11institutetext: Departamento de Ciencias Físicas, Facultad de Ciencias
Exactas, Universidad Andres Bello, Fernández Concha 700, Las Condes, Santiago,
Chile 22institutetext: Vatican Observatory, Vatican City State, V-00120, Italy
33institutetext: Centro de Astronomía (CITEVA), Universidad de Antofagasta,
Av. Angamos 601, Antofagasta, Chile 44institutetext: Millennium Institute of
Astrophysics, Santiago, Chile 55institutetext: Observatorio Astronómico,
Universidad Nacional de Córdoba, Laprida 854, Córdoba, Argentina
66institutetext: Institute of Astronomy, University of Cambridge, Madingley
Rd, Cambridge CB3 0HA, UK 77institutetext: INAF-Osservatorio Astronomico di
Capodimonte, Salita Moiariello 16, 80131, Naples, Italy

# Confirmation and physical characterization of the new bulge globular cluster
Patchick 99 from the VVV and Gaia surveys

E.R. Garro 11 D. Minniti M. Gómez 112211 J. Alonso-García 3344 T. Palma 55 L.
C. Smith 66 V. Ripepi 77

(Received 25 August 2020; Accepted 2 March 2021)

###### Abstract

Context. Globular clusters (GCs) are recognised as important tools to
understand the formation and evolution of the Milky Way (MW) because they are
the oldest objects in our Galaxy. Unfortunately, the known sample in our MW is
still incomplete, especially towards the innermost regions, due to the high
differential reddening, extinction, and stellar crowding. Therefore, the
discovery of new GC candidates and the confirmation of their true nature are
crucial for the census of the MW GC system.

Aims. Our main goal is to confirm the physical nature of two GC candidates:
Patchick 99 and TBJ 3. They are located towards the Galactic bulge. We use
public data in the near-infrared (IR) passband from the VISTA Variables in the
Via Láctea Survey (VVV), VVV eXtended Survey (VVVX) and the Two Micron All Sky
Survey (2MASS) along with deep optical data from the Gaia Mission DR 2, in
order to estimate their main astrophysical parameters, such as reddening and
extinction, distance, total luminosity, mean cluster proper motions, size,
metallicity and age.

Methods. We investigate both candidates at different wavelengths, allowing us
to discard TBJ3 as a possible GC. We use near-IR $(K_{s}\ vs.\ (J-K_{s}))$ and
optical $(G\ vs.\ (BP-RP))$ colour-magnitude diagrams (CMDs) in order to
analyse Patchick 99. First, we decontaminate CMDs, following a statistical
procedure, as well as selecting only stars which have similar proper motions
(PMs) and are situated within $3^{\prime}$ from the centre. Mean PMs are
measured from Gaia DR 2 data. Reddening and extinction are derived by adopting
optical and near-IR reddening maps, and then we use them to estimate the
distance modulus and the heliocentric distance. Metallicity and age are
evaluated by fitting theoretical stellar isochrones.

Results. Reddening and extinction values for Patchick 99 are
$E(J-K_{s})=(0.12\pm 0.02)$ mag and $A_{K_{s}}=(0.09\pm 0.01)$ mag from the
VVV data, whereas we calculate $E(BP-RP)=(0.21\pm 0.03)$ mag and
$A_{G}=(0.68\pm 0.08)$ mag from Gaia DR 2 data. We use those values and the
magnitude of the RC to estimate the distance, finding a good agreement between
the near-IR and optical measurements. In fact, we obtain $(m-M)_{0}=(14.02\pm
0.01)$ mag, equivalent to a distance $D=(6.4\pm 0.2)$ kpc in near-IR and
$(m-M)_{0}=(14.23\pm 0.1)$ mag and so $D=(7.0\pm 0.2)$ kpc in optical. In
addition, we derive the metallicity and age for Patchick 99 using our distance
and extinction values and fitting PARSEC isochrones. We find $[Fe/H]=(-0.2\pm
0.2)$ dex and $t=(10\pm 2)$ Gyr. The mean PMs for Patchick 99 are
$\mu_{\alpha}=(-2.98\pm 1.74)$ mas $yr^{-1}$ and $\mu_{\delta}=(-5.49\pm
2.02)$ mas $yr^{-1}$, using the Gaia DR 2 data. They are consistent with the
bulge kinematics. We also calculate the total luminosity of our cluster and
confirm that it is a low-luminosity GC, with $M_{K_{s}}=(-7.0\pm 0.6)$ mag.
The radius estimation is performed building the radial density profile and we
find its angular radius $r_{P99}\sim 10^{\prime}$. We also recognise seven RR
Lyrae star members within 8.2 arcmin from the Patchick 99 centre, but only
three of them have PMs matching the mean GC PM, confirming the distance found
by other methods.

Conclusions. We found that TBJ 3 shows mid-IR emissions that are not present
in GCs. Hence, we discard TBJ 3 as GC candidate and we focus our work on
Patchick 99. We conclude that Patchick 99 is an old metal-rich GC, situated in
the Galactic bulge. TBJ 3 is a background galaxy.

###### Key Words.:

Galaxy: bulge – Galaxy: center – Galaxy: stellar content – Stars Clusters:
globular – Infrared: stars – Surveys

##  1 Introduction

Globular Clusters (GCs) represent the most ancient stellar systems in the
Milky Way (MW). For this reason, they are excellent tracers of the formation
and chemodynamical evolution of the Galaxy, going from the outer halo to the
innermost Galactic regions. In addition, they survived different dynamical
processes, like dynamical friction, disk and bulge shocking, evaporation that
also led some ancient GCs to destruction (Weinberg 1994). Today, we
distinguish different kinds of GCs, depending on their metallicities, ages and
orbits. The metal-poor GCs could be the oldest population (Barbuy et al. 2006,
2009). More precisely, in the Galactic bulge, they may constitute the first
stellar generation whose origin could coincide with the formation of the bulge
itself or before the actual configuration of the bulge/bar component, examples
are NGC 6522, NGC 6626 and HP 1 (Kerber et al. 2018, 2019). On the other hand,
the metal-rich GCs may be the second generation in the MW (Muratov & Gnedin
2010). Furthermore, recent studies have demonstrated that some other luminous
GCs may be actually fossil relics of accreted dwarf galaxies, such as Terzan 5
(Ferraro et al. 2009), Liller 1 (Ferraro et al. 2020), Terzan 9 (Ernandes et
al. 2019) and also Omega Centauri (Lee et al. 1999; Sollima et al. 2008).
Further, with the advent of Gaia Mission and deep surveys, it is now possible
to associate individual GCs with specific merger events (e.g., Vasiliev 2019;
Myeong et al. 2019; Massari et al. 2019).

However, it seems that the frequency of the MW’s GCs is lower than the
Andromeda’s GC system (van den Bergh 2010) and the MW’s open clusters. Indeed,
the total number of Galactic GCs (Harris et al. 2013) is about half those of
our neighbour Andromeda (Barmby & Huchra 2001; Huxor et al. 2014), and the
number of Galactic open clusters is vastly higher than that of GCs (Bonatto &
Bica 2010). There are a number of issues that make their detection a
challenging task, certainly we are still missing a number of them due to high
stellar extinction and stellar crowding and very low-luminosities of some of
them. Nevertheless, in the last decades, the number of star cluster candidates
reported in the literature has increased significantly. Indeed, recent near-
infrared (IR) surveys, like the Two Micron All Sky Survey (2MASS; Skrutskie et
al. 2006) and the VISTA Variables in the Via Láctea Survey (VVV; Minniti et
al. 2010), and mid-IR survey, like Wide-field Infrared Survey Explorer (WISE;
Wright et al. 2010) have allowed us to search for new GC candidates. In
combination with near/mid-IR surveys, optical surveys are fundamental to
finding new GCs, like Gaia Mission DR 2 (Gaia Collaboration et al. 2018),
which provides optical photometry and proper motions (PMs). Thus, new GC
candidates are being discovered, in particular in the Galactic bulge, using
both the optical and near-IR images in order to help complete the census of
GCs in the MW. Examples of recently discovered GCs toward to innermost
Galactic regions are VVV-CL001 (Minniti et al. 2011), VVV-CL002 and VVV-CL003
(Moni Bidin et al. 2011), Camargo 1102, 1103, 1104, 1105 and 11061 (Camargo
2018), Camargo 1107, 1108 and 1109 (Camargo & Minniti 2019), and several tens
of candidates in Minniti et al. (2017, 2017a, 2017b) and Palma et al. (2019).
Even so, the nature of some new GC candidates has yet to be confirmed and main
parameters have still to be estimated.

In this paper, we focus on two recently discovered and not yet well
characterised GC candidates, Patchick99 and TBJ 3 (a.k.a. TJ 3). We are
studying for GC candidates in order to investigate the GC luminosity function
in the Galactic bulge, especially at very faint luminosities. For that
purpose, Patchick 99 and TBJ 3 were selected from Bica et al. (2018), who
compiled a list of 10,000 good star cluster candidates in the MW, as they had
not been studied before. We use a combination of optical and near-IR
catalogues, aiming on having a more complete information about them. Our main
goals are to confirm their true nature as GC and estimate their main
parameters for the first time, such as reddening, distance modulus,
heliocentric distance, size, total luminosity, mean cluster proper motions,
metallicity and age.

In Section 2, the observational data are briefly described. In Section 3 we
focus our attention to TBJ 3. Section 4 presents the methods used to estimate
the physical parameters and the resulting values for Patchick 99. In Section
5, a summary and conclusions are drawn.

##  2 Observational Data

The effects of dust become increasingly prominent towards the centre of our
Galaxy. Thus, we have to overcome problems like high differential reddening
and extinction (e.g., Alonso-García et al. 2017), which introduce systematic
errors in the determination of distances. Additionally, uncertainties in the
extinction values affect the estimation of ages when isochrones are used.
Also, the stellar density is very high in the inner regions and crowding does
not always allow to resolve individual stars. Further uncertainties are due to
the foreground and background field contamination that may alter the
measurement of the total luminosity. For these reasons, we use combinations of
optical and near-IR data in order to substantially reduce uncertainties in the
measurements and obtain robust results.
We analyse especially red clump (RC) and red giant branch (RGB) stars, since
they are very bright in infrared and they allow us to estimate accurate values
of reddening, extinction, distance and size for our GC candidates.

Below a short description of the observational material is provided.

