# Strains and pseudo-magnetic fields in circular graphene rings

Nima Abedpour  School of Physics, Institute for Research in Fundamental
Sciences, IPM, Tehran 19395-5531, Iran  Reza Asgari<EMAIL_ADDRESS>School of
Physics, Institute for Research in Fundamental Sciences, IPM, Tehran
19395-5531, Iran  F. Guinea  Instituto de Ciencia de Materiales de Madrid,
CSIC, Sor Juana Inés de la Cruz 3, E-28049 Madrid, Spin

###### Abstract

We demonstrate that circular graphene ring under a shear stress displays
strong pseudo-magnetic fields. We calculate the pseudo-magnetic field both
from continuum elasticity theory as well as molecular dynamics simulations.
Stable wrinkles are induced by shear deformations and lead to enhancement of
the pseudo-magnetic field. The strong pseudo-magnetic field found here can be
observed by imaging graphene flake at the atomic level e.g. through scanning
tunneling microscope.

###### pacs:

61.48.Gh, 81.40.Jj, 07.55.Db

##  I introduction

Graphene has recently attracted intensive interest as a promising candidate
material for the new generation of electronics and spintronics geim . One of
the exciting physics on graphene is strain exerted on graphene samples
vozmediano ; pereira ; guinea1 . It was proposed that strain can be utilized
to generate various basic elements for all-graphene electronics pereira .

Semiconductor quantum rings have been investigated by a number of groups bayer
; ribeiro ; Retal08 ; YPSR010 ; Hetal10 . In graphene ring, the spectrum
reveals signatures of effective time-reversal symmetry breaking, in which the
spectra are most naturally interpreted in terms of effective magnetic flux
contained in the ring, even when no real flux is present. Quantum rings can be
considered as prototypical devices in mesoscopic physics, as they show one of
the most basic coherence effects, namely the Aharonov-Bohm effect,
oscillations of the transmission as a function of the magnetic flux through
the ring. The reason for these oscillations is the phase difference between
electrons traveling along the different arms of the ring. Furthermore,
Benjamin and Pachos benjamin  proposed creating a ring of single-layer
graphene in which they induce d-wave superconductivity via the proximity
effect or directly make graphene superconducting by doping. The quantum qubits
would be built around the $\pi$-junction that naturally occurs in graphene and
would not require bilayer structures. The Aharonov Bohm oscillations, on the
other hand, has been observed Hetal10  in a graphene ring, consisting of a
planar honeycomb lattice of carbon atoms in a ring shape, by changing the
voltage applied to the side gate or the back gate.

When the graphene sheet is under tension, the side contacts induce a long-
range elastic deformation which acts as a pseudo-magnetic field for its
massless charge carriers suzuura ; maes . This is because strain changes the
bonds length between atoms and affects the way electrons move among them. The
pseudo-magnetic field would reveal itself through its effects on electron
orbits. The tension can be generated either by the electrostatic force of the
underlying gate fogler  by interaction of graphene with the side walls bunch ,
as a result of thermal expansion bao  or by quench height fluctuations guinea
. A particular strain geometry in graphene could lead to a uniform pseudo-
magnetic field and might open up interesting applications in graphene nano-
electronics with real magnetic fields low . Mechanical strains can introduce
new environments in studying novel physics of graphene.

It is commonly believed that strains have important influence on the
electronic structure of graphene neto . A graphene ring is a particularly
convenient geometry. Strains can possibly be manipulated efficiently in
samples with good adhesion to the substrate, such as graphene layers grown
epitaxially on SiC. Recently, the physical properties of graphene when its
hexagonal lattice is stretched out of equilibrium have been investigated by
many groups exp ; teague . Scanning tunneling microscopy studies on graphene
surface have indeed revealed a correlation between local strain and tunneling
conductance teague . Motivated by experiments pointing to a remarkable
stability of graphene with large strains, we have carried out theoretical
analysis and the molecular dynamics simulation to explore the pseudo-magnetic
field in strained graphene ring.

In this papers we focus on a particular aspect of the physics of graphene ring
namely the appearance of gauge fields and its corresponding pseudo-magnetic
fields which arise in the circular graphene ring when a shear force is applied
to its boundary. The common belief is that the morphologies of graphene
surface under strain are negligible. The aim of this paper is to show that,
contrary to these expectations, the structure of a deformed surface can lead
to a strong pseudo-magnetic field.

The paper is organized as follows. In Sec. II we introduce our model and
formalism. In Sec. III, our numerical results for the strain and the pseudo-
magnetic field for deformed graphene ring are presented. Finally, we conclude
in Sec. IV with a brief summary.

##  II Model and Theory

We analyze both analytically and numerically the strains and pseudo-magnetic
fields in circular graphene ring under a shear stress. A shear force is
applied to the boundary, inducing shear deformations inside graphene ring.

The properties of graphene presented lie on the special character of its low
energy excitations obeying a two dimensional massless Dirac equation. The
graphene ring with valley degree of freedom, $\tau=\pm 1$ for the inequivalent
$K$ and $K^{\prime}$ valleys, is modelled by the massless Dirac Hamiltonian in
the continuum model slon ; haldane

${\cal H}_{0}=\hbar v_{\rm
F}\tau\left(\sigma_{1}\,k_{1}+\sigma_{2}\,k_{2}\right)$

where $k_{i}$ is an envelope function momentum operator, $v_{\rm F}\simeq
10^{6}m/s$ is the Fermi velocity, and $\sigma_{i}=x,y,z$ are the Pauli
matrices that act on the sublattice pseudospin degree of freedom. Hence it is
important to establish the robustness of the low energy description under
small lattice deformations. The concepts of gauge fields and covariant
derivatives can be translated into the language of differential geometry,
based on differential forms. When ideal graphene is distorted, the effective
Hamiltonian will be changed into

$\displaystyle H=v_{\rm F}~{}(~{}\bf{p}-e~{}\bf{A})\cdot\bf{\sigma}$ (1)

where the induced vector potential field is defined through the deformations
of sample and $\bf{p}$ is the momentum in the polar coordinate. The induced
gauge fields can be calculated through the following expressions suzuura

$\displaystyle v_{\rm F}eA_{x}$ $\displaystyle=$ $\displaystyle\hbar
g_{2}(u_{xx}-u_{yy})$ (2) $\displaystyle v_{\rm F}eA_{y}$ $\displaystyle=$
$\displaystyle-2\hbar g_{2}u_{xy}$

with

$\displaystyle g_{2}$ $\displaystyle=$
$\displaystyle\frac{3\kappa\beta}{4}~{}t$ (3) $\displaystyle\kappa$
$\displaystyle=$ $\displaystyle\frac{\sqrt{2}\mu}{2B}$

where $t\approx 2.7$ eV is the nearest-neighbor hopping parameter and
$\beta=\partial\ln(t)/\partial\ln(a)\simeq 2$ is the electron Grüneisen
parameter. For the shear $\mu$ and bulk moduli, $B$ we have used zakharchenko
the values $\mu=9.95$ eV $\AA^{-2}$ and $B=12.52$ eV $\AA^{-2}$. We thus find
that $\kappa\approx 0.56$. Once the induced gauge filed is obtained, the
pseudo-magnetic field can be calculated by $\bm{B}=\nabla\times\bm{A}$.

When the graphene is deformed due to the force exerts on the boundaries, the
strain tensor can be calculated from

$\displaystyle
u_{\alpha\beta}=\frac{\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha}}{2}+\frac{\partial_{\alpha}h\partial_{\beta}h}{2}$
(4)

where $\bm{u}$ is the atomic displacement field and $h$ is the out of plane
displacement. The contribution of the out-of-plane atomic displacements are
noticeable in our numerical calculations.

