# The spectral radius and the distance spectral radius of complements of block
graphs

Xu Chen1 İD , Dongjun Fan1, Rongxiao Shao1, Guoping Wang2111Corresponding
author. Email<EMAIL_ADDRESS>
1\. School of Statistics and Data Science, Xinjiang University of Finance and
Economics,
Ürümqi, Xinjiang 830012, P.R.China;
2\. School of Mathematical Sciences, Xinjiang Normal University,
Ürümqi, Xinjiang 830017, P.R.China

Abstract. In this paper, we determine the graphs whose spectral radius and
distance spectral radius attain maximum and minimum among all complements of
clique trees. Furthermore, we also determine the graphs whose spectral radius
and distance spectral radius attain minimum and maximum among all complements
of block graphs, respectively.

Key words:

Spectral radius; Distance spectral radius; Complements; Clique trees; Block
graphs.

MR(2020) Subject Classification:

05C12, 05C50, 05C69

##  1 Introduction

The adjacency matrix of $G$ is $A(G)=(a_{ij})_{n\times n}$, where $a_{ij}=1$
if $v_{i}$ is adjacent to $v_{j}$, and otherwise $a_{ij}=0$. Since $A(G)$ is a
real and symmetric matrix, its eigenvalues can be arranged as
$\lambda_{1}(A(G))\geq\lambda_{2}(A(G))\geq\cdots\geq\lambda_{n}(A(G))$, where
eigenvalue $\lambda_{1}(A(G))$ is called the spectral radius. Let
$d_{G}(v_{i},v_{j})$ be the least distance between $v_{i}$ and $v_{j}$ in $G$.
Then the distance matrix of $G$ is $D(G)=(d_{ij})_{n\times n}$, where
$d_{ij}=d_{G}(v_{i},v_{j})$. Since $D(G)$ is an non-negative real symmetric
matrix, its eigenvalues can be arranged
$\lambda_{1}(D(G))\geq\lambda_{2}(D(G))\geq\cdots\geq\lambda_{n}(D(G))$, where
eigenvalue $\lambda_{1}(D(G))$ is called the distance spectral radius. The
complement of graph $G=(V(G),E(G))$ is denoted by $G^{c}=(V(G^{c}),E(G^{c}))$,
where $V(G^{c})=V(G)$ and $E(G^{c})=\\{xy\notin E(G):x,y\in V(G)\\}$. The
spectral radius and distance spectral radius of complements of graphs have
been studied, see references [8, 1, 5, 2, 3, 4, 6, 7].

Let $G$ be a connected simple graph. A cut vertex of a connected graph $G$ is
a vertex whose deletion results in a disconnected graph. A clique of a graph
is a set of mutually adjacent vertices. A block of $G$ is a maximal connected
subgraph of $G$ that has no cut vertex. If each block of graph is a clique,
then the graph is called clique tree. In this paper, we determine the unique
graphs whose spectral radius and distance spectral radius attain maximum and
minimum among all complements of clique trees. Furthermore, we also determine
the unique graphs whose spectral radius and distance spectral radius
respectively attain minimum and maximum among all complements of block graphs.

##  2 The spectral radius of complements of clique trees

Suppose $G$ is a connected simple graph with the vertex set
$V(G)=\\{v_{1},v_{2},\cdots,v_{n}\\}$. If two vertices $u$ and $v$ are
adjacent, then we write $uv$. Let $x=(x_{1},x_{2},\cdots,x_{n})^{T}$, where
$x_{i}$ corresponds to $v_{i}$, i.e., $x(v_{i})=x_{i}$ for $i=1,2,\cdots,n$.
Then

$x^{T}A(G)x=2\sum_{v_{i}v_{j}\in G}{x_{i}x_{j}}$ (1)

The neighbor $N_{G}(v)$ of the vertex $v$ of $G$ is the set of the vertices
which are adjacent to $v$. Suppose that $x$ is an eigenvector of $A(G)$
corresponding to the eigenvalue $\lambda$. Then for $v_{i}\in V(G)$, we have

$\lambda x_{i}=\sum_{v_{j}\in N_{G}(v_{i})}x_{j},~{}for~{}i=1,2,\cdots,n.$ (2)

Suppose $C_{T}$ is a clique tree of order $n$ with $=s$ cliques such that some
two cut vertices are not adjacent. If a clique $K$ of $C_{T}$ contains exactly
one cut vertex $v$, then we call $K$ the end clique, and call $v$ the end cut
vertex. Suppose $\widetilde{v}$ is another cut vertex of $C_{T}$. Let

$\begin{split}\widetilde{C}_{T}=C_{T}-\\{vu|~{}u\in
V(K)\\}+\\{\widetilde{v}u|~{}u\in V(K)\setminus{v}\\}.\end{split}$

We called the graph $\widetilde{C}_{T}$ obtained from $C_{T}$ by moving the
clique $K$ from $v$ to $\widetilde{v}$.

We denote by $d(G)$ the diameter of $G$ which is farthest distance between all
pairs of vertices.

######  Lemma 2.1.

Let $C_{T}$ and $\widetilde{C}_{T}$ be two clique trees of order $n$. Set
$x=(x_{1},x_{2},\cdots,x_{n})^{T}$ to be a perron vector of $A(C_{T}^{c})$
with respect to $\lambda_{1}(A(C_{T}^{c}))$. If $x(v)\geq x(\widetilde{v})$,
then $\lambda_{1}(A(C_{T}^{c}))\leq\lambda_{1}(A(\widetilde{C}_{T}^{c}))$ with
equality if and only if $\widetilde{v}=v$.

###### Proof.

Since $C_{T}$ contains two cut vertices which are not adjacent, $d(C_{T})>3$,
and so $d(\widetilde{C}_{T})\geq 3$. Note that $A(G^{c})+A(G)=J_{n}-I_{n}$.
Let $J_{n}$ be the matrix of order $n$ whose all entries are $1$, and let
$I_{n}$ be the identity matrix of order $n$. Recall
$\lambda_{1}(A(C_{T}^{c}))$ is a spectral radius of $C_{T}^{c}$. That is each
entry of $x$ is positive. Since $x(v)\geq x(\widetilde{v})$, by equation (1)
we have

$x^{T}A(C_{T})x=2\sum_{v_{i}v_{j}\in C_{T}}{x_{i}x_{j}}\geq
2\sum_{v_{i}v_{j}\in\widetilde{C}_{T}}{x_{i}x_{j}}=x^{T}A(\widetilde{C}_{T})x.$

Then

$\begin{split}\lambda_{1}(A(C_{T}^{c}))&=x^{T}A(C_{T}^{c})x\\\
&=x^{T}(J_{n}-I_{n})x-x^{T}A(C_{T})x\\\ &\leq
x^{T}(J_{n}-I_{n})x-x^{T}A(\widetilde{C}_{T})x\\\
&=x^{T}A(\widetilde{C}_{T}^{c})x\end{split}$

By Rayleigh’s theorem we have
$x^{T}A(\widetilde{C}_{T}^{c})x\leq\lambda_{1}(A(\widetilde{C}_{T}^{c}))$.
Thus, we have
$\lambda_{1}(A(C_{T}^{c}))\leq\lambda_{1}(A(\widetilde{C}_{T}^{c}))$.

We suppose for a contradiction that $\widetilde{v}\neq v$. Note that
$\lambda_{1}(A(\widetilde{C}_{T}^{c}))=\lambda_{1}(A(C_{T}^{c}))$. By equation
(2) we have

$\begin{split}0&=|\lambda_{1}(A(\widetilde{C}_{T}^{c}))x(\widetilde{v})-\lambda_{1}(A(C_{T}^{c}))x(\widetilde{v})|\\\
&=|\big{(}(A(\widetilde{C}_{T})-A(C_{T}))x\big{)}(\widetilde{v})|.\end{split}$

Whereas
$|\big{(}(A(\widetilde{C}_{T})-A(C_{T})x\big{)}(\widetilde{v})|=\sum_{u\in
K\setminus{\\{v\\}}}x(u)>0$. This contradiction shows that the necessity
holds. ∎

Suppose $C_{T}$ contains two nonadjacent cut vertices. Then $d(C_{T})>3$. Let
$w$ and $w^{\prime}$ be two cut vertices contained in the clique $K$. Write
the clique $K^{\prime}$ containing $w^{\prime}$. Moving all cliques except
$K^{\prime}$ from their cut vertices to $w$ in $C_{T}$, we get a graph
isomorphic to the clique tree $\mathbb{S}(s-2,1)$ which exactly contains two
end cut vertices $w$ and $w^{\prime}$, and $s-1$ end cliques, as illustrated
in Figure 1.

Fig. 1. $\mathbb{S}(s-2,1)$.

######  Theorem 2.2.

Suppose $C_{T}$ is a clique tree of order $n$ with $s$ cliques such that some
two cut vertices are not adjacent. Set $x=(x_{1},x_{2},\cdots,x_{n})^{T}$ to
be a perron vector of $A(C_{T}^{c})$ with respect to
$\lambda_{1}(A(C_{T}^{c}))$. Then
$\lambda_{1}(A(C_{T}^{c}))\leq\lambda_{1}(A(\mathbb{S}^{c}(s-2,1)))$.

###### Proof.

Let $x(w)$ be the minimum modulus among all cut vertices of $C_{T}$. From the
above construction of $\mathbb{S}(s-2,1)$ and the equation (1) we have

$x^{T}A(C_{T})x=\sum_{v_{i}v_{j}\in E(C_{T})}x_{i}x_{j}\geq\sum_{v_{i}v_{j}\in
E(\mathbb{S}(s-2,1))}x_{i}x_{j}=x^{T}A(\mathbb{S}(s-2,1))x.$

Since $C_{T}$ contains two cut vertices which are not adjacent, $d(C_{T})\geq
4$, and so $d(C_{T})>d(\mathbb{S}(s-2,1))=3$. From Lemma 2.1 we have

$\begin{split}\lambda_{1}(A(C_{T}^{c}))&=x^{T}(J_{n}-I_{n})x-x^{T}A(C_{T})x\\\
&\leq x^{T}(J_{n}-I_{n})x-x^{T}A(\mathbb{S}(s-2,1))x\\\
&=x^{T}A(\mathbb{S}^{c}(s-2,1))x.\end{split}$

By Rayleigh’s theorem we have $\lambda_{1}(\mathbb{S}^{c}(s-2,1))\geq
x^{T}A(\mathbb{S}^{c}(s-2,1))x$. Thus, we have
$\lambda_{1}(A(C_{T}^{c}))\leq\lambda_{1}(\mathbb{S}^{c}(s-2,1))$. ∎

Let $\mathbb{C_{T}}_{n,d}$ denote the set of all clique trees of diameter $d$
with $s$ cliques on $n$ vertices.

######  Lemma 2.3.

Let $d\geq 3$. Then

$\mathop{\max}\limits_{C_{T}\in\mathbb{C_{T}}_{n,d}}\lambda_{1}(A(C_{T}^{c}))\geq\mathop{\max}\limits_{C_{T}\in\mathbb{C_{T}}_{n,d+1}}\lambda_{1}(A(C_{T}^{c})).$

###### Proof.

Set $x=(x_{1},x_{2},\cdots,x_{n})^{T}$ to be a perron vector of $D(C_{T}^{c})$
with respect to $\lambda_{1}(C_{T}^{c}))$. Let $x(v)\geq x(\widetilde{v})$,
and move the clique $K$ from cut vertex $v$ to $\widetilde{v}$, and then move
some one clique from their cut vertex $v^{\prime}$ ($x(v^{\prime})\geq
x(\widetilde{v})$) to $\widetilde{v}$ in $C_{T}$, we get a graph isomorphic to
the clique tree $\widetilde{C}_{T}$. Continue the above process in
$\widetilde{C}_{T}$, we until get a graph $C_{T}^{\prime}$ whose diameter less
than $d$. By applying Lemma 2.1 we obtain that the maximum modulus of
$\lambda_{1}(A(C_{T}^{c}))$ is less than the maximum modulus of
$\lambda_{1}(A(C_{T}^{\prime c}))$. Thus, the result is clear. ∎

$\mathbb{P}_{n_{1},n_{2},\cdots,n_{s}}$ is called a clique path [3] if each
edge of the path $P_{s+1}$ of order $s+1$ is replaced by a clique $K^{i}$ such
that $V(K^{i})\cap V(K^{i+1})=v_{i}$ for $i=1,2,\cdots,s-1$ and $V(K^{i})\cap
V(K^{j})=\emptyset$ for $j\neq i-1,i+1$ and $2\leq i\leq s-1$. If two graphs
$G$ and $H$ are isomorphic, then we write $G\cong H$. Applying repeatedly
Lemma 2.3 we have the following theorem.

######  Theorem 2.4.

Suppose $C_{T}$ is a clique tree of order $n$ with $s$ cliques such that some
two cut vertices are not adjacent. Then

$\lambda_{1}(A(C_{T}^{c}))\geq\lambda_{1}(A(\mathbb{P}^{c}_{n_{1},n_{2},\cdots,n_{s}}))$

with equality if and only if
$C_{T}\cong\mathbb{P}_{n_{1},n_{2},\cdots,n_{s}}$.

We denote by $T(n-3,1)$ the tree obtained from the path $P_{3}$ by appending
$n-3$ vertices to some one end of $P_{3}$. The following result are two
special cases of Theorems 2.2 and 2.4.

