# How many phases nucleate in the
bidimensional Potts model?

Federico Corberi1 1Dipartimento di Fisica E. R. Caianiello and INFN, gruppo
collegato di Salerno, Università di Salerno, via Giovanni Paolo II 132, 84084
Fisciano (SA), Italy
2Sorbonne Université, CNRS UMR 7589, Laboratoire de Physique Théorique et
Hautes Energies,
4 Place Jussieu, 75252 Paris Cedex 05, France
3Institut Universitaire de France, 1, rue Descartes, 75231 Paris Cedex 05,
France
4Imperial College London, Chain Building, 7 Imperial College Rd, Kensington,
London, SW7 2AZ, UK
5Max Planck Institute for Mathematics in the Sciences, Inselstra$\beta$e 22,
04103 Leipzig Germany Leticia F. Cugliandolo2,3 1Dipartimento di Fisica E. R.
Caianiello and INFN, gruppo collegato di Salerno, Università di Salerno, via
Giovanni Paolo II 132, 84084 Fisciano (SA), Italy
2Sorbonne Université, CNRS UMR 7589, Laboratoire de Physique Théorique et
Hautes Energies,
4 Place Jussieu, 75252 Paris Cedex 05, France
3Institut Universitaire de France, 1, rue Descartes, 75231 Paris Cedex 05,
France
4Imperial College London, Chain Building, 7 Imperial College Rd, Kensington,
London, SW7 2AZ, UK
5Max Planck Institute for Mathematics in the Sciences, Inselstra$\beta$e 22,
04103 Leipzig Germany Marco Esposito4 1Dipartimento di Fisica E. R. Caianiello
and INFN, gruppo collegato di Salerno, Università di Salerno, via Giovanni
Paolo II 132, 84084 Fisciano (SA), Italy
2Sorbonne Université, CNRS UMR 7589, Laboratoire de Physique Théorique et
Hautes Energies,
4 Place Jussieu, 75252 Paris Cedex 05, France
3Institut Universitaire de France, 1, rue Descartes, 75231 Paris Cedex 05,
France
4Imperial College London, Chain Building, 7 Imperial College Rd, Kensington,
London, SW7 2AZ, UK
5Max Planck Institute for Mathematics in the Sciences, Inselstra$\beta$e 22,
04103 Leipzig Germany
Onofrio Mazzarisi1,5 and Marco Picco2 1Dipartimento di Fisica E. R. Caianiello
and INFN, gruppo collegato di Salerno, Università di Salerno, via Giovanni
Paolo II 132, 84084 Fisciano (SA), Italy
2Sorbonne Université, CNRS UMR 7589, Laboratoire de Physique Théorique et
Hautes Energies,
4 Place Jussieu, 75252 Paris Cedex 05, France
3Institut Universitaire de France, 1, rue Descartes, 75231 Paris Cedex 05,
France
4Imperial College London, Chain Building, 7 Imperial College Rd, Kensington,
London, SW7 2AZ, UK
5Max Planck Institute for Mathematics in the Sciences, Inselstra$\beta$e 22,
04103 Leipzig Germany

###### Abstract

We study the kinetics of the two-dimensional $q>4$-state Potts model after a
shallow quench slightly below the critical temperature and above the pseudo
spinodal. We use numerical methods and we focus on intermediate values of $q$,
$4<q\leq 100$. We show that, initially, the system evolves as if it were
quenched to the critical temperature. The further decay from the metastable
state occurs by nucleation of $k$ out of the $q$ possible phases. For a given
quench temperature, $k$ is a logarithmically increasing function of the system
size. This unusual finite size dependence is a consequence of a scaling
symmetry underlying the nucleation phenomenon for these parameters.

##  1 Introduction

When a control parameter is changed in such a way that a thermodynamical
system undergoes a first-order phase transition one observes hysteresis and
metastability. These phenomena are widespread in nature [1] and, particularly,
in many areas of physics [2, 3, 4, 1, 5]. Besides, metastability plays a
prominent role also in biological systems such as, for instance, proteins and
nucleic acids [6, 7].

The simplest example of metastability is, perhaps, a uniaxial ferromagnet in
an external field, whose paradigmatic modelisation is the Ising model. Upon
field reversal, provided that its magnitude is sufficiently small and the
temperature is subcritical, the magnetisation remains at the pre-reversal
value for a certain time, the lifetime of the metastable state, before
flipping to the new equilibrium value. The phenomenon can be ascribed to the
competition between surface tension and bulk energy and is accounted for, at
least at a simple semi-quantitative level, by classical nucleation theory [1,
8, 9, 10]: a nucleus of the equilibrium phase in a metastable sea is unstable
unless its size exceeds a critical value. If it is not the case it disappears.
Then, in this picture, the lifetime of the metastable state is the time needed
to nucleate, by thermal fluctuations, a critical nucleus (notice that in this
context critical does not refer to the criticality of the state at $T_{c}$).
Elaborating on these ideas, more refined theories of nucleation [11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22] have been developed, and they describe the
phenomenon with good accuracy.

In the previous example, the metastable and equilibrium states are associated
to the two ergodic components which characterise the system al low
temperatures. However, the situation is not as well understood in systems with
more ergodic components, such as those described at the simplest level by the
ferromagnetic Potts model [23]. This model is defined in terms of lattice
discrete variables taking $q\geq 2$ possible values, sometimes called colours,
and the interaction favours equal colour on nearest neighbour sites. For
$q=2$, the Ising model is recovered.

In the absence of external fields, the Potts model undergoes a phase
transition between a high temperature disordered phase and a low temperature,
symmetry broken, ordered one at a $q$-dependent critical temperature
$T_{c}(q)$. In two dimensions, the character of the transition changes
crossing $q=4$ [24, 25]: for $q\leq 4$ it is continuous (second order), while
for $q>4$ it is discontinuous (first order) and metastability is found.
Specifically, upon quenching from $T>T_{c}$ to $T\lesssim T_{c}$ a finite-size
system remains for some time in the disordered state before starting the
evolution towards the final equilibrium state. The usual interpretation is
that the metastable state is the (single) ergodic component at $T>T_{c}$ while
the target equilibrium state is one of the $q$ symmetry-related ergodic
components at $T<T_{c}$. Hence, at variance with the field-driven transition
in the Ising model, nucleation occurs towards a $q$-degenerate final state,
and one speaks of multi-nucleation. Another important difference is that the
transition is temperature driven.

To determine the lifetime of the metastable state, its dependence on the
system size, the dynamic escape from it, and the nature of the nucleation
process is a difficult and much debated issue [26, 27, 28, 29]. In particular,
it was argued [30, 31, 32, 33] that both the lifetime of the metastable state
and temperature range below $T_{c}$ where it takes place shrink when the
system size increases. This intriguing feature, which has no analog in the
discontinuous field-driven transition of the Ising model, poses yet unresolved
questions on the nature of the metastable state in the thermodynamic limit. In
a previous paper [34], some of us addressed this problem analytically with a
perturbation scheme valid for large $q$. It turns out that, for $q\to\infty$,
a long lived metastable state exists for quenches to final temperatures
$T>T_{\ell}$, where $T_{\ell}\to T_{c}/2$. In [34] a rather precise
description of the metastable state was given, both from the microscopic and
thermodynamic points of view. However, a similar characterisation of the
escape kinetics is still missing for finite $q$. Specifically, while the
nature of the metastable state can be well described in such an analytical
framework, its lifetime (for finite $q$) has not been determined yet and its
very existence in the thermodynamic limit cannot be proved. Related to this
issue, the order in which the two limits of large $q$ and large system size
should be taken remains to be clarified.

In this paper we study the process whereby the disordered metastable state is
run away and how the target equilibrium state is approached in the square
lattice Potts model with $q>4$, after shallow quenches from high to low
temperatures close to $T_{c}$. We use Monte Carlo numerical simulations with
the Metropolis rule for different values of $q$ and linear system sizes $L$.
We choose $q$ values which are larger than $4$, $q=9,24,100$, but not too
large so that we can observe nucleation in a relatively wide temperature
interval below $T_{c}$, see [34]. In a companion paper by some of us [35] the
related issue of deep quenches below the pseudo-spinodal of the same model for
much larger $q$ values, closer to the ideal $q\to\infty$ limit, is tackled and
the full parameter dependence of the curvature-driven growing length in the
coarsening regime is determined.

Our main result is the characterisation of the multi-nucleation process. We
show that, soon after the quench, the system evolves as if it were at the
critical temperature. Later, multi-nucleation takes over. We prove that for a
given system size, the number $k$ of phases that nucleate is a logarithmically
increasing function of $L$, see Eq. (15). In addition, we also study the
dependence of $k$ on $q$ and $T$. Phases which do not nucleate disappear from
the system immediately after the metastable state is escaped. Those which
nucleate start competing, resulting in a coarsening process [36, 37, 38, 39,
40, 35, 41, 42, 43, 44, 45, 46] which leads, at some point, to the progressive
elimination of the less represented colours until symmetry is definitively
broken, one single colour survives, and equilibrium is attained (at not too
low temperatures; otherwise, blocked states of the kind discussed in [36, 37,
38, 39, 40, 35] can stop the evolution).

