# Energetics of twisted elastic filament pairs

Julien Chopin,1 Animesh Biswas2 and Arshad Kudrolli2 1 Instituto de Física,
Universidade Federal da Bahia, Salvador-BA 40170-115, Brazil
2Department of Physics, Clark University, Worcester, Massachusetts 01610, USA

###### Abstract

We investigate the elastic energy stored in a filament pair as a function of
applied twist by measuring torque under prescribed end-to-end separation
conditions. We show that the torque increases rapidly to a peak with applied
twist when the filaments are initially separate, then decreases to a minimum
as the filaments cross and come into contact. The torque then increases again
while the filaments form a double helix with increasing twist. A nonlinear
elasto-geometric model that combines the effect of geometrical nonlinearities
with large stretching and self-twist is shown to capture the evolution of the
helical geometry, the torque profile, and the stored energy with twist. We
find that a large fraction of the total energy is stored in stretching the
filaments, which increases with separation distance and applied tension. We
find that only a small fraction of energy is stored in the form of bending
energy, and that the contribution due to contact energy is negligible. Our
study highlights the consequences of stretchablility on filament twisting
which is a fundamental topological transformation relevant to making ropes,
tying shoelaces, actuating robots, and the physical properties of entangled
polymers.

##  I Introduction

Twisting and braiding filaments are relevant to making ropes, tangling, and
are an important construct in biomaterials [1, 2, 3, 4, 5, 6, 7, 8]. It is an
important step in tying a knot, where the contact structure for filaments
meeting even at $90^{o}$ can be quite complex [9]. A common element in windup
toys (see Fig. 1(a)), elastomeric filaments are being developed towards
applications in novel energy storage and rapid mechanical actuation [10, 11,
12, 13]. The twist of a single filament itself can be quite subtle [14]. While
the filament deforms uniformly when twist is applied at its ends about its
central axis, a helical instability occurs above a critical twist depending on
the applied tension, which causes the central line of the filament to follow a
helical path about the central axis [15, 16]. The pitch cannot be less than
the diameter of the filament limiting the phase space of possible winding
angles that a filament can have [17, 18]. Although polymer filaments are
typically hyperelastic, their analysis has often treated them as being
inextensible [19, 20, 21]. The effective radius and contraction of filament
bundles have been measured as a function of applied twist and shown to deviate
from models based on volume conservation [22]. Braiding and torque of extended
and twisted DNA pairs was investigated by Charvin, et al. [23] with
experiments and models which mainly considered bending energy.

Here, we investigate the energy stored in a pair of elastic filaments by
measuring and analyzing the torque as a function of applied twist with an
elasto-geometric model. This system is not only important in its own right,
but also forms a basis to understand the topology and energetics of multi-
filament bundles where geometric frustration leads to further complexity in
arriving at their configurations with twist [24, 21]. A priori one may expect
energy contributions from bending, stretching, and twisting of each filament,
and the filament-filament contact. We consider a regime where the material
deformations themselves can be considered as linear or neo-Hookean, but where
the geometry of the configurations leads to a complex response. We show that
energy is essentially stored in twisting and stretching the filaments which
vary in relative magnitude depending on the system geometry and applied twist.
The contribution of bending and inter-filament contact are found to be
negligible over the observed twist angles.

Figure 1: (a) A rubber band airplane powered by elastic energy. (b) Images of
the filaments twisted about the central axis while clamped a fixed distance
$L/L_{0}=1.14$ and $W/D_{0}=16$ apart. (c) Schematic of the filament geometry
before the filaments come in contact and after contact when a helical bundle
forms at the center. (d) The measured evolution of the bundle size $L_{B}/L$
for various $W$ and comparison with Eq. (20). (e) The bundle radius as a
function of twist normalized by the radius $R_{360}$ at $\theta=360^{\circ}$
and Eq. (2), where $\lambda$ is obtained using the elasto-geometric model.

##  II Experiments

We consider two elastic filaments with radius $R_{0}$ (diameter
$D_{0}=2R_{0}$) and relaxed length $L_{0}$, that are initially parallel and
separated by a distance $W$ and clamped at their ends as shown in Fig. 1(b).
The filaments can be also pre-stretched by a distance $\Delta L$, resulting in
a length $L=L_{0}+\Delta L$. The filaments are twisted through end-to-end
angle $\theta$ about the central axis which parallels the initial orientation
of the filaments. The filaments are composed of silicone (USA Industrials) and
have a circular cross section of radius $R_{0}=1.27$ mm. We perform
measurements of torque with a Mark-10 MR50-10Z torque sensor fixed to the top
clamp. The bottom clamp is rotated through a prescribed angle $\theta$ with a
computer controlled microstepper motor, and images are taken with a 9.6
megapixel Pixelink PL-D729MU camera while $\theta$ is incremented in
$1^{\circ}$ steps.

Figure 1(b) shows a sequence of images as the filaments are twisted around
each other 5 times when $L=17.2$ cm and $W=4$ cm. As shown schematically in
Fig. 1(c), the filaments can be represented as straight lines before contact
at twist angle $\theta_{c}$ if the curvature due to bending at the clamps is
negligible, and then as a double helix connected to essentially straight
filament segments after contact. We measure and plot the helical bundle length
$L_{B}$ in Fig. 1(d), and the helical bundle radius $R$ normalized by the
radius $R_{360}$ after one full twist corresponding to $\theta=360^{\circ}$ in
Fig. 1(e). (As a consequence of geometry, this radius can be seen to be the
same as the radius of the filament if the contact deformation is negligible.)
We observe that the double helix grows in length while its radius decreases
slowly with increasing $\theta$ depending on $W$.

These changes in the geometry can be seen to have significant effects on the
torque required to twist the filaments. Figure 2(a) shows the torque $M$
measured as a function of $\theta$ corresponding to $L/W=4.2$ and
$W/D_{0}=16$. We observe that the torque increases rapidly before decreasing
to a minimum when the filaments come in contact, and then rises again. The
appearance of a peak may seem surprising, but it has been seen previously in
twisted DNA [23]. A similar feature has been also noted in a twisted sheet,
besides further non-monotonous behavior at higher twists due to shape
transformations [25]. When $W\approx D_{0}$, the peak is absent and a
monotonic increasing torque is observed (see Inset to Fig. 2(a)). As
illustrated in Fig. 2(b), decreasing pre-stretch is observed to lead to an
overall decrease in $M$, but the peak is present even when $L\simeq L_{0}$,
and the overall variation of $M$ with $\theta$ is similar for all the cases.

