47.14% | 49.70% | 50.18% | 52.13%

Although initially designed for images, our method can also generalize to
video tasks. Here we evaluate our SACC with 11-class low-light video action
recognition. We gather about 800 low-light video clips from ARID (Xu et al.,
2020a). Normal light data consists of 2.6k normal light video clips from
HMDB51 (Kuehne et al., 2011), UCF101 (Soomro et al., 2012), Kinetics-600 (Kay
et al., 2017), and Moments in Time (Monfort et al., 2020). The action
recognizer is I3D (Carreira and Zisserman, 2017) based on 3D-ResNet
(Feichtenhofer et al., 2019). We report the top-1 accuracy.

We extend SACC to videos by merging all frames into a big image, predicting a
unified $g$ for the whole video clip, then applying $g$ to each frame. Since
$g$ is temporally shared, temporal consistency can be naturally maintained.

In Table 8, video enhancement methods StableLLVE (Zhang et al., 2021) and
SMOID (Jiang and Zheng, 2019) can benefit action recognition but attain
limited performance gain. Our SACC instead increases the accuracy by about 5%,
showing the superiority of our method for videos.

###  6.5. Optical Flow Estimation in the Dark

Table 9. Adaptive low-light optical flow estimation results. | Exp-2 | Exp-3 | Exp-4 | Exp-5 | AVG
---|---|---|---|---|---
Original | 10.21 | 11.90 | 14.36 | 17.82 | 13.57
Zheng et al. (Zheng et al., 2020) | 9.04 | 8.81 | 8.78 | 9.31 | 8.99
SACC (Ours) | 6.70 | 7.03 | 7.57 | 8.47 | 7.44

Last but not least, we explore a task that differs a lot from classification
and detection: optical flow estimation. The VBOF (Zheng et al., 2020) dataset
contains 10k image pairs under various brightness levels. We select Exp-2 to
Exp-5 subsets as the adaptation target domain, where Exp-5 is the darkest. The
ground truth flow fields are originally estimated by FlowNet 2.0 (Ilg et al.,
2017) on the sharp and normal light Exp-1 subset, we re-calculate them with a
more state-of-the-art approach GMA (Jiang et al., 2021a). The baseline to
adapt is PWC-Net (Sun et al., 2018). Performance is evaluated by end-point
error (EPE).

Zheng et al. (Zheng et al., 2020) proposed to adapt optical flow models by
simulating the noise of dark images and synthesizing a low-light training
dataset. However, the synthesis process only considers signal distribution and
neglects machine vision, and thus the performance gain is worse than ours as
shown in Table 9. Furthermore, as shown in Fig. 7, our SACC is robust to
different input illumination levels.

###  6.6. Subjective Human Vision

We further analyze the relationship between human and machine vision. We
benchmark two non-reference image quality assessment (NR-IQA) scores: NIQE
(Mittal et al., 2013) and SSEQ (Liu et al., 2014) in Table 7. Although our
SACC only considers high-level tasks, our NR-IQA scores still rank high among
all methods. As shown in Fig. 6, our enhanced result has less artifacts and
better color balance.

The correlation coefficient of NIQE/SSEQ and the average mAP of all detectors
is -0.69/-0.84. This experiment indicates that there is indeed a correlation
between human and machine vision, but good visual quality does not always
guarantee good high-level performance, which validates our motivation for
designing enhancement models tailored for machine vision.

##  7\. Conclusion

Towards high-level vision, we propose deep concave curve, which migrates
illumination from low to normal light through a self-supervised strategy. Our
model architecture and learning strategy cooperate with each other, which form
a powerful normal-to-low light adaptation framework. Experimental results
demonstrate the superiority of our framework.

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