#  RRab variables with identical light-curve shapes at different pulsation
periods

Johanna Jurcsik,1 Áron Juhász1,2
1Konkoly Observatory, ELKH CSFK, H-1121, Budapest, Konkoly Thege Miklós út
15-17., Hungary
2Eötvös Loránd University, H-1117, Budapest, Pázmány Péter sétány 1/a, Hungary
E-mail<EMAIL_ADDRESS>
(Accepted 2022 September 20. Received 2022 September 12; in original form 2022
July 19)

###### Abstract

In this paper, we report on the detection of RRab stars with quasi-identical-
shape light curves but period differences as large as $0.05-0.21$ d using the
Galactic bulge data of the OGLE-IV survey. We have examined stars with shorter
periods than the Oosterhoff I ridge of the bulge. These stars generally have
smaller amplitudes and larger Fourier phase-differences than the typical bulge
RRab stars have at the same period. Many of these "anomalous" stars have good-
quality light curves without any sign of the Blazhko modulation. Examining
their Fourier parameters revealed that several of these stars show very
similar light-curve to the typical bulge RR Lyrae. We found hundreds of quasi-
identical-shape light-curve pairs with different periods between the
"anomalous"- and the "normal"-position RRab stars based on the OGLE $I$-band
data. The OGLE $V$-band, and the archive VVV and MACHO surveys $K_{s}$-, $b$\-
and $r$-band data of these stars were also checked for light-curve-shape
similarity. Finally, 149 pairs with identical-shape light curves in each
available photometric band were identified. Calculating the physical
properties of the variables using empirical formulae, on average, $-0.5$ dex,
$-0.13$ mag, 0.67, and 165 K differences between the [Fe/H], $M_{V}$,
$R/R_{\odot}$, and $T_{\mathrm{eff}}$ values of the members of the pairs are
derived, being the short-period stars less metal-poor, fainter, smaller and
hotter than the long-period variables. To explain the existence of variables
with different physical properties and pulsation periods but with identical-
shape light curves is a challenging task for modelling.

###### keywords:

stars: variables: RR Lyrae - Galaxy: bulge - stars: fundamental parameters -
techniques: photometric

††pubyear: 2022††pagerange: RRab variables with identical light-curve shapes
at different pulsation periods–References

##  1 Introduction

Pulsating stars have always been central among astronomy targets, because
their global physical parameters (mass, luminosity, effective temperature,
chemical composition) and their internal structures determine the emitted and
detectable light and light variations. Accordingly, most of their physical
properties are deducible from different-band photometric time-series data,
which are available today in large amounts thanks to the continuous
development of the photometric techniques.

The periods and the light-curve amplitudes/shapes of RR Lyrae (RRL) stars
depend on the physical properties of the stars as it was already shown by the
early radiative and convective model results (Christy, 1966; Bono &
Stellingwerf, 1994). N. Simon and co-workers were the very first who found
relations between the observed $\varphi_{31}$ Fourier parameter of the light
curve and the physical properties of first-overtone RRL stars using
hydrodynamic models (see e.g., Simon & Clement, 1993). The frequency dependent
radiative transfer calculations of nonlinear convective models published by
Dorfi & Feuchtinger (1999) showed in detail how the Fourier parameters of the
light curve vary with varying physical properties of the stars in different
photometric bands.

Among the observable parameters of the light variation of classical pulsating
stars like RRLs and Cepheids, the pulsation period is the determinant one.
Pulsation model calculations show how the pulsation period depends on the
fundamental physical parameters of the variables (see e.g., Marconi &
Degl’Innocenti, 2007), and restrict the possible domains of the physical
parameters of the stars. The period identifies the pulsation mode in most
cases; in addition, using the period-luminosity (P-L) and the period-
luminosity-colour (P-L-C) relations it is the primary indicator of the
luminosity, and, as a consequence, of the distance.

The shape of the light curve also holds information on the basic specifics
(chemical composition, luminosity, effective temperature, stellar radius,
surface gravity) of the variables. In eliciting this information, both
hydrodynamic modelling and empirical methods are successful.

However, after the pioneering work of (Simon & Davis, 1983), very few attempts
were made to model the light variations in different photometric bands with
sufficient accuracy (see e.g., Dorfi & Feuchtinger, 1999; Marconi et al.,
2005; Das et al., 2018). Recently, the fundamental parameters of Cepheids and
RRL stars in the Magellanic Clouds and the Galactic bulge were derived using
nonlinear 1D convective models by Bellinger et al. (2020). Nevertheless, the
fine structure of the light curve was not considered in this study. Some basic
light-curve parameters, viz. period, skewness, acuteness and amplitudes were
actually used to match the observations to the model predictions using the
artificial intelligence neural network training method.

The empirical relations utilising the period and light-curve-shape parameters
of RRL stars (e.g., Fourier amplitudes and epoch independent phase
differences) yield estimates on the [Fe/H], the absolute $V$ magnitude
($M_{V}$), and the effective temperature ($T_{\mathrm{eff}}$) with similar
accuracy as spectroscopic studies (e.g., Jurcsik & Kovács, 1996; Jurcsik,
1998; Kovács & Walker, 2001; Nemec et al., 2013; Dékány et al., 2021). The
existence of these relations is a natural consequence of the fact that the
surface gravity and effective temperature variations along the pulsation
cycle, which affect the radiative and convective energy transport, depend on
the global physical properties of the star.

RR Lyrae stars are supposed to constitute mono-metallic samples in most of the
globular clusters. The Fourier parameters of their light curves show
progressions with increasing pulsation periods (see e.g. Bramich et al., 2011
for M72 and Jurcsik et al., 2017 for M3). In the case of Cepheids, a
continuous change of the light-curve shape (Hertzsprung-progression) is also
detected (see e.g. Kovács & Buchler, 1989).

Based on these observations and model predictions, one would assume that the
light-curve shapes of RRL stars, neither of similar nor of different
metallicities, are completely identical at different pulsation periods.
Although the existence of RRL stars with identical-shape light curves at
different pulsation periods has not been proved or refused either
theoretically or by observations, the listed direct and indirect information
seems to contradict their existence.

Several plots of different Fourier-parameter combinations of the Galactic
bulge RRab stars (fundamental-mode RRLs) were shown in the papers of Prudil &
Skarka (2017, hereafter P17) and Prudil et al. (2019a). Looking at these
Fourier parameters versus period plots, there are several stars in the sample
that look to be outliers. These stars are usually below the Oosterhoff I (OoI)
ridge in the period-amplitude plot. They also appear at shorter periods, on
the left side of the OoI ridge, on the period versus Fourier phase
differences, and on period versus amplitude-ratio diagrams. Most of these
stars were identified by P17 to show the Blazhko effect, which distorts the
light curve, but there are hundreds of such stars among the bona fide non-
Blazko RRab variables selected by P17.

Possible explanations for the anomalous positions of these stars are as
follows:

  * •

Blazhko modulation, which was hidden in the shorter database analysed by P17,

  * •

blending or some other photometric defect,

  * •

significantly different physical parameters (e.g., [Fe/H]) of these stars as
for the bulk of the bulge RRLs.