The elastic energy stored in the filaments can be obtained by integrating the
torque over the twist angle, i.e. $E_{tot}=\sum_{0}^{\theta}M\Delta\theta$,
where $\Delta\theta$ corresponds to $1^{\circ}$ in our experiments. The
corresponding data are plotted in Fig. 2(c) and show that $E_{tot}$ increases
monotonically with $\theta$. Further, it can be observed that the energy
increases rapidly with increasing $W$ before filaments come in contact, and
then increases more slowly after they cross. Thus, the total stored energy can
be increased significantly by simply separating the filaments for the same
amount of elastic material.

Figure 2: (a) The measured (marker) torque versus twist angle. Comparison with
Neo-Hookean model (thin line). Inset: The measured torque when $W=2R_{0}$
compared with model. (b) Measured and calculated torque versus twist angle for
2-filament twist for $W=4$ cm for various pre-stretch conditions $L/L_{0}$.
The peak before filaments come in contact increases with $L$. The calculated
torque using stretching, twist and bending energy assuming Neo-Hookean model
with no fit parameters. Good overall agreement is observed. The measurement
errors are indicated by bars on the last set of markers. (c) The measured
(markers) and calculated (lines) energy as a function of twist for various
$W$. The stored energy increases with separation distance.

##  III Elasto-geometric Model

Figure 3: (a) Nominal stress $\sigma_{N}$ as a function of $\lambda$ is
sublinear for large enough extensions. (b) $\sigma_{N}$ scales linearly as a
function of $\lambda-\frac{1}{\lambda^{2}}$,, where $\lambda=L/L_{0}$ since
$\mathcal{L}=L$ as the filaments are not twisted during these measurements.
The linear fit confirms the neo-Hookean nature of the filament, and yields
shear modulus $\mu=1.43\pm 0.1$ MPa.

We now develop a model which can capture the geometry of the twisted filament,
the torque, and the stored elastic energy as a function of twist angle. An
established method to understand such problems is to consider the elastic
energy stored in the system, and consider its variation with appropriate
stretch parameter to obtain the force or torque required as a function of
applied boundary conditions [26].

Figure 3(a) shows the applied nominal stress $\sigma_{N}=F/S_{0}$, where $F$
is the applied longitudinal force and $S_{0}=(\pi/4)D_{0}^{2}$, versus
measured stretch $\lambda=L/L_{0}$ along the length of the filaments. We
observe that the stress increases linearly initially, but then becomes weakly
nonlinear. This is a hallmark of a neo-Hookean response. Then, the strain-
energy function [27]:

$\displaystyle\mathcal{W}=\frac{1}{2}\mu\left(\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-3\right),\quad\lambda_{1}\lambda_{2}\lambda_{3}=1,$
(1)

where $\lambda_{i}$ are the principal stretches and $\mu$ is the shear
modulus. For uniaxial stretching, $\lambda_{1}=\lambda$ along the filament
axis, $\lambda_{2}=\lambda_{3}=1/\sqrt{\lambda}$. Hence, the radius $R$ of a
stretched filament is reduced due to volume conservation and is given by:

$\displaystyle R=R_{0}/\sqrt{\lambda}.$ (2)

Accordingly, for a uniaxially stretched filament, the nominal stress
$\sigma_{N}$ is given by [27],

$\displaystyle\sigma_{N}=\mu\left(\lambda-\frac{1}{\lambda^{2}}\right).$ (3)

In Fig. 3(b), we find that the data agrees with the neo-Hookean model with
$\mu=1.43\pm 0.1$ MPa over the typical stretches experienced in our system.

We parameterize the kinematics of the filament pairs to obtain the stretch as
a function of twist. Before contact ($\theta<\theta_{c}$), the shapes of the
filaments are modeled by straight lines connecting the top and bottom clamps
(see Fig. 1(c)). One may expect a region of order $R_{0}$ from the clamp to be
curved, but we neglect this as being relatively small, as can be noted from
Fig. 1(b). After contact ($\theta\geq\theta_{c}$), as seen from Fig. 1(b), the
filament geometry can be divided into a central helicoidal section of length
$L_{B}$, and two triangular sections next to the clamps of height $L_{T}$ in
total, with $L=L_{B}+L_{T}$. The central shape where the filaments are in
enduring contact is modeled as a double helix of length $L_{B}$, radius $R$,
and helix angle $\beta$. Therefore, the curvature in the helix is given by
[28]:

$\displaystyle\kappa=\frac{1}{R}\frac{\tan^{2}\beta}{1+\tan^{2}\beta},$ (4)

with $\tan\beta=R(\theta-\theta_{c})/L_{B}=\Delta q/l_{B}$, $\Delta
q=R(\theta-\theta_{c})/L$ and $l_{B}=L_{B}/L$. The torsion $\tau$ is assumed
to be homogeneous along the entire filament and given by
$\tau=\theta/\mathcal{L}$ where $\mathcal{L}$ is the curvilinear length of a
filament.

As can be seen from the top view in Fig. 1(c), the distance between the
filaments in the $yz$-plane is $W\cos\theta/2$. Thus, contact occurs when
$W\cos\theta_{c}/2=2R$, yielding

$\displaystyle\theta_{c}=2\arccos\left(\frac{2R}{W}\right)$ (5)

for $R/W<1/2$. If the filaments are initially close to each other, i.e.
$W\approx D_{0}$, $\theta_{c}$ varies rapidly with pre-stretch and is given by
$\theta_{c}\approx 2\sqrt{\lambda-1}$.

After parametrizing the shape, we can evaluate the filament deformation using
the longitudinal stretch factor

$\displaystyle\lambda=\mathcal{L}/L_{0}.$ (6)

Before contact, as the filaments are modeled by straight lines, $\mathcal{L}$
is given as the distance between the top and bottom clamps, hence:

$\displaystyle\lambda=\lambda_{0}\sqrt{1+\left(\frac{W}{L}\sin\frac{\theta}{2}\right)^{2}}\,,\,\theta<\theta_{c},$
(7)

where $\lambda_{0}=L/L_{0}$ is the stretch factor before twist where
$\theta=0$.

