# Hydrodynamic effects in the capture of rod-like molecules by a nanopore

Radost Waszkiewicz  Institute of Theoretical Physics, Faculty of Physics,
University of Warsaw<EMAIL_ADDRESS>Maciej Lisicki  Institute of
Theoretical Physics, Faculty of Physics, University of Warsaw
<EMAIL_ADDRESS>
###### Abstract

In the approach of biomolecules to a nanopore, it is essential to capture the
effects of hydrodynamic anisotropy of the molecules and the near-wall
hydrodynamic interactions which hinder their diffusion. We present a detailed
theoretical analysis of the behaviour of a rod-like molecule attracted
electrostatically by a charged nanopore. We first estimate the time scales
corresponding to Brownian and electrostatic translations and reorientation. We
find that Brownian motion becomes negligible at distances within the pore
capture radius, and numerically determine the trajectories of the nano-rod in
this region to explore the effects of anisotropic mobility. This allows us to
determine the range of directions from the pore in which hydrodynamic
interactions with the boundary shape the approach dynamics and need to be
accounted for in detailed modelling.

††: J. Phys.: Condens. Matter

  * August 28, 2024

##  1 Introduction

Nanopore sequencing is now a well-established technique for the determination
of structure of biomolecules [1, 2], such as DNA [3], RNA [4], or proteins
[5]. The molecules, which are typically slender filaments, are
electrophoretically transported to the nanopore and then translocated through
an orifice. The process of passage or threading, controlled by a combination
of electric [6, 7], electrokinetic [8], entropic [9], osmotic [10, 11] and
mechanical forces [12], is now well understood and explored.

However, the approach to the pore is described in less detail. Available
models characterise the dynamics of the molecule by its diffusion coefficient
$D$ and electrophoretic mobility $\mu_{e}$. This simplified approach has
proved useful to establish the general properties of the system. In Ref. [13],
Grosberg and Rabin determined the concentration of DNA near the pore using the
Smoluchowski equation formalism. Qiao et al. defined the capture radius of the
pore [14], being the distance at which thermal fluctuations become comparable
to the electrostatic potential energy, which bounds the region of attraction
of the pore. In the following works, they extended this notion by introducing
the orientational capture radius [15], being the range at which the electric
field strongly orients the colloidal rods. These models, however, neglect the
anisotropy of the particles, and of the hydrodynamic interactions with the
wall which hinder diffusion at close distances. Hydrodynamic effects are known
to alter the trajectories of close sedimenting particles [16, 17] by coupling
to their inherent shape anisotropy and lead to a general slow-down of
colloidal dynamics [18].

In this contribution, we fill this gap by formulating a detailed theoretical
approach which accounts for anisotropic diffusivity of a model nano-rod both
due to its non-isotropic shape and due to the particle-wall flow-mediated
interactions. We first use this model to determine the time scales
corresponding to the subsequent phases of motion of a nano-rod approaching a
pore: purely Brownian motion far from the pore, electric field-induced
translation and reorientation, and the wall influence region. Then we provide
a quantitative insight by solving the equations of motion numerically for a
collection of initial positions and orientation. We describe the resulting
trajectories and mechanisms shaping the motion. This allows us to determine
the wall influence region.

We focus on microparticles which can be modelled as stiff rods. This is
appropriate for biomolecules of length $L$ shorter than their persistence
length $L_{p}$. Examples of such nano-rods include dsDNA shorter than
$L_{p}\approx 50\ \text{nm}$ (or ca. $150bp$) or $fd$-viruses [19], with
$L=880\ \text{nm}$ and $L_{p}=2.8\ \mu\text{m}$, translocating through solid-
state pores [20]. For longer molecules, the effects of elasticity and changing
conformation can lead to coiling [21] or knotting [12, 22] and must be taken
into account for proper modelling.

The paper is organised as follows. First, we describe the general model in
Sec. 2.1, specifying the form of electrostatic interactions in Sec. 2.2 and
hydrodynamic interactions in Sec. 2.3. The following Sec. 3 presents our
results, divided into scaling insights in Sec. 3.1, and a numerical analysis
of the trajectories in Sec. 3.2. We summarise our conclusions in Sec. 4.

##  2 Model

###  2.1 Motion of the nanorod

We consider an ellipsoidal rod of length $L$ with its centre located at a
point $\bm{r}$, as sketched in Fig. 1. The aspect ratio of the rod is $p=10$.
The director of the rod is a unit vector $\bm{u}$. The nanopore is at the
centre of a Cartesian lab coordinate system, with the $z$ axis being normal to
the wall, defined by the $xy$ plane. Thus, the rod is at a distance
$H=\bm{r}\cdot\bm{e}_{z}$ from the wall and its inclination angle $\theta$ is
determined by $\cos\theta=\bm{u}\cdot\bm{e}_{z}$.

The rod is charged and moving in an electric field generated by the nanopore.
The field exerts an electrostatic force, $\bm{F}_{e}$ and torque,
$\bm{T}_{e}$, on the rod. On the other hand, the suspending fluid reacts to
the motion of the particle by exerting a frictional force, $\bm{F}_{h}$, and
torque, $\bm{T}_{h}$. Because the flow is characterised by a small Reynolds
number, the lack of inertia yields the following equations of motion of the
particle

$\displaystyle\bm{F}_{e}+\bm{F}_{h}$ $\displaystyle=\bm{0},$ (1)
$\displaystyle\bm{T}_{e}+\bm{T}_{h}$ $\displaystyle=\bm{0},$ (2)

which determine the translational and rotational velocity of the particle,
$\bm{V}$ and $\bm{\Omega}$, respectively. These are then used to evaluate the
trajectory and orientation according to

$\displaystyle\frac{\mathrm{d}\bm{r}}{\partial t}$ $\displaystyle=\bm{V},$ (3)
$\displaystyle\frac{\partial\bm{u}}{\partial t}$
$\displaystyle=\bm{\bm{\Omega}}\times\bm{u}.$ (4)

Figure 1: Sketch of a nanorod close to a nanopore denoted by a red dot. The
pore generates a radial electric field, modelled by a point charge placed in
its location at the origin. The configuration of the rod is given by its
position $\bm{r}$ and orientation $\bm{u}$, which corresponds to an
inclination angle $\theta$. In addition, we denote by $\alpha$ the polar angle
at which the rod is seen from the pore.

###  2.2 Electrostatic interactions with a nanopore

Following previous works [15], we model the interaction between a pore and a
rod using the Coulomb potential from a point source. We assume that in the
capture process rods are uniformly charged with an effective electrophoretic
charge $Q$ [13]. We will measure the strength of the electric field by the
capture radius $\lambda_{e}$: a distance where the potential energy $Q\Psi$ of
the rod is comparable to the thermal fluctuations, so
$Q\Psi(\lambda_{e})=k_{B}T$, with $k_{B}$ being the Boltzmann constant and $T$
the temperature. In this setting, the electric field at a location $\bm{r}$
can be written as

$\bm{E}(\bm{r})=-\frac{k_{B}T\lambda_{e}}{Q}\frac{\bm{\hat{r}}}{r^{2}},$ (5)

where the hat denotes a unit vector. The potential energy of the particle in
the electric field is given by an integral along the rod

${\Psi}_{e}=\frac{k_{B}T\lambda_{e}}{L}\int_{-L/2}^{L/2}\frac{\mathrm{d}s}{r(s)},$
(6)

where $r(s)$ represents distance from the pore to the rod element $s$. From
this we can obtain exact integral expressions for the force $\bm{F}_{e}$ and
torque $\bm{T}_{e}$ acting on the rod by appropriate differentiation. Since in
the remainder of the paper we will be interested in intermediate particle-pore
distances, when $L/r\ll 1$, we expand them and to leading order we find

$\displaystyle\bm{F}_{e}$
$\displaystyle=-k_{B}T\lambda_{e}\frac{\bm{\hat{r}}}{r^{2}}+\mathcal{O}({(L/r)^{2}}),$
(7) $\displaystyle\bm{T}_{e}$
$\displaystyle=-\frac{k_{B}T\lambda_{e}L^{2}}{4}\frac{(\bm{\hat{r}}.\bm{u})(\bm{\hat{r}}\times\bm{u})}{r^{3}}+\mathcal{O}({(L/r)^{4}}).$
(8)

The force is radially attracting the rod and falls off as $r^{-2}$. When rod
is oriented at an angle to the direction towards the pore, uneven force
distribution generates a torque which reorients the rod, forcing it to point
towards the pore. This torque is proportional to the electric field gradient,
thus it scales as $r^{-3}$.

###  2.3 Near-wall mobility of a rod-like particle

On the colloidal length and time scales, relevant for nanopore experiments,
the flow field $\bm{v}(\bm{r})$ around a particle is described by the
stationary Stokes equations [23]

$\eta{\nabla}^{2}\bm{v}(\bm{r})-{\nabla}p(\bm{r})=-\bm{f}(\bm{r}),\qquad{\nabla}\cdot\bm{v}(\bm{r})=0,$
(9)

where $\bm{f}(\bm{r})$ is the force density the particle exerts on the fluid,
and $p(\bm{r})$ is the modified pressure field. If a particle is moving in a
quiescent fluid, the frictional force and torque are linearly related to its
translational and angular velocities, $\bm{V}$ and $\bm{\Omega}$, via the
friction (or resistance) tensor [23, 24]

$\begin{pmatrix}\bm{F}_{h}\\\
\bm{T}_{h}\end{pmatrix}=-\begin{pmatrix}\bm{\zeta}^{tt}&\bm{\zeta}^{tr}\\\
\bm{\zeta}^{rt}&\bm{\zeta}^{rr}\\\ \end{pmatrix}\begin{pmatrix}\bm{V}\\\
\bm{\Omega}\end{pmatrix}.$ (10)

The indices $tt$ and $rr$ above denote the translational and rotational parts,
respectively, while the tensors $\bm{\zeta}^{tr}$ and $\bm{\zeta}^{rt}$
describe the translation-rotation coupling.