The paper is organised as follows. In Sec. 2 we introduce the Potts model and
its dynamics. We also define the relevant observables that will be further
considered. In Sec. 3 we provide a general description of the whole dynamical
process, from the instant of the quench, to the escape from metastability and
further on. Section 4 is the core of the article and contains a thorough
discussion of the multi-nucleation process. Finally, Sec. 5 concludes the
paper.

##  2 Model and observables

The Hamiltonian of the Potts model [23] reads

$H[\\{s_{i}\\}]=-J\sum_{\langle ij\rangle}\delta_{s_{i}s_{j}}\;,$ (1)

where $J>0$ is a coupling constant and $\langle ij\rangle$ are nearest-
neighbour couples on a two-dimensional lattice of linear dimension $L$ which
we take to be a square one. $\delta_{ab}$ is the Kronecker delta and $s_{i}$
is an integer variable ranging from 1 to $q\geq 2$. We consider periodic
boundary conditions. In the sum we count each bond once, and for this geometry
the energy is bounded between $-2JL^{2}$ and $0$. The critical temperature is
exactly known in $d=2$ [23]

$k_{B}T_{c}(q)=\frac{J}{\ln\left(1+\sqrt{q}\right)}$ (2)

and the phase transition is second (first) order for $q\leq 4$ ($q>4$) [47].
Notice that for finite systems of size $L$ there are corrections to the value
of $T_{c}(q)$ given for infinite size in Eq. (2) according to
$(T_{c}(L,q)-T_{c}(q))/T_{c}(q)\simeq L^{-d}$ [48]. Henceforth we will set
$k_{B}=J=1$.

We consider the dynamics where single spins are updated according to a random
sequence with a probability $w(s,s^{\prime})$ to change from an initial value
$s$ to a final one $s^{\prime}$. We will use the Metropolis form
$w(s,s^{\prime})=\min[1,\exp(-\Delta E/T)]$, where $\Delta E$ is the energy
variation associated to the flip. A unit time, or Montecarlo step, is elapsed
after $L^{2}$ attempted moves.

In the following we will consider a protocol in which the system, initially
prepared in an infinite temperature equilibrium state, is suddenly quenched at
time $t=0$ to a low temperature $T<T_{c}$. The dynamical evolution has been
studied in previous works [35, 36, 37, 43, 41, 42, 44, 45, 46, 38, 39, 40]
mainly focusing on the coarsening regime. Here we are more interested in
metastability and multi-nucleation, which clearly arise at $T\lesssim T_{c}$.

After the quench the system relaxes to a low energy state and the energy
density $e(t)=\langle H[\\{s_{i}\\}](t)\rangle/L^{2}$ approaches the final
equilibrium value $e(\infty)$. Here and in the following the non equilibrium
average $\langle\dots\rangle$ is taken over thermal histories and initial
conditions. In order to quantify the relaxation, therefore, we will consider
the energy density excess

$\Delta e(t)=e(t)-e(\infty).$ (3)

In a state with well formed domains, connected regions with the same spin
state, the excess energy is stored on domain walls, and $\Delta e(t)$ is
proportional to the density of interfacial spins. Since for non fractal
aggregates this quantity is, in turn, inversely proportional to the typical
size of the domains, from $\Delta e(t)$ one can infer such characteristic
length as

$R(t)=\Delta e(t)^{-1}.$ (4)

We will discuss below another determination of a typical length scale, that
should behave in a similar way.

A fundamental quantity, usually considered in coarsening processes, is the
equal time spin-spin correlation function. Specifically, we define the
quantity

$C_{ij}(r,t)=\left.\dfrac{\sum\limits_{n=1}^{q}[\langle\delta_{s_{i}(t),n}\delta_{s_{j}(t),n}\rangle-\langle\delta_{s_{i}(t),n}\rangle\langle\delta_{s_{j}(t),n}\rangle]}{\left(\sum\limits_{n=1}^{q}[\langle\delta^{2}_{s_{i}(t),n}\rangle-\langle\delta_{s_{i}(t),n}\rangle^{2}]\right)^{1/2}\left(\sum\limits_{n=1}^{q}[\langle\delta^{2}_{s_{j}(t),n}\rangle-\langle\delta_{s_{j}(t),n}\rangle^{2}]\right)^{1/2}}\right|_{|\vec{r}_{i}-\vec{r}_{j}|=r}\\!\\!\\!\\!\\!\\!\\!.$
(5)

Notice that, due to isotropy and homogeneity, $C_{ij}(r,t)$ only depends on
the distance $r=|\vec{r}|$, where $\vec{r}$ is the vector joining $i$ and $j$,
and should be independent of $i$ and $j$. Hence, further on we will denote it
$C(r,t)$ and, enforcing this symmetry, we will rather compute the spatial
average of the quantity in Eq. (5), namely $L^{-2}\sum_{ij}C_{ij}(r,t)$. In
this way, we will improve the statistics. It is clear that, on the one hand,
at $r=0$ (namely $i=j$) $C(r=0,t)=1$ and, on the other hand, at very large
distance, $C(r\to\infty,t)=0$ because of the expected factorisation of the
first term in the numerator. Defining

$\sigma^{2}_{n}(t)=\langle\delta^{2}_{s_{i}(t),n}\rangle-\langle\delta_{s_{i}(t),n}\rangle^{2},$
(6)

Eq. (5) can be re-written as

$C(r,t)=\frac{\sum\limits_{n=1}^{q}\sigma^{2}_{n}(t)C_{n}(r,t)}{\sum\limits_{n=1}^{q}\sigma^{2}_{n}(t)},$
(7)

where

$C_{n}(r,t)=\dfrac{\langle\delta_{s_{i}(t),n}\delta_{s_{j}(t),n}\rangle-\left\langle\delta_{s_{i}(t),n}\right\rangle\langle\delta_{s_{j}(t),n}\rangle}{\sigma^{2}_{n}(t)}$
(8)

is the correlation function restricted to the colour $n$. Equation (7)
transparently expresses the fact that the unrestricted correlation is the
weighted average of the restricted ones, the weights being the variances of
the Boolean variables associated to $s_{i}$, see Eq. (6).

When dynamical scaling holds, $C(r,t)$ takes the form

$C(r,t)=g\left[\frac{r}{{\cal R}(t)}\right]\;,$ (9)

where $g$ is a scaling function and ${\cal R}(t)$ a characteristic size with
the meaning of the typical domain’s linear dimension. From the correlation
function one can extract such a size in different ways. For instance, one can
use the momenta as

${\cal
R}(t)=\left[\frac{\sum_{r}r^{m}\,C(r,t)}{\sum_{r}C(r,t)}\right]^{\frac{1}{m}}\;.$
(10)

In case of dynamical scaling, determinations with different values of $m$
provide proportional results (however large $m$ values are numerically
problematic since the noisy large-$r$ tails of $C(r,t)$ are heavily weighted).
Similarly, one can use the half-height width to define ${\cal R}(t)$ as

$C({\cal R}(t),t)=\frac{1}{2}\;.$ (11)

In the following we will use this determination which is easier. Analogously,
the typical size of domains of a specific colour ${\cal R}_{n}(t)$ can be
defined replacing $C$ with $C_{n}$ in Eq. (11):

$C_{n}({\cal R}_{n}(t),t)=\frac{1}{2}\;.$ (12)

Let us mention that Eq. (12) cannot be used to determine ${\cal R}_{n}$ in two
situations: i) The corresponding colour is not present in the system. In this
case $C_{n}\equiv 0$, Eq. (12) loses its meaning, and ${\cal R}_{n}$ could be
taken to be identically zero. ii) The colour has invaded the whole system.
Also in this case one has $C_{n}\equiv 0$, because of the subtraction term in
Eq. (8), and one can assume that the size of the domain of colour $n$ is the
system size $L$. In general, immediately before one of the two situations i),
ii) occur, one finds a very small value of ${\cal R}_{n}$. Hence it must be
kept in mind that finding a small value of ${\cal R}_{n}$ does not necessarily
mean that the corresponding colour is present only in small domains, but it
could as well be that it has flooded the system. Which case applies must be
ascertained differently, for instance, by visual inspection of the
configurations. This will be important further on.

Labelling of the $q$ possible values of the spins will be done in the
following in a dynamical way: at any time we order the colours according to
their abundances, from the most to the least represented, and label them with
$n$ increasing from $1$ to $q$. This method is useful, particularly in the
late stage of the dynamics where, as we will discuss, a clear hierarchy of
abundances sets in. We have checked that some exchanges in the ordering occur
during evolution for the $q$ values we use here but these are rare. Roughly
speaking, a hierarchy of colours is installed when the phases nucleate.

