# Structure, hydrolysis and diffusion of aqueous vanadium ions from Car-
Parrinello molecular dynamics

Zhen Jiang  Department of Chemical and Biomolecular Engineering, University of
Nebraska-Lincoln, Lincoln, NE 68588, USA  Konstantin Klyukin  Department of
Chemical and Biomolecular Engineering, University of Nebraska-Lincoln,
Lincoln, NE 68588, USA  Vitaly Alexandrov<EMAIL_ADDRESS>Department of
Chemical and Biomolecular Engineering, University of Nebraska-Lincoln,
Lincoln, NE 68588, USA  Nebraska Center for Materials and Nanoscience,
University of Nebraska-Lincoln, Lincoln, NE 68588, USA

###### Abstract

A molecular level understanding of the properties of electroactive vanadium
species in aqueous solution is crucial for enhancing the performance of
vanadium redox flow batteries (RFB). Here, we employ Car-Parrinello molecular
dynamics (CPMD) simulations based on density functional theory to investigate
the hydration structures, first hydrolysis reaction and diffusion of aqueous
V2+, V3+, VO2+, and VO${}_{2}^{+}$ ions at 300 K. The results indicate that
the first hydration shell of both V2+ and V3+ contains six water molecules,
while VO2+ is coordinated to five and VO${}_{2}^{+}$ to three water ligands.
The first acidity constants (p$K_{\mathrm{a}}$) estimated using metadynamics
simulations are 2.47, 3.06 and 5.38 for aqueous V3+, VO${}_{2}^{+}$ and VO2+,
respectively, while V2+ is predicted to be a fairly weak acid in aqueous
solution with a p$K_{\mathrm{a}}$ value of 6.22. We also show that the
presence of chloride ions in the first coordination sphere of the aqueous
VO${}_{2}^{+}$ ion has a significant impact on water hydrolysis leading to a
much higher p$K_{\mathrm{a}}$ value of 4.8. This should result in a lower
propensity of aqueous VO${}_{2}^{+}$ for oxide precipitation reaction in
agreement with experimental observations for chloride-based electrolyte
solutions. The computed diffusion coefficients of vanadium species in water at
room temperature are found to increase as V3+ $<$ VO${}_{2}^{+}$ $<$ VO2+ $<$
V2+ and thus correlate with the simulated hydrolysis constants, namely, the
higher the p$K_{\mathrm{a}}$ value, the greater the diffusion coefficient.

††preprint: AIP/123-QED

##  I INTRODUCTION

There is a pressing demand to substitute fossil fuels with renewable energy
sources such as solar and wind, but their intermittent and unpredictable
nature calls for the development of advanced electrical energy storage
technologies. In this regard, redox flow batteries (RFB) are one of the most
promising solutions for large-scale grid energy storage that offer a number of
advantages over their solid-state counterparts including the decoupling of
power and energy, rapid response times, long cycling lifespan and the
capability to store up to multimegawatt-hours of electrical energy.Weber  _et
al._ (2011); Alotto, Guarnieri, and Moro (2014); Soloveichik (2015)

Among various RFB, all-vanadium RFB offer some additional appealing properties
such as high electrochemical stability, reversibility and low cross-
contamination by crossover between electrode compartments.Skyllas-Kazacos  _et
al._ (1986); Kear, Shah, and Walsh (2012); Parasuraman  _et al._ (2013) The
operation of all-vanadium RFB is based on the electrochemical reactions of the
VO2+/VO${}_{2}^{+}$ redox couple in the catholyte and the V2+/V3+ in the
anolyte, and the typical supporting electrolyte is a sulfuric acid solution.
One of the major drawbacks of the current vanadium RFB systems is their low
specific energy density ($<$30 Wh/kg), as compared to other rechargeable
battery types such as lithium-ion batteries (up to 200 Wh/kg), at a narrow
window of operating temperatures (10-40 ∘C) due to various precipitation
reactions.Huang and Wang (2015) Increasing the energy density by simply making
a more concentrated vanadium solution results in poor electrolyte stability
and the formation of precipitates by all four vanadium cations that limits
their aqueous concentrationsDing  _et al._ (2013). For instance, the stability
of VO${}_{2}^{+}$ in sulfuric acid is limited by V2O5 precipitation at high
temperatures ($>$ 40 ∘C), and although the VO${}_{2}^{+}$ solubility can be
increased by adding more sulfuric acid to the solution, the increase in acid
concentration is limited by the stability of VO2+ species that are present in
the same electrolyte tank.Leung  _et al._ (2012) Overall, the presence of all
four electroactive vanadium species in the electrolyte is expected to lead to
a very rich aqueous chemistry with various hydrolysis, polymerization and
precipitation reactions, as demonstrated by the corresponding phase
diagrams.Greenwood and Earnshaw (1998) Their equilibria depend strongly on the
concentrations of both cationic and anionic species, pH, temperature and the
charge/discharge state of the flow battery.Skyllas-Kazacos, Peng, and Cheng
(1999) Such a complex interplay between various factors makes the design of
vanadium RFB electrolytes very challenging and requires a systematic approach
in studying elementary reactions involved in these processes.

In the past years, there have been great efforts to improve the energy density
of the vanadium RFB by modifying the chemistry of vanadium electrolyte
solutions. For example, it was demonstrated that the use of a mixed sulfate-
chloride electrolyte results in a 70% increase in energy capacity over a
broader operating temperature window as compared to the typical sulfate based
electrolyte.Li  _et al._ (2011) Through a combination of the nuclear magnetic
resonance (NMR) spectroscopy and density functional theory (DFT)
calculationsLi  _et al._ (2011); Vijayakumar  _et al._ (2011) it was shown
that this improvement is most likely due to the formation of soluble
VO2Cl(H2O)2 species that impedes V2O5 precipitation. However, the use of
halides in the electrolyte may lead to safety issues because of the formation
of chlorine and bromine vapors at elevated temperatures. Therefore, some other
studies focused on the use of safer organic and inorganic precipitation
inhibitors to stabilize vanadium solutions.Roe, Menictas, and Skyllas-Kazacos
(2016); Chang  _et al._ (2012)

One of the mechanisms underlying precipitation of metal oxides from the
aqueous phase involves deprotonation of the first-shell water molecules and
subsequent polymerization reactions via the formation of bridging oxygens or
hydroxyls between the metal ions. In the case of vanadium electrolytes, it is
believed that the formation of insoluble V2O5 occurs through this mechanism
from VO2(H2O)3+ species present in solution.Li  _et al._ (2011) Therefore, it
is indispensable to understand the thermodynamics and kinetics of the
hydrolysis of the vanadium aqua species.

The transport of vanadium ions in the electrolyte and across the membrane is
another key property of the vanadium RFB. The increase in concentration of
vanadium ions in the electrolyte should lead to saturation and undesirable
precipitation reactions. In addition, although the problem of cross-
contamination is substantially diminished in the case of all-vanadium RFB due
to the use of the same element for the redox-active couples, their transport
during long-term use can be significant. In experiments it is often difficult
to deconvolute the contributions from different vanadium species to the
overall ionic transport, especially due to various possible accompanying
chemical reactions between vanadium ions, anionic species in the electrolyte
and membrane constituents.Luo  _et al._ (2012); Knehr and Kumbur (2012) Thus,
it is important to examine transport properties and how they vary depending on
the solution chemistry under well controlled conditions.

In the past, there were a few first-principles studies aiming to investigate
the structural and thermodynamic properties of aqueous vanadium cations. While
some investigations were limited to modeling hydrated vanadium species in the
gas phase,Aakesson  _et al._ (1994); Vijayakumar  _et al._ (2010); Asmis and
Sauer (2007) very few took into account the influence of aqueous solution
under elevated temperatures.Sepehr and Paddison (2013, 2015) For example, the
hydration structure of all four vanadium cations (V2+, V3+, VO2+, and
VO${}_{2}^{+}$) was recently determined employing the gas-phase ab initio
calculations, whereas the thermodynamics of solvation was considered within
the quasi-chemical theory of solvation.Sepehr and Paddison (2013) This study
was able to provide basic information on the structural parameters of the most
stable hydrous vanadium cations at zero temperature in the absence of aqueous
solution. The effect of temperature and solvent on the structure and stability
of solvated VO${}_{2}^{+}$ was previously considered within the Car-Parrinello
molecular dynamics (CPMD) methodology.Sadoc  _et al._ (2007) Specifically, the
study demonstrated that the penta-coordinated VO2(H2O$)_{3}^{+}$ complex
should be more predominant than hexa-coordinated VO2(H2O)${}_{4}^{+}$ complex
in aqueous solution at room temperature, in agreement with the gas-phase
calculations.Sepehr and Paddison (2013) Another CPMD study has focused on the
evaluation of solvent effects on 51V NMR chemical shifts and also provided
some useful information on the VO${}_{2}^{+}$ speciation in aqueous solution
and water dynamics around the vanadium center, albeit at the short timescale
of 2 ps.Bühl and Parrinello (2001)

In this paper, we present a systematic CPMD-based investigation of the
hydration shell geometries, first hydrolysis constants (p$K_{\mathrm{a}}$) and
diffusion coefficients of all four electroactive vanadium cations (V2+, V3+,
VO2+, and VO${}_{2}^{+}$) in aqueous solution. The remainder of the paper is
organized as follows. First, the computational methodology and details of the
simulations are described. Second, the obtained results along with a
discussion of the atomic structure, water dynamics, first hydrolysis reaction
and diffusion of aqueous vanadium species are presented. At the end, main
conclusions of the work are provided.