If a particle is moving under the action of a known force and torque, a
complementary problem is formulated using the mobility tensor ${\bm{\mu}}$
which is an inverse of the friction tensor

$\bm{\mu}=\begin{pmatrix}\bm{\mu}^{tt}&\bm{\mu}^{tr}\\\
\bm{\mu}^{rt}&\bm{\mu}^{rr}\end{pmatrix}=\begin{pmatrix}\bm{\zeta}^{tt}&\bm{\zeta}^{tr}\\\
\bm{\zeta}^{rt}&\bm{\zeta}^{rr}\end{pmatrix}^{-1}=\bm{\zeta}^{-1}.$ (11)

Finally, the mobility tensor is related to the diffusion matrix $\bf{D}$ by
the fluctuation-dissipation theorem

$\bm{D}=k_{B}T\bm{\mu}.$ (12)

Thus, the diffusive properties of the particle are completely determined by
its hydrodynamic mobility.

For an axisymmetric particle, the friction (and mobility) tensors have a high
degree of symmetry. In a bulk system, the configuration of a particle is given
by its axial unit vector $\bm{u}$, and the friction tensor can be explicitly
written as

$\displaystyle\bm{\zeta}^{tt}$
$\displaystyle=\zeta^{t}_{\parallel}\bm{u}\bm{u}+\zeta^{t}_{\perp}(\bm{1}-\bm{u}\bm{u}),$
(13) $\displaystyle\bm{\zeta}^{rr}$
$\displaystyle=\zeta^{r}_{\parallel}\bm{u}\bm{u}+\zeta^{r}_{\perp}(\bm{1}-\bm{u}\bm{u}),$
(14) $\displaystyle\bm{\zeta}^{tr}$ $\displaystyle=\bm{\zeta}^{rt}=\bm{0},$
(15)

using only four coefficients. For ellipsoids, analytical formulae are
available for the bulk diffusion tensor and are listed in A. Otherwise,
efficient schemes for the calculation of bulk hydrodynamics properties of
macromolecules modelled as collections of beads, such as Hydro++ [25, 26],
GRPY [27], or Hydromultipole [28, 24] are also available.

The presence of a confining boundary changes this situation, since the
hydrodynamic tensors now depend both on the distance to the boundary, and on
the orientation of the nano-rod with respect to the surface. The friction
tensors of a near-wall particle, $\bm{\zeta}_{w}$, may be written as:

$\bm{\zeta}_{w}=\bm{\zeta}_{0}+\bm{\Delta}\bm{\zeta}_{w},$ (16)

where $\bm{\zeta}_{0}$ is the bulk resistance tensor, and the last term is a
wall-induced correction. An analytical leading-order approximation to
$\bm{\Delta}\bm{\zeta}_{w}$, with the expansion parameter $L/H$ being the
ratio of the size of the particle, $L$, to the wall-particle distance $H$, was
derived previously by some of us [29]. Earlier works provide the components of
the diffusion tensor for very slender filaments close to walls only for
particular alignments [30, 31]. The treatment proposed in Ref. [29] allows for
an efficient calculation of the diffusion tensor of a slender rod-like
particle for moderate and large wall-particle distances. By inverting
$\bm{\zeta}_{w}$ from Eq. (16), we arrive at a convenient approximation to the
near-wall mobility $\bm{\mu}_{w}=\bm{\zeta}_{w}^{-1}$, which will serve as the
starting point for present work. The correction terms have the following form

$\displaystyle\bm{\Delta}\bm{\zeta}^{tt}_{w}=$
$\displaystyle-\frac{\mathbf{A}_{1}}{2H}+\frac{\mathbf{A}_{2}}{(2H)^{2}}+\mathcal{O}(H^{-3}),$
(17) $\displaystyle\bm{\Delta}\bm{\zeta}^{tr}_{w}=$
$\displaystyle-\frac{\mathbf{B}}{(2H)^{2}}+\mathcal{O}(H^{-3}),$ (18)
$\displaystyle\bm{\Delta}\bm{\zeta}^{rt}_{w}=$
$\displaystyle-\frac{\mathbf{B}^{\mathrm{T}}}{(2H)^{2}}+\mathcal{O}(H^{-3}),$
(19) $\displaystyle\bm{\Delta}\bm{\zeta}^{rr}_{w}=$
$\displaystyle-\frac{\mathbf{C}}{(2H)^{3}}+\mathcal{O}(H^{-4}).$ (20)

The tensors $\mathbf{A}_{1,2}$, $\mathbf{B}$, $\mathbf{C}$ above are derived
from the multipole expansion of the Blake tensor [32] (Oseen tensor for the
wall-bounded geometry) and they depend on the bulk components of the friction
tensor of a rod-like particle and its orientation angle $\theta$ but not on
the wall-particle distance. For completeness, we write them explicitly in B.

Notably, there are other strategies for tackling the problem of near-wall
mobility or various levels of accuracy, such as boundary integral equations
[33] or finite element method simulations [34]. Approximate bead-model
numerical schemes on a similar level of accuracy to the analytical correction
above supplemented by lubrication treatment of close configurations have been
introduced by Swan and Brady [35]. More accurate multipole expansion
approaches have also been developed [36]. However, comparisons of the
analytical correction presented above in Eqs. (17)–(20) with accurate bead-
model hydromultipole scheme [36] for a rod of aspect ratio $p=10$ have shown
the validity of the correction for distances up to $H/L\sim 1$, provided that
the rod is far from touching the wall, in which case lubrication corrections
become important. Thus, in this contribution we will use the approximate
correction, bearing in mind that the analysed model has been developed for
moderate particle-nanopore distances. Here, we restrict our attention to the
semi-analytical scheme presented above, since the dynamics occur mostly in the
range of wide separation of the particle and the wall.

##  3 Results

###  3.1 Scaling analysis

The length of the particle is $L$. For dsDNA the length of $L=100\ bp$
corresponds to ca. $34\ \text{nm}$. Another length scale in the problem is the
electrostatic capture radius ${\lambda_{e}}$, which at room temperature is in
the range of micrometers [15], so $\lambda_{e}/L\gg 1$.

Table 1: Time scales in the system Symbol | Scaling | Description
---|---|---
$\tau$ | $\displaystyle{6\pi\eta L^{4}}/{k_{B}T\lambda_{e}}$ | Basic time scale
$\tau_{B}$ | $\displaystyle{6\pi\eta L^{3}}/{k_{B}T}$ | Brownian (diffusive) time
$\tau_{e}^{t}$ | $\displaystyle{6\pi\eta L^{2}r^{2}}/{k_{B}T\lambda_{e}}$ | Electrostatic translation time
$\tau_{e}^{r}$ | $\displaystyle{6\pi\eta Lr^{3}}/{k_{B}T\lambda_{e}}$ | Electrostatic rotation time
$\tau_{w}$ | $\displaystyle{6\pi\eta H^{2}r^{2}}/{k_{B}T\lambda_{e}}$ | Wall-induced rotation time

The interplay between different time scales in the problem determines the
ranges in which different mechanisms of capture dominate. We list them all for
convenience in Table 1. Far enough from the pore, the dynamics are purely
Brownian, and the relevant time scale is $\tau_{B}\sim
L^{2}/\langle{D^{t}}\rangle\approx 6\pi\eta L^{3}/k_{B}T$, where
$\left\langle{D^{t}}\right\rangle$ is the average diffusion coefficient of the
nano-rod. Rotational Brownian motion occurs on the same time scale
$1/\left\langle{D^{r}}\right\rangle\sim 8\pi\eta L^{3}/k_{B}T$. In the
presence of an electrostatic force (7) driving the translational motion, the
velocity scales as $V\sim\left\langle{D^{t}}\right\rangle\lambda_{e}/r^{2}\sim
k_{B}T\lambda_{e}/6\pi\eta Lr^{2}$. Thus the relevant translational time scale
is $\tau^{t}_{e}\sim L/V\approx 6\pi\eta L^{2}r^{2}/k_{B}T\lambda_{e}$. The
electrostatic torque falls off quicker with distance, so the relevant rotation
time scale derived from it becomes $\tau^{r}_{e}\approx 6\pi\eta
L^{2}r^{3}/k_{B}T\lambda_{e}$. By comparing the Brownian and electrostatic
rotational time scale, we recover the scaling for the orientational capture
radius defined in Ref. [15]. We note that in the mobility (or diffusion)
matrix $\bm{\mu}$, the translational elements scale as $\eta L$, the coupling
terms as $\eta L^{2}$, and the rotational terms as $\eta L^{3}$. This scaling
changes close to the wall, where an additional length scale, the wall-particle
distance $H$, comes into play. When the rod comes close to the wall,
reorientation due to hydrodynamic interactions with the boundary becomes
important, with the relevant time scale $\tau_{w}$ derived from the scaling
form of the equation
$\bm{\Omega}=\bm{\mu}^{rt}\cdot\bm{F}_{e}+\bm{\mu}^{rr}\cdot\bm{T}_{e}$. By
comparing different time scales we determine 4 general regimes of motion,
sketched in Fig. 2. The furthest region is Brownian, but closer to the pore
when $\tau_{B}\sim\tau^{t}_{e}$, translational motion is driven by
electrostatic forces but rotational motion remains Brownian. Moving even
closer, electrostatic torque dominates over thermal reorientation. The
boundaries between these regions are determined by the radial distance from
the pore. However, in the vicinity of the wall, the diffusivity and hence
mobility are generally hindered by hydrodynamic interactions with the wall,
which become the dominant driving mechanism.

Figure 2: The division of near-pore space into regions colored by dominant
terms determining the dynamics of the rod. The pore is located at the origin,
and the wall coincides with the bottom border of the graph. Closest to the
surface, wall interaction terms are the most important. When the particle is
moving further away from the wall, we expect concentric regions of
electrostatically determined dynamics, electrostatic translations with
Brownian rotations, and fully Brownian dynamics. This corresponds to the
intuition that electrostatic torque decays faster than force when moving away
from the pore. The boundaries between subsequent regions are obtained by
comparing the respective times scales of motion. Expressions used to determine
the time scales are collected for convenience in Table 1. For this calculation
we assumed $\lambda_{e}=10^{3}L$.

We thus conclude that for the analysis of the behaviour of a field-driven
nano-rod, since we focus on the dynamics in the range $L<r\ll\lambda_{e}$,
Brownian motion may be neglected, as it would only influence the initial
orientations with which the rod would enter the area dominated by
electrostatic interactions. It is essential, however, to retain the
hydrodynamic anisotropy of the rod, as it influences both the translational
and rotational motion in the presence of a strong electric field.