###  2.1 Infrared Surveys: the VVV, VVVX and 2MASS

We use deep near-IR data from the VVV and VVVX Survey (Minniti et al. 2010;
Minniti 2018), acquired with the VISTA InfraRed CAMera (VIRCAM) at the 4.1m
wide-field Visible and Infrared Survey Telescope for Astronomy (VISTA; Emerson
& Sutherland 2010) at ESO Paranal Observatory. VVV and VVVX data are reduced
at the Cambridge Astronomical Survey Unit (CASU; Irwin et al. 2004) and
further processing and archiving is performed with the VISTA Data Flow System
(VDFS; Cross et al. 2012) by the Wide-Field Astronomy Unit and made available
at the VISTA Science Archive111http://horus.roe.ac.uk/vsa/. We use preliminary
data from VIRAC (Smith et al. 2018) version 2, described in detail in Smith et
al. (in preparation).In summary, VIRAC v2 is based on a psf fitting reduction
of VVV and VVVX images using DoPhot (Schechter et al. 1993; Alonso-García et
al. 2012, 2018). Their astrometry are calibrated to the Gaia DR 2 (Gaia
Collaboration et al. 2018) astrometric reference system, and their photometry
are calibrated against 2MASS using a globally optimised model of frame-by-
frame zero points plus an illumination correction. The VVV and, in particular,
the VVVX Surveys are sampling the Galactic bulge and the Southern Galactic
Plane, using the $J$ (1.25 $\mu m$), $H$ (1.64 $\mu m$)), and $K_{s}$ (2.14
$\mu m$)) near-IR passbands, aiming to better disentangle and characterise the
stellar populations located at these low latitude regions. The VISTA Telescope
tile field of view is 1.501 deg2, hence 196 tiles are needed to map the bulge
area and 152 tiles for the disk. Adding some X and Y overlap between tiles for
a smooth match, the area of our unit tile covered twice is 1.458 deg2.
Specifically, Patchick 99 is located in the VVV tile b251, whereas TBJ 3 is
situated in the VVVX tile b462.

In addition, we use 2MASS data (Skrutskie et al. 2006; Cutri et al. 2003). It
is a near-IR Survey of the sky in $J$ ($1.25\ \mu m$), $H$ ($1.65\ \mu m$) and
$K_{s}$ ($2.17\ \mu m$) passbands. We adopt it specifically to extend the RGB
VVV colour-magnitude diagrams (CMDs), as these stars are saturated for
$K_{s}<11$ mag in VVV images. We downloaded the 2MASS data from the VizieR
Online Data Catalogue 222https://vizier.u-strasbg.fr/viz-bin/VizieR, and we
merge it with the VVV catalogue, including all stars located within
$10^{\prime}$ from the targets centres. However, since the magnitude scale is
different in the two photometric systems, we have transformed the 2MASS
photometry into the VVV magnitude scale 333http://casu.ast.cam.ac.uk/surveys-
projects/vista/technical/photometric-properties.

###  2.2 Optical Surveys: Gaia DR 2

The Gaia astrometric mission maps the MW at optical wavelengths, with the aim
of revealing motions, composition, formation and evolution of the Galaxy. It
provides very accurate position, PMs and radial velocity measurements for
sources both in our Galaxy and throughout the Local Group. We use Gaia DR 2
data (Gaia Collaboration et al. 2018) for both stellar cluster candidates,
downloaded from the VizieR Online Data Catalogue. Gaia DR 2 data have been
prepared by the Gaia Data Processing and Analysis Consortium (DPAC) and
contain G-band magnitudes for over $1.6\times 10^{9}$ sources with $G<21$ mag
and broadband colours $G_{BP}$ (330-680 nm) and $G_{RP}$ covering (630-1050
nm) for $1.4\times 10^{9}$ sources. PM components in equatorial coordinates
are available for $1.3\times 10^{9}$ sources, with an accuracy of 0.06 $mas\
yr^{-1}$, 0.2 $mas\ yr^{-1}$, and 1.2 $mas\ yr^{-1}$, for sources with $G<15$
mag, $G\sim 17$ mag, and $G\sim 20$ mag, respectively. We use the parallax
(plx) values to make the first cut in such a way to exclude clear nearby
foreground stars (those with large errors might scatter to $plx<0.5$ mas). In
our analysis, we did not make any Gaia photometric or colour cuts.

##  3 TBJ 3: background galaxy

TBJ 3 is situated at Equatorial coordinates $R.A.=17h18m14s$ and
$DEC=-27d54m15s$ (J2000) and Galactic coordinates $l=357^{\circ}.9299$ and
$b=5^{\circ}.0084$. It was mentioned for the first time as a GC candidate in
the list of Terzan & Bernard (1978) and Terzan & Ju (1980) with the name TJ 3.
Subsequently, it was spectroscopically studied by Bica et al. (1998), who
rename it as TBJ 3. We believe it to be the same object, although in those
works the Equatorial coordinates are $R.A.(1950)=17h15m06s$ and
$DEC(1950)=-27^{\circ}51^{\prime}$, that correspond to
$R.A.(J2000)=17h17m05.17s$ and $DEC(J2000)=-27d47m49.22s$, different from
those considered in this paper.

From Figure 1, it is clear that there is an object at these coordinates.
Analysing it at different wavelengths and by looking especially at the WISE
satellite images, we noted that TBJ 3 shows diffuse mid-IR emission, which is
not present in GCs. This feature, as well as the oblate and very elliptical
shape, lead us to discard TBJ 3 as a possible GC candidate, and consider this
to be a background galaxy.

Figure 1: Finding charts of TBJ 3, adopting different Surveys: 2MASS K-band
(a), ALLWISE w3 (b), ALLWISE w4 (c), Spitzer MIPS24 (d).

##  4 Confirmation of Patchick 99 as a new GC

Figure 2: Finding chart of Patchick 99 showing from left to right the central
$1.6^{\prime}\times 1.1’$ optical colour images from PanStarrs, and near-IR
colour images from the VVV and 2MASS surveys. Figure 3: Observed VVV near-IR
CMDs of a $3^{\prime}$ radius region centred on Patchick 99 (left), of a
background region with similar area (middle), and statistically decontaminated
CMD (right) following the procedure described by Palma et al. (2016, 2019). We
note the clear presence of the cluster RC in the decontaminated diagram.

Patchick 99 is located in the Galactic bulge at Equatorial coordinates
$R.A.=18h15m47s$ and $DEC=-29d48m46s$ (J2000) and Galactic coordinates
$l=2^{\circ}.4884$ and $b=-6^{\circ}.1452$ (Figure 2). We found this cluster
during our VVV search (Minniti et al. 2017, 2018; Minniti 2018), but later
realized that an object at that position had been reported before as part of
the deep sky hunters (DHS) survey of the DSS and 2MASS images (Kronberger et
al. 2006, 2012), so their discovery and nomenclature take precedence. This
cluster, which is named after the researcher who discovered it, was then
listed in the compilation of Bica et al. (2018), but has not been studied in
detail, and we therefore decided to investigate its nature and measure its
physical parameters with the VVV data.

###  4.1 Statistical and PM decontamination procedures

Contamination by Galactic bulge and disk stars is relevant, so we have made
some selections in order to decontaminate the CMDs. As mentioned in Section
2.2, we consider only stars that have $plx<0.5$ mas, in order to exclude
foreground stars. Moreover, Figure 3 shows the VVV near-IR CMDs of a
$3^{\prime}$ radius region centred on Patchick 99 compared with the CMD of a
nearby background region, and with the statistically decontaminated CMD. This
last CMD was constructed following the procedure described by Piatti & Bica
(2012) as applied by Palma et al. (2016, 2019) and Minniti et al. (2017). We
notice clearly the presence of the cluster RGB and RC that remain in the
decontaminated diagram, suggesting that a distance of $3^{\prime}$ from the
centre is an excellent compromise for our selection. However, there are some
blue stars outside of the cluster RGB that probably belong to the foreground
MW disk population that persist in this CMD. Therefore, we decided to use the
PMs as an independent method to decontaminate the CMD for this cluster, since
stars with similar PMs ensure belonging to the same cluster. In fact, we
compared the vector PM (VPM) diagram for our cluster, including stars in the
$3^{\prime}$ radius region and for the field stars, located between $4<r<5$
arcmin away from the cluster centre (Figure 4). Visually, the main difference
between the two PM distributions is that in the case of our cluster selection,
this one shows two over-densities, suggesting a bimodal distribution, unlike
the field stars that show a shallower homogeneous distribution when we move
away from the centre of the cluster, indicating an unimodal distribution. In
order to confirm our statement we have carried out the Gaussian mixture
modelling (GMM) analysis (Muratov & Gnedin 2010), which helps to give us the
statistical significance of an unimodal versus a bimodal distribution. We have
performed the GMM analysis with the heteroscedastic statistic for each sample.
In summary, we perform the GMM analysis for our three subsamples: the field
stars located between $4<r<5$ arcmin away from the Patchick 99 centre, the
position selection including all stars situated within $3^{\prime}$ from the
cluster centre, and our PM selection. Firstly, we do this for field stars and
position selection in order to obtain the statistical significance for the two
Gaussian distributions shown in Figure 4. Subsequently, we run the GMM
algorithm to distinguish the two Gaussian peaks shown in Figure 5. The
statistical parameters are shown in Tables 2 and 3. For a mixture of two
normal distributions the means and standard deviations along with the mixing
parameter (D, separation of the means relative to their widths) are usually
used, as a total of five parameters. Ashman et al. (1994) noted that $D>2$ is
required for a clean separation between the modes, while if the GMM method
detects two modes, but they are not separated enough ($D<2$), then such a
split is not meaningful.
Applying the GMM analysis to our samples we noted that they may show a bimodal
distribution, but with a low significance level. Doing a deep analysis of the
GMM resulting parameters, we can confirm that the distribution of field stars
located away from the cluster centre is unimodal since the $D<2$ and the
standard deviation of the second mode is of $\sigma_{2}\approx 14$, which is
not physical. About the $3^{\prime}$ radius region sample, the analysis
becomes more complicated. We believe that the two peaks have a very small
separation and the method is not able to distinguish both properly, indeed
comparing the parameters of the first Gaussian with the second one we can
appreciate that the first dominates over the second. On the other hand, the
GMM method on the PM selection indicates a bimodal distribution, since $D>2$.
This may be misleading to interpret since we should have expected an unimodal
distribution for the stars in a cluster. However, we have to highlight that
any sample will be contaminated by foreground stars that will have different
velocity (and PM) distribution. Thus, having two peaks in the sample of PM
selection is not unexpected. In any case, this does not represent a problem
since the second mode is very narrow and is not dominant over the first,
meaning that our final sample has got a very small contamination.