After contact, the inserted twist in the helix is
$\Delta\theta=\theta-\theta_{c}$ and the arc length of the helix is given by

$\displaystyle\mathcal{L}_{B}=L\sqrt{l_{B}^{2}+\Delta q^{2}},$ (8)

with $\Delta q=\Delta\theta R/L$, and $l_{B}=L_{B}/L$. Outside the helical
region, the filaments are not in contact and are assumed to be straight lines.
The filaments form two triangles of height $L_{T}/2=(L-L_{B})/2$ each and base
size $W$ in that region. We can calculate the triangle side length
$\mathcal{L}_{T}$ by evaluating the distance between the clamped end of one of
the filaments, say the right one, and the point at which the filament enters
into the helix at $(L_{B}/2,R\cos(\Delta\theta/2),R\sin(\Delta\theta/2)$.
Thus,
$(\mathcal{L}_{T}/2)^{2}=(L_{T}/2)^{2}+(W/2\cos(\theta/2)-R\cos(\Delta\theta/2))^{2}+(W/2\sin(\theta/2)-R\sin(\Delta\theta/2))^{2}$.
Using Eq. (5) and trigonometric identities, we have

$\displaystyle\mathcal{L}_{T}=L\sqrt{l_{T}^{2}+\Delta w^{2}},$ (9)

where $l_{T}=L_{T}/L$, and $\Delta w=\sqrt{W^{2}-4R^{2}}/L$.

From the total arc length of the filament given by
$\mathcal{L}=\mathcal{L}_{B}+\mathcal{L}_{T}$ and Eq. (6), we obtain

$\displaystyle\lambda=\lambda_{0}\sqrt{l_{T}^{2}+\Delta
w^{2}}+\lambda_{0}\sqrt{l_{B}^{2}+\Delta q^{2}}\,,\,\theta\geq\theta_{c}.$
(10)

Thus, $\lambda$ is completely known once $l_{T}$ and $l_{B}$ are determined.
Since $1=l_{B}+l_{T}$, we only need to determine, say, $l_{B}$ which can be
found by minimizing the total elastic energy $E_{tot}$.

In the two-filament system, the total energy can be written as

$E_{tot}=E_{t}+E_{s}+E_{b}+E_{c},$ (11)

where $E_{t}$ is the energy associated with twisting the filaments about their
axis, $E_{s}$ is the energy associated with stretching the filaments, $E_{b}$
is the energy associated with bending the filaments into the helical shapes
that they adopt with twist, and $E_{c}$ is the energy due to filaments contact
contractions. The twist energy of the two filaments is given by [29],

$\displaystyle E_{t}=\frac{1}{2}\mu V\frac{q_{0}^{2}}{\lambda},$ (12)

where $q_{0}=R_{0}\theta/L_{0}$. $V=\pi R_{0}^{2}L_{0}$ is the volume in the
zero-load configuration. The twist-induced stretching energy is given by [30]

$\displaystyle E_{s}=\mu
V\left(\lambda^{2}+\frac{2}{\lambda}-3\right)-E_{s}^{\circ},$ (13)

where $E_{s}^{\circ}=\mu
V\left(\lambda_{0}^{2}+\frac{2}{\lambda_{0}}-3\right)$ is the stretching
energy after pre-stretching. With this definition, the total energy is zero
after pre-stretching, consistent with our definition
$E_{tot}=\sum_{0}^{\theta}M\Delta\theta$. The bending energy is given by [31],

$\displaystyle E_{b}=\frac{3}{8}\mu V\,\lambda_{0}(\kappa R)^{2}l_{B},$ (14)

where the $\lambda$-dependence of $R$ is given by Eq. (2). The inter-filament
contact energy $E_{c}$ can be estimated by assuming the curvature is small
enough that the contact can considered as a straight line. The Hertz contact
energy of two parallel cylinders pressed against each other, forming a contact
area of length $L_{B}$, is given by:

$\displaystyle E_{c}$ $\displaystyle=$
$\displaystyle\frac{3}{2}S_{0}\mu\left(\frac{\delta
R}{R}\right)^{2}\frac{l_{B}}{\lambda_{0}},$ (15)

where $S_{0}=\pi R_{0}^{2}$. To obtain the indentation $\delta R$ of the two
filaments upon self-contact, we consider a force equilibrium between the
normal inward force $\mu S_{0}(\lambda-1/\lambda^{2})\kappa$ and the Hertz
resisting force $3\pi\mu\delta R\lambda$ (neglecting logarithmic correction),
which gives:

$\displaystyle\frac{\delta
R}{R}=\frac{1}{3}(\lambda-\frac{1}{\lambda^{2}})(\tau R)^{2}.$ (16)

After substituting in Eq. (15), this yields the Hertz contact energy
$E_{c}=\frac{3}{2}S_{0}\mu\left(\frac{\delta R}{R}\right)^{2}L_{B}$, thus

$\displaystyle
E_{c}=\frac{1}{3}V\mu\left(\lambda-\frac{1}{\lambda^{2}}\right)^{2}(\tau
R)^{4}l_{B},$ (17)

where we use $V=S_{0}L_{0}$ and $l_{B}=L_{B}/L$.

Then, we find numerically the bundle length $l_{B}$ which minimizes $E_{tot}$
and compare its evolution as a function of $\theta$ for various $W$ with
experiments in Fig. 1(d). We obtain good agreement without fitting parameters.
Further, $R$ as a function of $\theta$ is calculated using Eq. (2) and found
to also be in overall good agreement with the measured values (see Fig. 1(e)).
Thus, the geometry of the twisted filaments can be captured by the elasto-
geometric model by simply using energy minimization.

Having found $l_{B}$, we evaluate $E_{tot}$, and plot it as a function of
$\theta$ for the three different $W$ in Fig. 2(c). We observe that it follows
the measured values without any fitting parameters within experimental error.
Finally, we obtain the torque profile by numerical differentiation since
$M=\partial E_{tot}/\partial\theta$ to directly compare with the measurements.
Figure 2(a) shows the comparison for $W=4$ cm and for $W=D_{0}$ in the Inset
to Fig. 2(a), where quantitative agreement can be observed in each case
without fitting parameters. Thus, we have a fully tested model of the
geometry, torque and elastic energy stored in twisted two-filament pairs with
neo-Hookean material properties.