For quantitative calculations, we choose dimensionless units with the basic
length $L$. In the presence of an electric field, the more appropriate time
unit is $\tau=\tau_{B}L/\lambda_{e}=6\pi\eta L^{4}/k_{B}T\lambda_{e}$. For
dsDNA mentioned above, this time scale is of the order of $10^{-7}\ \text{s}$.
With this choice, we can write the force and torque acting on the rod as

$\bm{F}_{e}=-\frac{\bm{\hat{r}}}{r^{2}},\qquad\bm{T}_{e}=-\frac{(\bm{\hat{r}}\cdot\bm{u})(\bm{\hat{r}}\times\bm{u})}{r^{3}}.$
(21)

###  3.2 Discussion of trajectories

Before focusing on the numerical solutions of the full set of equations, it is
informative to consider a very simple case of an anisotropic particle subject
to a central force without the wall influence and with no external torque. In
this case, the particle’s mobility tensor has the form as in Eq. (13), with
two coefficients $\mu_{\parallel}$, $\mu_{\perp}$. For very slender rods, we
have $\mu_{\parallel}\approx 2\mu_{\perp}$.

We can describe such a situation by taking a coordinate system centred at the
pore, and oriented with the principal axes of the body ($\parallel,\perp)$.
Within this parametrisation, the equations of motion of the centre of the rod
$(r_{\parallel},r_{\perp})$ are

$\displaystyle\frac{\partial r_{\parallel}}{\partial t}$
$\displaystyle=\mu_{\parallel}\frac{F(r)}{r}r_{\parallel}$ (22)
$\displaystyle\frac{\partial r_{\perp}}{\partial t}$
$\displaystyle=\mu_{\perp}\frac{F(r)}{r}r_{\perp}$ (23)

By displaying the equations in this form, it is immediately clear that the
function $F(r)$ only influences the time dependence and has no bearing on the
trajectory. For bounded $F$, there is a hyperbolic fixed point at the origin
with two important trajectories intersecting at it: $r_{\parallel}=0$ and
$r_{\perp}=0$. The first corresponds to slower, sideways motion and the second
to faster, axial motion. We conclude that a particle with different values of
the drag coefficients along different axes will almost always approach the
fixed point along the slowest axis, which in the case of a rod-like colloids
means that the particles would most often approach the pore broadside. This
conclusion holds sway whenever the torques acting on the rod are negligible
and thus it concisely describes the initial dynamics when starting far enough
from the pore. Approaching the slow rectilinear trajectory is accomplished by
hydrodynamic ’gliding’ effect where velocity stays at an angle to the force in
the initial stages of motion. Moreover, we predict the existence of two types
of trajectories – concave and convex – depending on whether rod initially
points towards or away from the wall compared to the pore direction.

Figure 3: Trajectories of the article centre depending on the initial
orientation of the particle. Due to differences of drag coefficients in the
direction along and across the particle, the velocity is never aligned with
the drag force. This leads to gliding, either higher above the wall or closer
to the wall depending on the orientation, importantly ruling out a straight
path towards the pore.

To explore the dynamics driven by the interplay between electrostatic and
hydrodynamic forces acting on the nanorod, we integrate the deterministic
equations of motion numerically. As argued before, we neglect the influence of
Brownian motion in the range of distances under consideration. The equations
of motion for the system are

$\displaystyle\begin{pmatrix}\bm{V}\\\ \bm{\Omega}\end{pmatrix}$
$\displaystyle=\bm{\mu}_{w}\begin{pmatrix}\bm{F}_{e}\\\
\bm{T}_{e}\end{pmatrix},$ (24)

which together with Eqs. (3) and (4) allow us to calculate the trajectories of
nano-rods. We consider an ellipsoidal rod of aspect ratio $p=10$. We
integrated the equations of motion using NDSolve command of Mathematica 12.1
with Method -> {"EquationSimplification" -> "Residual"} option enabled to deal
with algebraic-differential nature of the equations.

Figure 4: Top: Four sample trajectories with the particle orientation sketched
at equal translation intervals. The particles start from an initial position
$(35,25)$ at four different initial orientations ($\theta_{0}=k\pi/36$, with
$k=3,11,22,34$), depending on which they follow convex or concave gliding
trajectories. Bottom: Inclination angles corresponding to the trajectories
plotted above against normalized time. Reorientation rapidly accelerates in
the later stages of motion because the initial offset due to the sideways
glide close to the starting point has increasingly larger effect on the
approach angle.

We present the resulting trajectories in Fig. 3, starting from a point
$\bm{r}_{0}=(35,25)$ above the wall, thus seen at an angle $\alpha_{0}\approx
2\pi/9$ from the nanopore. Rods starting at different orientations glide
sideways with respect to the field direction due to their shape (and therefore
also mobility) anisotropy. For the initial angle $\theta_{0}<\alpha_{0}$, the
trajectories are convex and they approach the vicinity of the pore from below
the direction $\alpha_{0}=\theta_{0}$. The initial inclination
$\theta_{0}>\alpha_{0}$ leads to a concave path gliding above the $\bm{r}_{0}$
direction. The shape of the trajectories is also dependent on the aspect ratio
of the nano-rod, which determines the parallel and perpendicular friction
(mobility) coefficients. At large distances, the wall does not influence the
observed dynamics. A closer look into the particles’ orientations, sketched in
Fig. 4 for four chosen initial orientations ($\theta_{0}=k\pi/36$, with
$k=3,11,22,34$), reveals a strong alignment with the field lines, predicted by
earlier works neglecting hydrodynamic anisotropy [15]. The time dependence of
the rods’ orientation shows a systematic change, the rate of which increases
when approaching the pore due to the increasingly strong attractive force and
aligning torque. At the considered separations from the pore, we see no
pronounced effect of the translation-rotation coupling due to the wall, which
in this case is too weak to compete with field-driven motion. It would come
into significance at near-touching configurations where we also expect the
detailed geometry of the pore to matter.

To test how strong the alignment with the field lines is, in Fig. 5 we present
results of numerical simulations for a spectrum of initial orientation angles
$\theta_{0}$ for rods released far away from the pore at three representative
values of the polar angle $\alpha_{0}$. For each starting configuration, we
determine the final orientation $\theta_{f}$ and plot the final angle to the
local field line, $|\alpha_{f}-\theta_{f}|$, at the final position of the
particle. For most initial values of the approach angle, conclusions are
similar to those from models neglecting wall interaction terms - the final
deviation angle from the field lines direction is typically of the order of
$\pi/50$. For small angles $\alpha_{0}$, however, the relationship between the
initial inclination and the final angle to field lined is asymmetric due to
wall interactions, which increase the offset to the field lines. For example,
we observe angles greater than $\pi/16$ in the region $(-\pi/8,\pi/8)$. When
interpreting this result, it is notable that in 3D under uniform distribution
of charged particles over the hemisphere of possible initial directions, small
angles $\alpha_{0}$ occur much more often (proportionally to
$\cos\alpha_{0}$), meaning that such effect can hinder the capture of a
substantial number of particles. The area of the region with
$\alpha_{0}<\pi/8$ accounts for nearly 40% of the considered hemisphere. Thus
it remains important in a high percentage of capture events to properly
resolve hydrodynamic interactions with the confining boundary.

Notably, we have chosen the initial orientations to lie within the $xz$ plane.
By solving the full, three-dimensional equations of motion (3), (4) and (24),
we confirm that the qualitative characteristics of motion remain unchanged
even when the initial orientation of the rod has an out-of-plane component.
The reorientation in the region far away from the wall remains mostly
Brownian, while in the near-pore region the strong alignment mechanism brings
the dynamics to a plane to which we restrict our attention from the beginning.
We thus do not see any changes as compared to the axisymmetric configurations
analysed here.

Figure 5: The final angle between the rod and the field line plotted against
the initial inclination angle at different starting positions far away from
the pore with different polar angles: $\alpha_{0}=\pi/8$ – close to the wall,
$\alpha_{0}=\pi/4$ – intermediate, $\alpha_{0}=3\pi/8$ – far from the wall.
Trajectories with $\theta_{0}<\alpha_{0}$ are convex and initially glide
towards the wall, and trajectories with $\theta_{0}>\alpha_{0}$ are concave
and initially glide away from the wall. Importantly, when the two angles are
similar, $\theta_{0}\approx\alpha_{o}$, a straight trajectory is unstable
because of drag anisotropy and leads to relatively large values of the final
angle between the field lines and the rod, $|\theta-\alpha|$. Additionally,
for small $\alpha_{0}$ this relationship is asymmetric due to wall influence
where the hydrodynamic torque from wall drag competes with electrostatic
reorientation in the late stages of movement.

##  4 Conclusions

We have presented an analysis of motion of a colloidal nano-rod driven by the
electric field of a nanopore in a viscous fluid with a particular focus on the
inclusion of hydrodynamic interactions and a detailed analysis of the
different regimes of motion. The nanopore is modelled as a point charge which
attracts a uniformly charged rod-like particle. Basing on scaling arguments,
we identified the various time scales of motion and demonstrated that far away
from the pore the motion of the particle is purely Brownian but as soon as it
reaches the electrostatic capture radius $\lambda_{e}$ it is systematically
attracted towards the pore. Its initial dynamics are then governed by an
electrostatic force driving its translational motion, with the velocity
resulting from the balance of this force and the fluid drag force. Since the
latter is anisotropic, and depends on the orientation of the particle, the
motion resembles sideways gliding towards the pore. At closer distances, the
electrostatic torque becomes pronounced and strongly aligns the rods with the
electric field lines, as reported previously [15]. However, earlier studies
neglected the role of hydrodynamic interactions both on the level of the
particle anisotropy, and the wall-induced increase of friction.

In this contribution, we have outlined a theoretical approach which takes into
account both the anisotropy of the particle and the wall hindrance effect.
Supported by scalings, we explored the trajectories at intermediate distances,
when Brownian motion can be neglected, but the rod is far enough from the pore
to disregard the field and flow effects of the pore geometry. At large and
moderate distances, we have employed an approximate analytical scheme, in
which the friction tensor of a colloid close to the wall can be decomposed
into its bulk value, encoding the particle anisotropy, and a wall-induced
correction. This allowed us to formulate a deterministic system of equations
which can be solved for arbitrary initial position and orientation of the
nano-rod.