Successively, we calculate the mean cluster PMs (adopting the
$\sigma$-clipping method) measured from Gaia DR 2, yielding
$\mu_{\alpha}=(-2.98\pm 1.74)$ mas $yr^{-1}$ and $\mu_{\delta}=(-5.49\pm
2.02)$ mas $yr^{-1}$, which are in good agreement with the PM values found by
the GMM method. After that, we selected as cluster members only stars within
$3\ mas\ yr^{-1}$, following the procedure of Garro et al. (2020), indicated
by the black circle drawn in Figure 4. Also, the size of the circle is set
taking into consideration the typical size of GC members in the VPM diagram
for other well studied MW bulge GCs. Additionally, in Figure 5 we show the PM
distributions in a wide range ($-20<\mu_{\alpha}<15$ mas yr-1 and
$-20<\mu_{\delta}<10$ mas yr-1), considering only stars situated within
$3^{\prime}$ from the Patchick 99 centre (blue histogram), attempting to
minimise field stars. The resulting star sample after the PM cut is shown with
the red histogram. Each histogram is constructed considering the sum of all
the combined values per unit bin, selecting a fixed bin size of $0.8$ mas yr-1
(value that allowed us to visually distinguish the two peaks of the Gaussian
distributions). We can appreciate double peaked distributions for both blue
histograms, with a very little separation (as expected from the GMM method),
suggesting that we have at least two overlapping populations: the bulge field
stars and the star cluster. Indeed, fitting two Gaussian distributions we
assign a narrower and more peaked distribution for Patchick 99 (green curve)
and a wider distribution for Bulge field stars (grey curve). The total
distribution (black curve) is obtained by coadding the two Gaussians, in order
to fit the entire sample.
Figure 6 displays the PM decontaminated CMDs for VVV and Gaia data. We can
recognise that the main GC features, like RGB and RC, remain but, in
comparison with the statistical decontamination (Figure 3), the foreground
disk population is now better removed.

Figure 4: Top panel: VPM diagram from Gaia DR 2 dataset for star members
located within $3^{\prime}$ from the Patchick 99 centre. The black circle
indicates the position of the cluster selection within $3\ mas\ yr^{-1}$.
Bottom panel: VPM diagram for field stars located at $4.8^{\prime}$ from the
Patchick 99 centre. In each panel, colour represents density, with magenta the
highest, followed by blue, green and red.

Figure 5: PM distributions in RA (top panel) and in DEC (bottom panel), for
stars located within $3^{\prime}$ from the Patchick 99 centre (blue histogram)
and for our cluster PM selection (red histogram). Grey and green curves are
the Gaussian distributions for the Bulge field stars and Patchick 99 stars,
respectively. The total distribution, obtained by coadding the two Gaussian
distributions, is fitted as black curve.

Figure 6: Observed (left panels) and PM decontaminated (right panels) CMDs for
the globular cluster Patchick 99 , employing the near-IR VVV (top panels) and
optical Gaia DR 2 data (bottom panels).

Following these procedures, we have obtained the decontaminated near-IR
($K_{s}$ versus $J-K_{s}$) and optical ($G$ versus $BP-RP$) CMDs for our
cluster, as shown in the right panels of Figures 6 and in Figure 7. We use
those CMDs to derive the cluster’s main parameters. Both diagrams clearly show
the RC at $K_{s}=(12.5\pm 0.04)$ mag and $G=(15.5\pm 0.05)$ mag, suggesting
that P99 is a metal-rich GC. It is crucial to use RC stars in this kind of
studies, because these stars are very good distance indicators.

###  4.2 Estimation of Physical Parameters for Patchick 99

Nowadays, a wide range of reddening maps already exist, such as Schlafly &
Finkbeiner (2011), Gonzalez et al. (2011, 2018), Minniti et al. (2018), Soto
et al. (2019), Saito et al. (2020). We estimate the reddening and the
extinction toward this GC following Ruiz-Dern et al. (2018) and we benefit
from the clear position of the RC in the CMDs. We adopt the intrinsic RC
magnitude $M_{K_{s}}=(-1.605\pm 0.009)$ mag and the intrinsic colour
$(J-K_{s})_{0}=(0.66\pm 0.02)$ mag. Using the latter, we calculate a color
excess of $E(J-K_{s})=(0.12\pm 0.02)$ mag, in accordance with the adopted
reddening map, and the extinction $A_{K_{s}}=0.72\times E(J-K_{s})=(0.09\pm
0.01)$ mag (Cardelli et al. 1989). Consequently, we use these values in order
to determine distances, so we measure a distance modulus $(m-M)_{0}=(14.02\pm
0.01)$ mag equivalent to a heliocentric distance $D=(6.4\pm 0.2)$ kpc.
Similarly, for optical wavebands we identify again RC stars and adopt the
intrinsic RC magnitude $M_{G}=(0.459\pm 0.009)$ mag (Ruiz-Dern et al. 2018).
The near-IR extinction $A_{K_{s}}=(0.09\pm 0.01)$ mag corresponds to the
optical extinction $A_{G}=0.86\times A_{V}=(0.68\pm 0.08)$ mag, deriving also
$E(BP-RP)=A_{G}/3.12=(0.21\pm 0.03)$ mag (Cardelli et al. 1989; O’Donnell
1994). Therefore, the resulting cluster distance modulus from the Gaia optical
photometry is $(m-M)_{0}=(14.23\pm 0.1)$ mag and the distance $D=(7.0\pm 0.2)$
kpc, agrees within 3$\sigma$ with our near-IR distance.

Figure 7: The VVV near-IR (left panel) and Gaia DR 2 optical (right panel) PM
decontaminated CMD for the globular cluster Patchick 99. The red dotted line
is fitted by a PARSEC isochrone with age 10 Gyr and metallicity $[Fe/H]=-0.2$
dex. The magenta points represent the position of RR Lyrae star members in the
near-IR CMD.

Using our estimates of the reddening and distance we were able to fit
PARSEC444http://stev.oapd.inaf.it/cgi-bin/cmd isochrones (Bressan et al. 2012;
Marigo et al. 2017) to the CMDs of Patchick 99. This allows us to have
estimation of the metallicity and the age of the GC, finding $[Fe/H]=(-0.2\pm
0.2)$ dex and $t=(10\pm 2.0)$ Gyr, respectively. In order to obtain a more
robust result, we have also derived the metallicity from the slope of the RGB
($\alpha=0.11\pm 0.01$) using the calibration of Cohen et al. (2016), finding
$[Fe/H]=-0.3\pm 0.2$ dex, in agreement with our original estimate. Figure 7
shows a good fit of the isochrones with the above mentioned values of
metallicity and age in both our optical and near-infrared CMDs. We determined
the errors in age and in metallicity by changing the isochrones at a fixed
metallicity and a fixed age, respectively, until they do not fit the cluster
sequence simultaneously in the near-IR and optical CMDs. Therefore, it is
clear that although age is one of the fundamental parameters to characterise
GCs, its correct determination is still one of the most complicated tasks to
perform, especially in environments with high stellar density and with
differential reddening, such as the Galactic bulge. This is mainly due to
difficulties to reach the main sequence turn-off (MS-TO) point. We can
appreciate that problem in Figure 8, which displays the VVV and 2MASS PM
decontaminated CMDs. We include 2MASS cluster members with $K_{s}<11$ mag, as
they are more sensitive to the metallicity variations (e.g. , McQuinn et al.
2019). We can see that variations in metallicity have an especially strong
effect along the RGB and RC, and vice-versa, we can notice evident changes in
age in the MS-TO region. For this reason our age should be taken as a crude
estimate, since we are unable to measure it precisely.

Figure 8: Fitting of isochrones, in the VVV CMDs, at fixed metallicity of
$[Fe/H]=-0.2$ dex and variation in ages t=8, 9, 10, 11, 13 Gyr (right panel);
while at fixed age of t=10 Gyr and different metallicities $[Fe/H]=0,\ -0.1,\
-0.2,\ -0.3,\ -0.4$ dex (left panel). Black points represent the VVV cluster
members, whereas blue points are those from 2MASS catalogue.

All of the previously derived parameters, including position, metallicity and
kinematics confirm that Patchick 99 is a bulge GCs, but if we consider its
age, it could be even younger than typical metal-rich GCs present in the
Galactic bulge, which have ages between 12 and 13 Gyr (e.g., Carretta et al.
2000. However, many studies have demonstrated that GCs show a relatively wide
range of ages in the bulge (Zoccali et al. 2003; Clarkson et al. 2011; Valenti
et al. 2013; Bernard et al. 2018).

Figure 9 shows the Gaia+VVV optical-near-IR colour-colour diagram (CCD) for
the whole catalogue without selection cuts, and for only cluster members of
Patchick 99 with $K_{s}<15$ mag. We can clearly notice that the colour
distribution reveals a tighter sequence for star members (circles, right
panel) than for the field stars (squares, left panel). In particular, we can
appreciate that there is a little differential extinction, in the right panel
of Figure 9. Based on the high resolution reddening map by Surot et al. 2020,
the mean reddening is $E(J-K_{s})=0.05$ mag in this field, which differs from
our reddening value by $\Delta(E(J-K_{s}))=0.07$ mag at most. Hence, we
believe that our estimation is in good agreement with the adopted map,
discarding this as a significant problem.

Finally, we derive the total luminosity of our cluster, using the catalogue
with only cluster members. Once the flux for each star is measured, we derive
the absolute magnitude from the total flux, obtaining $M_{K_{s}}=(-7.0\pm
0.6)$ mag, equivalent to $M_{V}=(-4.5\pm 0.8)$ (for a typical GC colour of
$(V-K_{s})=(2.5\pm 0.5)$ mag), indicating that it is a very faint GC. This
value is an underestimate of the total luminosity, since the faintest stars
are missing. Regarding this, we compared Patchick 99 luminosity with that
other known GCs similar to Patchick 99 in metallicity: NGC 6553
($[Fe/H]=-0.18$ dex; $D=6.01$ kpc; $M_{V}=-7.77$ mag), NGC 6528
($[Fe/H]=-0.11$ dex; $D=7.9$ kpc; $M_{V}=-6.57$ mag) and Terzan 5
($[Fe/H]=-0.23$ dex; $D=5.9$ kpc; $M_{V}=-7.42$ mag). Using the same method,
and also spanning on the same magnitude range ($10.7<K_{s}<17.2$), we derive
the integrated near-infrared absolute magnitude for each GC, finding $M_{Ks}\
(NGC\,6553)=-9.5$ mag, $M_{Ks}\ (NGC\,6528)=-8.0$ mag and $M_{Ks}\
(Ter5)=-9.6$ mag. We converted them in $V-$band and scaled to the $M_{V}$
values from the 2010 version of the Harris (1996) catalogue. Therefore,
estimating the $M_{V}$ differences and assuming the similarities between the
GCs, we derive that the total luminosity for Patchick 99 is $M_{V}=-5.2$ mag,
$\sim 2.2$ less luminous of the MW GC luminosity function peak
($M_{V}=(-7.4\pm 0.2)$ mag from Harris 1991; Ashman & Zepf 1998)

All these parameters are summarised in Table 1.