##  II COMPUTATIONAL DETAILS

Car-Parrinello molecular dynamics (CPMD)Car and Parrinello (1985) simulations
were carried out for V2+, V3+, VO2+ and VO${}_{2}^{+}$ cations in aqueous
solution in the canonical ensemble at 300 K employing the pseudo-potential
plane-wave density functional theory (DFT) formalism as implemented in the
NWChem code. Valiev  _et al._ (2010) Each cation was placed in a cubic box of
length 13 Å subject to periodic boundary conditions filled with 74 H2O
molecules to obtain the water density of approximately 1 g$/$cm3. The charges
on the vanadium cations were neutralized by a uniform negative charge
background. Exchange and correlation were treated using the Perdew-Burke-
Ernzerhof (PBE) functionalPerdew, Burke, and Ernzerhof (1996) within the
generalized gradient approximation. To describe electron-ion interactions the
norm-conserving pseudopotentials modified into a separable form according to
Kleinman and Bylander were employed.Kleinman and Bylander (1982) Troullier-
Martins pseudo-potentialsTroullier and Martins (1991) were used for vanadium
and Hamann-type pseudo-potentialsHamann, Schlüter, and Chiang (1979); Hamann
(1989) were used for oxygen and hydrogen. The kinetic energy cuttoffs of 60
and 120 Ry were applied to expand the Kohn-Sham electronic wave functions and
charge density, respectively. Note that similar energy cutoffs were previously
employed in CPMD simulations with the same pseudopotentials to examine
hydration structures and stability of various cationic species in aqueous
solutions,Vuilleumier and Sprik (2001); Sadoc  _et al._ (2007); Nichols  _et
al._ (2008) as well as for estimating p$K_{\mathrm{a}}$ values from ab initio
metadynamics.Tummanapelli and Vasudevan (2014) The Brillouin zone was sampled
with the ${\Gamma}$ point in all simulations. The Nose-Hoover thermostatNosé
(1984); Hoover (1985) was used to maintain the system at a temperature of 300
K and the hydrogen atoms were replaced with deuterium to facilitate numerical
integration. A fictitious electronic mass of 600 au and a simulation time step
of ${\delta t}$ = 5 au (0.121 fs) were set. Each system was initially
equilibrated during 3.6 ps using a QM/MM potential,Cauët  _et al._ (2010)
followed by a subsequent CPMD equilibration for 6 ps. The total production
times were 21 ps for aqueous V2+, V3+, VO2+ and 30 ps for aqueous
VO${}_{2}^{+}$. The collection time was longer in the case of VO${}_{2}^{+}$
because the structure of its first hydration shell changed at about 6.5 ps
during simulation, which will be discussed below. Configurations from the
post-equilibration CPMD simulations used for further analysis were saved at
time intervals of 10${\delta}$t.

To compute the free energy profiles of hydrolysis reaction ($\Delta F$) at 300
K for all four aqueous vanadium cations we employed ab initio metadynamics
technique.Micheletti, Laio, and Parrinello (2004); Laio and Gervasio (2008) To
describe the H transfer from the water molecules in the first coordination
sphere of vanadium ions we selected the following function
$\xi(|\mathbf{r}_{\mathrm{OH}}|)$ for the coordination number as the
collective variable (CV):

$\xi(|\mathbf{r}_{\mathrm{OH}}|)=\sum^{N_{H}}_{i=1}\frac{1}{1+\exp[k(|\mathbf{r}_{\mathrm{OH_{i}}}|-r_{\mathrm{cut}})]}$
(1)

Here, $N_{H}$ is the total number of protons, $k$ an arbitrary positive
constant used to reproduce the equilibrium coordination number (set to 10 Å-1)
and $r_{\mathrm{cut}}$ the O$-$H cutoff distance (set to 1.38 Å). This choice
of the collective variable to describe the first deprotonation reaction was
previously shown to provide good results for other metal ions in aqueous
solutions.Odoh, Bylaska, and De Jong (2013) Metadynamics simulations at 300 K
were performed using equilibrated geometries of aqueous vanadium ions picked
from the last block of the CPMD production runs. The height and width of the
repulsive Gaussian hills were set to 0.0001 au (0.063 kcal/mol) and 0.07,
respectively. Gaussian hills were added to the potential every 100$\delta$t
for V3+(aq) and VO${}_{2}^{+}$(aq) and 50$\delta$t for V2+(aq) and VO2+(aq).
The time interval was shorter for V2+(aq) and VO2+(aq) since their
deprotonation energies are expected to be greater than those for V3+(aq) and
VO${}_{2}^{+}$(aq). Once the free energy profiles are simulated, the first
hydrolysis constant (p$K_{\mathrm{a}}$) can be computed by using the standard
relation $\mathrm{p}K_{\mathrm{a}}=\Delta F/2.303RT$. A more detailed
discussion of the computational parameters employed in metadynamics
simulations can be found in the Supporting Material.

##  III RESULTS AND DISCUSSION

###  III.1 Hydration Shell Geometry.

Figure 1: Partial V$-$O radial distribution function (RDF) gV-O(r) and running coordination number nV-O(r) for aqueous V2+ (left panel) and V3+ (right panel). Table 1: Average distances R (in Å) and angles O-V-O (in degrees) for the first hydration shell of V2+ and V3+ in water and their comparison with previous gas-phase calculations and experimental data. RV-O stands for the distance between vanadium ions and oxygen atoms of the first-shell water molecules, RO-H corresponds to the O$-$H bond length of water ligands, OeqVOax and OaxVOax are bond angles where ”ax” denotes an axial and ”eq” equatorial first-shell water molecule (see Figure 2). | Method |  aN${}_{H_{2}O}$ | RV-O | RO-H |  ${\angle}$OeqVOax,eq |  ${\angle}$OaxVOax
---|---|---|---|---|---|---
V2+ | this work | 6 | 2.23 | 0.97 | 89.75 | 170.91
|  bB3LYP$/$6-311+G∗∗Sepehr and Paddison (2013) | 6 | 2.21 | - | - | -
| HF$/$cECPAakesson  _et al._ (1994) | 6 | 2.20 | - | - | -
|  dXRDMontgomery  _et al._ (1967) | 6 | 2.15 | - | 90.70 | -
|  eXRDCotton  _et al._ (1986) | 6 | 2.13 | - | 89.98 | 179.18
V3+ | this work | 6 | 2.10 | 0.99 | 90.03 | 171.98
| B3LYP$/$6-311+G∗∗Sepehr and Paddison (2013) | 6 | 2.09 | - | - | -
| HF$/$ECPAakesson  _et al._ (1994) | 6 | 2.06 | - | - | -
|  fXRDCotton  _et al._ (1984) | 6 | 2.00 | 1.00 | 91.74 | -
Figure 2: Snapshot from the CPMD trajectory of the first coordination sphere
of aqueous V2+ and V3+.

V2+(aq) and V3+(aq). We start by analyzing the atomic structure of vanadium
ions in aqueous solution based on our CPMD results and Figure 1 shows the
radial distribution function (RDF) and running coordination numbers of aqueous
V2+ and V3+ ions. For both ions two distinct peaks in the RDF can be
identified. The well-defined first peak is in the 2.0 $-$ 2.5 Å range for V2+
and in the 1.9 $-$ 2.3 Å range for V3+, corresponding to the first hydration
shell. It is seen from the running coordination numbers that the first
hydration shell of both V2+ and V3+ has six water molecules, in agreement with
previous experimentalVan Eldik  _et al._ (1992) and ab initio studies.Sepehr
and Paddison (2013); Aakesson  _et al._ (1994) The second shell peak is
characterized by a broader distribution between 3.7 and 5.1 Å with an average
V$-$O distance of 4.3 Å for V2+(aq) and 4.2 Å for V3+(aq). Table LABEL:tab1
lists the average structural parameters for the first hydration shell along
with the data from previous studies, while Figure 2 serves as a visual
representation of the first shell structure for both V2+ and V3+ ions. The
first shell is characterized by rather regular octahedral geometry around the
vanadium centers with four water molecules in equatorial plane and two water
molecules in axial direction. As seen from Table LABEL:tab1, the average V$-$O
distance is by about 0.1 Å shorter for V3+(aq) reflecting its larger ionic
size and stronger electrostatic attraction as compared to V2+. The
corresponding ${\angle}$OVO bond angles are found to be similar for V2+ and
V3+. Overall, both the V$-$O bond lengths and ${\angle}$OVO bond angles are
very similar to those obtained in previous computational studiesAakesson  _et
al._ (1994); Sepehr and Paddison (2013) and agree well with available
experimental data.Montgomery  _et al._ (1967); Cotton  _et al._ (1986, 1984)