##  IV Energy components

We examine the various energy components to understand their relative
contributions due to geometry and pre-stress. Figure 4(a-c) shows $E_{tot}$
for the three different $W$. We observe that the contribution of twisting
energy to total energy is larger than stretching when the filaments are close
together, but stretching makes an increasingly greater contribution with
increasing $W$. Furthermore, it can be noted that bending contributes a little
to the total energy, and that the contribution of contact energy is in fact
smaller than the line width, so we have not plotted it.

To tease out the twist and stretching energy contributions to the total energy
further, we plot the ratio $E_{t}/E_{tot}$ and $E_{s}/E_{tot}$ in Fig. 5 for
the various $W$ and $\Delta L/L_{0}$. We observe that a large fraction of the
total energy is stored in the twist mode when pre-stretch $\Delta L/L_{0}$ is
small, but an increasing fraction is in the stretch mode as the filaments are
stretched about the axis of rotation. In other words, increasing $W$ and
$\Delta L$ for the same length of filament $L_{0}$ leads to increasing
$E_{tot}$, with increasing fraction stored as stretching energy. This suggests
a maximization strategy for total stored energy per unit filament length, by
increasing pre-stretch and separation distance between twisted filaments one
increase store energy for the same amount of material and number of twist
turns.

Figure 4: (a-c) The contribution of twisting, bending, and stretching energies
to total stored energy as a function of twist for various $W$. The contact
energy contribution is negligible and thus not plotted. ($\Delta
L/L_{0}=0.14$, same as in Fig. 2(c)).  Figure 5: Energy fraction for various
$W$ and $\lambda$ stored in twist (a,b), and in stretching (c,d). While most
of the total energy is in the form of twist at lowest $\Delta L/L_{0}$ and
$W$, an increasing fraction is stored in stretching the filaments with twist.
Energy stored in stretching the filaments dominates as $W$ and $\Delta
L/L_{0}$ are increased.

##  V Analytical formulas for energy and torque

We next examine some analytical limits of our elasto-geometric model in order
to provide formulas of energy and torque in the case of twisted elastic
filaments.

###  V.1 Filament pairs in contact in the Hookean limit

When the two filaments are initially in contact, the calculation of $\lambda$
is straightforward as they are modeled by stretched helices, yielding
$\lambda=\lambda_{0}\sqrt{1+(R\theta/L)}$ which corresponds to Eq. (10) with
$L=L_{B}$ and $\Delta w=l_{T}=0$. In the Hookean limit, for $\Delta
L/L_{0}\sim(R_{0}/L_{0})^{2}\ll 1$, we have
$\lambda\approx\lambda_{0}+\frac{1}{2}\left(R_{0}/L_{0}\right)^{2}\theta^{2}$
and $\lambda_{0}\approx 1+\Delta L/L_{0}$. Thus,

$\displaystyle\frac{M}{\mu
V}=\left(\frac{R_{0}}{L_{0}}\right)^{2}\left(1+6\frac{\Delta
L}{L_{0}}\right)\theta.$ (18)

Therefore, the stretching of the filament introduces an additional term in the
torque which scales in the Hookean limit as $(R_{0}/L_{0})^{2}(\Delta
L/L_{0})$.

We compare the torques obtained using these analytical calculations assuming
Hookean-response and those assuming neo-Hookean response in Fig. 6(a). We
observe that these calculations agree for sufficiently small $\Delta L/L_{0}$,
but deviations are observed as $\Delta L/L_{0}$ is increased to 20%.

Figure 6: Torque as a function $\theta$ after contact calculated from the Neo-
Hookean model (solid lines) and the Hookean limit given by Eq. (22) (dashed
lines). (a) The Neo-Hookean and Hookean curves agree for small $\Delta
L/L_{0}$. (b,c) The two estimates agree for sufficiently small $\Delta
L/L_{0}$ for small enough $L/W$ (b), but disagree for large $W/L_{0}=1/5$ (c).

###  V.2 Filament pairs with separation

####  V.2.1 Before Contact

Since the filaments are described by straight lines, $\kappa=0$, and thus
$E_{B}=E_{c}=0$. Thus, a general expression for the torque can be obtained
using the chain rule $M=\frac{\partial E_{tot}}{\partial\theta}=\frac{\partial
E_{tot}}{\partial\lambda}\frac{\partial\lambda}{\partial\theta}$, with
$E_{tot}=E_{t}+E_{s}$. Then,

$\frac{M}{\mu
V}=\left(1-\frac{1}{\lambda^{3}}\left(1+\frac{1}{4}\left(\frac{R_{0}}{L_{0}}\right)^{2}\right)\right)\frac{\partial\lambda^{2}}{\partial\theta}+\frac{1}{\lambda}\left(\frac{R_{0}}{L_{0}}\right)^{2}\theta,$

Then, using Eq. 7, we have, in the Hookean limit

$\displaystyle\frac{M}{\mu
V}=\frac{3}{2}\left(\frac{W}{L_{0}}\right)^{2}\frac{\Delta
L}{L_{0}}\sin\theta+\left(\frac{R_{0}}{L_{0}}\right)^{2}\theta.$ (19)

Thus, the ratio between the stretch and twist contributions is given by

$\approx\frac{1}{2}\frac{\Delta L}{L_{0}}\left(\frac{W}{R_{0}}\right)^{2}$.
Therefore, as may be expected for sufficient large pre-stretch $\Delta
L/L_{0}\gg(R_{0}/W)^{2}$, stretching dominates over twisting contribution for
$\theta<\theta_{c}$.

####  V.2.2 After Contact

We can solve the problem of finding the bundle length $L_{B}$ asymptotically
after contact, and show that $\partial E_{tot}/\partial L_{B}=0$ reduces to
$\partial\lambda/\partial L_{B}=0$. Consistent with numerical analysis which
finds bending and contact contributions to be small, they are not included. We
will check a posteriori that the bending contribution to the total energy is
negligible in the limit $L_{0}\gg\theta R$. Using Eq. (10), the minimization
scheme yields after some algebra:

$\displaystyle l_{B}=1-l_{T}=\frac{\Delta q}{\Delta w+\Delta q}.$ (20)

Substituting Eq. (20) in Eq. (10),

$\displaystyle\lambda=\lambda_{0}\,\sqrt{1+\left(\Delta w+\Delta
q\right)^{2}}.$ (21)