For starting points at a large polar angle $\alpha$, we find that gliding
trajectories are governed by the shape anisotropy of the rod, and the wall
plays no significant role. However, for smaller approach angles, hydrodynamic
interactions with the wall significantly alter the angle at which the rod
approaches the near-pore region. We have shown the extent of this region to be
as large as $\pi/8$ which accounts for nearly 40% of spherical area in 3D,
thus signifying the importance of wall effects in the proper modelling of
dynamics in confined geometry.

Our work shows, basing on a scaling analysis, that it is justified to neglect
the role of Brownian motion in the near-pore region. Previous works on a
similar system included Brownian motion on the level of translational motion
only, by imposing a constant diffusion coefficient of the particle [15].
However, in order to properly resolve the question of Brownian displacements
and rotations, one should should account for two facts: a) the non-spherical
shape of the rod which makes its diffusion anisotropic even in a bulk system,
and b) the presence of the wall which renders the anisotropic diffusion tensor
of the particle a wall-particle distance-dependent and introduces translation-
rotation coupling. Even then, we would expect the effect of Brownian motion to
be pronounced outside the near-pore region which is dominated by electrostatic
interactions. Including these effects would be an interesting direction of
future research.

The analysis of trajectories close to the pore remains a separate issue, since
its non-planar geometry has a significant influence on the trajectories, both
by changing the structure of the electric field, which can no longer be
modelled by a point charge, and by the different character of hydrodynamic
interactions, where the pore opening shapes the lubrication flow and friction
landscape for the colloid. For an insight into these dynamics, detailed models
of the pore structure should be implemented.

## Acknowledgments

We thank Piotr Szymczak for helpful discussions. The work of RW and ML was
supported by the National Science Centre of Poland grant Sonata to ML no.
2018/31/D/ST3/02408.

##  Appendix A Bulk friction tensors of an ellipsoid

The elements of the bulk friction tensor for an ellipsoidal colloid are known
analytically [23]. For a prolate spheroid with a long axis $a=L/2$ and a short
axis $c$, corresponding to an eccentricity $e=\sqrt{a^{2}-c^{2}}/a$, the bulk
friction coefficients in $\bm{\zeta}_{0}$ can be written as

$\displaystyle\zeta^{t}_{\parallel}$ $\displaystyle=(6\pi\eta
a)\frac{8}{3}e^{3}\Bigl{[}-2e+\left(1+e^{2}\right)\ell\Bigr{]}^{-1},$ (25)
$\displaystyle\zeta^{t}_{\perp}$ $\displaystyle=(6\pi\eta
a)\frac{16}{3}e^{3}\Bigl{[}2e+\left(3e^{2}-1\right)\ell\Bigr{]}^{-1},$ (26)
$\displaystyle\zeta^{r}_{\parallel}$ $\displaystyle=(8\pi\eta
a^{3})\frac{4}{3}e^{3}(1-e^{2})\Bigl{[}2e-\left(1-e^{2}\right)\ell\Bigr{]}^{-1},$
(27) $\displaystyle\zeta^{r}_{\perp}$ $\displaystyle=(8\pi\eta
a^{3})\frac{4}{3}e^{3}(2-e^{2})\Bigl{[}-2e+\left(1+e^{2}\right)\ell\Bigr{]}^{-1},$
(28) $\displaystyle\zeta^{dr}$ $\displaystyle=(\pi\eta
a^{3})\frac{16}{3}e^{5}\Bigl{[}-2e+\left(1+e^{2}\right)\ell\Bigr{]}^{-1},$
(29) $\displaystyle\ell$ $\displaystyle=\log\left(\frac{1+e}{1-e}\right).$
(30)

The last coefficient, $\zeta^{dr}$, links the stresslet (symmetric dipole
moment) on the spheroid with the rate-of-strain tensor of an external flow.

##  Appendix B Wall correction terms

The correction terms are described in detail in Ref. [29]. It is most
convenient to specify the components of the near-wall friction tensor in a
body-fixed frame of reference. It is defined by a set of basis vectors
$\left\\{\bm{u},\bm{u}_{\perp 1},\bm{u}_{\perp 2}\right\\}$, where $\bm{u}$ is
the director along the long axis of the nano-rod, $\bm{u}_{\perp
1}=(\bm{e}_{z}\times\bm{u})/\left|\bm{e}_{z}\times\bm{u}\right|$ is parallel
to the wall and perpendicular to the particle axis, and $\bm{u}_{\perp
2}=\bm{u}_{\perp 1}\times\bm{u}$ completes the orthonormal basis. We can write
the tensors in Eqs. (17)–(20) explicitly in the body-fixed frame RW. For
convenience, we define the following shorthand notation

$\text{s}\equiv\sin\theta,\qquad\text{c}\equiv\cos\theta.$ (31)

For the translational part (17), we find the correction’s angular dependence
as

$\displaystyle\mathbf{A}_{1}=-\frac{3}{16\pi\eta}\begin{pmatrix}(\zeta^{t}_{\parallel})^{2}(1+\text{c}^{2})&0&-\zeta^{t}_{\parallel}\zeta^{t}_{\perp}\text{s}\,\text{c}\\\
0&(\zeta^{t}_{\perp})^{2}&0\\\
-\zeta^{t}_{\parallel}\zeta^{t}_{\perp}\text{s}\,\text{c}&0&(\zeta^{t}_{\perp})^{2}(1+\text{s}^{2})\end{pmatrix},$
(32)
$\displaystyle\mathbf{A}_{2}=\frac{9}{256\pi^{2}\eta^{2}}\begin{pmatrix}A_{\parallel}&0&A_{\parallel\perp}\\\
0&A_{\perp 1}&0\\\ A_{\parallel\perp}&0&A_{\perp 2}\end{pmatrix}.$ (33)

with the coefficients

$\displaystyle A_{\perp 1}$ $\displaystyle=(\zeta^{t}_{\perp})^{3},$ (34)
$\displaystyle A_{\parallel}$
$\displaystyle=(\zeta^{t}_{\parallel})^{3}(1+\text{c}^{2})^{2}+\zeta^{t}_{\perp}(\zeta^{t}_{\parallel})^{2}\text{s}^{2}\text{c}^{2},$
(35) $\displaystyle A_{\perp 2}$
$\displaystyle=(\zeta^{t}_{\perp})^{3}(1+\text{s}^{2})^{2}+(\zeta^{t}_{\perp})^{2}\zeta^{t}_{\parallel}\text{s}^{2}\text{c}^{2},$
(36) $\displaystyle A_{\parallel\perp}$
$\displaystyle=-\zeta^{t}_{\parallel}\zeta^{t}_{\perp}[\zeta^{t}_{\parallel}(1+\text{c}^{2})+\zeta^{t}_{\perp}(1+\text{s}^{2})]\text{s}\,\text{c}.$
(37)

The translation-rotation coupling part reads

$\displaystyle\mathbf{B}=\frac{3\zeta^{dr}}{26\pi\eta}\begin{pmatrix}0&\zeta^{t}_{\parallel}(1+\text{c}^{2})\text{s}&0\\\
0&0&\zeta^{t}_{\perp}\text{c}\\\
0&-\zeta^{t}_{\perp}(1+\text{s}^{2})\text{c}&0\end{pmatrix}.$ (38)

Finally, the rotational part $\mathbf{C}$ in Eq. (20) has the form

$\displaystyle\mathbf{C}=$
$\displaystyle-\frac{1}{16\pi\eta}\begin{pmatrix}(\zeta^{r}_{\parallel})^{2}(5-3\text{c}^{2})&0&3\zeta^{r}_{\parallel}\zeta^{r}_{\perp}\text{s}\,\text{c}\\\
0&5(\zeta^{r}_{\perp})^{2}&0\\\
3\zeta^{r}_{\parallel}\zeta^{r}_{\perp}\text{s}\,\text{c}&0&(\zeta^{r}_{\perp})^{2}(5-3\text{s}^{2})\end{pmatrix}+$
(39)
$\displaystyle\frac{3\zeta^{dr}}{16\pi\eta}\begin{pmatrix}0&0&\zeta^{r}_{\parallel}\text{s}\,\text{c}\\\
0&-2\zeta^{r}_{\perp}(1-2\text{c}^{2})&0\\\
\zeta^{r}_{\parallel}\text{s}\,\text{c}&0&2\zeta^{r}_{\perp}\text{c}^{2}\end{pmatrix}+$
$\displaystyle-\frac{3(\zeta^{dr})^{2}}{16\pi\eta}\begin{pmatrix}0&0&0\\\
0&3+\text{c}^{2}-\text{c}^{4}&0\\\ 0&0&1+2\text{c}^{2}\end{pmatrix}.$

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$\gtrsim\frac{r\tilde{\delta}}{N\delta}$. Also, there are
$\sim|\mathcal{T}|/N$ thick tubes
$\mathbf{T}_{\tilde{\delta}}\in\mathcal{T}^{\tilde{\delta}}$.