###  II.1 Analytical expression for pseudo-magnetic filed

We consider the case of a graphene ring of radius $R$ and width $W=R-R_{1}$
where the clamped outer and inner boundaries are circles of radius $R$ and
$R_{1}$, respectively. Additionally, we assume that the out-of-plane atomic
displacements being zero, $h=0$. The displacement at the outer boundary is
$u_{\theta}(R,\theta)=U_{\theta},u_{r}(R,\theta)=0$, which the shear
deformation at the boundary is defined by $U_{\theta}$, and at the inner
boundary we have $u_{\theta}(R_{1},\theta)=u_{r}(R_{1},\theta)=0$. The
displacements in the graphene ring are

$\displaystyle u_{r}(r,\theta)$ $\displaystyle=0$ $\displaystyle
u_{\theta}(r,\theta)$
$\displaystyle=U_{\theta}\left[-\frac{R_{1}^{2}R}{r(R^{2}-R_{1}^{2})}+\frac{Rr}{R^{2}-R_{1}^{2}}\right]~{}.$
(5)

The second term in the expression for $u_{\theta}(r,\theta)$ is a pure
rotation, required to satisfy the boundary conditions. The only non-zero
component of the strain tensor is $u_{r\theta}$. The strain at radius $r$ is

$u_{r\theta}(r,\theta)=\frac{1}{2}\left(\frac{\partial u_{\theta}}{\partial
r}+\frac{1}{r}\frac{\partial
u_{r}}{\partial\theta}-\frac{u_{\theta}}{r}\right)=\frac{U_{\theta}R_{1}^{2}R}{r^{2}(R^{2}-R_{1}^{2})}$
(6)

The maximum strain at the boundary is actually given by
$\bar{u}=U_{\theta}R/(R^{2}-R_{1}^{2})$. Using polar coordinates $(r,\theta)$,
Eq. (2) can be rewritten as

$\displaystyle A_{r}=$ $\displaystyle\Phi_{0}$
$\displaystyle\frac{c\beta}{a}[(\frac{\partial u_{r}}{\partial
r}-\frac{u_{r}}{r}-\frac{1}{r}\frac{\partial
u_{\theta}}{\partial\theta})\cos(3\theta)$ $\displaystyle+$
$\displaystyle(-\frac{\partial u_{\theta}}{\partial
r}+\frac{u_{\theta}}{r}-\frac{1}{r}\frac{\partial
u_{\theta}}{\partial\theta})\sin(3\theta)]~{},$ $\displaystyle A_{\theta}=$
$\displaystyle\Phi_{0}$ $\displaystyle\frac{c\beta}{a}[(-\frac{\partial
u_{\theta}}{\partial r}+\frac{u_{\theta}}{r}-\frac{1}{r}\frac{\partial
u_{r}}{\partial\theta})\cos(3\theta)$ (7) $\displaystyle+$
$\displaystyle(-\frac{\partial u_{r}}{\partial
r}+\frac{u_{r}}{r}+\frac{1}{r}\frac{\partial
u_{\theta}}{\partial\theta})\sin(3\theta)]~{},$

where $\Phi_{0}=h/2e$ is the quantum unit of magnetic flux and
$c=\sqrt{3}\kappa/(2\pi)$ is a constant. Furthermore, the induced gauge field
in the polar coordinate is given by

$\displaystyle A_{r}=\Phi_{0}\frac{c\beta
U_{\theta}}{a}\frac{2R_{1}^{2}R}{r^{2}(R^{2}-R_{1}^{2})}\sin(3\theta)$
$\displaystyle A_{\theta}=\Phi_{0}\frac{c\beta
U_{\theta}}{a}\frac{2R_{1}^{2}R}{r^{2}(R^{2}-R_{1}^{2})}\cos(3\theta)$ (8)

Eventually, the pseudo-magnetic field acting on the electrons is simply given
by

$B(r,\theta)=\Phi_{0}\frac{c\beta}{a}\frac{8U_{\theta}R_{1}^{2}R}{r^{3}(R^{2}-R_{1}^{2})}\cos(3\theta)$
(9)

It would be worthwhile mentioning that the pseudo-magnetic field diverges near
the clamped sites as $B(r)\approx 8c\Phi_{0}\beta\bar{u}R_{1}^{2}/(ar^{3})$.

###  II.2 Molecular dynamics simulation

We used Molecular Dynamics simulation (MD) to simulate a suspended circular
graphene ring, consists of a planar honeycomb lattice of carbon atoms in a
ring shape. Rigidly clamped boundary conditions were employed. We simulated
the system at different temperatures by employing Nosè-Hoover thermostat to
help the system reach equilibrium at a given temperature. Our present results
are limited to $T=50K$. In this work, we used both the Brenner’s bond-order
potential brenner ; khodemoon  incorporating the second nearest neighbors
interactions through their dependence on the bond angles and the second-
generation reactive empirical bond-order (REBO) potential rebo  for the
carbon-carbon interactions (for more details see Appendix A). It is well
established that MD simulations predict the correct mechanical properties and
the form of the structures with carbon atoms by using the Brenner potential.

The number of carbon atoms in our simulations is $5125$ corresponding to
graphene ring of radius $R=7$ nm and width $W=4$ nm. At the beginning of
calculations, we simulate the circular graphene ring at a given temperature
with circular clamped boundary conditions. After reaching a stable
configuration, we rotate the outer boundary of graphene ring about
$\theta_{0}=12^{\circ}$ as the system shown in Fig. 1( left panel). For later
purposes, we name the system as case one. In this case, the displacement of
the boundary atoms is about $U_{\theta}=R\pi\theta_{0}/180$. For the sake of
comparison, we would like to consider another system in which the inner
boundary is pulled down. In the later case, in order to avoid from strong
wrinkles, we reduce the rotation of the outer boundary about $6^{\circ}$ and
then pull slowly down the central part of graphene sheet about $z=-1.7$nm as
shown in Fig. 1( right panel). Again, the latter system is labeled as case
two.

##  III Numerical Results

From the analytical calculations, the pseudo-magnetic field obtained in the
graphene ring geometry where $h=0$ is shown in Fig. 2. For given the circular
radius, $B=7.89\times 10^{7}\cos(3\theta)/{r_{0}}^{3}$ in units of tesla where
$r_{0}$ scaled in units of Angstrom. The maximum field occurs in the vicinity
of the inner boundary and it decreases like $1/r^{3}$. Although the analytical
model described above can explain the behavior of the strains and magnetic
field, we point out that the wrinkle structures represent an important feature
of the results.

We use two different systems in the numerical simulations. We first study a
graphene ring which is placed in the $x-y$ plane and set at a given
temperature by allowing the surface fluctuations in the third dimension, and
then the outer boundary of graphene ring is rotated while the inner boundary
is clamped as is shown in Fig. 1 ( left panel). In case one, we have seen
noticeable wrinkle structures around the inner boundary along the
perpendicular direction (z-axis). In case two, on the other hand, the graphene
ring surface is smoother, but the average carbon-carbon distance is longer
than the one obtained in case one. In the following, we will describe two
cases and find the pseudo-magnetic fields emerge due to the strains. In our MD
simulations, the equations of motion are integrated by the Verlet algorithm
with a time step of $0.5$ fs. Our numerical results show that the formation of
the wrinkles are quantitatively sensitive to the details of the potential. We
simulate a system with two mentioned potentials and find that the order of the
pseudo-magnetic fields are the same ( for more details, see Appendix A ).
Therefore, we mainly use the Brenner potential brenner  to calculate the
pseudo-magnetic filed.

###  III.1 Deformed structure

As it can be seen from the left panel of Fig 1, the main part of the
deformation occurs in the area near the inner boundary and makes noticeable
fluctuations along the $z$-direction. Due to the appearance of wrinkles in the
system, the strain increases near the inner boundary and reduces in the far
regions. In contrast to the analytical calculations, the strain does not
behave as $r^{-2}$. The analytical model does not describe well a system with
out of plane wrinkles.

In the right panel of Fig 1, we show case two, where the static wrinkles are
approximately washed out and seemingly the surface of graphene ring is much
smoother than case one. The structure of surface displacements can also be
seen in Fig 3. This figure clearly shows that the morphology of the surfaces
are enormously different and we expect to have a stronger pseudo-magnetic
fields for case one with respect to case two.