######  Theorem 2.5.

Suppose $T$ is a tree of order $n$ with $d(T)>3$. Then we have

$\lambda_{1}(A(P_{n}^{c}))\leq\lambda_{1}(A(T^{c}))\leq\lambda_{1}(A(T^{c}(n-3,1))).$

The equality holds if and only if $T\cong P_{n}$ and $T\cong T(n-3,1)$.

###### Proof.

Note that $T$ has exactly $n-1$ cliques. Thus, by Theorems 2.2 and 2.4 we
obtain that $\mathbb{P}_{n_{1},n_{2},\cdot,n_{n-1}}$ and $\mathbb{S}(n-2,1)$
are respectively $P_{n}$ and $T(n-3,1)$ such that $s=n-1$. Then the result is
clear. ∎

##  3 The spectral radius of the complements of block graphs

Let $B$ be a block graph of order $n$ with blocks $B^{1},B^{2},\cdots,B^{s}$.
Replacing each block $B^{i}$ of $B$ by clique $K^{i}$ of order $|V(K^{i})|$,
we get a graph isomorphic to the clique tree $C_{B}$. We denote by
$\mathbb{B}_{n,d}$ the set of all block graphs of diameter $d$ with $s$ blocks
on $n$ vertices. Set $x=(x_{1},x_{2},\cdots,x_{n})^{T}$ to be a perron vector
of $A(B^{c})$ with respect to $\lambda_{1}(A(B^{c}))$.

######  Lemma 3.1.

Suppose $B$ is a block graph of order $n$ with $s$ blocks whose diameter
$d\geq 3$. Then

$\mathop{\max}\limits_{B\in\mathbb{B}_{n,d+1}}\lambda_{1}(A(B^{c}))\geq\mathop{\max}\limits_{B\in\mathbb{B}_{n,d}}\lambda_{1}(A(B^{c})).$

###### Proof.

Deleting some edges of some one block in $B$, and then connecting the above
process, we will get a new block graph $B^{\prime}$ whose diameter greater
than $d$. Set $x=(x_{1},x_{2},\cdots,x_{n})^{T}$ to be a perron vector of
$A(B^{c})$ with respect to $\lambda_{1}(A(B^{c}))$. From the equation (1) we
have

$x^{T}A(B)x=2\sum_{v_{i}v_{j}\in B}{x_{i}x_{j}}\geq 2\sum_{v_{i}v_{j}\in
B^{\prime}}{x_{i}x_{j}}=x^{T}A(B^{\prime})x.$

Then

$\begin{split}\lambda_{1}(A(B^{c}))&=x^{T}(J_{n}-I_{n})x-x^{T}A(B)x\\\ &\leq
x^{T}(J_{n}-I_{n})x-x^{T}A(B^{\prime})x\\\ &=x^{T}A(B^{\prime
c})x.\end{split}$

By Rayleigh’s theorem we have $\lambda_{1}(A(B^{\prime c}))\geq
x^{T}A(B^{\prime c})x$, and so
$\lambda_{1}(A(B^{c}))\leq\lambda_{1}(A(B^{\prime c}))$. Thus, the result is
clear. ∎

Connecting all pairs of vertices of each block which are not adjacent in $B$,
and then applying Lemma 3.1, we will get a graph isomorphic to $C_{B}$. Then
we have the following result.

######  Lemma 3.2.

Let $B$ and $C_{B}$ be two graphs of order $n$. Set
$x=(x_{1},x_{2},\cdots,x_{n})^{T}$ to be a perron vector of $A(B^{c})$ with
respect to $\lambda_{1}(A(B^{c}))$. If $B$ contains some two cut vertices
which do not belong to the same block, then

$\lambda_{1}(A(B^{c}))\geq\lambda_{1}(A(C_{B}^{c})).$

The equality holds if and only if $B\cong C_{B}$.

Combining Theorem 2.4 and Lemma 3.2 we have the following result.

######  Theorem 3.3.

Suppose $B$ is a block graph of order $n$ with $s$ blocks such that some two
cut vertices do not belong to the same block. Then

$\lambda_{1}(A(B^{c}))\geq\lambda_{1}(A(\mathbb{P}^{c}_{n_{1},n_{2},\cdots,n_{s}})).$

The equality holds if and only if
$B\cong\mathbb{P}^{c}_{n_{1},n_{2},\cdots,n_{s}}$.

##  4 The distance spectral radius of the complements of clique trees

Suppose the matrices $A=(a_{ij})_{n\times n}$ and $B=(b_{ij})_{n\times n}$.
Then we write $A=B$ if $a_{ij}=b_{ij}$, and $A\geq B$ if $a_{ij}\geq b_{ij}$.
The below Lemma 4.1 reflects the relationship of $D(G^{c})$ and $A(G)$.

######  Lemma 4.1 (6, Lemma 2.1).

Suppose $G$ is a simple graph on $n$ vertices whose diameter $d(G)$ is greater
than two. Then we have

  1. (I.)

when $d(G)>3$, $D(G^{c})=J_{n}-I_{n}+A(G)$.

  2. (II.)

when $d(G)=3$, $D(G^{c})\geq J_{n}-I_{n}+A(G)$.

######  Lemma 4.2.

Let $C_{T}$ and $\widetilde{C}_{T}$ be two clique trees of order $n$. Set
$x=(x_{1},x_{2},\cdots,x_{n})^{T}$ to be a perron vector of $D(C_{T}^{c})$
with respect to $\lambda_{1}(D(C_{T}^{c}))$. If $x(\widetilde{v})\geq x(v)$,
then $\lambda_{1}(D(C_{T}^{c}))\leq\lambda_{1}(D(\widetilde{C}_{T}^{c}))$ with
equality if and only if $\widetilde{v}=v$.

###### Proof.

Since $C_{T}$ contains two cut vertices which are not adjacent, $d(C_{T})>3$,
and so $d(\widetilde{C}_{T})\geq 3$. By Lemma 4.1 and equation (1) we have

$\begin{split}\lambda_{1}(D(\widetilde{C}_{T}^{c}))-\lambda_{1}(D(C_{T}^{c}))&\geq
x^{T}(D(\widetilde{C}_{T}^{c})-D(C_{T}^{c}))x\\\
&=x^{T}(A(\widetilde{C}_{T})-A(C_{T}))x\\\
&=2(x(\widetilde{v})-x(v))\sum_{u\in K\setminus{\\{v\\}}}x(u)\\\ &\geq
0\end{split}$

Thus, we have
$\lambda_{1}(D(C_{T}^{c}))\leq\lambda_{1}(D(\widetilde{C}_{T}^{c}))$.

We suppose for a contradiction that $\widetilde{v}\neq v$. Note that
$\lambda_{1}(D(\widetilde{C}_{T}^{c}))=\lambda_{1}(D(C_{T}^{c}))$. From
equation (2) we have

$\begin{split}0&=|\lambda_{1}(D(\widetilde{C}_{T}^{c}))x(\widetilde{v})-\lambda_{1}(D(C_{T}^{c}))x(\widetilde{v})|\\\
&=|\big{(}(A(\widetilde{C}_{T})-A(C_{T}))x\big{)}(\widetilde{v})|.\end{split}$

Whereas
$|\big{(}(A(\widetilde{C}_{T})-A(C_{T})x\big{)}(\widetilde{v})|=\sum_{u\in
K\setminus{\\{v\\}}}x(u)>0$. This contradiction shows that the necessity
holds. ∎

The proof is similar to the proof of Lemma 2.4. Applying repeatedly Lemma 4.2
we have the following result.

######  Lemma 4.3.

Let $d\geq 3$. Then

$\mathop{\max}\limits_{C_{T}\in\mathbb{C_{T}}_{n,d}}\lambda_{1}(D(C_{T}^{c}))\geq\mathop{\max}\limits_{C_{T}\in\mathbb{C_{T}}_{n,d+1}}\lambda_{1}(D(C_{T}^{c})).$

Applying repeatedly Lemma 4.3 we have the following result.

######  Lemma 4.4.

Suppose $C_{T}$ is a clique tree of order $n$ with $s$ cliques such that some
two cut vertices are not adjacent. Then

$\lambda_{1}(D(C_{T}^{c}))\geq\lambda_{1}(D(\mathbb{P}^{c}_{n_{1},n_{2},\cdots,n_{s}}))$

with equality if and only if
$C_{T}\cong\mathbb{P}_{n_{1},n_{2},\cdots,n_{s}}$.

######  Theorem 4.5.

Suppose $C_{T}$ is a clique tree of order $n$ with $s$ cliques such that some
two cut vertices are not adjacent. Set $x=(x_{1},x_{2},\cdots,x_{n})^{T}$ to
be a perron vector of $D(C_{T}^{c})$ with respect to
$\lambda_{1}(D(C_{T}^{c}))$. Then
$\lambda_{1}(D(C_{T}^{c}))\leq\lambda_{1}(D(\mathbb{S}^{c}(s-2,1)))$.

###### Proof.

Let $x(w)$ be the maximum modulus among all cut vertices of $C_{T}$. From the
above construction of $\mathbb{S}(s-2,1)$ and the equation (1) we have

$x^{T}A(C_{T})x=\sum_{v_{i}v_{j}\in E(C_{T})}x_{i}x_{j}\leq\sum_{v_{i}v_{j}\in
E(\mathbb{S}(s-2,1))}x_{i}x_{j}=x^{T}A(\mathbb{S}(s-2,1))x.$

Since $C_{T}$ contains two cut vertices which are not adjacent, $d(C_{T})\geq
4$, and so $d(C_{T})>d(\mathbb{S}(s-2,1))=3$. From Lemma 4.1 we have

$\begin{split}\lambda_{1}(D(C_{T}^{c}))&=x^{T}D(C_{T}^{c})x\\\
&=x^{T}(J_{n}-I_{n})x+x^{T}A(C_{T})x\\\ &\leq
x^{T}(J_{n}-I_{n})x+x^{T}A(\mathbb{S}(s-2,1))x\\\
&=x^{T}D(\mathbb{S}^{c}(s-2,1))x.\end{split}$

By Rayleigh’s theorem we have $\lambda_{1}(\mathbb{S}^{c}(s-2,1))\geq
x^{T}D(\mathbb{S}^{c}(s-2,1))x$. Thus,
$\lambda_{1}(D(C_{T}^{c}))\leq\lambda_{1}(\mathbb{S}^{c}(s-2,1))$. ∎

The proof is similar to the proof of Lemma 2.5, and the following result are
two special cases of Theorems 4.4 and 4.5.

######  Theorem 4.6.

Suppose $T$ is a tree of order $n$ with $d(T)>3$. Then we have

$\lambda_{1}(D(P^{c}_{n}))\leq\lambda_{1}(D(T^{c}))\leq\lambda_{1}(D(T^{c}(n-3,1))).$

The equality holds if and only if $T\cong P_{n}$ and $T\cong T(n-3,1)$.

##  5 The distance spectral radius of the complements of block graphs

######  Lemma 5.1.

Let $B$ and $C_{B}$ be two graphs of order $n$. Set
$x=(x_{1},x_{2},\cdots,x_{n})^{T}$ to be a perron vector of $D(B^{c})$ with
respect to $\lambda_{1}(D(B^{c}))$. If $B$ contains some two cut vertices
which do not belong to the same block, then

$\lambda_{1}(D(B^{c}))\leq\lambda_{1}(D(C_{B}^{c})).$

The equality holds if and only if $B\cong C_{B}$.

###### Proof.

Connecting all pairs of vertices in $B^{i}$ ($i=1,2,\cdots,n$) which are not
adjacent in $B$, we get a graph isomorphic to $C_{B}$. Obviously, from
equation (1) we have $x^{T}A(B)x=\sum_{v_{i}v_{j}\in
E(B)}x_{i}x_{j}\leq\sum_{v_{i}v_{j}\in E(C_{B})}x_{i}x_{j}=x^{T}A(C_{B})x$.

Since $B$ contains some two cut vertices which do not belong to the same
block, we have $d(B)\geq d(C_{B})>3$. From Lemma 4.1 we have

$\begin{split}\lambda_{1}(D(B^{c}))&=x^{T}D(B^{c})x\\\
&=x^{T}(J_{n}-I_{n})x+x^{T}A(B)x\\\ &\leq
x^{T}(J_{n}-I_{n})x+x^{T}A(C_{B})x\\\ &\leq x^{T}D(C_{B}^{c})x.\end{split}$

By Rayleigh’s theorem we have $\lambda_{1}(C_{B}^{c})\geq x^{T}D(C_{B}^{c})x$.
Then $\lambda_{1}(D(B^{c}))\leq\lambda_{1}(D(C_{B}^{c}))$.

Suppose for a contradiction that $B\not\cong C_{B}$. Note that
$\lambda_{1}(D(B^{c}))=\lambda_{1}(D(C_{B}^{c}))$. Then we have

$\begin{split}0&=\lambda_{1}(D(C_{B}^{c}))-\lambda_{1}(D(B^{c}))\\\
&=x^{T}(A(C_{B})-A(B))x\\\
&=\sum_{v_{i}v_{j}\in(E(C_{B})-E(B))}x_{i}x_{j}.\end{split}$

By hypothesis we have $E(C_{B})-E(B)\neq\emptyset$, and so
$\sum_{v_{i}v_{j}\in(E(C_{B})-E(B))}x_{i}x_{j}>0$. Thus, the contradiction
shows that the necessity holds. ∎

Combining Theorem 4.5 and Lemma 5.1 we get the following result.