By Proposition 4.4 applied at scale $\tilde{\delta}$ with
$\mathbf{r}\gtrsim\frac{r\tilde{\delta}}{N\delta}$ we find that the number of
essentially distinct balls $B(c_{p},\tilde{\delta})$ is

$\lessapprox\mathcal{S}\mathcal{T}(\tilde{\delta})\tilde{\delta}\frac{|\mathcal{T}^{\tilde{\delta}}|^{3}}{(r\tilde{\delta}/N\delta)^{3}}\sim\mathcal{S}\mathcal{T}(\tilde{\delta})\tilde{\delta}(\delta/\tilde{\delta})^{3}\frac{|\mathcal{T}|^{3}}{r^{3}}.$

Since, trivially, each $B(c_{p},\tilde{\delta})$ contains at most
$(\tilde{\delta}/\delta)^{2}$ $\delta$-squares, it follows that (recalling
that $|\mathcal{T}|\lesssim\delta^{-1}$)

(8.73)
$|\bigcup_{\mathbf{p}_{\rho}}\mathcal{P}_{\Delta,\mathbf{p}_{\rho}}|\lessapprox(\tilde{\delta}/\delta)^{2}\mathcal{S}\mathcal{T}(\tilde{\delta})\tilde{\delta}(\delta/\tilde{\delta})^{3}\frac{|\mathcal{T}|^{3}}{r^{3}}\sim\mathcal{S}\mathcal{T}(\tilde{\delta})\frac{|\mathcal{T}|^{2}}{r^{3}}.$

Also, combining this with (8.67), (8.70), and then with (8.72) we get

$|\mathcal{P}_{2}|\lessapprox(\rho/\delta)^{h(\epsilon)^{3}}(\Delta/\delta)^{\sqrt{h(\epsilon)}}\mathcal{S}\mathcal{T}(\tilde{\delta})\frac{|\mathcal{T}|^{2}}{r^{3}}\lessapprox(\tilde{\delta}/\delta)^{o_{\epsilon}(1)}\mathcal{S}\mathcal{T}(\tilde{\delta})\frac{|\mathcal{T}|^{2}}{r^{3}}.$

From the three inequalities in (8.72) we deduce that

(8.74)
$\tilde{\delta}/\delta\gtrsim(\rho_{0}/\delta)^{\frac{1}{4}h(\epsilon)^{7/4}}.$

To summarize, if we are in Case 2(a), we get

$|\mathcal{P}_{2}|\lessapprox(\tilde{\delta}/\delta)^{b_{\epsilon}}\max_{\tilde{\delta}\gtrsim\delta(\rho_{0}/\delta)^{c_{\epsilon}}}\mathcal{S}\mathcal{T}(\tilde{\delta})\frac{|\mathcal{T}|^{2}}{r^{3}},$

for some $b_{\epsilon}\to 0$, as $\epsilon\to 0$, and with
$c_{\epsilon}=\frac{1}{4}h(\epsilon)^{7/4}$. This verifies (8.55).

Case 2(b): Suppose Item $(1)$ of Theorem 5.3 happens for at least half of (and
we will casually assume this holds for each)
$\mathbf{p}_{\rho}\in\mathcal{P}_{\rho,gen}$. We no longer need to be precise
with quantifying the $o_{\epsilon}(1)$ exponents. As a general rule,
$(\rho_{0}/\delta)^{o_{\epsilon}(1)}$-losses are acceptable in this case.

We will derive two estimates for $|\mathcal{P}_{2}|,$ and then we derive an
upper bound for $W$.

First estimate:

Using (8.65), (8.66) and (8.67) we get

(8.75)
$|\mathcal{P}_{2}|\frac{r}{\tilde{r}}\lessapprox(\rho/\delta)^{o_{\epsilon}(1)}W|\mathcal{U}|\lessapprox(\rho/\delta)^{o_{\epsilon}(1)}W\rho_{0}^{-1}\mathcal{S}\mathcal{T}(\rho_{0})\frac{|\mathcal{T}|}{\tilde{r}^{3}},$

where the last estimate is as in Case 1. Thus

(8.76)
$|\mathcal{P}_{2}|\lessapprox\mathcal{S}\mathcal{T}(\rho_{0})(\rho_{0}/\delta)^{o_{\epsilon}(1)}W\rho_{0}^{-1}\frac{|\mathcal{T}|}{{r}\tilde{r}^{2}}.$

Second estimate:

We double count incidences between mini-segments
$\mathcal{U}_{1}=\cup_{\mathbf{p}_{\rho}\in\mathcal{P}_{\rho,gen}}\mathcal{U}_{1}[\mathbf{p}_{\rho}]$
and the squares in
$\cup_{\mathbf{p}_{\rho}\in\mathcal{P}_{\rho,gen}}\mathcal{P}[\mathbf{p}_{\rho}]$.
Each $p$ is incident to each $U_{1}\in\mathcal{U}_{1,p}$.

Recall that $|\mathcal{U}_{1,p}|=|\mathcal{T}_{p}^{\delta/\rho}|$, and this
number is independent of $p$. Let $\tilde{r}_{1}$ be such that

$|\mathcal{U}_{1,p}|=r/\tilde{r}_{1}.$

It follows from (8.67) that

(8.77)
$|\mathcal{P}_{2}|{r}/{\tilde{r}_{1}}\lesssim(\rho_{0}/\delta)^{o_{\epsilon}(1)}W|\mathcal{U}_{1}|.$

Recall our notation
$\mathcal{T}[\mathbf{p}]:=\\{T\in\mathcal{T}:T\cap\mathbf{p}\neq\emptyset\\}$.
Subject to only a logarithmic loss, we may assume
$|\mathcal{T}[\mathbf{p}_{\rho}]|_{\rho}\sim\mathbf{r}$ for each
$\mathbf{p}_{\rho}\in\mathcal{P}_{\rho,gen}$ and some fixed $\mathbf{r}$. More
precisely, the value $\mathbf{r}$ corresponds to a subcollection of
$\mathcal{P}_{\rho,gen}$ that contains a logarithmic fraction of
$\mathcal{P}_{2}$.

By Proposition 4.4

$|\mathcal{P}_{\rho,gen}|\lessapprox\mathcal{S}\mathcal{T}(\rho)\rho\frac{|\mathcal{T}|_{\rho}^{3}}{\mathbf{r}^{3}}.$

We get that

$|\\{(\mathbf{p}_{\rho},\mathbf{T}_{\rho})\in\mathcal{P}_{\rho,gen}\times\mathcal{T}^{\rho}:\;\mathbf{T}_{\rho}\cap\mathbf{p}_{\rho}\not=\emptyset\\}|\lessapprox\mathcal{S}\mathcal{T}(\rho)\rho\frac{|\mathcal{T}|_{\rho}^{3}}{\mathbf{r}^{2}}\sim\mathcal{S}\mathcal{T}(\rho)\rho\frac{|\mathcal{T}|_{\delta}^{2}\cdot|\mathcal{T}|_{\rho}}{|\mathcal{T}[\mathbf{p}_{\rho}]|_{\delta}^{2}}.$

Let us explain the $\sim$ from above. Recall we assumed $\mathcal{T}$ is
uniform. The collection of tubes $T$ in $\mathcal{T}[\mathbf{p}_{\rho}]$ is
the disjoint union of the tubes $\mathcal{T}[\mathbf{T}_{\rho}]$ inside the
$\sim\mathbf{r}$ fat tubes $\mathbf{T}_{\rho}$ intersecting
$\mathbf{p}_{\rho}$. On the dual side (where uniformization is performed) the
tubes in any given $\mathcal{T}[\mathbf{T}_{\rho}]$ coincide with the
$\delta$-squares inside a fixed $\rho$-square. Their number is either zero or
$|\mathcal{T}|_{\delta}/|\mathcal{T}|_{\rho}$. Thus

$|\mathcal{T}[\mathbf{p}_{\rho}]|_{\delta}\sim\mathbf{r}|\mathcal{T}|_{\delta}/|\mathcal{T}|_{\rho}.$

We conclude that

(8.78)
$|\\{(\mathbf{p}_{\rho},T)\in\mathcal{P}_{\rho,gen}\times\mathcal{T}:\;T\cap\mathbf{p}_{\rho}\not=\emptyset\\}|\lessapprox\mathcal{S}\mathcal{T}(\rho)\rho\frac{|\mathcal{T}|_{\delta}^{3}}{|\mathcal{T}[\mathbf{p}_{\rho}]|_{\delta}^{2}}.$

We now double count the intersections between the collection $\mathcal{U}_{1}$
and the tubes in $\mathcal{T}$. Each $U_{1}\in\mathcal{U}_{1}$ must be in some
$\mathcal{U}_{1,p}$, and thus it is contained in
$|\mathcal{T}_{p}|/|\mathcal{T}_{p}^{\delta/\rho}|=\tilde{r}_{1}$ tubes
$T\in\mathcal{T}_{p}$. Thus

$\tilde{r}_{1}|\mathcal{U}_{1}|\lesssim|\\{(U_{1},T)\in\mathcal{U}_{1}\times\mathcal{T}:\;U_{1}\subset
T\\}|\lessapprox\mathcal{S}\mathcal{T}(\rho)\rho\frac{|\mathcal{T}|_{\delta}^{3}}{|\mathcal{T}[\mathbf{p}_{\rho}]|_{\delta}^{2}}.$

This gives an upper bound for $|\mathcal{U}_{1}|$. Combining this with (8.77)
we find

(8.79)
$|\mathcal{P}_{2}|\lessapprox(\rho_{0}/\delta)^{o_{\epsilon}(1)}\mathcal{S}\mathcal{T}(\rho)\frac{W\rho}{r}\frac{|\mathcal{T}|_{\delta}^{3}}{|\mathcal{T}[\mathbf{p}_{\rho}]|_{\delta}^{2}}.$

Taking the geometric average of 8.76 and 8.79 we find

(8.80)
$|\mathcal{P}_{2}|\lessapprox[\mathcal{S}\mathcal{T}(\rho_{0})\mathcal{S}\mathcal{T}(\rho)]^{1/2}(\rho_{0}/\delta)^{o_{\epsilon}(1)}(\rho/\rho_{0})^{1/2}W\frac{|\mathcal{T}|_{\delta}^{2}}{r\tilde{r}|\mathcal{T}[\mathbf{p}_{\rho}]|_{\delta}}.$

Upper bound for $W$:

Since Item (1) of Theorem 5.3 holds for
$\mathbf{p}_{\rho}\in\mathcal{P}_{\rho,gen}$ we have

$|\mathcal{P}\cap\mathbf{p}_{\rho}|\lessapprox(\delta/\rho)^{t_{1}+o_{\epsilon}(1)}\frac{|\bar{\mathcal{U}}_{1}[\mathbf{p}_{\rho}]|^{2}}{(r/\tilde{r}_{1})^{2}}.$

Then double counting incidences between squares and mini-segments shows that

(8.81)
$W\lessapprox(\frac{\delta}{\rho})^{t_{1}+o_{\epsilon}(1)}\frac{|\bar{\mathcal{U}}_{1}[\mathbf{p}_{\rho}]|}{r/\tilde{r}_{1}}\lessapprox(\frac{\delta}{\rho})^{t_{1}+o_{\epsilon}(1)}\frac{|\mathcal{T}[\mathbf{p}_{\rho}]|_{\delta}}{r}.$

The second inequality uses again the fact that each mini-segment is contained
in at least $\tilde{r}_{1}$ tubes $T$. Tubes corresponding to distinct mini-
segments lying inside $\mathbf{p}_{\rho}$ must themselves be distinct.