Figure 1: ( Color online) A representative atomic configuration in MD
simulations for circular graphene ring at $T=50$ K. Left panel: case one in
which the rotation of the inner boundary is $12^{\circ}$ in $x-y$ plane. Right
panel, case two in which the inner boundary has been rotated about $6^{\circ}$
and afterwards the central part has been pulled down about $z=-1.7$ nm .

It should be noticed that the structure of the simulated samples weakly depend
on temperature up to $T=300$K.

###  III.2 Pseudo-magnetic field

Since the structure of wrinkles, after the relaxation of graphene’s shape, is
static, we can calculate the gauge fields and also the pseudo-magnetic fields
from Eq. (2).

Our numerical results for the induced gauge field calculated from the strains
are illustrated in Fig. 4 for two mentioned cases in the $x-y$ plane. We take
a time average over the atom positions in order to pass over the thermal
fluctuation effects and study only the wrinkling structures. The length value
of the vector denotes the absolute value of the gauge field. For case one, the
induced gauge fields are not uniform in the area which wrinkles appear.

The pseudo-magnetic field can be calculated from the induced gauge field and
its distributions are demonstrated in Fig. 5. It is worth mentioning that the
non-zero pseudo-magnetic field, $B_{z}$ almost occurs along the wrinkles,
however its value and structure depends on each wrinkle structure. This is due
to the fact that the structure of the wrinkles is different along different
lattice directions such as armchair and zigzag. The value of the pseudo-
magnetic field that increases with increasing curvature of the wrinkles mostly
occur around the central part.

The structure of the pseudo-magnetic field along one of wrinkles, which is
indicated in the inset figure, is illustrated in Fig 6 (left panel) for case
one. Notice that $B_{z}$ decreases for $x<0$ along the wrinkle. The curve can
be fitted quite well with the expression $\exp(-\alpha x)$ where
$\alpha=1.2\pm 0.1$, which displays fast decaying of the pseudo-magnetic field
close to the outer boundary. The pseudo-magnetic field behaves randomly for
$x>0$. The latter behavior is due to the fact that the path is no longer along
a specific wrinkle. As it can be observed from the results, the pseudo-
magnetic field value is immense. Similar massive pseudo-magnetic fields raised
by highly strained nanobubbles that form when graphene is grown on a platinum
surface have been measured by Levy et al. levy .

For case two, which wrinkles are disappeared, the symmetry of the graphene is
clearly induced in the $B_{z}$, (see Fig 5, right panel ).

In order to have a better understanding regarding the structure of the pseudo-
magnetic fields, we calculate the pseudo-magnetic fields for two cases along
the azimuthal angel at given radius, $r$. Fig. 7, shows the pseudo-magnetic
field $B_{z}$ as a function of $\theta$ around a ring for radius $r=4$ nm in
case one (left panel) and moreover the pseudo-magnetic field is shown at
$r=6.5$ nm for case two (right panel). In the latter case, we have seen the
behavior most likely as $B_{z}\propto\cos(3\theta)$. In this figure, we have
plotted function, which is proportional to $\cos(3\theta)$ as solid lines.
Apparently the result in the left panel refers to case one behaves similarly
to case two with respect to the azimuthal variable, however there are
detectable fluctuations due to the appearance of the wrinkles.

The analytical expression of the pseudo-magnetic field acting on the electrons
is $B(r,\theta)\propto\frac{1}{r^{3}}\cos(3\theta)$ when atomic out-of-plane
displacements are ignored. The expression was obtained for a system that lies
in the $x-y$ plane, $h=0$, where there are no wrinkle structures on the flake.
Since case two, which has less wrinkle structures, is somehow similar to the
aforementioned system, we have found numerically the $\cos(3\theta)$
dependence for the pseudo-magnetic field. However due to the fact that the
inner boundary is pulled down, the pseudo-magnetic field behaviors differently
as a function of $r$.

##  IV Conclusions

In summary, we have investigated the strains and pseudo-magnetic fields in
circular graphene ring under a shear stress. We find, from the elastic theory,
the induced gauge filed as function of the maximum strain at the boundary,
$\bar{u}$. The magnitude of the pseudo-magnetic field near the boundary is
$B\approx 4\sqrt{3}\Phi_{0}\beta\kappa\bar{u}/(\pi aR)$. Moreover, The field
diverges near the clamped sites as $B(r)\propto 1/r^{3}$, where $r$ is the
distance to the site while the strains diverge as $1/r^{2}$. From numerical
simulation results, we find wrinkles structures on graphene flakes and the
pseudo-magnetic field is the same order of magnitude that obtained by
analytical calculations. We also find that the wrinkle structures represent an
important feature of the pseudo-magnetic field. In addition, it is also shown
that the pseudo-magnetic field behaves like as $\cos(3\theta)$. These results
are essential for understanding the electronic properties of graphene ring and
its strain engineering for potential applications.

##  V ACKNOWLEDGMENTS

We thank A. Fognini and A. Naji for their useful comments. This research was
supported in part by the Project of Knowledge Innovation Program (PKIP) of
Chinese Academy of Sciences, Grant No. KJCX2.YW.W10. F. G. is supported by by
MICINN (Spain), grants FIS2008-00124 and CONSOLIDER CSD2007-00010.

##  Appendix A  Empirical potentials

We used both the Brenner’s bond-order potential brenner  incorporating the
second nearest neighbors interactions through their dependence on the bond
angles and the second-generation reactive empirical bond-order (REBO)
potential rebo  for the carbon-carbon interactions. The latter potential is
based on the empirical bond-order formalism and allows for covalent bond
binding and breaking with associated changes in the atomic hybridization.
Consequently, such a classical potential allows to model complex chemistry in
large many-atom systems. The Brenner bond-order potential can be written in
the following general form for the binding energy,

$E_{b}=\sum_{i}~{}\sum_{j>i}~{}\\{V^{R}(r_{ij})-b_{ij}V^{A}(r_{ij})\\}$ (10)

The first term is repulsive, and the second one is attractive. $r_{ij}$ is the
distance between pairs of nearest-neighbor atoms $i$ and $j$. Although this
expression is a simple sum over bond energies, it is not a pair potential
since the $b_{ij}$, which is called the bond-order factor, is in essence a
many-body factor. The many-body nature of $b_{ij}$ makes the bond energy
depend on the local environment of the bond. This feature allows the Brenner
potential to predict correct geometries and energies for many different carbon
structures. The empirical bond-order function used here is written as a sum of
terms $b_{ij}=[b_{ij}^{\sigma-\pi}+b_{ji}^{\sigma-\pi}]/2+b_{ij}^{\pi}$ where
values for the functions $b_{ij}^{\sigma-\pi}$ and $b_{ji}^{\sigma-\pi}$
depend on the local coordination and bond angles for atoms $i$ and $j$ ,
respectively. The first term is a function of the bond angles similar to that
in the Brenner potential brenner  while the second term incorporates the third
nearest neighbors via a bond-order term associated with the dihedral angles
and becomes nonzero upon bending of the graphene sheets. The values for all
the parameters used in our calculation for the potentials can be found in
Refs. brenner, ; rebo,  and are therefore not listed here.

We used the Brenner potential khodemoon  and alternatively the REBO potential
by using Large-scale Atomic or Molecular Massively Parallel Simulator package
lammps  ( LAMMPS) for carrying out the molecular dynamic simulations in this
work. We calculated the pseudo-magnetic fields for the two different
potentials along the azimuthal angel at given radius, $r\simeq 40$Å . Fig. 8,
shows the pseudo-magnetic fields $B_{z}$ as a function of $\theta$ along a
circular ring of radius $r=4$ nm. Note that the order of the pseudo-magnetic
field is the same in both cases and moreover, we see the same envelop behavior
at long-wavelength region. However, they exhibit different oscillations modes
at short wavelength due to the way in which the bond-order ( the dihedral
angle) is handled in the REBO potential.