Figure 9: Gaia+VVV optical-near-IR CCDs for Patchick 99, as density plot. On
the left: squares are the observed stars, using the entire catalogue without
selection cuts; on the right: circles represent star members of Patchick 99,
considering bright stars previously selected. At the top-left of the panel,
the red arrow represents the reddening vector.

###  4.3 The derivation of the surface density profile

A detailed description of the structural parameters of Patchick 99 would need
a careful treatment of completeness as well as relatively large numbers of
member stars from which a radial profile can be derived. Traditionally, King
profiles (King 1962) have been fitted to a significant fraction of MW old
clusters, with however more and more evidence that other profiles might
provide better fits, especially when extragalactic or massive GCs are included
(e.g., Mackey & Gilmore (2003), de Boer et al. (2019a)). Although rich
clusters allow for model-dependent sizes and structural properties, our sample
of member stars is not enough to adequately address a comparison among the
different models available. Instead, a King function is assumed to be a good
approximation for the radial distribution of the stellar density.

The correct determination of the cluster centre has a fundamental importance,
since an error in its position can lead to altered results regarding the
density profile. For this reason, we have used the K-Means algorithm in Python
to determine the centre in a robust way from our catalogue of cluster member
stars. In summary, this is a machine-learning algorithm that searches for
subgroups (or literally ”clusters”) in a given parameter space among a sample
of data points. We checked visually that the resulting centres were close to
the expected density peaks and we found that the initial position
($R.A.=18h15m47s$ and $DEC=-29d48m46s$ (J2000)) coincides almost-perfectly
with the resulting values of the centre, with a small separation of $\Delta
RA\approx 2^{\prime\prime}$ and $\Delta DEC\approx 5^{\prime\prime}$.
Starting from these new coordinates, we divided our sample of member stars
into ten circular annuli, out to a radius of $1.5^{\prime}$, in steps of
$0.15^{\prime}$. We do not need to apply geometric corrections as the
resulting circular annuli are within the FOV of our catalogue. We performed
the radial density profile of Patchick 99 (see Figure 10) dividing the total
number of stars in each radial bin by the area of that ring in order to derive
the density in each bin. Poisson errors are also shown. We note that the error
bars are relatively large, mainly due to the low-number statistics in each
radial bin, especially in the innermost point (see Figure 10). No background
has been subtracted since we are dealing with a list that is expected to be
free from foreground stars and other contaminants.

We have overplotted a family of King profiles adopting different values of
core radius ($r_{c}=1.8^{\prime}$, $1.5^{\prime}$, $1.2^{\prime}$), but
keeping the tidal radius fixed at $10^{\prime}$. The choice to use that value
of tidal radius is justified following de Boer et al. (2019b) (see their
Figures 13 and 14), where a clear correlation between the tidal radius and
galactocentric radius of a given GC is illustrated. As shown in Figure 10, the
observational points are best represented by $r_{c}=1.2^{\prime}$, which is
equivalent to $r_{c}\sim 1.1$ pc. Also, we consider the tidal radius
$r_{t}=10^{\prime}$ as the size of the cluster, which corresponds to a
physical size of $9.3$ pc.

Figure 10: Radial surface density of Patchick 99 (black points), taking into
consideration only PM selected member stars. The coloured lines are King model
profiles that best reproduce observational data, with a fixed value of the
tidal radius $10^{\prime}$ and different values of core radius, in cyan
$r_{c}=1.8^{\prime}$, in red $r_{c}=1.5^{\prime}$ and in blue
$r_{c}=1.2^{\prime}$ Poisson errors are shown as error bars.

###  4.4 The RR Lyrae stars in Patchick 99

RR Lyrae variable stars represent a very old-population in our Galaxy, found
in the Galactic halo and bulge. In particular, according to Pietrukowicz et
al. (2015) their spatial density ranges between 80 and 440 RRL/deg2 in the
bulge.
They are also considered powerful tools to reveal the nature of a GC candidate
because if detected in a stellar cluster, RR Lyrae stars guarantee that it is
an old GC. Although these variables are used as excellent tracers of old and
metal-poor populations, there are observational evidences that suggest the
presence of RR Lyrae also in metal-rich GCs, such as NGC 6388 ($[Fe/H]=-0.44$
dex) and NGC 6441 ($[Fe/H]=-0.46$ dex) (Pritzl et al. 2002; Clementini et al.
2005), and NGC 6440 $[Fe/H]=-0.36$ dex), all of which are somewhat more metal-
rich than Patchick 99. Additionally, they are good reddening and distance
indicators.

For these reasons, we have searched for these variable stars and we have
detected 21 RR Lyrae stars within 12 arcmin from the Patchick 99 centre,
matching the OGLE (Soszyński et al. 2014), the VVV (Majaess et al. 2018) and
the Gaia (Clementini et al. 2019) catalogues. They are listed in Table 4,
where we summarised: identification, location, period, photometry in different
filters, heliocentric distance, distance from the centre of the GC, type for
each source and their PMs.

Initially we constructed their phased light curves, in order to discriminate
their pulsation mode. We identify 13 fundamental mode pulsators (ab-type RR
Lyrae stars) with long periods ($P\gtrsim 0.4$ days) and large amplitudes, and
8 first-overtone pulsators (c-type RR Lyrae stars), with short period
($P\lesssim 0.4$ days) and small amplitudes. The RRc and RRab pulsators also
differ in the morphology of their light curves, since the RRab’s have
typically asymmetrical light curves, unlike RRc’s. Figure 11 displays the
Amplitude-Period diagram (so-called Bailey diagram, Bailey et al. 1919) in
such a way to highlight that differences. Indeed, we consider the Bailey
diagram a solid tool for verifying our estimations.

Figure 11: Bailey diagram for detected 21 RR Lyrae stars (black points),
situated within 12 arcmin from the Patchick 99 centre, while we highlight the
RR Lyrae member stars with magenta points.

Our main goal is to identify which RR Lyrae stars are real members of Patchick
99. Firstly, we measured their projected distance from the centre using the
centroid of each variable stars. Based on these distances and considering the
size of the cluster, we believe that seven RR Lyrae stars are members of
Patchick 99, as shown in Figure 12 and evidenced by a star symbol in Table 4.
In addition, we compare their projected distance with the resulting
heliocentric distance (Figure 12), confirming again the membership. In detail,
we used the period-luminosity (PL) relation in $K_{s}$-band following Muraveva
et al. (2015) (with RRc stars “fundamentalized” by adding $0.127$ to the
logarithm of the period) in order to derive their heliocentric distance. Once
selected the RR Lyrae members, we estimate the mean distance, finding a value
of $D=7.06\pm 0.48$ kpc, consistent with the heliocentric distances obtained
from the VVV and Gaia photometries (see Section 4), since these values differ
by less than 2$\sigma$. On the other hand, we discard the others as members
since they have heliocentric distances larger than those found using VVV near-
IR photometry, or they are located outside our assumed tidal radius. However,
we can restrict the RR Lyrae sample to the ones that have PMs matching the
mean GC PM within the errors, we find that only 3 of them fulfil this
requirement: OGLE-BLG-RRLYR-35312, 35476 and 35459, favouring a short distance
of $6.3\pm 0.5$ kpc in the mean. Additionally, Figure 12 does not show OGLE-
BLG-RRLYR-35364 and OGLE-BLG-RRLYR-35381, since their distance values are very
large and probably they are located in the Galactic halo or in the Sagittarius
dwarf galaxy. We also tested two other PL relations from Gaia Collaboration et
al. (2017) and Navarrete et al. (2017), yielding slightly larger distance
values. However, we use the comparison between the three PL relations, in
order to assign a mean distance error of 0.5 kpc. Once absolute magnitudes are
obtained, we also build-up the PL relation diagram (Figure 13). This allowed
us to confirm again the membership, since we find a stronger relation when the
least-square fit is used on the star cluster members (red line), rather than
on the whole sample (blue line). We also show the VVV phased light curves for
each RR Lyrae member (Figure 14). We note that the light curve profile for
OGLE-BLG-RRLYR-35459 is slightly worse than the others, since this variable
appears a bit blended.
Moreover, we derive the expected spatial density from Pietrukowicz et al.
(2015) and Navarro et al. (2020). Specifically, following Navarro et al.
(2020), we are able to predict a mean density of 4 field RR Lyrae type-ab in
the same field of Patchick 99. We found 21 variables in total, including 13
RRab. Hence, this value represents a clear excess over the background ($\sim
4.5\sigma$ detection).

Figure 12: Spatial distribution of RR Lyrae variable stars detected within
$12^{\prime}$ from the centre of Patchick 99. The magenta points represent the
seven RR Lyrae stars considered cluster members, whereas we highlight the PM
members with magenta squares. The distance errorbars are of 0.5 kpc for each
point. The dotted line depicts the heliocentric distance at $D=(6.4\pm 0.2)$
kpc derived from the VVV near-IR photometry. Figure 13: $M_{Ks}-\log(P)$
diagram for all RR Lyrae sample (black points) and RR Lyrae star members
(magenta points, while the squares represent the PM variable members). Blue
and red lines represent the least-squares fit for all RR Lyrae sample and
cluster members, respectively.