##  3 The dynamical process

In the following we will discuss the results of the numerical simulations of
the Potts model introduced above. We will mainly consider three $q$ values,
$q=9$, 24, 100, as paradigms of small, intermediate and large $q$ behaviors.
Other $q$ values have also been studied and we sometimes include them in the
presentation. These systems are quenched to final temperatures close to the
critical one, $T_{c}(q)$, in order to appreciate the metastable state and its
lifetime. In most of our simulations we set $T=0.9912\cdot T_{c}(q)$ for
$q=9$, $T=0.98\cdot T_{c}(q)$ for $q=24$, and $T=0.95\cdot T_{c}(q)$ for
$q=100$. These choices are motivated by the requirement to have a comparable
nucleation time $\tau(q,T)$ (see discussion below, and Fig. 1) for the three
reference values of $q$. Notice that fixing $\tau(q,T)$ does not correspond to
fixing $(T_{c}(q)-T)/T_{c}(q)$. Let us also remind that, as discussed below
Eq. (2), $T_{c}(q)$ is the critical temperature of the infinite system. In
order to investigate the finite-size effects, a central issue of this paper,
we will used systems in the range of sizes $L\in[200,1000]$ (only for
qualitative illustrations, such as the snapshots in Fig. 2, we consider
systems as small as $L=80$). Comparing the value of the equilibrium coherence
length $\xi$ at $T_{c}(q)$ given in [49] with the system sizes we use, one
concludes that $L/\xi$ is at least 13 (in the worst case $L=200$ with $q=9$).
Therefore our studies are expected to be basically free from the usual
equilibrium finite size effects occurring when $L$ is smaller or comparable to
$\xi$. This, however, does not exclude other finite size effect of non-
equilibrium nature, which are indeed a central point of this article, as we
will explain below.

(a) (b)

Figure 1: (a) The time evolution of growing length $R$ for $q=9,24,100$ (see
the key). The linear system size is $L=700$. The dashed segment shows the
asymptotic curvature driven growth $t^{1/2}$. In the inset the $T/T_{c}$
dependence of $R$ in the case $q=24$ and $L=700$ is shown. (b) $R(t)$ for
$q=24$ and different linear system sizes $L=80,92,110,160,300,700,1000$ (the
last two curves superimpose). The dashed vertical segments indicate the times
at which the snapshots of Fig. 2 were taken.

A first qualitative description of the kinetic process after a quench stems
from the behavior of $R(t)$, the typical length scale extracted from the
analysis of the excess energy as in Eq. (4), which is shown in Fig. 1. With
some quantitative differences, the same kind of pattern is observed for any
value of $q$ (see also [37, 44] and [50] where this quantity, for more $T$ and
$q$ values, respectively, was shown). After a short transient, one can clearly
identify three extended regimes. The first one is associated to $R$ growing
slowly or staying approximately constant, $R(t)\simeq R_{m}$. This plateau,
which is flatter and longer as $T\to T_{c}^{-}$ at fixed $q$ or as
$q\to\infty$ at fixed $(T-T_{c})/T_{c}$, is a clear manifestation of
metastability. In this time lag the kinetics is useless because the system is
confined in a local free energy minimum and the trapping barrier is not yet
jumped over. In order to do it, critical nuclei, namely ordered domains of a
sufficiently large size, must develop. However, this is an activated process
which requires a certain time $\tau(q,T)$, the nucleation time. At
$t\ll\tau(q,T)$ the system is still in a rather disordered state, visually not
too different from the initial one, see the snapshots taken at $t=800$ in Fig.
2. Some small ordered domains can be spotted but they are too tiny to start
nucleation.

Figure 2: Snapshots of a typical evolution of the $q=24$ Potts model quenched
to $T=0.98\cdot T_{c}$ for different lattice sizes. It is possible to
appreciate that the number of phases that nucleate increases with the linear
size of the system: $k=1$ for $L=80$, $k=2$ for $L=90$, $k=3$ for $L=110$ and
$k=7$ for $L=160$, all at $t=10^{5}$.

Actually, this metastable state is similar to the equilibrium one at $T_{c}$,
in a sense that we are going to specify below. We illustrate this point by
means of the correlation function defined in Eq. (5). In Fig. 3 this quantity
is plotted against $r$ during the evolution after a quench. Let us focus, to
start with, on panel (a), where curves for a quench to $T=0.99\cdot T_{c}$
(continuous lines) are compared to those, taken at the same times, for a
quench to $T=T_{c}$ (dashed ones). In the latter case one sees that, as time
goes on, the correlation extends to larger values of $r$ until at $t\simeq
t_{eq}(T_{c})\simeq 10^{3}$ the curves start to superimpose, signalling that
equilibrium has been reached. At short times, a similar pattern is displayed
by the curves of the subcritical quench (continuous ones) which stay close to
the ones previously discussed. At longer times, for $t>t_{eq}(T_{c})$, the
correlations move to the right further than those of the quench to $T_{c}$ and
there is tendency to accumulate on a curve somewhat broader than the
equilibrium one at $T_{c}$. This occurs roughly in the range of times
$t\in[10^{3},10^{4}]$ which corresponds to the lifetime of the metastable
state. In this time range the correlation is almost time-independent at short
$r$, whereas some evolution can be spotted at large $r$. This signals that the
metastable state is not really stable, and that a prodrome of its failure,
which can be interpreted as the build up of critical nuclei, is occurring at
large distances. Finally, roughly for $t>10^{4}$, a quick growth of
correlations is observed, unveiling that metastability is over and that the
next stage, where ordering takes place, has been entered. In this case, the
classification of the behaviour of the correlation into three stages (initial
evolution, stasis, late evolution) is not sharp because, as we discuss below,
$T$ is relatively far from $T_{c}$.

(a) (b) (c)

Figure 3:  The space-time correlation function $C(r,t)$ is plotted with
continuous lines against $r$ for a quench of the model with $q=24$, $L=700$,
from infinite temperature to $T=0.99\cdot T_{c}$ (a), $T=0.995\cdot T_{c}$
(b), $T=0.999\cdot T_{c}$ (c). Different curves correspond to different times
exponentially spaced (from bottom to top $t=2,\ 4,\ 7,\ 15,\ 31,\ 64,\ 137,\
291,\ 618,\ 1316,\ 2802,\ 5968,\ 12709,\ 27066,\ 57642,\ 122762,\ 261451,\
556822,\ 1185886$). The crosses refer to a quench to $T=T_{c}$. See the text
for a discussion.

Let us now have a look at the other two panels in Fig. 3. In the central one
(b) the quench is closer to $T_{c}$ and the accumulation of curves in the
metastable state is clearer than in the left panel. Furthermore, the locus
where superposition occurs is also closer to the equilibrium curve at $T_{c}$
with respect to what is seen in the left panel. In the right panel (c) the
subcritical quench is so close to $T_{c}$ that the departure from the
metastable state cannot even be measured within the simulated times. Besides
that, one sees a pattern similar to the one observed in the other panels, with
the notable difference that the correlations in the metastable state are
almost indistinguishable from the ones of equilibrium at $T_{c}$. This
clarifies our statement that the metastable state is similar to the
equilibrium one at $T_{c}$: the closer to $T_{c}$ the quench is (from below),
the longer the life-time of the metastable state and the more similar to
equilibrium at $T_{c}$ it is. More precisely, indicating with ${\cal
S}_{T}(t)$ a one-time observable measured in a system quenched to $T\lesssim
T_{c}$, one has $\lim_{T\to T_{c}^{-}}{\cal S}_{T}(t)={\cal S}_{eq,T_{c}}$,
where the latter is the measurement of ${\cal S}$ made in equilibrium at
$T_{c}$. Clearly, this cannot be true for any function ${\cal S}$ of the state
of the system, particularly if it is weighting hugely the large distance
features, but the statement is expected to be correct for most quantities of
physical interest.

The nucleation time $\tau(q,T)$ can be roughly identified as the time when the
plateau is over and $R$ jumps rather abruptly to a higher value, see Fig. 1.
This is a violent process corresponding to the fast invasion of available
space by the super-critical nuclei, as it can be seen from the snapshots in
Fig. 2 corresponding to $t=4\cdot 10^{4}$ and $t=6\cdot 10^{4}$. It was shown
in [50] that $R(t)$ increases exponentially in this time domain. Such speedy
relaxation becomes sharper upon increasing $q$.

The duration $\tau(q,T)$ of the metastable state increases as $T\to
T_{c}^{-}$, as it can be seen in the inset of Fig. 1(a). In the same picture
we can also observe that the plateau value $R_{m}$ is rather $T$-independent
while, instead, there is a clear dependence on $q$. In [50] it was found that
this value is well approximated by $R_{m}\propto[\ln(q-4)]^{-1}$ for small $q$
($q\lesssim 50$) and then, increasing further $q$, it saturates to a value of
order one. This dependence can be understood recalling that $R_{m}$ is the
inverse energy density excess (Eq. [4]). The energy density $e_{m}$ in the
metastable state was computed in [34] and goes to zero for large $q$ as
$e_{m}\simeq-a(T)q^{-1/2}$, where $a(T)>0$ is a temperature dependent factor.
On the other hand the asymptotic equilibrium quantity $e(\infty)<0$ becomes
independent on $q$ [52] for large $q$. Therefore we conclude that $\Delta
e_{m}=e_{m}-e(\infty)$ approaches a large-$q$ value from below, as observed.
Given the choice of temperatures made in Fig. 1(a) (so as to have roughly
$\tau(q,T)$ fixed) it is clear that, working instead at a constant $T$ or even
at a finite fraction of $T_{c}$, i.e. $T=x\cdot T_{c}$, the lifetime of
metastability increases with $q$. Similarly, we have also noticed that the
temperature range where metastability occurs widens for larger $q$.
Specifically, indicating with $T_{\ell}$ the lower temperature $T$ where
metastability is still observed we find that $(T_{c}-T_{\ell})/T_{c}$
increases with $q$.