######  Theorem 5.2.

Suppose $B$ is a block graph of order $n$ with $s$ blocks such that some two
cut vertices do not belong to the same block. Then

$\lambda_{1}(D(B^{c}))\leq\lambda_{1}(D(\mathbb{S}^{c}(s-2,1))).$

The equality holds if and only if $B\cong\mathbb{S}(s-2,1)$.

## References

  * [1] G. Jiang, G. Yu, W. Sun, Z. Ruan. The least eigenvalue of graphs whose complements have only two pendent vertices[J]. Applied Mathematics and Computation, 2018, 331: 112-119.
  * [2] G. Yu, Y. Fan, M. Ye, The least signless Laplacian eigenvalue of the complements of unicyclic graphs[J]. Applied Mathematics and Computation, 2017, 306: 13-21.
  * [3] H. Lin, S. Drury, The distance spectrum of complements of trees[J]. Linear Algebra and its Applications, 2017, 530: 185-201.
  * [4] R. Qin, D. Li, Y. Chen, J. Meng, The distance eigenvalues of the complements of unicyclic graphs[J]. Linear Algebra and its Applications, 2020, 598: 49-67.
  * [5] S. Li, S. Wang, The least eigenvalue of the signless Laplacian of the complements of trees[J]. Linear Algebra and its applications, 2012, 436(7): 2398-2405.
  * [6] X. Chen, G. Wang. The distance spectrum of the complements of graphs of diameter greater than three[J]. Indian Journal of Pure and Applied Mathematics, 2022, https://doi.org/10.1007/s13226-022-00315-9.
  * [7] X. Chen, G. Wang. The distance spectrum of the complements of graphs with two pendent vertices[J], Indian Journal of Pure and Applied Mathematics, 2022, https://doi.org/10.1007/s13226-022-00322-w.
  * [8] Y. Fan, F. Zhang, Y. Wang, The least eigenvalue of the complements of trees[J]. Linear algebra and its applications, 2011, 435(9): 2150-2155.


Permanent address: ]INFN Sezione di Lecce - Via Arnesano - 73100 Lecce, Italy

LUNA Collaboration

# Direct measurement of the 13C$(\alpha$,n$)^{16}$O cross section into the
$s$-process Gamow peak

G.F. Ciani  Gran Sasso Science Institute, Viale F. Crispi 7, 67100 L’Aquila,
Italy  INFN, Laboratori Nazionali del Gran Sasso (LNGS), 67100 Assergi, Italy
Institute for Nuclear Research (ATOMKI), PO Box 51, HU-4001 Debrecen, Hungary
L. Csedreki  Gran Sasso Science Institute, Viale F. Crispi 7, 67100 L’Aquila,
Italy  INFN, Laboratori Nazionali del Gran Sasso (LNGS), 67100 Assergi, Italy
Institute for Nuclear Research (ATOMKI), PO Box 51, HU-4001 Debrecen, Hungary
D. Rapagnani  Università di Napoli “Federico II”, 80126 Napoli, Italy  INFN,
Sezione di Napoli, 80126 Napoli, Italy  M. Aliotta  SUPA, School of Physics
and Astronomy, University of Edinburgh, EH9 3FD Edinburgh, United Kingdom  J.
Balibrea-Correa  Università di Napoli “Federico II”, 80126 Napoli, Italy
INFN, Sezione di Napoli, 80126 Napoli, Italy  F. Barile  Università degli
Studi di Bari, 70121 Bari, Italy  INFN, Sezione di Bari, 70125 Bari, Italy  D.
Bemmerer  Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstr. 400, 01328
Dresden, Germany  A. Best  Corresponding author<EMAIL_ADDRESS>Università di Napoli “Federico II”, 80126 Napoli, Italy  INFN, Sezione di
Napoli, 80126 Napoli, Italy  A. Boeltzig  Università di Napoli “Federico II”,
80126 Napoli, Italy  INFN, Sezione di Napoli, 80126 Napoli, Italy  C. Broggini
INFN, Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy  C.G. Bruno
SUPA, School of Physics and Astronomy, University of Edinburgh, EH9 3FD
Edinburgh, United Kingdom  A. Caciolli  INFN, Sezione di Padova, Via F.
Marzolo 8, 35131 Padova, Italy  Università degli Studi di Padova, Via F.
Marzolo 8, 35131 Padova, Italy  F. Cavanna  INFN, Sezione di Torino, Via
Pietro Giuria, 1, 10125 Torino, Italy  T. Chillery  SUPA, School of Physics
and Astronomy, University of Edinburgh, EH9 3FD Edinburgh, United Kingdom  P.
Colombetti  INFN, Sezione di Torino, Via Pietro Giuria, 1, 10125 Torino, Italy
P. Corvisiero  Università degli Studi di Genova, 16126 Genova, Italy  INFN,
Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy  S. Cristallo  INAF,
Osservatorio Astronomico d’Abruzzo, Teramo, Italy  INFN, Sezione of Perugia,
Via A. Pascoli snc, 06123 Perugia, Italy  T. Davinson  SUPA, School of Physics
and Astronomy, University of Edinburgh, EH9 3FD Edinburgh, United Kingdom  R.
Depalo  Università degli Studi di Padova, Via F. Marzolo 8, 35131 Padova,
Italy  INFN, Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy  A. Di
Leva  Università di Napoli “Federico II”, 80126 Napoli, Italy  INFN, Sezione
di Napoli, 80126 Napoli, Italy  Z. Elekes  Institute for Nuclear Research
(ATOMKI), PO Box 51, HU-4001 Debrecen, Hungary  F. Ferraro  Università degli
Studi di Genova, 16126 Genova, Italy  INFN, Sezione di Genova, Via Dodecaneso
33, 16146 Genova, Italy  E. Fiore  Università degli Studi di Bari, 70121 Bari,
Italy  INFN, Sezione di Bari, 70125 Bari, Italy  A. Formicola  Corresponding
author<EMAIL_ADDRESS>INFN, Laboratori Nazionali del Gran Sasso
(LNGS), 67100 Assergi, Italy  Zs. Fülöp  Institute for Nuclear Research
(ATOMKI), PO Box 51, HU-4001 Debrecen, Hungary  G. Gervino  Università degli
Studi di Torino, 10125 Torino, Italy  INFN, Sezione di Torino, Via P. Giuria
1, 10125 Torino, Italy  A. Guglielmetti  Università degli Studi di Milano,
20133 Milano, Italy  INFN, Sezione di Milano, Via G. Celoria 16, 20133 Milano,
Italy  C. Gustavino  INFN, Sezione di Roma La Sapienza, Piazzale A. Moro 2,
00185 Roma, Italy  Gy. Gyürky  Institute for Nuclear Research (ATOMKI), PO Box
51, HU-4001 Debrecen, Hungary  G. Imbriani  Università di Napoli “Federico
II”, 80126 Napoli, Italy  INFN, Sezione di Napoli, 80126 Napoli, Italy  M.
Junker  INFN, Laboratori Nazionali del Gran Sasso (LNGS), 67100 Assergi, Italy
M. Lugaro  Institute of Physics, ELTE Eötvös Loránd University, Budapest,
Hungary  Konkoly Observatory, Research Centre for Astronomy and Earth
Sciences, MTA Centre for Excellence, Budapest, Hungary  P. Marigo  INFN,
Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy  Università degli
Studi di Padova, Via F. Marzolo 8, 35131 Padova, Italy  E. Masha  Università
degli Studi di Milano, 20133 Milano, Italy  INFN, Sezione di Milano, Via G.
Celoria 16, 20133 Milano, Italy  R. Menegazzo  INFN, Sezione di Padova, Via F.
Marzolo 8, 35131 Padova, Italy  V. Mossa  INFN, Sezione di Bari, 70125 Bari,
Italy  F.R. Pantaleo  Università degli Studi di Bari, 70121 Bari, Italy  INFN,
Sezione di Bari, 70125 Bari, Italy  V. Paticchio  INFN, Sezione di Bari, 70125
Bari, Italy  R. Perrino  [  INFN, Sezione di Bari, 70125 Bari, Italy  D.
Piatti  INFN, Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy
Università degli Studi di Padova, Via F. Marzolo 8, 35131 Padova, Italy  P.
Prati  Università degli Studi di Genova, 16126 Genova, Italy  INFN, Sezione di
Genova, Via Dodecaneso 33, 16146 Genova, Italy  L. Schiavulli  Università
degli Studi di Bari, 70121 Bari, Italy  INFN, Sezione di Bari, 70125 Bari,
Italy  K. Stöckel  Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstr.
400, 01328 Dresden, Germany  Technische Universität Dresden, Institut für
Kern- und Teilchenphysik, Zellescher Weg 19, 01069 Dresden, Germany  O.
Straniero  INAF, Osservatorio Astronomico d’Abruzzo, Teramo, Italy  INFN,
Laboratori Nazionali del Gran Sasso (LNGS), 67100 Assergi, Italy  T. Szücs
Institute for Nuclear Research (ATOMKI), PO Box 51, HU-4001 Debrecen, Hungary
M.P. Takács  Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstr. 400,
01328 Dresden, Germany  Technische Universität Dresden, Institut für Kern- und
Teilchenphysik, Zellescher Weg 19, 01069 Dresden, Germany  F. Terrasi
Università degli Studi della Campania “L. Vanvitelli” (Caserta), 81100
Caserta, Italy  INFN, Sezione di Napoli, 80126 Napoli, Italy  D. Vescovi
INFN, Sezione of Perugia, Via A. Pascoli snc, 06123 Perugia, Italy  Goethe
University, Max-von-Laue-Strasse 1, Frankfurt am Main 60438, Germany  S.
Zavatarelli  INFN, Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy

###### Abstract

One of the main neutron sources for the astrophysical $s$-process is the
reaction 13C$(\alpha$,n$)^{16}$O, taking place in thermally pulsing Asymptotic
Giant Branch stars at temperatures around 90 MK. To model the nucleosynthesis
during this process the reaction cross section needs to be known in the
150-230 keV energy window (Gamow peak). At these sub-Coulomb energies cross
section direct measurements are severely affected by the low event rate,
making us rely on input from indirect methods and extrapolations from higher-
energy direct data. This leads to an uncertainty in the cross section at the
relevant energies too high to reliably constrain the nuclear physics input to
$s$-process calculations. We present the results from a new deep-underground
measurement of 13C$(\alpha$,n$)^{16}$O, covering the energy range 230-300 keV,
with drastically reduced uncertainties over previous measurements and for the
first time providing data directly inside the $s$-process Gamow peak. Selected
stellar models have been computed to estimate the impact of our revised
reaction rate. For stars of nearly solar composition, we find sizeable
variations of some isotopes, whose production is influenced by the activation
of close-by branching points that are sensitive to the neutron density, in
particular the two radioactive nuclei 60Fe and 205Pb, as well as 152Gd.

Low-mass Asymptotic Giant Branch (AGB) stars are major production sites of
heavy elements in the Universe (for a recent review see [1]). Their interior
contains a carbon-oxygen core surrounded by a thin He-rich mantel and a H-rich
envelope. Periodically, these stars undergo thermonuclear instabilities, He-
burning flashes, called thermal pulses (TPs). Each He-flash generates a large
convective zone that mixes the C produced by the triple-$\alpha$ reaction up
to the top of the He mantel. Later on, the shell-H burning, always active at
the base of the envelope, dies down and, in turn, the external convection
penetrates the He-rich mantel. This phenomenon, which moves the
nucleosynthesis yields up to the stellar surface, is called the _third dredge
up_. As early recognized, the creation of a 13C-pocket within the He-rich
mantel, through the reaction 12C(p,$\gamma)^{13}$N$(\beta^{+})^{13}$C, is a
byproduct of these recursive mixing episodes [2, 3]. Such a thin pocket (a few
$10^{-5}$M⊙ of 13C) harbors one of the most important nucleosynthesis sites in
the Universe. During the period between two TPs, the temperature attains about
90 MK and 13C is activated as a neutron source through the reaction
13C$(\alpha$,n$)^{16}$O. This process provides a relatively slow neutron flux
($\approx 10^{7}$ neutrons/s/cm2) for about $10^{4}$ years each time. Starting
from seed nuclei in the iron region, this neutron flux slowly builds up heavy
elements along the line of stability [4]. This $s$-process (_slow-neutron-
capture_) is responsible for the production of about half of all the heavy
elements (A$\geq 90$) in the Universe.

In order to constrain this important nucleosynthesis process, the cross
section of the 13C$(\alpha$,n$)^{16}$O neutron source needs to be known in the
astrophysical energy window (the Gamow peak) around $E_{0}$= 150-230 keV111All
energies in this manuscript are center-of-mass energies, unless explicitly
stated otherwise.. The available direct cross section data in the lower energy
interval $280<E<350$ keV [5, 6] are affected by uncertainties $\geq 40\%$. Any
effort of pushing direct reaction measurements to lower energies is basically
rendered futile by the steep drop of the cross section and the presence of
natural and instrumental backgrounds. Extrapolation of the cross section into
the Gamow peak is further complicated by a broad state at $E_{x}=6356(8)$ keV
($J^{\pi}=\frac{1}{2}^{+}$) just near the $\alpha$-threshold in 17O
(S${}_{\alpha}=6359$ keV [7]). The cross section evaluation in the Gamow peak
requires a careful matching between the tail of this near-threshold state and
the higher-energy experimental data. While this state has been the focus of
great experimental attention over the past years [8, 9, 10, 11, 12] its
influence remains a major source of uncertainty for the $s$-process [13] and
the need for more cross section data to fill the gap has been frequently
expressed [14, 13].