Figure 3: Partial V$-$O radial distribution function (RDF) gV-O(r) and running coordination number nV-O(r) for aqueous VO2+. Table 2: The same as in Table LABEL:tab1 for aqueous VO2+ (see Figure 4 for visual representation). | Method | N${}_{H_{2}O}$ | R${}_{V-O_{ax}}$ |  aR${}_{V-O_{ax}^{w}}$ | R${}_{V-O_{eq}}$ |  ${\angle}$OaxVOeq |  ${\angle}$OaxVO${}_{ax}^{w}$
---|---|---|---|---|---|---|---
VO2+ | this work | $5$ | $1.72$ | $2.24$ | $2.11$ | $97.40$ | $169.61$
| B3LYP$/$6-311+G∗∗Sepehr and Paddison (2013) | $5$ | 1.57 | 2.36 | 2.12 | - | -
|  bDFT/QZ4PVijayakumar  _et al._ (2010) | $5$ | $1.57$ | 2.36 | 2.12 | - | -
|  cXRDBallhausen, Djurinskij, and Watson (1968) | $5$ | $1.59$ | 2.22 | $2.04$ | 98.00 | 174.20
|  dLAXSKrakowiak, Lundberg, and Persson (2012) | $5$ | $1.62$ | $2.20$ | $2.03$ | - | -
Figure 4: Snapshot from the CPMD trajectory of the first coordination sphere
of aqueous VO2+.

VO2+(aq). The RDF and running coordination number of VO2+(aq) are shown in
Figure 3. The first lower peak in the RDF curve is in the range of 1.6$-$1.8 Å
and corresponds to the O atom of the VO2+ cation, while the second taller peak
between 1.8$-$2.4 Å is associated with the oxygen atoms from the first
hydration shell. Therefore, based on the nV-O(r) curve there are five water
molecules in the first hydration shell of VO2+(aq). The average first-shell
structural parameters along with some literature data are listed in Table
LABEL:tab2, while the visual representation of the first hydration geometry
with the corresponding definitions is given in Figure 4. The average bond
length between vanadium and oxygen atom of the VO2+ cation (the maximum of the
first peak in the RDF at 1.72 Å) is by about 0.5 and 0.4 Å shorter than the
bond distances between V2+ and V3+ ions and the oxygen from the first-shell
water molecules, respectively. The geometry of the aqua-VO2+ complex is a
slightly more distorted octahedron than the ones observed for V2+ and V3+ with
a larger average ${\angle}$OaxVOeq of 97.4∘.

The longest bond distance of about 2.24 Å is found to be between the vanadium
and axial water oxygen (R${}_{V-O^{w}_{ax}}$). The average V$-$O bond length
in the equatorial plane (R${}_{V-O_{eq}}$) is 2.11 Å being very close to that
of V3+(aq). The values of the V$-$Oweq and V$-$Oeq bond distances reflect the
stronger attraction force of VO2+ to the five coordinated water molecules than
that of V2+, whereas it is weaker than for V3+.

Figure 5: CPMD trajectory snapshots of the first coordination sphere of
aqueous VO2(H2O)${}_{4}^{+}$ (left panel) and VO2(H2O)${}_{3}^{+}$ (right
panel).

Figure 6: Partial V$-$O radial distribution function (RDF) gV-O(r) and running
coordination number nV-O(r) for two stable states of VO2+(aq):
VO2(H2O)${}_{4}^{+}$ (left panel) and VO2(H2O)${}_{3}^{+}$ (right panel) which
geometries are shown in Figure 5. Figure 7: Variation of the bond distance
between V and Oesc, where Oesc atom belongs to the labile water molecule
escaping from the first to the second shell of VO2(H2O)${}_{4}^{+}$(aq) during
CPMD simulation. Figure 8: Free energy profile of the reaction
VO2(H2O)${}_{4}^{+}$ ${\rightleftharpoons}$ VO2(H2O)${}_{3}^{+}$ \+ H2O in
aqueous solution, where $\xi=1$ corresponds to the reactant
VO2(H2O)${}_{4}^{+}$ and $\xi=0$ to the product VO2(H2O)${}_{3}^{+}$ \+ H2O.

VO${}_{2}^{+}$(aq) In the case of VO${}_{2}^{+}$ ion we observe two stable
hydration shell structures during CPMD simulation (Figure 5). The starting
geometry corresponds to VO2(H2O)${}_{4}^{+}$ complex with four water ligands
in the first shell as seen from the RDF and running coordination number (see
left panel of Figure 6). After approximately 6.5 ps one water molecule is
expelled from the first to the second coordination sphere of VO${}_{2}^{+}$ as
shown by the variation of the corresponding V$-$O distance as a function of
time in Figure 7 to form a more stable trigonal bipyramidal
VO2(H2O)${}_{3}^{+}$ complex. Previously, the trigonal bipyramidal structure
was estimated to be about 8.6 kcal$/$mol more stable than the hexagonal aqua
complex at zero temperature using the gas-phase calculations at the
B3LYP$/$6-311+G∗∗ level.Sepehr and Paddison (2013) To get some insight into
the energetics of this chemical transformation between two complexes in
solution at 300 K, we use metadynamics simulations to find that the
VO2(H2O)${}_{3}^{+}$ species is indeed by about 10.6 kcal/mol more stable with
the energy barrier of its formation from VO2(H2O)${}_{4}^{+}$ of only 3.2
kcal/mol (Figure 8). Thus, our results support that the tri-aqua
VO${}_{2}^{+}$ complex should be dominant VO${}_{2}^{+}$ species in the
aqueous phase at 300 K.

Table 3: The same as in Table LABEL:tab1 for two stable aqua complexes of VO${}_{2}^{+}$ (see Figure 5 for visual representations). | Methodref. | N${}_{H_{2}O}$ | R${}_{V-O_{ax}}$ | R${}_{V-O_{ax}^{w}}$ | R${}_{V-O_{eq}}$ |  ${\angle}$OaxVOax |  ${\angle}$OaxVOeq |  ${\angle}$OaxVO${}_{ax}^{w}$
---|---|---|---|---|---|---|---|---
VO2(H2O)${}_{4}^{+}$ | this work | 4 | $1.73$ | $2.27$ | $2.08$ | $105.54$ | $93.05$ | $162.41$
| B3LYP$/$6-311+G∗∗Sepehr and Paddison (2013) | 4 | $1.61$ | $2.32$ | $2.10$ | - | - | -
|  aBP86/6-31G∗/all-electronBühl and Parrinello (2001) | 4 | $1.61$ | $2.28$ | $2.13$ | - | - | -
|  bLAXSKrakowiak, Lundberg, and Persson (2012) | 4 | $1.63$ | $2.23$ | $2.01$ | - | - | -
VO2(H2O)${}_{3}^{+}$ | this work | 3 | $1.72$ | $2.12$ | $2.05$ | $108.77$ | $97.96$ |  c143.30, 106.31
| B3LYP$/$6-311+G∗∗Sepehr and Paddison (2013) | 3 | $1.60$ | $2.11$ | - | - | - | -
| BP86/6-31G∗/all-electronBühl and Parrinello (2001) | 3 | $1.61$ | $2.10$ | $2.13$ | - | - | -
|  dB3LYP/QZ4PVijayakumar  _et al._ (2011) | 3 | $1.64$ | $2.13$ | $2.10$ | - | - | -

Table LABEL:tab3 lists the average structural parameters for
VO2(H2O)${}_{4}^{+}$ and VO2(H2O)${}_{3}^{+}$ along with the available data
from previous studies, while Figure 5 provides representative snapshots from
the CPMD trajectories for the two first-shell geometries. We find that the
average values of R${}_{V-O_{eq}}$ and R${}_{V-O_{ax}}$ for
VO2(H2O)${}_{4}^{+}$ are effectively the same as those of
VO2(H2O)${}_{3}^{+}$. As for the ${\angle}$OVO bond angles in
VO2(H2O)${}_{3}^{+}$, two different values (143.30∘ and 106.31∘) are reported
to show the average ${\angle}$OaxVO${}_{ax}^{w}$ corresponding to the Oax atom
above and below the paper plane, respectively. Thus, combining these two
values with 108.77∘ of ${\angle}$OaxVOax, the sum is equal to 358.38∘
indicating that all three axial water oxygen atoms and central vanadium cation
shown in Figure 5 are in the same plane. The average ${\angle}$OaxVOeq is
equal to 97.96∘ showing that the VO2(H2O)${}_{3}^{+}$ structure is not a
perfect trigonal bipyramidal.