##  Appendix B Simulation methods

Computer simulations generates very detailed information at the microscopic
level and the conversion of this information into macroscopic level is the
province of statistical mechanics. Molecular dynamics, on the other hand, is
an important tool to investigate the microscopic behaviors by integrating the
motions of particles or particle clusters. In the molecular dynamics, the
trajectories of atoms are determined by numerically solving Newton’s equations
of motion for a many-body interacting systems, where forces between the
particles and their potential energy are defined by certain force fields. We
used NVT ensemble where the system is isolated from changes in moles (N),
volume (V) and temperature (T). In NVT, the energy of endothermic and
exothermic processes is exchanged with a thermostat. A variety of thermostat
methods is available to add and remove energy from the boundaries of a MD
system in a nearly realistic way, approximating the canonical ensemble. On the
other hand, a micro-canonical molecular dynamics trajectory may be seen as an
exchange of potential and kinetic energy, with total energy being conserved.
For every time step, each particle’s position and velocity may be integrated
with a method such as Verlet. Given the initial positions and velocities, all
future positions and velocities can be calculated. If there is a large enough
number of atoms, statistical temperature can be estimated from the
instantaneous temperature, which is found by equating the kinetic energy of
the system to $nk_{B}T/2$ where $n$ is the number of degrees of freedom of the
system.

We simulated the system at nonzero temperatures by employing a Nosé-Hoover
thermostat and the time step is taken as $0.5$ fs. At the beginning of the
simulation, we consider a circular graphene ring incorporating atoms which are
located in the $x-y$ plane. The initial structures are firstly optimized
giving the carbon-carbon bond length of $1.45$ Å and all samples are initially
relaxed at a desired temperature for a duration $1.5$ ns. In atomic positions
at the boundaries, we enforce $z=0$ to prevent atomic motion along the $z$
direction and furthermore, after some simulation run-times, we set
$x=y=$const. to clamp atoms.

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  * (28) ”LAMMPS Molecular Dynamics Simulator”. Sandia National Laboratories.

Figure 2: (Color online) Pseudo-magnetic field, given by Eq. 9 in units of
tesla induced by shear strains in a circular graphene ring. Here, $R=70$,
$R_{1}=30$Åand $\theta=12^{\circ}$.

Figure 3: (Color online) Projected atomic displacements for the case one, left
panel and case two, right panel.

Figure 4: (Color online) Projected the gauge filed, $A_{\mu}$ for two
mentioned cases. Notice that the length value of the vector denotes the
absolute value of the gauge filed.

Figure 5: (Color online) Distribution of the pseudo-magnetic field, $B_{z}$ on
graphene ring for two mentioned cases. Green, red and black colors correspond
to positive, negative and zero values of the pseudo-magnetic field,
respectively. For the case one, the dark-green(or dark-red) correspond to the
strongest pseudo-magnetic filed which is about $1000T$ and for the case case
two, it is about $200T$

Figure 6: (Color online) Pseudo-magnetic field, $B_{z}$ on a specific
direction indicates in the inset figure, left panel, and along $y$-direction
passing the origin of the ring, right panel.

Figure 7: (Color online) Pseudo-magnetic field, $B_{z}$ as a function of
azimuthal angle for case one at $r=4$ nm, left panel and $r=6.5$nm for case
two( right panel). Solid line curve denotes a function which is proportional
to $\cos(3\theta)$. Figure 8: (Color online) Pseudo-magnetic field, $B_{z}$ as
a function of azimuthal angle at $T=50K$ along a circular ring of radius $4$
nm when the Brenner potential brenner  and the REBO potential rebo  are used.
Their long-wavelength behavior is very similar to the form
$B_{z}\propto\cos(3\theta)$, however their short wavelength behavior is
different due to the way in which the bond-order is handled in the REBO
potential. The outer boundary of graphene ring, in both cases, is rotated
about $\theta_{0}=8^{\circ}$.


# Style Spectroscope: Improve Interpretability and Controllability through
Fourier Analysis

Zhiyu Jin
Department of Computer Science
Fudan University
<EMAIL_ADDRESS>
&Xuli Shen
Department of Computer Science
Fudan University
<EMAIL_ADDRESS>
&Bin Li
Department of Computer Science
Fudan University
<EMAIL_ADDRESS>
&Xiangyang Xue
Department of Computer Science
Fudan University
<EMAIL_ADDRESS>

###### Abstract

Universal style transfer (UST) infuses styles from arbitrary reference images
into content images. Existing methods, while enjoying many practical
successes, are unable of explaining experimental observations, including
different performances of UST algorithms in preserving the spatial structure
of content images. In addition, methods are limited to cumbersome global
controls on stylization, so that they require additional spatial masks for
desired stylization. In this work, we provide a systematic Fourier analysis on
a general framework for UST. We present an equivalent form of the framework in
the frequency domain. The form implies that existing algorithms treat all
frequency components and pixels of feature maps equally, except for the zero-
frequency component. We connect Fourier amplitude and phase with Gram matrices
and a content reconstruction loss in style transfer, respectively. Based on
such equivalence and connections, we can thus interpret different structure
preservation behaviors between algorithms with Fourier phase. Given the
interpretations we have, we propose two manipulations in practice for
structure preservation and desired stylization. Both qualitative and
quantitative experiments demonstrate the competitive performance of our method
against the state-of-the-art methods. We also conduct experiments to
demonstrate (1) the abovementioned equivalence, (2) the interpretability based
on Fourier amplitude and phase and (3) the controllability associated with
frequency components.

##  1 Introduction

Style transfer deals with the problem of synthesizing an image which has the
style characteristics from a style image and the content representation from a
content image. The seminal work [Gatys et al., 2016] uses Gram matrices of
feature maps to model style characteristics and optimizes reconstruction
losses between the reference images and stylized images iteratively. For the
purpose of gaining vivid visual styles and less computation cost, more trained
feed-forward networks are proposed [Wang et al., 2020, Li and Wand, 2016,
Johnson et al., 2016, Sheng et al., 2018, Li et al., 2017a, Chen et al., 2017,
Dumoulin et al., 2017]. Recent works focus on arbitrary style transfer [Park
and Lee, 2019, Chen et al., 2021a, b, Chandran et al., 2021], or artistic
style [Chen et al., 2021b, Liu et al., 2021, Chen et al., 2021c]. These works
capture limited types of style and cannot well generalize to unseen style
images [Hong et al., 2021].

To obtain the generalization ability for arbitrary style images, many methods
are proposed for the task of universal style transfer (UST). Essentially, the
main challenge of UST is to properly extract the style characteristics from
style images and transfer them onto content images without any prior knowledge
of target style. The representative methods of UST consider various notions of
style characteristics. For example, AdaIN [Huang and Belongie, 2017] aligns
the channel-wise means and variances of feature maps between content images
and style images, and WCT [Li et al., 2017b] further matches up the covariance
matrices of feature maps by means of whitening and coloring processes, leading
to more expressive colors and intensive stylization.

While these two approaches and their derivative works show impressive
performances on stylization, they behave differently in preserving the
structure of content images. It is observed that the operations performed by
AdaIN can do better in structure preservation of content images while those of
WCT might introduce structural artifacts and distortions. Many follow-up works
focus on alleviating the problem of WCT [Li et al., 2018, Chiu and Gurari,
2022, Yoo et al., 2019], but seldom can analytically and systematically
explain what makes the difference. In the field of UST, we need an analytical
theory to bridge algorithms with experimental phenomena for better
interpretability, potentially leading to better stylization controls. To this
end, we resort to apply Fourier transform for deep analysis, aiming to find
new equivalence in frequency domain and bring new interpretations and
practical manipulations to existing style transfer methods.

In this work, we first revisit a framework by [Li et al., 2017b] which unifies
several well-known UST methods. Based on the framework, we derive an
equivalent form for it in the frequency domain, which has the same simplicity
with its original form in the spatial domain. Accordingly, the derived result
demonstrates that these UST methods perform a uniform transformation in the
frequency domain except for the origin. Furthermore, these UST methods
transform frequency components (excluding the zero-frequency component) and
spatial pixels of feature maps in an identical manner. Thus, these UST methods
perform manipulations on the whole frequency domain instead of specific
subsets of frequencies (either high frequencies or low frequencies).