Combining (8.80) with (8.81) we find

$\displaystyle|\mathcal{P}_{2}|$
$\displaystyle\lessapprox[\mathcal{S}\mathcal{T}(\rho_{0})\mathcal{S}\mathcal{T}(\rho)]^{1/2}\frac{|\mathcal{T}|^{2}}{r^{3}}\frac{r}{\tilde{r}}(\frac{\delta}{\rho})^{t_{1}}(\frac{\rho}{\rho_{0}})^{1/2+o_{\epsilon}(1)}$
$\displaystyle\lessapprox[\mathcal{S}\mathcal{T}(\rho_{0})\mathcal{S}\mathcal{T}(\rho)]^{1/2}(\rho_{0}/\rho)^{t_{1}-\frac{1}{2}}(\rho_{0}/\delta)^{o_{\epsilon}(1)}\frac{|\mathcal{T}|^{2}}{r^{3}}.$

The last inequality is due to (8.58). This verifies (8.55) in Case 2(b).

##  9\. proof of Theorem 1.2

We will rely on the framework introduced in [3], which we briefly recall
below.

Fix a smooth $\psi:\mathbb{R}^{2}\to\mathbb{R}$ satisfying $0\leq\psi\leq
1_{B(0,1)}$. Write $\psi_{\delta}(\xi)=\delta^{-2}\psi(\frac{\xi}{\delta})$
and $\mu_{\delta}=\mu*\psi_{\delta}$. Then (1.5) can be reformulated
equivalently as

(9.82)
$\|\widehat{\mu_{1/R}}\|_{L^{6}(\mathbb{R}^{2})}^{6}\lesssim_{\epsilon}R^{2-2s-\frac{s}{4}+\epsilon}.$

Since $\mu_{1/R}$ is supported on the $1/R$-neighborhood of $\Gamma$, we
partition this neighborhood into essentially rectangular regions $\theta$ with
dimensions roughly $R^{-1/2}$ and $1/R$. We also partition the
$1/R^{1/2}$-neighborhood into essentially rectangular regions $\tau$ with
dimensions roughly $R^{-1/4}$ and $R^{-1/2}$. Each $\theta$ sits inside some
$\tau$.

Let $D,M,P$ be dyadic parameters, with $D$ and $P$ integers. We decompose
$\mu$

$\mu=\sum_{D,M,P}\mu_{D,M,P},$

with each $\mu_{D,M,P}$ satisfying the following properties:

(P1) for each $\theta$, either $\mu_{D,M,P}(\theta)=\mu(\theta)\sim MR^{-s}$,
or $\mu_{D,M,P}(\theta)=0$. We call $\theta$ active if it falls into the first
category.

(P2) each $\tau$ contains either $\sim P$ or no active $\theta$. We call
$\tau$ active if it fits into the first category.

(P3) there are $\sim D$ active $\theta^{\prime}s$, or equivalently, there are
$\sim D/P$ active $\tau^{\prime}s$.

We note that these properties together with (1.4) force the following
inequalities

(9.83) $M\lesssim R^{s/2}$ (9.84) $DM\lesssim R^{s}$ (9.85) $MP\lesssim
R^{3s/4}.$

For the rest of the argument we fix $D,M,P$ and let
$F=R^{s-2}\widehat{\mu_{D,M,P}*\psi_{1/R}}$ be the $L^{\infty}$ upper
normalized version, $\|F\|_{\infty}\lesssim 1$. Invoking the triangle
inequality, it suffices to prove that

(9.86) $\|F\|_{L^{6}}^{6}\lesssim_{\epsilon}R^{4s-10-\frac{s}{4}+\epsilon}.$

Write $\mu_{\theta}$ for the restriction of $\mu$ to an active $\theta$. Let
$F_{\theta}=R^{s-2}\widehat{\mu_{\theta}*\psi_{1/R}}$ be the Fourier
restriction of $F$ to $\theta$, so that $F=\sum_{\theta}F_{\theta}$. Then

$F_{\theta}=\sum_{T\in\mathcal{T}_{\theta}}\langle F_{\theta},W_{T}\rangle
W_{T}+O(R^{-100}).$

Each $T$ is a rectangle (referred to as tube) with dimensions $R^{1/2}$ and
$R$, with the long side pointing in the direction normal to $\theta$. The term
$O(R^{-100})$ is negligible, and can be dismissed. Its role is to ensure that
all $T$ sit inside, say, $[-R,R]^{2}$. The wave packet $W_{T}$ is $L^{2}$
normalized, has spectrum inside (a slight enlargement of) $\theta$, and is
essentially concentrated spatially in (a slight enlargement) of $T$. Thus,
with $\chi_{T}$ being a smooth approximation of $1_{T}$, we have

$|W_{T}|\lesssim R^{-3/4}\chi_{T}.$

Given the dyadic parameter $\lambda$ we write

$\mathcal{T}_{\lambda,\theta}=\\{T\in\mathcal{T}_{\theta}:\;|\langle
F_{\theta},W_{T}\rangle|\sim\lambda\\}.$

We have

(9.87) $\lambda\leq\lambda_{max}\sim MR^{-\frac{5}{4}}.$

(9.86) boils down to proving

(9.88) $\|\sum_{T\in\cup_{\theta}\mathcal{T}_{\lambda,\theta}}\langle
F_{\theta},W_{T}\rangle
W_{T}\|_{L^{6}}^{6}\lesssim_{\epsilon}R^{4s-10-\frac{s}{4}+\epsilon}.$

For each rectangle $B$ with dimensions $\Delta$ and $R$, and with orientation
identical to that of the tubes in $\mathcal{T}_{\theta}$, we have the estimate

(9.89) $|\\{T\in\mathcal{T}_{\lambda,\theta}:\;T\subset
B\\}|\lesssim(\frac{\Delta}{\sqrt{R}})^{1-s}\frac{MR^{\frac{s-5}{2}}}{\lambda^{2}}.$

In particular,

(9.90)
$|\cup_{\theta}\mathcal{T}_{\lambda,\theta}|\lesssim\frac{MD}{\lambda^{2}R^{2}}.$

Each arc of length $r\lesssim 1$ on $\Gamma$ intersects at most

(9.91) $\lesssim(r\sqrt{R})^{s}\frac{R^{s/2}}{M}$

many $\theta$.

We derive two estimates.

First estimate: We first use decoupling ([2]) into caps $\theta$ (combined
with property (P3)), then (9.90) to get

$\displaystyle\|\sum_{\theta\text{
active}}\sum_{T\in\mathcal{T}_{\lambda,\theta}}\langle F_{\theta},W_{T}\rangle
W_{T}\|_{6}^{6}$
$\displaystyle\lesssim_{\epsilon}R^{\epsilon}D^{2}\sum_{\theta\text{
active}}\|\sum_{T\in\mathcal{T}_{\lambda,\theta}}\langle
F_{\theta},W_{T}\rangle W_{T}\|_{6}^{6}$
$\displaystyle\lesssim_{\epsilon}R^{\epsilon}D^{2}R^{-9/2}\lambda^{6}\sum_{\theta\text{
active}}\|\sum_{T\in\mathcal{T}_{\lambda,\theta}}\chi_{T}\|_{6}^{6}$
$\displaystyle\lesssim_{\epsilon}R^{\epsilon}D^{2}R^{-9/2}\lambda^{6}R^{3/2}|\cup_{\theta}\mathcal{T}_{\lambda,\theta}|$
$\displaystyle\lesssim_{\epsilon}R^{\epsilon}\frac{MD^{3}\lambda^{4}}{R^{5}}.$

Using (9.87) we conclude with

(9.92) $\|\sum_{\theta\text{
active}}\sum_{T\in\mathcal{T}_{\lambda,\theta}}\langle F_{\theta},W_{T}\rangle
W_{T}\|_{6}^{6}\lesssim_{\epsilon}R^{\epsilon}\frac{M^{5}D^{3}}{R^{10}}.$

Second estimate: We recall the following bush estimate proved in [3], that
uses decoupling into the larger caps $\tau$.

######  Proposition 9.1.

Let $\mathcal{B}$ be a collection of tubes
$T\in\cup_{\theta}\mathcal{T}_{\lambda,\theta}$ intersecting a
$\sqrt{R}$-square $q$, such that each $\tau$ contains either $\sim P^{\prime}$
or no $\theta$. Assume there are $\sim r^{\prime}/P^{\prime}$ caps $\tau$ from
the first category. Then

(9.93) $\|\sum_{T\in\mathcal{B}}\langle F_{\theta},W_{T}\rangle
W_{T}\|_{L^{6}(q)}^{6}\lesssim_{\epsilon}R^{\epsilon-7/2}(r^{\prime})^{3}(P^{\prime})^{2}\lambda^{6}.$

Now, for each dyadic $r\geq 1$, let

$\mathcal{Q}_{r}=\\{q:\;q\text{ intersects }\sim r\text{ tubes in
}\cup_{\theta}\mathcal{T}_{\lambda,\theta}\\}.$

To estimate $|\mathcal{Q}_{r}|$, we note that, due to (9.89) and (9.91), the
rescaled copies of the tubes $T\in\cup_{\theta}\mathcal{T}_{\lambda,\theta}$
satisfy the hypotheses of Proposition 4.2 with $\delta=R^{-1/2}$,
$K_{1}\sim\frac{R^{s/2}}{M}$ and
$K_{2}\sim\frac{MR^{\frac{s-5}{2}}}{\lambda^{2}}.$ Combining this proposition
with (9.90) shows that

(9.94)
$|\mathcal{Q}_{r}|\lesssim_{\epsilon}\frac{R^{\frac{1}{2}+\epsilon}}{r^{3}}(\frac{R^{s-\frac{5}{2}}}{\lambda^{2}})^{2}\frac{MD}{\lambda^{2}R^{2}}.$

For each $q\in\mathcal{Q}_{r}$, the bush
$\mathcal{B}(q)=\\{T\in\cup_{\theta}\mathcal{T}_{\lambda,\theta}:\;q\cap
T\not=\emptyset\\}$ can be partitioned into $\lessapprox 1$ many bushes with
data $(P^{\prime},r^{\prime})$ satisfying $P^{\prime}\leq P$, $r^{\prime}\leq
r$. Thus, combining (9.93) with (9.94) we find