Finally, the magenta points in Figure 7 define the position of the seven RR
Lyrae stars in the VVV CMD. Their near-IR magnitudes are on average much
fainter than the RC ($\sim 1-1.5$ mag). However, both theory and observations
suggest that the RR Lyrae indeed should be fainter than the RC in the near-IR
CMDs and also indicate that more metal-rich RR Lyrae are fainter than metal-
poor ones. For example, Cassisi et al. (2004) explored the predicted behavior
of the pulsators as a function of the horizontal branch morphology and over
the metallicity range Z=0.0001 to 0.006, finding that the RR Lyrae luminosity
decreases with increasing metallicity. Further, Marconi et al. (2018)
constructed new sets of helium-enhanced (Y = 0.30, Y = 0.40) nonlinear, time-
dependent convective hydrodynamical models of RR Lyrae stars covering a broad
range in metal abundances (Z = 0.0001 – 0.02), finding that an increase in
helium content from the canonical value (Y = 0.245) to Y = 0.30 – 0.40 causes
a simultaneous increase in stellar luminosity and in pulsation period (see
their Figure 1 and 2).
Also, from the observational point of view, we have compared the location of
our RR Lyrae with those confirmed PM members of the cluster NGC 6441 from
Alonso-García et al. (in preparation). The RR Lyrae stars in Patchick 99 fall
in the expected region of the near- IR CMD for this metal-rich cluster. In any
case, spectroscopic metallicities are needed to confirm if this is the most
metal-rich GC known with RR Lyrae.

##  5 Summary and Conclusions

We have investigated the GC candidates Patchick 99 and TBJ 3 in order to
understand their real nature.
They are analysed in optical and near-IR wavelengths with the goal of
determining their physical parameters. The visual examination of TBJ 3 at
various wavelengths confirms that it is not a GC but from its mid-IR emission
we conclude that TBJ 3 is a background galaxy.
Therefore, we have focused our work on Patchick 99. We have measured its
reddening, extinction, distance modulus and heliocentric distance, size,
proper motion, total luminosity, metallicity and age. We summarised all those
values in Table 1.
Even though the field contamination is severe, we used two different methods
(statistical and PM decontamination) recovering the same results, with the GC
RGB and RC clearly present. We conclude that Patchick 99 is a metal-rich GC,
located in the Galactic bulge at a heliocentric distance of $D=6.6\pm 0.6$
kpc, obtained as an average distance of three-independent methods (Gaia DR 2
and VVV photometries, and three RR Lyrae star members). We define it as a
genuine GC because both CCD and CMDs yield a population with an age of 10 Gyr.
These results are confirmed by the presence of seven RR Lyrae stars, located
at $d<12^{\prime}$ from the cluster centre and at the same distance of the
cluster from the Sun. However, analysing their PM we can confirm that three of
them are surely member of our cluster: OGLE-BLG-RRLYR-35312, 35476 and 35459,
since their PMs match the mean cluster PM within the errors. Patchick 99 may
be the most metal-rich globular cluster with RR Lyrae, to be confirmed by
spectroscopic observations. As mentioned previously, even if position,
metallicity and kinematics confirm that it is a bulge GC, the age of Patchick
99 is a few Gyr younger than the typical age of metal-rich GCs ($\sim 12-13$
Gyr) in the same regions. In any case, it is even important to underline that
our estimation of age is quite rough and we need deeper observations to be
more accurate.
Another important conclusion is that Patchick 99 is a low-luminosity GC
($M_{Ks}=-7.0$ mag), suggesting that many other faint GCs are still to be
revealed in the bulge of the Milky Way.

###### Acknowledgements.

We gratefully acknowledge the use of data from the ESO Public Survey program
IDs 179.B-2002 and 198.B-2004 taken with the VISTA telescope and data products
from the Cambridge Astronomical Survey Unit. ERG acknowledges support from an
UNAB PhD scholarship. D.M. acknowledges support by the BASAL Center for
Astrophysics and Associated Technologies (CATA) through grant AFB 170002. D.M.
and M.G. are supported by Proyecto FONDECYT No. 1170121. J.A.-G. acknowledges
support from Fondecyt Regular 1201490 and from the ANID’s Millennium Science
Initiative $ICN12_{\\_}009$, awarded to the Millennium Institute of
Astrophysics (MAS). This work has made use of the University of
Hertfordshire’s high-performance computing facility.

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Table 1: Main physical parameters for Patchick 99 determined using near-IR and optical datasets. Physical parameters | Patchick 99
---|---
RA (J2000) | 18h15m47s
DEC (J2000) | -29d48m46s
Latitude | $2^{\circ}.4884$
Longitude | $-6^{\circ}.1452$
$\mu_{\alpha}$ [mas/yr] | $-2.98\pm 1.74$
$\mu_{\delta}$ [mas/yr] | $-5.49\pm 2.02$
$A_{K_{s}}$ [mag] | $0.09\pm 0.01$
$A_{G}$ [mag] | $0.68\pm 0.08$
$E(J-K_{s})$ [mag] | $0.12\pm 0.02$
$E(BP-RP)$ [mag] | $0.21\pm 0.03$
$(m-M)_{0}$ [mag] | $14.02\pm 0.04$
$D_{mean}$ [kpc] | $6.6\pm 0.6$
$M_{K_{s}}$ [mag] | $-7.0\pm 0.6$
$M_{V}$ [mag] | $-5.2$
$[Fe/H]$ [dex] | $-0.2\pm 0.2$
Age [Gyr] | $10.0\pm 2$
$r_{c}$ [arcmin] | 1.2 (1.1 pc)
$r_{t}$ [arcmin] | 10 (9.3 pc)
Table 2: GMM analysis for each sub-sample (field stars, position and PM selection) applied to PM in RA. The subscripts $1$ and $2$ indicate the first and second mode of the Gaussian distribution. | $\mu_{1}$ | $\sigma_{1}$ | $\mu_{2}$ | $\sigma_{2}$ | D
---|---|---|---|---|---
| [mas yr-1] | [mas yr-1] | [mas yr-1] | [mas yr-1] |
Field stars | $-2.11\pm 0.11$ | $4.41\pm 1.98$ | $-1.26\pm 0.52$ | $13.99\pm 1.98$ | $0.08\pm 0.05$
Position sel. | $-1.96\pm 0.58$ | $4.08\pm 2.8$ | $-2.54\pm 0.55$ | $10.71\pm 3.16$ | $0.07\pm 0.06$
PM selection | $-3.35\pm 0.13$ | $1.42\pm 0.05$ | $-0.73\pm 0.14$ | $0.53\pm 0.08$ | $2.46\pm 0.09$
Table 3: GMM analysis for each sub-sample (field stars, position and PM selection) applied to PM in DEC. The subscripts $1$ and $2$ indicate the first and second mode of the Gaussian distribution. | $\mu_{1}$ | $\sigma_{1}$ | $\mu_{2}$ | $\sigma_{2}$ | D
---|---|---|---|---|---
| [mas yr-1] | [mas yr-1] | [mas yr-1] | [mas yr-1] |
Field stars | $-4.61\pm 0.11$ | $4.4\pm 0.2$ | $-3.53\pm 0.57$ | $13.63\pm 0.85$ | $0.11\pm 0.06$
Position sel. | $-4.51\pm 0.11$ | $3.52\pm 0.21$ | $-4.09\pm 0.41$ | $7.05\pm 0.54$ | $0.08\pm 0.07$
PM selection | $-6.13\pm 0.39$ | $1.35\pm 0.16$ | $-3.52\pm 0.41$ | $0.8\pm 0.2$ | $2.35\pm 0.13$
Table 4: Detected RR Lyrae stars within $12^{\prime}$ from the Patchick 99
centre. The symbol $\bigstar$ indicates the star members of the cluster, while
the double symbols $\bigstar\bigstar$ highlight the variable stars that have
PM matching the mean GC PM within the errors.

SourceID | RA | DEC | P | V | $K_{s}$ | J | $D_{RRL}$ | $d_{centre}$ | Type | $\mu_{\alpha}$ | $\mu_{\delta}$
---|---|---|---|---|---|---|---|---|---|---|---
| (J2000) | (J2000) | [day] | [mag] | [mag] | [mag] | [kpc] | [arcmin] |  | [mas yr${}^{-}1$] | [mas yr${}^{-}1$]
OGLE-BLG-RRLYR-35312★★ | 273.8098 | -29.8325 | 0.4864334 | 15.47 | 13.84 | 14.10 | 6.1 | 8.2 | RRab | -3.866 | -3.070
OGLE-BLG-RRLYR-35348 | 273.8499 | -29.9502 | 0.2734252 | 16.91 | 15.57 | 15.93 | 12.2 | 10.1 | RRc | -0.453 | -4.932
OGLE-BLG-RRLYR-35447★★ | 273.9633 | -29.9070 | 0.2879753 | 15.93 | 14.58 | 14.82 | 7.9 | 5.8 | RRc | 6.448 | -14.647
OGLE-BLG-RRLYR-35476★ | 274.0053 | -29.9212 | 0.5301811 | 15.48 | 13.77 | 14.16 | 6.1 | 7.4 | RRab | -4.627 | -4.028
OGLE-BLG-RRLYR-35557 | 274.1133 | -29.8866 | 0.6141217 | 16.09 | 14.26 | 14.64 | 8.3 | 11.0 | RRab | 0.210 | -10.364
OGLE-BLG-RRLYR-15997 | 273.7959 | -29.7434 | 0.2933517 | 16.18 | 14.79 | 15.13 | 8.7 | 9.9 | RRc | -3.290 | -9.259
OGLE-BLG-RRLYR-16094★ | 273.9599 | -29.7208 | 0.5121471 | 15.88 | 14.27 | 14.66 | 7.6 | 5.6 | RRab | 4.453 | -3.489
OGLE-BLG-RRLYR-16108★ | 273.9885 | -29.7488 | 0.5217077 | 15.90 | 14.25 | 14.75 | 7.6 | 4.6 | RRab | 1.692 | 1.172
OGLE-BLG-RRLYR-16115 | 274.0016 | -29.6816 | 0.3376222 | - | 14.83 | 15.08 | 9.4 | 8.6 | RRc | -4.906 | -8.455
OGLE-BLG-RRLYR-35292 | 273.7955 | -29.8620 | 0.2931980 | 16.64 | 15.06 | 15.37 | 9.8 | 9.5 | RRc | -3.943 | -4.551
OGLE-BLG-RRLYR-35355 | 273.8588 | -29.9444 | 0.5942311 | 19.28 | 17.05 | 17.53 | 29.4 | 9.5 | RRab | - | -
OGLE-BLG-RRLYR-35364 | 273.8741 | -29.8688 | 0.5927029 | 18.35 | 16.51 | 16.85 | 22.9 | 5.5 | RRab | -4.234 | -3.179
OGLE-BLG-RRLYR-35381 | 273.8937 | -29.8050 | 0.6040448 | 16.61 | 14.76 | 15.22 | 10.3 | 3.2 | RRab | -1.456 | -0.187
OGLE-BLG-RRLYR-35382★ | 273.8947 | -29.8940 | 0.6214080 | 15.72 | 13.91 | 14.50 | 7.1 | 5.8 | RRab | 2.041 | 1.390
OGLE-BLG-RRLYR-35384 | 273.8970 | -29.7807 | 0.7768699 | 16.00 | 14.17 | 14.67 | 8.9 | 3.5 | RRab | -3.910 | -5.198
OGLE-BLG-RRLYR-35390 | 273.9015 | -29.6619 | 0.2519186 | 15.26 | 13.98 | 14.25 | 5.7 | 9.4 | RRc | -5.090 | 3.183
OGLE-BLG-RRLYR-35431 | 273.9381 | -29.9478 | 0.6426197 | 15.90 | 14.17 | 14.60 | 8.1 | 8.1 | RRab | 0.943 | -1.826
OGLE-BLG-RRLYR-35459★★ | 273.9749 | -29.8401 | 0.3754443 | 15.50 | 13.98 | 14.30 | 6.6 | 2.4 | RRc | -3.172 | -8.688
OGLE-BLG-RRLYR-35533 | 274.0821 | -29.7864 | 0.3104254 | 16.34 | 14.96 | 15.25 | 9.6 | 8.3 | RRc | -1.772 | -4.132
OGLE-BLG-RRLYR-35535 | 274.0874 | -29.8810 | 0.4767085 | 16.07 | 14.37 | 14.77 | 7.6 | 9.4 | RRab | -5.984 | -2.042
OGLE-BLG-RRLYR-35571 | 274.1270 | -29.8545 | 0.5653381 | 16.34 | 14.52 | 14.83 | 9.0 | 11.2 | RRab | -2.534 | -3.027

