aainstitutetext: Department of Mathematical Sciences, Durham University,
UKbbinstitutetext: African Institute for Mathematical Sciences, Muizenberg,
South Africaccinstitutetext: Osmanağa Mahallesi Vahapbey Sokak No:8 Kadiköy,
Istanbul 34714, Turkeyddinstitutetext: Faculty of Physics, Astronomy and
Applied Computer Science, Jagiellonian University, Kraków,
Polandeeinstitutetext: Institute of Physics, University of Oldenburg,
Oldenburg D-26111, Germany

# Collisions of weakly-bound kinks in the Christ-Lee model

Patrick Dorey c Anastasia Gorina d Tomasz Romańczukiewicz e and Yakov Shnir
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(April 2022)

###### Abstract

We investigate soliton collisions a one-parameter family of scalar field
theories in 1+1 dimensions which was first discussed by Christ and Lee Christ-
Lee . The models have a sextic potential with three local minima, and for
suitably small values of the parameter its kinks have an internal structure in
the form of two weakly-bound subkinks. We show that for these values of the
parameter kink collisions are best understood as an independent sequence of
collisions of these subkinks, and that a static mode analysis is not enough to
explain resonant structures emerging in this model. We also emphasise the role
of radiation and oscillon formation in the collision process.

##  1 Introduction

One of the most fascinating features of soliton solutions in non-integrable
classical field theories Manton:2004tk ; Vachaspati ; Shnir2018  is the
remarkable interplay between perturbative and non-perturbative dynamics, see
for example kevrekidis2019dynamical . It gives rise to an interesting give-
and-take mechanism of reversible energy exchange, which in particular results
in the emergence of intricate resonance structures in collisions of solitons,
such as kinks and antikinks in the 1+1 dimensional $\phi^{4}$ model
Campbell:1983xu ; Goodman:2005 ; Makhankov:1978rg ; Moshir:1981ja ;
Anninos:1991un , and in the $\phi^{6}$ and other polynomial models
Dorey:2011yw ; Weigel:2013kwa ; Gani:2014gxa ; Demirkaya:2017euk ;
Khare:2014kva ; Romanczukiewicz:2018gxb ; Bazeia:2023qpf .

Numerical modeling of inelastic kink-antikink collisions reveal various
mechanisms of the reversible energy transfer between the translational mode of
the kink and its internal mode Campbell:1983xu ; Anninos:1991un ; Goodman:2005
; Moshir:1981ja ; Sugiyama:1979mi ; the energy can be also stored in
excitations of scalar modes trapped by the kink-antikink pair Dorey:2011yw ;
Weigel:2013kwa ; Gani:2014gxa , or in narrow quasi-normal modes Dorey:2017dsn
; Campos:2019vzf . Similar patterns have also been observed in resonant kink-
impurity interactions Kivshar:1991zz ; malomed2  and in boundary scattering of
kinks Arthur:2015mva ; Dorey:2015sha ; Lima:2018mzy .

The collisional dynamics of the kinks becomes more involved as the number of
modes localized on the kink increases Demirkaya:2017euk ; Dorey:2021mdh , or
in the scattering of wobbling kinks AlonsoIzquierdo:2020hyp . The situation
can be even more complicated in deformed models which support false vacua
Dorey:2019uap ; Simas:2016hoo ; Ashcroft:2016tgj ; Gomes:2018heu ;
Adam:2019uat ; Adam:2019prh . Deformations of the potential may result in
appearance of subkinks, which can be considered as weakly bound kinks
interpolating between the true and false vacua sanati1999half ;
Mendonca:2015nka ; Demirkaya:2017euk ; Zhong:2019fub ; Dorey:2021mdh . In such
cases, as we shall explore in the following, the collision of the solitons can
be considered as a multi-step process, further complicated by the fact that
the initial impact of the forward pair of subkinks produces a burst of scalar
radiation which deflects the trajectories of the second pair of subkinks and
affects the outcome of the subsequent collisions111Discussions of how
radiation pressure (positive and negative) influences the motion of kinks can
be found in Forgacs:2013oda ; Forgacs:2008az ; Romanczukiewicz:2003tn . .
Among other things this results in curious spine-like structures in plots of
the final state of collisions as a function of the deformation parameter and
initial velocity Dorey:2021mdh ; similar patterns can be seen in figure 6
below.

In our previous work Dorey:2021mdh  we briefly discussed this effect,
alongside a plethora of other interesting phenomena, in the deformed sine-
Gordon model of Dorey:2019uap  near the strong-deformation limit where it
becomes a $\phi^{6}$-like model with three degenerate vacua. In this paper we
revisit this pattern in another model, first studied in a different context by
Christ and Lee in 1975 Christ-Lee .

Our paper is structured as follows. In the next section we introduce the model
and describe its static solutions and the fluctuations about these
configurations. In section 3 we discuss various regimes of kink - antikink
collisions. There special attention is paid to collisions of weakly bound
pairs of subkinks and related radiation effects. Collisions between these
subkinks and bions also play a role and these are given separate study in
section 4. We give our conclusions and further remarks in the final section.

##  2 The model

We consider a model with the Lagrangian density that was discussed in Christ-
Lee

$\mathcal{L}=\frac{1}{2}\phi_{t}^{2}-\frac{1}{2}\phi_{x}^{2}-U(\phi,\epsilon)\,,$
(1)

where, setting the parameters $g$ and $\mu$ in Christ-Lee  to $1$ and $2$
respectively, the scalar potential is

$U(\phi,\epsilon)=\frac{1}{2(1+\epsilon^{2})}(\epsilon^{2}+\phi^{2})(1-\phi^{2})^{2}$
(2)

Figure 1 shows this potential for various values of $\epsilon$.

Figure 1: The potential $U(\phi)$ of the model for different values of
parameter $\epsilon$.

For $\epsilon=0$ the potential has three global minima at
$\phi\in\\{-1,0,1\\}$ and the model is the triply-degenerate $\phi^{6}$ model.
Increasing $\epsilon$ breaks the degeneracy and the vacuum at $\phi=0$ becomes
false. For $\epsilon>1/\sqrt{2}$ this false vacuum is lost, and the model has
only two local minima; in the limit $\epsilon\to\infty$ it becomes the well-
known $\phi^{4}$ model. The masses of small perturbations around the vacua are

$m_{1}=m_{-1}=2,\qquad m_{0}=\sqrt{\frac{1-2\epsilon^{2}}{1+\epsilon^{2}}}$
(3)

To discuss the limit $\epsilon\to 0$ it will sometimes be convenient to swap
$\epsilon$ for an alternative parameter $\beta$, with

$\beta(\epsilon)=\log\left(1+\frac{1}{\epsilon}\right),\qquad\epsilon(\beta)=\frac{e^{-\beta}}{1-e^{-\beta}}\,.$
(4)

This maps $(0,\infty)\ni\epsilon\to\beta\in(\infty,0)$.

###  2.1 Static solutions

For all $\epsilon>0$ the kink profile, interpolating between the vacua at $-1$
and $+1$, is known analytically:

$\Phi_{K}(x)=\frac{\epsilon\sinh(x)}{\sqrt{1+\epsilon^{2}\cosh^{2}(x)}}\,.$
(5)

Figure 2 plots some representative examples. At $\epsilon=0$ the middle vacuum
at $0$ becomes degenerate with the outer two and this solution is trivial.
Smaller $\phi^{6}$ kinks $\phi_{(-1,0)}$ and $\phi_{(0,1)}$ exist instead,
interpolating between $-1$ and $0$, and $0$ and $1$ respectively, where

$\phi_{(0,1)}(x)=\sqrt{\frac{1+\tanh x}{2}}=\frac{1}{\sqrt{1+\exp(-2x)}}$ (6)

and $\phi_{(-1,0)}(x)=-\phi_{(0,1)}(-x)$. These smaller kinks emerge as
$\epsilon\to 0$ through the splitting of the $\Phi_{K}$ kink into two
increasingly-separated $\phi^{6}$-like subkinks, as can be seen by rewriting
$\Phi_{K}$ as

$\begin{split}\Phi_{K}(x)&=\frac{\epsilon\exp(x)}{2\sqrt{1+\epsilon^{2}\cosh^{2}(x)}}-\frac{\epsilon\exp(-x)}{2\sqrt{1+\epsilon^{2}\cosh^{2}(x)}}\\\\[3.0pt]
&=\phi_{\epsilon}(x-\overline{x}_{0}(\epsilon))-\phi_{\epsilon}(-(x-\overline{x}_{0}(\epsilon)))\end{split}$
(7)

where

$\phi_{\epsilon}(x)=\frac{1}{\sqrt{(1+{\textstyle\frac{1}{4}}\epsilon^{2}\exp(-2x))^{2}+\exp(-2x)}}$
(8)

and

$\overline{x}_{0}(\epsilon)=\log\left(\frac{2}{\epsilon}\right)\approx\beta+\log(2).$
(9)

As $\epsilon\to 0$, $x_{0}$ grows, $\phi_{\epsilon}(x)\to\phi_{(0,1)}(x)$ and
the two subkinks emerge, positioned at $x=\pm\overline{x}_{0}$:

$\Phi_{K}(x)\sim\phi_{(0,1)}(x-\overline{x}_{0}(\epsilon))-\phi_{(0,1)}(-(x-\overline{x}_{0}(\epsilon))).$
(10)

Alternatively, one can find the same expression by balancing the forces
between the subkinks and the false vacuum as discussed, for example, in
Dorey:2021mdh .

Figure 2: Profiles of the stable static solutions. Dashed lines show the
approximation (10).