In order to find some special cases showing the temporal appearance of the
Blazhko phenomenon (see e.g. V114/M3 and OGLE-BLG-RRLYR-07605 Jurcsik et al.,
2013; Soszyński et al., 2014), we investigated these outlying-position stars.
In several cases, the photometry seems to be affected by crowding, and the
light curve is very noisy. However, many of these stars have typical RRL-like
shapes and good-quality light curves with no indication of any modulation or
photometric defect. The positions of these stars on the plots showing
different combinations of the Fourier parameters but the period do not
indicate any systematically anomalous behaviour. They fit the overall patterns
traced out by the normal OoI/OoII RRLs of the bulge. Consequently, these stars
seem to be ordinary RRL stars of probable different physical properties than
most RRL stars in the bulge.

As the shape of the light variation depends on the physical properties of the
star, it may be assumed that the light curves of these stars have to be
somewhat different from the light curves of the normal bulge sample stars.
However, comparing the light curves of the "anomalous"-position and the normal
bulge RRab stars we noticed that their shapes are identical within the
uncertainty limits in several cases. Only the periods of these stars differ
noticeably.

In the present paper, we aim to show that RRab stars with quasi-identical
light-curve shapes do exist at significantly different pulsation periods.

##  2 Data and method

The fourth survey of the Optical Gravitational Lensing Experiment (OGLE IV)
provides light curves of over 38000 RRL stars in and towards the Galactic
bulge (Soszyński et al., 2014). As the basis of the present work, we used the
updated OGLE IV catalogue, which includes observations until the 2017
observing season.

P17 identified 3341 variables showing Blazhko modulation using a selected
sample of variables from the 2014 issue of the OGLE IV catalogue. Omitting
these Blazhko stars, the uncertain Blazhko candidates and also the period-
changing ones from the P17 catalogue, and applying the criteria given in E.qs.
1.-4. in Prudil et al. (2019a) to exclude possible foreground stars, we
remained at 4862 bona fide non-Blazhko stars of the Galactic bulge RRab
sample. This is the primary data set for our study. Prudil et al. (2019a),
studying the Oosterhoff properties of the bulge RRL stars, defined a similar
sample, but they considered only variables with observation in the Vista
Variables in the Via Lactea (VVV) survey as well, and this was not a criterion
in our selection. However, the OGLE $V$, the Macho instrumental $b$ and $r$
(Alcock et al., 1999) and the VVV $K_{s}$-band (Minniti et al., 2010) data, if
available, were also used for the stars in our interest.

The amplitudes of the public VVV $K_{s}$-band light curves derived from
different-size aperture photometry (Dékány et al., 2018; Hajdu et al., 2020)
are slightly different, therefore, there is some ambiguity in the amplitudes
of the $K_{s}$ light curves. Typically, we used $K_{s}$-band photometry
showing light curves with the smallest residual scatter.

Hereafter, the RRL stars from the bulge OGLE sample (OGLE-BLG-RRLYR-NNNNN) are
referred to as their number, NNNNN, alone.

Period04 (Lenz & Breger, 2005) software was used for frequency analysis to
generate Fourier parameters and their errors.

Figure 1: Light-curve-parameter spaces of the OGLE $I$ data of the Galactic
bulge non-Blazhko RRab stars are shown in the panels. $A_{I}$, $R_{31}$ and
$\phi_{31}$ versus the period are shown in the left-hand panels. The right-
hand panels display the relations between the $A_{I}$, $R_{31}$ and
$\phi_{31}$ parameters. Grey dots, red crosses and blue x symbols denote the
total sample of stars, the anomalous (A) and the normal (N) samples of RRab
stars selected for the analysis, respectively. Figure 2: One example of the
pair-selection results is shown. The phase- and magnitude-matched $I$\- and
$V$-band light curves of the anomalous variable, 05507 (red crosses), are
compared with the light curves of its five best normal pairs (blue x symbols)
in the top and bottom panels, respectively. These five pairs are supposed to
have similar light curves based on their $I$-band Fourier parameters, and the
differences between their periods are larger than 0.05 d. The names of the
selected normal stars are given at the top of each panel. Only the last two
normal stars (11463 and 25979) are accepted as of probable identical shapes
with 05507. The other three are rejected: the first and the third pairs
suggest amplitude differences in the $V$-band, and the light curves of 31505
are too noisy.

##  3 Selection of the shorter-period, "anomalous", and the normal samples of
RRab stars

First, the sample of the "anomalous" RRab stars was selected based on their
positions on the period - $A_{I},R_{21},R_{31},\phi_{21},\phi_{31}$ plots,
using the data given in the OGLE database. Any star with light-curve
parameters below the lower, left-side boundary defined by the main sample of
RRab stars on any of these plots was regarded as "anomalous" compared to the
bulk of the bulge variables.

As our aim was to find similar-shaped RRLs to the "anomalous" stars among the
normal bulge variables, at second, variables that did not fit the global
trends defined by the bulge sample in the plots of any combinations of the
light-curve parameters without the period (i.e.
$A_{I},R_{21},R_{31},\phi_{21},\phi_{31}$) were skipped. This restriction
ensures that the light-curve shape of a star in the remaining sample does not
differ significantly from the light curves of normal bulge RRLs.

Variables with noisy $I$-band light curves were omitted.

The image charts provided by the OGLE team were visually inspected for any
contamination of the neighbouring stars, but no apparent reason for any
photometric defect was detected in any of the selected stars. However, we have
to note that this process does not exclude the possibility of unresolved
blends in some cases.

At the end, we arrived at a sample of 312 "anomalous" (A) stars, whose light
curves were supposed to be similar to the light curves of other RRab stars in
the bulge, but their periods were systematically shorter.

To find light curves similar to these "anomalous" stars, the total sample of
bulge RRLs are also cleaned from outliers. Variables with parameters not
fitting the parameter spaces outlined by the bulk of the bulge sample are
ignored. This process led to the selection of the normal (N) sample of 3068
RRab stars located at proper positions in any parameter space, including the
period.

The total sample of 4862 non-Blazhko stars listed by P17, and the selected A
and N samples of the variables are plotted by different symbols in the
different light-curve-parameter plots shown in Fig. 1.

##  4 Searching for identical-shape light curves in the two samples

To this end, we searched for the smallest values of the square roots of the
sums of the normalised quadratic differences between the
$A_{I},R_{21},R_{31},\phi_{21},\phi_{31}$ parameters of the $I$-band light
curves of the A- and the N-sample stars. As the light curves of variables with
similar periods may also be very similar, the search was restricted to
variables with period differences larger than 0.05 days.

For each variable of sample A, we selected the first five stars with the
smallest differences from sample N. The OGLE $I$\- and $V$-band light curves
of these pairs were then fitted in magnitude and phase, and their similarity
was checked visually to find pairs of identical shapes, even in the smallest
features (bumps, humps, etc.) of the light curves.

An example of this process is shown in Fig. 2. The $I$ and $V$ band light
curves of the five best pairs are compared for the 05507 "anomalous" RRL. The
normalised quadratic differences between the parameters of the pairs increase
from left to right. Although it is somewhat subjective which pairs are
selected as candidates for identical respect to their light-curve shapes, we
tried to be as rigorous as possible and skipped every pair with any small
differences in their light curves. In the case of 05507, the agreement between
the phase- and magnitude-adjusted light curves seems to be satisfactory for
two of its five best pairs.

Figure 3: The same plots as in Fig. 1 are shown for the 95 anomalous (red
crosses) RRab stars and their 116 normal pairs (blue crosses) in the figure.
The pairs are connected by grey lines. The period differences of the A (short-
period) and the N (long-period) members of the pairs are larger than 0.05 d,
but the light curves of their available observations do not indicate any
differences in their shapes.