The fast process associated to the steep increase of $R(t)$ ends at a time
$t_{coal}$ when nuclei of different colour come in contact, a configuration
that can be observed in Fig. 2 at times $t=8\cdot 10^{4}$ and/or $t=10^{5}$
(except for the smallest size $L=80$ where one single colour nucleates, see
the discussion in Sec. 4). At this point a coarsening phenomenon sets in
produced by the competition among domains, as shown in Fig. 2 and reported in
[36, 37, 43, 41, 42, 44, 45, 46, 38] and, for the closely related vector Potts
(or clock) model, in [51]. In this kind of evolution one expects dynamical
scaling and $R(t)\sim t^{1/2}$, which is indeed roughly noted in Fig. 1 at
very long times (for $q=100$ this behavior is likely to be reached on times
longer than the simulated ones). Since this regime is not the focus of this
paper we content with such a semi-quantitative indication.

Before closing this section let us comment on the fact that the data for $R$
are free from finite-size effects until domains coarsen up to a length
comparable to the system size, which occurs at a time $t_{fs}(L)$. This can be
seen in Fig. 1 (b) where one sees that the curves depart from the one
corresponding to the larger system size ($L=1000$) when $R(t)\simeq 0.1\cdot
L$. For instance, this occurs at a time $t_{fs}(L=300)$ of order $6\cdot
10^{5}$, and around $t_{fs}(L=160)\simeq 10^{5}$. However, the fact that this
particular quantity ($R$) does not feel the size of the system before
$t_{fs}(L)$ does not mean that the dynamics is globally free from finite-size
effects in this time domain. Actually, in Fig. 2 one clearly sees that the
configurations at a given time look very different in systems of different
sizes. The next section is devoted to the discussion of this feature.

##  4 Multi-nucleation

In the introduction we have briefly discussed the fact that, at variance with
the Ising field driven first order transition, in the thermal one the role of
the system size is important. The reason is that $\tau(q,T)$ and the pseudo-
spinodal temperature depend on system size. In the previous section we have
already anticipated that, besides this feature, also the nature of the dynamic
escape from the metastable state changes significantly with the system size,
as a visual inspection of Fig. 2, where the same quench of the $q=24$ model is
operated on systems of different sizes, clearly displays. Specifically, when a
relatively small size is considered ($L=80$, upper row) a single critical
droplet forms and grows, invading the whole system (last two snapshots). This
is what we call mono-nucleation, or 1-nucleation. We have checked that such
phenomenon is observed rather independently of the thermal realisation of the
process. Notice that the dominating colour (orange) did not seem to be the
favoured one before its outburst (one would have rather predicted the violet
or yellow colour as better candidates), a fact that shows the rapidity of
critical nucleation.

In the second row the behaviour of a slightly larger system ($L=92$) is shown.
This small size increase is sufficient to modify qualitatively the situation,
in that there are now two nucleating colours, the red and the violet ones.
Then, in this case, we have bi-nucleation, or 2-nucleation. Again, this
feature is rather independent of the thermal realisation. After nucleation,
surface tension closes the domain of the minority colour (last snapshot),
symmetry is definitely broken and equilibrium is attained, but this occurs on
much larger times than those of nucleation.

Increasing further the system size as in the two rows below, the fate of a
system changes again and one observes 3-nucleation and 6-nucleation. Notice
that, after nucleation, the competition among domains of different colours
leads to a progressive elimination of their number. Increasing further $L$
(not shown) one can observe $k$-nucleation with $k$ up to $q$ ($q=24$ in this
example).

(a) (b) (c)

Figure 4: The typical lengths ${\cal R}_{n}$ of the various colours are
plotted against time for the model with $q=9$ quenched to $T=0.9912\cdot
T_{c}$. The log-log scale is the same for the three panels. The black dashed
line is the average ${\cal R}(t)$. The system sizes are $L=300$ (a), $L=700$
(b) and $L=1000$ (c). Each curve represents an average over $500$ non-
equilibrium realisations.

In order to discuss the multi-nucleation process and its dependence on the
system size at a more quantitative level we consider the typical lengths of
the various colours, ${\cal R}_{n}(t)$, which are shown for three different
system sizes in Fig. 4. From this figure one clearly sees that ${\cal R}_{n}$
for the various colours initially grow, up to a characteristic time
$t_{n}^{*}$ when they reach a maximum ${\cal R}^{*}_{n}$ and then shrink. This
means that domains of the corresponding colour grow, reach a maximum size, and
then collapse and disappear. For the prevailing colour (and possibly the next
ones in the hierarchy also reaching domain sizes comparable to the system size
itself) denoted by #1 in Fig. 4, the interpretation is different, since the
final decrease of ${\cal R}_{1}$ must be associated to the invasion of the
whole space, according to what discussed in case ii) at the end of Sec. 2.

(a) (b)

(c) (d)

Figure 5: The maximum of ${\cal R}_{n}(q,T,L)$, called ${\cal
R}^{*}_{n}(q,T,L)$, is plotted in log-log scales against the rescaled size
$L/L_{n}(q,T)$ (see the text) for $q=9$ (a), $q=24$ (b) and $q=100$ (c). The
system sizes are $L=200,\ 250,\ 300,\ 400,\ 500,\ 600,\ 700,\ 800$ and
different colours correspond to different $L$ data. The dashed black lines are
the linear behavior $y\sim x$. The inset in (b) contains the unscaled data
(${\cal R}^{*}_{n}(q,T,L)$ vs. $L$) for $q=24$ and various values of $n$ (from
the most abundant phase on the top, to the less abundant on the bottom). In
(d) the dependence of $L_{n}$ on the colour $n$ is plotted for the three
values of $q$, in a semilog plot.

From the comparison between the three different system sizes represented in
Fig. 4, one can make some remarks. First, it is clear that for small sizes
some colours do not nucleate, whereas they do in larger systems. For instance,
colour #9, represented in blue, does not show the exponential increase typical
of nucleation for $L=300$ and ${\cal R}_{9}$ does not go beyond $\sim 1$,
whereas it does grow significantly in the larger systems with $L=700$ and
$L=1000$. This indicates that ${\cal R}_{n}$ is a quantity one can use to
establish the number of nucleating colours. Secondly, from Fig. 4 one sees
that both the peak time $t_{n}^{*}$ and its height ${\cal R}^{*}_{n}(q,T,L)$
are increased in a larger system. Exploiting these two observations we
developed a method to determine the number of nucleating phases as a function
of $L$ and $q$, that we now describe.

First, we measure ${\cal R}_{n}^{*}(q,T,L)$ for all $n=1,\dots,q$ and
different system sizes. This information is summarised for $q=24$ in the inset
of Fig. 5 (b). The first three panels of this figure, (a)-(c), which refer to
different values of $q$, show that the data very clearly obey a scaling form

${\cal R}^{*}_{n}(q,T,L)=f_{q}\left[\frac{L}{L_{n}(q,T)}\right],$ (13)

where $L_{n}(q,T)$ is a fitting parameter which is plotted in Fig. 5 (d). A
similar pattern is found for other values of $q$, not reported in the figure.
Notice that imposing collapse of the curves according to Eq. (13) only fixes
$L_{n}(q,T)$ up to an arbitrary multiplicative constant. We fix it by asking
that the steep part of $f_{q}(x)$ be centred at $x=1$.

The scaling function $f_{q}(x)$ stays small for $x\ll 1$, it suddenly
increases around $x=1$ (the larger the $q$, the steeper the growth) and it
then behaves as $f_{q}(x)\sim x$ for large $x$ (see the dashed lines in Fig. 5
(a)-(c)). This latter trend indicates that, for each colour such that the
representative point lies in this sector, the maximum size of the domains is
triggered linearly by the system size, as it usually happens for a finite size
effect in a standard (e.g. binary) coarsening system (see also the discussion
at the end of this section). Hence, we argue that points belonging to this
sector of $x$, i.e. $x>1$, correspond to colours that, given the system size
$L$, have nucleated. Analogously, points in the small $x$ sector, i.e. $x<1$,
where $f_{q}(x)$ takes an approximately constant small value, correspond to
colours that did not succeed nucleating. On the basis of this, we conclude
that $x>1$, namely $L>L_{n}(q,T)$, is the condition for the $n$-th colour to
enter the coarsening stage. In particular, one can interpret $L_{1}(q,T)$ as
the smallest system size in order to have competition between domains of
different colour. This means that such a size must host the largest domain
present at $t_{coal}$ when coarsening starts. Hence we conclude that
$L_{1}(q,T)$ has the physical meaning of the typical size of the largest
droplets in the system at the coalition time $t_{coal}$. This can be
appreciated in Fig. 6 where a ruler of length $L_{1}(q,T)$ is plotted on top
of the configurations of the model at $t=t_{coal}$.