The mass of the full ($\epsilon>0$) kink is equal to

$M(\epsilon)=\frac{1}{2}-\frac{\epsilon^{2}}{4}+\frac{\epsilon^{2}(4+\epsilon^{2})}{\sqrt{1+\epsilon^{2}}}{\mathop{\mathrm{asinh}}}(1/\epsilon)=\frac{1}{2}+\epsilon^{2}\log(\epsilon)+\mathcal{O}(\epsilon^{2})$
(11)

Given that the mass of a single $\phi^{6}$ kink is $1/4$, this shows that for
small values of $\epsilon$ the mass of the full kink exceeds the mass of two
$\phi^{6}$ kinks by an amount $\epsilon^{2}\log(\epsilon)$, which can be
attributed to the energy of the stretch of false vacuum lying between its two
subkinks.

The presence of the false vacuum also allows for unstable infinite-energy
static lump-like solutions, or sphalerons222The name is borrowed from
Manton:1983nd ; Klinkhamer:1984di , where saddle-point solutions of this type
were discussed in the context of the Weinberg-Salam Theory.:

$\Phi_{S}(x)=\pm ab\sqrt{\frac{1-t^{2}}{b^{2}-a^{2}t^{2}}}\qquad\text{ where
}\qquad t=\tanh\frac{abx}{\sqrt{1+\epsilon^{2}}}$ (12)

and

$a=\sqrt{\frac{1}{2}\left(2-\epsilon^{2}-\epsilon\sqrt{4+\epsilon^{2}}\right)},\qquad
b=\sqrt{\frac{1}{2}\left(2-\epsilon^{2}+\epsilon\sqrt{4+\epsilon^{2}}\right)}\,.$
(13)

Requiring $a$ to be real restricts $0<\epsilon<1/\sqrt{2}$ or
$\beta>\log(1+\sqrt{2})\approx 0.881374$. At the critical value
$\epsilon=1/\sqrt{2}$ the false vacuum ceases to exist and the unstable lumps
disappear.

For small values of $\epsilon$ the unstable lump resembles a bound subkink –
antisubkink pair, submerged in the false vacuum which balances their mutual
attraction:

$\Phi_{S}(x)\approx\sqrt{\frac{1+\tanh(x+\overline{x}_{1})}{2}}+\sqrt{\frac{1-\tanh(x-\overline{x}_{1})}{2}}-1\,.$
(14)

The value of $\overline{x}_{1}$ can be found in a similar way as for the kink.
The full formula is rather complicated, but the leading behaviour for small
$\epsilon$ is

$\overline{x}_{1}(\epsilon)=\frac{1}{2}\log\left(\frac{2}{\epsilon}\right)\approx\frac{1}{2}\beta+\frac{1}{2}\log(2).$
(15)

As illustrated in figure 3, this approximates the position very well for
$\epsilon\lesssim 0.1$ .

Figure 3: Profiles of the unstable static solutions. Dashed lines correspond
to the approximation (14) and (15).

###  2.2 Linearization

The linearized potential for fluctuations about a static solution $\phi$ is

$V(x)=\frac{1}{1+\epsilon^{2}}\left(15\phi^{4}+(6\epsilon^{2}-12)\phi^{2}-2\epsilon^{2}+1\right).$
(16)

Figure 4 shows these potentials for a single kink $\Phi_{K}(x)$ at various
values of $\epsilon$, while figure 5 shows the frequencies of the
corresponding bound modes as a function of $\beta$. Due to the translational
symmetry there is always a zero mode. For $\beta=0$ there is a single bound
mode with frequency $\omega_{1}=\sqrt{3}$, which reflects the fact that for
$\beta=0$ the model is the $\phi^{4}$ model. For $\beta=0$ there is one more
(nonintegrable) mode exactly at the threshold $\omega=2$ which becomes a
proper bound mode for $\beta>0$. At values of $\beta$ equal to $0.73289$,
$0.98518$, $1.27641$, $1.61620$, $\ldots$ new modes appear below the
threshold.

As $\beta$ increases the frequencies of all modes decrease. The frequency of
the first oscillating bound mode tends to $0$, while the frequencies of all
other modes stay above the mass of the false vacuum $m_{0}(\epsilon)$. This is
consistent with the association of the lowest mode for large values of $\beta$
with translational oscillations of the weakly bounded subkinks. The higher
modes correspond to excitations trapped between the subkinks, with frequencies
high enough to propagate in the false vacuum but still too low to propagate in
the true vacuum.

Figure 4: Linearized fluctuation potentials for the stable static solutions at
different values of $\epsilon$. Figure 5: Frequencies of the internal modes.

##  3 The small $\epsilon$ regime

For small values of $\epsilon$, or equivalently large values of $\beta$, it is
useful to rewrite the potential as

$U(\phi,\epsilon)=\frac{1}{2}\phi^{2}(\phi^{2}-1)^{2}-\frac{\epsilon^{2}}{2(1+\epsilon^{2})}(\phi^{2}-1)^{3}.$
(17)

In this regime the field’s evolution is almost entirely governed by the first
term, which corresponds to the ordinary $\phi^{6}$ equation of motion.
However, the $\phi^{6}$ model has very different static solutions than the
$\epsilon>0$ Christ-Lee model. Thus we should expect that a collision between
a kink and an antikink for $\epsilon>0$ would be very similar, not to a kink-
antikink collision in the $\phi^{6}$ model, but rather to the collision of two
$\phi^{6}$ kinks and two $\phi^{6}$ antikinks with suitably-tuned initial
separations to match the particular value of $\epsilon$ under consideration.
Figures 6 and 7 test this idea. Figure 6 shows the value of the field
$\phi(0,T)$ after time $T=700$ for the Christ-Lee model with initial
separation between the centres of the kink and antikink $2x_{0}=2\cdot 40$,
and initial velocities $\pm v_{i}$. Figure 7 shows instead the value of
$\phi(0,T)$ obtained using $\epsilon=0$ equation of motion, for the same
initial conditions as figure 6. Note that for $\epsilon=0$ the subkinks become
independent and the static kink solutions for any particular value of
$\epsilon>0$ (comprising two weakly bound subkinks) are not static solutions
for the $\epsilon=0$ equation of motion. But for sufficiently small values of
$\epsilon$ the repulsion between subkinks can be neglected for the relatively
short time scales over which the collisions take place.

The match between the two scans for $\beta\gtrsim 5$ justifies the
approximation, and indeed beyond $\beta\approx 10$ they become
indistinguishable. Most notably the spine structure is the same and the blue
“stream” around $v_{i}\approx 0.289$ is clearly visible on both plots. We will
argue that these features can be understood using the nontrivial internal
structure of the kinks.

Figure 6: Late-time field values $\phi(0,700)$ for $K\bar{K}$ collisions with
initial velocities $\pm v_{i}$ in the Christ-Lee model. The pattern of spines
for $\beta\gtrsim 4$ is similar to that found in a deformed sine-Gordon model
in Dorey:2021mdh . Figure 7: Late-time field values $\phi(0,700)$ for the same
Christ-Lee initial conditions as in figure 6, but evolving the field using the
$\phi^{6}$ ($\epsilon=0$) equation of motion.

To this end we first recall how the kinks and antikinks of the $\phi^{6}$
model scatter Dorey:2011yw , bearing in mind that these correspond to subkinks
of the Christ-Lee model. There are two distinct processes, depending on
whether the vacuum at infinity is $0$ or $\pm 1$. The first, which we will
call type I, corresponds to a $(0,1)+(1,0)$ collision and results either in a
single bounce for $v_{i}>v_{cr}^{I}\approx 0.289$, or annihilation (with the
formation of a centrally-located bion, seen also in figure 8b) for
$v_{i}<v_{cr}^{I}$. By contrast, a $(1,0)+(0,1)$ collision, which we will call
type II, can involve not only the direct formation of a bion but also a
variety of resonant multi-bounce scenarios, leading to a fractal-like
structure for velocities between $0.0228<v_{i}<0.0457$. (For further work on
the $\phi^{6}$ model see Weigel:2013kwa ; Takyi:2016tnc ; Gani:2014gxa  and
also Adam:2022mmm , where a detailed study of the collective coordinate
description of the model was undertaken, confirming the role of the collective
bound mode trapped between the $K\bar{K}$ pair in resonant type II
scattering.)

Returning to the Christ-Lee model and looking at a $K\bar{K}$ collision for
$\beta=8$ (figure 8a) we can indeed discern four subcollisions. The first,
around $t=88$, is of type I. The critical value $v_{i}=v_{cr}^{I}=0.289$ for
type I collisions is visible on the scans shown in figures 6 and 7 as the
narrow (in $v_{i}$) blue “streams’ stretching over a wide range of $\beta$. In
cases such as that shown in figure 8a where the initial type I collision
results in a bounce, two outgoing subkinks are produced which then collide
with the two incoming outer subkinks (at $t\approx 111$ in figure 8a). These
collisions are topologically of type II, but since they take place in the
presence of radiation escaping from the first collision we will denote them by
II∗ with ∗ indicating the additional radiation. Note that while, as mentioned
above, pure type II collisions in the $\phi^{6}$ model have a fractal-like
structure for velocities between $0.0228<v_{i}<0.0457$, in the current model
the first collision results in escaping subkinks only for velocities larger
than $0.289$, and the second collisions therefore take place with velocities
which are higher the the upper critical velocity for type II collisions. In
the absence of radiation, we would expect that the second collisions would
always result in bounces. The fourth collision ($t=127$) is of type I∗ because
the subkinks after a previous bounce collide again. But this time the
collision takes place in the presence of radiation from the earlier
collisions, which can affect the outcome. As we will see shortly, the
radiation can also affect the trajectories of the subkinks in between these
collisions.