###  III.2 Metadynamics Simulations of First Acidity Constant

Understanding the hydrolysis reactions of vanadium species in the aqueous
phase is of great significance for improving the stability of supporting
electrolytes in vanadium RFB. For example, it is believed that the solvated
VO${}_{2}^{+}$ can be converted to insoluble V2O5 according to the following
deprotonation and condensation reactions:Vijayakumar  _et al._ (2011)

$\mathrm{[VO_{2}(H_{2}O)_{3}]^{+}{\rightleftharpoons}VO(OH)_{3}+H_{3}O^{+}}$
(2) $\mathrm{2VO(OH)_{3}{\rightleftharpoons}V_{2}O_{5}\cdot 3H_{2}O}$ (3)

Depending on solution chemistry, pH and thermodynamic conditions, the
deprotonated species can form new vanadium complexes and precipitates that
will limit stability and redox characteristics of vanadium ions in aqueous
solution.Li  _et al._ (2011) Therefore, it is important to understand the
propensity of different vanadium species for deprotonation under ambient
conditions. Here, we analyze the mechanism and kinetics of the first
hydrolysis reaction across all four vanadium species using the metadynamics
simulations. For example, the first deprotonation reaction of [VO2(H2O)3]+
complex in water can be described as:

$\mathrm{[VO_{2}(H_{2}O)_{3}]^{+}+H_{2}O{\rightleftharpoons}VO_{2}(H_{2}O)_{2}(OH)+H_{3}O^{+}}$
(4)

First of all, we note that the proton transfer from the first to the second
hydration shell did not result in any significant transformation of the
vanadium complexes except some reasonable changes in interatomic distances and
angles. The vanadium ions did not change their coordination number as no water
molecule departed from the first shell during hydrolysis. It is expected that
the higher charged V3+(aq) will undergo the hydrolysis reaction easier being a
much stronger acid in aqueous solution than other vanadium species. The
measured p$K_{\mathrm{a}}$ value of V3+(aq) is indeed smaller than those of
the other cations and our calculations support this observation (see Table 4).

Figure 9: Free energy profiles of the first deprotonation reaction for aqueous V2+, V3+, VO2+ and VO${}_{2}^{+}$. $\xi$ $=$ 1 corresponds to the product and $\xi$ $=$ 2 to the reactant state. Table 4: Entropy corrected free energies ${\Delta}$F (in kcal/mol) and p$K_{\mathrm{a}}$ values of aqueous V2+, V3+, VO2+ and VO${}_{2}^{+}$. |  ${\Delta}$F | p$K_{\mathrm{a}}$ | p$K_{\mathrm{a}}$ (exp.)
---|---|---|---
V2+ | $8.54$ | $6.22$ | -
V3+ | $3.39$ | $2.47$ | 2.8 Walker, Newman, and Enache (2012), $2.9$Yatsimirskir and Vailèv (1960); McCleverty and Meyer (2004)
VO2+ | $7.38$ | $5.38$ | 5.35Francavilla and Chasteen (1975), 5.66Henry, Mitchell, and Prue (1973), 5.67McCleverty and Meyer (2004), 6.0Rossotti (1955)
VO${}_{2}^{+}$ | $4.20$ | $3.06$ | 3.2McCleverty and Meyer (2004); Yuchi, Yamada, and Tanaka (1979)
Figure 10: Free energy profiles of the first deprotonation reaction for
aqueous VO2Cl(H2O)2 versus [VO2(H2O)3]+ species.

Figure 9 shows the simulated free energy profiles of the deprotonation
reaction as a function of the coordination number $\xi$ of a specific first-
shell water oxygen atom, as defined by Equation 1. In this figure $\xi\approx
2$ corresponds to the reactant, taken as the zero reference point, and
$\xi\approx 1$ corresponds to the product state with one proton departed from
the first-shell water molecule to form a hydroxyl attached to a vanadium ion.
Thus, the free energy difference between these two states can be directly
related to p$K_{\mathrm{a}}$ (see Table 4). It is seen that V3+(aq) is the
strongest acid with $\Delta F=F(\xi=1.01)-F(\xi=1.95)=4.46$ kcal/mol. Since
any of the six water molecules from the first shell could participate in the
hydrolysis reaction an entropic energy correction of $-TS=-300k_{b}\ln
6=-1.07$ kcal/mol must be added to the free energy. This gives the final
estimate of $\Delta$F to be 3.39 kcal/mol yielding a p$K_{\mathrm{a}}$ value
of 2.47. The VO${}_{2}^{+}$(aq) is characterized by a slightly larger
$\Delta$F value of 4.86 kcal/mol. Since we found that the VO${}_{2}^{+}$ ion
prefers to be coordinated with three water molecules, it has the smallest
entropic correction to the free energy among all vanadium ions of only
$-k_{b}T\ln 3$ = -0.66 kcal/mol resulting in a p$K_{\mathrm{a}}$ value of
3.06. The p$K_{\mathrm{a}}$ value of VO2+(aq) is calculated to be 5.38, in
very good agreement with the 5.3$-$6.0 range in experimental
values.Francavilla and Chasteen (1975); Rossotti (1955); Henry, Mitchell, and
Prue (1973); McCleverty and Meyer (2004) The calculated value is larger than
that of the V3+ due to the presence of oxygen ion in VO2+(aq) that makes the
effective charge on the vanadium center smaller thus strengthening the O$-$H
bond of the dissociating water molecule. The p$K_{\mathrm{a}}$ value of
V2+(aq) is predicted to be the largest being 6.22. The less effective charge
transfer from the first-shell water molecule to the V2+ ion should make the
release of proton less favorable, which is consistent with the longest average
bond length between vanadium and oxygen atom of the first-shell water molecule
observed for V2+ among all vanadium ions. The low propensity of V2+(aq) for
hydrolysis agrees well with the experimental observations of good stability of
V2+ at high V2+ concentrations.Li  _et al._ (2011) Overall, our estimated
p$K_{\mathrm{a}}$ values are a little smaller than the experimental values
showing a reasonable agreement given the approximations made in our
simulations.

We next examine the influence of chloride ions from the electrolyte solution
on the deprotonation reaction of aqueous VO${}_{2}^{+}$. It was previously
demonstrated experimentally that the use of an aqueous electrolyte containing
a mixture of H2SO4 and HCl acids can considerably inhibit the rate of V(V)
precipitation reaction leading to a higher thermal stability of V(V) species
in mixed-acid electrolyte solutions.Li  _et al._ (2011) This effect was
attributed to the presence of chloride ions in the first shell of the aqueous
VO${}_{2}^{+}$ ion that should diminish the water deprotonation reaction. A
recent ab initio study has corroborated that the probability of the formation
of VO2Cl(H2O)2 species is very high and they are energetically favored over
[VO2(H2O)3]+ complexes in a mixed chloride-sulfate solution.Bon  _et al._
(2016) To quantitatively estimate the propensity of the VO2Cl(H2O)2 complex
for deprotonation reaction, we perform metadynamics simulations of the first
acidity constant of an aqueous solution containing VO2Cl(H2O)2 species. To
this end, we first carry out additional 2.4 ps CPMD equilibration starting
from the last CPMD block for VO${}_{2}^{+}$(aq) with one water molecule
substituted by a chloride ion. Figure 10 shows the free energy profiles for
the chloride-free and chloride-containing species and the pKa value of 4.8
estimated for VO2Cl(H2O)2 is indeed much greater than that of [VO2(H2O)3]+
(3.06) thus supporting experimental findings for chloride-containing
electrolyte solutions.Li  _et al._ (2011)

###  III.3 Diffusion Coefficients

The vanadium ion migration is central to the operation of vanadium RFB. It is
believed that the crossover of vanadium ions across the membrane may play a
considerable role in causing the capacity loss over long-term charge-discharge
cycling.Sun  _et al._ (2010) Moreover, the increase in concentration of
vanadium ions due to their net transfer may lead to undesirable precipitation
reactions. Therefore, it is of great importance to provide detailed vanadium
ion transport information. Here, we estimate the diffusion coefficients of all
four vanadium ions in water based on the previously generated CPMD
trajectories. Diffusion coefficients are calculated from a linear fit to the
mean square displacement over time according to the relation

$6Dt=<|\mathbf{r}(t)-\mathbf{r}(0)|^{2}>$ (5)

where $\mathbf{r}$($t$) is the position of the vanadium center at time $t$ and
$D$ is the diffusion coefficient. The time interval used for the fit was 20 ps
for each ion, but we note that in the case of VO${}_{2}^{+}$ a more distant
time interval (starting from $t$ = 7.5 ps) was chosen as it corresponds to the
more energetically favorable complex with three water ligands in the first
coordination shell. We also note that our evaluation of the diffusion
coefficients serves only as a very approximate measure due to short simulation
times, however, similar time scales were previously used to estimate diffusion
coefficients at room temperature.Allesch  _et al._ (2004) The details of the
fits along with the corresponding plots are provided in the Supporting
Material, while the computed diffusion coefficients are presented in Table 5
along with experimental estimates.