Secondly, through the lens of the Fourier transform, we further explore the
relation of Fourier phase and amplitude with key notions in style transfer,
and then we present new interpretations based on the equivalence we have. On
one hand, we prove that a content reconstruction loss between two feature maps
reaches a local minimum when they have identical Fourier phase, which implies
that Fourier phase of feature maps contributes to the structure of stylized
results. On the other hand, we prove that the Fourier amplitude of feature
maps determines the diagonals of their Gram matrices, which implies that
Fourier amplitude contributes to the intensity information of stylized images.
Next, We demonstrate that AdaIN does preserve the Fourier phase of feature
maps while WCT does not, and we interpret the different behaviors between the
UST methods in structure preservation as a consequence of their different
treatment with the Fourier phase of feature maps.

Thirdly, based on the connection we establish between style transfer and
Fourier transfer, we propose two manipulations on the frequency components of
feature maps: 1) a phase replacement operation to keep phase of feature maps
unchanged during stylization for better structure preservation. 2) a feature
combination operation to assign different weights to different frequency
components of feature maps for desired stylization. We then conduct extensive
experiments to validate their efficacy.

The contributions of this paper are summarized as follows:

  * •

Equivalence We present a theoretically equivalent form for several state-of-
the-art UST methods in the frequency domain and reveal their effects on
frequencies. We conduct corresponding experiments to validate the equivalence.

  * •

Interpretability We connect Fourier amplitude and phase with key notions in
style transfer and present new interpretations on different behaviors of UST
methods. The interpretations are validated by experiments.

  * •

Controllability We propose two manipulations for structure preservation and
desired stylization. We have experimental validation for their efficacy and
controllability.

##  2 Preliminaries

###  2.1 Fourier transform

The Fourier transform has been widely used for the analysis of the frequency
components in signals, including images and feature maps in the shallow layers
of neural networks. Given an image $F\in\mathbb{R}^{C\times H\times W}$, the
discrete Fourier transform (DFT) [Jenkins and Desai, 1986] decomposes it into
a unique representation $\mathcal{F}\in\mathbb{C}^{C\times H\times W}$ in the
frequency domain as follows:

$\displaystyle\mathcal{F}_{u,v}=\sum^{H-1}_{h=0}\sum^{W-1}_{w=0}F_{h,w}e^{-j2\pi(u\frac{h}{H}+v\frac{w}{W})},\quad
j^{2}=-1,$ (1)

where $(h,w)$ and $(u,v)$ are the indices on the spatial dimensions and the
frequency dimensions, respectively. Since images and feature maps consist of
multiple channels, we here apply the Fourier transform upon each channel
separately and omit the explicit notation of channels. Each frequency
component $\mathcal{F}_{u,v}$ can be decomposed into amplitude
$|\mathcal{F}_{u,v}|$ and phase $\angle\mathcal{F}_{u,v}$:

$\displaystyle|\mathcal{F}_{u,v}|$
$\displaystyle=\sqrt{\left(\mathcal{R}_{u,v}\right)^{2}+\left(\mathcal{I}_{u,v}\right)^{2}},\quad\angle\mathcal{F}_{u,v}=\texttt{atan2}(\mathcal{I}_{u,v},\mathcal{R}_{u,v}),$
(2)

where $\mathcal{R}_{u,v}$ and $\mathcal{I}_{u,v}$ are the real part and the
imaginary part of the complex frequency component $\mathcal{F}_{u,v}$,
respectively. Intuitively, as for images, amplitude carries much of intensity
information, including the contrast or the difference between the brightest
and darkest peaks of images, and phase crucially determines the spatial
content of images [Gonzalez and Woods, 2008].

###  2.2 A unified framework for universal style transfer

To better demonstrate the connection between style transfer and the Fourier
transform, a unified framework of different style transfer methods is
preferred to serve as a bridge. Given a content image $I^{c}$ and a style
image $I^{s}$, we denote the feature maps of $I^{c}$ and $I^{s}$ as
$F^{c}\in\mathbb{R}^{C\times H^{c}\times W^{c}}$ and
$F^{s}\in\mathbb{R}^{C\times H^{s}\times W^{s}}$ respectively, where $C$
denotes the number of channels, $H^{c}$ $(H^{s})$ the height and $W^{c}$
$(W^{s})$ the width. For a majority of UST methods, their goal is to transform
the content image feature maps $F^{c}$ into stylized feature maps $F^{cs}$,
whose first-order and second-order statistics are aligned with those of the
style image feature maps $F^{s}$. Accordingly, their methods mainly depend on
the corresponding channel-wise mean vectors $\mu^{c},\mu^{s}\in\mathbb{R}^{C}$
and the covariance matrices $\Sigma^{c},\Sigma^{s}\in\mathbb{R}^{C\times C}$
of $F^{c}$ and $F^{s}$, respectively.

A framework is proposed in [Lu et al., 2019] for unifying several well-known
methods under the same umbrella. Specifically, each pixel $F^{c}_{h,w}$ of
$F^{c}$ is first centralized by subtracting the mean vector $\mu^{c}$, where
$h$ and $w$ are indices on spatial dimensions. Then the framework linearly
transforms $F^{c}_{h,w}$ with the transformation matrix
$T\in\mathbb{R}^{C\times C}$ and re-centers $F^{c}_{h,w}$ by adding the mean
vector $\mu^{s}$ of the style. Each pixel $F^{cs}_{h,w}\in\mathbb{R}^{C}$ of
stylized feature maps can be represented as follows:

$\displaystyle F^{cs}_{h,w}=T\left(F^{c}_{h,w}-\mu^{c}\right)+\mu^{s},$ (3)

where the transformation matrix $T$ has multiple forms based on a variety of
configurations of different methods. We here demonstrate the relation between
the unified framework and several methods in details.

  1. 1.

AdaIN In Adaptive Instance Normalization (AdaIN) [Huang and Belongie, 2017],
the transformation matrix
$T=\texttt{diag}\left(\Sigma^{s}\right)/\texttt{diag}\left(\Sigma^{c}\right)$,
where $\texttt{diag}\left(\Sigma\right)$ denotes the diagonal matrix of a
given matrix $\Sigma$ and $/$ denotes the element-wise division. Because of
the characteristics of diagonal matrices, only the means and variances within
each single feature channel of $F^{cs}$ are matched up to those of $F^{s}$,
ignoring the correlation between channels.

  2. 2.

WCT Instead of shifting a single set of intra-channel statistics, [Li et al.,
2017b] proposes a Whitening and Coloring Transform (WCT) that focuses further
on the alignment of covariance matrices. Similar with AdaIN, the
transformation matrix for WCT is
$T={\left(\Sigma^{s}\right)}^{\frac{1}{2}}{\left(\Sigma^{c}\right)}^{-\frac{1}{2}}$,
leading to well-aligned second-order statistics.

  3. 3.

LinearWCT While WCT generates stylized images more expressively, it is still
computationally expensive because of the high dimensions of feature maps in
neural networks. [Li et al., 2019] proposes LinearWCT to use light-weighted
neural networks to model the linear transformation $T$ by optimizing the Gram
loss, known as a widely-used style reconstruction objective function:

$\displaystyle
T=\mathop{\arg\min}_{T}\|F^{cs}{F^{cs}}^{\top}-F^{s}{F^{s}}^{\top}\|_{F}^{2},$
(4)

where $F^{cs}{F^{cs}}^{\top}$ is the Gram matrix for $F^{cs}$ and
$\|\cdot\|_{F}^{2}$ denotes the squared Frobenius norm of the differences
between given matrices.

  4. 4.

OptimalWCT Similarly, [Lu et al., 2019] proposes OptimalWCT to derive a
closed-form solution for $T$ without the help of optimization process:

$\displaystyle
T={\left(\Sigma^{c}\right)}^{-\frac{1}{2}}\left({\left(\Sigma^{c}\right)}^{\frac{1}{2}}{\Sigma^{s}}{\left(\Sigma^{c}\right)}^{\frac{1}{2}}\right)^{\frac{1}{2}}{\left(\Sigma^{c}\right)}^{-\frac{1}{2}}.$
(5)

Their method reaches the theoretical local minimum for the content loss
$\mathcal{L}_{c}=\|F^{c}-F^{cs}\|_{F}^{2}$, which is widely-used in style
transfer [Huang and Belongie, 2017, Gatys et al., 2016, Lu et al., 2019] for
structure preservation of content images.