Figure 14: Phased light curves for the RR Lyrae star members of Patchick 99
GC, classified as ab-type with $P>0.4$ days and c-type with $P<0.4$ days.


11institutetext: Institute for Mathematics, Brandenburg University of
Technology Cottbus-Senftenberg, 03046 Cottbus, Germany
11email<EMAIL_ADDRESS>22institutetext: Forschungszentrum
Jülich GmbH, Jülich Supercomputing Centre,
Wilhelm-Johnen-Straße, 52425 Jülich, Germany
22email<EMAIL_ADDRESS>33institutetext: University of Applied
Sciences Aachen, Faculty of Medical Engineering and Technomathematics,
Heinrich-Mußmann-Straße 1, 52428 Jülich, Germany
44institutetext: UMIT TIROL – Private University for Health Sciences and
Health Technology, Eduard-Wallnöfer-Zentrum 1, 6060 Hall/Tyrol, Austria
44email<EMAIL_ADDRESS>
# An Approach to Colour Morphological Supremum Formation using the LogSumExp
Approximation

Marvin Kahra 11 Michael Breuß 11 Andreas Kleefeld 2233 and Martin Welk 44

###### Abstract

Mathematical morphology is a part of image processing that has proven to be
fruitful for numerous applications. Two main operations in mathematical
morphology are dilation and erosion. These are based on the construction of a
supremum or infimum with respect to an order over the tonal range in a certain
section of the image. The tonal ordering can easily be realised in grey-scale
morphology, and some morphological methods have been proposed for colour
morphology. However, all of these have certain limitations.

In this paper we present a novel approach to colour morphology extending upon
previous work in the field based on the Loewner order. We propose to consider
an approximation of the supremum by means of a log-sum exponentiation
introduced by Maslov. We apply this to the embedding of an RGB image in a
field of symmetric $2\times 2$ matrices. In this way we obtain nearly
isotropic matrices representing colours and the structural advantage of
transitivity. In numerical experiments we highlight some remarkable properties
of the proposed approach.

###### Keywords:

mathematical morphology colour image matrix-valued image symmetric matrix
transitivity

##  1 Introduction

Mathematical morphology is a theory used to analyse spatial structures in
images. Over the decades it has developed into a very successful field of
image processing, see e.g. [14, 15, 16] for an overview. Morphological
operators basically consist of two main components. The first of these is the
structuring element (SE), which is characterised by its shape, size and
position. These in turn can be divided into two types of SEs, flat and non-
flat, cf. [9]. A flat SE basically defines a neighbourhood of the central
pixel where appropriate morphological operations are performed, while a non-
flat SE also contains a mask with finite values that are used as additive
offsets. The SE is usually implemented as a window sliding over the image. The
second main component is used to perform a comparison of values within an SE.
Two basic operations in mathematical morphology are dilation and erosion,
where a pixel value is set to the maximum and minimum, respectively, of the
discrete image function within the SE. Many morphological filtering procedures
of practical interest, such as opening, closing or top hats, can be formulated
by combining dilation and erosion operations. Since dilation and erosion are
dual operations, it is often sufficient to restrict oneself to one of the two
when constructing algorithms.

Let us briefly discuss how the operations of dilation and erosion can be
applied to colour morphology, as it is the underlying concept for our further
considerations. As already mentioned, the most important operation in
morphology is to perform a comparison of the tonal values, or in our case the
colour values, within the SE. For the simpler application areas such as grey
value morphology, one can act directly on complete lattices in order to obtain
a total order of the colour values, cf. [4, 16]. In the case of colour
morphology, this is no longer the case, as there is no total order of the
colour values. For this reason, corresponding half-orders and different basic
structures are used, cf. [3]. The first approach that could be used for this
would be to regard each colour channel of an image as an independent image and
to perform grey value morphology on each of them. The other approach, which is
more popular, uses a vector space structure where each colour is considered as
a vector in an underlying colour space. This can take the form of [1], [2] or
[11]. We will take the latter approach, but use symmetric matrices instead of
vectors. We will order the elements by means of a half-order, namely the
Loewner order (see [6]).

To calculate the basic operations of colour morphology, the supremum or
infimum is determined instead of the maximum or minimum. There are also
different approaches how to choose the supremum of symmetric matrices. To give
some examples, we mention here the nuclear norm, the Frobenius norm and the
spectral norm. For a comparison of these norms we refer to the work [19] by
Welk, Kleefeld and Breuß.

In this paper, however, we want to consider a different approach, namely the
approximation of the supremum by means of a log-sum exponentiation. The reason
why we follow this approach is that it is an approximation which already gave
promising results in the work by Kahra, Sridhar and Breuß (see [10]) for grey
scale images in connection with a fast Fourier transform. This approach was
also used for colour images in [18], but only in the above described sense of
a channel-wise structuring of colour images. Another connection worth
mentioning is with the paper of Burgeth, Welk, Feddern and Weickert (see [8]),
where root and power were used for symmetric positive semidefinite matrices
instead of logarithm and exponential function. However, our approach here has
the advantage of being transitive.

This paper will be the first step for transferring the LogSumExp approach to
colour morphology with tonal vectors/matrices. The goal is to present a first
introduction to the characterisation of this approach for tonal value matrices
and a comparison with some of the other already existing approaches.

##  2 General Definitions

###  2.1 Colour Morphology

In order to make this paper self-contained, we briefly recall some basics of
mathematical morphology in general and colour morphology in particular.

For this purpose, we first consider a two-dimensional, discrete image domain
$\Omega\subseteq\mathbb{Z}^{2}$ and a single-channel grey-scale image, which
is explained by a function $f:\Omega\rightarrow[0,255]$. In the case of non-
flat morphology, the SE can be represented as a function
$b:\mathbb{Z}^{2}\rightarrow\mathbb{R}\cup\\{-\infty\\}$ with

$\displaystyle
b(\boldsymbol{x}):=\begin{cases}\beta(\boldsymbol{x}),\quad&\boldsymbol{x}\in
B_{0},\\\ -\infty,&\text{otherwise},\end{cases}\quad
B_{0}\subset\mathbb{Z}^{2},$

where $B_{0}$ is a set centred at the origin. In the case of a flat filter
(flat morphology), it is simply the special case $\beta(\boldsymbol{x})=0$.
The most elementary operations of mathematical morphology are dilation

$\displaystyle(f\oplus
b)(\boldsymbol{x}):=\max_{\boldsymbol{u}\in\mathbb{Z}^{2}}{\\{f(\boldsymbol{x}-\boldsymbol{u})\>+\>b(\boldsymbol{u})\\}}$

and erosion

$\displaystyle(f\ominus
b)(\boldsymbol{x}):=\min_{\boldsymbol{u}\in\mathbb{Z}^{2}}{\\{f(\boldsymbol{x}+\boldsymbol{u})\>-\>b(\boldsymbol{u})\\}}.$

With these two operations, many other operations can be defined that are of
great interest in practice, for example opening $f\circ b=(f\ominus b)\oplus
b$ and closing $f\bullet b=(f\oplus b)\ominus b$.

We now come to our real area of interest, namely colour morphology. This is
similar to the greyscale morphology already shown, with the difference that we
no longer have just one channel, but three. There are many useful formats for
expressing this, see [17]. A classic approach in this sense is the channel-by-
channel processing of an image with the RGB colour model, see e.g. [18] for a
recent example of channel-wise scheme implementation. However, instead of RGB
vectors, we will use symmetric, positive semi-definite $2\times 2$ matrices.
For this we assume that the colour values are already normalised to the
interval $[0,1]$. First, we transfer this vector into the HCL colour space by
means of $M=\max\\{r,g,b\\}$, $m=\min\\{r,g,b\\}$, $c=M-m$,
$\ell=\frac{1}{2}(M+m)$ and

$\displaystyle h=\begin{cases}\frac{g-b}{6c}\mod 1,\quad&M=r,\\\
\frac{b-r}{6c}+\frac{1}{3}\mod 1,\quad&M=g,\\\ \frac{r-g}{6c}+\frac{2}{3}\mod
1,\quad&M=b.\end{cases}$

Then we replace the luminance $\ell$ with $\tilde{\ell}=2\ell-1$ and consider
the quantities $c$, $2\pi h$ and $\tilde{\ell}$ as radial, angular and axial
coordinates of a cylindrical coordinate system, respectively. This represents
a bijection onto the bi-cone, which in turn is interpreted with Cartesian
coordinates by $x=c\cdot\cos(2\pi h)$, $y=c\cdot\sin(2\pi h)$ and
$z=\tilde{\ell}$. Finally, we map these Cartesian coordinates onto a symmetric
matrix as follows:

$\displaystyle\boldsymbol{A}:=\frac{\sqrt{2}}{2}\begin{pmatrix}z-y&x\\\
x&z+y\end{pmatrix},$

where the complete transformation process is a bijective mapping, see [6].