$\|\sum_{T\in\cup_{\theta}\mathcal{T}_{\lambda,\theta}}\langle
F_{\theta},W_{T}\rangle
W_{T}\|_{L^{6}(\cup_{q\in\mathcal{Q}_{r}q})}^{6}\lesssim_{\epsilon}R^{\epsilon-7/2}r^{3}P^{2}\lambda^{6}R^{\frac{1}{2}}(\frac{R^{s-\frac{5}{2}}}{\lambda^{2}})^{2}\frac{MD}{\lambda^{2}R^{2}}\frac{1}{r^{3}}=R^{2s+\epsilon}\frac{MDP^{2}}{R^{10}}.$

Since there are $\lessapprox 1$ dyadic values of $r$, we get our second
estimate

(9.95) $\|\sum_{T\in\cup_{\theta}\mathcal{T}_{\lambda,\theta}}\langle
F_{\theta},W_{T}\rangle
W_{T}\|_{L^{6}}^{6}\lesssim_{\epsilon}R^{2s+\epsilon}\frac{MDP^{2}}{R^{10}}.$

When $M\sim R^{s/2}$, (9.95) together with (9.84) and (9.85) imply that
$MDP^{2}\lesssim R^{\frac{3s}{2}}$, leading to the extra saving $R^{s/4}$
(that means, the exponent $\frac{7s}{2}-10+\epsilon$) in (9.88). But when $M$
is small, (9.95) by itself gives no useful upper bound. Instead, taking the
geometric average of (9.92) and (9.95) we find

$\|\sum_{T\in\cup_{\theta}\mathcal{T}_{\lambda,\theta}}\langle
F_{\theta},W_{T}\rangle
W_{T}\|_{L^{6}}^{6}\lesssim_{\epsilon}R^{s+\epsilon-10}(MD)^{2}MP\lesssim_{\epsilon}R^{4s-10-\frac{s}{4}+\epsilon}.$

The last inequality follows from (9.84) and (9.85). The extreme case for which
our argument is tight is when $M\sim R^{\frac{3s}{8}}$, $D\sim
R^{\frac{5s}{8}}$, $P\sim R^{\frac{3s}{8}}$.

######  Remark 9.2.

When the measure is $AD$-regular, $M$ is forced to be $\sim R^{s/2}$.
Moreover, as explained in [3], there is an extra gain in the bush inequality
(9.93) that comes from estimating non trivially the energy of flat AD-regular
sets. As a result, the exponent in (1.5) for such measures can be lowered to
$2-2s-\frac{s}{2}-\beta$, for some unspecified $\beta>0$. Similarly, in the
context of Theorem 1.3 we get

$\mathbb{E}_{3,\delta}(S)\lesssim\delta^{-\frac{7s}{2}+\beta}.$

## References

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# Hadamard Layer to Improve Semantic Segmentation

###### Abstract

The Hadamard Layer, a simple and computationally efficient way to improve
results in semantic segmentation tasks, is presented. This layer has no free
parameters that require to be trained. Therefore it does not increase the
number of model parameters, and the extra computational cost is marginal.
Experimental results show that the new Hadamard layer substantially improves
the performance of the investigated models (variants of the Pix2Pix model).
The performance’s improvement can be explained by the Hadamard layer forcing
the network to produce an internal encoding of the classes so that all bins
are active. Therefore, the network computation is more distributed. In a sort
that the Hadamard layer requires that to change the predicted class, it is
necessary to modify $2^{k-1}$ bins, assuming $k$ bins in the encoding. A
specific loss function allows a stable and fast training convergence.

Index Terms—  Semantic segmentation, Hadamard codification. Conditional
generative network. Pix2Pix model.

##  1 Introduction

Many problems in computer vision can be defined as image-to-image translation
with an input image into a corresponding output image. A clear example is
semantic image segmentation, where each pixel in an input image is assigned to
a class label; in a certain way, we predict an output image that is less
complex than the input, even if the input image is a medical image or a street
image seen by an autonomous car. Recently, this task has been solved with
different strategies using convolutional neural networks (CNN), i.e., [1, 2],
even though the main trouble of predicting pixels from pixels does not change.

An alternative to solved image segmentation using CNN is Conditional
Generative Adversarial Networks (cGAN), an image-to-image translation network.
Pix2Pix, proposed by Isola et al. [3], was the first demonstration of cGANs
successfully generating nearly discrete segmentation labels rather than
realistic images.

In this work, we present the Hadamard layer. This layer enforces each class
label to be encoded to have an equal number of $+1$s and $-1$s, unlike the
classic one-hot encoding. With that, we increase the Hamming distance between
the class labels and obtain better segmentation results. We think this happens
because to attack one-hot encoding requires changing the response in a bin to
flip the assigned class, which explains the susceptibility to adversary
attacks by neural networks in classification tasks. On the other hand, using
the Hadamard encoding requires modifying half of the bins to change a label.
In this way, the network requires a more distributed activation to produce a
label.

We present experiments on modified cGANs to evaluate the performance of this
new layer. In those experiments, we noted a performance increment without
sacrificing training time.

Fig. 1: Shows a Hadamard Layer embedded in the generator of a Pix2Pix
architecture.

##  2 Related Work

###  2.1 Semantic Segmentation

Several convolutional NN architectures for semantic segmentation have been
reported in recent years. Among them, popular architectures include UNet [4],
which was initially developed for BioMedical Image Segmentation. Since its
publication in 2015, variants of the UNet have emerged for such purpose. Among
them, Pix2Pix-based models are computationally efficient methods that produce
good-quality segmentation. Pix2Pix consists of a stable Generative Adversarial
Networks (GANs) strategy for training a UNet. The advantage of UNet resides in
its simplicity: it consists of a single UNet (hourglass stage). We chose the
Pix2Pix model to introduce our Hadamard Layer in this work. Thus, we evaluate
the performance impact of the proposed Hadamard Layer in different
implementations, such as the classic UNet, the ResUNet, the VGG-UNet, and the
UNet3+ variants. We briefly describe such variants below.

A UNet comprises two stages: a contraction path (encoder) and an expansion one
(decoder). The encoder captures the image’s context using a stack of
convolutional and max pooling layers. The decoder transform combines and
expands the extracted features to construct the desired output. ResUNet [5]
takes the performance gain of Residual networks and uses it with the UNet.
This architecture was developed by Zhengxin Zhang et al. and initially used
for road extraction from high-resolution aerial images in remote sensing image
analysis. Later, it was adopted by researchers for multiple other
applications, such as brain tumor segmentation, human image segmentation, and
many more. Although this is a capable network, it has a slightly large number
of parameters. VGG-UNet [6] is another variant of UNet model that combines a
pretrained backbone as the encoder. In this proposal, the decoder consists of
five up-sampling blocks and uses a symmetric expanding path to enable precise
localization segmentation to detect corrosions from steel bridges and cracks
from rubber bearings under limited conditions. More recently, UNet3+ [7] was
reported as the best variant for image segmentation: full-scale skip
connections incorporate low-level details with high-level semantics from
feature maps in different scales and full-scale deep supervision that helps to
learn hierarchical representations from the full-scale aggregated feature
maps. The growing popularity of the UNet3+ is due to its superior performance
compared with other reported variants as Attention UNet [8], PSPNet [9], and
DeepLab [10], which were considered SOTA.

Despite the advances mentioned in this problem, these solutions are not yet
perfect, which is why in this work, we propose a change that can be applied to
any neural network that brings a notable benefit. To understand it better, we
will first introduce related theory.

Fig. 2: Shows some examples of test images from the CelebAMask-HQ dataset: (a)
Input image and (b) target segmentation map. The reminder columns depict the
results: (c-d) UNet model, (e-f) ResUNet, (g-h) VGG-UNet, and (i-j) UNet3$+$;
where columns (c, e, g, i) correspond to models trained with one-hot
codification and columns (d, f, h, j) to models trained with Hadamard layer
codification.

###  2.2 Linear Error-Correction Codes

Information transmission is a well-understood area of mathematics where the
central idea is to increase the Hamming distance between codewords by adding
redundancy to a message to ensure that even if noise flips some random bits,
the message is still legible. The solution to this problem is using linear
error-correcting codes like Hamming codes [11] or Reed-Solomon codes[12]. In
this work, we selected a particular case of Reed-Solomon codes, the Hadamard
codes, for our proposal.

We recognize that Hadamard codes are suitable to be incorporated into neural
network architectures, given previous works that show the effectiveness of
Hadamard codes in the classification of deep features [13] and as a defense
strategy against multiple adversary attacks [14]. Nevertheless, this approach
has not been explored in semantic segmentation. Hadamard codes are easy to
construct; assume we need codes of length $2^{k}$, or in other words, there
are at most $2^{k}$ possible classes in our segmentation problem; they can be
fewer. The Hadamard codes are the rows of the matrix $H_{2^{k}}$, defined
recursively as

$H_{2^{k}}=\begin{bmatrix}[r]H_{2^{k-1}}&H_{2^{k-1}}\\\
H_{2^{k-1}}&-H_{2^{k-1}}\end{bmatrix},$ (1)

with

$H_{2}=\begin{bmatrix}[r]1&1\\\ 1&-1\end{bmatrix}.$ (2)

The above procedure for computing Hadamard matrices is named Silvester’s
construction and produces symmetric matrices [15]. For example, using this
construction for eight classes, we have

$H_{8}=\begin{bmatrix}[r]1&\;1&1&1&1&1&1&1\\\ 1&-1&1&-1&1&-1&1&-1\\\
1&1&-1&-1&1&1&-1&-1\\\ 1&-1&-1&1&1&-1&-1&1\\\ 1&1&1&1&-1&-1&-1&-1\\\
1&-1&1&-1&-1&1&-1&1\\\ 1&1&-1&-1&-1&-1&1&1\\\ 1&-1&-1&1&-1&1&1&-1\\\
\end{bmatrix}.$ (3)

In the examples above, we can observe that any pair of codes in a $H_{n}$
Hadamard matrix are at a distance ${2^{k-1}}$; this property allows us to
define a more demarcated space between classes, which we consider essential
for image segmentation.

