$\displaystyle\,\left[\frac{\partial}{\partial\tau_{0}},I_{1}\right]v_{\lambda}+\left[\frac{\partial}{\partial\tau_{1}},\left[\frac{\partial}{\partial\tau_{0}},X^{\circ}\right]\right]v_{\lambda}$
$\displaystyle=$
$\displaystyle\,\left[\frac{\partial}{\partial\tau_{1}},\left[\frac{\partial}{\partial\tau_{0}},\hat{I}_{1}+X^{\circ}\right]\right]v_{\lambda}+\frac{\partial
A^{\alpha}}{\partial\tau_{0}}\frac{\partial v_{\lambda}}{\partial
t^{\alpha,1}}+\frac{\partial B^{\alpha}}{\partial\tau_{0}}\frac{\partial
v_{\lambda}}{\partial t^{\alpha,0}}.$

Due to Lemma 4.10 we only need to verify the following identity:

$\left[\frac{\partial}{\partial\tau_{1}},D\right]v_{\lambda}+\frac{\partial
A^{\alpha}}{\partial\tau_{0}}\frac{\partial v_{\lambda}}{\partial
t^{\alpha,1}}+\frac{\partial B^{\alpha}}{\partial\tau_{0}}\frac{\partial
v_{\lambda}}{\partial t^{\alpha,0}}=0,$

which can be checked directly from the definition of $D$. Similarly we can
prove that

$\left[\frac{\partial}{\partial\tau_{0}},I_{1}\right]\sigma_{\lambda,0}+\left[\frac{\partial}{\partial\tau_{1}},\left[\frac{\partial}{\partial\tau_{0}},X^{\circ}\right]\right]\sigma_{\lambda,0}=0.$

The proposition is proved. ∎

It remains to prove the second identity of (3.17).

######  Lemma 4.12.

The following identities hold true:

$\displaystyle\left[I_{1},\frac{\partial}{\partial\tau_{1}}\right]v_{\lambda}=$
$\displaystyle\,\sum_{s\geq 0}G^{\gamma}_{s}\frac{\partial}{\partial
v_{\gamma}^{(s)}}\frac{\partial
v_{\lambda}}{\partial\tau_{1}}+F^{\gamma}_{s}\frac{\partial}{\partial\sigma_{\gamma,0}^{s}}\frac{\partial
v_{\lambda}}{\partial\tau_{1}}$
$\displaystyle-A^{\alpha}\frac{\partial}{\partial\tau_{1}}\frac{\partial
v_{\lambda}}{\partial
t^{\alpha,1}}-B^{\alpha}\frac{\partial}{\partial\tau_{1}}\frac{\partial
v_{\lambda}}{\partial t^{\alpha,0}}+\frac{\partial
B^{\alpha}}{\partial\tau_{1}}\frac{\partial v_{\lambda}}{\partial
t^{\alpha,0}},$
$\displaystyle\left[I_{1},\frac{\partial}{\partial\tau_{1}}\right]\sigma_{\lambda,0}=$
$\displaystyle\,\sum_{s\geq 0}G^{\gamma}_{s}\frac{\partial}{\partial
v_{\gamma}^{(s)}}\frac{\partial\sigma_{\lambda,0}}{\partial\tau_{1}}+F^{\gamma}_{s}\frac{\partial}{\partial\sigma_{\gamma,0}^{s}}\frac{\partial\sigma_{\lambda,0}}{\partial\tau_{1}}$
$\displaystyle-A^{\alpha}\frac{\partial}{\partial\tau_{1}}\frac{\partial\sigma_{\lambda,0}}{\partial
t^{\alpha,1}}-B^{\alpha}\frac{\partial}{\partial\tau_{1}}\frac{\partial\sigma_{\lambda,0}}{\partial
t^{\alpha,0}}+\frac{\partial
B^{\alpha}}{\partial\tau_{1}}\frac{\partial\sigma_{\lambda,0}}{\partial
t^{\alpha,0}},$

where $G^{\gamma}_{s}$ and $F^{\gamma}_{s}$ are differential polynomials
defined by

$G_{s}^{\gamma}=\left[\frac{\partial}{\partial\tau_{1}},\hat{I}_{1}\right]v_{\gamma}^{(s)}+\partial_{x}^{s}\left(I_{1}v_{\gamma}\right),\quad
F_{s}^{\gamma}=\left[\frac{\partial}{\partial\tau_{1}},\hat{I}_{1}\right]\sigma_{\gamma,0}^{s}+\partial_{x}^{s}\left(I_{1}\sigma_{\gamma,0}\right).$

######  Lemma 4.13.

The differential polynomials $G^{\gamma}_{s}$ and $F^{\gamma}_{s}$ defined in
Lemma 4.12 have the following expressions:

$\displaystyle G_{s}^{\gamma}=$
$\displaystyle\,A^{\alpha}\partial_{x}^{s}\frac{\partial v_{\gamma}}{\partial
t^{\alpha,1}}+B^{\alpha}\partial_{x}^{s}\frac{\partial v_{\gamma}}{\partial
t^{\alpha,0}},$ $\displaystyle F_{s}^{\gamma}=$
$\displaystyle\,A^{\alpha}\partial_{x}^{s}\frac{\partial\sigma_{\gamma,0}}{\partial
t^{\alpha,1}}+B^{\alpha}\partial_{x}^{s}\frac{\partial\sigma_{\gamma,0}}{\partial
t^{\alpha,0}}$
$\displaystyle+\delta_{s,0}\left(\frac{3}{2}+\mu_{\gamma}\right)\left(\frac{\partial\sigma_{\gamma,1}}{\partial\tau_{2}}-\sigma_{\beta,1}\eta^{\alpha\beta}\left(\frac{1}{2}+\mu_{\alpha}\right)\frac{\partial\sigma_{\gamma,1}}{\partial
t^{\alpha,0}}\right).$

###### Proof.

By using the definition of $G^{\gamma}_{s}$ we can obtain the identity

$G^{\gamma}_{s+1}-\partial_{x}G^{\gamma}_{s}=\frac{\partial}{\partial\tau_{1}}\left(\left[\hat{I}_{1},\partial_{x}\right]v_{\gamma}^{(s)}\right)+\left[\partial_{x},\hat{I}_{1}\right]\frac{\partial
v_{\gamma}^{(s)}}{\partial\tau_{1}}.$

Therefore it follows from Lemma 4.8 that

$G^{\gamma}_{s+1}-\partial_{x}G^{\gamma}_{s}=-\partial_{x}A^{\alpha}\partial_{x}^{s}\frac{\partial
v_{\gamma}}{\partial
t^{\alpha,1}}-\partial_{x}B^{\alpha}\partial_{x}^{s}\frac{\partial
v_{\gamma}}{\partial t^{\alpha,0}}.$

Hence $G^{\gamma}_{s}$ can be solved recursively starting from the initial
condition

$G^{\gamma}_{0}=I_{1}v_{\gamma}-\hat{I}_{1}\frac{\partial
v_{\gamma}}{\partial\tau_{1}}=A^{\alpha}\frac{\partial v_{\gamma}}{\partial
t^{\alpha,1}}+B^{\alpha}\frac{\partial v_{\gamma}}{\partial t^{\alpha,0}}.$

The differential polynomials $F_{s}^{\gamma}$ can be computed similarly. The
lemma is proved. ∎

######  Proposition 4.14.

The second identity of the closedness condition (3.17) holds true, i.e.,

$\left[\frac{\partial}{\partial\tau_{1}},I_{1}\right]=0.$

###### Proof.

The proof of the proposition is straightforward by combining the results of
Lemma 4.12 and Lemma 4.13. ∎

###  4.3. Vanishing of the genus one obstruction

In this subsection, we will work with the canonical coordinates of the
Frobenius manifold. Let us start by recalling some useful formulae related to
the canonical coordinates. For the details one may refer to the work [5, 12].

We denote by $u^{1},\cdots,u^{n}$ the local canonical coordinates of a
semisimple Frobenius manifold $M$ and denote by $(u^{i};\theta_{i})$ the
corresponding coordinates of $\hat{M}$. Under this system of local
coordinates, the bihamiltonian structure (2.12) takes the form (2.6), i.e.,

$P_{0}^{[0]}=\frac{1}{2}\int\sum_{i,j=1}^{n}\left(\delta_{i,j}f^{i}\theta_{i}\theta_{i}^{1}+A^{ij}\theta_{i}\theta_{j}\right),\quad
P_{1}^{[0]}=\frac{1}{2}\int\sum_{i,j=1}^{n}\left(\delta_{i,j}u^{i}f^{i}\theta_{i}\theta_{i}^{1}+B^{ij}\theta_{i}\theta_{j}\right).$

Introduce functions

(4.12) $\psi_{i1}=\frac{1}{\sqrt{f^{i}}},$

where the sign of the square root can be arbitrarily chosen, and define

$\psi_{i\alpha}=\frac{1}{\psi_{i1}}\frac{\partial v_{\alpha}}{\partial
u^{i}},\quad\gamma_{ij}=\frac{1}{\psi_{j1}}\frac{\partial\psi_{i1}}{\partial
u^{j}},\quad i\neq j.$

Then it is proved in [5] that

(4.13) $\frac{\partial v_{\alpha}}{\partial
u^{i}}=\psi_{i1}\psi_{i\alpha},\quad\frac{\partial u^{i}}{\partial
v^{\alpha}}=\frac{\psi_{i\alpha}}{\psi_{i1}},$ (4.14)
$\frac{\partial\psi_{i\alpha}}{\partial u^{k}}=\gamma_{ik}\psi_{k\alpha},\quad
i\neq k;\quad\frac{\partial\psi_{i\alpha}}{\partial u^{i}}=-\sum_{k\neq
i}\gamma_{ik}\psi_{k\alpha}.$

Let $V_{ij}=\mu_{\alpha}\eta^{\alpha\beta}\psi_{i\alpha}\psi_{j\beta},$ then
we have the identity

(4.15) $\gamma_{ij}(u^{j}-u^{i})=V_{ij}.$

Now we are going to describe the deformation problem given in Sect. 3.3 in
terms of the canonical coordinates. Introduce the odd variables $\theta_{i,m}$
by the recursion relations (2.15), then it follows from the the relation
(2.16) that the equations (3.8) and (3.9) can be represented in the form

(4.16) $\displaystyle\frac{\partial u^{i}}{\partial s_{2}}=\ $ $\displaystyle
a^{\alpha\beta}\left(f_{\beta,0}\frac{\partial u^{i}}{\partial
t^{\alpha,1}}+f_{\alpha,1}\frac{\partial u^{i}}{\partial
t^{\beta,0}}\right)+b^{\alpha\beta}\left(f_{\beta,0}\frac{\partial
u^{i}}{\partial t^{\alpha,0}}+f_{\alpha,0}\frac{\partial u^{i}}{\partial
t^{\beta,0}}\right)$ $\displaystyle+Xu^{i}+\mathcal{L}_{2}u^{i},$ (4.17)
$\displaystyle\frac{\partial\theta_{i,0}}{\partial s_{2}}=\ $ $\displaystyle
a^{\alpha\beta}\left(f_{\beta,0}\frac{\partial\theta_{i,0}}{\partial
t^{\alpha,1}}+f_{\alpha,1}\frac{\partial\theta_{i,0}}{\partial
t^{\beta,0}}\right)+b^{\alpha\beta}\left(f_{\beta,0}\frac{\partial\theta_{i,0}}{\partial
t^{\alpha,0}}+f_{\alpha,0}\frac{\partial\theta_{i,0}}{\partial
t^{\beta,0}}\right)$
$\displaystyle+X\theta_{i,0}+\sum_{j}A^{j}_{i}\theta_{j,2}+B^{j}_{i}\theta_{j,1}+\mathcal{L}_{2}\theta_{i,0}.$