In order to provide direct data at low energies to better constrain the cross
section inside the Gamow peak the LUNA collaboration has performed an
intensive experimental campaign at the deep-underground accelerator LUNA400
[15] inside the Gran Sasso National Laboratory (LNGS). The LNGS neutron
background flux ($\approx 10^{-7}$n/cm2/s) [16]) is dominated by the natural
radioactivity of the surrounding rock and it is up to four orders of magnitude
lower than on the surface of the Earth, improving the sensitivity over
previous studies.

The experimental setup and the target analysis are described in detail in
refs. [17, 18, 19], following is a brief summary. The accelerator provided a
He+ beam on target of up to 150 $\mu$A with $\alpha$ energies of 305-400 keV,
corresponding to c.m. energies in the range 233-306 keV. Deposited charges
vary from 15 C at the highest to 90 C at the lowest energy. The beam impinged
upon water-cooled targets made of an 99% enriched 13C evaporated on a 0.2 mm
thick Ta backing [17].

Near the target, the beam passed a liquid nitrogen cooled shroud and an
electrically insulated collimator at a negative voltage of 300 V (to suppress
the effects of secondary electrons). The neutrons produced by the
13C($\alpha$,n)16O reaction, with energies around 2.4 MeV, were thermalized in
a polyethylene moderator and detected by 18 3He-filled proportional counters
with stainless steel housing. Two geometrical detector configurations (a
vertical and a horizontal orientation) were used to optimize the absolute
neutron detection efficiency, $34\pm 3$% and $38\pm 3$%, respectively [18].
Moreover, the measurement reproducibility was checked by separately analysing
datasets acquired with the two detector configurations at the same energy,
finding agreement in the final results. The detector signals were digitized
with a 100 MHz sampling rate and analyzed offline to suppress the internal
background from $\alpha$-decays in the steel housing of the counters using a
custom pulse-shape discrimination (PSD) technique [20]. The PSD in combination
with the borated polyethylene shielding (5 and 10 inches, depending on the
setup) around the detector as well as the underground location results in a
total background rate inside the neutron signal region of $1.2\pm 0.1$
counts/hour for both configurations, leading to an improvement by more than
two orders of magnitude compared with previous experiments.

Blank backings were irradiated obtaining a background level of 1.3$\pm$ 0.2
counts/hour, in agreement with the environmental one. The conclusion was that
beam induced background was negligible. The evaporated 13C targets were
characterized in term of homogeneity and thickness immediately after the
evaporation at ATOMKI by means of the 13C(p,$\gamma)^{14}$N resonance at
$E_{\rm{r,lab}}$= 1748 keV. The average target thickness at resonance energy
was 5 keV, corresponding to 170 nm. Because of the cross section reduction of
one order of magnitude, the differential neutron yield becomes negligible from
reactions occurring beyond 150 nm inside the target, so all the targets can be
considered to be of the same effective thickness. At LUNA, target quality was
frequently checked (every 1.5 C accumulated charge) using direct $\gamma$-ray
measurements of the 13C(p,$\gamma)^{14}$N reaction [17].

The cross section was calculated as follows:

$Y=\frac{\eta(E_{\alpha})}{Qe}\int^{E_{\rm\alpha}}_{E_{\alpha}-\Delta
E}\frac{\sigma(E)}{\epsilon_{\rm{eff}}(E)}dE\quad,$ (1)

where $Y$ is the number of detected neutrons per projectile (PSD corrected),
$\eta$ the neutron detection efficiency, $Qe$ is the incident number of
particles on the target, $E_{\alpha}$ the beam energy,
$\epsilon_{\rm{eff}}(E)$ the effective stopping power and $\Delta E$ the
projectile energy loss in the target.

The maximum accumulated $\alpha$-charge on each target was limited to 3 C,
corresponding to at most a 30% degradation.The lowest energies $E=245$ and
$E=233$ keV required special attention, as the statistics collected during a
single run was insufficient to obtain a reliable estimate of the cross
section. Two independent approaches were used in the analysis. In the first
one, all runs at the same energy with similar target degradation levels were
summed together and the cross sections for each “subset” were calculated and
combined to the cross section for this energy.In the second one, the two
lowest energies were also analyzed using a Bayesian approach [21], see the
supplemental material for details.

In Figure 1, the 233 keV cross sections as a function of target degradation
are compared with the Bayesian results. The mean values are in agreement; the
“grouping” method was used for the extraction of the cross sections presented
later in this work.

Figure 1: Comparison of the cross section at 233 keV for different target
degradation and the Bayesian results. The solid and dashed lines represent the
weighted average of the groups and the corresponding uncertainty,
respectively.

The experimental results are summarized in Table 1, where we also show the
$S$-factor222The astrophysical $S$-factor is defined as
$\sigma(E)=S(E)\frac{1}{E}e^{-2\pi\eta}$ and is commonly used to remove the
strong energy dependence in the cross section due to the Coulomb repulsion.
$\eta=0.1575Z_{a}Z_{b}\sqrt{\frac{1}{E}\frac{M_{a}M_{b}}{M_{a}+M_{b}}}$ is the
Sommerfeld parameter. after correcting for the electron screening effect333A
screening potential of 937 V was used for the calculations: the corresponding
correction is less then 10% for all the energies. (bare in Table 1), following
refs. [22, 23, 24]. The Table includes statistical and systematic
uncertainties. The latter are: 3% for the charge integration, 5% for stopping
power calculated with SRIM-2010 [25] and about 8% for the detection
efficiency. The uncertainty of the beam energy is around 300 eV, and the beam
energy spread is less than 100 eV [15].

Table 1: Experimental and electron screening-corrected $S$-factors. The data format is data$\pm$stat$\pm$syst, where syst is a common systematic uncertainty of 10 $\%$ (further details in the text). Energy |  $S$-factor |  $S$-factor
---|---|---
[keV] | [105MeV b] | [105MeV b]
|  | (bare)
306 | 8.06 $\pm$ 0.18 $\pm$ 0.8 | 7.61 $\pm$ 0.17 $\pm$ 0.8
298 | 8.1 $\pm$ 0.3 $\pm$ 0.8 | 7.6 $\pm$ 0.3 $\pm$ 0.8
291 | 7.3 $\pm$ 0.3 $\pm$ 0.7 | 6.8 $\pm$ 0.3 $\pm$ 0.7
283 | 8.6 $\pm$ 0.3 $\pm$ 0.9 | 8.0 $\pm$ 0.3 $\pm$ 0.8
275 | 9.2 $\pm$ 0.6 $\pm$ 0.9 | 8.6 $\pm$ 0.5 $\pm$ 0.9
260 | 8.7 $\pm$ 0.8 $\pm$ 0.9 | 8.1 $\pm$ 0.7 $\pm$ 0.8
245 | 11.7 $\pm$ 1.7 $\pm$ 1.2 | 10.8 $\pm$ 1.5 $\pm$ 1.1
233 | 12.7 $\pm$ 2.3 $\pm$ 1.3 | 11.6 $\pm$ 2.1 $\pm$ 1.2

The new data extend the energy range covered by direct cross-section
measurements into the $s$-process Gamow peak, but an extrapolation towards
zero energy, also taking into account the near-threshold state ($E_{x}=6356$
keV, J${}^{\pi}=\frac{1}{2}^{+}$), is still required. This was done with an
R-matrix analysis using the code Azure2 [26]. The low-energy cross section is
dominated by two broad states, the already mentioned near-threshold state and
a $\frac{3}{2}^{+}$, $E_{x}=7215$ keV one. Narrow resonances in the energy
range covered by the analysis ($E<1.2$ MeV) do only have very localized
effects on the cross sections and were omitted. As the threshold state was
assumed to be $\alpha$-bound, an asymptotic normalization coefficient instead
of an $\alpha$ partial width was used [10]. Energy and width or asymptotic
normalization constant (ANC) of the threshold state were kept fixed (note that
they were varied in the Monte Carlo analysis, see below). Channel radii of
4.15 fm and 6.684 fm were used for the neutron and $\alpha$ channels,
respectively. In addition to the cross sections from Heil and Drotleff, the
data by Harissopulos [27] were included in the analysis, as they cover a wider
energy range and help better constrain the $\approx$ 800 keV resonance. A
normalization factor of 1.37 was applied to the latter to match the absolute
scales of the different data sets. Two high-energy poles were included to take
into account the influence of higher-lying resonances on the cross section.
Their widths and that of the $E_{x}$ = 7215 keV state were kept free. The
parameters resulting from the R-matrix fit are given in Table 2. The top panel
of Figure 2 shows the experimental $S$ factors and the R-matrix fits with
(solid line) and without (dashed line) our new LUNA data.

Table 2: R-matrix parameters and data sets. $E_{x}$ | $\Gamma_{n}$ |  $\Gamma_{\alpha}|$ANC | Dataset | Norm.
---|---|---|---|---
$[$keV$]$ |  $[$keV$]$ |  $[$keV]$|$[fm-1/2 $]$
6356 | 124 | $5.44\times 10^{90}$ | Drotleff [5] | 1
7215 | 305.3 | $9.75\times 10^{-2}$ | Heil [6] | 1
15000 | $2.42\times 10^{4}$ | $6.04\times 10^{5}$ | Harissopulos [27] | 1.37
15000 | $4.33\times 10^{2}$ | $6.02\times 10^{5}$ | This work | 1

The uncertainties in the final cross section were investigated using a Monte
Carlo (MC) approach. The ANC and the neutron partial width of the threshold
state, as well as the absolute scales of the four experimental data sets, were
randomly sampled from a Gaussian distribution according to their respective
uncertainties. To be conservative, a 12% uncertainty was assumed for the
absolute scale of each data set. 30000 MC variations were evaluated and the
R-matrix cross sections of each run were saved for later processing. The
density map of the results and traditional 1 $\sigma$ contours are displayed
in Figure 2. As mentioned above, the cross sections from [27] are rescaled to
match the [5] and [6] data, as in other recent papers. However, there are no
strong motivations for doing so, and one could choose to instead base the
normalization on the Harissopulos et al. data (as suggested by a recent
measurement [28]). To investigate the effect of the two different
normalizations we performed R-matrix calculation using data by Harissopulos et
al. as a reference for the normalization of [5] and [6]. Inside the Gamow
peak, the effect is only of the order of 5%, increasing towards higher
energies. The absolute scale of the normalization of the historical data is
still a matter for debate. Therefore in the MC procedure, only for the sake of
the determination of a lower limit, we also considered the case (for half of
the total trials) of using unscaled Harissopulos et al. [27] data, while the
[5] and [6] normalizations were changed accordingly. It is worth noting that
this additional source of uncertainty contributes only marginally, about 5%,
at the $s$-process energies, where the cross section is well constrained by
the present data, while it has a larger impact at higher energies444New
measurements of the higher-energy cross section are planned at various
facilities worldwide, including with the upcoming LUNA MV accelerator at the
LNGS[29], addressing this uncertainty.. As will be discussed below, the
establishment of a reliable lower limit is crucial for the determination of
possible nucleosynthesis variations. In total, three different rates were
calculated: “LUNA”, using the R-matrix best fit including the new cross
sections, “no-LUNA”, the best fit without the new data, and “low-LUNA” using
the 5th percentile of the fit adopting the Harissopulos normalization. The
“no-LUNA”, and “low-LUNA” cover approximately the $\pm$95% percentiles around
the best fit.

(a)

(b)

Figure 2: Top: Astrophysical $S$-factor of 13C$(\alpha$,n$)^{16}$O. The lines
show the results of two R-matrix analyses, with and without the new data. The
Harissopulos dataset was normalized according to value in Table 2. The blue
curve is the “Gamow peak” at 90 MK and the right y axis refers to its relative
scale. Bottom: Monte Carlo probability density for the $S$-factor.

Finally, the astrophysical reaction rate $R=N_{A}\langle\sigma v\rangle$ as a
function of stellar temperature was calculated (in units of
$\text{cm}^{3}\text{mol}^{-1}\text{s}^{-1}$) by integration of the R-matrix
cross section:

$R=\frac{3.7318}{T_{9}^{3/2}}\sqrt{\frac{M_{0}+M_{1}}{M_{0}M_{1}}}\times\int_{0}^{\infty}E\sigma(E)e^{-11.605E/T_{9}}dE\enspace,$
(2)

where $T_{9}$ is the temperature in GK, the energy is given in MeV, the cross
section $\sigma$ in b and the $M_{i}$ are the atomic masses of the reaction
partners. Probability density functions for the rate were generated based on
the results of the various MC cross sections. The contributions from the
narrow resonances were as usual summed to the final rate. Tabulated results
are shown in the supplemental material accompanying this Letter.