Table 5: Calculated diffusion coefficients along with available experimental data for aqueous V2+, V3+, VO2+ and VO${}_{2}^{+}$ (in cm${}^{2}/$s$\times$10-6). | V2+ | V3+ | VO2+ | VO${}_{2}^{+}$
---|---|---|---|---
this work | 6.85 | 0.95 | 4.22 | 4.03
aexp.Kim, Lee, and Park (2001) | 5.89 | - | - | -
bexp.Gattrell  _et al._ (2004) | - | - | (3.00 ${\pm}$ 0.30) | 2.00
cexp.Oriji, Katayama, and Miura (2005) | 1.10 | 0.57 | 1.00 | 1.00

As it is seen from Table 5, our calculated diffusion coefficients for the four
vanadium species in water follow the trend V3+ $<$ VO${}_{2}^{+}$ $<$ VO2+ $<$
V2+. The available experimental values are rather scattered due to various
experimental conditions and techniques used to estimate the diffusion
coefficients. It is interesting to note that here we observe the same trend as
we found for the first deprotonation reaction (p$K_{\mathrm{a}}$) indicating
that the strongest acid, V3+(aq), exhibits the slowest diffusion, whereas the
weakest acid, V2+(aq), is characterized by the most rapid diffusion. This
correlation can be rationalized in terms of the weak interaction between V2+
and the surrounding water molecules leading to the largest p$K_{\mathrm{a}}$
value among all vanadium species and, as a result, to the fastest diffusion.

##  IV CONCLUSIONS

Properties of the V2+/V3+ and VO2+/VO${}_{2}^{+}$ redox couples play a central
role in all-vanadium redox flow batteries. Here, we employed Car-Parrinello
molecular dynamics simulations to investigate the atomic structure, first
hydrolysis reaction and diffusion of all four vanadium ions in water at 300 K.
Our simulations show that both V2+ and V3+ cations exist as stable hexa-aqua
octahedral complexes in solution, whereas VO2+ is found to have five and
VO${}_{2}^{+}$ three water ligands in the first hydration shell. The
octahedral VO2(H2O)${}_{4}^{+}$ complex was found to be by about 10.6 kcal/mol
less stable than the trigonal bipyramidal VO2(H2O)${}_{3}^{+}$. The average
distance between the vanadium ion and the oxygen atom from the first-shell
water molecules RV-O changes as V3+(aq) $\approx$ VO${}_{2}^{+}$(aq) $<$
VO2+(aq) $<$ V2+(aq).

The free energy profiles of the first hydrolysis reaction obtained from Car-
Parrinello metadynamics simulations were used to estimate the first acidity
constant (p$K_{\mathrm{a}}$) across all aqueous vanadium species. During the
proton transfer from a coordinated water molecule to a second-shell water
molecule all water molecules remain in the first coordination sphere of the
vanadium center. Our simulated p$K_{\mathrm{a}}$ values are found to be 2.47,
3.06 and 5.38 for aqueous V3+, VO2+ and VO${}_{2}^{+}$, respectively, being in
good agreement with available experimental data. The aqueous V2+ is predicted
to be a fairly weak acid with a p$K_{\mathrm{a}}$ value of 6.22. The high
p$K_{\mathrm{a}}$ value found for V2+(aq) reflects its low propensity for
hydrolysis and precipitation reactions and this is consistent with
experimental observations of good stability of V2+ at high V2+ concentrations.
We also show that the complexation of aqueous VO${}_{2}^{+}$ with a chloride
ion leads to a much higher p$K_{\mathrm{a}}$ value of 4.8 and thus should
substantially inhibit the water deprotonation reaction and precipitation of
the V(V) oxide in full agreement with experimental findings for vanadium ions
in mixed sulfate-chloride solutions.

The computed diffusion coefficients of V3+, VO${}_{2}^{+}$, VO2+ and V2+ in
water at 300 K are 0.95, 4.03, 4.22 and 6.85 $\times$ 10-6 cm2/s,
respectively, and thus follow the same trend as the first acidity constants,
V3+(aq) $<$ VO${}_{2}^{+}$(aq) $<$ VO2+(aq) $<$ V2+(aq), with the strongest
acid exhibiting slowest diffusion.

##  V Supplementary Material

See supplementary material for the details on ab initio metadynamics
simulations including the choice of the number of Gaussian functions used to
obtain free-energy profiles of the deprotonation reaction (Section I). Section
II provides the details on the estimation of the diffusion coefficients along
with the plots of the mean square displacement as a function of time (Figure
S6).

###### Acknowledgements.

V.A. gratefully acknowledges financial support from the startup package and
the Layman seed grant provided by the University of Nebraska Foundation. The
Holland Computing Center at the University of Nebraska-Lincoln is acknowledged
for providing computational resources.

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# A Pixel-Level Meta-Learner
for Weakly Supervised Few-Shot Semantic Segmentation

Yuan-Hao Lee Fu-En Yang Yu-Chiang Frank Wang
Graduate Institute of Communication Engineering, National Taiwan University,
Taiwan, R.O.C.
ASUS Intelligent Cloud Services, Taiwan, R.O.C.
{r07942074, f07942077<EMAIL_ADDRESS>
###### Abstract

Few-shot semantic segmentation addresses the learning task in which only few
images with ground truth pixel-level labels are available for the novel
classes of interest. One is typically required to collect a large mount of
data (i.e., base classes) with such ground truth information, followed by
meta-learning strategies to address the above learning task. When only image-
level semantic labels can be observed during both training and testing, it is
considered as an even more challenging task of weakly supervised few-shot
semantic segmentation. To address this problem, we propose a novel meta-
learning framework, which predicts pseudo pixel-level segmentation masks from
a limited amount of data and their semantic labels. More importantly, our
learning scheme further exploits the produced pixel-level information for
query image inputs with segmentation guarantees. Thus, our proposed learning
model can be viewed as a pixel-level meta-learner. Through extensive
experiments on benchmark datasets, we show that our model achieves
satisfactory performances under fully supervised settings, yet performs
favorably against state-of-the-art methods under weakly supervised settings.

##  1 Introduction

Recent advances in deep convolutional neural networks (CNNs) [19] have
significantly improved the performances of several computer vision tasks.
Among them, semantic segmentation aims at predicting class labels for each
pixel in an image, with applications ranging from autonomous driving to
medical imaging. With the help of CNNs, state-of-the-art semantic segmentation
models including FCN [25], SegNet [1] and DeepLabs [5, 6, 7, 8] have all
achieved very promising results and been successfully applied to the above
applications. However, these models are generally trained in a fully
supervised manner, requiring a huge amount of training data with pixel-level
annotation for each category of interest. This substantially limits the
scalability and practicality of these models, as collecting densely labeled
data would be very time-consuming.

Figure 1: Illustration of weakly supervised few-shot segmentation. With only
semantic labels but not pixel-level masks observed during few-shot training
and testing, semantic segmentation of particular image categories can be
achieved. Note that a 2-way 3-shot scheme is depicted.

Extended from the learning task of semantic segmentation and few-shot
learning, _few-shot semantic segmentation_ considers a more challenging
setting in which only a few images are with ground-truth pixel-level labels
for the (novel) classes of interest. In order to realize the learning of few-
shot segmentation models, existing methods typically adopt meta-learning
schemes [13, 30, 34], utilizing support and query images sampled from base
categories (i.e., those with a sufficient amount of training data) for
performing pixel-wise classification. Recent methods like AMP [38], PANet
[46], FWB [27], PFENet [42] and ASGNet [20] choose to extract prototypes from
support-set images using their ground truth masks, and expect such prototypes
to sufficiently describe the associated semantic category. Nevertheless, these
existing methods require pixel-level ground truth labels for each image of the
base categories, and cannot be easily extended to weakly supervised settings.

In order to alleviate the requirement of annotating pixel-level ground truth
label information, we consider an even more challenging yet practical setting
of weakly supervised few-shot segmentation, which requires only image-level
labels collected for images in both base and novel categories, as depicted in
Figure 1. As shown in the figure, we aim at inferring pixel-level labels from
image-level labels in few-shot settings, followed by a meta-learning scheme
enforced at pixel-level for segmentation purposes.

With this goal in mind, we propose a novel learning scheme in this paper,
focusing on deriving a pixel-level meta learner for weakly supervised few-shot
semantic segmentation. During the meta-training stage, our proposed learning
framework observes only image-level labels and utilizes Classification
Activation Maps (CAM) [59, 35] for prediction of pseudo pixel-level labels for
each input image. While the class label embedding [2, 26] is exploited to
bridge the gap between image and pixel-level information, our proposed method
does not encounter any information leak since the class labels of both base
and novel categories are not present in those of CAM.

Under the guidance of the produced pseudo pixel-level labels, we uniquely
reinterpret the original problem as a _pixel-wise few-shot classification_
task. That is, we view each pixel in support/query set images as individual
samples, and turn the proposed model into a pixel-level meta learner for few-
shot semantic segmentation. As confirmed later by our experiments, our model
not only achieves satisfactory performances on standard fully supervised few-
shot semantic segmentation tasks, it would perform favorably against several
state-of-the-art approaches on benchmark datasets under weakly supervised
settings.

The contributions of this work are summarized below:

  * •

We address weakly supervised few-shot semantic segmentation, which requires
only image-level labels during both training and testing.

  * •

We bridge the gap between image and pixel-level labels using classification
activation maps with no information leak, which allows prediction of pseudo
pixel-level labels in weakly supervised settings.

  * •

We uniquely design a pixel-level meta learner which enforces segmentation
consistency across support and query set images during few-shot semantic
segmentation. The proposed model can be realized in both fully supervised and
weakly supervised settings.