In the canonical coordinates, the Hamiltonian operator $\mathcal{P}_{1}$ of
$P_{1}$ has the form

$\mathcal{P}_{1}^{ij}=u^{i}f^{i}\delta_{ij}\partial_{x}+\frac{1}{2}\partial_{x}\left(u^{i}f^{i}\right)\delta_{ij}+B^{ij}+Q^{ij}\partial_{x}^{3}+\cdots$

where $B^{ij}$ is defined in (2.5) and $Q^{ij}$ is given by the following
formula (c.f., e.g., [13])

(4.18)
$Q^{ij}=3c_{i}\left(f^{i}\right)^{2}\delta_{ij}+\frac{1}{2}\left(u^{i}-u^{j}\right)\left(f^{j}\partial_{j}f^{i}c_{i}-f^{i}\partial_{i}f^{j}c_{j}\right),$

here $c_{i}$ is the $i$-th central invariant. Let us also denote the
evolutions of $\theta_{i}$ along the flows $\frac{\partial}{\partial
t^{\alpha,p}}$ by the following equations:

$\frac{\partial\theta_{i}}{\partial
t^{\alpha,p}}=T_{\alpha,p}^{i}\theta_{i}^{1}+\cdots+K_{\alpha,p;j}^{i}\theta_{j}^{3}+\cdots,\quad
T_{\alpha,p}\in\hat{\mathcal{A}}^{0}_{0},\
K_{\alpha,p;j}^{i}\in\hat{\mathcal{A}}^{0}_{3},$

here and henceforth we omit the terms that do not contribute to the relevant
computations given later. Note that the coefficients of $\theta_{j}^{1}$ of
the leading term of $\frac{\partial\theta_{i}}{\partial t^{\alpha,p}}$ are
zero for $j\neq i$, this is due to the fact that the leading term of the flow
$\frac{\partial}{\partial t^{\alpha,p}}$ is diagonal, one may refer to [9] for
details.

Let us turn to the proof of the vanishing of the genus one obstruction. Due to
the closedness condition (3.15), we can choose a derivation $X^{\circ}$ such
that when we take $X=X^{\circ}$ in (4.16) and (4.17) we have

$\left[\frac{\partial}{\partial\tau_{0}},\frac{\partial}{\partial
s_{2}}\right]=0.$

We require that the leading term of $X^{\circ}$ is given by the genus zero
Virasoro symmetry, hence it follows from the result of [11] that the actions
of $X^{\circ}$ on $u^{i}$ and $\theta_{i}$ take the form

$\displaystyle X^{\circ}u^{i}$
$\displaystyle=(u^{i})^{3}+\sum_{j}L^{i}_{j}u^{j,2}+\cdots,\quad
L^{i}_{j}\in\hat{\mathcal{A}}^{0}_{0},$ $\displaystyle X^{\circ}\theta_{i}$
$\displaystyle=\sum_{j}M^{j}_{i}\theta_{j}+\sum_{j}J^{j}_{i}\theta_{j}^{2}+\cdots,\quad
M^{j}_{i},J^{j}_{i}\in\hat{\mathcal{A}}^{0}_{0}.$

######  Lemma 4.15.

We have the following identity:

$\displaystyle J^{i}_{i}-L^{i}_{i}=$
$\displaystyle\,\frac{1}{f^{i}}2c_{0}(u^{i})^{2}Q^{ii}-\sum_{j}\frac{A^{j}_{i}}{f^{j}}(u^{i}+u^{j})Q^{ij}-\sum_{j}\frac{B^{j}_{i}}{f^{j}}Q^{ij}$
$\displaystyle-b^{\beta,q}_{2;1,0}K^{i}_{\beta,q;i}-a^{\alpha,p;\beta,q}_{2}\left(\left(f_{\beta,q}^{\prime}\right)_{0}K^{i}_{\alpha,p;i}+\left(f_{\alpha,p}^{\prime}\right)_{0}K^{i}_{\beta,q;i}\right),$

where $b^{\beta,q}_{2;\alpha,p}$ and $a^{\alpha,p;\beta,q}_{2}$ are the
constants that appear in the operator $\mathcal{L}_{2}$, and
$\left(f_{\alpha,p}^{\prime}\right)_{0}$ denotes the differential degree zero
component of $f_{\alpha,p}^{\prime}$.

###### Proof.

We can prove this lemma by computing the coefficient of $\theta_{i}^{3}$ of
differential degree 3 component of the left hand side of the equation

$\left[\frac{\partial}{\partial\tau_{0}},\frac{\partial}{\partial
s_{2}}\right]u^{i}=0.$

The lemma is proved. ∎

######  Remark 4.1.

Note that $c_{0}$ is the arbitrary constant that appears in the operator
$\mathcal{L}_{2}$, and $c_{1},\cdots,c_{n}$ are the central invariants.

According to the discussion given in Sect. 3.3, we need to compute the
cohomology class of the differential degree 3 component of the derivation
$\left[\frac{\partial}{\partial\tau_{1}},\mathcal{C}\right]=I_{1}-\left[\frac{\partial}{\partial\tau_{1}},X^{\circ}\right]$.
Due to Theorem 3.2, it suffices to prove that the coefficient of
$\theta_{i}^{3}$ of the differential degree 3 component of

$I_{1}u^{i}-\left[\frac{\partial}{\partial\tau_{1}},X^{\circ}\right]u^{i}$

vanishes. By using Lemma 4.15 and a straightforward computation, we obtain the
following lemma.

######  Lemma 4.16.

The coefficient of $\theta_{i}^{3}$ of the differential degree 3 component of
$I_{1}u^{i}-\left[\frac{\partial}{\partial\tau_{1}},X^{\circ}\right]u^{i}$
reads

(4.19)
$\displaystyle\sum_{j}Q^{ij}\left(M^{i}_{j}+A^{i}_{j}(u^{i})^{2}+B^{i}_{j}u^{i}\right)+E^{3}Q^{ii}-(6+c_{0})(u^{i})^{2}Q^{ii}$
$\displaystyle+3Q^{ii}b^{\beta,q}_{2;1,0}T_{\beta,q}^{i}+3Q^{ii}a^{\alpha,p;\beta,q}_{2}\left(\left(f_{\beta,q}^{\prime}\right)_{0}T_{\alpha,p}^{i}+\left(f_{\alpha,p}^{\prime}\right)_{0}T_{\beta,q}^{i}\right),$

here $E^{3}$ is the cubic of the Euler vector field, which is given by

$E^{3}=\sum_{i}(u^{i})^{3}\frac{\partial}{\partial u^{i}}.$

######  Lemma 4.17.

We have the identities

$\displaystyle M^{i}_{i}+A^{i}_{i}(u^{i})^{2}+B^{i}_{i}u^{i}$ $\displaystyle=$
$\displaystyle\,(3+c_{0})(u^{i})^{2}-\frac{1}{f^{i}}E^{3}f^{i}-b^{\beta,q}_{2;1,0}T_{\beta,q}^{i}-a^{\alpha,p;\beta,q}_{2}\left(\left(f_{\beta,q}^{\prime}\right)_{0}T_{\alpha,p}^{i}+\left(f_{\alpha,p}^{\prime}\right)_{0}T_{\beta,q}^{i}\right),$

and $M^{j}_{i}+A^{j}_{i}(u^{j})^{2}+B^{j}_{i}u^{j}=0$ for $i\neq j$.

###### Proof.

We can prove this lemma by computing the coefficient of $\theta_{j}^{1}$ of
the differential degree 1 component of the left hand side of the equation

$\left[\frac{\partial}{\partial\tau_{0}},\frac{\partial}{\partial
s_{2}}\right]u^{i}=0.$

The lemma is proved. ∎

In order to prove the vanishing of the genus one obstruction, we need to
verify that the expression (4.19) vanishes. To this end, we only need to
check, due to Lemma 4.17 and the expression (4.18) for $Q^{ii}$, the following
identity:

$E^{3}f^{i}+2f^{i}\left(b^{\beta,q}_{2;1,0}T_{\beta,q}^{i}+a^{\alpha,p;\beta,q}_{2}\left(\left(f_{\beta,q}^{\prime}\right)_{0}T_{\alpha,p}^{i}+\left(f_{\alpha,p}^{\prime}\right)_{0}T_{\beta,q}^{i}\right)\right)=3(u^{i})^{2}f^{i}.$

######  Proposition 4.18.

For $m\geq-1$, we have

$E^{m+1}f^{i}+2f^{i}\left(b^{\beta,q}_{m;1,0}T_{\beta,q}^{i}+a^{\alpha,p;\beta,q}_{m}\left(\left(f_{\beta,q}^{\prime}\right)_{0}T_{\alpha,p}^{i}+\left(f_{\alpha,p}^{\prime}\right)_{0}T_{\beta,q}^{i}\right)\right)=(1+m)(u^{i})^{m}f^{i},$

here $E^{m+1}$ is the $(m+1)$-th power of the Euler vector field $E$ which is
given by

$E^{m+1}=\sum_{i}(u^{i})^{m+1}\frac{\partial}{\partial u^{i}}.$

###### Proof.

Let us consider the following generating functions:

$\displaystyle\sum_{m\geq-1}\frac{1}{\lambda^{m+2}}E^{m+1}f^{i}=\sum_{j}\frac{1}{\lambda-u^{j}}\frac{\partial
f^{i}}{\partial u^{j}};$
$\displaystyle\sum_{m\geq-1}\frac{1}{\lambda^{m+2}}(1+m)(u^{i})^{m}f^{i}=\frac{f^{i}}{(\lambda-u^{i})^{2}};$
$\displaystyle\sum_{m\geq-1}\frac{1}{\lambda^{m+2}}\left(b^{\beta,q}_{m;1,0}T_{\beta,q}^{i}+a^{\alpha,p;\beta,q}_{m}\left(\left(f_{\beta,q}^{\prime}\right)_{0}T_{\alpha,p}^{i}+\left(f_{\alpha,p}^{\prime}\right)_{0}T_{\beta,q}^{i}\right)\right)$
$\displaystyle=$
$\displaystyle\,\frac{1}{2}\frac{1}{(\lambda-u^{i})^{2}}+\sum_{j\neq
i}\frac{\psi_{j1}}{\psi_{i1}}\frac{V_{ij}}{(\lambda-u^{i})(\lambda-u^{j})},$

where the last generating function was computed in [12], one may refer to
Lemma 3.10.18 and the proof of Theorem 3.10.29 of [12] for details. Then the
proposition is proved by using (4.12), (4.14) and (4.15). ∎

Finally we have the following theorem.