Note that $\theta_{c}$, $\Delta q$, and $dw$ should depend on $\lambda$
through $R$. To obtain a closed form of expression for $\lambda$, we take
$R\approx R_{0}/\sqrt{\lambda_{0}}$. This gives an accurate expression if
$\lambda$ remains close to $\lambda_{0}$, which occurs when $\Delta w$ and
$\Delta q$ are small. Now, we evaluate the bending energy to check that it is
indeed negligible in the limit $R\theta/L\ll 1$ and $W/L\ll 1$. Using Eqs (4),
(14) and (21), we find that $\kappa R\approx(\Delta q+\Delta w)^{2}$ and
$E_{b}\sim(\Delta q+\Delta w)^{3}\Delta q$. Interestingly, we find that the
bending energy increases with $W$ because the curvature increase with $W$
overcompensates for the decrease in $l_{B}$ (see Eq. 20). Then, using Eqs.
(12) and (13), we find that $E_{s}\sim(\Delta q+\Delta w)^{2}$, $E_{t}\sim
q_{0}^{2}$, and $E_{b}\sim(\Delta q+\Delta w)^{3}\Delta q\ll E_{s},E_{t}$ when
$\Delta q,\Delta w\ll 1$. For five turns and $W=40\,$mm, $\Delta q\approx 1/3$
and $\Delta w\approx 1/5$ are not much smaller than unity which explains why
bending energy is predicted to be a fraction of the twisting and stretching
energy. Thus, bending energy can be neglected at a small enough twist angle.

Finally, the torque after self-contact is given by substituting in Eq. (21) in
Eq. (12) and Eq. (13). Using the chain rule as before, we have in the Hookean
limit

$\displaystyle\frac{M}{\mu
V}=\left(\frac{R_{0}}{L_{0}}\right)\left(6\frac{\Delta L}{L_{0}}\left(\Delta
w+\Delta q\right)+q_{0}\right).$ (22)

We compare these asymptotic calculations carried out in the case of Hookean-
response with numerical calculations with the full elasto-geometric model in
cases where the filaments are separate in Fig. 6(b,c). One can note from Fig.
6(b) that the spacing between the filament $W>2R_{0}$ introduces an additional
term in the torque, proportional to $\Delta w$. This term introduces an offset
in the torque independent of $\theta$ and scales as $(RW/L_{0}^{2})\,(\Delta
L/L_{0})$. Further comparing the two calculations for small $\Delta L/L_{0}$,
we observe deviations for sufficiently large $W/L_{0}$ (see Fig. 6(c)). This
can be understood from the fact that the filaments are stretched significantly
as $\theta$ is increased to $180^{\circ}$.

##  VI Conclusions

In summary, we investigated the torque required to twist a pair of extensible
filaments that can be modeled as a neo-Hookean material. Geometry is shown to
play an important role in determining the torque required to twist the
filament pair. By performing torque measurements and calculating them using an
energy based approach, we have developed a deeper understanding of how energy
is distributed in a twisted filament pair. We find that a surprisingly large
amount of energy is stored in stretching. In fact, it is typically greater
than in the twisting component for sufficiently large separation distances or
pre-stretching.

The overall torque profile, namely the rapid initial increase and peak, and
then a slower increase is a feature of the geometry of the system. These
features are true not only for nonlinear neo-Hookean materials, but for
Hookean materials as well. We provide analytical formulas in the Hookean limit
which can prove useful in calculating the energy in twisted filaments which
are increasingly being considered in energy storage applications. Further, our
results further invite critical examination of analysis of twisted polymeric
systems which focus on bending while considering the elastic energy stored in
the system, and neglect stretching [23, 21].

Acknowledgements

This work was supported under U.S. National Science Foundation grant
DMR-2005090.