Figure 4: Two examples for the quasi identical-shape light curves at different
pulsation periods are shown. The top and bottom panels display the light
curves and the Fourier solution results for the 12322-12865 and the
14347-11518 pairs, respectively. The OGLE number and the pulsation period of
the stars are given at the top of the light-curve panels. The light curves of
the short- and the long-period members of the pairs are over-plotted on each
other by red crosses and blue x symbols in different order in the two left-
side panels. The phase matched, mean-magnitude corrected values (arbitrarily
shifted) are shown. The fitted light curves are drawn in the third panels. The
right-hand panels compare the Fourier parameters of the two stars. The
$3\sigma$ formal errors of the parameters and the equality lines are also
drawn in in these plots.

Notwithstanding that the $V$ and $I$ band OGLE light curves of the pairs
selected at this step do not show any noticeable differences, because of their
different periods, we have guessed that at shorter or longer wavelengths some
systematic differences may occur. Therefore, the MACHO and the VVV data, if
they are available and are of good enough quality are also checked for
similarity. Unfortunately, the MACHO data of most of the stars in question are
too noisy to draw any conclusion in these bands, but differences between the
$K_{s}$-band light curves of several pairs are indeed detected.

The MACHO $b$-band is somewhat bluer than the photometric band, and the MACHO
$r$-band is between the $V$ and $I$ bands (Alcock et al., 1999). Therefore,
slight differences between the $b$-band light curves of variables with
identical $I$ and $V$ light-curve shapes may occur, but differences in their
$r$-band light curves are not supposed to be detected. However, there are some
pairs that do not show any differences in the $V$\- and $I$-bands but both the
MACHO $b$ and $r$ light curves of one or both members of the pair have
different shape and/or amplitude. Some photometric defects and the increased
scatter of the MACHO light curves mimicking some differences in the shapes of
the light curves may be the reason for this behaviour. The MACHO observations
were obtained about twenty years earlier than the OGLE IV measurements, and
the multi-mode and the modulation properties of RRL stars may change abruptly
on a timescale of years resulting in a change in the shape and or the
amplitude of the mean light curve (see e.g., Soszyński et al., 2014).
Therefore, the possibility that the character of the light curve of one of the
stars has changed between the MACHO and OGLE observations cannot be excluded
either. Whatever the reason for the contradicting behaviour of the MACHO light
curves of these pairs, they were also removed from our final sample of pairs.

Finally, we remained at 149 quasi-identical shape, but different-period pairs
between 95 and 116 RRLs of the A and N samples. Good accuracy VVV and MACHO
data are available for 106 and 11 of these pairs, respectively. The positions
of these pairs are shown and connected by lines in Fig. 3. This figure
displays the same plots as shown in Fig. 1. The short- (A) and long-period (N)
members of the pairs are denoted by the "s" and "l" subscripts in the
following.

Fig. 4 shows examples of the similarity of the light curves of the pairs. The
light curves of the short-period and the long-period members of the pairs are
over-plotted on each other in different order in the first two panels, and the
third panels show the synthetic light curves of the stars based on
appropriate-order Fourier fits to the data. The orders of the fits are
typically 10, 10, 10, 16 and 5 in the $b$-, $V$-, $r$-, $I$,- and
$K_{s}$-bands, respectively, but higher or lower orders are used in some cases
if necessary. The right-hand panels of the figures compare the low-order
Fourier parameters of the two stars.

The period differences of the two pairs shown in Fig. 4 (12322-12865 and
14347-11518) are 0.13 d and 0.10 d, but no sign of any differences in their
light curves are observed in any of the photometric bands shown.

Table 1: The identification of the 149 pairs of RRab stars with quasi identical-shape light curves at different periods and their photometric data used. The full table is available as a supplementary material to this paper. IDs | Perids | IDl | Periodl | $b$ | $V$ | $r$ | $I$ | $K_{s}$
---|---|---|---|---|---|---|---|---
00234 | 0.56378 | 02039 | 0.68431 | - | + | - | + | -
00234 | 0.56378 | 13632 | 0.63782 | - | + | - | + | -
00265 | 0.57375 | 08810 | 0.62815 | - | + | - | + | +
00265 | 0.57375 | 12865 | 0.63272 | - | + | - | + | +
00268 | 0.47287 | 08672 | 0.53916 | - | + | - | + | -
……

Table 1 lists the identification of the 149 pairs detected to have similar-
shape light curves at different periods and the photometric bands of their
observations utilized.

In the course of the selection process of the pairs, we also detected several
pairs with light curves showing minor differences in amplitude and/or shape
and yet their low-order Fourier components do not show noticeable differences.

An example is shown in Fig. 5, which displays similar plots as shown in Fig.
4. Nonetheless that the low-order Fourier components of the two stars are very
similar (right-hand panels in the top figure), a close inspection of the light
curves (bottom panels) reveals small but characteristic differences e.g.,
between the shapes of the bumps and in the amplitudes. Actually, a normal pair
(or pairs) with such small differences is/are found nearly to all the RRLs of
the A sample.

As the automation-based algorithms working on mass photometry data use
typically limited numbers of parameters, several such quasi-twin pairs of
different period variables occurring at any period are supposed to be detected
in these databases.

Figure 5: An example of very small differences between the light curves of
different-period stars. The top panels display the same plots as shown in Fig.
4, and the bottom panels display the OGLE $I$ and $V$ data of the two stars on
larger plots. The Fourier parameters of the pair do not indicate any
significant differences, however, the light curves are slightly different in
amplitude and in the shape of the bump on the descending branch.

##  5 Physical parameters of the pairs

The light-curve parameters are good indicators of the fundamental physical
properties of RRL stars.

The physical parameters ([Fe/H], $M_{V},R/R_{\odot},T_{\mathrm{eff}}$) of the
stars are estimated using the formulae given in Smolec (2005); Dékány et al.
(2021); Kovács & Walker (2001); Marconi et al. (2015) and Jurcsik (1998),
respectively. To calculate [Fe/H] both the Smolec (2005) formula transformed
to the UVES metallicity scale of globular clusters as given in Jurcsik et al.
(2021, 2022) and the formula given by Dékány et al. (2021) recently, are
applied. As a consequence, two values for the $R$ and $T_{\mathrm{eff}}$
values are derived depending on which metallicities are used. Following
Arellano Ferro et al. (2010) 0.41 mag is taken for the zero point of the
Kovács & Walker (2001) $M_{V}$ formula. To convert [Fe/H] to $Z$, when
calculating the radius, the [$\alpha$/Fe] elemental ratio was taken to be 0.3
based on the bulge sample of metal-poor red giants and clump stars
([Fe/H]$<-0.5$) published by Duong et al. (2019). This $\alpha$-element
enhancement is somewhat larger than obtained for field RRL stars by Crestani
et al. (2021) ([$\alpha/{\mathrm{Fe}}]=0.2$ at [Fe/H]$=-1.0$) recently, most
probably because of the different chemical evolution of the galactic disk and
the bulge. Note also that, actually, the $\alpha$-element enhancement also
depends on the [Fe/H], however, this dependency has no significant effect on
the results shown here. The results obtained using the Dékány et al. (2021)
[Fe/H] formula are denoted by an asterisk hereafter.

Table 2 lists, the derived [Fe/H], $M_{V},R/R_{\odot},T_{\mathrm{eff}}$ values
of the short- and long-period members of the pairs.

As the pulsation period is the most dominant parameter in each of the
relations and that there are no significant differences between any other
Fourier parameters of the pairs, it is not surprising that systematic
differences between the calculated physical parameters of the members of the
pairs are detected.