######  Theorem 4.19.

There exists a unique $X\in\mathrm{Der}^{\partial}(\hat{\mathcal{A}})^{0}$
such that the following flow defines a symmetry of the deformed Principal
Hierarchy which is a deformation of the genus zero Virasoro symmetry
$\frac{\partial}{\partial s_{2}}$ of the Principal Hierarchy:

(4.20) $\displaystyle\frac{\partial v_{\lambda}}{\partial s_{2}}=$
$\displaystyle\,a^{\alpha\beta}\left(f_{\beta,0}\frac{\partial
v_{\lambda}}{\partial t^{\alpha,1}}+f_{\alpha,1}\frac{\partial
v_{\lambda}}{\partial
t^{\beta,0}}\right)+b^{\alpha\beta}\left(f_{\beta,0}\frac{\partial
v_{\lambda}}{\partial t^{\alpha,0}}+f_{\alpha,0}\frac{\partial
v_{\lambda}}{\partial t^{\beta,0}}\right)$
$\displaystyle+Xv_{\lambda}+\mathcal{L}_{2}v_{\lambda},$ (4.21)
$\displaystyle\frac{\partial\sigma_{\lambda,0}}{\partial s_{2}}=$
$\displaystyle\,a^{\alpha\beta}\left(f_{\beta,0}\frac{\partial\sigma_{\lambda,0}}{\partial
t^{\alpha,1}}+f_{\alpha,1}\frac{\partial\sigma_{\lambda,0}}{\partial
t^{\beta,0}}\right)+b^{\alpha\beta}\left(f_{\beta,0}\frac{\partial\sigma_{\lambda,0}}{\partial
t^{\alpha,0}}+f_{\alpha,0}\frac{\partial\sigma_{\lambda,0}}{\partial
t^{\beta,0}}\right)$
$\displaystyle+X\sigma_{\lambda,0}+\left(\frac{5}{2}+c_{0}+\mu_{\lambda}\right)\sigma_{\lambda,2}+N^{\zeta}_{\lambda}\sigma_{\zeta,1}+\mathcal{L}_{2}\sigma_{\lambda,0},$

where $N^{\zeta}_{\lambda}\in\hat{\mathcal{A}}^{0}_{\geq 0}$ is the
differential polynomial described in Lemma 4.3.

###  4.4. Lifting to the tau-covers

In order to lift the symmetry (4.20) to the tau-cover of the deformed
Principal Hierarchy, we first need to rewrite (4.20) in terms of the normal
coordinates $w^{1},\cdots,w^{n}$ of $M$. We start by proving the following
lemmas.

######  Lemma 4.20.

Let $g_{\lambda}\in\hat{\mathcal{A}}^{0}_{\geq 1}$ be the differential
polynomials given in (2.31) which satisfy the identities

$h_{\lambda,0}=v_{\lambda}+\partial_{x}g_{\lambda},\quad\lambda=1,\cdots,n.$

Then we have:

  1. (1)

$g_{1}=0$.

  2. (2)

For any $\alpha=1,\cdots,n$ and $p\geq 0$,

$\Omega_{\alpha,p;\lambda,0}=\frac{\delta h_{\alpha,p+1}}{\delta
v^{\lambda}}+\frac{\partial g_{\lambda}}{\partial t^{\alpha,p}}.$

###### Proof.

Due to Proposition 2.17, we have $D_{X_{1,0}}=\partial_{x}$ and
$X_{1,0}=-[H_{1,0},P_{0}]$, from which it follows that

$v_{\alpha}=\frac{\delta H_{1,0}}{\delta v^{\alpha}}.$

By taking $\alpha=1$, we obtain the first property by using the definition
(2.24). The second one is obvious due to Theorem 2.19. The lemma is proved. ∎

######  Lemma 4.21.

There exists a derivation
$X^{\circ}\in\mathrm{Der}^{\partial}(\hat{\mathcal{A}})$ such that its leading
term is given by the Virasoro symmetry $\frac{\partial}{\partial s_{2}}$ of
the Principal Hierarchy, and it satisfies the equations
$\left[\frac{\partial}{\partial\tau_{0}},X^{\circ}\right]=I_{0}$ and

(4.22) $\displaystyle X^{\circ}h_{\lambda,0}+\hat{I}_{1}h_{\lambda,0}=$
$\displaystyle\,a^{\alpha\beta}\left(f_{\alpha,1}^{\prime}\Omega_{\beta,0;\lambda,0}+f_{\beta,0}^{\prime}\Omega_{\alpha,1;\lambda,0}+\frac{\partial^{2}\Omega_{\alpha,1;\beta,0}}{\partial
t^{\lambda,0}\partial t^{1,0}}\right)$
$\displaystyle+b^{\alpha\beta}\left(f_{\alpha,0}^{\prime}\Omega_{\beta,0;\lambda,0}+f_{\beta,0}^{\prime}\Omega_{\alpha,0;\lambda,0}+\frac{\partial^{2}\Omega_{\alpha,0;\beta,0}}{\partial
t^{\lambda,0}\partial t^{1,0}}\right)$
$\displaystyle+b^{\beta,q}_{2;1,0}\Omega_{\beta,q;\lambda,0}+b^{\beta,q}_{2;\lambda,0}\Omega_{\beta,q;1,0}+2c_{2;\lambda,0;1,0}.$

###### Proof.

We first find a particular solution $\tilde{X}^{\circ}$ of the equation
$\left[\frac{\partial}{\partial\tau_{0}},X\right]=I_{0}$, then we modify it by
a solution $\tilde{\mathcal{C}}$ of the homogeneous equation
$\left[\frac{\partial}{\partial\tau_{0}},{\mathcal{C}}\right]=0$ such that
$X^{\circ}:=\tilde{X}^{\circ}+\tilde{\mathcal{C}}$ satisfies (4.22).

Let us define $\tilde{X}^{\circ}\in\mathrm{Der}^{\partial}(\hat{\mathcal{A}})$
as follows:

$\displaystyle\tilde{X}^{\circ}v_{\lambda}=$ $\displaystyle\
a^{\alpha\beta}\left(f_{\alpha,1}^{\prime}\frac{\delta h_{\beta,1}}{\delta
v^{\lambda}}+f_{\beta,0}^{\prime}\frac{\delta h_{\alpha,2}}{\delta
v^{\lambda}}\right)+b^{\alpha\beta}\left(f_{\alpha,0}^{\prime}\frac{\delta
h_{\beta,1}}{\delta v^{\lambda}}+f_{\beta,0}^{\prime}\frac{\delta
h_{\alpha,1}}{\delta v^{\lambda}}\right)$
$\displaystyle+b^{\beta,q}_{2;1,0}\frac{\delta h_{\beta,q+1}}{\delta
v^{\lambda}}+b^{\beta,q}_{2;\lambda,0}\frac{\delta h_{\beta,q+1}}{\delta
v^{1}}+2c_{2;\lambda,0;1,0};$
$\displaystyle\tilde{X}^{\circ}\sigma_{\lambda,0}=$ $\displaystyle\
a^{\alpha\beta}\left(\frac{\partial
f_{\alpha,1}}{\partial\tau_{0}}\frac{\delta h_{\beta,1}}{\delta
v^{\lambda}}+\frac{\partial f_{\beta,0}}{\partial\tau_{0}}\frac{\delta
h_{\alpha,2}}{\delta v^{\lambda}}\right)+b^{\alpha\beta}\left(\frac{\partial
f_{\alpha,0}}{\partial\tau_{0}}\frac{\delta h_{\beta,1}}{\delta
v^{\lambda}}+\frac{\partial f_{\beta,0}}{\partial\tau_{0}}\frac{\delta
h_{\alpha,1}}{\delta v^{\lambda}}\right)$
$\displaystyle+b^{\beta,q}_{2;\lambda,0}\frac{\partial
f_{\beta,q}}{\partial\tau_{0}}-\left(\frac{5}{2}+\mu_{\lambda}\right)\sigma_{\lambda,2}-N^{\zeta}_{\lambda}\sigma_{\zeta,1}.$

Firstly, from the definition of $N^{\zeta}_{\lambda}$ it follows that
$\tilde{X}^{\circ}$ is indeed local. We also note that the leading terms of
$\tilde{X}^{\circ}v_{\lambda}$ and $\tilde{X}^{\circ}\sigma_{\lambda,0}$
coincide with the local terms of the Virasoro symmetry $\frac{\partial
v_{\lambda}}{\partial s_{2}}$ and $\frac{\partial\sigma_{\lambda,0}}{\partial
s_{2}}$ of the Principal Hierarchy. By a direct computation, it is easy to
check that
$\left[\frac{\partial}{\partial\tau_{0}},\tilde{X}^{\circ}\right]=I_{0}$.

Next we want to determine
$\tilde{\mathcal{C}}\in\mathrm{Der}^{\partial}(\hat{\mathcal{A}})$ such that
$X^{\circ}=\tilde{X}^{\circ}+\tilde{\mathcal{C}}$ satisfies (4.22). By using
the definition of $\tilde{X}^{\circ}v_{\lambda}$ and Lemma 4.8, 4.20, it is
straightforward to show that

(4.23)
$\tilde{\mathcal{C}}(h_{\lambda,0})=a^{\alpha\beta}\frac{\partial^{2}\Omega_{\alpha,1;\beta,0}}{\partial
t^{\lambda,0}\partial
t^{1,0}}+b^{\alpha\beta}\frac{\partial^{2}\Omega_{\alpha,0;\beta,0}}{\partial
t^{\lambda,0}\partial
t^{1,0}}-\partial_{x}\hat{I}_{1}(g_{\lambda})-\partial_{x}\tilde{X}^{\circ}(g_{\lambda}),$

which uniquely determines the actions $\tilde{\mathcal{C}}v_{\lambda}$.

Finally we need to check such $\tilde{\mathcal{C}}$ satisfies
$\left[\frac{\partial}{\partial\tau_{0}},\tilde{\mathcal{C}}\right]=0$. By
using the next lemma, we know that it suffices to show that
$\int\tilde{\mathcal{C}}v_{\lambda}$ = 0, which is obvious from (4.23) and
(2.31). The lemma is proved. ∎

######  Lemma 4.22.

Let $U_{1},\cdots,U_{n}$ be differential polynomials with
$U_{\lambda}\in\hat{\mathcal{A}}^{0}_{\geq 2}$. Then there exists
$\mathcal{C}\in\mathrm{Der}^{\partial}(\hat{\mathcal{A}})^{0}_{\geq 1}$ such
that $\left[\frac{\partial}{\partial\tau_{0}},\mathcal{C}\right]=0$ and
$\mathcal{C}v_{\lambda}=U_{\lambda}$ if and only if $\int U_{\lambda}=0$.

###### Proof.

Let $\mathcal{C}\in\mathrm{Der}^{\partial}(\hat{\mathcal{A}})^{0}$ be a
derivation such that
$\left[\frac{\partial}{\partial\tau_{0}},\mathcal{C}\right]=0$. Then by using
the triviality of the variational Hamiltonian cohomology $H^{0}_{\geq
1}\bigl{(}\mathrm{Der}^{\partial}(\hat{\mathcal{A}})\bigr{)}$, we know the
existence of a certain
$\mathcal{K}\in\mathrm{Der}^{\partial}(\hat{\mathcal{A}})^{-1}$ such that
$\left[\frac{\partial}{\partial\tau_{0}},\mathcal{K}\right]=\mathcal{C}$. Let
us denote $\mathcal{K}\sigma_{\lambda,0}=V_{\lambda}\in\hat{\mathcal{A}}^{0}$.
Then it is easy to see that

$\mathcal{C}v_{\lambda}=\left[\frac{\partial}{\partial\tau_{0}},\mathcal{K}\right]v_{\lambda}=\mathcal{K}\sigma_{\lambda,0}^{1}=\partial_{x}V_{\lambda}.$

Therefore we have $\int U_{\lambda}=\int\partial_{x}V_{\lambda}=0$.