Figure 2: Comparisons of different _few-shot_ semantic segmentation schemes.
(a) _Fully Supervised_ [38, 46, 54, 27, 42]: $M$ required in meta-training and
meta-testing; (b) _Loosely Weakly Supervised_ [37]: both $M$ and $c$ available
during meta-training, while only $c$ observed in meta-testing; (c) _Weakly
Supervised_ : only image-level labels $c$ available in both phases.

##  2 Related Works

Semantic Segmentation. The task of semantic segmentation aims at performing
pixel-level classification for each image. Most recent methods involve the use
of deep convolutional neural networks (e.g., FCN [25]). Following works
include SegNet [1], PSPNet [58], U-Net [32], DeepLabs [5, 6, 7, 8] and FastFCN
[47]. A notable improvement is achieved by embedding features that contain
multi-scale context information by applying dilated convolution [53, 5] or
spatial pyramid pooling [58, 6]. These methods, however, are trained in a
fully supervised manner and require a huge amount of pixel-level labels, which
are time-consuming and laborious to obtain.

Weakly Supervised Semantic Segmentation. To alleviate the need for densely
annotated ground truth data, some recent works address semantic segmentation
in weak supervision, which utilize multiple-instance learning [43, 44], graph
[55, 49, 28] and self-training based [50, 51, 56, 48] techniques. However,
such weakly supervised methods cannot be easily extended to few-data or open-
set scenarios.

Few-Shot Semantic Segmentation. Few-shot learning aims at learning models
which would generalize to categories with only a limited amount of labeled
data [12, 11]. Meta-learning [13, 30, 34] has been widely applied for this
task, with the core idea of adapting the learning scheme from base to novel
categories. For example, metric-based meta-learning algorithms such as
ProtoNet [40] and RN [41] learn feature embeddings that exhibit proper
distance metrics for classification and generalization. Few-shot semantic
segmentation, on the other hand, aims at generalizing the ability of pixel-
level classification across categories, while only a limited number of images
are with ground truth pixel-level labels. OSLSM [36] is the first proposed
method to tackle this problem, leveraging information learned from support-set
images and outputs parameters for query image segmentation. PL [9] adopts
metric learning methods [40] to extract prototypes of each semantic class, and
measures their distances between feature maps of query images; CANet [54] adds
an iterative optimization module to refine the predicted results; PFENet [42]
generates additional prior masks to enrich the extracted features.
Moreover, PANet [46], FWB [27] and CRNet [23] propose to further leverage
information from the support set by performing segmentation in the reversed
direction (i.e., segmentation of the support set) for improved model learning.
To exploit knowledge from the foreground objects, DAN [45] and SimPropNet [14]
introduce attention mechanisms, PMMs [52] and ASGNet [20] employ multiple
prototypes for a single category, and PPNet [24] proposes part-aware
prototypes to capture fine-grained features. While some of the existing few-
shot semantic segmentation methods present results in weak supervision
settings (e.g., use of bounding boxes or scribbles [29, 46, 54] as guidance),
they cannot produce satisfactory performance with only image-level annotation
observed. More recently, [31, 4, 37] follow the few-shot setting using image-
level supervision, but they still require collection of ground truth pixel-
level masks for base-class images during meta-training. As depicted in Figure
2 and Table 1, to the best of our knowledge, we are the first to tackle few-
shot semantic segmentation using only image-level annotations during both
(meta) training and testing stages.

Figure 3: Illustration of pixel-level pseudo label generation. Given an input
$I$ and its image-level label $c$, the CAM module extracts heatmaps
$\mathbf{T}$ for each training class $c_{k}$ of CAM (not in
$\mathcal{C}_{\text{base}}$ nor $\mathcal{C}_{\text{novel}}$). $\mathbf{w}$
denotes the visual similarity for the heatmap of each CAM category, which is
measured by the distance between the word embeddings of the associated class
labels. The output heatmap $\overline{T}$ is converted into the pseudo mask
$\tilde{M}$ for $I$ via saliency gating. Note that $\pi$ represents the word
embedding function.

Method | Weak Supervision | Few-Shot Setting
---|---|---
DeepLab [5] | - | -
EDAM [48] | $\checkmark$ | -
PFENet [42] | - | $\checkmark$
Co-att [37] | $\triangle$ | $\checkmark$
Ours | $\checkmark$ | $\checkmark$

Table 1: Comparisons of different semantic segmentation methods. Existing
methods cope with either weak supervision or few-shot settings, while ours
combine both. Note that [37] still requires full supervision during training,
as detailed in Figure 2(b).

##  3 Proposed Method

###  3.1 Notation and Problem Formulation

For the sake of completeness, we define the notations which will be used in
this paper. The semantic classes are denoted as $\mathcal{C}$, which are split
into two disjoint subsets: base categories $\mathcal{C}_{\text{base}}$ and
novel categories $\mathcal{C}_{\text{novel}}$ (i.e.,
$\mathcal{C}_{\text{base}}\cup\mathcal{C}_{\text{novel}}=\mathcal{C}$ and
$\mathcal{C}_{\text{base}}\cap\mathcal{C}_{\text{novel}}=\varnothing$). Note
that the novel categories are with only a few samples available during
training (typically less than 5 per class). For each input RGB image
$I\in\mathbb{R}^{H\times W\times 3}$, we denote its image-level labels as
$\bm{c}\subseteq\mathcal{C}$, and the associated ground truth pixel-level
semantic mask as $M\in\\{\bm{c}\cup{\mbox{\ooalign{0\cr\kern 0.0pt\raise
3.22223pt\hbox{$\cdot$}\cr\vphantom{0}}}}\\}^{H\times W}$, where  0 $\cdot$
indicates the background pixels. For each training episode, we sample a
support/query pair from the image dataset
$\mathcal{D}_{\text{train}}=\\{(S_{j},Q_{j})\\}^{n_{\text{train}}}_{j=1}$ that
contain the same set of base categories, while the testing episodes consist of
those from $\mathcal{C}_{\text{novel}}$ (i.e.,
$\mathcal{D}_{\text{test}}=\\{(S_{j},Q_{j})\\}^{n_{\text{test}}}_{j=1}$).

In standard fully supervised $N$-way $K$-shot settings, each support set
$S_{j}=\\{(I^{s}_{i},M^{s}_{i})\\}^{N\times K}_{i=1}$ contains $N\times K$
image/label example pairs ($K$ pairs from each of $N$ categories), and the
query sets are denoted as $Q_{j}=\\{(I^{q}_{i},M^{q}_{i})\\}^{n_{q}}_{i=1}$
where $n_{q}$ is the number of query images in a single episode. As for the
weakly supervised setting considered in this work, only image-level labels are
available during training and testing, so that support/query sets are denoted
as $S_{j}=\\{(I^{s}_{i},\bm{c}^{s}_{i})\\}^{N\times K}_{i=1}$ and
$Q_{j}=\\{(I^{q}_{i},\bm{c}^{q}_{i})\\}^{n_{q}}_{i=1}$, respectively. Our
proposed weakly supervised learning framework is able to produce pseudo masks
of each image given only its class label (as depicted in Figure 3), followed
by our pixel-level meta-learner for few-shot semantic segmentation (see Figure
4 for the complete framework).

###  3.2 Pixel-Level Pseudo Label Generation

In the weakly supervised scenario, only the class labels are available for the
image data during both training and inference. Thus, we first present a module
that is able to generate pseudo pixel-level semantic masks, guiding the
following segmentation process. As shown in Figure 3, this pseudo pixel-level
label generation can be viewed as the process of semantics-oriented heatmap
extraction. As adopted in previous works like [18], Classification Activation
Maps (CAM) [59] have been utilized to localize discriminative regions in
images that are informative in classifying image-level labels. In order to
produce the heatmap for the input image $I$ as its pseudo pixel-level labels,
we follow [46, 54] and apply a VGG-16 [39] network as our CAM backbone. It is
worth noting that, the CAM backbone is pre-trained on a reduced subset of
ImageNet [33], in which the images do not belong to either the base or novel
categories in our segmentation task. This would minimize possible leakage of
semantic information.

With CAM obtained, we extract per-class heatmaps $\mathbf{T}$ of the input
image $I$:

$\displaystyle\text{CAM}(I)=\mathbf{T}=[T_{1},T_{2},\ldots,T_{N_{\text{CAM}}}]\text{,}$
(1)

where $T_{k}\in[0,1]^{H\times W}$ denotes the heatmap of the $k$-th image
class in CAM, whereas $N_{\text{CAM}}$ is the total number of pre-training
image categories.