The average differences between the parameters
($Param_{\mathrm{l}}-Param_{\mathrm{s}}$) of the 149 pairs are:
$\overline{\Delta{P}}=0.09$ d, s.d. 0.03;
$\overline{\Delta{\mathrm{[Fe/H]}}}=-0.42$, s.d. 0.14;
$\overline{\Delta{\mathrm{[Fe/H]}}^{*}}=-0.59$, s.d. 0.20;
$\overline{\Delta{M_{V}}}=-0.13$ mag, s.d. 0.04;
$\overline{\Delta{R/R_{\odot}}}=0.63$, s.d. 0.21;
$\overline{\Delta{R/R_{\odot}}^{*}}=0.72$, s.d. 0.24;
$\overline{\Delta{T_{\mathrm{eff}}}}=-169$ K, s.d. 66;
$\overline{\Delta{T_{\mathrm{eff}}}^{*}}=-162$ K, s.d. 64.

Supposing that the spread in the masses of RRL stars in the sample does not
exceed 0.02 $M_{\odot}$, as a result of the systematically larger radius of
the long-period stars, their surface gravity is expected to be about 0.1 dex
smaller than the log$g$ of their short-period counterparts.

Being the long-period member of the pairs the brighter, the larger and the
smaller surface gravity star, the possibility that they are more evolved
horizontal-branch stars cannot be excluded. However, the $R$, $M_{V}$ and
log$g$ differences are partly/mostly the consequences of the chemical
composition and period differences between the two samples. Therefore, the
differences in the evolutionary status of the variables may not be indeed
significant.

The difference between the observed mean $V$ and $I$ magnitudes of the two
samples are $-0.21$ mag and $-0.24$ mag, being the short-period stars 0.2 mag
fainter than their long-period counterparts, on average. This is in relatively
good agreement with the calculated $-0.13$ mag mean $M_{V}$ differences of the
pairs. However, taking into account the large and heterogeneous reddening of
the bulge, and its extension, this agreement might just be accidental. Despite
this, the agreement between the observed and the measured brightness
differences between the two samples indicates that there is no systematic
difference between the locations of the two samples, i.e., both the short- and
the long-period members of the pairs belong to the galactic Bulge population
of RRL stars.

Marconi et al. (2018) investigating the impact of the He content on the
absolute magnitudes in different bands and on the period of RRL stars, derived
a formula that connects the mean $M_{V}$ of RRLs to their [Fe/H] and $Y$
contents. Using the $M_{V}$ and the [Fe/H] values of the stars to estimate
$Y$, the mean differences between the $Y$ values of the short- and the long-
period members of the pairs are 0.000 and 0.006 depending on which $I$-band
metallicity formula are used. The He content of the more metal-poor, long-
period sample is marginally, if at all, larger than those of the short-period
sample. The estimated mean $Y$ values of the short and the long-period samples
are 0.207/0.208 and 0.211/0.217 using the Dékány et al. (2021) and the Smolec
(2005) [Fe/H] formula, respectively. Taking into account the uncertainties of
the method the agreement with the canonical value of $Y=0.24$ of RRL stars is
satisfactory.

Table 2: Physical parameters of the pairs derived from the Fourier parameters of the light curves. Results using the Dékány et al. (2021) $I$-band [Fe/H] formula are denoted by an asterisk. The full table is available as a supplementary material to this paper. IDs | Pers | IDl | Perl | ${\mathrm{[Fe/H]}}_{\mathrm{s}}$ | ${{\mathrm{[Fe/H]}}_{\mathrm{s}}^{*}}$ | ${\mathrm{[Fe/H]}}_{\mathrm{l}}$ | ${{\mathrm{[Fe/H]}}_{\mathrm{l}}^{*}}$ | ${R/R_{\odot}}_{\mathrm{s}}$ | ${{R/R_{\odot}}_{\mathrm{s}}^{*}}$ | ${R/R_{\odot}}_{\mathrm{l}}$ | ${{R/R_{\odot}}_{\mathrm{l}}^{*}}$ | $M_{V_{\mathrm{s}}}$ | $M_{V_{\mathrm{l}}}$ | $T_{{\mathrm{eff}_{\mathrm{s}}}}$ | ${T_{{\mathrm{eff}_{\mathrm{s}}}}}^{*}$ | $T_{{\mathrm{eff}_{\mathrm{l}}}}$ | ${T_{{\mathrm{eff}_{\mathrm{l}}}}}^{*}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
00234 | 0.56378 | 02039 | 0.68431 | $-0.70$ | $-0.75$ | $-1.22$ | $-1.48$ | 5.17 | 5.19 | 5.99 | 6.13 | 0.72 | 0.56 | 6502 | 6502 | 6325 | 6337
00234 | 0.56378 | 13632 | 0.63782 | $-0.70$ | $-0.75$ | $-1.07$ | $-1.26$ | 5.17 | 5.19 | 5.70 | 5.80 | 0.72 | 0.62 | 6502 | 6502 | 6404 | 6413
00265 | 0.57375 | 08810 | 0.62815 | $-0.84$ | $-0.91$ | $-1.07$ | $-1.24$ | 5.28 | 5.32 | 5.65 | 5.74 | 0.71 | 0.63 | 6466 | 6466 | 6361 | 6369
00265 | 0.57375 | 12865 | 0.63272 | $-0.84$ | $-0.91$ | $-1.10$ | $-1.28$ | 5.28 | 5.32 | 5.69 | 5.78 | 0.71 | 0.63 | 6466 | 6466 | 6340 | 6348
00268 | 0.47287 | 08672 | 0.53916 | $-0.63$ | $-0.72$ | $-0.96$ | $-1.17$ | 4.69 | 4.73 | 5.17 | 5.27 | 0.80 | 0.68 | 6661 | 6661 | 6549 | 6560
… | … | … | … | … | … | … | … | … | … | … | … | … | … | … | … | … | …

##  6 Differences in the light curve parameters of the pairs

Despite the identical-looking light curves of the selected pairs, the
differences between some of the Fourier parameters of the two stars are larger
than $3\sqrt{{\sigma_{param_{l}}}^{2}+{\sigma_{param_{s}}}^{2}}$ in several
cases. The errors of the Fourier parameters are calculated using the Monte-
Carlo routine of Period04.

Fig. 6 shows an example. The light curves of these stars seems to be
absolutely equiform, however, the
$a_{1},R_{21},\varphi_{21},\varphi_{31},\varphi_{41}$ parameter-differences
are as large as 15, 6, 9, 12, and 7 times of their estimated $\sigma$
uncertainty, respectively.

We conclude that the calculated errors underestimate the true uncertainty of
the parameters significantly, especially for the small-scatter, large data-
number $I$-band data.

Moreover, the low-order Fourier parameters do not reflect the differences in
the fine details of the light curves (bump/hump features etc.) reliably.

This is why, in the course of the selection process, we relied on careful
visual inspection of the light curves rather than selecting the pairs based on
the errors of the Fourier parameter differences.

Notwithstanding that the selected pairs are supposed to have no differences in
shape, their Fourier parameters may not be completely identical. It is also
checked whether there are any systematics in these small differences. Table 3
summarises the statistical properties of the low-order Fourier-parameter
differences for the 149 pairs. The average $\sigma$ errors of the Fourier-
parameter differences and their mean, median and s.dev. values are given in
the four lines of Table 3.