Conversely, if $U_{\lambda}=\partial_{x}V_{\lambda}$ for some
$V_{\lambda}\in\hat{\mathcal{A}}^{0}$, we can define a derivation
$\mathcal{C}\in\mathrm{Der}^{\partial}(\hat{\mathcal{A}})^{0}$ by

$\mathcal{C}v_{\lambda}=\partial_{x}V_{\lambda},\quad\mathcal{C}\sigma_{\lambda,0}=\frac{\partial
V_{\lambda}}{\partial\tau_{0}},$

then it is easy to check that
$\left[\frac{\partial}{\partial\tau_{0}},\mathcal{C}\right]=0$ and the lemma
is proved. ∎

Now let us denote $\mathcal{C}=X-X^{\circ}$, where the derivation $X$ is
described in Theorem 4.19 and $X^{\circ}$ satisfies (4.22). Then we can
rewrite the Virasoro symmetry in terms of the normal coordinates as follows:

$\displaystyle\frac{\partial w_{\lambda}}{\partial s_{2}}=$ $\displaystyle\
a^{\alpha\beta}\left(f_{\beta,0}\frac{\partial w_{\lambda}}{\partial
t^{\alpha,1}}+f_{\alpha,1}\frac{\partial w_{\lambda}}{\partial
t^{\beta,0}}\right)+b^{\alpha\beta}\left(f_{\beta,0}\frac{\partial
w_{\lambda}}{\partial t^{\alpha,0}}+f_{\alpha,0}\frac{\partial
w_{\lambda}}{\partial t^{\beta,0}}\right)$
$\displaystyle+a^{\alpha\beta}\left(f_{\alpha,1}^{\prime}\Omega_{\beta,0;\lambda,0}+f_{\beta,0}^{\prime}\Omega_{\alpha,1;\lambda,0}+\frac{\partial^{2}\Omega_{\alpha,1;\beta,0}}{\partial
t^{\lambda,0}\partial t^{1,0}}\right)$
$\displaystyle+b^{\alpha\beta}\left(f_{\alpha,0}^{\prime}\Omega_{\beta,0;\lambda,0}+f_{\beta,0}^{\prime}\Omega_{\alpha,0;\lambda,0}+\frac{\partial^{2}\Omega_{\alpha,0;\beta,0}}{\partial
t^{\lambda,0}\partial t^{1,0}}\right)$
$\displaystyle+b^{\beta,q}_{2;1,0}\Omega_{\beta,q;\lambda,0}+b^{\beta,q}_{2;\lambda,0}\Omega_{\beta,q;1,0}+2c_{2;\lambda,0;1,0}+\mathcal{C}(h_{\lambda,0})+\mathcal{L}_{2}w_{\lambda}.$

According to Lemma 4.22, there exists
$Q_{\lambda}\in\hat{\mathcal{A}}^{0}_{\geq 1}$ such that
$\mathcal{C}(h_{\lambda,0})=\partial_{x}Q_{\lambda}$. In particular, the fact
that

$\frac{\partial}{\partial t^{\gamma,0}}\frac{\partial w_{\lambda}}{\partial
s_{2}}=\frac{\partial}{\partial t^{\lambda,0}}\frac{\partial
w_{\gamma}}{\partial s_{2}}$

gives the equation

$\frac{\partial Q_{\lambda}}{\partial t^{\gamma,0}}=\frac{\partial
Q_{\gamma}}{\partial t^{\lambda,0}},$

which means that $\int Q_{1}$ is a conserved quantity of the flows
$\frac{\partial}{\partial t^{\lambda,0}}$. If the dimension of the Frobenius
manifold $\dim M\geq 2$, then it follows from Lemma 4.12 and Theorem A.2 of
[9] that there exists $Q\in\hat{\mathcal{A}}^{0}_{\geq 0}$ such that
$Q_{\lambda}=\frac{\partial Q}{\partial t^{\lambda,0}}$.

######  Theorem 4.23.

Let $\mathcal{Z}$ be a tau-function of the tau-cover (2.28) of the deformed
Principal Hierarchy. Then there exists a differential polynomial
$O_{2}\in\hat{\mathcal{A}}^{0}_{\geq 0}$ which yields a symmetry of the tau-
cover given by

$\frac{\partial\mathcal{Z}}{\partial
s_{2}}=L_{2}^{even}\mathcal{Z}+O_{2}\mathcal{Z},$

where the evolutions of $f_{\alpha,p}$ and $w_{\lambda}$ along the flow
$\frac{\partial}{\partial s_{2}}$ are defined by

$\frac{\partial f_{\alpha,p}}{\partial s_{2}}=\frac{\partial}{\partial
t^{\alpha,p}}\frac{\partial\log Z}{\partial s_{2}},\quad\frac{\partial
w_{\lambda}}{\partial s_{2}}=\frac{\partial}{\partial t^{1,0}}\frac{\partial
f_{\lambda,0}}{\partial s_{2}}.$

###### Proof.

When $\dim M\geq 2$ we take $O_{2}=Q$ which is just given above. When $\dim
M=1$, $O_{2}$ is given in (3.26). The theorem is proved. ∎

Now let us proceed to determine all other Virasoro symmetries
$\frac{\partial}{\partial s_{m}}$ of the tau-cover of the deformed Principal
Hierarchy. In what follows we will denote by $\mathcal{F}=\log\mathcal{Z}$ and
denote

$\displaystyle\mathcal{G}_{m}=a_{m}^{\alpha,p;\beta,q}\left(\frac{\partial\mathcal{F}}{\partial
t^{\alpha,p}}\frac{\partial\mathcal{F}}{\partial
t^{\beta,q}}+\frac{\partial^{2}\mathcal{F}}{\partial t^{\alpha,p}\partial
t^{\beta,q}}\right)+c_{m;\alpha,p;\beta,q}t^{\alpha,p}t^{\beta,q};$
$\displaystyle\mathcal{L}_{m}^{even}=b^{\beta,q}_{m;\alpha,p}t^{\alpha,p}\frac{\partial}{\partial
t^{\beta,q}},\quad m\geq-1.$

It is proved in [9] that

$\frac{\partial\mathcal{F}}{\partial
s_{-1}}=\mathcal{G}_{-1}+\mathcal{L}_{-1}^{even}\mathcal{F}$

induces a symmetry of the tau-cover of the deformed Principal Hierarchy. On
the other hand, by using Theorem 4.23 we know that

$\frac{\partial\mathcal{F}}{\partial
s_{2}}=\mathcal{G}_{2}+O_{2}+\mathcal{L}_{2}^{even}\mathcal{F}$

also induces a symmetry of the tau-cover of the deformed Principal Hierarchy.
From the commutation relation

$[L_{-1}^{even},L_{2}^{even}]=-3L_{1}^{even},$

it follows that

$\left[\frac{\partial}{\partial s_{-1}},\frac{\partial}{\partial
s_{2}}\right]\mathcal{F}=3\left(\mathcal{G}_{1}+\mathcal{L}_{1}^{even}\mathcal{F}\right)+\frac{\partial
O_{2}}{\partial s_{-1}}-\mathcal{L}_{-1}^{even}O_{2}.$

Since $\frac{\partial O_{2}}{\partial s_{-1}}-\mathcal{L}_{-1}^{even}O_{2}$ is
a differential polynomial, we can define

$\frac{\partial\mathcal{F}}{\partial
s_{1}}=\frac{1}{3}\left[\frac{\partial}{\partial
s_{-1}},\frac{\partial}{\partial
s_{2}}\right]\mathcal{F}=\mathcal{G}_{1}+O_{1}+\mathcal{L}_{1}^{even}\mathcal{F},\quad
O_{1}=\frac{1}{3}\left(\frac{\partial O_{2}}{\partial
s_{-1}}-\mathcal{L}_{-1}^{even}O_{2}\right).$

Then it is obvious that $\frac{\partial\mathcal{F}}{\partial s_{1}}$ induces a
symmetry of the tau-cover of the deformed Principal Hierarchy. Similarly we
define the symmetry

$\frac{\partial\mathcal{F}}{\partial
s_{0}}=\frac{1}{2}\left[\frac{\partial}{\partial
s_{-1}},\frac{\partial}{\partial
s_{1}}\right]\mathcal{F}=\mathcal{G}_{0}+O_{0}+\mathcal{L}_{0}^{even}\mathcal{F},\quad
O_{0}=\frac{1}{2}\left(\frac{\partial O_{1}}{\partial
s_{-1}}-\mathcal{L}_{-1}^{even}O_{1}\right).$

Now let us define $\frac{\partial\mathcal{F}}{\partial s_{m}}$ for $m\geq 3$
recursively in the following way. Assume we have defined

$\frac{\partial\mathcal{F}}{\partial
s_{m}}=\mathcal{G}_{m}+O_{m}+\mathcal{L}_{m}^{even}\mathcal{F}$

such that it induces a symmetry of the tau-cover of the deformed Principal
Hierarchy for $m\geq 2$. Then we have

$\frac{\partial}{\partial s_{1}}\frac{\partial\mathcal{F}}{\partial
s_{m}}=\frac{\partial\mathcal{G}_{m}}{\partial s_{1}}+\frac{\partial
O_{m}}{\partial
s_{1}}+\mathcal{L}_{m}^{even}\left(\mathcal{G}_{1}+O_{1}+\mathcal{L}_{1}^{even}\mathcal{F}\right).$

It follows from the definition of $\mathcal{G}_{m}$ that

$\displaystyle\frac{\partial\mathcal{G}_{m}}{\partial s_{1}}=$
$\displaystyle\,\frac{\partial}{\partial
s_{1}}\left(a_{m}^{\alpha,p;\beta,q}\left(\frac{\partial\mathcal{F}}{\partial
t^{\alpha,p}}\frac{\partial\mathcal{F}}{\partial
t^{\beta,q}}+\frac{\partial^{2}\mathcal{F}}{\partial t^{\alpha,p}\partial
t^{\beta,q}}\right)+c_{m;\alpha,p;\beta,q}t^{\alpha,p}t^{\beta,q}\right)$
$\displaystyle=$
$\displaystyle\,a_{m}^{\alpha,p;\beta,q}\left(f_{\alpha,p}\frac{\partial}{\partial
t^{\beta,q}}(\mathcal{G}_{1}+O_{1}+\mathcal{L}_{1}^{even}\mathcal{F})+f_{\beta,q}\frac{\partial}{\partial
t^{\alpha,p}}(\mathcal{G}_{1}+O_{1}+\mathcal{L}_{1}^{even}\mathcal{F})\right)$
$\displaystyle+a_{m}^{\alpha,p;\beta,q}\frac{\partial^{2}}{\partial
t^{\alpha,p}\partial
t^{\beta,q}}(\mathcal{G}_{1}+O_{1}+\mathcal{L}_{1}^{even}\mathcal{F})$
$\displaystyle=$
$\displaystyle\,a_{m}^{\alpha,p;\beta,q}\left(f_{\alpha,p}\frac{\partial
O_{1}}{\partial t^{\beta,q}}+f_{\beta,q}\frac{\partial O_{1}}{\partial
t^{\alpha,p}}+\frac{\partial^{2}O_{1}}{\partial t^{\alpha,p}\partial
t^{\beta,q}}\right)+\cdots,$

here $\cdots$ stands for remaining terms that do not contain $O_{1}$. In a
similar way we can compute $\frac{\partial}{\partial
s_{m}}\frac{\partial\mathcal{F}}{\partial s_{1}}$. By using the commutation
relation

$[L_{1}^{even},L_{m}^{even}]=(1-m)L_{m+1}^{even}$

we obtain the following result:

$\displaystyle\left[\frac{\partial}{\partial s_{1}},\frac{\partial}{\partial
s_{m}}\right]\mathcal{F}=$ $\displaystyle\
(m-1)\left(\mathcal{G}_{m+1}+\mathcal{L}_{m+1}^{even}\mathcal{F}\right)$
$\displaystyle+\frac{\partial O_{m}}{\partial
s_{1}}-a_{1}^{\alpha,p;\beta,q}\left(f_{\alpha,p}\frac{\partial
O_{m}}{\partial t^{\beta,q}}+f_{\beta,q}\frac{\partial O_{m}}{\partial
t^{\alpha,p}}+\frac{\partial^{2}O_{m}}{\partial t^{\alpha,p}\partial
t^{\beta,q}}\right)-\mathcal{L}_{1}^{even}O_{m}$ $\displaystyle-\frac{\partial
O_{1}}{\partial
s_{m}}+a_{m}^{\alpha,p;\beta,q}\left(f_{\alpha,p}\frac{\partial
O_{1}}{\partial t^{\beta,q}}+f_{\beta,q}\frac{\partial O_{1}}{\partial
t^{\alpha,p}}+\frac{\partial^{2}O_{1}}{\partial t^{\alpha,p}\partial
t^{\beta,q}}\right)+\mathcal{L}_{m}^{even}O_{1}.$