##  3 Method

In this section, we first show an equivalent form of the framework in the
frequency domain. In this way, all the methods based on the framework in
Section 2.2 can be interpreted as effecting on the frequency domain. We
further connect amplitude and phase with existing concepts in the context of
style transfer, and explain why AdaIN preserves the structure of content
images better while WCT might not. Finally, we propose two operations for
better structure preservation and desired stylization.

###  3.1 The equivalent form of the framework in the frequency domain

We theoretically analyze the unified framework from the angle of 2-D DFT. We
denote the DFT of $F^{cs}$ as $\mathcal{F}^{cs}\in\mathbb{C}^{C\times
H^{c}\times W^{c}}$, where $\mathbb{C}$ is the set of complex numbers.
According to the unified framework in Eq. (3), we can derive each complex
frequency component $\mathcal{F}^{cs}_{u,v}$ as:

$\displaystyle\mathcal{F}^{cs}_{u,v}$
$\displaystyle=\sum^{H^{c}-1}_{h=0}\sum^{W^{c}-1}_{w=0}F^{cs}_{h,w}e^{-j2\pi(u\frac{h}{H^{c}}+v\frac{w}{W^{c}})}=\sum^{H^{c}-1}_{h=0}\sum^{W^{c}-1}_{w=0}[T(F^{c}_{h,w}-{\mu}^{c})+{\mu}^{s}]e^{-j2\pi(u\frac{h}{H^{c}}+v\frac{w}{W^{c}})}$
(6)
$\displaystyle=T\underbrace{\sum^{H^{c}-1}_{h=0}\sum^{W^{c}-1}_{w=0}F^{c}_{h,w}e^{-j2\pi(u\frac{h}{H^{c}}+v\frac{w}{W^{c}})}}_{\begin{cases}H^{c}W^{c}\mu^{c},&\text{if
}u=v=0;\\\
\mathcal{F}^{c}_{u,v},&\text{else}\end{cases}}+[\mu^{s}-T\mu^{c}]\underbrace{\sum^{H^{c}-1}_{h=0}\sum^{W^{c}-1}_{w=0}e^{-j2\pi(u\frac{h}{H^{c}}+v\frac{w}{W^{c}})}}_{\begin{cases}H^{c}W^{c},&\text{if
}u=v=0;\\\ 0,&\text{else}\end{cases}}$
$\displaystyle=\begin{cases}H^{c}W^{c}\mu^{s},&\text{if }u=v=0;\\\
T\mathcal{F}^{c}_{u,v},&\text{else}\end{cases}\qquad=\begin{cases}\left(\frac{H^{c}W^{c}}{H^{s}W^{s}}\right)\mathcal{F}^{s}_{0,0},&\text{if
}u=v=0;\\\ T\mathcal{F}^{c}_{u,v},&\text{else}\end{cases},$

where $u$ and $v$ are indices upon the frequency dimensions, and
$\mathcal{F}^{c}$ and $\mathcal{F}^{s}$ are the DFTs of $F^{c}$ and $F^{s}$,
respectively. According to the Fourier transform in Eq. (1),
$\mathcal{F}_{0,0}=\sum^{H-1}_{h=0}\sum^{W-1}_{w=0}F_{h,w}=HW\mu$. Thus, we
have
$\mathcal{F}^{cs}_{u,v}=H^{c}W^{c}\mu^{s}=\left(\frac{H^{c}W^{c}}{H^{s}W^{s}}\right)\mathcal{F}^{s}_{0,0}$
when $u=v=0$. Therefore, in the frequency domain, style transfer methods based
on the unified framework are simple linear transformations on
$\mathcal{F}^{c}$ except for the zero-frequency component
$\mathcal{F}^{c}_{0,0}$, which is replaced with the re-scaled zero-frequency
component of $\mathcal{F}^{s}$.

From Eq. (6), we find that each individual frequency component (excluding the
zero-frequency component) has an identical linear transformation with pixels
on the feature maps. In this way, there is no entanglement between different
frequencies in the process of style transfer. Thus, it is feasible to treat
and manipulate each frequency component of $\mathcal{F}^{cs}$ as an individual
for practical usage. Therefore, we justify the claim that mainstream methods
in Section 2.2 for UST are not sole transfer on specific subsets of
frequencies (either high frequencies or low frequencies), but essentially on
the whole frequency domain.

###  3.2 Connections and interpretations: amplitude and phase

To better bridge style transfer with the Fourier transform, we connect phase
and amplitude with a reconstruction loss and the Gram matrix in style
transfer, respectively.

Phase and the content loss We here demonstrate the relation between phase and
the content loss, which widely serves as a construction loss for optimizing
the differences of spatial arrangement between stylized images $I^{cs}$ and
content images $I^{c}$. Given their feature maps
$F^{cs},F^{c}\in\mathbb{R}^{C\times H\times W}$, corresponding DFTs
$\mathcal{F}^{cs},\mathcal{F}^{c}$, Fourier amplitude
$|\mathcal{F}^{cs}|,|\mathcal{F}^{c}|$ and Fourier phase
$\angle\mathcal{F}^{cs},\angle\mathcal{F}^{c}$, the content loss between
$F^{cs}$ and $F^{c}$ can be derived as:

$\displaystyle\mathcal{L}_{c}$
$\displaystyle=\|F^{cs}-F^{c}\|_{F}^{2}=\sum_{k=0}^{C}\sum_{h=0}^{H}\sum_{w=0}^{W}\left(F^{cs}_{k,h,w}-F^{c}_{k,h,w}\right)^{2}=\frac{1}{HW}\sum_{k=0}^{C}\sum_{u=0}^{H}\sum_{v=0}^{W}|\mathcal{F}^{cs}_{k,u,v}-\mathcal{F}^{c}_{k,u,v}|^{2}$
(7)
$\displaystyle=\frac{1}{HW}\sum_{k=0}^{C}\sum_{u=0}^{H}\sum_{v=0}^{W}\left[|\mathcal{F}^{cs}_{k,u,v}|^{2}+|\mathcal{F}^{c}_{k,u,v}|^{2}-2|\mathcal{F}^{cs}_{k,u,v}||\mathcal{F}^{c}_{k,u,v}|\cos\left(\angle\mathcal{F}^{cs}_{k,u,v}-\angle\mathcal{F}^{c}_{k,u,v}\right)\right],$

where the second equality is held by the Parseval’s theorem and $k$, $(h,w)$
and $(u,v)$ are indices on channels, spatial dimensions and frequency
dimensions, respectively. When $F^{cs}$ is optimized for the content loss,
since $|\mathcal{F}^{cs}_{k,u,v}|$ and $|\mathcal{F}^{c}_{k,u,v}|$ are non-
negative numbers, the content loss $\mathcal{L}_{c}$ reaches a local minimum
when $\angle\mathcal{F}^{cs}_{k,u,v}=\angle\mathcal{F}^{c}_{k,u,v}$ for all
$(k,u,v)$. Furthermore, whenever $\angle\mathcal{F}^{cs}_{k,u,v}$ gets closer
to $\angle\mathcal{F}^{c}_{k,u,v}$, the content loss decreases, demonstrating
the crucial role of phase of feature maps in determining the spatial
information of corresponding decoded images. Therefore, we can interpret the
structure preservation abilities of methods from the perspective of Fourier
phase. Furthermore, we can manipulate Fourier phase for better performances in
structure preservation.