The detected mean/median values of the parameter differences are close to
their typical $\sigma$ errors, however, this very small, but systematic
offsets are statistically significant with the exceptions of the total
amplitude ($A_{I}$) and the $R_{31}$, $R_{41}$ amplitude ratios. These small
offsets indicate that, in spite of the very careful visual inspection, the
light curves of the members of the pairs are still not completely identical.
However, these differences are actually below the detection limit in
individual cases and are much smaller to have any influence on the calculated
physical parameters of the stars.

Figure 6: An example for light curves looking to be identical but with Fourier-parameter differences significantly larger than their $3\sigma$ estimates. The two light curves are the $I$-band data of 08273 (red) and 11173 (blue) over-plotted on each other in different order. The period difference of this pair is 0.088 d. Table 3: Statistics of the $I$-band Fourier-parameter differences of the 149 pairs. | ${{A_{I}}}$ | ${{a_{1}}}$ | ${R_{21}}$ | ${R_{31}}$ | ${R_{41}}$ | ${R_{51}}$ | $\phi_{21}$ | $\phi_{31}$ | $\phi_{41}$ | $\phi_{51}$
---|---|---|---|---|---|---|---|---|---|---
$\overline{\sigma_{\Delta}}$ |  | .0005 | .004 | .004 | .004 | .004 | .011 | .019 | .035 | .063
meanΔ | .000 | $-.001$ | .005 | .000 | .000 | .002 | $-.004$ | .016 | $-.032$ | $-.043$
medianΔ | .000 | $-.001$ | .005 | .000 | .000 | .002 | $-.001$ | .017 | $-.020$ | $-.038$
s.d.Δ | .012 | .004 | .005 | .006 | .007 | .006 | .021 | .027 | .058 | .112

##  7 Discussion / Conclusions

The results documented in the previous sections have shown that the light-
curve shape of RRab stars is not a unique function of the pulsation period,
instead, RRab stars with quasi identical-shape light curves even in different
wavelength bands do exist at different pulsation periods.

It is a natural conclusion that this phenomenon may be connected to the
Oosterhoff dichotomy and the period shift effect (Lee & Carney, 1999; Sandage,
2004; Kunder & Chaboyer, 2009), at least in part. Although the Oo dichotomy
has been suggested to occur simply as a consequence either of statistics or of
systematics affecting previous investigations based on homogeneous
spectroscopic studies of large sample of field RRL star in the recent papers
by Fabrizio et al. (2019, 2021), the two Oo groups can be clearly identified
in the Galactic bulge. Moreover, the spatial and kinematical distributions and
the mean physical properties ([Fe/H], $T_{\mathrm{eff}}$) of the two Oo
populations are different in the bulge sample (Prudil et al., 2019a, b).

Most of the variables of the short-period sample are OoI-type variables while
many of their long-period pairs belong to the OoII class according to their
positions shown in Fig. 3.

The ratio of the mean differences between the log$P$ and the [Fe/H] values of
the similar-shaped light-curve pairs is $-0.118$. This is in excellent
agreement with the $-0.116$ period-shift value given by Sandage (1982).

OoI stars are less metal-poor than OoII stars, typically. If the light curves
of an OoI and an OoII star are very similar, then their Fourier amplitudes and
phase-difference values have to be similar, too. Only their periods differ.
Taking into account that the photometric metallicity formulae estimate the
[Fe/H] as a function of $P$, $A$ and $\varphi_{31}$, in principle, it can
happen that at different periods, the light curve parameters of different
metallicity OoI and OoII RRL stars are identical.

However, to our best knowledge no detection of completely identical-shape
light curves of OoI- and OoII-type stars has been ever shown.

Figure 7: Comparison of the $V$\- and $I$-band light curves of similar
amplitude stars in an OoI (V15/M3) and an OoII-type (V8/M53) globular
clusters. The [Fe/H] of these clusters are $-1.46$ and $-2.00$, and the period
difference between the two stars is 0.085 d. The amplitudes of the stars are
the same in both photometric bands, however, their light-curve structure shows
significant differences.  Figure 8: Comparison of the light curves of Oo-type
I and Oo-type II RRab stars in M3. The differences between the light-curve
shapes of similar-amplitude stars at different periods are evident.

The possible similarity of the light-curve shapes of OoI- and OoII-type RRL
stars of similar amplitudes at different periods are checked using two
examples of globular cluster variables. The light curves of variables in the
M3 (OoI, [Fe/H]$=-1.46$) and the M53 (OoII, [Fe/H]=$-2.00$) globular clusters
are compared, and also the light curves of some OoI and OoII stars in M3,
despite the fact that the metallicities, in the latter case, are supposed to
be the same. The data published in Jurcsik et al. (2017) and Bramich et al.
(2011) are used for the M3 and M53 RRLs, respectively. Figs. 7 and 8 show the
results. The period differences of the OoI and OoII stars compared in these
figures are $0.07-0.11$ d, the amplitudes are the same both in the $V$ and in
the $I$ bands, but significant differences in the light-curve shapes are
evident in each plot.

These trial checks do not rule out the possibility that similar light-curve-
shape OoI and OoII RRLs may occur in globular-cluster data but they show that
the existence of these pairs cannot be explained simply by the Oo dichotomy.

Notwithstanding that the physical properties determined from the light-curve
parameters in Sect. 5 hold some uncertainties, the detected mean differences
between the physical properties of the long- and the short- period sequences
are statistically significant on a 149-element sample.

The estimated physical parameters help to identify what type of RRLs are
supposed to show identical-shape light curves at different pulsation periods.

Measuring these differences ($Param_{\mathrm{l}}-Param_{\mathrm{s}}$) as the
function of their period differences we obtain the following connections:
${\Delta{\mathrm{[Fe/H]}}}=-4.51{\Delta{P}}-0.02$, s.d. 0.03;
${\Delta{\mathrm{[Fe/H]}}^{*}}=-6.33{\Delta{P}}-0.02$, s.d. 0.04;
${\Delta{M_{V}}}=-1.54{\Delta{P}}+0.27P_{\mathrm{l}}-0.16$, s.d. 0.01;
${\Delta{R/R_{\odot}}}=6.90{\Delta{P}}+0.02$, s.d. 0.02;
${\Delta{R/R_{\odot}}^{*}}=7.91{\Delta{P}}+0.02$, s.d. 0.02;
${\Delta{T_{\mathrm{eff}}}=-1851\Delta{P}}$, s.d. 34;
${\Delta{T_{\mathrm{eff}}^{*}}}=-1766{\Delta{P}}$, s.d. 35.

The physical parameter differences of the similar-amplitude OoI and OoII
globular-cluster variables shown in Figs. 7 and 8 do not fit these relations.
Supposing that the M3 globular cluster is mono-metallic, no [Fe/H] difference
between the OoI and the OoII-type variables of the cluster is probable. In the
case of the other example, notwithstanding that the estimated $M_{V}$ and $R$
differences of the compared M3 and M53 variables fit these relations, the
differences between their [Fe/H] and ${T_{\mathrm{eff}}}$ values are twice as
large as the relations given above would indicate.

However, even if these parameter-difference relations are met, we cannot be
sure that there is no hidden extra parameter that accounts for the identical-
shape light curves of these stars.