In the $s$-process in AGB stars [30, 3], shortly after a third-dredge-up
episode, a 13C-pocket forms within the He-rich mantel. During the following
interpulse period, this region heats up and, around $\sim$ 80-100 MK, the
13C$(\alpha$,n$)^{16}$O reaction starts to release neutrons. The high neutron
exposure ($\sim 0.4$ mb-1 in solar metallicity stars) coupled to a low neutron
density (a few $10^{6}$ n/cm3) are the two major features of the resulting
$s$-process nucleosynthesis. The first ensures a neutron flux over a time long
enough to produce a large overabundance of all the elements belonging to the
main component of the $s$-process ($A>90$), while the second favors
$\beta^{-}$ decays over neutron captures at the various branching points of
the $s$-process path. In most cases, the 13C-pocket is fully consumed during
the interpulse period. However, if a small amount of 13C survives (i.e., if
the reaction rate is low enough), it will be engulfed into the convective
shell powered by the subsequent thermal pulse and burned at a higher
temperature ($\sim 200$ MK) [31]. This second (convective) neutron burst is
characterized by a higher neutron density ($>10^{9}$ n/cm3) but much lower
neutron exposure, than the main (radiative) event. As a consequence, it does
not modify the bulk of stellar yields, but may affect some key isotopes at the
$s$-process branching points [32]. Extant models of low-mass AGB stars have
shown that this second neutron burst may occur during the first few thermal
pulses, in stars with metallicity Z$\geq 0.01$ and that its efficiency is
sensitive to the adopted 13C$(\alpha$,n$)^{16}$O reaction rate [14].

In order to evaluate the impact of the new reaction rate on the $s$-process,
we have calculated three models of an AGB star with mass M=2 M⊙, metallicity
Z$=0.02$ and Y$=0.27$, under the three different assumptions (“LUNA” and the
$\sim\pm$ 95 % values corresponding to “no-LUNA” and “low-LUNA”) for the
13C$(\alpha$,n$)^{16}$O rate mentioned above. In all the three models, some
13C survives at the end of the first two interpulse periods and is burned at
high temperature in the convective thermal pulse. Stronger effects of this
second neutron burst are expected for the low-LUNA case. The results are
compared in Figure 3. More details on the stellar models are given in the
supplemental material, which includes Refs. [33, 34, 35].

Figure 3: Percentage variations of the mass fractions of heavy isotopes in the
low-LUNA model with respect to the NO-LUNA model. In both models, the final
surface composition is the cumulative result of 11 third dredge up episodes.
The 60Fe variation (+102%) is out of the scale and marked by an arrow.

Most of the nuclei belonging to the main component are depressed when the
13C$(\alpha$,n$)^{16}$O rate is lowered. This is the natural consequence of
the suppression of the radiative $s$-process. The reduction of the surface
abundances is stronger for the heavier isotopes ($A>130$). The heavy-s (Ba,
La, Ce, Nd, Sm) to light-s (Sr, Y, Zr) abundance ratio is an important
spectroscopic index, often used to probe the mean neutrons to seeds ratio of
an $s$-process site [36]. At the end of the AGB phase, the [heavy-s/light-s]
ratio of the low-LUNA model is $\sim 6\%$ smaller than in the no-LUNA model.
The abundance variations are generally small ($\leq 10\%$), with interesting
exceptions, i.e., 60Fe, 152Gd and 205Pb. All these isotopes are sensitive to
the neutron density because of the existence of close-by branching points.
152Gd cannot be produced by the r-process and, except for a small p-process
contribution, is mainly synthesized by the $s$-process. On the other hand,
60Fe and 205Pb are short-lived radioactive isotopes that were found to be
alive in the early solar system [37]. The 60Fe is produced when the neutron
density is high enough to allow neutron captures at the 59Fe branching point
(half-life 44.5 d). Therefore, its final abundance is enhanced in case of the
activation of the second (convective) neutron burst. With respect to the no-
LUNA model, we find that the 60Fe final mass fraction is a 30% higher, in the
LUNA model, and a factor of 2 higher, in the low-LUNA model. On the contrary,
the production of both 152Gd, and 205Pb requires low neutron density, while
they are mainly destroyed in case of high neutron density. Indeed, the first
isotope is bypassed by the $s$-process when the 151Sm may capture a neutron
before decaying into 151Eu. Similarly, the 205Pb production is suppressed when
the 204Tl branch is open. Therefore, their production is reduced when the
13C$(\alpha$,n$)^{16}$O rate is lowered, as it happens for all the other
isotopes of the main component, and a further decrease of their abundances
occurs in case of a more efficient second (convective) neutron burst. As a
result, the final mass fractions of both isotopes are reduced by 15% at the
95% lower bound.

To conclude, the present work reports a much improved calculation of the
13C$(\alpha$,n$)^{16}$O reaction rate at $T\sim 90$ MK, for the first time
based on direct data inside of the Gamow window. The reduced uncertainty will
help our understanding of the $s$-process branching points that are sensitive
to the neutron density. We find that the new low-energy cross-section
measurements imply sizeable variations of the 60Fe, 152Gd and 205Pb yields.
Other isotopes, whose production or destruction are influenced by close-by
branching points, such as the two neutron-magic nuclei 86Kr and 87Rb as well
as 96Zr, are less affected by a variation of the 13C$(\alpha$,n$)^{16}$O
reaction rate, mainly because of their higher initial (solar) abundance.
However, we cannot exclude that larger changes may occur in models with
different initial mass and composition. For this reason, a more extended set
of AGB models is required to accurately evaluate the general impact on the
galactic chemical evolution.

###### Acknowledgements.

This work has been supported by the INFN. D. Ciccotti of the LNGS proved to be
invaluable over the course of the campaign. We acknowledge the support of the
mechanical workshops of INFN-LNGS and INFN-Na. Financial support is
acknowledged as follows. D. V.: the German-Israeli Foundation (GIF No.
I-1500-303.7/2019); A. Be, a. Bo, A. D.L., G. I. and D. R. received funding
from the European Research Council (ERC No.852016); A. Be. and J. B.-C.: the
University of Naples-Compagnia di San Paolo grant STAR-2016; L. Cs.,T.S, F.Zs,
Gy.Gy: the Hungarian National Research Development and Innovation Office NKFIH
(contract numbers PD129060, K120666 and K134197); T.S.: the János Bolyai
research fellowship of the Hungarian Academy of Sciences and ÚNKP-20-5-DE-297
New National Excellence Program of the Ministry of Human Capacities of
Hungary;D.B.: DFG (BE 4100/4-1); M. A., C.G.B., T.C., T. D.: STFC-UK. R.D. and
D.P. acknowledge funding from the Italian Ministry of Education, University
and Research (MIUR) through the ‘Dipartimenti di eccellenza’ project Science
of the Universe. A. Be. is grateful to Richard J. deBoer for discussions about
the R-matrix analysis.

## References

  * Sneden et al. [2008] C. Sneden, J. J. Cowan, and R. Gallino, Annu. Rev. Astron. Astrophys. 46, 241 (2008), URL https://doi.org/10.1146/annurev.astro.46.060407.145207.
  * Straniero et al. [2006] O. Straniero, R. Gallino, and S. Cristallo, Nucl. Phys A 777, 311 (2006), ISSN 0375-9474, URL http://www.sciencedirect.com/science/article/pii/S037594740500028X.
  * Gallino et al. [1998] R. Gallino, C. Arlandini, M. Busso, M. Lugaro, C. Travaglio, O. Straniero, A. Chieffi, and M. Limongi, Astrophys. J. 497, 388 (1998).
  * Käppeler et al. [2011] F. Käppeler, R. Gallino, S. Bisterzo, and W. Aoki, Rev. Mod. Phys. 83, 157 (2011).
  * Drotleff et al. [1993] H. W. Drotleff, A. Denker, H. Knee, M. Soine, G. Wolf, J. W. Hammer, U. Greife, C. Rolfs, and H. P. Trautvetter, Astrophys. J. 414, 735 (1993).
  * Heil et al. [2008] M. Heil, R. Detwiler, R. E. Azuma, A. Couture, J. Daly, J. Görres, F. Käppeler, R. Reifarth, P. Tischhauser, C. Ugalde, et al., Phys. Rev. C 78, 025803 (2008).
  * Tilley et al. [1993] D. R. Tilley, H. R. Weller, and C. M. Cheves, Nucl. Phys. A564, 1 (1993).
  * Kubono et al. [2003] S. Kubono, K. Abe, S. Kato, T. Teranishi, M. Kurokawa, X. Liu, N. Imai, K. Kumagai, P. Strasser, M. H. Tanaka, et al., Phys. Rev. Lett. 90, 062501 (2003).
  * Johnson et al. [2006] E. D. Johnson, G. V. Rogachev, A. M. Mukhamedzhanov, L. T. Baby, S. Brown, W. T. Cluff, A. M. Crisp, E. Diffenderfer, V. Z. Goldberg, B. W. Green, et al., Phys. Rev. Lett. 97, 192701 (2006), URL https://link.aps.org/doi/10.1103/PhysRevLett.97.192701.
  * Avila et al. [2015] M. L. Avila, G. V. Rogachev, E. Koshchiy, L. T. Baby, J. Belarge, K. W. Kemper, A. N. Kuchera, and D. Santiago-Gonzalez, Phys. Rev. C 91, 048801 (2015), URL https://link.aps.org/doi/10.1103/PhysRevC.91.048801.
  * Trippella and Cognata [2017] O. Trippella and M. L. Cognata, Astrophys. J. 837, 41 (2017), URL http://stacks.iop.org/0004-637X/837/i=1/a=41.
  * Keeley et al. [2018] N. Keeley, K. W. Kemper, and K. Rusek, Eur. Phys. J. A 54, 71 (2018), URL https://doi.org/10.1140/epja/i2018-12507-5.
  * deBoer et al. [2020] R. J. deBoer, C. R. Brune, M. Febrarro, J. Görres, I. J. Thompson, and M. Wiescher, Phys. Rev. C 101, 045802 (2020), URL https://link.aps.org/doi/10.1103/PhysRevC.101.045802.
  * Cristallo et al. [2018] S. Cristallo, M. L. Cognata, C. Massimi, A. Best, S. Palmerini, O. Straniero, O. Trippella, M. Busso, G. F. Ciani, F. Mingrone, et al., Astrophys. J. 859, 105 (2018), URL http://stacks.iop.org/0004-637X/859/i=2/a=105.
  * Formicola et al. [2003] A. Formicola, G. Imbriani, M. Junker, D. Bemmerer, R. Bonetti, C. Broggini, C. Casella, P. Corvisiero, H. Costantini, G. Gervino, et al., Nucl. Instr. Meth. A 507, 609 (2003).
  * Best et al. [2016] A. Best, J. Görres, M. Junker, K.-L. Kratz, M. Laubenstein, A. Long, S. Nisi, K. Smith, and M. Wiescher, Nucl. Instr. Meth. A 812, 1 (2016), ISSN 0168-9002.
  * Ciani et al. [2020] G. F. Ciani, L. Csedreki, J. Balibrea-Correa, A. Best, M. Aliotta, F. Barile, D. Bemmerer, A. Boeltzig, C. Broggini, C. G. Bruno, et al., Eur. Phys. J. A 56, 75 (2020), URL https://doi.org/10.1140/epja/s10050-020-00077-0.
  * Csedreki et al. [2021] L. Csedreki, G. Ciani, J. Balibrea-Correa, A. Best, M. Aliotta, F. Barile, D. Bemmerer, A. Boeltzig, C. Broggini, C. Bruno, et al., Nucl. Instr. Meth. A 994, 165081 (2021), ISSN 0168-9002, URL https://www.sciencedirect.com/science/article/pii/S0168900221000656.
  * G.F.Ciani [2019] G.F.Ciani, Ph.D. thesis, Gran Sasso Science Insitute $\&$ SISSA (2019).
  * Balibrea-Correa et al. [2018] J. Balibrea-Correa, G. Ciani, R. Buompane, F. Cavanna, L. Csedreki, R. Depalo, F. Ferraro, and A. Best, Nucl. Instr. Meth. A 906, 103 (2018), ISSN 0168-9002, URL http://www.sciencedirect.com/science/article/pii/S0168900218309318.
  * Caldwell et al. [2009] A. Caldwell, D. Kollár, and K. Kröninger, Comput. Phys. Commun 180, 2197 (2009), ISSN 00104655.
  * Strieder et al. [2012] F. Strieder, B. Limata, A. Formicola, G. Imbriani, M. Junker, D. Bemmerer, A. Best, C. Broggini, A. Caciolli, P. Corvisiero, et al., Physics Letters B 707, 60 (2012), ISSN 0370-2693, URL http://www.sciencedirect.com/science/article/pii/S0370269311014869.
  * Bracci et al. [1990] L. Bracci, G. Fiorentini, V. Melezhik, G. Mezzorani, and P. Quarati, Nucl. Phys A 513, 316 (1990).
  * Huang et al. [1976] K.-N. Huang, M. Aoyagi, M. H. Chen, B. Crasemann, and H. Mark, At. Data Nucl. Data Tables 18, 243 (1976).
  * Ziegler et al. [2010] J. F. Ziegler, M. Ziegler, and J. Biersack, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 268, 1818 (2010), ISSN 0168-583X, 19th International Conference on Ion Beam Analysis, URL https://www.sciencedirect.com/science/article/pii/S0168583X10001862.
  * Azuma et al. [2010] R. Azuma, E. Uberseder, E. Simpson, C. Brune, H. Costantini, R. de Boer, J. Görres, M. Heil, P. LeBlanc, C. Ugalde, et al., Phys. Rev. C 81, 045805 (2010).
  * Harissopulos et al. [2005] S. Harissopulos, H. W. Becker, J. W. Hammer, A. Lagoyannis, C. Rolfs, and F. Strieder, Phys. Rev. C 72, 062801(R) (2005).
  * Febbraro et al. [2020] M. Febbraro, R. J. deBoer, S. D. Pain, R. Toomey, F. D. Becchetti, A. Boeltzig, Y. Chen, K. A. Chipps, M. Couder, K. L. Jones, et al., Physical Review Letters 125, 062501 (2020).
  * Ferraro et al. [2021] F. Ferraro, G. F. Ciani, A. Boeltzig, F. Cavanna, and S. Zavatarelli, Front. Astron. Space Sci. 7, 119 (2021), ISSN 2296-987X, URL https://www.frontiersin.org/article/10.3389/fspas.2020.617946.
  * Straniero et al. [1995] O. Straniero, R. Gallino, M. Busso, A. Chieffi, C. M. Raiteri, M. Limongi, and M. Salaris, Astrophys. J. 440, L85 (1995).
  * Cristallo et al. [2009] S. Cristallo, O. Straniero, R. Gallino, L. Piersanti, I. Domínguez, and M. T. Lederer, Astrophys. J. 696, 797 (2009).
  * Bisterzo et al. [2015] S. Bisterzo, R. Gallino, F. Käppeler, M. Wiescher, G. Imbriani, O. Straniero, S. Cristallo, J. Görres, and R. J. deBoer, Mon. Not. R. Astron. Soc. 449, 506 (2015).
  * Cristallo et al. [2011] S. Cristallo, L. Piersanti, O. Straniero, R. Gallino, I. Domínguez, C. Abia, G. Di Rico, M. Quintini, and S. Bisterzo, Astrophys. J. , Suppl. Ser. 197, 17 (2011), eprint 1109.1176.
  * Vescovi et al. [2020] D. Vescovi, S. Cristallo, M. Busso, and N. Liu, Astrophys. J. Lett. 897, L25 (2020), eprint 2006.13729.
  * Karakas and Lugaro [2016] A. I. Karakas and M. Lugaro, Astrophys. J. 825, 26 (2016), eprint 1604.02178.
  * Busso et al. [1999] M. Busso, R. Gallino, and G. J. Wasserburg, Annu. Rev. Astron. Astrophys. 37, 239 (1999).
  * Wasserburg et al. [2006] G. J. Wasserburg, M. Busso, R. Gallino, and K. M. Nollett, Nucl.Phys.A 777, 5 (2006).