For velocities smaller than $v_{cr}^{I}$, a bion is formed after the first
subcollision (figure 8b). Therefore instead of three subsequent subkink-
subantikink subcollisions only a subcollision between a bion and two incoming
subkinks takes place during the initial set of interactions.

In the following sections we will take a closer look at the new types of
collisions and the role of background radiation.

a) b)

Figure 8: Maps of two basic types of collision for large $\beta$. Examples for
$\beta=8$ and $v_{i}=0.36$ (a) and $v_{i}=0.26$ (b).

###  3.1 Spines as a result of radiation pressure

In situations such as that shown in figure 8a, the first three collisions
produce a large amount of radiation which affect the subsequent behaviour.
This can be seen on all collision maps. In the $\phi^{6}$ model radiation
exerts a pressure on solitons, pushing them in one direction only, no matter
from which side of the soliton the radiation came Romanczukiewicz:2017hdu . In
fact, small fluctuations always act in such a way that the vacuum with smaller
mass grows. In the current model this means that for large values of $\beta$
the radiation pressure acts in the opposite direction to the vacuum pressure.
Furthermore, at least for short time dynamics, the vacuum pressure can be
neglected for large values of $\beta$. In contrast, we would expect that the
last collision can be highly influenced by radiation pressure.

a) b)

Figure 9: A long false window (or false island) close to the base of a spine,
for $\beta=9.7600017$ and $v_{i}=0.31808$. In this and subsequent figures the
left-hand plot shows the field values, while the right-hand plot shows the
logarithm of the energy density.

Figure 9 shows a collision for large $\beta=9.7600017$, very close to the edge
of a spine. After the second subkink collisions at $t\approx 130$ the outer
subkinks bounce back, and then accelerate further after the collision. This is
due to the background radiation which acts to expand $\phi=0$ vacuum. (Another
possible cause of acceleration could be the presence of the inner kinks after
the last collision, but they do not approach the outer kinks closely enough to
interact efficiently.) After the fourth subkink collision at $t\approx 150$,
the initially outgoing inner subkinks are deflected back towards the centre of
collision, mostly by radiation. This acceleration weakens with time and only
when the subkinks approach each other to a separation of around $3$ or $4$ do
they attract each other strongly and form a bion. This weakening acceleration
can be explained by radiation escaping from the region so that only very weak
fluctuations remain.

False vacuum pressure will ultimately cause the outer kinks to return to the
central region, but over a very long time scale $T\sim e^{2\beta}$, well
beyond the end of our simulation. This can be estimated as follows. The false
vacuum force is equal to the potential difference between the true and the
false vacua, which is

$F=\Delta
U=\frac{\epsilon^{2}}{2(1+\epsilon^{2})}\leadsto\frac{1}{2}\epsilon^{2}\,.$
(18)

This approximately constant force turns back the subkink with initial velocity
$v_{0}$ and forces it to recollide after time

$T=\frac{2v_{0}M}{F}\approx\frac{4v_{0}M}{\epsilon^{2}}=\frac{v_{0}}{\epsilon^{2}}\approx
v_{0}e^{2\beta}.$ (19)

This time can be even longer when radiation effects are taken into account.

Given the immense time scale needed for recollisions at these large values of
$\beta$ and the even longer time bions need to decay, we stress that figures 6
and 7 are only late-time approximations to the final state of the system. The
soliton resolution conjecture tao2009solitons ; Bizon:2021ose  implies that
the final state consists only of solitons with a variety of velocities and
dispersing radiation, and the non-zero force between solitons in the model
means that this state can contain at most one of these solitons can be static.
A single static soliton would break the $x\to-x$ symmetry of our initial
conditions, so all solitons in the final state must have non-zero velocity and
ultimately leave the neighbourhood of the origin. Hence the only possible
limiting values of $\phi(0,t)$ as $t\to\infty$ are those of the true vacua,
that is $\pm 1$, meaning that all points on our scans should be either white
or blue if they were to show the limiting field values as $t\to\infty$. As
will be discussed in section 4 below, the bion-subkink re-collisions are
chaotic and it can be hard to predict at which of these true vacua the field
$\phi(0,t\to\infty)$ will end. However, from the energetic point of view full
annihilation is more likely and therefore blue should dominate the lower parts
of figures 6 and 7. Note that there is a clear distinction at the intermediate
time scales shown in the figures between the yellow-green regions where a
central and more slowly-decaying bion has been formed which is still in the
process of decaying, and the regions which are already blue, where energy has
been exported from the central region by the emission of a pair of bions.
These blue regions were called pseudowindows in our earlier paper
Dorey:2021mdh , and it would be interesting to give them a more mathematically
precise definition.

a) b)

Figure 10: A type I kink collision in the $\phi^{6}$ model with above-critical
initial velocity $v_{i}=0.3$.

a) b)

Figure 11: The same collision as shown in figure 10, with additional radiation
in the form of two wavepackets with $\omega=2.2$, $s=50$, $\alpha=0$,
$a=0.00125$, $A=0.028$ in the initial data.

To assess the effect of perturbations such as radiation on subkink collisions,
we returned to the $\phi^{6}$ model and took initial data corresponding to a
type I collision with and without an additional perturbing term, setting

$\phi(x,0)=\left[\phi_{I}(x,t)+\delta\phi(x,t)\right]_{t=0}\,,~{}~{}\phi_{t}(x,0)=\left[{\textstyle\frac{\partial}{\partial
t}}(\phi_{I}(x,t)+\delta\phi(x,t))\right]_{t=0}$ (20)

with

$\phi_{I}(x,t)=-\phi_{(0,1)}(\gamma(x-x_{0}-v_{i}t))-\phi_{(0,1)}(-\gamma(x+x_{0}+v_{i}t))+1$
(21)

being the initial data for a type I collision in the $\phi^{6}$ model, and
$\delta\phi(x,t)$ being the possible perturbation.

Figure 10 shows the unperturbed ($\delta\phi=0$) collision for $v_{i}=0.3$,
$x_{0}=10$, and figure 11 shows the same collision with the addition of two
travelling wave packets of amplitudes $A$ and widths $1/\sqrt{a}$ into the
initial data:

$\delta\phi(x,t)=Ae^{-a(x+vt-s)^{2}}\sin(\omega
t+k(x{-}s)+\alpha)+Ae^{-a(x-vt+s)^{2}}\sin(\omega t-k(x{+}s)+\alpha)$ (22)

where $\omega=\sqrt{k^{2}+1}$, $\pm s$ are the locations of the wavepackets at
$t=0$, $\mp v=\mp d\omega/dk$ are their group velocities, and $\alpha$ is a
phase. The wave packets reach the central region just after the initial
collision and push the outgoing subkinks back towards the centre, but after
passing by the acceleration vanishes. This models quite well the trajectories
of the inner subkinks after the last collision shown in figure 9.

Similar results can be found when instead of moving wave packets the initial
simulation starts with a gaussian perturbation at the centre (figure 12)

$\delta\phi=-Ae^{-bx^{2}}.$ (23)

For the example shown we again set $x_{0}=10$, but took $v_{i}=0.28$, which is
below the critical velocity, and chose $A=0.028$ and $b=0.1$.

In the simulations shown in figures 8 and 9 a we see that the first collision
generates outgoing waves which then reflect from the outer subkinks before the
second and the last collisions and turn back as wave packets. Some residual
radiation also stays near the centre as low frequency slowly moving waves.
Therefore the overall effect is rather a superposition of an almost stationary
wave and wave packets coming from a distance.

Moreover, the force exerted by the waves is not constant on short time scales,
and the phase of the wave packet can influence the outcome of the evolution
(c.f. figure 13). Small differences in the phase can result in ultimate
annihilation of the inner subkinks or their escape. We think that exactly this
mechanism is responsible for the creation of spines. Velocities of individual
subkinks measured after the collision plotted in figure 14 show great
similarity to figure 13.

a) b)

Figure 12: (a) A $\phi^{6}$ type I collision just below the critical velocity
for $v_{i}=0.28$ and $x_{0}=10$, (b) the same collision but with a gaussian
perturbation at centre $\delta\phi=-0.028e^{-0.1x^{2}}$ for $v_{i}=0.28$.

At this point it is worth mentioning that double kink collisions were also
extensively studied in the double sine-Gordon model CAMPBELL1986165 ;
Campos:2021mkn ; Gani:2017yla . However, the double sine-Gordon model
interpolates between two versions of the integrable sine-Gordon model, as a
certain parameter $R$, corresponding to half the separation between the
subkinks, varies from $0$ to $\infty$ Campos:2021mkn . One of the consequences
of integrability is the fact that solitons do not lose energy during
collisions, which translates into the vanishing of the critical velocity both
at $R=0$ and also as $R\to\infty$. This is one reason why spines are not
observed in the double sine-Gordon model for collisions of unexcited kinks,
even in the large $R$ limit.

a) b)

Figure 13: Final velocities for type I collisions of two $\phi^{6}$ kinks with
perturbed initial data. Plot (a) shows the effect of incoming wave packets
with $\omega=2.2$, $A=0.028$, $s=50$ for different phases $\alpha$, while plot
(b) shows the effect of a gaussian perturbation at the centre with $A=0.028$,
$b=0.1$ for different initial locations $\pm x_{0}$ of the $\phi^{6}$ kinks.
In both cases the $A=0$ curve shows the unperturbed situation. Figure 14:
Final velocity of subkinks as a result of the collision for Christ-Lee model
for $\beta=10$. For $v_{i}<0.28$ the gain in velocity is due to the radiation.