Summarising our findings, we identified pairs of RRab stars with different
pulsation periods and different physical properties but with identical-shape
light curves according to all the available photometric observations. To show
such identical-shape light variations the complete dynamical structures of
these stars have to be very similar. However, the different metallicity, and
consequently the opacities of the atmosphere of these stars make it very
unlikely. The shocks, forming the shapes of the bump, and the hump features of
the light curves are not expected to behave so similarly in RRLs with
noticeably different fundamental parameters, either. Quoting Bono et al.
(2020): "The change in the physical structure also implies variation in the
pulsation properties." Therefore, to explain the light curve similarity of
these pairs is a challenging task for modelling RRL pulsation.

The OGLE $I$-band data yield very accurate light curves but the $V$-band data
are sparse, and the VVV and the MACHO light curves have large scatter and some
amplitude ambiguity. Therefore, time-series observations in different bands of
some of the best candidate twins would be essential to strengthen the results
published in this paper. Observations at shorter wavelengths were especially
very informative, however, taking into account the large reddening of the
bulge stars, there are limitations in obtaining these data.

##  8 Acknowledgement

The authors thank Gergely Hajdu for valuable comments and for providing us the
corrected VVV photometry of the studied variables. The research leading to
these results has been supported by the Hungarian National Research,
Development and Innovation Office (NKFIH) grants NN-129075 and K-129249. We
gratefully thank the work of the OGLE team, this paper is based primarily on
the OGLE-BULGE-RRL data. We also acknowledge the data taken with the VISTA
telescope from the ESO Public Survey program ID 179.B-2002. This paper
utilises public domain data obtained by the MACHO Project, jointly funded by
the US Department of Energy through the University of California, Lawrence
Livermore National Laboratory under contract No. W-7405-Eng-48, by the
National Science Foundation through the Center for Particle Astrophysics of
the University of California under cooperative agreement AST-8809616, and by
the Mount Stromlo and Siding Spring Observatory, part of the Australian
National University.

##  9 Data Availibility Statment

The paper utilises photometric data of the OGLE-IV, the VVV and the MACHO
surveys, which are publicly available.

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# Reconstruction Distortion of Learned Image Compression with Imperceptible
Perturbations

Yang Sui
Rutgers University
&Zhuohang Li
Vanderbilt University
&Ding Ding
Tencent America
&Xiang Pan
Tencent America
&Xiaozhong Xu
Tencent America
&Shan Liu
Tencent America
&Zhenzhong Chen
Wuhan University
The work of Y. Sui was done during the internship at Tencent America.The work
of Z. Li was done during the visit at Tencent America.The work of Z. Chen was
done during the visit at Tencent.

###### Abstract

Learned Image Compression (LIC) has recently become the trending technique for
image transmission due to its notable performance. Despite its popularity, the
robustness of LIC with respect to the quality of image reconstruction remains
under-explored. In this paper, we introduce an imperceptible attack approach
designed to effectively degrade the reconstruction quality of LIC, resulting
in the reconstructed image being severely disrupted by noise where any object
in the reconstructed images is virtually impossible. More specifically, we
generate adversarial examples by introducing a Frobenius norm-based loss
function to maximize the discrepancy between original images and reconstructed
adversarial examples. Further, leveraging the insensitivity of high-frequency
components to human vision, we introduce Imperceptibility Constraint (IC) to
ensure that the perturbations remain inconspicuous. Experiments conducted on
the Kodak dataset using various LIC models demonstrate effectiveness. In
addition, we provide several findings and suggestions for designing future
defenses.

##  1 Introduction

Learned Image Compression (LIC) has recently gained tremendous success in
transmitting images under bit-rate limits because of its superior performance
over traditional image compression. Specifically, LIC frameworks [1, 2, 11, 6]
exhibit significant rate-distortion (R-D) performance and outperform standard
image compression methods such as JPEG [16], JPEG2000 [15], and BPG [4].
Generally, the fundamental LIC structure leverages the auto-encoder framework,
which adopts a Deep Neural Network (DNN)-based non-linear transformation with
an encoder for image compression and a decoder for reconstruction.

Despite demonstrating a high recovery ability, the DNN-based encoder and
decoder also bring concern about robustness. Adversarial attacks [14, 8] are a
type of security threat against DNN-based systems where the attacker injects
specially crafted perturbations to images to construct adversarial examples,
which are natural-looking images that can cause misclassification to DNN
models. Originally discovered in image classification tasks [14, 8],
adversarial examples have then attracted great research attention in many
fields of computer vision, including object detection and semantic
segmentation [17], facial recognition [7], and visual question answering [9].

While the robustness of downstream tasks has been extensively investigated [5,
10], the robustness of LIC, which is evaluated in terms of image
reconstruction quality instead of classification accuracy, has received little
attention. As depicted in Fig. 1, a typical LIC system can accurately
reconstruct an original (unperturbed) image. However, the LIC is considered to
be not robust if an attacker can introduce small perturbations into the
original image to significantly disrupt the reconstructed image, resulting in
the main object in the reconstructed image being unrecognizable.

In this paper, we propose to investigate the robustness of LIC by attacking
the image reconstruction process. The main idea is to solve an optimization
problem to minimize the adversarial perturbation while maximizing the
Frobenius norm-based loss metric between the original and reconstructed
images. However, the adversarial perturbations generated with unconstrained
Frobenius norm-based loss are always sensitive to human eyes. To improve the
imperceptibility of the generated adversarial images, we draw insights from
the observation that high-frequency components are less perceptible to human
vision [12], and consider generating perturbations from a frequency
perspective by introducing a Discrete Cosine Transform (DCT)-based
Imperceptibility Constraint (IC) into the adversarial loss function, rendering
the perturbations more unnoticeable by human perception. Our contributions can
be summarized as follows: ❶ We conduct a systematic investigation on the
robustness of LIC by launching a series of attacks that disrupts the image
reconstruction process by introducing a Frobenius norm-based loss with IC. ❷
Our experiments demonstrate that our proposed attack can disrupt LIC while
maintaining the imperceptibility of the induced perturbations. ❸ Based on our
experiments, we provide several intriguing findings and potential insights
regarding designing robust LICs.

##  2 Preliminary

###  2.1 Learned Image Compression

Given represent the non-linear encoder $g_{a}(\cdot)$ and decoder
$g_{s}(\cdot)$, let $\boldsymbol{\mathcal{\mathcal{X}}}$ and
$\boldsymbol{\mathcal{\mathcal{\hat{X}}}}$ denote the original input and
reconstructed images, and $\boldsymbol{\mathcal{\mathcal{Y}}}$ and
$\boldsymbol{\mathcal{\mathcal{\hat{Y}}}}$ denote the pre-quantized and
quantized latent representation, respectively. The image compression process
is formulated as follows:

$\displaystyle\boldsymbol{\mathcal{\mathcal{Y}}}=g_{a}(\boldsymbol{\mathcal{\mathcal{X}}}),\enspace\boldsymbol{\mathcal{\mathcal{\hat{Y}}}}=\text{AD}(\text{AE}(Q(\boldsymbol{\mathcal{\mathcal{Y}}}))),\enspace\boldsymbol{\mathcal{\mathcal{\hat{X}}}}=g_{s}(\boldsymbol{\mathcal{\mathcal{\hat{Y}}}}),$
(1)

where $Q(\cdot)$ is the quantization operation, and AE and AD represent the
arithmetic encoding and decoding processes, respectively. The reconstructed
image $\boldsymbol{\mathcal{\mathcal{\hat{X}}}}$ is the output of the
corresponding (inverse) transform. In addition, a hyper-prior is used as side
information to reduce the bit-rate.