Figure 6: Snapshots at $t=10^{5}$ of the $q=24$ Potts model quenched to
$T=0.98\cdot T_{c}$. This is approximately the time $t_{coal}$ when
coarsening, that is competition between touching domains, starts (see Fig. 2).
The black ruler is the length $L_{1}(q,T)$. One observes $k$-nucleation with
$k=1$ for $L=80$, $k=2$ for $L=92$, $k=3$ for $L=110$ and $k=7$ for $L=160$.

We also observe that the small-$x$ plateau value of $f_{q}(x)$ decreases with
increasing $q$. This can be explained recalling that points laying there are
representative of colours that cannot nucleate, and arguing that, as such,
their properties are basically those of the metastable state. The latter, in
turn, is akin to the equilibrium state at $T_{c}$, as discussed, whose
coherence length decreases upon increasing $q$.

Looking at the plot of $L_{n}(q,T)$ in Fig. 5 (d), one finds an exponential
dependence

$L_{n}(q,T)=\ell(q,T)\,e^{n/{\cal N}(q)}.$ (14)

This law describes well all the data for $q=9,24$, and the large $n$ region
for $q=100$ (with fitting parameters $\ell(q,T)=59.43$, 82.97, 200.00 and
${\cal N}(q)=3.546$, 11.848, 67.568 for $q=9,24,100$, respectively).
Furthermore, we found an analogous behaviour for other values of $q$ not
portrayed in Fig. 5. Repeating the procedure above for different temperatures
we find that ${\cal N}$ is, within errors, independent of $T$. This is shown
in the inset of Fig. 7 (a) for a fixed value of $q$, $q=24$ in this case. It
seems that the behaviour reported can be ascribed more to some random effect,
or errors, rather than to a genuine dependence (notice also that the
fluctuations of ${\cal N}$ are only around 10% of ${\cal N}$ itself). Similar
results are found for other values of $q$. This explains the parameter
dependencies written in Eq. (14).

As shown in Fig. 7 (a) ${\cal N}(q)$ turns out to increase in a roughly linear
way with $q$. In this case, at variance with the $T$ dependence discussed
before, the variation of ${\cal N}(q)$ is monotonic and important, as compared
to ${\cal N}(q)$ itself. Moving to $\ell(q,T)$, its dependence on $T/T_{c}(q)$
(at fixed $q=24$) can be appreciated in inset of the right panel of Fig. 7,
where one finds that $\ell(q,T)$ increases in a roughly linear way with $T$.
On the other hand, with the choice of $T$ made in our simulations, which as
discussed earlier correspond to fix the lifetime $\tau(q,T)$ of the metastable
state, the dependence of $\ell(q,T)$ on $q$, shown in the right panel of Fig.
7, cannot be straightforwardly interpreted.

(a) (b)

Figure 7: (a) The dependence of ${\cal N}$ introduced in eq. (14), on $q$ by
choosing $T$ at a $q$-dependent value such that the lifetime $\tau(q,T)$ of
the metastable state is kept constant. In the inset the dependence of ${\cal
N}$ on $T/T_{c}(q)$, for $q=24$, is displayed, showing that it is not
significative (notice the narrow interval on the $y$-axis). (b) The dependence
of $\ell$, also defined in eq. (14), on $q$ with the same choice of a
$q$-dependent $T$ as in (a). In the inset, the variation of $\ell$ as a
function of $T/T_{c}(q)$ for $q=24$.

We now discuss the meaning of the length $\ell(q,T)$ appearing in Eq. (14). We
consider an intermediate range of values of $q$ such that the following two
conditions are obeyed: a) $q$ is sufficiently small as to have the exponential
form (14) valid down to $n=1$ (as already commented, this is true for $q=9,24$
but not for $q=100$) and b) ${\cal N}(q)\gg 1$. For example, the two
conditions are reasonably obeyed for $q=24$, since ${\cal N}(q=24)=11.848\gg
1$. In such an intermediate range of $q$, after Eq. (14) one has
$\ell(q,T)\simeq L_{1}(q,T)$, the typical size associated to the prevailing
colour whose interpretation was discussed above. Hence $\ell(q,T)$ amounts to
the typical size of the largest droplets in the system at the coalition time
$t_{coal}$. For $q$ outside such intermediate range, e.g. for large $q$, we
expect $\ell(q,T)$ to have a related interpretation, even though we cannot
derive it directly from Eq. (14). Let us also observe that from Fig. 7 one
extrapolates $\ell(q,T)\to\infty$ for $T\to T_{c}$, implying that the
metastable state becomes the stable equilibrium one, as expected.

Coming to the issue of multi-nucleation, applying the argument above to
$L_{n}(q,T)$, we can say that the $n$-th colour has nucleated if $L>L_{n}$.
Using Eq. (14) we arrive at the conclusion that, given a system size $L$, one
has $k$-nucleation with

$k(q,T,L)=\left\\{{\cal
N}(q)\ln\left[\frac{L}{\ell(q,T)}\right]\right\\}_{\in[1,q]},$ (15)

where the notation $\\{Z\\}_{\in[1,q]}=\max\\{1,\min\\{q,Z\\}\\}$ simply
accounts for the constraint $k\in[1,q]$. One can easily check that, using the
values of ${\cal N}(q)$ and $\ell(q,T)$ in this equation, one can correctly
predict the number of nucleating colours observed in the various cases of Fig.
2. Notice however that the domain of validity of this equation is inherited by
that of Eq. (14), hence if $q$ is large it holds true only for sufficiently
large values of $k$.

This result shows that, for a given $q$, the number of nucleating phases grows
only logarithmically with the system size. When the thermodynamic limit is
taken from the onset, all phases nucleate. Also, if the size of the system is
large enough, i.e. $L\gg\ell(q,T)$, increasing $q$ reduces the number of
nucleating phases. Let us also notice that Eq. (15) informs us on the number
of nucleating phases, but does not predict the time needed to nucleate.
Clearly, if this time grew towards infinity, no nucleation would be found. In
the large $q$-limit, it was also observed that a different mechanism takes
over, see for instance Fig. 6(c)(d) in [35], with, possibly, a different
scaling of $k$ with $L$ that still needs to be investigated in depth.

The discussion above shows that the multi-nucleation dynamics of the two
dimensional Potts model is interested by peculiar finite size effects, rather
different from the usual ones arising in the phase-ordering of binary systems.
In the latter, there are only two competing colours and the growth of the
domains does not feel the finiteness of the system until some time
$t_{end}(L)$, when the domains’ size becomes comparable to $L$. At this point
the coarsening process is interrupted and the system enters a final stage
whereby equilibrium is approached. For times $t\ll t_{end}$ any (intensive)
measurement does not depend on the value of $L$. In the Potts model, one
observes an analogous behaviour when observing a quantity like $R$, as we
discussed at the end of Sec. 3. This is also true also for the sizes ${\cal
R}_{1}$ and ${\cal R}_{2}$ of the two prevailing colours. However, the number
$k$ of coarsening phases depends on $L$ up to a value of $L$ as large as
$L_{q}$, given in Eq. (14). Notice that this characteristic size diverges both
as $T\to T_{c}$ for a given $q$ and as $q\to\infty$ for a given $T$, as
discussed above. Worst, the sizes ${\cal R}_{n}$ ($n\in[3,q]$) of the domains
of the minority colours, do depend on $L$ even if measured at times $t\ll
t_{end}$ (see Fig. 4). This strange behaviour is perhaps at the origin of many
controversies on the size dependence of the metastable dynamics.

One might wonder how it may be possible that the unrestricted quantity $R$ be
size-independent whereas the restricted ones ${\cal R}_{n}$ do depend on $L$,
given that $R$ is morally the average of the ${\cal R}_{n}$s. This occurs
because $R$ only fixes the average size of domains but not their colour, as it
is schematically illustrated in Fig. 8. The configuration on the left shows a
portion of a system of size $L_{A}$ at a time $t$ where six phases are
present. On the right, instead, an analogous system with a different size
$L_{B}<L_{A}$ is represented at the same time $t$. The represented portion of
the first system equals the total size of the second. It can be seen that,
although two phases (the green and the brown one) are absent in the smaller
lattice (hence the number of colours is $L$ dependent), the size of the
domains does not change (hence is $L$ independent).

Figure 8: Two pictorial configurations of the bidimensional Potts model on the
square lattice. The system on the right has a size $L_{B}$ and the picture
represents the whole lattice. The one on the left has a larger size
$L_{A}>L_{B}$ but in the picture only a part of it, of size $L_{B}$, is shown.
The two systems are characterised by the same size of the domains, but in the
right one the green and brown phases have disappeared.

##  5 Discussion

In this paper we studied the multi-nucleation phenomenon after a shallow
quench of the Potts model with $4<q\leq 100$ to $T\lesssim T_{c}(q)$. A
related study of the model with $q\gg 100$ addressing deep quenches to $T\ll
T_{c}(q)$ is presented in a companion article by some of us [35].