Interpretations on structure preservation Based on the equivalent form in Eq.
(6) and the relation between Fourier phase and the content loss, we can give
interpretations to different behaviors of methods in structure preservation.
Concerning AdaIN and WCT as instances of the equivalent framework in the
frequency domain, we have $\mathcal{F}^{cs}_{u,v}=T\mathcal{F}^{c}_{u,v}$ when
$(u,v)\neq(0,0)$. Note that for AdaIN, the transformation matrix
$T=\texttt{diag}\left(\Sigma^{s}\right)/\texttt{diag}\left(\Sigma^{c}\right)\in\mathbb{R}^{C\times
C}$ is a real diagonal matrix, which has the same scaling upon the real part
and the imaginary part of $\mathcal{F}^{c}$. As a result, AdaIN can preserve
the phase in each feature channel and keep the content loss of feature maps in
a local minimum. While WCT provides a non-diagonal matrix
$T={\left(\Sigma^{s}\right)}^{\frac{1}{2}}{\left(\Sigma^{c}\right)}^{-\frac{1}{2}}$
for transformation, the information between different channels is consequently
entangled, the phase of each channel is disturbed and the content loss after
the process of WCT is likely to increase much more than the one after the
process of AdaIN. Therefore, WCT needs more efforts to preserve the spatial
information of content images, resulting in its less appealing performances in
structure preservation.

Amplitude and Gram matrices We theoretically demonstrate the connection
between the Fourier amplitude of feature maps and their Gram matrices. Given
feature maps $F\in\mathbb{R}^{C\times H\times W}$, corresponding Fourier
amplitude $|\mathcal{F}|$ and Fourier phase $\angle\mathcal{F}$ of their DFT
$\mathcal{F}$, the pixels of the Gram matrix $FF^{\top}$ can be derived as:

$\displaystyle\left(FF^{\top}\right)_{c_{1},c_{2}}$
$\displaystyle=\sum_{h=0}^{H}\sum_{w=0}^{W}F_{c_{1},h,w}F_{c_{2},h,w}=\frac{1}{HW}\sum_{u=0}^{H}\sum_{v=0}^{W}\mathcal{F}_{c_{1},u,v}\mathcal{F}^{*}_{c_{2},u,v}$
(8)
$\displaystyle=\frac{1}{HW}\sum_{u=0}^{H}\sum_{v=0}^{W}|\mathcal{F}_{c_{1},u,v}||\mathcal{F}_{c_{2},u,v}|\cos\left(\angle\mathcal{F}_{c_{1},u,v}-\angle\mathcal{F}_{c_{2},u,v}\right),$

where $c_{1},c_{2}$ are indices on the channels, $(*)$ represents complex
conjugate and the second equality is held by the Parseval’s theorem. Since
$\left(FF^{\top}\right)_{c_{1},c_{2}}$ is a real number, we omit the imaginary
part in the final step. In a special case where $c_{1}$ equals $c_{2}$,
$\left(FF^{\top}\right)_{c_{1},c_{2}}$ equals
$\frac{1}{HW}\sum_{u=0}^{H}\sum_{v=0}^{W}|\mathcal{F}_{c_{1},u,v}|^{2}$. This
indicates that the sum of the square of amplitude components directly
determines the diagonals of Gram matrices. Each elements on the diagonal of
Gram matrices represents infra-channel second-order statistics of feature
maps, measuring the intensity of information in each channel. Therefore, if we
only manipulate the Fourier phase of the DFTs of feature maps and keep the
Fourier amplitude unchanged, it can be expected that the intensity
presentations of the corresponding decoded images are roughly the same. For
detailed proof, see the Supplementary Materials.

###  3.3 Manipulations on stylized feature maps in the frequency domain

The equivalent form in Eq. (6) and abovementioned connections enable further
manipulations for better structure preservation or desired stylization. We
propose two simple operations upon frequency components of feature maps, which
we call phase replacement and frequency combination.

Phase replacement Given the DFT of the content feature maps $\mathcal{F}^{c}$
and the DFT of the stylized feature maps $\mathcal{F}^{cs}$, we calculate the
phase of $\mathcal{F}^{c}$, denoted as
$\angle\mathcal{F}^{c}\in\left[0,2\pi\right)^{C\times H^{c}\times W^{c}}$ and
the amplitude of $\mathcal{F}^{cs}$, denoted as
$|\mathcal{F}|^{cs}\in\mathbb{R}_{+}^{C\times H^{c}\times W^{c}}$, where
$\mathbb{R}_{+}$ denotes the set of non-negative real numbers. We then
reconstruct $\mathcal{F}^{cs}$ as:

$\displaystyle\mathcal{F}^{cs}_{u,v}$
$\displaystyle=|\mathcal{F}|^{cs}_{u,v}\odot\cos\angle\mathcal{F}^{c}_{u,v}+j|\mathcal{F}|^{cs}_{u,v}\odot\sin\angle\mathcal{F}^{c}_{u,v},$
(9)

where $\cos$ and $\sin$ are element-wise operators on vectors (e.g.,
$\cos\phi=[\cos\phi^{1},...,\cos\phi^{C}]$), $\odot$ is the element-wise
multiplication and $j$ is the imaginary unit. Based on the connections
established in Section 3.2, when we replace the phase of $\mathcal{F}^{cs}$
with $\angle\mathcal{F}^{c}$, the content loss between $F^{cs}$ and $F^{c}$ is
reduced and in this way, the structure of content images is more preserved in
$F^{cs}$. In addition, since the amplitude of $\mathcal{F}^{cs}$ is not
changed, the diagonal of $F^{cs}{F^{cs}}^{\top}$ stays unchanged and so does
the basic intensity information of the stylized results. A similar amplitude
transferring method is proposed in [Yang and Soatto, 2020] for semantic
segmentation, which shares resembling views on phase and amplitude with ours.

Frequency combination To accommodate different requirements from users,
appropriate control on the stylization is needed for practical usage. Plenty
of works for style transfer use linear combination of content feature maps
$F^{c}$ and stylized feature maps $F^{cs}$ as shown in Eq. (10):

$\displaystyle F^{cs}=\alpha F^{cs}+(1-\alpha)F^{c},$ (10)

where $\alpha$ is the weight for controlling on the stylization. In this way,
all the global characteristics of images (e.g., the sharp edges of trees and
the smooth background of sky) are combined uniformly. While in most cases,
users are expecting for customized global changes on images (e.g., having the
details of trees less stylized but keeping the sky moderately stylized). Since
high frequencies determine the details and low frequencies determine the
overview of images, we can accommodate the customized needs of users with
combination of frequencies in different proportions.

Given the DFT of content feature maps $\mathcal{F}^{c}$ and the DFT of
stylized feature maps $\mathcal{F}^{cs}$, we first rearranges their frequency
components with the zero-frequency components in the center point
$(u_{0},v_{0})$, following a common technique in digital image processing. In
this way, the frequency components close to $(u_{0},v_{0})$ are low-frequency
components whereas the rest of components represent high frequencies. Next, we
combine $\mathcal{F}^{cs}$ and $\mathcal{F}^{c}$ using a weighting function
$\alpha:\mathbb{R}^{2}\rightarrow[0,1]$:

$\displaystyle\mathcal{F}^{cs}_{u,v}=\alpha(u,v)\mathcal{F}^{cs}_{u,v}+[1-\alpha(u,v)]\mathcal{F}^{c}_{u,v},$
(11)

where $\alpha$ serves as the stylization weighting function dependent on the
indices $(u,v)$. For example, if users want to have the details less stylized,
higher frequencies of $\mathcal{F}^{cs}$ need to be less weighted, and
accordingly a lower value of $\alpha$ can be set for $(u,v)$ indexing higher
frequencies. In practice, the function $\alpha$ is set to be controlled by a
hyper-parameter $\sigma$:

$\displaystyle\alpha(u,v)=\exp{\left[-\frac{\left(u-u_{0}\right)^{2}+\left(v-v_{0}\right)^{2}}{\sigma}\right]},$
(12)

where $\sigma$ represents the degree for combining the low frequencies of
$\mathcal{F}^{cs}$. When $\sigma$ gets larger, the value of $\alpha(u,v)$
increases for every $(u,v)$. In this way, more low frequencies of
$\mathcal{F}^{cs}$ (indexed by $(u,v)$ close to $(u_{0},v_{0})$ ) are
gradually kept.