# One-shot Face Reenactment Using Appearance Adaptive Normalization

Guangming Yao1, Yi Yuan 1, Tianjia Shao∗2, Shuang Li3, Shanqi Liu4, Yong Liu4,
Mengmeng Wang4, Kun Zhou2
Both authors contributed equally to this research.Correspongding author

###### Abstract

The paper proposes a novel generative adversarial network for one-shot face
reenactment, which can animate a single face image to a different pose-and-
expression (provided by a driving image) while keeping its original
appearance. The core of our network is a novel mechanism called appearance
adaptive normalization, which can effectively integrate the appearance
information from the input image into our face generator by modulating the
feature maps of the generator using the learned adaptive parameters.
Furthermore, we specially design a local net to reenact the local facial
components (i.e., eyes, nose and mouth) first, which is a much easier task for
the network to learn and can in turn provide explicit anchors to guide our
face generator to learn the global appearance and pose-and-expression.
Extensive quantitative and qualitative experiments demonstrate the significant
efficacy of our model compared with prior one-shot methods.

Figure 1: Generated examples by our method. The source image provides the
appearance and different driving images provide different expressions and head
poses. The reenacted face has the same appearance as the source and the same
pose-and-expression as the driving. Both the source and driving images are
unseen in the training stage.

## Introduction

In this paper we seek a one-shot face reenactment network, which can animate a
single source image to a different pose-and-expression (provided by a driving
image) while keeping the source appearance (i.e identity). We start with the
perspective that a face image can be divided into two parts, the pose-and-
expression and the appearance, which is also adopted by previous work (Zhang
et al. 2019). In face reenactment, the transferring of pose-and-expression is
relatively easy because the training data can cover most possible poses and
expressions. The main challenge of face reenactment is how to preserve the
appearances of different identities. This insight motivates us to design a new
architecture, which exploits a novel mechanism called the appearance adaptive
normalization, to better control the feature maps of the face generator for
the awareness of the source appearance. In general, the appearance adaptive
normalization can effectively integrate the specific appearance information
from the source image into the synthesized image, by modulating the feature
maps of the face generator. Especially, the appearance adaptive normalization
learns specific adaptive parameters (i.e., mean and variance) from the source
image, which are utilized to modulate feature maps in the generator. In this
way, the face generator can be better aware of the appearance of the source
image and effectively preserve the source appearance.

The appearance adaptive normalization is inspired by recent adaptive
normalization methods (Huang and Belongie 2017; Park et al. 2019), which
perform cross-domain image generation without retraining for a specific
domain. This attribute makes adaptive normalization potentially suitable for
one-shot face reenactment, in which each identity could be seen as a domain.
However, there exists a key challenge to apply these adaptive normalization
methods to face reenactment. That is, these existing adaptive normalization
methods are all designed to deal with the pixel-aligned image-to-image
translation problems. For example, in (Park et al. 2019) they propose
spatially-adaptive normalization for synthesizing photorealistic images given
an input semantic layout. However, in the scenario of face reenactment, the
source and driving images are not pixel-aligned. Such pixel mis-alignment
makes it difficult to optimize the adaptive normalization layers during
training in existing methods. Consequently, the existing methods will yield
distorted images after reenactment, and we will show it in the experiments. To
tackle this challenge, one key insight of our work is that instead of learning
individual adaptive parameters for different adaptive normalization layers
using independent architectures, we can use a unified network to learn all the
adaptive parameters from the source image in a global way. The benefit of such
paradigm is, by jointly learning the adaptive parameters, the different
adaptive normalization layers can be globally modulated rather than being
modulated locally. In this way, we can effectively optimize the adaptive
normalization layers and control the feature maps of face generator to keep
the source appearance. Specifically, we design a simple but effective skip-
connected network to predict the adaptive parameters from the source image,
which can explicitly promote the relations within adaptive parameters for
different adaptive normalization layers, and thus effectively propagate the
appearance information throughout the network during reenacting.

Figure 2: The architecture of generator of our proposed method.

We make another key observation that, compared with reenacting the whole faces
with largely varying appearances and expressions, reenacting the local facial
components (i.e., eyes, nose, and mouth) is a much easier task for the network
to learn. It is because the space of appearance and pose-and-expression is
significantly reduced for these local regions. To this end, we can learn the
reenactment of these local regions first, which can in turn provide explicit
anchors to guide our generator to learn the global appearance and pose-and-
expression. Especially, the landmarks are utilized to locate the source and
target positions of each face component, so the network only needs to learn
the reenactment of these components locally. After local reenacting, the
synthesized face components are transformed to the target positions and scales
with a similarity transformation and fed to the global generator for the
global face synthesis.

In summary, we propose a novel framework for one-shot face reenactment, which
utilizes appearance adaptive normalization to better preserve the appearance
during reenacting and local facial region reenactment to guide the global
synthesis of the final image. Our model only requires one source image to
provide the appearance and one driving image to provide the pose-and-
expression, both of which are unseen in the training data. The experiments on
a variety of face images demonstrate that our method outperforms the state-of-
the-art one-shot methods in both objective and subjective aspects (e.g.,
photo-realism and appearance preservation).

The main contributions of our work are:

  * 1)

We propose a novel method for one-shot face reenactment, which animates the
source face to another pose-and-expression while preserving its original
appearance using only one source image. In particular, we propose an
appearance adaptive normalization mechanism to better retain the appearance.

  * 2)

We introduce the reenactment of local facial regions to guide the global
synthesis of the final reenacted face.

  * 3)

Extensive experiments show that our method is able to synthesize reenacted
images with both high photo-realism and appearance preservation.

## Related Work

### Face Reenactment

Face reenactment is a special conditional face synthesis task that aims to
animate a source face image to a pose-and-expression of driving face. Common
approaches to face reenactment could be roughly divided into two categories:
many-to-one and many-to-many. Many-to-one approaches perform face reenactment
for a specific person. ReenactGAN (2018) utilizes CycleGAN (2017) to convert
the facial boundary heatmaps between different persons, and hence promote the
quality of the result synthesized by an identity-specific decoder. Face2Face
(2016) animates the facial expression of source video by swapping the source
face with the rendered image. The method of Kim et al. (2018) can synthesize
high-resolution and realistic facial images with GAN. However, all these
methods require a large number of images of the specific identity for training
and only reenact the specific identity. On the contrary, our method is capable
of reenacting any identity given only a single image without the need for
retraining or fine-turning.

To extend face reenactment to unseen identities, some many-to-many methods
have been proposed recently. Zakharov et al. (2019) adopt the architecture of
BigGAN (2018) and fashional meta-learning, which is capable of synthesizing a
personalized talking head with several images, but it requires fine-tuning
when a new person is introduced. Zhang et al. (2019) propose an unsupervised
approach to face reenactment, which does not need multiple poses for the same
identity. Yet, the face parsing map, an identity-specific feature, is utilized
to guide the reenacting, which leads to distorted results when reenacting a
different identity. Geng et al. (2018) introduce warp-guided GANs for single-
photo facial animation. However, their method needs a photo with frontal pose
and neutral expression, while ours does not have this limitation. (Pumarola et
al. 2018) generates a face guided by action units (1978), which makes it
difficult to handle pose changes. X2Face (2018) is able to animate a face
under the guidance of pose, expression, and audio, but it can not generate
face regions that do not exist in original images. MonkeyNet (2019a) provides
a framework for animating general objects. However, the unsupervised keypoints
detection may lead to distorted results in the one-shot case. MarioNetTe
(2020) proposes the landmark transformer to preserve the source shape during
reenactment, but it does not consider how to retain the source appearance. Yao
et al. (2020) introduce graph covolutional network to learn better optical
flow, which helps method to yield better results. Different from previous
many-to-many methods, our goal is to synthesize a high-quality face image, by
learning the appearance adaptive parameters to preserve the source appearance
and utilizing the local component synthesis to guide the global face
synthesis.

### Adaptive Normalization

The idea of adapting features to different distributions has been successfully
applied in a variety of image synthesis tasks (Huang and Belongie 2017; Park
et al. 2019). The adaptive normalization normalizes the feature to zero mean
and unit deviation first, and then the normalized feature is denormalized by
modulating the feature using the learned mean and standard deviation. In
conditional BN (Courville 2017), the fixed categorical images are synthesized
using different parameters of the normalization layers for different
categories. However, unlike the categorical image generation with fixed
categories, the number of identities is unknown in the one-shot face
reenactment. AdaIN (Huang and Belongie 2017) predicts the adaptive parameters
for style transfer, which is spatially sharing. However, it is insufficient in
controlling the global appearance, since the facial appearance is spatially
varying. SPADE (Park et al. 2019) deploys a spatially varying normalization,
which makes it suitable for spatially varying situations. However, SPADE (Park
et al. 2019) is designed for the pixel-aligned image translation task which
uses independent blocks to locally predict the adaptive parameters for
different layers. In face reenactment, the source and driving images are not
pixel-aligned, which makes it difficult to locally optimize the different
adaptive normalization layers. Hence, we propose the appearance adaptive
normalization mechanism to globally predict adaptive parameters of different
layers using a skip-connected network, which better promotes the relations
within the adaptive parameters for different layers during transferring.

## Methodology

For convenience, we denote the images in the dataset as
${I_{i}^{j}}_{i=1,…,N_{j}}^{j=1,…,M}$, where $j$ denotes the identity index
and $i$ denotes the image index of identity $j$. $M$ is the number of
identities and $N_{j}$ is the number of images of identity $j$.
$S_{i}^{j}\in\mathbb{R}^{68\times H\times W}$ denotes the corresponding
heatmaps for the 68 facial landmarks of $I_{i}^{j}\in\mathbb{R}^{3\times
H\times W}$, where $H$ and $W$ are the image height and width.

### Overview

Our method is a generative adversarial method. We adopt a self-supervised
approach to train the network in an end-to-end way, where the driving image
$I_{d}$ has the same identity as $I_{s}$ in the training stage (i.e., two
frames from a video). The landmark transformer (Ha et al. 2020) is utilized to
improve the identity preservation. Fig.2 shows the architecture of the
proposed generator, which takes as input the source image $I_{s}$ and the
driving image $I_{d}$. Our generator is composed of 4 sub-nets, and all the 4
sub-nets are jointly trained in an end-to-end way. First, to preserve the
source appearance, we send $I_{s}$ to the appearance extractor to learn the
appearance adaptive parameters $\Theta$ as well as the encoded appearance
feature $F_{a}$, as shown at the top of Fig. 2. Second, to estimate the facial
movements from the source image to the driving pose-and expression, the flow
estimation module estimates the optical flow $F_{sd}$ from $I_{s}$ to $I_{d}$
, which is then utilized to warp the encoded appearance feature, as shown in
the middle of Fig. 2. Third, the local net is deployed to reenact the local
facial regions, which provides essential anchors to guide the subsequent
synthesis of the whole face, as shown at the bottom of Fig. 2. Finally, the
fusion net fuses the adaptive parameters $\Theta$, the reenacted local face
regions $\hat{I}_{d}^{local}$ and the warped appearance feature $\hat{F}_{a}$,
to synthesize the reenacted face. By modulating the distribution of feature
maps in the fusion net using the appearance adaptive parameters, we let
$F_{sd}$ determine the pose-and-expression, and $F_{a}$ and $\Theta$ retain
the appearance.