###  3.2 Further radiation effects

At large values of $\beta$ the two subkinks making up a full kink are weakly
bound, separated by a large region of false vacuum. The vacuum pressure trying
to collapse this region is very small in comparison with other effects such as
radiation pressure, and can be ignored over typical collision timescales. As
evidenced by the match between figures 6 and 7 for $\beta\gtrsim 5$, the
motion of subkinks in this regime is very close to that of full kinks in the
$\phi^{6}$ model. Radiation pressure in the $\phi^{6}$ model was studied in
Romanczukiewicz:2017hdu , where it was shown that both negative and positive
radiation pressure can be exerted on a single $\phi^{6}$ kink. Since the
vacuum at $\phi=0$ has the lower mass, radiation coming from the vacua at
$\phi=\pm 1$ to that at $\phi=0$ causes negative radiation pressure, while
radiation moving from the $\phi=0$ vacuum to those at $\phi=\pm 1$ causes
positive radiation pressure. In other words radiation in the $\phi^{6}$ model
always pushes subkinks so as to expand the region of $\phi=0$ vacuum at the
expense of $\phi=\pm 1$ vacua.

These radiation effects persist away from the $\beta=\infty$ ($\phi^{6}$)
limit and provided $\beta$ is large enough they can have a decisive role in
determining the outcome of collision processes. This is illustrated in figures
15 and 16. The initial velocity is above the critical velocity of a type I
collision, so the first subkink collision produces a pair of subkinks moving
away from each other, together with some radiation. This radiation reaches the
outer subkinks (which act like tampers) at $t\approx 100$ and reflects back,
reversing the direction of motion of the inner kinks and causing them to
recollide before they get the chance to interact with the outer subkinks. This
time their velocity is less than the type I critical velocity, and therefore
they form a bion. Soon after that the outer kinks, slightly slowed down by
radiation, collide with this bion and two bions are ejected from the centre,
transporting most of the energy away from the central region.

Figure 15: Field values for a kink collision at large $\beta$:
$\beta=28.199532$, $v_{i}=0.291247$. Figure 16: The logarithm of energy
density for the collision shown in figure 15, showing the importance of
radiation in this regime.

Radiation can also have an effect on a single kink, as illustrated in figure
17. Effectively it acts as a force pushing the subkinks away from each other
no matter where the radiation comes from, meaning that an incident burst of
radiation can break the subkinks apart, provided $\beta$ is large enough.
Figure 17 shows the effect of a right-moving wave train with frequency
$\omega=3$ and amplitude $A$, supported at $t=0$ on $[-\infty,-20)$, on a
single kink with $\beta=3$ (left plot) and $\beta=5$ (right plot). For
$\beta=3$ only small oscillations in the kink width could be observed even for
the relatively-large amplitude $A=0.1$, though the kink as a whole undergoes a
negative radiation pressure. By contrast, for $\beta=5$, which is within the
regime where subkinks behave as independent $\phi^{6}$ kinks, an incident wave
train with the smaller amplitude $A=0.05$ has no problem separating the
subkinks. Over an exponentially-large timescale at least as great as that
given by equation (19) above (which ignores residual radiation pressure), the
extremely-weak false vacuum pressure will cause these subkinks to recollide,
but over the timescales shown in the plot, the lower (negative spatial
coordinates) subkink trajectory is precisely matched by that of the
corresponding single $\phi^{6}$ kink subject to the same incident radiation,
as would be expected from our general considerations.

Figure 17:  Radiation incident on a single kink for $\beta=3$ (left) and
$\beta=5$ (right). In both cases the radiation frequency $\omega=3$; on the
left the amplitude $A=0.1$, while on the right $A=0.05$.

##  4 Kink-bion-antikink collisions for $\epsilon=0$

As seen in the last section, in the large $\beta$ regime a centrally located
bion formed from a pair of subkinks can play an important role as an
intermediate state in collision processes. This bion may then be impacted by
other subkinks, either during the initial set of collisions as seen in figure
8 b, or in the far future of processes such as those shown in figures 8 a and
9, where false vacuum pressure will ultimately force the return of the outer
subkinks to the central region. These impacts are well modelled by kink-bion-
antikink collisions in the $\phi^{6}$ model. This section is devoted to a
study of such processes.

Bions formed in the $\phi^{6}$ model after a type I collision such as that
shown in figure 12 a have surprisingly uniform properties over a wide range of
initial velocities, from near zero up to the critical velocity. Their
amplitudes do not decay steadily (so they are not pure oscillons333Some
authors reserve the word oscillon only to almost periodic solutions in time,
in contrast to the quasi-periodic nature of bions created from kink-antikink
collisions. In this paper, however, we will use the terms interchangeably.)
but rather have beats which range between $A\sim(0.55-0.65)\pm 0.02$ over the
time-scale of the simulation. A typical example is shown in figure 18. In the
spectrum there are three dominating frequencies. At early stages, $500<t<1500$
these are $\omega=0.75$, $0.50$, $2.26=3\times 0.75$. At longer times the
frequencies change but the rate of change slows down and the peaks are better
defined, at $\omega=0.79$, $0.59$, $2.38=3\times 0.79$. Note that
$2\omega_{1}-\omega_{2}\approx 1$ in both cases, where $\omega_{1}$ is the
dominant frequency, and $\omega_{2}$ corresponds to the highest peak below
$\omega_{1}$. The time between the first and the second bounce in the
collision during which the bion is formed increases as the velocity approaches
the critical value. But after the second bounce, a bion with properties
described above is formed.

The bions are formed around the $\phi=0$ vacuum and because of the symmetry of
the potentials of the Christ-Lee (and $\phi^{6}$) models they can only have
odd harmonics. Therefore the radiation coming from the first propagating
harmonic comes from the third order of nonlinearity and as such is smaller
than the more-usually observed second harmonic radiation (as in the $\phi^{4}$
model) making central $\phi^{6}$ bions more stable (in the sense that the
radiation leading to oscillon decay is relatively weaker). Note that the
potentials around $\phi=\pm 1$ is not symmetric and therefore oscillons/bions
created around those vacua do possess all harmonics, including the even ones.

a) b)

Figure 18: Typical bion after the type I $\phi^{6}$ collision ($v=0.2)$: (a)
field at centre, (b) its power spectrum at different times.

Collisions between a bion and a pair of symmetrically-placed kinks in the
$\phi^{6}$ model depend strongly on velocity and initial separation of the
kinks, but the most important factor is actually the phase of the central bion
at the moment of collision. To model this we created a central oscillon from
gaussian initial conditions

$\phi_{O}(x,0)=Ae^{-bx^{2}}\qquad\phi_{O,t}(x,0)=0.$ (24)

Taking $A=0.7$ and $b=0.212$ we were able to create an oscillon with similar
properties to the bion resulting from a type I $K\bar{K}$ collision in the
$\phi^{6}$ model (figure 19). Far away from the oscillon, at positions $\pm
x_{0}$ , we added $(-1,0)$ and $(0,-1)$ kinks moving symmetrically towards the
centre with initial velocities $\mp v_{i}$. Then we evolved the system
according to the $\phi^{6}$ ($\epsilon=0$) equation of motion. Some examples
of possible solutions are shown in figure 20. For small velocities (figure 20
a) the radiation emitted from the oscillon pushed the kinks away and no
contact was made. For higher velocities the kinks could split the oscillon and
form two escaping bions (figure 20 b). Topologically if we can treat the
central oscillon as a tightly bound kink – antikink pair, then after the
collision with the incoming kinks different pairs are bound. Another
possibility is that the kinks bounce off the oscillon, and can either take
some of its energy and escape with much higher velocities than $v_{i}$ (figure
20 c) or can lose some of their kinetic energy to excite the oscillon and
produce radiation (figure 20 d). Note that in the last case the kinks are
clearly accelerated by radiation pressure after the interaction.

For each bion there exists a minimal velocity for the incoming kinks below
which the collision cannot take place. Assuming a constant radiation force we
could expect that the lower critical velocity would be proportional to
$\sqrt{x_{0}}$. The measured critical velocities for $x_{0}=15$ and $30$ were
$0.149$ and $0.186$ respectively which gives the ratio $1.25$ instead of
$\sqrt{2}\approx 1.41$. This discrepancy is not surprising, given that in our
simulations the force was not constant because the initial gaussian did not
have radiation tails instantly. These tails develop with time. Moreover, the
wavefront has a peak which pushes the kinks even more for a short time.
Finally, as the solitons slow down so does the radiation frequency observed in
the moving frame of the solitons, and the radiation pressure depends on
frequency.

a) b)

Figure 19: Evolution of an oscillon created by gaussian initial conditions.
(a) the field at centre, (b) measured frequency defined by time differences
between subsequent descending zeros of $\phi(0,t)$

a) b)
c) d)

Figure 20: Basic kink-bion-antikink collision scenarios for $x_{0}=30$. (a)
$v_{i}=0.15$, (b) $v_{i}=0.40$, (c) $v_{i}=0.41$, (d) $v_{i}=0.454$.