The main result of this paper, which is contained in Eq. (15), is that the
number of nucleating phases of the two-dimensional Potts model with not too
large $q$ increases logarithmically with the system size. The logarithmic
behaviour informs us that this feature cannot be related to the trivial
geometrical fact that a larger system can accommodate a larger number of
developing colours. If this were the origin, the snapshots in Fig. 2 should
display an equal number of nucleating phases in portions with equal area of
systems with different sizes. This, however, is not observed. For instance,
comparing the systems of relative double size $L=80$ (first line) and $L=160$
(last row), at time $t=80000$, one sees that there is only one nucleating
colour (orange, besides a remnant of the disordered phase) in the smallest
system, whereas there are typically more than one colour in a portion of area
$80\times 80$ of the system with $L=160$. This raises an important question
which, by the way, concerns also the size dependence of the metastable state
recalled in the introduction: how can a system with a very short coherence
length (of the order of the nuclei size, as it can also be appreciated from
the correlation function of Fig. 3) feel the boundary effects? Rephrased
differently, if nucleation is a local process, since correlations are short,
how could it be influenced by a global property such as the total size?

Here we provide a possible, although at the moment completely speculative,
answer. In a series of papers [53, 54, 55] some of us showed that in a rather
broad class of $2d$ binary models the fate of the system, namely which of the
two colours will eventually invade the sample, is decided at a very early
stage when a percolation cluster grows as large as the whole sample. Since
percolation is an uncorrelated phenomenon, this occurs when spatial
correlations are still minimal and the ferromagnetic domains microscopic.
However, such tenuous invisible structure determines the phase that will
eventually (at much longer times) win the competition simply by touching the
boundaries. Therefore, it may be conjectured that something similar – in a way
to be better investigated – also happens in the present case with $q$ colours.
Namely, a large but slender structure (not necessarily with a critical
percolation topology as in the case with $q=2$), which cannot be easily
associated to the pattern of the nucleating domains, could invade the sample
till its boundaries, and determine the fate of the colours. This seems to
agree with the observation, made regarding Fig. 3, that within the metastable
state $C(r,t)$ does not vary in time at small $r$, whereas it does so at large
distances. Which are the geometrical features to look for to identify such
large scale structure, if any, is still an open question.

The mechanism for ordering in systems with $q\gg 100$ is rather different from
the one discussed here. In these systems after an extremely long waiting
period (the longer the closer to $T_{c}(q)$ at fixed $q$, or the larger $q$ at
$T/T_{c}(q)$ fixed) the systems suddenly jump to a partially ordered state
with large and blocked domains. No “sand” (disordered regions between clusters
that can be seen in Fig. 2) lubricates the motion of the domain walls in this
case and the interfaces are mostly straight. Although the quench is to high
$T$, even close to $T_{c}(q)$, these blocked states resemble the ones at zero
temperature. We postpone the analysis of the ordering process under these
($q,T/T_{c}(q))$ conditions to a future publication. Still, an idea of the way
in which the typical length scale $R$ increases in time in this regime of
parameters can be reached by looking at Fig. 6 (c)-(d) in [35].

Acknowledgements We thank A. Vezzani for very useful comments, and F. Chippari
with whom two of us performed a parallel study of quenches towards parameters
farther away from $T_{c}(q)$ [35].

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# SS-SFR: Synthetic Scenes Spatial Frequency Response on Virtual KITTI and
Degraded Automotive Simulations for Object Detection

Daniel Jakab1,5 1Dept. of Electronic and Computer Engineering, University of
Limerick, Castletroy, Co. Limerick V94 T9PX, Ireland
2Faculty of Electrical Engineering & Information Technology , University of
Applied Sciences, Düsseldorf 40476, Germany
3Dept. of Electrical and Electronic Engineering, University of Galway, Galway
H91 TK33, Ireland
4Valeo Vision Systems, Tuam, Galway DY1 22DJ, Ireland
5Lero the Science Foundation Ireland Research Centre for Software, University
of Limerick, Limerick, V94 T9PX, Ireland
Alexander Braun2 1Dept. of Electronic and Computer Engineering, University of
Limerick, Castletroy, Co. Limerick V94 T9PX, Ireland
2Faculty of Electrical Engineering & Information Technology , University of
Applied Sciences, Düsseldorf 40476, Germany
3Dept. of Electrical and Electronic Engineering, University of Galway, Galway
H91 TK33, Ireland
4Valeo Vision Systems, Tuam, Galway DY1 22DJ, Ireland
5Lero the Science Foundation Ireland Research Centre for Software, University
of Limerick, Limerick, V94 T9PX, Ireland
Cathaoir Agnew1 1Dept. of Electronic and Computer Engineering, University of
Limerick, Castletroy, Co. Limerick V94 T9PX, Ireland
2Faculty of Electrical Engineering & Information Technology , University of
Applied Sciences, Düsseldorf 40476, Germany
3Dept. of Electrical and Electronic Engineering, University of Galway, Galway
H91 TK33, Ireland
4Valeo Vision Systems, Tuam, Galway DY1 22DJ, Ireland
5Lero the Science Foundation Ireland Research Centre for Software, University
of Limerick, Limerick, V94 T9PX, Ireland
Reenu Mohandas1,5 1Dept. of Electronic and Computer Engineering, University of
Limerick, Castletroy, Co. Limerick V94 T9PX, Ireland
2Faculty of Electrical Engineering & Information Technology , University of
Applied Sciences, Düsseldorf 40476, Germany
3Dept. of Electrical and Electronic Engineering, University of Galway, Galway
H91 TK33, Ireland
4Valeo Vision Systems, Tuam, Galway DY1 22DJ, Ireland
5Lero the Science Foundation Ireland Research Centre for Software, University
of Limerick, Limerick, V94 T9PX, Ireland
Brian Michael Deegan3,5 1Dept. of Electronic and Computer Engineering,
University of Limerick, Castletroy, Co. Limerick V94 T9PX, Ireland
2Faculty of Electrical Engineering & Information Technology , University of
Applied Sciences, Düsseldorf 40476, Germany
3Dept. of Electrical and Electronic Engineering, University of Galway, Galway
H91 TK33, Ireland
4Valeo Vision Systems, Tuam, Galway DY1 22DJ, Ireland
5Lero the Science Foundation Ireland Research Centre for Software, University
of Limerick, Limerick, V94 T9PX, Ireland
Dara Molloy4 1Dept. of Electronic and Computer Engineering, University of
Limerick, Castletroy, Co. Limerick V94 T9PX, Ireland
2Faculty of Electrical Engineering & Information Technology , University of
Applied Sciences, Düsseldorf 40476, Germany
3Dept. of Electrical and Electronic Engineering, University of Galway, Galway
H91 TK33, Ireland
4Valeo Vision Systems, Tuam, Galway DY1 22DJ, Ireland
5Lero the Science Foundation Ireland Research Centre for Software, University
of Limerick, Limerick, V94 T9PX, Ireland
Enda Ward4 1Dept. of Electronic and Computer Engineering, University of
Limerick, Castletroy, Co. Limerick V94 T9PX, Ireland
2Faculty of Electrical Engineering & Information Technology , University of
Applied Sciences, Düsseldorf 40476, Germany
3Dept. of Electrical and Electronic Engineering, University of Galway, Galway
H91 TK33, Ireland
4Valeo Vision Systems, Tuam, Galway DY1 22DJ, Ireland
5Lero the Science Foundation Ireland Research Centre for Software, University
of Limerick, Limerick, V94 T9PX, Ireland
Tony Scanlan1 1Dept. of Electronic and Computer Engineering, University of
Limerick, Castletroy, Co. Limerick V94 T9PX, Ireland
2Faculty of Electrical Engineering & Information Technology , University of
Applied Sciences, Düsseldorf 40476, Germany
3Dept. of Electrical and Electronic Engineering, University of Galway, Galway
H91 TK33, Ireland
4Valeo Vision Systems, Tuam, Galway DY1 22DJ, Ireland
5Lero the Science Foundation Ireland Research Centre for Software, University
of Limerick, Limerick, V94 T9PX, Ireland
Ciarán Eising1,5 1Dept. of Electronic and Computer Engineering, University of
Limerick, Castletroy, Co. Limerick V94 T9PX, Ireland
2Faculty of Electrical Engineering & Information Technology , University of
Applied Sciences, Düsseldorf 40476, Germany
3Dept. of Electrical and Electronic Engineering, University of Galway, Galway
H91 TK33, Ireland
4Valeo Vision Systems, Tuam, Galway DY1 22DJ, Ireland
5Lero the Science Foundation Ireland Research Centre for Software, University
of Limerick, Limerick, V94 T9PX, Ireland

###### Abstract

Automotive simulation can potentially compensate for a lack of training data
in computer vision applications. However, there has been little to no image
quality evaluation of automotive simulation and the impact of optical
degradations on simulation is little explored. In this work, we investigate
Virtual KITTI and the impact of applying variations of Gaussian blur on image
sharpness. Furthermore, we consider object detection, a common computer vision
application on three different state-of-the-art models, thus allowing us to
characterize the relationship between object detection and sharpness. It was
found that while image sharpness (MTF50) degrades from an average of
0.245cy/px to approximately 0.119cy/px; object detection performance stays
largely robust within 0.58%(Faster RCNN), 1.45%(YOLOF) and 1.93%(DETR) across
all respective held-out test sets.