###  2.2 The Loewner Order and the Decision of a Minimiser

Since there is no classical ordering of the elements in $\mathbb{R}^{3}$ or
$\mathbb{R}^{2\times 2}$, we need to ask when an element is larger or smaller
than another element. To answer this question, we need to use a weaker
definition of an ordering relation, i.e. a half-order. For this reason, we
resort to the promising Loewner order and the colour morphological processing
based on this, which was already presented by Burgeth and Kleefeld (see [7]).

For two symmetric matrices $\boldsymbol{A}$ and $\boldsymbol{B}$, the Loewner
order $\leq_{L}$ or $<_{L}$ is defined as

$\displaystyle\boldsymbol{A}<_{L}\boldsymbol{B}\leavevmode\nobreak\
(\boldsymbol{A}\leq_{L}\boldsymbol{B})\Longleftrightarrow\boldsymbol{B}-\boldsymbol{A}\text{
is positive (semi-)definite.}$

The problem with this semi-order is that it is not a lattice order [5] and
therefore it is not possible to find a unique matrix maximum or supremum. To
get around this problem, one needs another property to select a uniquely
determined matrix maximum from the convex set of symmetric matrices
$\mathcal{U}(\mathcal{X})$ that are upper bounds in the Loewner sense for the
multi-set
$\mathcal{X}=\\{\boldsymbol{X}_{1},\boldsymbol{X}_{2},\dots,\boldsymbol{X}_{n}\\}$
of given data matrices of the set $\text{Sym}(2)$ of symmetric real $2\times
2$ matrices with

$\displaystyle\mathcal{U}(\mathcal{X}):=\left\\{\boldsymbol{Y}\in\text{Sym}(2):\boldsymbol{X}\leq_{L}\boldsymbol{Y}\quad\forall\boldsymbol{X}\in\mathcal{X}\right\\},$

see [19]. For representation of this property, we use the function
$\varphi:\mathcal{U}(\mathcal{X})\rightarrow\mathbb{R}$, which is convex and
Loewner-monotone, i.e.

$\displaystyle\varphi(\boldsymbol{A})\leq\varphi(\boldsymbol{B})\quad\Longleftrightarrow\quad\boldsymbol{A}\leq_{L}\boldsymbol{B}.$

Furthermore, $\varphi$ should have a unique minimiser in
$\mathcal{U}(\mathcal{X})$, then we can define the $\varphi$-supremum of
$\mathcal{X}$ as said minimiser:

$\displaystyle\operatorname{\textup{Sup}}_{\varphi}(\mathcal{X}):=\operatorname*{arg\,min}_{\boldsymbol{Y}\in\mathcal{U}(\mathcal{X})}\varphi(\boldsymbol{Y}).$

The matrix supremum introduced in the paper [8] is based on the calculation of
the trace and is therefore also called trace-supremum. That is, one has
$\varphi(Y)=\tr Y$ and we get as supremum:

$\displaystyle\operatorname{\textup{Sup}}_{\tr}(\mathcal{X}):=\operatorname*{arg\,min}_{\boldsymbol{Y}\in\mathcal{U}(\mathcal{X})}\tr\boldsymbol{Y}.$

Based on the corresponding norms, the Frobenius supremum

$\displaystyle\operatorname{\textup{Sup}}_{2}(\mathcal{X}):=\operatorname*{arg\,min}_{\boldsymbol{Y}\in\mathcal{U}(\mathcal{X})}\sum_{\boldsymbol{X}\in\mathcal{X}}\norm{\boldsymbol{Y}-\boldsymbol{X}}_{2}$

and the spectral supremum

$\displaystyle\operatorname{\textup{Sup}}_{\infty}(\mathcal{X}):=\operatorname*{arg\,min}_{\boldsymbol{Y}\in\mathcal{U}(\mathcal{X})}\sum_{\boldsymbol{X}\in\mathcal{X}}\absolutevalue{\lambda_{1}(\boldsymbol{Y}-\boldsymbol{X})},$

where $\lambda_{1}(\boldsymbol{A})$ denotes the largest eigenvalue of
$\boldsymbol{A}$, were derived in [19]. At this point it should be noted that,
in the case of positive semidefinite matrices, all three norms are Schatten
norms $\norm{\,\cdot\,}_{p}$ for $p\in\\{1,2,\infty\\}$.

##  3 Mathematical Background of the Log-Exp-Supremum

In this section, we will give a brief introduction to the construction of a
characterisation for the so-called log-exp-supremum and summarise some of its
properties, but will refer to a future paper for the proofs.

Let the multi-set
$\mathcal{X}=\\{\boldsymbol{X}_{1},\dots,\boldsymbol{X}_{n}\\}$,
$n\in\mathbb{N}$, of symmetric real $2\times 2$ matrices represent our colour
values in the considered neighbourhood. Then, we define the log-exp-supremum
(LES) as

$\displaystyle\boldsymbol{S}:=\operatorname{\textup{Sup}}_{\text{LE}}(\mathcal{X}):=\lim_{m\rightarrow\infty}\left(\frac{1}{m}\log\sum_{i=1}^{n}\exp(m\boldsymbol{X}_{i})\right).$
(1)

For the calculation of the LES, we will use a characterisation of the form

$\displaystyle\boldsymbol{S}=\lambda_{1}\boldsymbol{u}_{1}\boldsymbol{u}_{1}^{\textup{T}}+\mu\boldsymbol{v}_{1}\boldsymbol{v}_{1}^{\textup{T}},$
(2)

where $\lambda_{1}$ is the largest eigenvalue of the matrices of $\mathcal{X}$
with the corresponding normalised eigenvector $\boldsymbol{u}_{1}$,
$\boldsymbol{v}_{1}$ is the normalised eigenvector perpendicular to
$\boldsymbol{u}_{1}$ and $\mu$ is the next largest eigenvalue of $\mathcal{X}$
with an eigenvector not aligned with $\boldsymbol{u}_{1}$ if $\lambda_{1}$ is
unique, otherwise $\mu=\lambda_{1}$ holds. At this point it should be noted
that $\mu$ does not have to be the corresponding eigenvalue to the eigenvector
$\boldsymbol{v}_{1}$. The characterisation is based on the spectral
decomposition of the input $\boldsymbol{X}_{i}$:

$\displaystyle\begin{split}&\boldsymbol{X}_{i}=\lambda_{i}\boldsymbol{u}_{i}\boldsymbol{u}_{i}^{\textup{T}}+\mu_{i}\boldsymbol{v}_{i}\boldsymbol{v}_{i}^{\textup{T}},\quad\lambda_{i}\geq\mu_{i},\quad\langle\boldsymbol{u}_{i},\boldsymbol{v}_{i}\rangle=0,\quad\absolutevalue{\boldsymbol{u}_{i}}=1=\absolutevalue{\boldsymbol{v}_{i}},\\\
&\boldsymbol{u}_{i}=(c_{i},s_{i})^{\textup{T}},\quad\boldsymbol{v}_{i}=(-s_{i},c_{i})^{\textup{T}},\quad
c_{i}=\cos(\varphi_{i}),\quad s_{i}=\sin(\varphi_{i}),\\\
&\varphi_{i}\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right],\quad
i=1,\dots,n,\end{split}$ (3)

where $\lambda_{i},\mu_{i}\in\mathbb{R}$ are the eigenvalues of
$\boldsymbol{X}_{i}$ and $\boldsymbol{u}_{i}\perp\boldsymbol{v}_{i}$ are the
unit eigenvectors to the corresponding eigenvalues. Additionally, we used for
the equation (2) that without loss of generality the matrix
$\boldsymbol{X}_{1}$ has (one of) the largest eigenvalues of $\mathcal{X}$.
The proof exploits the spectral decomposition (3) and certain properties of
the Rayleigh product and symmetric $2\times 2$ matrices. However, we will not
go into the proof of the LES characterisation (2) here, as it would strain the
scope of this paper too much. Instead, we intend to do this in a separate
paper and refer to the results of our experiments for now.

We now turn our attention to some of the properties of (1). First of all, we
need to clarify whether it is actually a supremum in the Loewner sense. In
fact, it is easy to show that it is an upper bound:

###### Lemma 1

The LES $\boldsymbol{S}$ according to (1) is an upper bound in the Loewner
sense for the given matrices $\mathcal{X}$.

###### Proof

Apparently applies

$\displaystyle\sum_{i=1}^{n}\exp(m\boldsymbol{X}_{i})\geq_{L}\exp(m\boldsymbol{X}_{j})\quad\forall
j\in\\{1,\dots,n\\},$

which in combination with the fact that the logarithm is an operator-monotone
function (see [12]), i.e.
$\boldsymbol{A}\leq_{L}\boldsymbol{B}\Longrightarrow\log\boldsymbol{A}\leq_{L}\log\boldsymbol{B}$
for symmetric positive definite matrices $\boldsymbol{A},\boldsymbol{B}$,
results in $\boldsymbol{S}\geq_{L}\boldsymbol{X}_{j}$ for all
$j\in\\{1,\dots,n\\}$. $\square$

We already mentioned in the previous section that it is necessary to have
another property to select a uniquely determined maximum from the convex set
$\mathcal{U}(\mathcal{X})$. This should be represented by a function
$\varphi$, which should have a unique minimiser in $\mathcal{U}(\mathcal{X})$.
Unfortunately, there is no such total ordering function $\varphi$ on
$\mathcal{U}(\mathcal{X})$ for which $\boldsymbol{S}$ is the unique minimiser.
However, by defining the $\boldsymbol{p}$-power upper bound cone as

$\displaystyle\mathcal{U}_{p}(\mathcal{X}):=\left(\mathcal{U}(\mathcal{X}^{p})\right)^{\frac{1}{p}}=\big{\\{}\boldsymbol{Y}\in\textup{Sym}(2):\boldsymbol{Y}^{p}\in\mathcal{U}(\mathcal{X}^{p})\big{\\}}$
(4)

for the element-wise application of the $p$-th power to the multi-set
$\mathcal{X}$ in form of
$\mathcal{X}^{p}:=\\{\boldsymbol{X}^{p}:\boldsymbol{X}\in\mathcal{X}\\}$, we
can slightly modify the problem to obtain a unique minimiser. To be exact, we
denote the intersection of all $p$-power upper bound cones as the super-upper
bound cone