Therefore we obtain the symmetry $\frac{\partial\mathcal{F}}{\partial
s_{m+1}}$ by defining

$\frac{\partial\mathcal{F}}{\partial
s_{m+1}}=\frac{1}{m-1}\left[\frac{\partial}{\partial
s_{1}},\frac{\partial}{\partial
s_{m}}\right]\mathcal{F}=\mathcal{G}_{m+1}+O_{m+1}+\mathcal{L}_{m+1}^{even}\mathcal{F},$

where $O_{m+1}$ is the differential polynomial given by

$\displaystyle O_{m+1}=$ $\displaystyle\ \frac{1}{m-1}\left(\frac{\partial
O_{m}}{\partial
s_{1}}-a_{1}^{\alpha,p;\beta,q}\left(f_{\alpha,p}\frac{\partial
O_{m}}{\partial t^{\beta,q}}+f_{\beta,q}\frac{\partial O_{m}}{\partial
t^{\alpha,p}}+\frac{\partial^{2}O_{m}}{\partial t^{\alpha,p}\partial
t^{\beta,q}}\right)-\mathcal{L}_{1}^{even}O_{m}\right)$
$\displaystyle-\frac{1}{m-1}\left(\frac{\partial O_{1}}{\partial
s_{m}}-a_{m}^{\alpha,p;\beta,q}\left(f_{\alpha,p}\frac{\partial
O_{1}}{\partial t^{\beta,q}}+f_{\beta,q}\frac{\partial O_{1}}{\partial
t^{\alpha,p}}+\frac{\partial^{2}O_{1}}{\partial t^{\alpha,p}\partial
t^{\beta,q}}\right)-\mathcal{L}_{m}^{even}O_{1}\right).$

Thus we have defined recursively an infinite set of symmetries of the tau-
cover of the deformed Principal Hierarchy. Their actions on $\mathcal{F}$ can
be represented by

$\frac{\partial\mathcal{F}}{\partial
s_{m}}=\mathcal{G}_{m}+O_{m}+\mathcal{L}_{m}^{even}\mathcal{F}.$

Next we show that we can further adjust $O_{m}$ by adding certain constants
such that these symmetries satisfy the Virasoro commutation relation

$\left[\frac{\partial}{\partial s_{k}},\frac{\partial}{\partial
s_{l}}\right]=(l-k)\frac{\partial}{\partial s_{k+l}},\quad k,l\geq-1.$

######  Lemma 4.24.

There is a unique choice of constants $\kappa_{m}$ for $m\geq-1$ such that the
flows

$\frac{\partial\mathcal{F}}{\partial
s_{m}}=\mathcal{G}_{m}+O_{m}+\kappa_{m}+\mathcal{L}_{m}^{even}\mathcal{F}$

satisfy the Virasoro commutation relation

(4.24) $\left[\frac{\partial}{\partial s_{k}},\frac{\partial}{\partial
s_{l}}\right]=(l-k)\frac{\partial}{\partial s_{k+l}},\quad k,l\geq-1.$

###### Proof.

Let us first fix an arbitrary choice of $O_{m}$ and denote

$\frac{\partial\mathcal{F}}{\partial\hat{s}_{m}}=\mathcal{G}_{m}+O_{m}+\mathcal{L}_{m}^{even}\mathcal{F}.$

Then we obtain the differential polynomials $\tilde{O}_{k+l}$ such that

$\left[\frac{\partial}{\partial\hat{s}_{k}},\frac{\partial}{\partial\hat{s}_{l}}\right]\mathcal{F}=(l-k)\left(\mathcal{G}_{l+k}+\tilde{O}_{l+k}+\mathcal{L}_{l+k}^{even}\mathcal{F}\right).$

But both $\frac{\partial}{\partial\hat{s}_{l+k}}$ and
$\left[\frac{\partial}{\partial\hat{s}_{k}},\frac{\partial}{\partial\hat{s}_{l}}\right]$
are symmetries of the tau-cover of the deformed Principal Hierarchy, so we
conclude that

$\frac{\partial\mathcal{F}}{\partial
s}:=\frac{\partial\mathcal{F}}{\partial\hat{s}_{k+l}}-\frac{1}{l-k}\left[\frac{\partial}{\partial\hat{s}_{k}},\frac{\partial}{\partial\hat{s}_{l}}\right]\mathcal{F}=O_{k+l}-\tilde{O}_{k+l}$

is also a symmetry of the tau-cover of the deformed Principal Hierarchy. The
action of this symmetry on the normal coordinates has the expression

$\frac{\partial w_{\lambda}}{\partial s}=\frac{\partial^{2}}{\partial
t^{\lambda,0}\partial t^{1,0}}\left(O_{k+l}-\tilde{O}_{k+l}\right).$

Thus $\frac{\partial w_{\lambda}}{\partial s}\in\hat{\mathcal{A}}_{\geq 2}$,
and therefore such a symmetry must vanish due to the result of the
bihamiltonian cohomology [7]. Hence we conclude that

$O_{k+l}-\tilde{O}_{k+l}=c_{k,l}$

for some constant $c_{k,l}$, and this means that

(4.25)
$\left[\frac{\partial}{\partial\hat{s}_{k}},\frac{\partial}{\partial\hat{s}_{l}}\right]=(l-k)\frac{\partial}{\partial\hat{s}_{k+l}}+c_{k,l},\quad
k,l\geq-1.$

Let us denote by $\mathfrak{W}_{1}$ the Lie algebra of formal vector fields on
a line, which is an infinite dimensional Lie algebra with a basis

$e_{m}=z^{m+1}\frac{d}{dz},\quad m\geq-1.$

Then the relation (4.25) implies that the Lie algebra
$\\{\frac{\partial}{\partial\hat{s}_{m}}\\}$ defines a central extension of
$\mathfrak{W}_{1}$. It is computed in [15] that
$H^{2}(\mathfrak{W}_{1},\mathbb{R})=0$ and hence every central extension is
trivial. Therefore we can modify each $O_{m}$ by adding an appropriate
constant $\kappa_{m}$ such that the modified flows

$\frac{\partial\mathcal{F}}{\partial
s_{m}}=\mathcal{G}_{m}+O_{m}+\kappa_{m}+\mathcal{L}_{m}^{even}\mathcal{F}$

satisfy the commutation relations (4.24). Moreover, the choice of $\kappa_{m}$
is unique since $H^{1}(\mathfrak{W}_{1},\mathbb{R})=0$ (see [15]). The lemma
is proved. ∎

Thus we have proved the following theorem.

######  Theorem 4.25.

For every tau-symmetric bihamiltonian deformation of the Principal Hierarchy
associated with a semisimple Frobenius manifold, the deformed integrable
hierarchy possesses an infinite set of Virasoro symmetries. The actions of
these symmetries on the tau function $\mathcal{Z}$ are represented by

$\frac{\partial\mathcal{Z}}{\partial
s_{m}}=L_{m}^{even}\mathcal{Z}+O_{m}\mathcal{Z},\quad m\geq-1,$

where $O_{m}\in\hat{\mathcal{A}}$ are certain differential polynomials, and
the flows $\frac{\partial}{\partial s_{m}}$ satisfy the commutation relations

$\left[\frac{\partial}{\partial s_{k}},\frac{\partial}{\partial
s_{l}}\right]=(l-k)\frac{\partial}{\partial s_{k+l}},\quad k,l\geq-1.$

######  Example 4.1.

Let $M$ be the 2-dimensional Frobenius manifold defined on the orbit space of
the Weyl group of type $B_{2}$. Its potential and Euler vector field are given
by

$F=\frac{1}{2}v^{2}u+\frac{4}{15}u^{5},\quad
E=v\partial_{v}+\frac{1}{2}u\partial_{u}.$

Here $v=v^{1}$ and $u=v^{2}$ are the flat coordinates of $M$. We denote by
$\sigma_{1}$ and $\sigma_{2}$ the dual coordinates of the fiber of $\hat{M}$,
then the bihamiltonian structure $(P_{0}^{[0]},P_{1}^{[0]})$ associated with
$M$ is given by

$P_{0}^{[0]}=\frac{1}{2}\int\sigma_{1}\sigma_{2}^{1}+\sigma_{2}\sigma_{1}^{1},\quad
P_{1}^{[0]}=\frac{1}{2}\int
8u^{3}\sigma_{1}\sigma_{1}^{1}+\frac{1}{2}u\sigma_{2}\sigma_{2}^{1}+2v\sigma_{1}\sigma_{2}^{1}-\frac{1}{2}v_{x}\sigma_{1}\sigma_{2}.$

Let us first write down the Virasoro symmetry $\frac{\partial}{\partial
s_{1}}$ of the tau-cover of the Principal Hierarchy associated with $M$. The
Virasoro operator $L_{1}^{even}$ has the expression

$L_{1}^{even}=\frac{3}{16}\frac{\partial^{2}}{\partial t^{1,0}\partial
t^{2,0}}+\mathcal{L}_{1}^{even},$

where

$\mathcal{L}_{1}^{even}=\sum_{p\geq
0}\left(p+\frac{1}{4}\right)\left(p+\frac{5}{4}\right)t^{1,p}\frac{\partial}{\partial
t^{1,p+1}}+\left(p+\frac{3}{4}\right)\left(p+\frac{7}{4}\right)t^{2,p}\frac{\partial}{\partial
t^{2,p+1}}.$

Then the action of $\frac{\partial}{\partial s_{1}}$ on the genus zero free
energy $\mathcal{F}^{[0]}$ of the tau-cover of the Principal Hierarchy is
given by

$\frac{\partial\mathcal{F}^{[0]}}{\partial
s_{1}}=\frac{3}{16}f_{1,0}f_{2,0}+\mathcal{L}_{1}^{even}\mathcal{F}^{[0]}.$

Consider the bihamiltonian structure $(P_{0},P_{1})$ of the Drinfel’d-Sokolov
hierarchy [3] associated with the untwisted affine Kac-Moody algebra
$B_{2}^{(1)}$. After performing a suitable Miura type transformation we have
$P_{0}=P_{0}^{[0]}$, and the Hamiltonian operator $\mathcal{P}_{1}$ of $P_{1}$
has the expression

$\mathcal{P}_{1}=\begin{pmatrix}8u^{3}\partial_{x}+12u^{2}u_{x}&v\partial_{x}+\frac{1}{4}v_{x}\\\
v\partial_{x}+\frac{3}{4}v_{x}&\frac{1}{2}u\partial_{x}+\frac{1}{4}u_{x}\end{pmatrix}+\varepsilon^{2}\begin{pmatrix}D_{1}&D_{2}\\\
D_{3}&D_{4}\end{pmatrix}+O(\varepsilon^{4}),$

where the differential operators $D_{i}$ are given by
$D_{2}=u\partial_{x}^{3}+\frac{3}{4}u_{x}\partial_{x}^{2}$,
$D_{4}=\frac{5}{8}\partial_{x}^{3}$ and

$\displaystyle D_{1}=$
$\displaystyle\,14u^{2}\partial_{x}^{3}+42uu_{x}\partial_{x}^{2}+\left(20u_{x}^{2}+16uu_{xx}+\frac{1}{2}v_{xx}\right)\partial_{x}+12u_{x}u_{xx}+6uu^{(3)}+\frac{1}{4}v^{(3)},$
$\displaystyle D_{3}=$
$\displaystyle\,u\partial_{x}^{3}+\frac{9}{4}u_{x}\partial_{x}^{2}+\frac{3}{2}u_{xx}\partial_{x}+\frac{1}{4}u^{(3)}.$

The bihamiltonian structure $(P_{0},P_{1})$ is a deformation of
$(P_{0}^{[0]},P_{1}^{[0]})$ with central invariants $c_{1}=\frac{1}{6}$,
$c_{2}=\frac{1}{12}$ (see [8, 21]), and it determines a unique deformation of
the Principal Hierarchy associated with $M$. We can find the Virasoro symmetry
$\frac{\partial}{\partial s_{1}}$ of the tau-cover of the deformed Principal
Hierarchy by using the results developed in the present paper. It turns out
that the action of $\frac{\partial}{\partial s_{1}}$ on the tau-function
$\mathcal{Z}$ can be represented by

(4.26) $\frac{\partial\mathcal{Z}}{\partial
s_{1}}=L_{1}^{even}\mathcal{Z}+\left(\frac{1}{2}u^{2}+\frac{1}{4}v+\frac{1}{4}\varepsilon^{2}u_{xx}\right)\mathcal{Z}.$

A similar result is also given in the Example 5.5 of [26] by using the Kac-
Moody-Virasoro algebra.