In figure 21 we show scans through a range of initial velocities $v_{i}$. The
upper plots show the field value at the centre and the lower plots show the
velocities of the escaping kink or subkink. The dashed black line corresponds
to $v_{i}=v_{f}$, allowing parameters to be found where the kinks gained speed
(scenario 20 c). Blue columns in the upper plots indicate two escaping bions
(scenario 20 b). Green-brown stripes indicate that an oscillon remained in the
central region after the collision which is marked by a blue stripe (scenario
20 d). Note that there is no blue stripe (collision) below a certain velocity
$0.149$ for the left plot with $x_{0}=15$ and $0.186$ for the right plot with
$x_{0}=30$. Note that these numbers are consistent with the critical
velocities for $x_{0}=15$ and 30 of 0.149 and 0.186 respectively that were
quoted earlier.

a) b)

Figure 21: Field at centre and escape velocity of a kink for the same central
oscillon $A=0.7$ and $b=0.212$ but for different initial separations
$x_{0}=15$ (plot a) and $x_{0}=30$ (plot b). The vertical dotted lines
labelled a-d on plot b indicate the velocities at which the field maps shown
in figure 20 were made.

Just above the velocity when the two escaping bions scenario (figure 20 b)
changes into kink reflection (figure 20 c) there is a significant peak in the
final velocity of the kinks. This shows that there is some resonant mechanism
of energy exchange between the oscillon and the kinks, which at least
initially leads to near-complete destruction of the central oscillon.

Further zooms near the edges of the green-brown columns in figure 21, shown in
figures 22 a and 22 b, reveal that the structure repeats itself to some
extent, exhibiting fractal-like features. In Figure 23 we present some typical
collision processes where after the initial impact a pair of oscillons is
created (corresponding to blue regions at intermediate times in the scans in
figures 21 and 22) which later annihilates into a kink-antikink pair escaping
to infinity and a central oscillon, corresponding to the green-brown columns
on the scans. The processes shown in figures 23 and 20 b, during which
incident kinks are converted into oscillons, are reminiscent of the mechanism
for resonant scattering of sine-Gordon solitons on a non-integrable boundary
discussed in Arthur:2015mva , where in some situations an incident kink can be
reflected as a breather.

Figure 22: Zooms reveal fractal-like features in the kink-bion-antikink
collisions for $A=0.7$, $b=0.212$ and $x_{0}=30$.

a) b)

Figure 23: Analogues of a long two-bounce window $x_{0}=30$, $v_{i}=0.4545$
(a), and a multi-bounce window $x_{0}=30$ and $v_{i}=0.312054$ (b).

As already mentioned, a crucial element determining the outcome of a kink-
bion-antikink collision is the phase of the central bion. To verify this we
estimate the corresponding resonant frequency from the scattering data, by
plotting $T_{n}=x_{0}/v_{n}$ vs $n$, where $n$ is a relative window number (as
counted from higher velocities to lower) and $v_{n}$ is a measured velocity
for the transition from the process of ejecting two oscillons (figure 20 b) to
subkink ejection with maximal velocity (figure 20 c). We expect that

$T_{n}=\frac{2\pi}{\omega}(n-n_{0}).$ (25)

For initial separations $x_{0}=15$ and $x_{0}=30$ the linear fits to the data
shown in figure 24 yield $\omega=0.6992\pm 0.0340$ and $\omega=0.6885\pm
0.0024$. These values correspond very well with the measured oscillon
frequencies at small times as shown in figure 19. The oscillon created has a
dominant frequency, but its evolution is not fully periodic, and the measured
frequency varies between 0.68 and 0.73.

Figure 24:  Transition collision time $T_{n}=x_{0}/v_{n}$ as a function of
relative window number for $x_{0}=15$ and $30$, along with the straight line
fits.

##  5 Conclusions

The main purpose of this work has been to extend the analysis of the
collisional dynamics of kinks in a 1+1 dimensional scalar theory to the case
of weakly bound pairs of kinks. We focused our analysis on the Christ-Lee
model, which, depending on the value of the parameter $\beta$, interpolates
between the $\phi^{4}$ model with two vacua ($\beta=0$), and the $\phi^{6}$
model with a triply-degenerate vacuum ($\beta\to\infty$). For sufficiently
large values of $\beta$ the system has a false vacuum at $\phi=0$ and the
static solutions of the model have an inner structure as a pair of weakly
bound subkinks. In addition to this, the presence of the false vacuum allows
for existence of unstable lumps (or sphalerons), which can be produced as
intermediate states in the collisions of the solitons.

The spectrum of linearized fluctuations about the kinks contains multiple
localized modes. Despite this, we observed that the excitation of these modes
plays little role in the collisional dynamics of kinks and antikinks as
$\beta\to\infty$. In this limit the subkinks can be treated as independent
solitons, and the collision becomes a multi-step process. Depending on the
impact velocity, the first collision of the advanced pair of subkinks may
produce a burst of radiation, a pair of outgoing subkinks or/and an oscillon.
In all cases the presence of background radiation affects the subsequent
collision of the second pair of subkinks. This mechanism is related to the
appearance of spine structures in the collisions of the solitons. The
resulting structure is rather less regular than those observed in previous
works, where internal modes controlled the appearance of the resonance
windows. It will therefore be a challenging question for future work to see
whether the spines can be understood within a collective coordinates approach.

## Acknowledgements

We would like to thank Nick Manton for interesting conversations and many
insightful remarks. The research of PED was supported in part by the National
Science Foundation under Grant No. NSF PHY-1748958, and in part from the STFC
under consolidated grant ST/T000708/1; he also wishes to thanks the African
Institute for Mathematical Sciences South Africa for hospitality while this
work was completed. TR wishes to thank National Science Centre, grant number
2019/35/B/ST2/00059 and the Priority Research Area under the program
Excellence Initiative – Research University at the Jagiellonian University in
Kraków. YS gratefully acknowledges the support of the Alexander von Humboldt
Foundation.

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# Multi-NeuS: 3D Head Portraits from Single Image with Neural Implicit
Functions

Egor Burkov1 Ruslan Rakhimov1 Aleksandr Safin1 Evgeny Burnaev12 Victor
Lempitsky3
1Skoltech 2AIRI 3Cinemersive Labs

###### Abstract

We present an approach for the reconstruction of textured 3D meshes of human
heads from one or few views. Since such few-shot reconstruction is
underconstrained, it requires prior knowledge which is hard to impose on
traditional 3D reconstruction algorithms. In this work, we rely on the
recently introduced 3D representation — neural implicit functions — which,
being based on neural networks, allows to naturally learn priors about human
heads from data, and is directly convertible to textured mesh. Namely, we
extend NeuS, a state-of-the-art neural implicit function formulation, to
represent multiple objects of a class (human heads in our case)
simultaneously. The underlying neural net architecture is designed to learn
the commonalities among these objects and to generalize to unseen ones. Our
model is trained on just a hundred smartphone videos and does not require any
scanned 3D data. Afterwards, the model can fit novel heads in the few-shot or
one-shot modes with good results.

|  |  |  |  |  |  |
---|---|---|---|---|---|---|---
[0.0cm 1.6cm]Input |  |  |  |  [0.0cm 1.6cm]Input |  |  |
Figure 1: Multi-NeuS can reconstruct a realistic textured 3D mesh of a head
from a single in-the-wild photo or painting despite significant domain gap
with our training data and the small number of identities in the training set.

##  1 Introduction

We consider the task of 3D portraiture, i.e. automatic acquisition of 3D
models of human heads that capture both the geometry and the texture. While a
number of learning-based methods for this task have been suggested [37, 14],
most of these methods require 3D scans or synthetic data for learning. Here,
we propose an alternative approach that learns to model human head shape and
appearance directly from a collection of RGB videos.

Our approach is based on a recent class of methods that use implicit
representations for shape and appearance such as the recently introduced NeuS
method [47] and very related approaches introduced in parallel with NeuS [34,
50, 22]. We introduce new and simple way to fit such models to individual
videos, while sharing a subset parameters, resulting in the approach that we
call Multi-NeuS.

We show that sharing the parameters across training videos facilitates
knowledge transfer to new individuals unseen during training. As a result,
Multi-NeuS achieves noteworthy data-efficiency (capable of learning a generic
human head model from the videos of as little as 103 individuals) and is fast
to train (takes only 24 hours on a single V100 GPU). After training, Multi-
NeuS can create convincing textured 3D head meshes from as little as a single
photograph (Figure 1).

Within this work, we investigate two parameter sharing patterns and sharing-
related regularizations that can be used within Multi-NeuS. These are (1)
sharing the parameters of layer subsets, as well as more sophisticated (2)
low-rank regularization on non-shared parameters. We assess the effect of the
sharing setting on the quality of the results.

Overall, in the experiments we show that our system is capable of creating
high-quality 3D portraits from few photographs, and reasonably good portraits
from single in-the-wild photographs. More generally, our approach proposes a
new way to do transfer learning within implicit shape and appearance modeling
frameworks, and we hope that our findings will boost future meta-learning
research involving implicit functions. To sum up, our contributions are:

  * •

We introduce a new type of 3D neural implicit architecture that can
efficiently fit to many objects of the same class simultaneously and recover
their surfaces, given sets of multi-view photos.

  * •

We devise a meta-learning pipeline for the above model that enables it to
reconstruct the textured 3D surface of an unseen object from one or few
images.

  * •

We demonstrate that our system can be applied to single-view reconstruction of
3D full head portraits, producing convincing 3D meshes from in-the-wild images
after being trained on just a hundred short smartphone videos.

##  2 Related Work

###  2.1 Neural Implicit 3D Reconstruction

Neural implicit functions have attracted a lot of attention recently, notably
as a flexible approach to represent 3D scenes with neural networks. Contrary
to traditional explicit 3D representations such as meshes, they are not
limited to a fixed resolution or topology, and, most importantly to us, can
naturally employ the power of modern neural network methods.