## References

  * [1] J. Bohr and K. Olsen.  The ancient art of laying rope.  EPL (Europhysics Letters), 93(6):60004, March 2011.
  * [2] Antoine Legrain, Erwin JW Berenschot, Leon Abelmann, José Bico, and Niels R Tas.  Let’s twist again: elasto-capillary assembly of parallel ribbons.  Soft matter, 12(34):7186–7194, 2016.
  * [3] Nicholas Weiner, Yashraj Bhosale, Mattia Gazzola, and Hunter King.  Mechanics of randomly packed filaments—the “bird nest” as meta-material.  Journal of Applied Physics, 127(5), 2020.
  * [4] Valentin M. Slepukhin, Maximilian J. Grill, Qingda Hu, Elliot L. Botvinick, Wolfgang A. Wall, and Alex J. Levine.  Topological defects produce kinks in biopolymer filament bundles.  Proceedings of the National Academy of Sciences, 118(15):e2024362118, 2021.
  * [5] Xiang Gao, Yifeng Hong, Fan Ye, James T. Inman, and Michelle D. Wang.  Torsional stiffness of extended and plectonemic dna.  Phys. Rev. Lett., 127:028101, Jul 2021.
  * [6] Thomas B. Plumb-Reyes, Nicholas Charles, and L. Mahadevan.  Combing a double helix.  Soft Matter, 18:2767–2775, 2022.
  * [7] Antoine Seguin and Jérôme Crassous.  Twist-controlled force amplification and spinning tension transition in yarn.  Physical Review Letters, 128(7):078002, 2022.
  * [8] Vishal P. Patil, Harry Tuazon, Emily Kaufman, Tuhin Chakrabortty, David Qin, Jörn Dunkel, and M. Saad Bhamla.  Ultrafast reversible self-assembly of living tangled matter.  Science, 380(6643):392–398, 2023.
  * [9] Paul Grandgeorge, Changyeob Baek, Harmeet Singh, Paul Johanns, Tomohiko G. Sano, Alastair Flynn, John H. Maddocks, and Pedro M. Reis.  Mechanics of two filaments in tight orthogonal contact.  Proceedings of the National Academy of Sciences, 118(15):e2021684118, 2021.
  * [10] Moshe Shoham.  Twisting Wire Actuator.  Journal of Mechanical Design, 127(3):441–445, 07 2004.
  * [11] James J. Guzek, Conrad Petersen, Stephane Constantin, and Hod Lipson.  Mini Twist: A Study of Long-Range Linear Drive by String Twisting.  Journal of Mechanisms and Robotics, 4(1), 02 2012.  014501\.
  * [12] C. H. Kwon, S. H. Lee, Y.-B. Choi, J. A. Lee, S. H. Kim, H. H. Kim, G. M. Spinks, G. G. Wallace, M. D. Lima, M. E. Kozlov, R. H. Baughman, and S. J. Kim.  High-power biofuel cell textiles from woven biscrolled carbon nanotube yarns.  Nat. Commun., 5(1):1–7, 2014.
  * [13] Carter S. Haines, Na Li, Geoffrey M. Spinks, Ali E. Aliev, Jiangtao Di, and Ray H. Baughman.  New twist on artificial muscles.  Proceedings of the National Academy of Sciences, 113:11709 – 11716, 2016.
  * [14] Nicholas Charles, Mattia Gazzola, and L Mahadevan.  Topology, geometry, and mechanics of strongly stretched and twisted filaments: solenoids, plectonemes, and artificial muscle fibers.  Physical review letters, 123(20):208003, 2019.
  * [15] John Thompson and A. Champneys.  From helix to localized writhing in the torsional post-buckling of elastic rods.  Proceedings: Mathematical, Physical and Engineering Sciences, 452:The Royal Society–, 01 1996.
  * [16] Basile Audoly.  CHAPTER 1: Introduction to the elasticity of rods, pages 1–24.  01 2016.
  * [17] Kasper Olsen and Jakob Bohr.  Geometry of the toroidal n-helix: optimal-packing and zero-twist.  New Journal of Physics, 14(2):023063, feb 2012.
  * [18] Gregory M. Grason.  Colloquium: Geometry and optimal packing of twisted columns and filaments.  Rev. Mod. Phys., 87:401–419, May 2015.
  * [19] S. Neukirch and G. van der Heijden.  Geometry and mechanics of uniform n-plies: From engineering ropes to biological filaments.  Journal of Elasticity, 69:41–72, 11 2002.
  * [20] Gregory M. Grason.  Defects in crystalline packings of twisted filament bundles. i. continuum theory of disclinations.  Phys. Rev. E, 85:031603, Mar 2012.
  * [21] Andreea Panaitescu, Gregory M. Grason, and Arshad Kudrolli.  Persistence of perfect packing in twisted bundles of elastic filaments.  Phys. Rev. Lett., 120:248002, Jun 2018.
  * [22] Jesse M. Hanlan, Gabrielle E. Davis, and Douglas J. Durian.  Twist and measure: characterizing the effective radius of strings and bundles under twisting contraction.  Soft Matter, 19:4315–4322, 2023.
  * [23] Gilles Charvin, Alexander Vologodskii, David Bensimon, and Vincent Croquette.  Braiding dna: Experiments, simulations, and models.  Biophysical Journal, 88:4124–36, 07 2005.
  * [24] Isaac R. Bruss and Gregory M. Grason.  Topological defects, surface geometry and cohesive energy of twisted filament bundles.  Soft Matter, 9:8327–8345, 2013.
  * [25] Julien Chopin and Arshad Kudrolli.  Tensional twist-folding of sheets into multilayered scrolled yarns.  Science Advances, 8(14), apr 2022.
  * [26] Basile Audoly and Yves Pomeau.  Elasticity and Geometry.  Oxford University Press, New York, 2010.
  * [27] Ray W. Ogden.  Non-Linear Elastic Deformations.  Dover, New York, 1984.
  * [28] Barrett O’neill.  Elementary differential geometry.  Elsevier, 2006.
  * [29] Ronald S. Rivlin and D.W. Saunders.  Large elastic deformations of isotropic materials vii. experiments on the deformation of rubber.  Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 243(865):251–288, 1951.
  * [30] R.S. Rivlin.  Large elastic deformations of isotropic materials. iii. some simple problems in cyclindrical polar co-ordinates.  Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 240(823):509–525, 1948.
  * [31] A. Ghatak and L. Mahadevan.  Solenoids and plectonemes in stretched and twisted elastomeric filaments.  Phys. Rev. Lett., 95:057801, Jul 2005.


# ViSeRet: A simple yet effective approach to moment retrieval via fine-
grained video segmentation

Aiden Seungjoon Lee
Twelve Labs
Seattle, U.S.A.
<EMAIL_ADDRESS>Hanseok Oh
KAIST
Seoul, Republic Of Korea
<EMAIL_ADDRESS>Minjoon Seo
KAIST
Seoul, Republic Of Korea
<EMAIL_ADDRESS>
###### Abstract

Video-text retrieval has many real-world applications such as media analytics,
surveillance and robotics. This paper presents the 1st place solution to the
video retrieval track of the ICCV VALUE Challenge 2021. We present a simple
yet effective approach to jointly tackle two video-text retrieval tasks (video
retrieval and video corpus moment retrieval) by leveraging the model trained
only on the video retrieval task. In addition, we create an ensemble model
that achieves the new state-of-the-art performance on all four datasets (TVr,
How2r, YouCook2r, and VATEXr) presented in the VALUE Challenge.

##  1 Introduction

The video retrieval track of the ICCV VALUE Challenge 2021 [6] tackles two
types of text-to-video tasks: video retrieval (VR) and video corpus moment
retrieval (VCMR). The goal of both tasks is to find the best video that
describes the given query(caption) sentence within an extensive video
database. For the VCMR task, the model should additionally output the start
and end time within the video, making the task more challenging.

The competition is comprised of four datasets: TVr, How2r, YouCook2r, and
VATEXr [4, 5, 12, 10]. TVr and How2r datasets are designed as VCMR tasks,
while YouCook2r and VATEXr datasets are designed as VR tasks. The metric for
both tasks is the average value of Recall$@$1, Recall$@$5, and Recall$@$10
(i.e., average recall). For the VCMR task, the retrieved video is considered
correct only if the temporal intersection over union (tIoU) between the ground
truth (GT) timestamp and the predicted timestamp is greater than 0.7 (tIoU $>$
0.7). The overall rank of the model is determined by averaging the rank of the
model’s performance on each dataset (i.e., average rank). Our contributions to
the competition are the following:

  * •

We propose a simple yet effective approach, ViSeRet (Video Segment Retrieval),
to tackle video moment localization retrieval (VCMR) task using the model
trained only for the video retrieval (VR) task. We find that our video
retrieval model alone with a simple heuristic segmenting technique can
outperform the previous state-of-the-art model optimized for the VCMR task.

  * •

An ensemble of ViSeRet and HERO [5] achieves the new state-of-the-art
performance on all four datasets in the VALUE Challenge Retrieval Track.