With the above per-class heatmaps observed, we next leverage the word
embedding features of each class label as intermediate representations, with
the associated similarity indicating the weight for each of the
$N_{\text{CAM}}$ categories. More specifically, as depicted in Figure 3, we
extract word embedding features of each class label in CAM and that of the
input image. We perform pairwise comparisons between these features to obtain
weighting factors $w_{k}$ for each CAM category $c_{k}$:

$\displaystyle w_{k}=d(\pi(c),\pi(c_{k}))^{-1}\text{,}$ (2)

where $\pi(\cdot)$ represents the word embedding function. We note that
$d(\cdot,\cdot)$ denotes the distance metric, and we use consine similarity in
our work. Thus, categories that are semantically similar with each other
(e.g., _goat_ /_sheep_) would result in a higher weight, and vice versa for
those that are dissimilar (e.g., _goat_ /_tree_). As a result, a weighted
heatmap $\overline{T}$ can be obtained by averaging the above $N_{\text{CAM}}$
heatmaps, which is calculated as

$\displaystyle\overline{T}=\mathbf{w}\cdot\mathbf{T}^{\top}=\sum\limits_{k=1}^{N_{\text{CAM}}}w_{k}T_{k}\text{.}$
(3)

As the CAM heatmaps identify only regions that are helpful in terms of
classification, more detailed structural information such as edges and
boundaries would not be well described. To this end, we impose a class-
agnostic saliency map on the weighted heatmap as a gating mechanism,
alleviating the presence of false-positive pixels in the generated pseudo
labels. Here we note that, to comply with our weakly supervised setting, the
saliency maps are obtained via a network pre-trained with only
foreground/background information without any categorical supervision. In
other words, minimized leakage of semantic information is also enforced at
this stage.

Figure 4: Architecture of our pixel-level meta-learner for weakly supervised
segmentation. During (meta) training, DeepLabv3+ extracts pixel-wise feature
maps $F^{s}$ and $F^{q}$ for support and query inputs, with the produced
pseudo pixel-level labels $\tilde{M}^{s}$ and $\tilde{M}^{q}$, respectively.
Our meta-learner encoder $E$ transforms the above $F$ into a latent space, in
which the pixel-level prototypical loss $\mathcal{L}_{\text{meta}}$ can be
calculated based on $\tilde{M}$ for segmentation purposes. During inference
(i.e., meta-testing), segmentation $\hat{M}^{q}$ for the query can be
performed by pixel-wise $k$-NN classification using $F^{s}$ and
$\tilde{M}^{s}$.

###  3.3 Pixel-Level Meta-Learner for Few-Shot Semantic Segmentation

We now detail our proposed meta-learning scheme for few-shot semantic
segmentation. Under the guidance of CAM, the pseudo pixel-level labels
obtained in Section 3.2 tend to contain only partial discriminative areas, and
may not sufficiently cover the entire foreground object. In order to learn
few-shot segmentation models with only such information from pseudo labels, we
present a unique pixel-level meta-learning framework, as illustrated in Figure
4. In our pixel-level meta-learner, we utilize DeepLabv3+ [8] as the feature
extractor, together with the introduced pseudo label generation and learning
modules for pixel-wise classification. It is worth noting that, the DeepLabv3+
is pre-trained on categories not appeared in $\mathcal{C}$, and this module
remains fixed throughout the meta-learning process. In other words, no
semantic information is leaked from its training stage. By removing the final
classification layers of DeepLabv3+, we obtain a pixel-wise feature map
$F^{s}\in\mathbb{R}^{H\times W\times d}$ from the support image $I^{s}$, where
$(H,W)$ is the original image dimension and $d$ is the feature channel size.

Next, we label each spatial location in $F^{s}$ using the generated support
pseudo mask $\tilde{M}^{s}$. By randomly sampling a fixed number of pixels
from each background/foreground category, the resulting pixel-wise features
are then collected into a set of _support pixel features_. It is worth noting
that, we choose not to use all the labeled pixel features. This is not only
because that such pseudo labels might not match the ground truth ones
(although not available), this sampling mechanism also makes the meta-learning
process more robust against weak labels.

Likewise, a pixel-wise feature map $F^{q}\in\mathbb{R}^{H\times W\times d}$ of
the query image $I^{q}$ is also obtained via the same DeepLabv3+ feature
extractor, resulting in a total of $H\times W$ $d$-dimensional _query pixel
features_. All support and query pixel features are then jointly embedded into
a latent space via the learnable encoder $E$. With the support pixel features
and pseudo labels obtained, we define the prototypes $p_{c}$ for each
associated category $c\in\\{\bm{c}^{s}\cup{\mbox{\ooalign{0\cr\kern
0.0pt\raise 3.22223pt\hbox{$\cdot$}\cr\vphantom{0}}}}\\}$ as

$\displaystyle
p_{c}=\frac{\sum_{l}E(F^{s}_{l})\mathds{1}[\tilde{M}^{s}_{l}=c]}{\sum_{l}\mathds{1}[\tilde{M}^{s}_{l}=c]}\text{,}$
(4)

(fully sup. / weakly sup.) | 1-shot | 5-shot
---|---|---
Method | Backbone | Split-0 | Split-1 | Split-2 | Split-3 | Mean | $\Delta$ | Mean | $\Delta$
1-way |  |  |  |  |  |  |  |  |
Co-att [37] | VGG-16 | 49.5 | 65.5 | 50.0 | 49.2 | 53.5 | — | 51.7 | —
AMP [38] | VGG-16 | 41.9 / 10.6 | 50.2 / 14.1 | 46.7 / 7.6 | 34.7 / 10.9 | 43.4 / 10.8 | 32.6 | 46.9 / 14.7 | 32.2
PANet [46] | VGG-16 | 42.3 / 25.7 | 58.0 / 33.4 | 51.1 / 28.8 | 41.2 / 20.7 | 48.1 / 27.1 | 21.0 | 55.7 / 37.7 | 18.0
PFENet [42] | ResNet-50 | 61.7 / 33.4 | 69.5 / 42.5 | 55.4 / 43.6 | 56.3 / 39.9 | 60.8 / 39.9 | 20.9 | 61.9 / 44.8 | 17.1
Ours | VGG-16 | 38.3 / 36.5 | 57.6 / 51.7 | 54.0 / 45.9 | 40.1 / 35.6 | 47.5 / 42.4 | 5.1 | 50.6 / 45.5 | 5.1
2-way |  |  |  |  |  |  |  |  |
PANet [46] | VGG-16 | 45.1† / 24.5 | 45.1† / 33.6 | 45.1† / 26.3 | 45.1† / 20.3 | 45.1 / 26.2 | 18.9 | 53.1 / 36.6 | 16.5
Ours | VGG-16 | 36.5 / 31.5 | 51.8 / 46.7 | 48.5 / 41.4 | 38.9 / 31.2 | 43.9 / 37.7 | 6.2 | 49.3 / 43.0 | 6.3

(a) PASCAL-$5^{i}$

(fully sup. / weakly sup.) | 1-shot | 5-shot
---|---|---
Method | Backbone | Split-0 | Split-1 | Split-2 | Split-3 | Mean | $\Delta$ | Mean | $\Delta$
1-way |  |  |  |  |  |  |  |  |
PANet [46] | VGG-16 | 20.9† / 12.7 | 20.9† / 8.7 | 20.9† / 5.9 | 20.9† / 4.8 | 20.9 / 8.0 | 12.9 | 29.7 / 13.9 | 15.8
Ours | VGG-16 | 26.0 / 24.2 | 14.5 / 12.9 | 20.0 / 17.0 | 18.3 / 14.0 | 19.7 / 17.0 | 2.7 | 27.0 / 17.5 | 9.5
2-way |  |  |  |  |  |  |  |  |
Ours | VGG-16 | 18.2 / 17.4 | 12.2 / 9.5 | 9.1 / 10.4 | 6.5 / 7.1 | 11.5 / 11.1 | 0.4 | 14.8 / 11.9 | 2.9

(b) MS COCOi

Table 2: Performance evaluation on (a) PASCAL-$5^{i}$ and (b) MS COCO in terms
of mean-IoU (Mean) and performance difference $\Delta$ due to change of
settings. The numbers before and after ‘/’ indicate results under fully and
weakly supervised settings, respectively. Note that [37] considers a loosely
weakly supervised setting and requires ground truth pixel-level masks during
training, while [42] utilizes a stronger backbone (ResNet-50) compared to
others (VGG-16).

where $l$ iterates over each pixel, and $\mathds{1}[\cdot]$ is an indicator
function which only outputs $1$ when the condition holds.

Inspired by [40, 9, 42], we advance the prototypical loss
$\mathcal{L}_{\text{meta}}$ as the objective during the meta-training stage,
which is calculated by accumulating the distance between each query pixel
(with pseudo labels) and its corresponding pixel-level prototype from the
support sets:

$\displaystyle\mathcal{L}_{\text{meta}}=-\frac{\sum_{c}\sum_{l}\exp(-d(E(F^{q}_{l}),p_{c}))\mathds{1}[\tilde{M}^{q}_{l}=c]}{\sum_{c}\sum_{l}\mathds{1}[\tilde{M}^{q}_{l}=c]}\text{.}$
(5)

During inference (i.e., meta-testing), the labels of the embedded query pixel
features are determined by performing pixel-wise $k$-nearest neighbors
classification, as depicted by the blue dotted arrows in Figure 4. The final
predicted query semantic mask $\hat{M}^{q}$ is then compared with the ground
truth $M^{q}$ for performance evaluation.