Neural radiance fields (NeRF) [30] and its extensions (e.g. [26]) model
density and emitted radiance with neural nets which are trained via
backpropagating through volumetric ray casting. Although NeRF achieves
impressive results in novel view synthesis, it is not designed for
reconstructing geometry. For instance, it was shown in [34] that meshes
directly obtained from NeRF density functions are full of artifacts.

A more “geometry-friendly” implicit approach is to model the object surface as
a zero-level set of implicitly defined occupancy [29] or signed distance
function (SDF) [35, 45], which goes back to the classical works on level set
reconstruction [8]. The isosurface can then be easily converted to a mesh via
marching cubes [28]. To train such models without 3D supervision, several
authors have done inverse rendering by modeling color or radiance similar to
NeRF and then applying some kind of ray marching [27, 33, 34, 22, 50, 47].
From these single-scene multi-view methods, we pick NeuS [47] as a base of our
multi-scene few-view method due to simplicity and code availability. We
revisit NeuS in more detail in Section 3.1.

###  2.2 Meta-Learning Neural Implicits

The meta-learning paradigm addresses (among other things) the few-shot problem
when given several training examples the network aims to achieve better
performance. The most common line of approaches is the optimization-based
approaches [9, 32] that learn the best weight initialization. For a deeper
meta-learning review we refer the reader to [17]. Regarding the application of
meta-learning to neural implicits, MetaSDF [42] exploits this idea to learn
the initialization of the SDF network, while the work [44] applies meta-
learning to a wider variety of signal types. Our work concentrates on human
body representation and uses shared network layers across different tasks
(with different people identities).

###  2.3 Few- or Single-View Head Reconstruction

Historically, directly fitting statistical 3D Morphable Models (3DMMs) to an
image has been a popular method to recover the 3D head shape [36, 20, 18, 12],
but 3DMMs are limited to coarse shape estimation, requiring separate steps of
reconstructing e.g. wrinkles [20] or hair [2]. In addition, 3DMMs are
constructed from 3D scans which might be hard to obtain for many classes.
Other more descriptive and flexible 3D representations include depth maps [40,
48, 41], regular meshes [43, 38], and volumetric grids [19], although many of
these approaches still rely on 3DMM in their intermediate steps. Two rare
examples of completely model-free methods that also reconstruct hair [41, 43]
are self-supervised GANs [21] that learn from unlabeled collections of images.
However, actual fitting to unseen images (e.g. GAN inversion) was not
demonstrated. More information on face/head reconstruction before the advent
of 3D neural implicit methods is available in recent comprehensive surveys
[31, 7].

Recently, several works have successfully applied neural implicit
representations to the head reconstruction task, but most of them either do
not reconstruct geometry directly (e.g. because of ill-suited NeRF
representation) or require complex datasets. Portrait-NeRF [11] is an early
attempt of meta-learning a single-view NeRF. The support of only slight
viewpoint changes has been demonstrated for this method. The i3DMM method [51]
introduced the first 3DMM to include hair. This method is based on SDFs and is
constructed from about 2000 3D scans of 64 people. H3D-Net [37] meta-learns
high-quality SDF representation of full static heads and supports
reconstruction from as few as three posed images (though feeding just one
image is also possible). The method is trained on a private dataset of 10,000
structured-light 3D scans. HeadNeRF [16] yields controllable NeRF portraits
conditioned on latent 3DMM vectors (identity, expression, albedo, and
illumination). It is a fully supervised approach, and authors were able to
train it in reasonable time thanks to a strategy that improves rendering
performance [15]. The authors of EG3D [5] went even further and trained a
StyleGAN2 [21] to yield volume-renderable 3D heads with very little
supervision (a similar idea was proposed in VolumeGAN [49] simultaneously).
Like HeadNeRF, EG3D can fit an arbitrary head photo by optimizing the latent
vector(s). Moreover, their paper demonstrates extracting meshes of convincing
quality. Still, this method is computationally $80\times$ more expensive to
train than Multi-NeuS, and it does not reconstruct parts of the head that are
further from the face due to the lack of dedicated background modeling.

Figure 2: Architecture of Multi-NeuS, a 3D neural implicit function that can
represent multiple objects of a class simultaneously (boxes depict fully
connected layers and their output dimensionalities; $\gamma$ is the positional
encoding function). Since some layers (blue) are shared between all scenes,
they can learn class priors to then transfer knowledge to novel scenes of the
same class, enabling few-shot reconstruction. The model is trained via
volumetric rendering and simple pixelwise loss, just like NeuS [47], but on a
dataset of multiple scenes. Afterwards, when fitting to an unseen object,
scene-specific layers (yellow, Section 3.3) are fitted first, and finally all
layers are fine-tuned together.

##  3 Method

###  3.1 Recap: NeuS Reconstruction

As our method builds upon NeuS [47], we start with the review of this method.
NeuS is a modification of NeRF [30] for non-transparent objects. It models the
object surface directly, thus allowing 3D surface reconstruction from images
using differentiable neural rendering. Specifically, the object surface in
NeuS is represented as the zero-level set of a signed distance function
$\left\\{x\in\mathbb{R}^{3}~{}|~{}\text{SDF}(x)=0\right\\}$, where SDF is
defined as signed distance to object surface and is modeled by a neural
network (a multi-layer perceptron). In addition, RGB radiance at any 3D point
is modeled by another neural net, and density is modeled as a bell-shaped
function of SDF that attains its maximum at zero, i.e. at the object surface.
Combining radiandce and density, differentiable volume rendering is run over
camera rays corresponding to pixels of input images. The optimization
algorithm forces the RGB results of ray integration to be similar to the
corresponding known pixel intensities by progressively tuning the weights of
neural networks. The loss function to optimize is a simple pixelwise mean
squared error combined with an eikonal regularization term that ensures
$\left\|\nabla\text{SDF}(x)\right\|=1$. After convergence, it is possible to
obtain object mesh via marching cubes [28] over $\text{SDF}(x)$, as well as to
synthesize novel views by volume rendering or any ray marching algorithm, such
as sphere tracing.

The multi-view captures may include distant background which is difficult to
represent by the above neural nets. Therefore, the object of interest is
considered to be within a unit sphere, and everything outside of that sphere
is modeled by a separate dedicated NeRF with the special parametrization of
coordinates [52]. To optimize this NeRF along with NeuS, extra ray points are
sampled outside of the unit sphere. A sufficiently large dataset lets such
tandem to disentangle background from the central object automatically,
without mask supervision.

NeuS achieves excellent results when applied to sets containing dozens of
images. Our goal is to create a NeuS-based system that can perform
reconstruction given a single image or very few images. This scenario is too
under-constrained for the original NeuS and will result in poor convergence.
To alleviate this, we narrow down the class of potential scenes to human heads
and pre-train our model on a dataset of multiple people, while facilitating
knowledge transfer to unseen people as discussed below.

###  3.2 Multi-NeuS

Our solution called Multi-NeuS is depicted in Figure 2. We upgrade NeuS so
that it can fit to $N$ scenes simultaneously. Our high-level idea is simple.
We create $N$ copies of scene-specific NeuS instances that share some of the
layers, while keeping other layers unshared (scene-specific). We then fit
these $N$ instances to the scenes simultaneously, while optionally imposing
additional structural regularization on scene-specific layers.

Naturally, we expect that during such fitting shared layers will tend to model
features useful to represent any object, while scene-specific layers combine,
refine and augment the output of shared layers to model a specific object. For
instance, a shared layer might model rough basic human head shapes, while the
following (scene-specific) layer may learn the weights with which to combine
those shapes, like in linear blend skinning models.

We experiment with two architectures for scene-specific layers that are
described below in Section 3.3. As shown in Figure 2, we use scene-specific
layers in the first halves of the SDF network and the radiance network, while
sharing all other layers. This choice is evaluated in Section 4.4.

Differently from NeuS, Multi-NeuS learns $N$ independent scene-specific
instances of background NeRFs. Also, we do not model view-dependent effects in
our architecture, effectively assuming that human heads do not produce
specular reflections. We find that on our dataset (which is captured in
scattered light), this does not hurt validation performance but significantly
reduces overfitting in few-shot mode especially when generalizing to new
lighting.

###  3.3 Scene-Specific Layers

Figure 3: The two architectures of scene-specific layers explored in our
paper, independent (a) and low-rank (b). They are fully connected layers whose
weights and biases $w\left(i\right)$ depend on scene index $i$. An independent
layer learns individual weights and biases for each of $N$ scenes, while a
low-rank layer learns $r$ copies and then linearly combines them with each
scene’s own learnable coefficients.

We use the scene (person) index $i\in\overline{1,N}$ to enumerate scene-
specific layer instances. Thus, by considering different instances within
scene-specific layers, the same network architecture models every object in
the dataset. In this work, we experiment with two architecture choices for
scene-specific layers (Figure 3), which we term independent and low-rank. They
are described below.

#### Independent layers

(Figure 3, a). This is a straightforward implementation where the scene-
specific layer has a dedicated set of weights and biases $w_{1},\dots,w_{N}$
for each scene. This architecture has large representational power but has
significant drawbacks.

First, during meta-learning, each $w_{i}$ receives infrequent weight updates
during learning. Thus, if a training minibatch includes pixels from few ($m\ll
N$) scenes, then sub-layers corresponding to all other scenes do not receive
any weight updates. Alternatively, a minibatch can be composed of random
pixels from the entire dataset ($m\approx N$). In this case, however,
$w_{i}$’s gradients become too noisy, coming from just few
($\approx\frac{N}{m}$) pixels, again leading to slow/poor convergence.