###  2.2 Adversarial Attack

Given a natural image
$\boldsymbol{\mathcal{\mathcal{X}}}\in\mathbb{R}^{H\times W\times C}$, the
corresponding label $k$, and a classification model $f(\cdot)_{i}$ which
predicts the probability of the image belonging to class $i$, the goal of
adversarial attacks is to craft an adversarial perturbation
$\boldsymbol{\mathcal{\mathcal{\delta}}}\in\mathbb{R}^{H\times W\times C}$ to
be added onto $\boldsymbol{\mathcal{\mathcal{X}}}$ so that it is misclassified
by $f(\cdot)$, which can be formulated as:

$\displaystyle\operatorname*{arg\,max}\limits_{i}f(\boldsymbol{\mathcal{\mathcal{X}}}+\boldsymbol{\mathcal{\mathcal{\delta}}})_{i}\neq
k,\quad\|\boldsymbol{\mathcal{\mathcal{\delta}}}\|_{p}\leq\epsilon,$ (2)

where $\epsilon$ controls the perturbation budget.

##  3 Distortion with Imperceptible Perturbation

Figure 1: Illustration of proposed adversarial attack with IC on LIC to
disrupt the reconstruction image. Top: The standard LIC process. Middle: The
proposed adversarial attack against LIC. Bottom: The proposed adversarial
attack with IC against LIC while ensuring the perturbation on the original
image remains more imperceptible.

###  3.1 Reconstruction Distortion Attack

Unlike the traditional adversarial attack, which aims to mislead the
classification model into predicting a wrong label, the adversarial attack on
LIC aims to introduce small noise to the original image so that the
reconstructed image is severely corrupted. This objective can be formulated
as:

$\displaystyle\operatorname*{arg\,max}\limits_{\boldsymbol{\mathcal{\mathcal{\delta}}}}\enspace\texttt{dis}(\boldsymbol{\mathcal{\mathcal{X}}},g_{s}(Q(g_{a}((\boldsymbol{\mathcal{\mathcal{X}}}+\boldsymbol{\mathcal{\mathcal{\delta}}}))))),\enspace\|\boldsymbol{\mathcal{\mathcal{\delta}}}\|_{p}\leq\epsilon,$
(3)

where $\texttt{dis}(\cdot,\cdot)$ denotes a distance function that computes
the distance between two tensors. In addition, we utilize differential
approximation quantization [13] to achieve gradient-based attacks. We note
that, in practice, the underlying technology for image compression is often
standardized or industrially recommended (such as JPEG compression) to ensure
compatibility across all scenarios. Therefore, we assume the attacker has
complete knowledge of the LIC and can launch the attack in a white-box manner.

###  3.2 Imperceptible Perturbations

As illustrated in Fig. 1 (middle), although the image can be heavily distorted
by solving the Eq. 3, in order to reduce the reconstructed image quality, a
significant amount of perturbation needs to be injected, which makes the
primary subject of the image barely recognizable and the attack easily
detectable by human inspectors.

Previous research [12] has demonstrated that high-frequency perturbations are
less noticeable. Typically, performing DCT on an image reveals that the low-
frequency components carry the major semantic information, while the high-
frequency components include the edge structural information. Human beings
tend to focus on semantic information, such as the main object in an image,
but ignore detailed information, particularly beneath edge structures. Hence,
human visual perception is typically not sensitive to these high-frequency
perturbations.

Motivated by the frequency perspective, we propose a DCT-based IC to generate
imperceptible high-frequency perturbations. In particular, IC encourages the
perturbation to mainly modify the high-frequency components of the original
images, while constraining the low-frequency components of the adversarial
images to remain consistent and close to those in the original images, which
can be formulated as:

$\displaystyle\mathcal{I}(\boldsymbol{\mathcal{\mathcal{X}}},\boldsymbol{\mathcal{\mathcal{X}}}+\boldsymbol{\mathcal{\mathcal{\delta}}})=\|\mathcal{T}(\boldsymbol{\mathcal{\mathcal{X}}})-\mathcal{T}(\boldsymbol{\mathcal{\mathcal{X}}}+\boldsymbol{\mathcal{\mathcal{\delta}}})\|_{F},$
(4)

where $\|\cdot\|_{F}$ denotes the Frobenius norm of the distance.
$\mathcal{T}(\cdot)$ denotes the function truncates the low-frequency band
based on DCT, which can be formulated as follows:

$\displaystyle\mathcal{T}(\boldsymbol{\mathcal{\mathcal{X}}})=\texttt{IDCT}(\boldsymbol{M}\odot\texttt{DCT}(\boldsymbol{\mathcal{\mathcal{X}}})),$
(5)

where $\odot$ denotes the Hadamard product (element-wise product).
$\boldsymbol{M}\in\mathbb{R}^{H\times W}$ is a binary mask applied to the
frequency domain of the tensor $\boldsymbol{\mathcal{\mathcal{X}}}$ after DCT
to constrain its frequency components. To make the perturbation imperceptible,
we mask out half of the components with lower frequencies and only preserve
the higher half of the frequency components.

By combining Eq. 3 and Eq. 4, the overall optimization objective is as
follows:

$\displaystyle\mathcal{L}(\boldsymbol{\mathcal{\mathcal{X}}},\boldsymbol{\delta})=$
$\displaystyle-\|\boldsymbol{\mathcal{\mathcal{X}}}-g_{s}(Q(g_{a}((\boldsymbol{\mathcal{\mathcal{X}}}+\boldsymbol{\mathcal{\mathcal{\delta}}}))))\|_{F}^{2}$
(6)
$\displaystyle+\eta\cdot\mathcal{I}(\boldsymbol{\mathcal{\mathcal{X}}},\boldsymbol{\mathcal{\mathcal{X}}}+\boldsymbol{\mathcal{\mathcal{\delta}}}),\quad\textrm{s.t.}\|\boldsymbol{\mathcal{\mathcal{\delta}}}\|_{\infty}\leq\epsilon,$

where $\eta$ controls the influence of IC.

##  4 Experiments

Setting. For LIC models, we train the Anchor [6] , Hyperprior [2], Factorized
[2], and Joint [11] LIC models following the CompressAI [3], to evaluate the
distortion. We use the differential approximation quantization from [13] to
execute the gradient-based attack. We set $\eta=10^{2}$ for solving Eq. 6. The
evaluations are conducted on the Kodak dataset using a single NVIDIA A100.

Metric. The effectiveness of the proposed attack is evaluated by Peak Signal-
to-Noise Ratio (PSNR) and Multi-Scale Structural Similarity Index (MS-SSIM). A
higher value of PSNR and MS-SSIM indicates better reconstruction quality.
“$\downarrow$ PSNR" and “$\downarrow$ MS-SSIM" denote the decrease in PSNR and
MS-SSIM, measured between adversarial examples and their reconstructed
counterparts, where a higher value indicates a more effective attack.

Figure 2: Visual results of our proposed reconstruction quality attack with
IC. The attack is evaluated on the Kodak dataset with four various LIC models
as victim models.