##  4 Experiments

Datasets. We follow the evaluation protocol in [36] and conduct experiments on
the PASCAL-$5^{i}$ dataset. It contains a total of 20 object categories from
the PASCAL VOC 2012 [10] and the extended SDS [15] datasets, which are evenly
divided into 4 splits ($i=0,1,2,3$). Additionally, we consider the MS COCO
2014 [21] dataset, which contains 80 object categories and thus is more
challenging. Following the settings of [46], we divide MS COCO into 4 splits
with 20 categories each. For both datasets, three of the splits are used as
base (training) classes, with the remaining one as novel (testing) classes in
each experiment.

Evaluation Metrics. We follow [36, 57, 46, 54, 27] and apply the mean
intersection over union (mean-IoU) as the evaluation metric, which computes
separate IoUs for each foreground category, and then averages them along with
the background class. Following the protocol in [46], the mean-IoU in each
evaluation is calculated by the average of 5 runs, with each run randomly
sampling 1,000 episodes from the testing set.

Figure 5: Example 1-way 1-shot segmentation results on PASCAL-$5^{i}$. We have
green, yellow and red pixels represent ground truth pixel-level foreground
labels, generated pseudo masks, and the predicted outputs, respectively. Note
that our weakly supervised setting does not observe ground truth pixel-level
labels for the support set images (in dotted frames). 22footnotetext: Split-
wise results not reported in the original paper.

Implementation Details. In our experiments, all images are normalized and
reshaped to $129\times 129$ pixels. For pseudo pixel-level label generation,
we choose the VGG-16 [39] network as the CAM backbone, with weights pre-
trained over a reduced subset of ILSVRC 2012 [33] with categories in
$\mathcal{C}_{\text{base}}$ and $\mathcal{C}_{\text{novel}}$ removed. For
semantic labels, we use the word embedding vectors pre-trained on Wikipedia
using fastText [3]. We apply DSS [16] to extract saliency maps for the gating
mechanism in Fig. 3, which is a model pre-trained over MSRA-B [22] without
categorical supervision.

For the segmentation model, we use a DeepLabv3+ [8] pre-trained over
irrelevant categories, which remains fixed throughout the meta-learning
process. The encoder $E$ is a multilayer perceptron with two hidden layers,
and the output dimension (for the latent space) is set to 64. All experiments
are implemented by PyTorch, and are run using a single NVIDIA Titan RTX
graphics card with 24GB of video memory.

###  4.1 Comparison with State-of-the-art Methods

Since no previous work was designed to address this weakly supervised few-shot
segmentation task, we choose to implement modified versions of state-of-the-
art supervised methods by replacing the ground truth masks for training with
our generated pseudo labels (i.e., as introduced in Section 3.2). As depicted
in Figure 4, our proposed framework can be trained in a fully supervised
fashion (i.e., using ground truth pixel-level masks during training). Thus, we
include this fully supervised version of our model as the performance upper
bounds. We compare our results mainly with methods using the same backbone
[37, 38, 46], with the exception of [42] for cross-backbone analysis.

PASCAL-$\bm{5^{i}}$. We first compare the performances of different methods on
PASCAL-$5^{i}$, with results listed in Table 2a. In the first row of this
table, we consider a loosely weakly supervised model of [37] (as illustrated
in Figure 2b). For other methods (including ours) in this table, we present
results under both fully and weakly supervised settings. While our fully
supervised model achieved comparable performance with state-of-the-art methods
in the standard 1-way 1-shot setting, our model reported a significant
improvement over PANet [46] by 15.3% (42.4% v.s. 27.1%) in the weakly
supervised setting. It is worth noting that PFENet [42] is trained using a
stronger backbone (ResNet-50), while all other methods (including ours)
utilize VGG-16. Nevertheless, our model still outperforms [42] by a
considerable margin of 2.5% in the weakly supervised setting. We also observe
that the performance drop between the two different settings of our model is
significantly less than the others. That is, when only image-level labels
(instead of ground truth pixel-level masks) are observed during both training
and testing, both PFENet [42] and PANet [46] suffered from a $>$20%
performance drop while only 5.1% was reported by our model.

MS COCO. As shown in Table 2b, despite the increased difficulty in few-shot
segmentation on MS COCO, our model is able to achieve satisfactory
performances under both fully and weakly supervised settings when comparing to
[46]. Specifically, we only observe a 2.7% performance drop between the two
settings on the 1-way 1-shot task, while that of [46] is 12.9%. The above
quantitative results support the use of our propose framework for solving few-
shot semantic segmentation, especially when only image-level labels can be
observed during both training and testing (i.e., the weakly supervised
setting).

Multi-way Segmentation. We now show that our model is applicable to the cases
when there is more than one foreground object category in an image, which is
more challenging since it requires more information to be learned in each
episode. In the lower parts of Table 2, we list the performances of different
methods under the 2-way setting (i.e., two types of foreground objects exist
in an image). It is worth noting that, while most existing methods [36, 38,
54, 27, 17, 42] are designed to tackle only 1-way segmentation, they typically
claim such extension can be realized by forward passing $K$ times with
additional decision rules. On the contrary, our method can directly produce
output labels of a multi-category image as the final classification by a
$k$-NN search. As shown in Table 2a, our model performed favorably against
previous methods by a margin of 11.5% on the PASCAL-$5^{i}$ 2-way 1-shot task.
To the best of our knowledge, we are the first to report results of 2-way
tasks for the MS COCO dataset, as shown in the last row of Table 2b.

Figure 6: Example failure cases for our weakly supervised model and the fully
supervised PFENet [42].

###  4.2 Analysis of Our Proposed Method

Ablation Study. As our meta-learning process is mainly achieved by the
episodic learning of encoder $E$, we now design a baseline version such that
it directly predicts the output mask by a $k$-NN search on the pixel features
encoded by the DeepLabv3+ backbone (i.e., without embedding into the latent
space via $E$). As shown in the first row of Table 3, the results were
severely degraded with a drop of up to 18%, if the model was not learned via
the meta-learning objectives. This further confirms that, the promising
performance achieved by our proposed method is not a direct result of using
particular strong backbones. Instead, it leverages spatial and structural
details captured by the backbone, upon which semantic and category-wise
information is reinforced via the meta-learning process.

Additionally, we provide results that are trained using pseudo masks generated
without saliency gating (as mentioned in the last step of Section 3.2). As
shown in the second row of Table 3, a slight decrease in mean-IoU (less than
5%) was observed, while still outperforming the baseline version by a large
margin. Thus, the use of saliency gating as the post-processing step for
pseudo pixel-level masks would be preferable but not critical.

Method | 1-shot | 5-shot
---|---|---
1-way | 2-way | 1-way | 2-way
xxxOurs w/o $E$xxx | 30.0 | 19.7 | 31.5 | 25.4
Ours w/o Saliency | 41.0 | 35.4 | 42.1 | 39.4
Full Version | 42.4 | 37.7 | 45.5 | 43.0

Table 3: Ablation study of our model on PASCAL-$5^{i}$ in mean-IoUs. Ours w/o
$E$ denotes our framework without meta-learner encoder $E$, while w/o Saliency
indicates our model without applying saliency gating to process the generated
pseudo masks.

Qualitative Analysis. For visual comparisons, we present qualitative results
of 1-way 1-shot segmentation on PASCAL-$5^{i}$ dataset in Figure 5. As
detailed in Section 3.1, our model is realized in the weakly supervised
setting and does not observe ground truth masks of support images during
training (i.e., column 1 in Figure 5). Instead, given each image and its
image-level label (i.e., class name), we first generate its pseudo mask, which
serves as the guidance for our meta-learning framework. As evident in column 2
of Figure 5, our generated pseudo masks do not cover the entire foreground
object, but instead contain only discriminative regions that are informative
in classifying image-level labels (e.g., muzzle of a horse, or wheels and
pedals of a motorcycle). Nevertheless, from this figure, we see that our
proposed model is able to predict masks for query images with satisfactory
performances (e.g., columns 4 and 6 in Figure 5). It is also worth noting that
no post processing is performed on our predicted results.

From the split-wise mean-IoUs listed in Table 2, we see that some data splits
would generally suffer from performance drops across different models
including ours. In Figure 6, we show failure segmentation example results by
our weakly supervised model and the fully supervised PFENet [42]. As evident
in this figure, the ground truth images of such categories generally contain
small foreground regions (e.g., bottle), or those that cannot be easily
distinguished from the background (e.g., chair). For such image categories,
training with few-shot samples would not be expected to address semantic
segmentation tasks well. This would be the limitation of developing solutions
for few-shot semantic segmentation.

##  5 Conclusion

In this paper, we proposed a unique learning framework for few-shot semantic
segmentation in a weakly supervised manner, which only observes image-level
labels during both training and testing. Under such weak supervision, our
proposed model serves as a pixel-level meta-learner, which produces pseudo
segmentation masks for guiding pixel-wise classification in a meta-learning
fashion. Our experiments confirmed the effectiveness of our model, which
achieved satisfactory results on two benchmark datasets under fully supervised
settings, while surpassing the state-of-the-art methods in weakly supervised
settings. Finally, we point out the challenge and limitation of the task of
few-shot semantic segmentation.

#### Acknowledgement

This work is supported in part by the Ministry of Science and Technology of
Taiwan under grants MOST 110-2634-F-002-036 and 110-2221-E-002-121.

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