#### Flow Estimation Module

Figure 3: The procedure of flow estimation module.

The procedure of flow estimation module is illustrated in Fig. 3. Firstly, we
estimate landmarks for $I_{s}$ and $I_{d}$ to obtain the source heatmap
$S_{s}$ and the driving heatmap $S_{d}$ respectively using OpenFace(Amos,
Ludwiczuk, and Satyanarayanan 2016). We then feed $S_{s}$ and $S_{d}$ into the
flow estimation net (FEN) to produce an optical flow
$F_{sd}\in\mathcal{R}^{2\times H\times W}$, representing the motion of pose-
and-expression. $F_{sd}$ is then utilized to warp the appearance feature
$F_{a}$. Bilinear sampling is used to sample $F_{sd}$ to the spatial size of
$F_{a}$. The warped $F_{a}$ is denoted as $\hat{F}_{a}$, which is subsequently
fed into the fusion net to synthesize the final reenacted face. Besides, we
also build the heatmaps of local regions for source and driving images based
on the landmarks, denoted as $S_{s}^{local}$ and $S_{d}^{local}$ respectively.
The architecture of FEN is an hourglass net (Yang, Liu, and Zhang 2017),
composed of several convolutional down-sampling and up-sampling layers.
Notably, large shape differences between the source identity and the driving
identity will lead to severe degradation of the quality of generated images,
which is also mentioned by (Wu et al. 2018). To deal with this issue, we
additionally adopt the landmark transformer (Ha et al. 2020), which edits the
driving heatmap $S_{d}$ so that $S_{d}$ has a shape close to $S_{s}$. For more
details, please refer to (Ha et al. 2020).

#### Local Net

The local net $G_{local}$ is built with the U-Net structure (Ronneberger,
Fischer, and Brox 2015). We reenact the left eye, right eye, nose and mouth
with 4 independent networks $G_{eyel}$, $G_{eyer}$, $G_{nose}$, and
$G_{mouth}$. Each of them is a U-Net with three down-convolution blocks and
three up-convolution blocks. The inputs of each local generator are
$I_{s}^{local}$, $S_{s}^{local}$ and $S_{d}^{local}$, where local refers to
the corresponding parts (i.e., left eye, right eye, nose and mouth) on the
image and heatmap. The reenacted face local regions serve as anchor regions
that can effectively guide the fusion net to synthesize the whole reenacted
face.

#### Appearance Extractor

The source image $I_{s}$ is fed into the appearance extractor $E_{a}(I_{s})$
for predicting the adaptive parameters $\Theta$ and the appearance feature
$F_{a}$. Here
$\Theta=\\{\theta_{i}=(\gamma_{i},\beta_{i}),i\in\\{1,2,...,N_{a}\\}\\}$,
where $i$ is the index of the adaptive normalization layer and $N_{a}$ denotes
the number of adaptive normalization layers in the fusion net. For a feature
map $F_{i}\in\mathcal{R}^{c\times h\times w}$ in the fusion net, we have the
corresponding $\gamma_{i},\beta_{i}\in\mathcal{R}^{c\times h\times w}$ to
modulate it. The encoded source appearance feature $F_{a}$ is warped to
$\hat{F}_{a}$ using the optical flow $F_{sd}$, and $\Theta$ and $\hat{F}_{a}$
are fed to the fusion net for face synthesis by controlling the distributions
of feature maps. We employ the U-net (2015) architecture for the appearance
extractor, because the skip-connection in appearance extractor can effectively
promote the relations between adaptive parameters.

#### Fusion Net

Figure 4: The fusion block of the proposed method.

The fusion net $\hat{I}_{d}=G_{f}(\hat{I}_{d}^{local},{\hat{F}_{a}},\Theta)$
aims to decode the reenacted local regions $I_{d}^{local}$ and the warped
appearance feature $\hat{F}_{a}$ to a reenacted face image $\hat{I}_{d}$ under
the control of adaptive parameters $\Theta$. $G_{f}$ is a fully convolutional
network, which performs decoding and up-sampling to synthesize the reenacted
face. $G_{f}$ consists of several fusion blocks to adapt the source
appearance, followed by several residual-connected convolution layers to
produce the final result. The architecture of fusion block is illustrated in
Fig. 4. $F_{i}$ denotes the input feature map of $i$-th fusion block,
$\gamma_{i}$ and $\beta_{i}$ denote the $i$-th adaptive parameters and
$FB_{i}$ denotes the $i$-th fusion block. Before fed into the fusion block,
the reenacted local regions $\hat{I}_{d}^{local}$ are similarly transformed to
the target scale-and-position. In this way, the aligned face regions provide
explicit anchors to the generator. These aligned $\hat{I}_{d}^{local}$ are
then resized to the same spatial size as $F_{i}$ using bilinear interpolation.
At last, $F_{i}$ and $\hat{I}_{d}^{local}$ are concatenated along the channel
axis and fed into next block of $G_{f}$. In this way, the formulation of
fusion block can be written as:

$F_{i+1}=FB_{i}([F_{i},\hat{I}_{d}^{local}],\gamma_{i},\beta_{i}).$ (1)

The core of our fusion net is the appearance adaptive normalization mechanism.
Specifically, the feature map is channel-wisely normalized by

$\mu_{c}^{i}=\frac{1}{NH^{i}W^{i}}\sum_{n,h,w}F_{n,c,h,w}^{i},$ (2)
$\sigma_{c}^{i}=\sqrt{\frac{1}{NH^{i}W^{i}}\sum_{n,h,w}[(F_{n,c,h,w}^{i})^{2}-(\mu_{c}^{i})^{2}]},$
(3)

where $F_{n,c,h,w}^{i}$ is the feature map value before normalization, and
$\mu_{c}^{i}$ and $\sigma_{c}^{i}$ are the mean and standard deviation of the
feature map in channel $c$. The index of the normalized layer is denoted as
$i$. Notably, the denormalization in adaptive normalization is element-wise,
where the normalized feature map is denormalized by

$\gamma_{c,h,w}^{i}\frac{F_{n,c,h,w}^{i}-\mu_{c}^{i}}{\sigma_{c}^{i}}+\beta_{c,h,w}^{i}.$
(4)

Here $\gamma_{c,h,w}^{i}$ and $\beta_{c,h,w}^{i}$ are the scale and bias
learned by the appearance extractor from $I_{s}$. Besides, instead of using
the transposed convolutional layer or the bilinear up-sampling layer followed
by a convolutional layer to expand the feature-map (Isola et al. 2017; Wang et
al. 2018), we adopt the pixel-shuffle (Shi et al. 2016) to upscale the feature
map.

### Discriminator

There are two discriminators in our method, a discriminator $D_{L}$ to
discriminate whether the reenacted image and the driving heatmap are matched
(pose-and-expression consistency) and a discriminator $D_{I}$ to discriminate
whether the source and reenacted image share the same identity (appearance
consistency). $D_{L}$ takes $\hat{I}_{d}$ and $S_{d}$ as input, while $D_{I}$
takes $\hat{I}_{d}$ and $I_{s}$ as input. $\hat{I}_{d}$ is concatenated with
$S_{d}$ or $I_{s}$ along the channel axis, before being fed into $D_{L}$ or
$D_{I}$ respectively. To generate a sharp and realistic-looking image, the
discriminators should have a large receptive field (Wang et al. 2018). In our
method, instead of using a deeper network with larger convolutional kernels,
we use a multi-scale discriminator (Wang et al. 2018) which can improve the
global consistency of generated images in multiple scales.

Figure 5: Qualitative comparison with state-of-the-art one-shot methods. Our
proposed method generates more natural-looking and sharp results compared to
previous methods.

### Loss Function

The total loss function is defined as:

$\begin{split}L_{total}=\arg\min_{G}\max_{D_{L},D_{I}}\lambda_{GAN}L_{GAN}+\lambda_{c}L_{c}\\\
+\lambda_{local}L_{local}.,\end{split}$ (5)

where $L_{c}$ denotes the content loss, $L_{GAN}$ denotes the adversarial loss
and $L_{local}$ denotes local region loss. The adversarial loss is the GAN
loss for $D_{L}$ and $D_{I}$:

$\begin{split}L_{GAN}=\mathbb{E}_{I_{s},\hat{I}_{d},S_{d}}[\log
D_{L}(I_{d},S_{d})+\log(1-D_{L}(\hat{I}_{d},S_{d}))]\\\
+\left.\mathbb{E}_{I_{s},\hat{I}_{d},I_{d}}[\log
D_{I}(I_{s},I_{d})+\log(1-D_{I}(I_{s},\hat{I}_{d},I_{d}))]\right..\end{split}$
(6)

The content loss is defined as:

$L_{c}=L_{1}(I_{d},\hat{I}_{d})+L_{per}(I_{d},\hat{I}_{d}),$ (7)

where $L_{1}(I_{d},\hat{I}_{d})$ is the pixel-wise L1 loss, measuring the
pixel distance between the generated image and the ground-truth image.
$L_{per}(I_{d},\hat{I}_{d})$ is the perceptual loss (Johnson, Alahi, and Fei-
Fei 2016), which has been shown to be useful for the task of image generation
(Ledig et al. 2017). We make use of the pre-trained VGG (Simonyan and
Zisserman 2014) to compute the perceptual loss, and $L_{per}$ is written as:

$L_{per}(I_{d},\hat{I}_{d})=\mathbb{E}_{i\in
X}[||\Phi_{i}(I_{d})-\Phi_{i}(\hat{I}_{d})||_{1}],$ (8)

where $X$ represents the layers we use in VGG and $\Phi_{i}(x)$ denotes the
feature map of the $i$-th layer in $X$.

The local region loss penalizes the perceptual differences between the
reenacted local regions and the local regions on the ground-truth and is
defined as:

$\begin{split}L_{local}&=L_{per}(I_{eyel},\hat{I}_{eyel})+L_{per}(I_{mouth},\hat{I}_{mouth})\\\
&+L_{per}(I_{nose},\hat{I}_{nose})+L_{per}(I_{eyer},\hat{I}_{eyer}).\end{split}$
(9)

## Experiments

### Implementation

The learning rate for the generator and discriminator are set to $2e^{-5}$ and
$1e^{-5}$ respectively. We use Adam (Kingma and Ba 2014) as the optimizer.
Spectral Normalization (Miyato et al. 2018) is utilized for each convolution
layer in the generator. We set $\lambda_{GAN}=10$, $\lambda_{c}=5$ and
$\lambda_{local}=5$ in the loss function. The Gaussian kernel variance of
heatmaps is 3.

### Datasets and Metrics

Both the FaceForensics++ (Rössler et al. 2019), VoxCeleb1(Nagrani, Chung, and
Zisserman 2017) and Celeb-DF (Li et al. 2020) datasets are used for
quantitative and qualitative evaluation. The OpenFace (Amos, Ludwiczuk, and
Satyanarayanan 2016) is utilized to detect the face and extract facial
landmarks. Following the work of MarionNetTe(2020), we adopt the following
metrics to quantitatively evaluate the reenacted faces of different methods.
We evaluate the identity preservation by calculating the cosine similarity
(CSIM) of identity vectors between the source image and the generated image.
The identity vectors are extracted by the pre-trained state-of-the-art face
recognition networks (Deng et al. 2019). To inspect the model’s capability of
properly reenacting the pose and expression of driving image, we calculate
PRMSE (Ha et al. 2020) and AUCON (Ha et al. 2020) between the generated image
and the driving image to measure the reenacted pose and expression
respectively.

### Quantitative and Qualitative Comparison

Model | CSIM$\uparrow$ | PRMSE$\downarrow$ | AUCON$\uparrow$
---|---|---|---
VoxCeleb1 (2017)
X2face | 0.689 | 3.26 | 0.813
NeuralHead-FF | 0.229 | 3.76 | 0.791
MarioNETte | 0.755 | 3.13 | 0.825
FirstOrder | 0.813 | 3.79 | 0.886
Ours | 0.823 | 3.26 | 0.831
Celeb-DF (2020)
X2face | 0.676 | 4.10 | 0.679
NeuralHead-FF | 0.511 | 6.09 | 0.747
MarioNETte | 0.650 | 3.98 | 0.714
FirstOrder | 0.687 | 3.15 | 0.839
Ours | 0.753 | 3.12 | 0.751

Table 1: Quantitative comparison in the self-reenactment setting. Up/down
arrows correspond to higher/lower values for better performance. Bold and
underlined numbers represent the best and the second-best values of each
metric respectively.