Keywords: Automotive Simulation, Object Detection, Gaussian Blur, Image
Sharpness, Modulation Transfer Function(MTF)

##  1 Introduction

Autonomous vehicles require large amounts of data to anticipate safety-
critical scenarios and corner cases on the road. However, even the largest and
most influential automotive datasets (such as BDD100k [Yu et al., 2020] and
KITTI [Geiger et al., 2013]) only capture a small subset of possible
scenarios. Simulation is promising as it is relatively feasible to set up an
environment that can reflect real life and be utilized for perception systems.
The main issue with simulation is that there has been little to no quality
evaluation of this data. There are considerable differences between simulation
and real life as discussed in a recent survey [Jakab et al., 2024a] where
automotive simulators lack the simulation of lenses from the real world. We
can refer to this concept as the ’syn-to-real-gap’ where the need to bridge
the gap between simulation and real life must be realised. As a means of
measuring the image quality of automotive simulation, we measure the
Modulation Transfer Function(MTF) curve where it falls off to 50%, also known
as MTF50 and is a typical measurement for cameras adopted by the industry
[Mueller et al., 2020, Burns et al., 2022]. MTF50 is a scalar value selected
from the MTF curve which indicates image sharpness in capturing details
through the lens of a camera. We use the sfrmat5 algorithm by Peter Burns with
updates made recently [Burns et al., 2022], a Matlab software script function,
which provides a Spatial Frequency Response (SFR) (otherwise known as the
Modulation Transfer Function) from digital image files using the Slanted Edge
Method. It follows the ISO12233:2023 standard recently published [Institute,
2023] to resolve photographic camera measurements [da Silva et al., 2021]. We
obtain the measurement of MTF50 directly from the simulation scenes of Virtual
KITTI. Recent work has shown that natural scenes can be measured in public
datasets to obtain the Edge Spatial Frequency Response (e-SFR), especially in
autonomous vehicles [Jakab et al., 2024b, Van Zwanenberg et al., 2021]. This
strategy is also known as Natural Scenes Spatial Frequency Response (NS-SFR)
[van Zwanenberg et al., 2023, van Zwanenberg, 2022, Van Zwanenberg et al.,
2021]. In this paper, we apply the NS-SFR approach to synthetic scenes in
simulation called Synthetic Scenes Spatial Frequency Response (SS-SFR) using
the Virtual KITTI dataset. The main contributions of this paper are:

  * •

Assess simulation as an alternative to real-world data collection.

  * •

Degrade simulation with Gaussian blur[Szeliski, 2011] to simulate the effects
of an out-of-focus lens.

  * •

Investigate how image sharpness is impacted in the degraded simulations.

  * •

Compare image sharpness to object detection to see how computer vision behaves
with degradation.

##  2 Related Work

Automotive simulation has recently emerged as a new field in the research of
autonomous vehicles. Blueprint measurements of road test tracks for autonomous
vehicles have been proven useful to reconstruct road surfaces and terrain in
ASAM OpenDRIVE simulations[of Standardization of Automation and Systems,
2024]. Simulations of this nature are also widely known as model-driven
simulations. As an example, virtual simulation models were publicly released
for ZalaZONE[Zal, 2020], an automotive proving ground for autonomous driving
research. These models can be exported to any model-driven simulator (i.e.
Unity, Unreal Engine, and Vires VTD, to name but a few). The latest research
shows a clear trend towards the use of simulation in computer vision models
[Gaidon et al., 2016, Cabon et al., 2020, Saad and Schneider, 2019, Carlson et
al., 2019]. The Virtual KITTI dataset, a model-driven simulation dataset
created in the Unity Simulator, was first released by [Gaidon et al., 2016].
This was one of the first automotive simulation datasets annotated for
computer vision. Qualitatively, the dataset lacked fundamental aspects of
photorealism but was sufficient for training algorithms. The visual quality of
automotive simulation has drastically improved since then [Cabon et al.,
2020]. Despite this, considerable differences remain compared to real-life. A
recent concept [Carlson et al., 2019] shows some work in degrading automotive
simulation using image augmentations to introduce realistic optical artefacts
for computer vision. However, there is no evaluation of image quality in this
work which suggests an inconclusive analysis. In [Mueller et al., 2020], a
framework is demonstrated to simulate and validate the realism of camera
simulations using the ISO12233 standard for camera measurement. ISO12233 is an
established method to measure the resolution of a camera system also known as
the SFR [Mueller et al., 2020]. This framework shows how closely simulations
can model the real-life lens from the laboratory [Zernike, 1934]). Inspired by
both sets of work [Carlson et al., 2019, Mueller et al., 2020], it is
interesting to further investigate how the ISO12233 standard with its latest
updates[Institute, 2023] can validate automotive simulation.

##  3 Methodology

(a)  Figure 1: Virtual KITTI Gaussian blur image degradation between
$\sigma=[1,2,3]$.

To understand how image sharpness behaves in simulation, we shall proceed to
do the following:

  1. 1.

Choose the clone camera perspectives from the latest version of Virtual KITTI,
consisting of 2,126 images in total [Cabon et al., 2020].

  2. 2.

Perform degradations by using a Gaussian blur filter [Szeliski, 2011]:

$G(x,y)=\frac{1}{2*\pi*\sigma^{2}}*e^{-\frac{x^{2}+y^{2}}{2*\sigma^{2}}}$ (1)

Where G(x,y) is the degraded two dimensional image in terms of x and y where
$\sigma=[1,2,3]$ and kernel size is determined by:

$k=6*\sigma+1$ (2)

As shown in Figure 1, Virtual KITTI is degraded in three iterations by
adjusting the value of $\sigma=[1,2,3]$ in Equation 1. Therefore, the kernel
sizes for each $\sigma$ value are $(k_{\sigma},k_{\sigma})=[(7,7),(13,13),\\\
(19,19)]$, respectively. This degradation approach uses the OpenCV
library[Ope, 2024] to apply the Gaussian blur filter on each image creating a
degraded variation of the original dataset.

  3. 3.

Apply NS-SFR as demonstrated in previous work [Jakab et al., 2024b, van
Zwanenberg et al., 2023] to isolate and extract regions of valid slanted edges
from Virtual KITTI. All four dataset variations were processed using all
scenes with the clone camera perspectives.

  4. 4.

Measure image sharpness using the ISO12233:2023 Slanted Edge Method (sfrmat5)
[Institute, 2023, Burns et al., 2022] on all extracted edge regions from the
images of Virtual KITTI before and after each of the three Gaussian
degradations. Results are averaged for each of the four dataset variations
across all clone perspectives of Virtual KITTI.

  5. 5.

We utilize annotated Virtual KITTI for object detection where the dataset is
split into 80% training, 10% validation, and 10% test. The same split is used
for all four variations of Virtual KITTI where each degradation split contains
the same images.

  6. 6.

Three different state-of-the-art object detection models are chosen for
training where the $1242\times 375$ resolution images are resized according to
default configurations for the models [Chen et al., 2019] with a batch size of
2 in each training run:

    1. (a)

Faster RCNN[Ren et al., 2017], images resized by default to a scale of
$1333\times 800$ (see default configuration [Chen et al., 2019]).

    2. (b)

YOLOF[Chen et al., 2021], images resized by default to $1333\times 800$ (see
default configuration
[Chen et al., 2019]).

    3. (c)

DETR[Carion et al., 2020], images resized by default between $1333\times 480$
and $1333\times 800$ (see training approach in the original work [Carion et
al., 2020]).

All three models are trained on an NVIDIA GeForce RTX 3080. As a method for a
controlled experiment, the same ResNet50[He et al., 2016] backbone is used for
all three networks. The stopping criteria for both the Faster RCNN and YOLOF
models are determined using ‘1x’ configuration from the MMDetection toolbox
[Chen et al., 2019] which has a 12 epoch training cycle and the learning rate
decays by a factor of 10 at the 8 and 11 epochs, respectively. For DETR, the
default learning policy of 150 epochs is used where the learning rate decays
by a factor of 10 at the 100th epoch. Results and training logs for 150 epochs
are provided in the original work [Carion et al., 2020]. Transformer models
typically take longer to converge whereas the YOLOF model can reach the same
performance with seven times fewer training epochs than DETR [Chen et al.,
2021].

##  4 Results

In this section, we present the SS-SFR of Virtual KITTI and object detection
performance by comparing image sharpness(MTF50) to mean Average
Precision(mAP).

###  4.1 Virtual KITTI SS-SFR

For SS-SFR measurements, the Virtual KITTI clone perspective was evaluated in
four variations:

  1. 1.

Original or baseline dataset(i.e. No degradation applied).

  2. 2.

Degradation for $\sigma=1,[(k_{\sigma},k_{\sigma})[=[(7,7)]$.

  3. 3.

Degradation for $\sigma=2,[(k_{\sigma},k_{\sigma})[=[(13,13)]$.

  4. 4.

Degradation for $\sigma=3,[(k_{\sigma},k_{\sigma})[=[(19,19)]$.

####  4.1.1 Horizontal Edges

Horizontal MTF50 (HMTF50) can be seen in Table 1, four different results can
be measured for MTF50. The baseline dataset was measured to be around
0.234cy/px. This is notably higher than KITTI measurements from previous work
where horizontal edges averaged between 0.20-0.22cy/px [Jakab et al., 2024b].
This indicates that Virtual KITTI tends to have sharper edges than KITTI
images taken from a real-life camera.