In practice, batching together pixels from many scenes is inefficient as it
requires to run $m\approx N$ layers in each forward pass, so in our
experiments we set $m=1$, i.e. we sample all pixels of a minibatch from just
one scene. We therefore use the Adam [23] optimizer but update moment
statistics for a scene-specific layer only when the corresponding scene
participates in the forward pass (known as “sparse/lazy Adam”).

Another related problem with independent layers is overfitting due to the
excessive number of parameters. This often leads to poor generalization to new
subjects. Our second architecture below is designed to alleviate this by a
built-in regularization.

#### Low-rank layers

(Figure 3, b). In this scheme, scene-specific layer’s weights and biases
$w\left(i\right)\in\mathbb{R}^{p}$ are not learnt directly. Instead, they are
computed as a linear combination of $r$ basis vectors $b_{1},\dots,b_{r}$:

$w\left(i\right)=\sum_{j=1}^{r}c_{ij}{b_{j}}~{},$ (1)

where $r$ is the layer’s rank. We learn both the basis vectors
$b_{j}\in\mathbb{R}^{p}$ and the linear combination coefficients
$c_{ij}\in\mathbb{R}$, where $i\in\overline{1,N}$ and $j\in\overline{1,r}$.
Thus, each scene-specific layer learns a single set of $r$ basis vectors for
the entire training dataset containing multiple scenes, and these vectors are
recombined with different weights to model different scenes, therefore a
separate set of $r$ coefficients is learned for each of the $N$ scenes. Such
low-rank factorization decreases the number of parameters significantly (by a
factor of several hundreds in our experiments), reducing overfitting.

###  3.4 Training

Multi-NeuS is applied in two stages: meta-learning and fitting.

In the initial meta-learning stage, we pre-train the whole architecture using
the same volumetric rendering procedure as in NeuS (Section 3.1) but on a
dataset of multi-view RGB images of $N$ scenes rather than a single scene. At
every optimization step, a minibatch of camera rays (or, equivalently, image
pixels) is sampled uniformly from eight random images of one random scene.
Eventually, Multi-NeuS estimates the 3D shape and texture of every scene
(subject) in the dataset.

After meta-learning, we can fit to new scenes starting from the pre-trained
initialization. This fitting stage is thus conducted to estimate the 3D shape
and the texture of a novel unseen object. This amounts to adding the new
$(N+1)$-st scene to the model. This time, we are given images of the new
subject (can be as few as one or two), their estimated camera parameters, and
their background segmentation masks.

The fitting process is performed in two steps: we first retrain the scene-
specific weights and then we fine-tune all weights to the new scene. The first
step begins with initializing scene-specific weights. We do it by simply
averaging these weights over $N$ scenes so that the $(N+1)$-st representation
in Multi-NeuS essentially represents “the average object in the dataset” as
learned by the scene-specific layers. That is, for independent layer we set
$w_{N+1}=\frac{1}{N}\sum_{i=1}^{N}w_{i}$, and for the low-rank layer
$c_{N+1}=\frac{1}{N}\sum_{i=1}^{N}c_{i}$. Note that in the low-rank layer the
basis weights $b_{1},\dots,b_{r}$ are not scene-specific but are in fact
shared by all scenes. Therefore, we do not optimize or reset them in the first
step of the fitting process. After optimizing the newly initialized scene-
specific weights in the first step, in the second step we “unfreeze” the
shared weights and optimize all weights while using a smaller learning rate.

The optimization during the fitting stage is performed in the same way as in
the meta-learning stage with two notable differences. Firstly, instead of
using a dedicated background NeRF [47], we explicitly estimate background
masks and optimize an additional loss that forces the SDF isosurface to match
these masks. The loss used in this case is the binary cross-entropy between
the accumulated density over a ray and the foreground mask value ($1$ if
object, $0$ if background). This is needed since we found that background
separation in the original NeuS works unsatisfactory in the few-shot regime.

The second modification is fine-tuning the camera parameters. This is needed
because camera estimates can be inaccurate, especially for in-the-wild images
from the Internet. To compensate for that, we backpropagate the losses into
the camera parameters and optimize them alongside the neural networks with a
$10\times$ smaller learning rate.

Please refer to Section 4.5 for additional details.

Input | H3D-Net | Ours | GT | Input | H3D-Net | Ours | GT
---|---|---|---|---|---|---|---
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
Figure 4: Single-view mesh reconstruction on the first four scenes of the H3DS
dataset. H3D-Net [37], a method related to ours, was designed for three-view
reconstruction but can also be evaluated in the one-shot mode. The H3D-Net
system was trained on 10,000 3D scans from the same distribution as these test
examples. Our method is trained on a hundred smartphone videos and still
matches the quality of H3D-Net, while demonstrating somewhat smaller identity
gap and less pronounced regression-to-mean effect.

##  4 Experiments

###  4.1 Datasets

Our training (meta-learning) dataset is a subset of SmartPortraits [24]. It
consists of 107 short ($\approx$ 25 seconds) smartphone videos of still people
with neutral pose and facial expression. Four of these (two female and two
male subjects) serve as the validation set. In each video, the distance to the
head ($\approx$ 1.5 m) and the elevation are roughly constant, while the
azimuth travels within $\pm 45^{\circ}$. From each video, we remove frames
with flash and randomly pick about 77 frames from the rest, shrinking the
entire dataset to 8256 images. We obtain camera parameters by running the
COLMAP structure-from-motion software [39] on these images. Finally, these
images are loosely cropped to head and shoulders using a face detector. Note
that we do not use any motion or depth information in our system.

Because Multi-NeuS takes in the absolute 3D coordinates, all scenes are
aligned against each other to minimize the relative difference between
objects. This helps our network not to spend capacity on modeling translations
and scaling, and thus to fit the training set easier. We accomplish
approximate alignment as follows. For each scene, we detect six prominent
facial landmarks in images [4]. We then triangulate the 2D landmarks to get
their 3D coordinates, and compute an optimal similarity transform $T$ [46]
that aligns these 3D coordinates with those estimated in the first scene of
SmartPortraits. Finally, $T$ is applied to all camera poses of the current
scene. We estimate and apply such similarity transform not only for
SmartPortraits but for every scene of every dataset used in this work.

We also validate on the H3DS dataset [37], which consists of ten individuals.
For each individual, the dataset offers a full head 3D scan (mesh) alongside
with 60 to 70 $360^{\circ}$ photos taken with varying lighting, and camera
parameters for these photos.

Besides, we provide qualitative results on several paintings and in-the-wild
photos found on the Web. To that end, we estimate camera parameters for a
single photo as follows. We detect the same six landmarks as above, but this
time obtain their approximate 3D coordinates in orthographic camera coordinate
system ([4] provides them directly). We assume that these coordinates are 3D
world coordinates, and that the image was taken with a telephoto lens with the
vertical field of view of $\approx 10^{\circ}$. These asssumptions allow us to
roughly recover the camera pose in world coordinates, namely via an algorithm
for the Perspective-n-Point (PnP) problem [6].

When fitting to any unseen pictures, we estimate background masks using an
off-the-shelf model [13] and manually refine them.

###  4.2 Single-View Geometry Reconstruction

By providing ground truth 3D scans, H3DS permits a quantitative comparison of
geometry reconstruction, so we use it to compare against H3D-Net [37], which
was tailored for this dataset. Although H3D-Net was demonstrated to
reconstruct from three or more views, it can fit to a single view as well, and
it has an advantage on the H3DS dataset since this model was trained on a
large dataset (10,000 scenes) from the same distribution.

The target metrics, as in [37], are unidirectional Chamfer distances in
millimeters from the predicted mesh to the ground truth, computed after rigid
alignment via ICP [3]. One metric is the distance computed over facial area
only, and the other one is computed over the entire ground truth mesh of a
head.

We compute 1-view metrics by reconstructing from left, right (azimuth $\approx
45^{\circ}$), and frontal views. We do not apply our method in few-shot
setting on H3DS because images in this dataset are taken with varying lighting
and exposure, lacking multi-view consistency required for Multi-NeuS.

We compare our best model (low-rank architecture, $r=1000$; evaluated in
Section 4.3) with H3D-Net in Table 1 and Figure 4. Our method practically
matches H3D-Net in reconstruction accuracy while learning from a different
dataset that has $100\times$ fewer identities and does not require 3D
scanning. Furthermore, rendered samples suggest that H3D samples look very
similar to each other, especially outside of the face region (the so-called
regression-to-mean effect) while our model predicts more “personalized”
shapes.

| face | head
---|---|---
Input view | F | L | R | mean | F | L | R | mean
H3D-Net, 3-view | - | - | - | 1.34 | - | - | - | 10.53
H3D-Net, 1-view | 1.82 | 1.83 | 1.91 | 1.85 | 13.83 | 13.01 | 12.51 | 13.12
Ours | 1.89 | 1.77 | 1.86 | 1.84 | 13.00 | 13.27 | 11.95 | 12.74
Table 1: Mesh reconstruction error on H3DS dataset. Lower is better, ”F/L/R”
are for ”frontal/left/right”. See Section 4.2 for details.

To demonstrate additional single-view geometry reconstruction, we show several
reconstructions of in-the-wild photographs and paintings in Figure 5.