##  3 Methodology

Figure 1 illustrates the use of the proposed _Hadamard Layer_ in the UNet
model to achieve a segmentation task. Our Hadamard Layer consists of two
operations. First, to multiply by $H^{\top}$ the UNet stage output’s
$\hat{y}_{c}\in\mathbb{R}^{H\times W\times 2^{k}}$. Second, to apply a
$softmax$ activation to enforce, even more, the output $\hat{y}$ to be more
informative and closer to the one-hot code:

$\displaystyle\hat{y}$ $\displaystyle=\mathcal{H}(\hat{y}_{c})$
$\displaystyle=\textrm{softmax}(H^{\top}\hat{y}_{c}).$ (4)

A valuable property of Hadamard matrices $H$ is that they are orthogonal:
$H\,H^{\top}\;=\;n\,I_{n}$; where $n$ is the order, $I_{n}$ is the $n\times n$
identity matrix and $H^{T}$ is the transpose of $H$. Since $H$ is orthogonal,
$y$ would be closer to a one-hot code if $y_{c}$ is similar to a hadamard
code; _i.e._ , $y_{c}$ would have all its entries activated.

Now, we introduce another essential ingredient for having successful training:
the loss functions for the cGAN. Then, let $(x,y)$ be the input data for our
UNet, $\hat{y}$ is the predicted segmentation, and $\alpha$ the output of the
Discriminator (matrix with dimensions $h\times w$) of the full cGAN. Each
element $\alpha_{ij}$ can be understood as the probability that the support
region of the analyzed pair, $(x,y)$ or $(x,\hat{y})$, is real. Then, we use
the Discriminator loss given by

$L_{D}(\hat{y})=S(\mathbf{1}|\alpha)+S(\mathbf{0}|\hat{\alpha})$ (5)

where $\alpha$ is the discriminator response to the real pair $(x,y)$, and
$\hat{\alpha}$ to the predicted pair $(x,\hat{y})$, and $S$ is the cross-
entropy loss

$S(\hat{z}|z)=-\frac{1}{N}\sum_{k}z_{k}\,\log\hat{z}_{k}$ (6)

On the other hand, the loss function for the Generator is

$L_{G}(\hat{z})=S(\mathbf{1}|\hat{\alpha})+\Lambda_{1}S(\hat{y}|y)\\\
+\Lambda_{2}MAE(\hat{y},y)+\Lambda_{3}MAE(\hat{y}_{c},y_{c});$ (7)

where the parameters $(\Lambda_{1},\Lambda_{2},\Lambda_{3})=(1000,100,250)$
weight the relative contributions of the loss terms, and

$MAE(\hat{z},z)=\sum_{k}|\hat{z}_{k}-z_{k}|$ (8)

is the mean absolute error.

Note that the loss (7) includes four terms. The first two terms are the
classical one: binary-cross-entropy loss to reward that the generator manages
to deceive the discriminator, and a multi-class cross-entropy loss to
encourage the predicted segmentation probabilistic map $\hat{y}$ to be similar
to the one-hot encoded ground truth $y$. The third additional loss term
reinforces the segmentation by including a $L_{1}$ penalty for the differences
between $\hat{y}$ and $y$. The last loss term promotes enforcing the input to
the Hadamard Layer be similar to the Hadamard code of the corresponding class.
We found that the last two loss terms greatly contribute to the stability and
convergence ratio of the entire cGAN. In this way, we train our network to
compute a realistic mask image and return the correct classes.

##  4 Experiments and Results

We evaluate our proposal in the face segmentation task. For this purpose, we
use the CelebAMask-HQ (annotation maps). We separated the images into 25,000
training images and 5,000 test images. We investigated the performance
improvement when the Hadamard Layers are used in different variants of the
Pix2Pix model. Although we have limited our study to Pix2Pix models, it is
essential to say that one can introduce the Hadamard Layer in other reported
architectures for semantic segmentation. However, such a study is outside the
scope of this work. Herein, we investigate the Pix2Pix variants using the
standard UNet, the ResUNet, the VGG-UNet, and the UNet3+ networks. Each
variant follows a training using one-hot and Hadamard coding for class labels
under the same number of steps.

Figure 2 depicts examples of predicted segmentation for the evaluated models.
We compute the segmentation maps $M$ with

$M(r,c)=\underset{k}{\mathrm{argmax}}\,\hat{y}_{k}(r,c)$ (9)

where $C_{k}\in{0,1,2,...,K}$ are class index in the matrix codes, $(r,c)$ are
the pixel at coordinates. We can observe the differences in segmentation for
each model in Fig. 2. In columns d, f, h, and j, the models that include the
Hadamard Layer, we can notice a more detailed segmentation when compared with
the respective results in columns c, e, g, and i, the models trained only with
one-hot encoding.

Table 1 resumes the metrics (Pixel Accuracy and the mean Class IoU) of the
predicted segmentation for 5,000 test images. In all cases, using a Hadamard
Layer gets better results than one-hot codification in these two metrics.
Indeed, the models UNet and ResUNet show a more noticeable improvement than
VGG-UNet and UNet3+ when using the Hadamard Layer. Layer.

Model | Pixel Acc. | Class IoU
---|---|---
One-Hot | Hadamard | One-Hot | Hadamard
UNet | $0.49$ | 0.69 | $0.12$ | 0.23
ResUNet | $0.53$ | 0.68 | $0.15$ | 0.28
VGG-UNet | $0.61$ | 0.62 | $0.16$ | 0.18
UNet3+ | $0.64$ | 0.67 | $0.22$ | 0.26
Table 1: Accuracy and mean IoU between one-hot and Hadamard codification:
CelebAMask-HQ dataset.

Next, Table 2 presents detailed results for two models, UNet and ResUNet. We
can observe some consistency in the IoU values for the class background and
face. Meanwhile, there is a quantitative difference favoring the models that
use the proposed Hadamard layer for the classes neck, mouth, lips, brow, and
hair.

Class | UNet | ResUNet
---|---|---
One-Hot | Hadamard | One-Hot | Hadamard
Background | $0.76$ | 0.79 | 0.81 | $0.79$
Face | $0.69$ | 0.76 | $0.79$ | 0.81
Neck | $0.29$ | 0.47 | $0.40$ | 0.55
Glasses | $0.03$ | $0.03$ | $0.00$ | 0.01
L-ear | $0.13$ | 0.14 | 0.22 | $0.19$
R-ear | $0.10$ | 0.11 | 0.20 | $0.19$
Mouth | $0.08$ | 0.14 | $0.09$ | 0.18
Nose | $0.10$ | 0.51 | $0.15$ | 0.61
U-Lip | $0.01$ | 0.21 | $0.03$ | 0.34
L-Lip | $0.02$ | 0.30 | $0.04$ | 0.41
R-Brow | $0.01$ | 0.18 | $0.02$ | 0.27
L-Brow | $0.01$ | 0.13 | $0.03$ | 0.23
L-Eye | $0.01$ | 0.11 | $0.01$ | 0.13
R-Eye | $0.00$ | 0.05 | $0.00$ | 0.06
Clothes | $0.00$ | 0.07 | $0.00$ | 0.08
Hair | $0.03$ | 0.47 | $0.02$ | 0.41
Earrings | $0.00$ | $0.00$ | $0.00$ | $0.00$
Necklace | $0.00$ | $0.00$ | $0.00$ | $0.00$
Hat | $0.00$ | $0.00$ | $0.00$ | $0.00$
Table 2: Pixel Accuracy and Class IoU between one-hot and Hadamard class
codification for test Images in CelebAMask-HQ dataset.

The Discriminator of all the evaluated models is of the PatchGAN type. The
Discriminator’s output is a matrix where each element evaluates the realness
of the support region (receptive field). According to Isola et al., for
generating images from semantic segmentations in the CitySky database, the
appropriate receptive field is $70\times 70$ pixels [3]. For our task, we
choose the receptive field size as that one produces the best results in the
model based on the simple UNet network with one-hot encoding. We conducted
this experiment on a subset of just 1000 training images and 100 testing
images; instead of the complete data set of 25,000/5000 for training/testing
images. The receptive field size turned out to be $30\times 30$. In Table 3,
we show the results for different receptive field sizes. Note that models with
the Hadamard layer perform better for larger receptive fields. However, we use
a patch size equal to $30\times 30$ for a fair comparison.

Patch Sz | Pixel Acc. | Class IoU
---|---|---
One-Hot | Hadamard | One-Hot | Hadamard
15x15 | $0.51$ | $0.66$ | $0.12$ | $0.21$
30x30 | $0.54$ | $0.64$ | $0.14$ | $0.20$
60x60 | $0.51$ | $0.66$ | $0.12$ | $0.19$
120x120 | $0.51$ | $0.67$ | $0.13$ | $0.21$
Table 3: Pixel Accuracy and Class IoU between one-hot and Hadamard class
codification for test Images in CelebAMask-HQ Mini dataset, with different
patch sizes in the Discriminator.

##  5 Conclusions and future work

We proposed the Hadamard Layer as a simple and computationally efficient way
to improve results in semantic segmentation tasks. The new layer is constant,
so it does not increase the number of model parameters. As test architecture,
we use different variants of the Pix2Pix model for the face segmentation task
using the CelebAMask-HQ database. The results show that the new Hadamard layer
substantially improves the performance of the investigated models. The metrics
evaluated are the simple accuracy (number of pixels correctly classified) and
the Intersection Over Union (IOU). The best performance of the intervened
models can be explained by the Hadamard layer forcing the network to produce
an encoding of the classes so that all bins are active. Consequently, the
network computation is more distributed. In a sort that the Hadamard layer
requires that to change the predicted class, it is necessary to modify
$2^{k-1}$ bins, assuming $k$ bins in the encoding. On the other hand, changing
the predicted class in one-hot encoding is enough to modify a single bin
(decrease the response of the most significant bin or increase the response of
any other.). Our future work will extend our evaluation to other databases and
different architectures and assess the performance of implicit coding against
adversary attacks.

Acknowledgments. This work was partly supported by the CONACYT (Mexico); Grant
CB 2017-2018-A1-S-43858.

## References

  * [1] Irem Ulku and Erdem Akagündüz,  “A survey on deep learning-based architectures for semantic segmentation on 2d images,”  Applied Artificial Intelligence, pp. 1–45, 2022.
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