Upon degrading the Virtual KITTI images notable image sharpness degradation
can be measured. With $\sigma=1$ and kernel size of $(k_{1},k_{1})=(7,7)$
MTF50 results degraded by 31.62%. With $\sigma=2$ and kernel size of
$(k_{2},k_{2})=(13,13)$ MTF50 results degraded by 49.15%. Finally, with
$\sigma=3$ and kernel size of $(k_{3},k_{3})=(19,19)$ MTF50 results degraded
produced the same percentage as for the previous degradation(i.e. 49.15%).
Please see Table 1 for HMTF50 results. For horizontal edges, both $\sigma=2$
and $\sigma=3$ columns generated the largest degradations compared to the
baseline indicating the little difference between both. The degradation of
edges follows that of a typical Gaussian blur with a radius of 2 which can be
measured around 0.117cy/px from a previous study[Ima, 2024].

Table 1: Comparison of degraded Virtual KITTI MTF50 measurements for both
horizontal and vertical edges to the baseline dataset.

sigma | kernel size | HMTF50 | % | VMTF50 | % | MTF50(mean) | %
---|---|---|---|---|---|---|---
baseline | baseline | 0.234 | - | 0.256 | - | 0.245 | -
1 | (7,7) | 0.160 | -31.62 | 0.159 | -37.89 | 0.160 | -34.69
2 | (13,13) | 0.119 | -49.15 | 0.122 | -52.34 | 0.120 | -51.02
3 | (19,19) | 0.119 | -49.15 | 0.119 | -53.52 | 0.119 | -51.43

  * •

H/VMTF50 = MTF50 for both horizontal(H) and vertical(V) edges where the mean
of both is taken as _MTF50(mean)_.

Table 2: Object Detection Results on all four degradations of Virtual KITTI.

Model | sigma | kernel size | mAP0.50:0.95(%) | mAP0.5(%) | mAP0.75(%) |  $mAP0.5:0.95_{s}$(%) |  $mAP0.5:0.95_{m}$(%) |  $mAP0.5:0.95_{l}$(%)
---|---|---|---|---|---|---|---|---
Faster RCNN | baseline | baseline | 69.45 | 81.24 | 76.46 | 51.28 | 86.76 | 90.20
1 | (7,7) | 69.05 | 80.54 | 75.57 | 51.13 | 86.31 | 90.68
2 | (13,13) | 69.06 | 80.59 | 75.22 | 51.02 | 86.27 | 90.20
3 | (19,19) | 69.17 | 81.75 | 75.16 | 56.09 | 85.56 | 90.02
YOLOF | baseline | baseline | 60.06 | 72.23 | 65.89 | 26.16 | 80.25 | 90.90
1 | (7,7) | 59.82 | 72.35 | 66.22 | 30.80 | 80.29 | 89.51
2 | (13,13) | 59.70 | 72.36 | 65.90 | 33.03 | 79.34 | 89.81
3 | (19,19) | 59.20 | 71.64 | 66.72 | 30.58 | 78.38 | 90.16
DETR | baseline | baseline | 61.91 | 86.33 | 65.90 | 43.95 | 72.95 | 91.09
1 | (7,7) | 62.28 | 85.82 | 67.19 | 41.83 | 73.04 | 90.41
2 | (13,13) | 61.15 | 84.49 | 63.79 | 46.18 | 70.54 | 90.46
3 | (19,19) | 61.10 | 85.32 | 65.32 | 42.32 | 70.55 | 89.12

  * •

Note: all mAP values are recorded between an Intersection over Union(IoU) of
0.5:0.95 except for mAP0.5(%) where IoU = 0.5 and mAP(0.75) where IoU = 0.75.
$mAP0.5:0.95_{x}$ indicates object area in $pixels^{2}$ where _x_ can be _s_ ,
_m_ or _l_. _s_ = small $\implies area<32^{2}$, _m_ = medium $\implies
32^{2}<area<96^{2}$ and _l_ = large $\implies area>96^{2}$[cocodataset.org,
nd].

####  4.1.2 Vertical Edges

(a) _Faster RCNN_

(b) _YOLOF_

(c) _DETR_

Figure 2: Mean MTF50 versus mean Average Precision with IoU of between 0.5 and
0.95 (mAP0.5:0.95) for the Virtual KITTI baseline and respective degradations
between $\sigma=[1,2,3]$. Mean MTF50 was calculated by averaging results
between HMTF50 and VMTF50 from Table 1 creating the following:
$MTF50(cy/px)=[0.119,0.120,0.160,0.245]$(see Table 1 for _MTF50(mean)_).

Similar to HMTF50, Vertical MTF50(VMTF50) can be seen in Table 1 where four
different results can be measured for MTF50. The baseline dataset was measured
to be around 0.256cy/px which is slightly higher than for horizontal edges.
With $\sigma=1$ and kernel size of $(k_{1},k_{1})=(7,7)$ MTF50 results
degraded by 37.89%. With $\sigma=2$ and kernel size of $(k_{2},k_{2})=(13,13)$
MTF50 results degraded by 52.34%. Finally, with $\sigma=3$ and kernel size of
$(k_{3},k_{3})=(19,19)$ MTF50 results degraded by 53.52%. Please see Table 1
for VMTF50 results. $\sigma=3$ generated the largest degradations compared to
the baseline.

###  4.2 Object Detection

Object detection results are shown in Table 2. As can be seen, the Faster RCNN
model produced the highest results where overall mAP0.5:0.95 was between
69.05-69.45%. The YOLOF model produced the lowest results where overall
mAP0.5:0.95 was between 59.20-60.06%. It is clear from Table 2 that the YOLOF
model did not perform as well with small objects where mAP0.5:0.95s was
between 26.16-33.03%. The Faster RCNN model was also 5.27% higher than YOLOF
for medium objects. After 150 epochs, DETR performance exceeded YOLOF by at
least 1.73% and mAP0.5:0.95 results were between 61.10-62.28%. DETR also had
the lowest results for medium objects where mAP0.5:0.95m was between
70.54-73.04%.

###  4.3 Discussion

While there are very small differences between both $\sigma=2$ and $\sigma=3$,
it can be concluded that $\sigma=3$ with $(k_{3},k_{3})=(19,19)$ provided the
largest degradation out of all variations (see Table 1). Object detection
results in Table 2, show that despite the degradation effects of Gaussian
blur, there is no substantial degradation in results and the models are
robust. For example, Figure 2 shows only small decreases in mAP0.5:0.95 of
approximately 0.58% for Faster RCNN (see Figure 2a), 1.45% for YOLOF (see
Figure 2b) and 1.93% for DETR (see Figure 2c) across all four variations of
the held-out test sets for Virtual KITTI. Faster RCNN is a two-stage object
detection architecture that predicts region proposals, allowing it to focus on
relevant regions in an image, leading to higher accuracy in detecting small
and dense objects in images. This is evident in the significant jump in mAP
for small objects(see Table 2). In contrast, YOLOF uses Uniform Matching to
achieve a balance on positive samples with sparse anchors where the
architecture bridges the gap in performance with Feature Pyramid Networks
(FPN) typically used with Faster RCNN. However, YOLOF struggles to detect
small objects closely packed together which is associated with the spatial
constraints of the YOLO algorithm. Using the ’1x’ configuration from
MMDetection[Chen et al., 2019], YOLOF can produce an overall mAP that is
within 1.04 mAP of DETR reflecting the observations of the original paper
where a relatively low number of epochs is required to match DETR performance
[Chen et al., 2021]. These findings might suggest that an out-of-focus camera
system, even with considerable image sharpness degradations, may not
significantly impact autonomous perception systems for object detection.

##  5 Conclusion

In this paper, we have investigated the impact of Gaussian blur on both image
sharpness and object detection performance of three different computer vision
models for the Virtual KITTI dataset. This is a crucial step in understanding
how image quality degradation might impact a scene in real life. Overall,
object detection results show a consistent trend in stable performance where
it was found that while image quality degrades from an average of 0.245cy/px
to approximately 0.119cy/px; object detection performance stays largely robust
where models trained on degraded images can generalize and learn Gaussian blur
from the images. However, a slight downward trend can be observed in some
cases where there is a decrease in values for the degraded images and the
baseline between the lowest and highest values. There are decreases of
approximately 0.58%(Faster RCNN), 1.45%(YOLOF), and 1.93%(DETR) across all
held-out test sets. Future work that can be considered is the impact of
Gaussian blur degradations in other applications such as Depth Estimation or
Instance Segmentation for Virtual KITTI. We also recognise that Gaussian blur
is a simplistic model of image degradation and other well-known degradations
exist such as the defocus blur corruptions of the Hendrycks and Dietterich
ImageNet-C dataset [Hendrycks and Dietterich, 2019]. In future work, we intend
to integrate more realistic models, such that we can examine the impact of
realistic camera degradations on computer vision performance. More realistic
optical degradation models can be, for example, an optical doublet or a
fisheye lens with ray-tracing performed and simulated to degrade images,
introducing realistic optical artefacts into simulation.

##  6 Acknowledgments

This work was supported with the financial support of the Science Foundation
Ireland grant 13/RC/2094_2. The authors of this work would like to thank Dr.
Eoin Martino Grua for his invaluable support and encouragement, shaping the
direction and the outcome of this study.

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