Figure 1: Our approach for tackling VCMR task using the model trained for VR
task.

##  2 ViSeRet Model

Figure 1 presents our approach. Our approach is to reformat the video corpus
moment retrieval(VCMR) task into the traditional video retrieval (VR) task,
and use the model trained for VR task to jointly solve the two tasks. Our
assumption is that the model maximized for the VR task should ideally perform
competitively in the VCMR task with a simple heuristic video segmentation
technique.

To be specific, we segment each video into several fixed-length (i.e., 5
seconds, 10 seconds) video clips, treat them as separate videos for our video
retrieval model to rank them, and use the pre-segmented timestamp information
to calculate the tIoU with the ground truth(GT) timestamp. For the VR task, we
simply use the entire video segment for the same video retrieval model to rank
them. We find that our video-retrieval-only approach outperforms the previous
state-of-the-art models trained on the VCMR task not only on the VR task
datasets (YouCook2r and VATEXr), but also in the VCMR task datasets (TVr and
How2r).

###  2.1 Model architecture

The model architecture of ViSeRet is illustrated in Figure 2. Our model is
inspired by the recent success of transferring the knowledge from a large-
scale image-text pre-trained model, such as CLIP [9], to video retrieval task
[2, 7].

Visual Encoding For each frame, we obtain two types of CLIP-based features:
precomputed CLIP feature and raw frame CLIP feature that is directly extracted
from the raw video. The difference between the two features is whether the
weight of the CLIP model is updated or not during the training. The
precomputed CLIP feature is obtained from a separate expert-like freezed CLIP
model while the raw frame CLIP feature is obtained through the CLIP model
where its weight is updated during the training process. We elementwise add
these two CLIP features and use it as the final frame-level visual features.
There are two reasons for using two types of CLIP features. First, the
elementwise-sum structure of the precomputed CLIP feature and the raw frame
CLIP feature creates a residual-connection-like structure for the CLIP model.
Second, we can robustly handle missing data where the original raw videos are
not obtainable due to copyright and privacy issues by replacing each raw frame
CLIP feature with a zero vector.

After the visual encodings are obtained for each frame, we create temporal
connections among them using Transformer and obtain a single video-level
visual encoding vector.

Subtitle Encoding We obtain a single BERT-based [1] subtitle encoding
vector([CLS] token) for the entire video.

Visual-Subtitle Fusion Encoding We elementwise add the visual encoding vector
with the subtitle encoding vector and use this vector as the final video
encoding. We find this simple approach enough to see a noticeable performance
difference. We leave it as future work to evaluate other fusion methods [3,
5].

Similarity Estimation We use CLIP’s text encoder to obtain the query (caption)
embedding vector. The similarity is calculated using cosine similarity between
the query (caption) embedding with the final video embedding.

###  2.2 Training

We train our model based on the segmented video-query(caption) pairs for VCMR
task datasets TVr and How2r, and the original video-query(caption) pairs for
YouCook2r and VATEXr(VR task) datasets. The videos are segmented based on
their ground truth (GT) start and end time steps for each query (caption)
annotation.

Figure 2: Our ViSeRet model architecture.

| TVr | How2r
---|---|---
Lengths(s) | 3$/$5$/$10$/$20$/$30$/$60 | 5$/$10$/$20$/$30
Strides(s) | 1$/$2$/$3$/$5$/$8$/$10 | 2$/$5$/$10$/$15
$\\#$ of videos | 2.2k | 1k
$\\#$ of video segments | 330k | 48k
% of annotations covered | 86% | 82%

Table 1: The lengths and strides we used for the TVr and How2r dataset to
segment the videos. Each video is segmented into the lengths and strides
specified in the table. For example, a 60 seconds long video in the How2r
dataset is segmented into [0$\sim$5s, 2$\sim$7s, 4$\sim$9s,…] video clips.

###  2.3 Inference

During the inference time, the model is given a query (caption) and a pool of
videos without any specific timestamp information. Therefore, for VCMR task
datasets, we segment each video into several fixed-length video clips (i.e.,
3s, 5s, 10s, …) and give these segmented videos as input to the model. The
specific video clip lengths are determined based on the timestamp distribution
of the validation data. We also make sure that the new pool of segmented
videos include at least 80% of the annotation’s ground truth timestamp-video
pairs that satisfy tIoU $>$ 0.7. The specific numbers that we used for the TVr
and How2r datasets are presented in Table 1.

This new pool of segmented videos allows the model trained for the VR task to
perform VCMR tasks by using the video segment’s timestamp information to
calculate the tIoU with the ground truth timestamp.

##  3 Experiments

###  3.1 Experiment Setups

The model is trained on each dataset(TVr, How2r, VATEXr, and YouCook2r)
independently using noisy contrastive estimation(NCE) loss function. The batch
size is 64, with the learning rate set to 1e-4. We set the maximum frame
length and subtitle token length to 64 and 512 and truncated the rest. We use
the pre-trained BERT-base model for subtitle encoding, CLIP ViT-B/32 model for
visual encoding, and CLIP text model for query(caption) encoding.

| TVr | How2r | YouCook2r | VATEXr
---|---|---|---|---
HERO (FT only) | 8.81 | 2.13 | 42.37 | 30.22
Ours (FT only) | 9.77 | 7.74 | 50.92 | 55.46
Relative Gain | +10.8% | +263.3% | +20.1% | +83.5%

Table 2: Our ViSeRet finetune only(FT only) model outperforms the baseline
HERO finetune only(FT only) model on all four datasets by a large margin.. The
units are measured in average recall (higher the better).

###  3.2 Single Model Results

Table 2 compares our finetune only(FT only) model with the baseline HERO model
on the VALUE datasets. For a fair comparison, we compare our model with the
finetune only (FT only) model. Our model exceeds the baseline model
performance on all four datasets with up to 263.3% relative performance gain
in the How2r dataset.

Table 3 compares the performance of our submitted single models with the
previous state-of-the-art, HERO (AT$\rightarrow$ST, PT$\rightarrow$FT). Our
single model achieves better performance on three out of the four VALUE
datasets (How2r, YouCook2r, VATEXr). Due to the time constraint, we only had
the chance to pretrain our model on the subset of the HowTo100m dataset [8,
11] and finetune it on the YouCook2r dataset. We leave it as future work to
perform large-scale pretraining on other external datasets and finetune the
model to each dataset.