Model | CSIM$\uparrow$ | PRMSE$\downarrow$ | AUCON$\uparrow$
---|---|---|---
Faceforensics++ (2019)
X2face | 0.604 | 9.80 | 0.697
NeuralHead-FF | 0.381 | 6.82 | 0.730
MarioNETte | 0.620 | 7.68 | 0.710
FirstOrder | 0.614 | 6.62 | 0.734
Ours | 0.658 | 7.04 | 0.706
Celeb-DF (2020)
X2face | 0.400 | 6.52 | 0.400
NeuralHead-FF | 0.352 | 8.30 | 0.480
MarioNETte | 0.460 | 5.16 | 0.662
FirstOrder | 0.432 | 6.10 | 0.500
Ours | 0.463 | 5.10 | 0.660

Table 2: Quantitative comparison of reenacting a different identity.

Table 1 lists the quantitative comparisons with existing one-shot reenactment
methods when reenacting the same identity, and Table 2 reports the evaluation
results when reenacting a different identity. It is worth mentioning that the
method that, following (Ha et al. 2020), we re-implement (Zakharov et al.
2019) using only the feed-forward network in the one-shot setting. Differ from
other competitors, FirstOrder (2019b) require two driving image to perform the
relative motion transfer, one image provide the initial driving pose-and-
expression and another one to provides the target driving pose-and-expression.
We use the source image to provide the initial driving pose-and-expression
when reenacting the same identity to perform the relative motion transfer, and
the absolute motion transfer is adopted when reenacting different identities
as the initial driving image is lacked for all competitors.

Notably, the results show that our method outperforms other methods in many
metrics, demonstrating our method can synthesize highly realistic faces while
effectively retaining the source appearance and faithfully reenacting the
pose-and-expression. Fig. 5 illustrates typical qualitative examples, all of
which are randomly selected from the testing set. We can see that X2face
(2018) is unable to generate face regions that do not exist in the source
images, so it may result in large artifacts. As the state of art, MarioNETte
(2020) can effectively preserve the source shape, but there may still be some
appearance artifacts in some regions. Our method fixes this issue by
introducing the appearance adaptive normalization and local region reenacting.

Figure 6: Comparison of our method with FSGAN(2019), source andn driving
images are cited from FSGAN(2019)

Figure 7: Comparison of our method with Zhang et al.(2019), source andn
driving images are cited from Zhang et al.(2019).

We also qualitatively compare our method with recently proposed methods of
Zhang et al. (2019) and FSGAN(2019), demonstrated in Fig. 6 and Fig. 7. We can
observe blurriness and color-inconsistency in the results of FSGAN(2019). Also
the images synthesized by Zhang et al. (2019) have distorted face shapes and
artifacts in boundaries, because Zhang et al. (2019) utilize the face parsing
map, which is an identity-specific feature, to guide the reenacting. On the
contrary, with the help of appearance adaptive normalization and local region
reenacting, our method can achieve more detailed and natural-looking results.

### Ablation Study

Figure 8: Qualitative results of the ablation study. Our full model leads to better results than other variants. Model | CSIM$\uparrow$ | PRMSE$\downarrow$ | AUCON$\uparrow$
---|---|---|---
\- local net | 0.615 | 7.293 | 0.698
\- AAN + SPADE | 0.558 | 11.030 | 0.660
Ours | 0.658 | 7.04 | 0.706
Table 3: Quantitative ablation study for reenacting a different identity on
the Faceforensics++.

To better evaluate the key components within our network, we perform the
ablation study by evaluating the following variants of our method:

  * •

$-LocalNet$. The local net is excluded from the full model.

  * •

$-AAN+SPADE$. To validate the effectiveness of appearance adaptive
normalization, we use the spatially-adaptive normalization to replace it, and
all the other components are the same as our model.

The qualitative results are illustrated in Fig. 8 and quantitative results are
listed in Table 3. We can see that our full model presents the most realistic
and natural-looking results. The local net can help reduce the pose-and-
expression error, as it explicitly provides anchors for local face regions to
guide the reenacting. The appearance adaptive normalization can effectively
improve image quality and reduce artifacts by globally modulating the
appearance features. Compared to the spatially-adaptive normalization (2019),
our appearance adaptive normalization can better preserve the source
appearance and leads to more realistic results. It validates our appearance
adaptive normalization is more suitable for face reenactment.

### Conclusion and Future Work

In the paper, we propose a novel method to deal with the challenging problem
of one-shot face reenactment. Our network deploys a novel mechanism called
appearance adaptive normalization to effectively integrate the source
appearance information into our face generator, so that the reenacted face
image can better preserve the same appearance as the source image. Besides, we
design a local net to reenact the local facial components first, which can in
turn guide the global synthesis of face appearance and pose-and-expression.
Compared to previous methods, our network exhibits superior performance in
different metrics.

## Acknowledgments

We thank anonymous reviewers for their valuable comments. This work is
supported by National Key R&D Program of China (2018YFB1004300), NSF China
(No. 61772462, No. U1736217) and the 100 Talents Program of Zhejiang
University.

## References

  * Amos, Ludwiczuk, and Satyanarayanan (2016) Amos, B.; Ludwiczuk, B.; and Satyanarayanan, M. 2016.  OpenFace: A general-purpose face recognition library with mobile applications.  Technical report, CMU-CS-16-118, CMU School of Computer Science.
  * Brock, Donahue, and Simonyan (2018) Brock, A.; Donahue, J.; and Simonyan, K. 2018.  Large Scale GAN Training for High Fidelity Natural Image Synthesis.  In _International Conference on Learning Representations_.
  * Courville (2017) Courville, A. C. 2017.  Modulating early visual processing by language.  In _NIPS_.
  * Deng et al. (2019) Deng, J.; Guo, J.; Xue, N.; and Zafeiriou, S. 2019.  Arcface: Additive angular margin loss for deep face recognition.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 4690–4699.
  * Friesen and Ekman (1978) Friesen, E.; and Ekman, P. 1978.  Facial action coding system: a technique for the measurement of facial movement.  _Palo Alto_ 3\.
  * Geng et al. (2018) Geng, J.; Shao, T.; Zheng, Y.; Weng, Y.; and Zhou, K. 2018.  Warp-guided GANs for single-photo facial animation.  In _SIGGRAPH Asia 2018 Technical Papers_ , 231. ACM.
  * Ha et al. (2020) Ha, S.; Kersner, M.; Kim, B.; Seo, S.; and Kim, D. 2020.  Marionette: Few-shot face reenactment preserving identity of unseen targets.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 34, 10893–10900.
  * Huang and Belongie (2017) Huang, X.; and Belongie, S. 2017.  Arbitrary Style Transfer in Real-Time with Adaptive Instance Normalization.  _2017 IEEE International Conference on Computer Vision (ICCV)_ doi:10.1109/iccv.2017.167.  URL http://dx.doi.org/10.1109/iccv.2017.167.
  * Isola et al. (2017) Isola, P.; Zhu, J.-Y.; Zhou, T.; and Efros, A. A. 2017.  Image-to-Image Translation with Conditional Adversarial Networks.  _2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ doi:10.1109/cvpr.2017.632.  URL http://dx.doi.org/10.1109/cvpr.2017.632.
  * Johnson, Alahi, and Fei-Fei (2016) Johnson, J.; Alahi, A.; and Fei-Fei, L. 2016.  Perceptual losses for real-time style transfer and super-resolution.  In _European conference on computer vision_ , 694–711. Springer.
  * Kim et al. (2018) Kim, H.; Carrido, P.; Tewari, A.; Xu, W.; Thies, J.; Niessner, M.; Pérez, P.; Richardt, C.; Zollhöfer, M.; and Theobalt, C. 2018.  Deep video portraits.  _ACM Transactions on Graphics (TOG)_ 37(4): 163.
  * Kingma and Ba (2014) Kingma, D. P.; and Ba, J. 2014.  Adam: A method for stochastic optimization.  _arXiv preprint arXiv:1412.6980_ .
  * Ledig et al. (2017) Ledig, C.; Theis, L.; Huszár, F.; Caballero, J.; Cunningham, A.; Acosta, A.; Aitken, A.; Tejani, A.; Totz, J.; Wang, Z.; et al. 2017.  Photo-realistic single image super-resolution using a generative adversarial network.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 4681–4690.
  * Li et al. (2020) Li, Y.; Yang, X.; Sun, P.; Qi, H.; and Lyu, S. 2020.  Celeb-df: A large-scale challenging dataset for deepfake forensics.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 3207–3216.
  * Miyato et al. (2018) Miyato, T.; Kataoka, T.; Koyama, M.; and Yoshida, Y. 2018.  Spectral normalization for generative adversarial networks.  _arXiv preprint arXiv:1802.05957_ .
  * Nagrani, Chung, and Zisserman (2017) Nagrani, A.; Chung, J. S.; and Zisserman, A. 2017.  Voxceleb: a large-scale speaker identification dataset.  _arXiv preprint arXiv:1706.08612_ .
  * Nirkin, Keller, and Hassner (2019) Nirkin, Y.; Keller, Y.; and Hassner, T. 2019.  Fsgan: Subject agnostic face swapping and reenactment.  In _Proceedings of the IEEE International Conference on Computer Vision_ , 7184–7193.
  * Park et al. (2019) Park, T.; Liu, M.-Y.; Wang, T.-C.; and Zhu, J.-Y. 2019.  Semantic image synthesis with spatially-adaptive normalization.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2337–2346.
  * Pumarola et al. (2018) Pumarola, A.; Agudo, A.; Martinez, A. M.; Sanfeliu, A.; and Moreno-Noguer, F. 2018\.  Ganimation: Anatomically-aware facial animation from a single image.  In _Proceedings of the European Conference on Computer Vision (ECCV)_ , 818–833.
  * Ronneberger, Fischer, and Brox (2015) Ronneberger, O.; Fischer, P.; and Brox, T. 2015.  U-Net: Convolutional Networks for Biomedical Image Segmentation.  _ArXiv_ abs/1505.04597.
  * Rössler et al. (2019) Rössler, A.; Cozzolino, D.; Verdoliva, L.; Riess, C.; Thies, J.; and Nießner, M. 2019.  FaceForensics++: Learning to Detect Manipulated Facial Images.  In _International Conference on Computer Vision (ICCV)_.
  * Shi et al. (2016) Shi, W.; Caballero, J.; Huszár, F.; Totz, J.; Aitken, A. P.; Bishop, R.; Rueckert, D.; and Wang, Z. 2016.  Real-time single image and video super-resolution using an efficient sub-pixel convolutional neural network.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 1874–1883.
  * Siarohin et al. (2019a) Siarohin, A.; Lathuilière, S.; Tulyakov, S.; Ricci, E.; and Sebe, N. 2019a.  Animating Arbitrary Objects via Deep Motion Transfer.  In _The IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_.
  * Siarohin et al. (2019b) Siarohin, A.; Lathuilière, S.; Tulyakov, S.; Ricci, E.; and Sebe, N. 2019b.  First Order Motion Model for Image Animation.  In _Conference on Neural Information Processing Systems (NeurIPS)_.
  * Simonyan and Zisserman (2014) Simonyan, K.; and Zisserman, A. 2014.  Very deep convolutional networks for large-scale image recognition.  _arXiv preprint arXiv:1409.1556_ .
  * Thies et al. (2016) Thies, J.; Zollhofer, M.; Stamminger, M.; Theobalt, C.; and Nießner, M. 2016\.  Face2face: Real-time face capture and reenactment of rgb videos.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2387–2395.
  * Wang et al. (2018) Wang, T.-C.; Liu, M.-Y.; Zhu, J.-Y.; Tao, A.; Kautz, J.; and Catanzaro, B. 2018\.  High-Resolution Image Synthesis and Semantic Manipulation with Conditional GANs.  _2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition_ doi:10.1109/cvpr.2018.00917.  URL http://dx.doi.org/10.1109/cvpr.2018.00917.
  * Wiles, Koepke, and Zisserman (2018) Wiles, O.; Koepke, A.; and Zisserman, A. 2018.  X2face: A network for controlling face generation using images, audio, and pose codes.  In _Proceedings of the European conference on computer vision (ECCV)_ , 670–686.
  * Wu et al. (2018) Wu, W.; Zhang, Y.; Li, C.; Qian, C.; and Change Loy, C. 2018.  Reenactgan: Learning to reenact faces via boundary transfer.  In _Proceedings of the European Conference on Computer Vision (ECCV)_ , 603–619.
  * Yang, Liu, and Zhang (2017) Yang, J.; Liu, Q.; and Zhang, K. 2017.  Stacked hourglass network for robust facial landmark localisation.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops_ , 79–87.
  * Yao et al. (2020) Yao, G.; Yuan, Y.; Shao, T.; and Zhou, K. 2020.  Mesh Guided One-shot Face Reenactment Using Graph Convolutional Networks.  _Proceedings of the 28th ACM International Conference on Multimedia_ doi:10.1145/3394171.3413865.  URL http://dx.doi.org/10.1145/3394171.3413865.
  * Zakharov et al. (2019) Zakharov, E.; Shysheya, A.; Burkov, E.; and Lempitsky, V. 2019.  Few-shot adversarial learning of realistic neural talking head models.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , 9459–9468.
  * Zhang et al. (2019) Zhang, Y.; Zhang, S.; He, Y.; Li, C.; Loy, C. C.; and Liu, Z. 2019.  One-shot Face Reenactment.  In _British Machine Vision Conference (BMVC)_.
  * Zhu et al. (2017) Zhu, J.-Y.; Park, T.; Isola, P.; and Efros, A. A. 2017.  Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networkss.  In _Computer Vision (ICCV), 2017 IEEE International Conference on_.