###  4.3 Effect of Number of Views and Layer Type

Although our primary aim is to reconstruct heads given just one image, our
method naturally benefits from additional views. We demonstrate this on the
validation subset of SmartPortraits. Similarly to H3DS, we restrict the scenes
to the views: left, right (with azimuths around $\pm 45^{\circ}$) and frontal.
Since 3D ground truth is not available in this case, we render two additional
control views ($\pm 20^{\circ}$) and compute masked PSNR against the ground
truth images corresponding to these two views (control images). These are in
turn averaged over four validation scenes.

|  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
[0.0cm 1.6cm]Input |  |  |  [0.0cm 1.6cm]Input |  |  |  [0.0cm 1.6cm]Input |  |

|  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
[0.0cm 1.6cm]Input |  |  |  [0.0cm 1.6cm]Input |  |  |  [0.0cm 1.6cm]Input |  |

Figure 5: Additional results for 3D reconstruction of in-the-wild photographs
and paintings. Our method is able to handle different hair styles and performs
reasonably well for images that are different from the training SmartPortraits
dataset. Note the back may have artifacts because this part of the head is
mostly absent in the training dataset due to very limited view angles.

We observe that during fitting to a novel person, optimizing camera parameters
provides additional degrees of freedom. This often leads to the person’s shape
in Multi-NeuS drifting away from its “canonical” position (Section 4.1) and
inflating the validation error, even when the reconstruction is good.
Moreover, the two control cameras might be estimated inaccurately during data
pre-processing. To address this, before reporting PSNR against control images,
we refine the control cameras’ poses and focal distances by optimizing for
PSNR.

Figure 6: Quality of novel view reconstruction depending on scene-specific
layer type (Section 3.3), measured on our validation subset of SmartPortraits.
Lower rank metamodels have fewer degrees of freedom and underfit during the
first step of fitting; higher rank models fit better and provide a more
convenient initialization for the second step of fitting (fine-tuning of all
weights).

Figure 6 compares how faithfully the novel views are reconstructed depending
on the number of input views (one, two, or three) and depending on the layer
type (independent or low-rank). In the case of a single input view, the metric
is averaged over reconstructions from left, right, and frontal views. In the
two views case, we take the left+right views as input.

Consider low-rank models. Clearly, at the first step of fitting (Figure 6,
left), when only linear combination coefficients $c_{N+1}$ and camera
parameters are optimized, the models underfit in the case of low ranks. When
the rank is very high (2000), the models start to overfit since the number of
parameters becomes excessive. This is additionally illustrated in Figure 7.

In all cases, the second step of fitting (Figure 6, right) where all
parameters are fine-tuned is necessary because low-rank coefficients have few
degrees of freedom. Multi-NeuS without the second step thus underfits the
input views. However, in the few-shot setting, optimizing the full network can
lead to severe overfitting. So the primary goal of the first fitting step is
to provide a good initialization for this second step. According to the
diagram, models with a reasonably high rank (e.g. 1000) provide best
initializations, but this comes at a cost of overfitting during fine-tuning
which may even decrease the overall PSNR by distorting the unseen areas of the
resulting shape. This is probably because high rank scene-specific layers gain
too much representational power and the shared layers are not forced as hard
to learn universal features, thus hampering generalization. An alternative
interpretation is that since we spend the same number of fine-tuning
iterations regardless of rank, early stopping might alleviate some overfitting
issues.

Figure 7: The first (coarse) step of fitting (i.e. shared layers frozen) for
various architectures of scene-specific layers. The scene is a subject from
the validation subset of SmartPortraits. The amount of overfitting can be
traced by looking at the right cheek and ear, which are invisible in the
source view. The independent architecture with overparametrized scene-specific
layers overfits already at this fitting stage. The low-rank variants become
better at fitting these hidden parts with increasing ranks and model the
texture better, but at some point ($r=2000$ in this case) get too many degrees
of freedom and start to overfit. $r$ = 1000 provides optimal reconstruction in
this case (and on average).

Another obvious observation is that with more views the effect of overfitting
decreases, and the advantage of higher-rank models becomes less pronounced
(e.g. a 50-rank model already does well for the three-view reconstruction). In
addition, the change in the number of views allows to assess the capacity of
scene-specific layers.

The model with independent scene-specific layers does not generalize well
because of the excessive capacity. Although it demonstrates larger PSNR than
50- and 150-rank models, it does so because its scene-specific layers are
usual linear layers which can fit the training view really well, while low-
rank models underfit. At the same time, the unseen parts in the validation
views already get distorted in the first step and this is why the second step
(fine-tuning) does not improve the score in this case.

Finally, to prove the necessity of shared architectures and meta-learning, we
compare to a simple baseline (Figure 6, extreme right) where a vanilla NeuS
(without view directions) is trained on a scene from SmartPortraits and is
then fine-tuned (transfer-learned) in a few-shot scenario to the target scene.
This is essentially equivalent to Multi-NeuS with $N=1$, i.e. with a pre-
training dataset of 1 scene. The score for this baseline was computed by
transferring from 4 different SmartPortraits training scenes (2 male, 2
female) and averaging the metric. Although NeuS typically fits better to a
single scene than Multi-NeuS to any of its meta-learning scenes, its few-shot
generalization ability is clearly lower compared to any version of Multi-NeuS.

###  4.4 Which Layers to Make Scene-Specific

| 1 step of fitting | full fitting
---|---|---
Number of views | 1 | 2 | 3 | 1 | 2 | 3
$\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare~{}\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare$ | did not converge
$\square\square\square\square\square\square\square\square\square~{}\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare$ | 17.61 | 19.35 | 20.78 | 18.19 | 19.21 | 23.67
$\square\blacksquare\square\square\blacksquare\square\square\blacksquare\square~{}\square\blacksquare\square\square\blacksquare$ | 20.49 | 23.10 | 23.71 | 20.76 | 24.03 | 25.25
$\square\blacksquare\square\blacksquare\square\blacksquare\square\blacksquare\square~{}\square\blacksquare\square\blacksquare\square$ | 21.48 | 23.72 | 24.18 | 21.26 | 24.19 | 25.16
$\square\square\square\square\square\blacksquare\blacksquare\blacksquare\blacksquare~{}\square\square\square\blacksquare\blacksquare$ | 21.75 | 23.94 | 24.29 | 20.78 | 24.31 | 25.15
$\blacksquare\blacksquare\blacksquare\blacksquare\square\square\square\square\square~{}\square\square\square\square\square$ | 21.85 | 23.85 | 24.56 | 21.84 | 24.28 | 25.23
$\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare~{}\square\square\square\square\square$ | 21.92 | 24.25 | 24.77 | 21.59 | 24.03 | 25.22
$\blacksquare\blacksquare\blacksquare\blacksquare\square\square\square\square\square~{}\blacksquare\blacksquare\square\square\square$ | 22.43 | 24.26 | 24.66 | 21.99 | 24.39 | 25.27
Table 2: Novel view reconstruction quality depending on the choice of layers
to replace with their scene-specific variants. Boxes depict sequential fully-
connected layers — like in NeuS, there are 9 layers that predict SDF, followed
by 5 layers that predict radiance. $\blacksquare$ means scene-specific layer,
$\square$ means shared, i.e. vanilla linear layer. Numbers are PSNR on
SmartPortraits, layer type is low-rank, $r$ = 1000.

In this section, we evaluate the exact choice to put scene-specific layers
into the first halves of the SDF network and the radiance network of vanilla
NeuS.

Some possible choices are listed in Table 2 and are evaluated for low-rank
shared layers with $r$ = 1000. Putting the scene-specific layers to the
radiance network only results in a constant SDF in the first stage of fitting.
This results in very low metrics. Results for other sharing patterns are
harder to interpret, but arguably good performance requires sufficient number
of scene-specific layers (having too few of them is detrimental). Furthermore,
at least some of these layers should be among the early processing layers.

###  4.5 Training Details

In both meta-learning and fitting, we use minibatches of 512 rays. In our
experiments, there are 610,000 optimization iterations during meta-learning,
12,000 iterations in the first stage of fine-tuning and 13,000 in the second
stage of fine-tuning. Pre-training (meta-learning) Multi-NeuS takes around 24
hours and fitting it to a novel subject takes about an hour on a single NVIDIA
V100 GPU. The learning rate is $1.8\cdot 10^{-4}$ in meta-learning, $4\cdot
10^{-4}$ in the first step of fitting and $6\cdot 10^{-5}$ in the second step.
We multiply the learning rate by $0.316$ every time the loss stops decreasing
(known as “reduce-on-plateau schedule”).

We optimize camera parameters similar to [25]. Specifically, we (1) multiply
initial camera rotation matrix by optimizable update matrix parametrized using
$\mathfrak{so}(3)$ Lie algebra, (2) add a optimizable residual to the
translation parameters, and (3) multiply focal length by an optimizable
scalar.

##  5 Discussion

We have presented Multi-NeuS – an approach for one- and few-shot 3D head
portrait reconstruction. The approach can reconstruct head portraits in the
form of surface mesh and texture. To enable the few-shot capability, we
propose and validate a very simple idea of taking a scene-specific deep
architecture (NeuS) and fitting it to multiple scenes, while sharing some
parameters across scenes. We show that despite simplicity, this idea is
sufficient to accomplish knowledge transfer from the training scenes to
previously unseen test scenes. We believe that this general idea might be
applicable beyond head portrait reconstruction to other classes (e.g. full-
body reconstruction) and architectures (e.g. different NeRF types).

Our approach has certain limitations. First, the models produced with our
approach come without rigging capability, and in the future it would be
interesting to extend our framework to address this. Furthermore, our models
still exhibit certain amount of overfitting that mostly results in artefacts
in the unobserved parts of the texture. This can be addressed with ad-hoc
inpainting procedures that exploit class-specific symmetries, or more
principled extensions of our method such as learned gradient descent [1, 10].
In general, the obvious way to further improve the quality of our method is by
expanding the training set (which currently has only 103 training sequences).
In particular including some sequences where the back of the head is observed
should significantly improve the quality of our reconstructions for those head
parts.

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