##  5\. Conclusion

In the present paper, we prove the existence of an infinite set of Virasoro
symmetries for a given tau-symmetric bihamiltonian deformation of the
Principal Hierarchy associated with a semisimple Frobenius manifold. These
symmetries can be represented in terms of the tau-function $Z$ of the
integrable hierarchy in the form

(5.1) $\frac{\partial Z}{\partial s_{m}}=L_{m}Z+O_{m}Z,\quad m\geq-1.$

Note that the differential polynomials $O_{m}$ depend on the choice of the
representative, in the equivalence class of Miura type transformations, of the
deformations of the bihamiltonian structure of hydrodynamic type
$(P_{0}^{[0]},P_{1}^{[0]})$. It is proved in [9] that, for two different
choices of representatives $(P_{0},P_{1})$ and
$(\tilde{P}_{0},\tilde{P}_{1})$, the corresponding normal coordinates

$w^{\alpha}=\eta^{\alpha\beta}\frac{\partial^{2}\log Z}{\partial
t^{1,0}\partial
t^{\beta,0}},\quad\tilde{w}^{\alpha}=\eta^{\alpha\beta}\frac{\partial^{2}\log\tilde{Z}}{\partial
t^{1,0}\partial t^{\beta,0}},\quad\alpha=1,\dots,n$

of the deformed Principal Hierarchy are related by a Miura type transformation

$\tilde{w}^{\alpha}=w^{\alpha}+\eta^{\alpha\beta}\frac{\partial^{2}G}{\partial
t^{1,0}\partial t^{\beta,0}},$

where $G\in\hat{\mathcal{A}}$ is a differential polynomial, and the tau-
functions are related by the equation

(5.2) $\tilde{Z}=\exp(G)Z.$

Conversely, any differential polynomial $G\in\hat{\mathcal{A}}$ defines a
Miura type transformation for the deformed bihamiltonian structure and the
integrable hierarchy in the manner described above.

After a Miura type transformation induced from (5.2), the Virasoro symmetries
(5.1) are transformed to the form

$\frac{\partial\tilde{Z}}{\partial
s_{m}}=L_{m}\tilde{Z}+\tilde{O}_{m}\tilde{Z},\quad m\geq-1,$

where the differential polynomials $\tilde{O}_{m}$ can be computed from
$O_{m}$ and $G$.

We are going to study the problem of linearization of Virasoro symmetries in
subsequent work, i.e., to study whether it is possible to find a suitable
differential polynomial $G$ such that all the functuons $\tilde{O}_{m}$
vanish.

Let us exam the possibility of linearizing the Virasoro symmetries given in
the example of Sect. 3.4 for the one-dimensional Frobenius manifold. We want
to find a certain Miura type transformation given by (5.2) which linearizes
the Virasoro symmetry (3.26) and leaves the expression of the Virasoro
symmetry

$\frac{\partial Z}{\partial s_{-1}}=L_{-1}^{even}Z$

unchanged. It follows from these requirements that the differential degree
zero component $G_{0}$ of $G$ must satisfy the equations

$\frac{\partial G_{0}}{\partial v}=0,\quad v^{3}\frac{\partial G_{0}}{\partial
v}=-\left(3c-\frac{3}{8}\right)\frac{v^{2}}{2},$

which do not possess any solution unless $c=\frac{1}{8}$. When
$c=\frac{1}{8}$, the linearized Virasoro symmetries for this example are well
known [25], and the central invariant of the corresponding deformed
bihamiltonian structure is $\frac{1}{3}c=\frac{1}{24}$.

We can do a similar computation for Example 4.1. We want to find a Miura type
transformation given by (5.2) to linearize the Virasoro symmetry (4.26) and to
preserve the expression of the Virasoro symmetry

$\frac{\partial Z}{\partial s_{-1}}=L_{-1}^{even}Z.$

Then the differential degree zero component $G_{0}$ of $G$ must satisfy the
equations

$\frac{\partial G_{0}}{\partial v}=0,\quad uv\frac{\partial G_{0}}{\partial
u}=-\left(\frac{1}{2}u^{2}+\frac{1}{4}v\right),$

which have no solution. Therefore the Virasoro symmetries given by the
bihamiltonian structure $(P_{0},P_{1})$ in this example cannot be linearized.

In general we have the following conjecture.

######  Conjecture 5.1.

The Virasoro symmetries for a given tau-symmetric bihamiltonian deformation of
the Principal Hierarchy associated with a semisimple Frobenius manifold is
linearizable if and only if the central invariants of the corresponding
deformed bihamiltonian structure are all equal to $\frac{1}{24}$.

We will study this conjecture and give a proof of it in the paper[18].

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Si-Qi Liu,

Department of Mathematical Sciences, Tsinghua University
Beijing 100084, P.R. China
<EMAIL_ADDRESS>
Zhe Wang,

Department of Mathematical Sciences, Tsinghua University
Beijing 100084, P.R. China
<EMAIL_ADDRESS>
Youjin Zhang,

Department of Mathematical Sciences, Tsinghua University
Beijing 100084, P.R. China
<EMAIL_ADDRESS>


# Remark on an inequality for closed hypersurfaces in complete manifolds with
nonnegative Ricci curvature

Xiaodong Wang  Department of Mathematics, Michigan State University, East
Lansing, MI 48824<EMAIL_ADDRESS>
###### Abstract.

We give a simple proof of a recent result due to Agostiniani, Fogagnolo and
Mazzieri [AFM].

The following result was proved by Agostiniani, Fogagnolo and Mazzieri [AFM].

######  Theorem 1.

Let $\left(M^{n},g\right)$ ($n\geq 3$) be a complete Riemannian manifold with
nonnegative Ricci curvature and $\Omega\subset M$ a bounded open set with
smooth boundary. Then

(1)
$\int_{\partial\Omega}\left|\frac{H}{n-1}\right|^{n-1}d\sigma\geq\mathrm{AVR}\left(g\right)\left|\mathbb{S}^{n-1}\right|,$

where $H$ is the mean curvature of $\partial\Omega$ and
$\mathrm{AVR}\left(g\right)$ is the asymptotic volume ratio of $M$. Moreover,
if $\mathrm{AVR}\left(g\right)>0$, equality holds iff $M\backslash\Omega$ is
isometric to
$\left([r_{0},\infty)\times\partial\Omega,dr^{2}+\left(\frac{r}{r_{0}}\right)^{2}g_{\partial\Omega}\right)$
with

$r_{0}=\left(\frac{\left|\partial\Omega\right|}{\mathrm{AVR}\left(g\right)\left|\mathbb{S}^{n-1}\right|}\right)^{\frac{1}{n-1}}$

In particular, $\partial\Omega$ is a connected totally umbilic submanifold
with constant mean curvature.

The proof in [AFM] is highly nontrivial. It is based on the study of the
solution of the following problem

$\left\\{\begin{array}[c]{cc}\Delta u=0,&\text{on }M\backslash\Omega\\\
u=1&\text{on \ \ }\partial\Omega\\\ u\left(x\right)\rightarrow 0&\text{as
}x\rightarrow\infty,\end{array}\right.$

which exists when $\mathrm{AVR}\left(g\right)>0$. The key step consists of
showing that, with $\beta\geq\left(n-2\right)/\left(n-1\right)$

$U_{\beta}\left(t\right)=t^{-\beta\left(\frac{n-1}{n-2}\right)}\int_{u=t}\left|\nabla
u\right|^{\beta+1}d\sigma$

is monotone in $t\in(0,1]$. The geometric inequality (1) then follows by
analyzing the asymptotic behavior of $U_{\beta}\left(t\right)$ as
$t\rightarrow 0$. It is a beautiful argument.

In this short note, we show that this theorem can be proved by standard
comparison methods in Riemannian geometry.

To prove the inequality (1), we assume, without loss of generality, that
$\Omega$ has no hole, i.e. $M\backslash\Omega$ has no bounded component. In
the following we write $\Sigma=\partial\Omega$ and let $\nu$ be the outer unit
normal along $\Sigma$. For each $p\in\Sigma$ let
$\gamma_{p}\left(t\right)=\exp_{p}t\nu\left(p\right)$ be the normal geodesic
with initial velocity $\nu\left(p\right)$. We define

$\tau\left(p\right)=\sup\left\\{L>0:\gamma_{p}\text{ is minimizing on
}\left[0,L\right]\right\\}\in(0,\infty].$

It is well known that $\tau$ is a continuous function on $\Sigma$ and the
focus locus

$C\left(\Sigma\right)=\left\\{\exp_{p}\tau\left(p\right)\nu\left(p\right):\tau\left(p\right)<\infty\right\\}$

is a closed set of measure zero in $M$. Moreover the map
$\Phi\left(r,p\right)=\exp_{p}r\nu\left(p\right)$ is a diffeomorphism from

$E=\left\\{\left(r,p\right)\in\Sigma\times[0,\infty):r<\tau\left(p\right)\right\\}$

onto $\left(M\backslash\Omega\right)\backslash C\left(\Sigma\right)$. And on
$E$ the pull back of the volume form takes the form
$d\mu=\mathcal{A}\left(r,p\right)drd\sigma\left(p\right)$. We will also
understand $r$ as the distance function to $\Sigma$ and it is smooth on
$M\backslash\Omega$ away from $C\left(\Sigma\right)$. By the Bochner formula
and nonnegative Ricci curvature condition

$\displaystyle 0$ $\displaystyle=\frac{1}{2}\Delta\left|\nabla
r\right|^{2}=\left|D^{2}r\right|^{2}+\left\langle\nabla r,\nabla\Delta
r\right\rangle+Ric\left(\nabla r,\nabla r\right)$
$\displaystyle\geq\frac{\left(\Delta
r\right)^{2}}{n-1}+\frac{\partial}{\partial r}\Delta r.$

In view of the initial condition $\Delta r|_{r=0}=H$, it is standard to deduce
from the above inequality $\tau\leq\frac{n-1}{H^{-}}$ and

$\frac{\mathcal{A}^{\prime}}{\mathcal{A}}=\Delta
r\leq\frac{\left(n-1\right)H}{n-1+Hr}$

This shows that the function

$\theta\left(r,p\right)=\frac{\mathcal{A}\left(r,p\right)}{\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}}$

is non-increasing in $r$ on $[0,\tau\left(p\right))$. As
$\theta\left(0,p\right)=1$, we obtain

$\mathcal{A}\left(r,p\right)\leq\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}.$

The above analysis is standard in Riemannian geometry. We also remark that
this argument involving the Bochner formula can be replaced by an argument
involving the index form along each individual geodesic $\gamma_{p}$. For more
details, cf. [P, S] or other books on Riemannian geometry.