$\displaystyle\mathcal{U}_{*}(\mathcal{X}):=\bigcap_{p>0}\mathcal{U}_{p}(\mathcal{X})$
(5)

and use this cone to find the minimiser instead of $\mathcal{U}(\mathcal{X})$.
In fact, following the reasoning in [8], one sees that
$\boldsymbol{S}\in\mathcal{U}_{p}(\mathcal{X})$ holds for any $p>0$, and
therefore $\boldsymbol{S}\in\mathcal{U}_{*}(\mathcal{X})$. By declaring the
function $\boldsymbol{\varphi}:\textup{Sym}(2)\rightarrow\mathbb{R}^{2}$ with
$\boldsymbol{\varphi}(\boldsymbol{A})=(\lambda_{1},\lambda_{2})$, where
$\lambda_{1},\lambda_{2}$ are the eigenvalues of
$\boldsymbol{A}\in\textup{Sym}(2)$, and endowing the $\mathbb{R}^{2}$ with the
lexicographic order

$\displaystyle(a,b)\prec(a^{\prime},b^{\prime}):\Longleftrightarrow(a<a^{\prime}\vee(a=a^{\prime}\wedge
b\leq b^{\prime}))$ (6)

one can show that the function $\boldsymbol{\varphi}$ is Loewner-monotone,
convex and $\boldsymbol{S}$ is its unique minimiser in
$\mathcal{U}_{*}(\mathcal{X})$. In addition, one can show the following lemma
for transitivity, which we will not do here, but refer to our experiments.

###### Lemma 2

The LES (1) is transitive, i.e. for multi-sets $\mathcal{X}$ and $\mathcal{Y}$
of symmetric $2\times 2$ matrices one has

$\displaystyle\operatorname{\textup{Sup}}_{\textup{LE}}(\mathcal{X}\cup\mathcal{Y})=\operatorname{\textup{Sup}}_{\textup{LE}}\big{(}\\{\operatorname{\textup{Sup}}_{\textup{LE}}(\mathcal{X}),\operatorname{\textup{Sup}}_{\textup{LE}}(\mathcal{Y})\\}\big{)}.$
(7)

##  4 Experiments

We divide this section into three parts with four experiments. First, we will
show that our new method yields correct results in the Loewner sense, as did
the previous Loewner methods. Then we will show how the approximations of
different morphological operations affect it. Our main focus will be on the
almost isotropic character and especially the transitivity compared to
previous approaches. For this, we will apply some elementary flat
morphological operations to synthetic and natural colour images of different
sizes. For the experiments we are using various structuring elements: a
$3\times 3$ square, a $5\times 5$ square and a $9\times 9$ square, all centred
on the middle pixel.

###  4.1 Correctness of the Loewner Order

For this we consider a simple $30\times 30$ image divided into blue (RGB =
(0,0,1)) and green (RGB = (0,1,0)) and perform a dilation and an erosion on it
with the $9\times 9$ SE, see Figure 1. To better distinguish the images, we
have placed a black frame around them. The white border in between does not
belong to the image.

Figure 1:  Comparison of different dilation and erosion methods of an
$30\times 30$ image with a $9\times 9$ SE. From left to right: Original image,
dilation with the new method, channel-wise dilation and erosion (new method or
channel-wise).

Since both colours are on opposite sides of the $x$-$y$-plane of the bi-cone
with respect to the $y$-coordinate, the maximum and minimum should be white
and black, respectively. The dilation with our approach correctly shows white
and the erosion black, as was to be expected from earlier Loewner approaches,
compare the work [7] of Burgeth and Kleefeld, since all colour channels are
considered simultaneously. In contrast, at least channel-wise dilation, i.e.
separate dilation for each colour channel, does not produce the desired white
according to the Loewner order, but a mixture of the blue and green tones.

###  4.2 Application of different Morphological Operations

We will use the $64\times 64$ pepper image and apply dilation, erosion,
closing and opening with the $3\times 3$ SE according to our approach, see
Figure 2. We immediately notice that most of the colours fade. As a result of
log-sum exponentiation, we basically only compare the leading eigenvalues with
different eigenvectors, and as $m$ increases, the leading terms become more
dominant, which results in higher eigenvalues and thus brighter colours. This
is especially true if the two largest eigenvalues in the neighbourhood under
consideration are already close to each other, which often happens with
smaller SEs and natural images. If the two largest eigenvalues $\lambda_{1}$
and $\lambda_{2}=\lambda_{1}+\varepsilon$ for a sufficiently small
$\varepsilon>0$ have different eigenvectors, we obtain with our usual notation

$\displaystyle\boldsymbol{S}=\lambda_{2}\boldsymbol{u}_{2}\boldsymbol{u}_{2}^{\textup{T}}+\lambda_{1}\boldsymbol{v}_{2}\boldsymbol{v}_{2}^{\textup{T}}=(\lambda_{1}+\varepsilon)\boldsymbol{u}_{2}\boldsymbol{u}_{2}^{\textup{T}}+\lambda_{1}\boldsymbol{v}_{2}\boldsymbol{v}_{2}^{\textup{T}}=\lambda_{1}\boldsymbol{I}+\varepsilon\boldsymbol{u}_{2}\boldsymbol{u}_{2}^{\textup{T}},$

where we set $\boldsymbol{u}_{2}=(1,0)^{\textup{T}}$ and
$\boldsymbol{v}_{2}=(0,1)^{\textup{T}}$. For example, the exact dilation in
Figure 2 has an average difference between the largest and second largest
eigenvalue of 0.0268 per pixel, while the largest and smallest eigenvalues of
the image are 0.5574 and $-0.7071$. To illustrate this further, we have
included in Figure 2 an approximate dilation with a scaling factor of $m=69$,
chosen because this is the largest factor that Matlab can compute without
error. The clear difference to the converged dilation is possibly due to two
reasons. Firstly, the factor is far from the limit value, but this cannot be
changed, at least with the current implementation in Matlab, for the reason
mentioned above. Secondly, it may be due to the fact that if the eigenvector
directions are close together, which is not unusual in natural images where
there are no fast or strong colour transitions, the convergence against the
converged matrix should be quite slow.

Figure 2:  Illustration of exact filtering results obtained with the new LES
method for different operations of a $64\times 64$ pepper image with a
$3\times 3$ SE. Top: From left to right: Original image, exact dilation and
exact erosion. Bottom: From left to right: Approximate dilation ($m=69$),
exact closing and exact opening.

###  4.3 Transitivity

For this experiment we again use the $64\times 64$ pepper image and apply a
dilation with the $5\times 5$ mask to it and compare the result with applying
a dilation with the $3\times 3$ mask twice for our method and Burgeth’s and
Kleefeld’s trace optimisation method, see Figure 3. By comparison we mean a
channel-wise comparison of the two images in the form of a difference from
which we take the absolute values. Our method should give the exact same image
due to transitivity and Burgeth’s and Kleefeld’s method [6] should give a
slightly different image.

Figure 3:  Comparison of differences in the channel-wise deviations between
the exact filtering results of the dilation with a $5\times 5$ SE and applying
a dilation with the $3\times 3$ SE twice using our new method and the method
of Burgeth and Kleefeld. Top: From left to right: Original image, dilation
with the $5\times 5$ SE by using our method and Burgeth’s and Kleefeld’s
method. Mid: From left to right: Differences in our method with absolute
values in the channels red, green and blue. Bottom: From left to right:
Differences in Burgeth’s and Kleefeld’s method with absolute values in the
channels red, green and blue.

In fact, with our method we do not see any differences in the colour channels.
In contrast, Burgeth’s and Kleefeld’s method shows clear deviations, which
occur across the entire image in all channels. In particular, the red colour
channel shows strong deviations of up to around 20 tonal values.

To examine this difference more closely, we look at how the error behaves when
we double the number of dilations. This means we compare a dilation with a
$9\times 9$ SE with a fourfold dilation with a $3\times 3$ SE, see Figure 4.
Because of the doubling of operations, we would expect the error for the
corresponding methods to double as well.

Figure 4:  Comparison of differences in the channel-wise deviations between
the exact filtering results of the dilation with a $9\times 9$ SE and fourfold
dilation with the $3\times 3$ SE using our new method and the method of
Burgeth and Kleefeld. Top: From left to right: Original image, dilation with
the $9\times 9$ SE by using our method and Burgeth’s and Kleefeld’s method.
Mid: From left to right: Differences in our method with absolute values in the
channels red, green and blue. Bottom: From left to right: Differences in
Burgeth’s and Kleefeld’s method with absolute values in the channels red,
green and blue.

We see a continuation of the previous behaviour in the individual channels,
which means that the transitivity of our method is preserved and that errors
still occur everywhere with Burgeth’s and Kleefeld’s method, but particularly
in the red colour channel. We see a further increase in the size of the
errors, but it is not clear whether the error has actually doubled.

To measure the error more accurately, we use the Frobenius norm, which we
apply to the channel-by-channel difference of the images, and then add this up
for the individual channels. In this way, we obtain the value 336.9095 for the
first case of a twofold dilation and the value 686.5281 for the second case of
a fourfold dilation, which is almost twice the error.

##  5 Conclusion and Future Work

In this work, we have used the colour morphology used by Burgeth and co-
authors, see [6, 7, 8], using Loewner order to introduce a new method for
determining a “maximum colour” in the sense of dilation. We applied the log-
sum exponentiation to approximate the supremum, which was introduced by Maslov
[13], to the symmetric $2\times 2$ matrices, which were created using a
bijective mapping from the HC$\tilde{\textup{L}}$ bi-cone. By using the
spectral decomposition of these matrices, we have managed to express the
result of the approximation using these spectral decompositions, which means
that it only depends on the input data. We conducted several experiments on
synthetic and natural images to demonstrate Loewner order compliance, how
different morphological operations affect our approach, and identified some
key factors that influence the approximation. In particular, we have shown
that our approach fulfils the property of transitivity, which is important for
lattices and orders, in contrast to Burgeth’s and Kleefeld’s method [7] or
other methods derived from it, cf. [19].

In future work, we will take a closer look at the construction briefly
introduced here and prove the properties presented. In this regard, we will
particularly examine the minimality of the approach in more detail and provide
a relaxation for the LES (1) so that it even depends continuously on the input
data.

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