###  3.3 Ensemble Model Results

Methodology Our ensemble model combines ViSeRet (single) and
HERO(AT$\rightarrow$ST, PT$\rightarrow$FT. For each dataset, we find the best
weight coefficient combination $\alpha$ and $\beta$ and multiply them to each
model’s prediction (logit) scores for each output. Then, we merge the outputs
of the two models and re-rank them based on the weighted scores. The best
weight coefficient $\alpha$ and $\beta$ are found through a grid search on the
validation data.

Results Table 3 presents the result of our submitted ensemble model. Our
ensemble model achieves 22.9% performance increase on average compared to our
single model, achieving the new state-of-the-art across all four datasets. In
addition, our ensemble model ranks the first place on How2r and VATEXr
dataset, the second place on YouCook2r dataset, and the third place on TVr
dataset, winning the video retrieval track for VALUE Challenge 2021 with the
mean rank of 1.75.

| TVr | How2r | YouCook2r | VATEXr
---|---|---|---|---
HERO(AT$\rightarrow$ST, PT$\rightarrow$FT) | 13.56 | 3.95 | 54.28 | 49.09
Ours (single) | 9.77 | 7.74 | 55.73 111The difference between our model results in Table 1 for the YouCook2 dataset is due to additional pretraining on the subset(25k videos) of the HowTo100m dataset. | 55.46
Ours (ensemble) | 14.18 | 7.74777Due to the time constraint, we didn’t have enough time to obtain the ensemble model’s result on How2r dataset. Instead, we submitted the same value from our single model result. | 62.72 | 58.03

Table 3: Our final submission results to the VALUE Challenge 2021 Leaderboard.
The units are measured in average recall (higher the better). Our ViSeRet-
ensemble model outperforms the previous state-of-the-art(SOTA) on all four
VALUE datasets.

##  4 Conclusion and Future Work

In this report, we show that our video retrieval model ViSeRet can perform
effectively on video moment localization retrieval (VCMR) tasks with a simple
heuristic segmenting technique. Our finetune only(FT only) model shows a
significant performance increase on all four datasets of VALUE Challenge
compared to the baseline HERO fine-tune only model with up to 263% relative
increase in How2r dataset. Our ensemble model that is composed of ViSeRet and
HERO achieves the new state-of-the-art on all four datasets, winning the video
retrieval track of ICCV Value Challenge 2021.

Future work includes a large-scale pretraining of ViSeRet model on other
external datasets. It is also worth noting that our model might benefit from
incorporating a separate reader model that extracts moments from the results
of ViSeRet. Due to the time constraint, we did not have enough time to
incorporate the reader model into the final submission, but we briefly
describe what we tried here.

Using the same embedding vectors of query and video as that of the video
retriever model, the reader model can easily compute query-video similarity
scores. On top of the scores we can use classifier layers, which can be
convolution 1D filters following HERO model’s reader setting [5] to modify the
retrieval model’s results. Because we found that using a retriever model with
segmented clips could result in better VCMR performance than baseline, we set
a threshold on the re-ranking score to choose whether to use the output of the
reader model. When re-rank score is above the threshold, which can be
interpreted as the video reader model being reliable, we use the reader-
predicted time span to extract a subset of the retriever’s outputs. Otherwise,
we use time segment information of the segmented video clips as the model
output instead. We expect that the reader model trained end-to-end with the
retriever can improve the end-to-end performance over the current proposed
model.

## References

  * [1] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.  Bert: Pre-training of deep bidirectional transformers for language understanding.  arXiv preprint arXiv:1810.04805, 2018.
  * [2] Maksim Dzabraev, Maksim Kalashnikov, Stepan Komkov, and Aleksandr Petiushko.  Mdmmt: Multidomain multimodal transformer for video retrieval.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, pages 3354–3363, June 2021.
  * [3] Valentin Gabeur, Chen Sun, Karteek Alahari, and Cordelia Schmid.  Multi-modal transformer for video retrieval.  In European Conference on Computer Vision (ECCV), pages 214–229, 2020.
  * [4] Jie Lei, Licheng Yu, Tamara L Berg, and Mohit Bansal.  Tvr: A large-scale dataset for video-subtitle moment retrieval.  In ECCV, 2020.
  * [5] Linjie Li, Yen-Chun Chen, Yu Cheng, Zhe Gan, Licheng Yu, and Jingjing Liu.  Hero: Hierarchical encoder for video+ language omni-representation pre-training.  In Empirical Methods in Natural Language Processing conference (EMNLP), September 2020.
  * [6] Linjie Li, Jie Lei, Zhe Gan, Licheng Yu, Yen-Chun Chen, Rohit Pillai, Yu Cheng, Luowei Zhou, Xin Eric Wang, William Yang Wang, et al.  Value: A multi-task benchmark for video-and-language understanding evaluation.  In 35th Conference on Neural Information Processing Systems (NeurIPS 2021) Track on Datasets and Benchmarks, August 2021.
  * [7] Huaishao Luo, Lei Ji, Ming Zhong, Yang Chen, Wen Lei, Nan Duan, and Tianrui Li.  Clip4clip: An empirical study of clip for end to end video clip retrieval.  arXiv preprint arXiv:2104.08860, 2021.
  * [8] Antoine Miech, Dimitri Zhukov, Jean-Baptiste Alayrac, Makarand Tapaswi, Ivan Laptev, and Josef Sivic.  Howto100m: Learning a text-video embedding by watching hundred million narrated video clips.  In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), October 2019.
  * [9] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al.  Learning transferable visual models from natural language supervision.  arXiv preprint arXiv:2103.00020, 2021.
  * [10] Xin Wang, Jiawei Wu, Junkun Chen, Lei Li, Yuan-Fang Wang, and William Yang Wang.  Vatex: A large-scale, high-quality multilingual dataset for video-and-language research.  In The IEEE International Conference on Computer Vision (ICCV), October 2019.
  * [11] Luowei Zhou, Jingjing Liu, Yu Cheng, Zhe Gan, and Lei Zhang.  Cupid: Adaptive curation of pre-training data for video-and-language representation learning.  arXiv preprint arXiv:2104.00285, 2021.
  * [12] Luowei Zhou, Chenliang Xu, and Jason J Corso.  Towards automatic learning of procedures from web instructional videos.  In AAAI Conference on Artificial Intelligence, pages 7590–7598, 2018.