##  4 Experiments

In this section, we first introduce our method specification and
implementation details. Then we compare our method with the state-of-the-art
style transfer methods in terms of visual effect, structure preservation and
computing time. Moreover, we conduct experiments to validate the equivalence
presented in Eq. (6), the interpretations on Fourier amplitude and phase
introduced in Section 3.2, and the efficacy of manipulations proposed in 3.3.
More qualitative results and implementation details are available in the
Supplementary Materials.

Method specification. Based on the equivalence and connections mentioned
above, the proposed method first performs a selected UST algorithm in the
frequency domain. In practice, we choose to implement our method in
conjunction with WCT because WCT produces expressive stylization in spite of
introduced distortions. To deal with the distortions, we adopt phase
replacement (PR) to substitute the Fourier phase of stylized feature maps with
that of content feature maps. Since PR can optimize the content loss to a
local minimum according to Eq. (7), the structure of content images is
preserved. Finally, the proposed method uses inverse discrete Fourier
transform to reverse the frequency components back to the spatial domain.

Implementation details. We adopt a part of the VGG-19 network [Simonyan and
Zisserman, 2015] (from the layer conv1_1 to the layer conv4_1) as our encoder.
The weights of our encoder are borrowed from ImageNet-pretrained weights,
following existing style transfer methods. We train our decoder for image
reconstruction by minimizing the $L_{2}$ reconstruction loss. During the
inference stage, we apply our method to feature maps in each layer of the
decoder. We choose MS-COCO dataset [Lin et al., 2014] and WikiArt dataset
[Nichol, 2016] as our content dataset and style dataset, respectively. Our
decoder is also trained on the content dataset, whose training images are
first resized into 512$\times$512 and randomly cropped into a size of
256$\times$256\. We run all the experiments on a single NVIDIA Tesla V100.

Figure 2: Synthesized results with replaced Fourier amplitude or phase. Table
1: The SSIM scores, the Gram loss and the time cost (seconds) for different
UST methods.

Method | AdaIN | WCT | LinearWCT | OptimalWCT | AvatarNet | SANet | Self-Contained | Ours
---|---|---|---|---|---|---|---|---
SSIM | 0.307 | 0.234 | 0.378 | 0.250 | 0.329 | 0.310 | 0.276 | 0.403
SSIM (w/ PR) | 0.307 | 0.251 | 0.427 | 0.263 | - | - | - | 0.438
Time | 0.0218 | 0.3167 | 0.0038 | 0.6247 | 3.027 | 0.0052 | 0.0513 | 0.098
Time (w/ PR) | 0.0226 | 0.3293 | 0.0045 | 0.6321 | - | - | - | 0.112

###  4.1 Performance comparison

Qualitative comparison In Figure LABEL:Fig1, we show some visualization
results of the qualitative comparison between the proposed methods and the
state-of-the-art UST methods (i.e., AdaIN [Huang and Belongie, 2017], WCT [Li
et al., 2017b], LinearWCT [Li et al., 2019], OptimalWCT [Lu et al., 2019],
SANet[Park and Lee, 2019], AvatarNet[Sheng et al., 2018] and Self-Contained
[Chen et al., 2020]). We observe that AdaIN roughly preserves the structure of
images, but often produces unappealing patterns on the edges (e.g., $2^{nd}$,
$3^{rd}$, and $5^{th}$ columns). WCT and OptimalWCT can produce intensive but
distorted artistic style and yield images less similar with content images in
structure (e.g., $2^{nd}$, $4^{th}$, and $5^{th}$ columns). LinearWCT roughly
preserves the spatial structure of content images, but the stylization is less
intensive (e.g., $1^{st}$, $2^{nd}$, and $4^{th}$ columns). Comparatively, the
proposed method performs well in the contrast and intensity of stylized
results (e.g., the misty sky and the red lips). The proposed method also well
preserves the spatial structure of content images, including the details (e.g.
the contours of architectures and hair) and the overview (e.g. the light and
shadow of cloudy sky and the human face) of images.

Quantitative comparison In addition to the comparison on visual effect, we
also conduct a quantitative comparison. Moreover, we implement PR on each
method and present corresponding quantitative scores. The Structural
Similarity Index (SSIM) between the stylized images and corresponding content
images is adopted as the metric for evaluating structure preservation. As
shown in Table 1, our method achieves the highest SSIM score and PR improves
SSIM scores of all methods except for AdaIN, since AdaIN does not change the
Fourier phase of feature maps during stylization. The improved SSIM validates
our interpretations on structure preservation behaviors between alogrithms.
Regarding the computing time, while our method needs to utilize Fourier
transform, it still has a competitive time cost compared with other methods.
It is noteworthy that our method has less additional computational cost if PR
is adopted, since it has already performed style transfer in the frequency
domain.

###  4.2 Equivalence, interpretations and manipulations

Validation of equivalence For AdaIN and WCT, we implement them in the
frequency domain based on Eq. (6), shown from the $3^{rd}$ to the $6^{th}$ row
in Figure LABEL:Fig1. It can be observed that these two implemented UST
algorithms in the frequency domain produce the same visual effect with
original algorithms. This observation validates the proposed equivalence.

Interpretations on amplitude and phase To validate the roles of amplitude and
phase, we replace the Fourier amplitude or phase of stylized feature maps in
each layer during the stylization and present the results in Figure 2. It can
be observed that feature maps with the same Fourier phase produce images with
highly similar spatial arrangements (e.g., the structure of leaves and the
light from the bottom in I.(c), I.(d) and I.(e)). This observation matches up
with our interpretations on phase, provided by its connection with the content
loss in Eq. (7). On the other hand, feature maps with same Fourier amplitude
produces images with highly similar contrast and intensities in colors (e.g.,
the same differences between the brightest pixels and the darkest pixels of
II.(c), II.(d) and II.(e)). This observation aligns with our interpretations
on amplitude, supported by its connection with Gram matrices in Eq. (8).

Figure 3: Results on the efficacy of phase replacement (abbr. PR).

Stylization manipulations First, we empirically display the effect of PR in
Section 3.3 for image stylization, whose results are shown in Figure 3. It can
be observed that for results without PR, the details (e.g., edges of the eyes
and the nose) and the overview (e.g., the sky and the sea) become messier and
more distorted, yielding unappealing distortions. The reason might be that PR
can preserve the phase of both high frequencies and low frequencies, which are
responsible for the spatial arrangement of the details and overview of images,
respectively.

Second, to demonstrate the manipulations of frequency combination (FC) in
Section 3.3, we present an example in Figure 4. We choose the weighting
function $\alpha$ in (12) and adjust the hyper-parameter $\sigma$ for
stylization controls. In Figure 4, with different value of $\sigma$, FC can
have the details less stylized (e.g., the colorful buildings in the $4^{th}$
column) while keeping the background moderately stylized (e.g., the sky with
sketch style in the $4^{th}$ column). Furthermore, FC can be customized for
various purposes and the linear combination in Eq. (10) can be viewed as an
instance of FC by setting the weighting function $\alpha(u,v)$ as a simple
scalar. Therefore, the controllability of FC is better than that of linear
combination.

Figure 4: Comparison between (c) linear combination and (d) frequency
combination. It is worth noting that the proposed method helps the background
remain stylized (e.g., the sky more sketch-stylized than (c)) while renders
the details more realistic (e.g., the buildings and the sailboat more
realistic than (c)). Note that the proposed method achieves this performance
without any spatial masks to identify where to stylize. The hyper-parameter
$\sigma$ for results in (d) is set to 0.99.

##  5 Conclusion

In this paper, we apply Fourier analysis to a unified framework of UST
algorithms. We present the equivalent form of the framework and reveal the
connections between the concepts of Fourier transform with those of style
transfer. We give interpretations on the different performances between UST
methods in structure preservation. We also present two operations for
structure preservation and desired stylization. Extensive experiments are
conducted to demonstrate (1) the equivalence between the framework and its
proposed form, (2) the interpretability prompted by Fourier analysis upon
style transfer and (3) the controllability through manipulations on frequency
components.

In the future, we will extend it to more semantic manipulations. Since our
work performs much better in structure preservation, it would be better to
consider the copyright protection before applying the proposed method.

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