Therefore for any $R>0$

$\displaystyle Vol\left\\{x\in M:d\left(x,\Omega\right)<R\right\\}=$
$\displaystyle\left|\Omega\right|+\int_{\Sigma}\int_{0}^{\min\left(R,\tau\left(p\right)\right)}\mathcal{A}\left(r,p\right)drd\sigma\left(p\right)$
$\displaystyle\leq$
$\displaystyle\left|\Omega\right|+\int_{\Sigma}\int_{0}^{\min\left(R,\tau\left(p\right)\right)}\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)$
$\displaystyle\leq$
$\displaystyle\left|\Omega\right|+\int_{\Sigma}\int_{0}^{\min\left(R,\tau\left(p\right)\right)}\left(1+\frac{H^{+}\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)$
$\displaystyle\leq$
$\displaystyle\left|\Omega\right|+\int_{\Sigma}\int_{0}^{R}\left(1+\frac{H^{+}\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)$
$\displaystyle=$
$\displaystyle\left|\Omega\right|+\frac{R^{n}}{n}\int_{\Sigma}\left(\frac{H^{+}\left(p\right)}{n-1}\right)^{n-1}d\sigma\left(p\right)+O\left(R^{n-1}\right).$

Dividing both sides by
$\left|\mathbb{B}^{n}\right|R^{n}=\left|\mathbb{S}^{n-1}\right|R^{n}/n$ and
letting $R\rightarrow\infty$ yields

$\mathrm{AVR}\left(g\right)\leq\frac{1}{\left|\mathbb{S}^{n-1}\right|}\int_{\Sigma}\left(\frac{H^{+}}{n-1}\right)^{n-1}d\sigma,$

which implies (1).

We now analyze the equality case. Suppose

(2)
$\mathrm{AVR}\left(g\right)=\frac{1}{\left|\mathbb{S}^{n-1}\right|}\int_{\Sigma}\left(\frac{H^{+}}{n-1}\right)^{n-1}d\sigma>0.$

It is clear from the proof that $\tau\equiv\infty$ on the open set
$\Sigma^{+}=\left\\{p\in\Sigma:H\left(p\right)>0\right\\}$. For any
$R^{\prime}<R$ we have

$\displaystyle Vol\left\\{x\in M:d\left(x,\Omega\right)<R\right\\}$
$\displaystyle=$
$\displaystyle\left|\Omega\right|+\int_{\Sigma^{+}}\int_{0}^{R}\mathcal{A}\left(r,p\right)drd\sigma\left(p\right)+\int_{\Sigma\backslash\Sigma^{+}}\int_{0}^{\min\left(R,\tau\left(p\right)\right)}\mathcal{A}\left(r,p\right)drd\sigma\left(p\right)$
$\displaystyle\leq$
$\displaystyle\left|\Omega\right|+\int_{\Sigma^{+}}\int_{0}^{R}\theta\left(r,p\right)\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)+\int_{\Sigma\backslash\Sigma^{+}}\int_{0}^{R}drd\sigma\left(p\right)$
$\displaystyle\leq$
$\displaystyle\left|\Omega\right|+\int_{\Sigma^{+}}\int_{R^{\prime}}^{R}\theta\left(r,p\right)\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)$
$\displaystyle+\int_{\Sigma^{+}}\int_{0}^{R^{\prime}}\theta\left(r,p\right)\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)+O\left(R\right)$
$\displaystyle\leq$
$\displaystyle\left|\Omega\right|+\int_{\Sigma^{+}}\theta\left(R^{\prime},p\right)\int_{R^{\prime}}^{R}\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)$
$\displaystyle+\int_{\Sigma^{+}}\int_{0}^{R^{\prime}}\theta\left(r,p\right)\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}drd\sigma\left(p\right)+O\left(R\right).$

Dividing both sides by
$\left|\mathbb{B}^{n}\right|R^{n}=\left|\mathbb{S}^{n-1}\right|R^{n}/n$ and
letting $R\rightarrow\infty$ yields

$\mathrm{AVR}\left(g\right)\leq\frac{1}{\left|\mathbb{S}^{n-1}\right|}\int_{\Sigma^{+}}\left(\frac{H\left(p\right)}{n-1}\right)^{n-1}\theta\left(R^{\prime},p\right)d\sigma\left(p\right).$

Letting $R^{\prime}\rightarrow\infty$ yields

$\mathrm{AVR}\left(g\right)\leq\frac{1}{\left|\mathbb{S}^{n-1}\right|}\int_{\Sigma^{+}}\left(\frac{H}{n-1}\right)^{n-1}\theta_{\infty}d\sigma,$

where
$\theta_{\infty}\left(p\right)=\lim_{r\rightarrow\infty}\theta\left(r,p\right)\leq
1$. As we have equality (2) we must have $\theta_{\infty}\left(p\right)=1$ for
a.e. $p\in\Sigma^{+}$. It follows that

$\mathcal{A}\left(r,p\right)=\left(1+\frac{H\left(p\right)}{n-1}r\right)^{n-1}\text{
on }[0,\infty)$

for a.e. $p\in\Sigma^{+}$. By continuity the above identity holds for all
$p\in\Sigma^{+}$.

Inspecting the comparison argument, we must have on
$\Phi\left([0,\infty)\times\Sigma^{+}\right)$

$\displaystyle D^{2}r$ $\displaystyle=\frac{\Delta
r}{n-1}g=\frac{H}{n-1+Hr}g,$ $\displaystyle Ric\left(\nabla r,\nabla r\right)$
$\displaystyle=0.$

As $Ric\geq 0$, it follows that $Ric\left(\nabla r,\cdot\right)=0$. From the
1st equation above $\Sigma^{+}$ is an umbilic hypersurface, i.e. the 2nd
fundamental form $\Pi=\frac{H}{n-1}g_{\Sigma^{+}}$. Working with an
orthonormal frame $\left\\{e_{0}=\nu,e_{1},\cdots,e_{n-1}\right\\}$ along
$\Sigma^{+}$ we have by the Codazzi equation, with $1\leq i,j,k\leq n-1$

$R\left(e_{k},e_{j},e_{i},\nu\right)=\Pi_{ij,k}-\Pi_{ik,j}=\frac{1}{n-1}\left(H_{k}\delta_{ij}-H_{j}\delta_{ik}\right).$

Taking trace over $i$ and $k$ yields

$-\frac{n-2}{n-1}H_{j}=Ric\left(e_{j},\nu\right)=0.$

As a result $H$ is locally constant on $\Sigma^{+}$. Therefore $\Sigma^{+}$
must be the union of several components of $\Sigma$. We know that $\Phi$ is a
diffeomorphism form $[0,\infty)\times\Sigma^{+}$ onto its image and the
pullback metric $\Phi^{\ast}g$ takes the following form

$dr^{2}+h_{r},$

where $h_{r}$ is a $r$-dependent family of metrics on $\Sigma^{+}$ and
$h_{0}=g_{\Sigma^{+}}$. We have

$D^{2}r=\frac{H}{n-1+Hr}g.$

In terms of local coordinates $\left\\{x_{1},\cdots,x_{n-1}\right\\}$ on
$\Sigma^{+}$ the above equation implies

$\frac{1}{2}\frac{\partial}{\partial r}h_{ij}=\frac{H}{n-1+Hr}h_{ij}.$

Therefore $h_{r}=\left(1+\frac{H}{n-1}r\right)^{2}g_{\Sigma^{+}}$. This proves
that $\Phi\left([0,\infty)\times\Sigma^{+}\right)$ is isometric to
$\left([r_{0},\infty)\times\Sigma^{+},dr^{2}+\left(\frac{r}{r_{0}}\right)^{2}g_{\Sigma^{+}}\right)$,
where $r_{0}=\frac{n-1}{H}$.

Since $M$ has nonnegative Ricci curvature and Euclidean volume growth, it has
only one end by the Cheeger-Gromoll theorem. Therefore $\Sigma^{+}$ is
connected and if $\Sigma$ has other components besides $\Sigma^{+}$, they all
bound bounded components of $M\backslash\Omega$.

If we have the stronger identity

$\mathrm{AVR}\left(g\right)=\frac{1}{\left|\mathbb{S}^{n-1}\right|}\int_{\Sigma}\left|\frac{H}{n-1}\right|^{n-1}d\sigma>0,$

inspecting the proof of the inequality (1) shows that we must have $H\geq 0$
on $\Sigma$. Then $\overline{\Omega}$ is compact Riemannian manifold with mean
convex boundary. It is a classic fact that its boundary must be connected, see
[I, K] or [HW] for an analytic argument. Therefore $\Sigma=\Sigma^{+}$ is
connected and $M\backslash\Omega$ is isometric to
$\left([r_{0},\infty)\times\Sigma,dr^{2}+\left(\frac{r}{r_{0}}\right)^{2}g_{\Sigma}\right)$.

Acknowledgement. I would like to thank Fengbo Hang for helpful discussions.

## References

  * [AFM] V. Agostiniani; M. Fogagnolo; L. Mazzieri. Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature. Invent. Math. 222 (2020), no. 3, 1033-1101.
  * [HW] F. Hang; X. Wang. Vanishing sectional curvature on the boundary and a conjecture of Schroeder and Strake, Pacific J. Math. 232 (2007), no. 2, 283-287.
  * [I] R. Ichida. Riemannian manifolds with compact boundary. Yokohama Math. J. 29 (1981), no. 2, 169-177.
  * [K] A. Kasue. Ricci curvature, geodesics and some geometric properties of Riemannian manifolds with boundary. J. Math. Soc. Japan 35 (1983), no. 1, 117-131.
  * [P] P. Petersen. Riemannian Geometry. Third edition. Graduate Texts in Mathematics, 171. Springer, Cham, 2016.
  * [S] T. Sakai. Riemannian geometry. Translations of Mathematical Monographs, 149. American Mathematical Society, Providence, RI, 1996.


# Selective Light Field Refocusing for Camera Arrays Using Bokeh Rendering and
Superresolution

Yingqian Wang, Jungang Yang, Yulan Guo, Chao Xiao, Wei An This work was
partially supported by the National Natural Science Foun-dation of China (Nos.
61401474, 61602499 and 61471371), the Hunan Pro-vincial National Science
Foundation (No. 2016JJ3025), the National Postdoc-toral Program for Innovative
Talents (No. BX201600172), and the China Postdoctoral Science Foundation.
(Corresponding author: Jungang Yang) Y. Wang, J. Yang, Y. Guo, C. Xiao, and W.
An are with the College of Electronic Science and Technology, National
University of Defense Technology, Changsha, China. (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>anwei@nudt.edu.cn). Y. Guo is also with the School of
Electronics and Communication Engineering, Sun Yat-sen University, Guangzhou,
China.

###### Abstract

Camera arrays provide spatial and angular information within a single
snapshot. With refocusing methods, focal planes can be altered after exposure.
In this letter, we propose a light field refocusing method to improve the
imaging quality of camera arrays. In our method, the disparity is first
estimated. Then, the unfocused region (bokeh) is rendered by using a depth-
based anisotropic filter. Finally, the refocused image is produced by a
reconstruction-based super-resolution approach where the bokeh image is used
as a regularization term. Our method can selectively refocus images with
focused region being super-resolved and bokeh being aesthetically rendered.
Our method also enables post-adjustment of depth of field. We conduct
experiments on both public and self-developed datasets. Our method achieves
superior visual performance with acceptable computational cost as compared to
other state-of-the-art methods. Code is available at
https://github.com/YingqianWang/Selective-LF-Refocusing.