Model-$\epsilon$ |  | Reconstructed Original Image | AE | Reconstructed AE |  $\Delta$ of AE vs. Reconstructed AE
---|---|---|---|---|---
| PSNR | MS-SSIM | PSNR | MS-SSIM | PSNR | MS-SSIM |  $\downarrow$ PSNR |  $\downarrow$ MS-SSIM
Anchor-32 |  | 32.99 | 0.9880 | 22.47 | 0.9112 | 7.500 | 0.2188 | 14.97 | 0.6923
Anchor-64 |  | 32.99 | 0.9880 | 17.63 | 0.8281 | 6.617 | 0.1816 | 11.01 | 0.6464
Hyperprior-32 |  | 36.48 | 0.9943 | 20.30 | 0.8919 | 12.05 | 0.6617 | 8.246 | 0.2301
Hyperprior-64 |  | 36.48 | 0.9943 | 15.15 | 0.8135 | 7.118 | 0.3608 | 8.031 | 0.4527
Factorized-32 |  | 33.29 | 0.9901 | 19.90 | 0.9020 | 7.614 | 0.4443 | 12.28 | 0.4570
Factorized-64 |  | 33.29 | 0.9901 | 14.99 | 0.8221 | 5.946 | 0.2868 | 9.050 | 0.5353
Joint-32 |  | 35.21 | 0.9911 | 21.38 | 0.9205 | 5.921 | 0.1633 | 15.46 | 0.7572
Joint-64 |  | 35.21 | 0.9911 | 17.52 | 0.8662 | 5.733 | 0.1335 | 11.78 | 0.7327

Table 1: Average PSNR and MS-SSIM of the reconstructed original images,
adversarial examples (AE), and reconstructed adversarial examples across 24
images from the Kodak dataset. The bold number denotes the average degradation
of PSNR and MS-SSIM between the adversarial examples and their reconstructed
counterparts by our proposed attack method.

Results for Reconstruction Distortion. We generate adversarial examples on the
four LIC models with perturbation budget $\epsilon=\\{32,64\\}$ and visual
results are presented in Fig. 2, which includes 8 images from the Kodak
dataset. As observed, the LIC successfully reconstructs the original image.
However, for the adversarial image, despite the perturbation being almost
imperceptible and unnoticeable, the reconstructed adversarial examples are
significantly disrupted. Table 1 presents the average PSNR and MS-SSIM of the
reconstructed original images, adversarial examples, and reconstructed
adversarial examples across 24 Kodak images. As shown in the Table, the
average PSNR and MS-SSIM of reconstructed adversarial examples, denoted by the
number with the underline, are extremely low, which verifies the effectiveness
of our proposed attack. We also evaluate the average degradation of PSNR and
MS-SSIM measured between the adversarial examples and their reconstructed
counterparts, marked as bold. Notably, the attack on the Joint model with
$\epsilon=32$ can achieve an MS-SSIM degradation of 0.7572 (decreased from
0.9205 to 0.1633). This demonstrates that the reconstructed images are almost
completely destroyed.

Figure 3: Impact of proposed IC. The visualization depicts perturbations
generated from attacks with or without IC. As shown in the figure, the
adversarial example generated with IC exhibits a more natural behavior
compared to those without IC.

| $\epsilon$ | Low | Middle | High | Average
---|---|---|---|---|---
w/o IC | 32 | 0.8268 | 0.8359 | 0.8374 | 0.8333
w/ IC | 32 | 0.9315 | 0.9015 | 0.9006 | 0.9112
w/o IC | 64 | 0.6820 | 0.6781 | 0.6873 | 0.6824
w/ IC | 64 | 0.8570 | 0.8147 | 0.8126 | 0.8281

Table 2: The MS-SSIM of adversarial examples compared to original images on
the low, middle, and high quality Anchor models with or without high-frequency
constraints. A higher MS-SSIM indicates that the adversarial perturbations are
more imperceptible.

Effect of High-frequency Perturbation. We evaluate the impact of our proposed
IC on the Anchor model, with a perturbation budget of $\epsilon=\\{32,64\\}$.
The low, middle, and high models are trained with quality level coefficients
$\lambda$ 0.0130, 0.0250, and 0.0483, respectively. The results of our
proposed attack on the high-quality model with $\epsilon=32$ in Fig. 3, which
contains three Kodak images. Table 1 presents the average MS-SSIM of the
adversarial examples across 24 images. As observed in the column of
adversarial examples and perturbations, LIC successfully disrupts the
reconstructed adversarial examples, while the adversarial examples generated
by our proposed IC achieve imperceptible perturbations. In contrast,
adversarial examples without the IC show conspicuous noises. As shown in Table
2, an attack with IC can increase the average MS-SSIM by 0.078 and 0.146,
respectively, demonstrating a substantial difference in image reconstruction
quality.

Figure 4: Perturbation of different quality levels.

Reconstruction Quality |  | Reconstructed Original Image | AE | Reconstructed AE |  $\Delta$ of AE vs. Reconstructed AE
---|---|---|---|---|---
| PSNR | MS-SSIM | PSNR | MS-SSIM | PSNR | MS-SSIM |  $\downarrow$ PSNR |  $\downarrow$ MS-SSIM
Anchor
Low |  | 32.01 | 0.9838 | 23.96 | 0.9315 | 11.57 | 0.4006 | 12.39 | 0.5309
Middle |  | 33.16 | 0.989 | 21.68 | 0.9015 | 5.510 | 0.1274 | 16.17 | 0.7741
High |  | 33.81 | 0.9914 | 21.78 | 0.9006 | 5.421 | 0.1285 | 16.35 | 0.7721
Average |  | 32.99 | 0.9880 | 22.47 | 0.9112 | 7.500 | 0.2188 | 14.97 | 0.6923
Factorized
Low |  | 29.97 | 0.9831 | 20.36 | 0.9002 | 7.786 | 0.4178 | 12.57 | 0.4824
Middle |  | 34.07 | 0.9929 | 19.76 | 0.8945 | 8.270 | 0.4779 | 11.49 | 0.4166
High |  | 35.84 | 0.9944 | 19.59 | 0.9115 | 6.787 | 0.4374 | 12.80 | 0.4741
Average |  | 33.29 | 0.9901 | 19.90 | 0.9020 | 7.614 | 0.4443 | 12.28 | 0.4577

Table 3: Results on LIC models with different quality levels.

Effect on LIC Models with Different Quality Levels. We further evaluate our
proposed attack method on Anchor and Factorized LIC models across low, middle,
and high reconstruction quality levels. Table 3 presents the results with
varying quality levels. For example, for the low, middle, and high-quality
models of Factorized, our proposed method consistently achieves an MS-SSIM
degradation of 0.4824, 0.4166, and 0.4741, respectively, averaging 0.4577. The
results illustrate that the proposed reconstruction distortion can affect all
quality levels.

Findings. Our experiments have led to several intriguing observations. ❶
Besides arbitrary noises, the generated adversarial perturbation also contain
certain irregular patterns. For instance, it can be observed in Fig. 3 that
there are small square patterns within each generated perturbation. We
hypothesize that these specific areas may have a significant impact on
reconstruction quality. Future work may investigate designing countermeasures
for detecting and defending the adversarial attack leveraging these patterns.
❷ Different LIC models demonstrate varying levels of robustness. From Fig. 2,
we find that Hyperprior and Joint appear to be more robust than others. Based
on this, we hypothesize that LIC models with higher-quality reconstruction
capabilities also have superior robustness.

##  5 Conclusion

In this paper, we explore the robustness of LIC by launching adversarial
quality attacks based on the Frobenius norm-based loss function to create
adversarial examples that maximize the deviation between the original and the
reconstructed images and introduce the IC to ensure the perturbations are
invisible to human perception. Experiments on the Kodak dataset and various
LIC models illustrate the effectiveness and reveal intriguing findings,
including irregular perturbation patterns and varying levels of robustness
across different LIC models.

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