###### Index Terms:

Bokeh, camera array, depth of field (DoF), light field (LF), refocusing,
super-resolution (SR).

##  I Introduction

Although miniature cameras (e.g., smartphone cameras) have become increasingly
popular in recent years, professional photographers still prefer traditional
digital single lens reflex (DSLR) to generate aesthetical photographs. Due to
the large-size image sensors and large-aperture lens, images produced by DSLRs
are high in resolution, high in signal-to-noise ratio (SNR), and can be
shallow in depth of field (DoF). That is, only a limited range of depths are
in focus, leaving objects in other depths suffering from varying degrees of
blurs (termed as bokeh). Using bokeh effect properly can blur out non-
essential elements in foreground or background, and guide the visual attention
of audiences to major objects in a photograph.

However, most professional DSLRs tend to be bulk and expensive due to their
high-quality lenses and sensors. Recently, with the rapid development of light
field (LF) imaging [1, 2, 3], it becomes possible to use low-cost cameras to
take photographs with a high quality as DSLRs. Several designs [4, 5, 6, 7, 8,
9, 10, 11] have been proposed to capture LF. Among them, plenoptic cameras [5]
place an array of microlens between main lens and image sensors to capture the
angular information of a scene. However, due to the limitation of sensor
resolution, one can only opt for a dense angular sampling [5], or a dense
spatial sampling [6]. In contrast, camera arrays [7, 8, 9, 10] set several
independent cameras in a regular grid, and therefore have an improved image
resolution. Wu et al. [12] claim that although early camera array systems [7,
8, 9] are bulk and hardware-intensive, these LF acquisition systems have a
bright prospective with camera miniaturization being exploited and small-size
camera arrays [10, 11] being developed. Compared to plenoptic cameras, camera
arrays have wider baselines and sparser angular samplings. Consequently,
traditional refocusing methods [8, 9, 13] face aliasing problems in the bokeh
regions.

LF reconstruction methods [2, 14, 15] are widely used to address the aliasing
problem. These methods interpolate novel views between existing views, and use
the synthetic LF to refocus sub-images. Specifically, methods [2, 14, 15] can
simultaneously increase the number of viewpoints and image resolution. Since
there is a trade-off between the density of views and computational cost,
methods [2, 14, 15] either suffer from a high computational burden, or remain
aliasing artifacts to some degree. Bokeh rendering methods [16, 17, 18, 19,
20] provide novel paradigms. Methods [16, 17] map source pixels onto circles
of confusion (CoC) [16] using the estimated depth information, and blend CoC
in the order of depth. Although [16, 17] can eliminate aliasing artifact, the
required depth sorting process is costly . Methods [18, 19, 20] filter images
according to the CoC size, and achieve a compromise between computational cost
and bokeh performance. In summary, methods [16, 17, 18, 19, 20] can improve
bokeh performance, but cannot increase image resolution.

In this paper, we propose an LF refocusing method to improve the imaging
quality of camera arrays. The outline of our method is illustrated in Fig. 1.
The main contributions of this paper can be summarized as follows: (1) Bokeh
rendering and super-resolution (SR) are selectively integrated into one
scheme, i.e., our method can simultaneously improve bokeh performance and
increase image resolution. (2) DoF can be adjusted in a post-processing mode,
which is important for producing aesthetical photographs on different scenes,
and it cannot be achieved by DSLRs or previous refocusing methods. (3) We
conducted experiments on a self-developed dataset. Images generated by our
method are more similar to those generated by a DSLR than existing methods.

Figure 1: Outline of our method. The disparity map is first estimated. The
bokeh is then rendered on the extracted center-view sub-image. Finally, the
refocused image is generated by a reconstruction-based SR approach where the
rendered bokeh image is used as a regularization term.

##  II Reconstruction-based SR model

Sub-images captured by camera arrays suffer from shearing shifts (caused by
parallax), blurring (caused by optical distortions), and down-sampling (caused
by low-resolution image sensors). Considering these factors, we formulate the
degradation model [21] of a camera array as

$\boldsymbol{y}_{k}=\boldsymbol{D}\boldsymbol{H}\boldsymbol{F}_{i,k}\boldsymbol{x}+\boldsymbol{n}_{k},$
(1)

where $\boldsymbol{x}$, $\boldsymbol{y}_{k}$, and $\boldsymbol{n}_{k}$ denote
high-resolution (HR) image, sub-image captured by the $k^{th}$ camera, and
noise of the $k^{th}$ sub-image, respectively. $N$ represents the number of
cameras in the array, and $n$ represents depth resolution (i.e., total number
of depth layers). $\boldsymbol{H}$, $\boldsymbol{D}$, and
$\boldsymbol{F}_{i,k}$ denote optical blurring, down-sampling, and shifts
(dependent of depth $i$ and viewpoint $k$), respectively. Since the main task
of SR reconstruction is to estimate $\boldsymbol{x}$ to suit the degradation
model in Eq. (1), we formulate the SR model as a task to minimize the
following function:

$\displaystyle\hat{\boldsymbol{x}}$
$\displaystyle=\underset{\boldsymbol{x}}{\arg\min}\\{\sum_{k=1}^{N}\left\|\left(\boldsymbol{1}-\boldsymbol{\omega}_{b}\right)\varodot\left(\boldsymbol{y}_{k}-\boldsymbol{D}\boldsymbol{H}\boldsymbol{F}_{i,k}\boldsymbol{x}\right)\right\|_{2}^{2}$
(2)
$\displaystyle+\lambda_{b}J_{b}\left(\boldsymbol{x}\right)+\lambda_{\textit{BTV}}J_{\textit{BTV}}\left(\boldsymbol{x}\right)\\},\qquad
i=1,2,\cdots,n.$

The first term in Eq. (2) denotes the distance between observations and an
ideal HR image. $J_{b}\left(\boldsymbol{x}\right)$ is the bokeh regularization
term and $J_{\textit{BTV}}\left(\boldsymbol{x}\right)$ is the bilateral total
variation (BTV) regularization term. Readers are referred to [21] for more
details of BTV regularization. $\lambda_{b}$ and $\lambda_{\textit{BTV}}$ are
regularization weights. $\boldsymbol{1}$ denotes a vector with all elements
equal to 1, and $\varodot$ denotes element-wise multiplication.
$\boldsymbol{\omega}_{b}$ is a depth-based and spatial-variant weight vector
where unfocused regions share larger values.
$J_{b}\left(\boldsymbol{x}\right)$ can be expressed as

$J_{b}\left(\boldsymbol{x}\right)=\left\|\boldsymbol{\omega}_{b}\varodot\left(\boldsymbol{x}-\boldsymbol{x}_{b}\right)\right\|_{2}^{2},$
(3)

where $\boldsymbol{x}_{b}$ represents the bokeh image. More details on
$\boldsymbol{x}_{b}$ and $\boldsymbol{\omega}_{b}$ will be presented in
Section III. We use the gradient descend approach [21] to approximate the
optimal solution. The settings of step size and number of iterations (NoI)
will be introduced in Section IV. Readers can refer to [22] for the
convergence issue of SR process.

##  III Bokeh Rendering

As described in Section II, the key step of our method is to generate the
bokeh image $\boldsymbol{x}_{b}$. Assuming that point $p$ is out of focus and
corresponds to a CoC in the image, the radius of CoC can be calculated as

$r=\left|\frac{f^{2}\left(\gamma_{f}-\gamma_{p}\right)}{2F\gamma_{p}\left(\gamma_{f}-f\right)}\right|,$
(4)

where $\gamma_{f}$ and $\gamma_{p}$ represent the depth of focus and the depth
of $p$, respectively. $f$ is the focal length, $F$ is the F-number of the
lens. From the LF model presented in [23], the depth can be expressed as
$\gamma=fB/d$, where $B$ is the length of the baseline, and $d$ represents the
disparity. Consequently, we can rewrite Eq. (4) as

$r=\left|\frac{f\left(d_{p}-d_{f}\right)}{2F\left(B-d_{f}\right)}\right|=K\left|d_{p}-d_{f}\right|.$
(5)

Note that, the radius of CoC is proportion to the absolute disparity
difference between point $p$ and the focus. Since $f$, $F$, $B$, and $d_{f}$
are constant during a bokeh rendering process, we consider
$K=f/2F\left(B-d_{f}\right)$ as bokeh intensity which represents the overall
degree of bokeh and reflects DoF. A large $K$ corresponds to a strong bokeh
and accordingly, a shallow DoF.

Figure 2: An illustration of the bokeh rendering process. To calculate the
intensity at Q, contributions from the surrounding CoC have to be combined.

We attribute bokeh rendering to anisotropic filtering. As shown in Fig. 2, $Q$
is surrounded by four CoC centered at $P_{1}$ to $P_{4}$. Since intensities in
a CoC are uniformly distributed, the contribution from $P_{i}$ to $Q$ can be
calculated as $I_{P_{i}\rightarrow Q}=\nicefrac{{I_{P_{{}_{i}}}}}{{\pi
r_{P_{i}}^{2}}},\>\left(\varphi_{P_{i}Q}\leq r_{P_{i}}\right)$, where
$r_{P_{i}}$ is the radius of CoC centered at $P_{i}$, $I_{P_{i}}$ is the
intensity at $P_{i}$ before bokeh rendering, $\varphi_{P_{i}Q}$ represents the
distance between $P_{i}$ and $Q$. Note that, $P_{i}$ has an impact on $Q$ only
if $Q$ is within the CoC of $P_{i}$, i.e., $\varphi_{P_{i}Q}\leq r_{P_{i}}$.
To calculate the intensity at $Q$, the contributions of surrounding points
have to be combined:

$I_{Q}=\sum_{P_{i}\in\varOmega_{Q}}\omega_{P_{i}}I_{P_{i}\rightarrow Q},$ (6)

where $\varOmega_{Q}=\left\\{P_{i}|\varphi_{P_{i}Q}\leq r_{max}\right\\}$
represents the set of points around $Q$, $r_{max}$ is the maximum radius of
CoC in the image. Note that, some points in $\varOmega_{Q}$ may have no
contribution to $Q$, e.g., point $P_{4}$ in Fig. 2. Taken this into
consideration, weight $\omega_{P_{i}}$ is calculated as

$\omega_{P_{i}}=\begin{cases}\nicefrac{{1}}{{\pi
r_{P_{i}}^{2}}},&\varphi_{P_{i}Q}\leq r_{P_{i}}\\\
0,&r_{P_{i}}\leq\varphi_{P_{i}Q}\leq r_{max}\end{cases}.$ (7)

An anisotropic filter can be used to render the extracted center view based on
Eqs. (6) and (7). By using bicubic interpolation, the bokeh image
$\boldsymbol{x}_{b}$ is generated by up-sampling the rendered center view to
the target resolution.

Another significant step is to calculate $\boldsymbol{\omega}_{b}$ used in
Eqs. (2) and (3). Since the blurring degree in an image is determined by the
radius of CoC, we calculate $\boldsymbol{\omega}_{b}$ in two steps.

Step 1: Normalize the radius to interval $\left[0,1\right]$ by
$\eta_{p}=\nicefrac{{\left(r_{p}-r_{min}\right)}}{{\left(r_{max}-r_{min}\right)}}$,
where $r_{min}$ is the minimum radius of CoC in the image.

Step 2: To divide $\eta_{p}$ into two parts (focus and bokeh) appropriately,
we use sigmoid function to transform $\eta_{p}$ into $\omega_{p}$ according to
$\omega_{p}=\nicefrac{{1}}{{\left(1+exp\left\\{-a\left(\eta_{p}-b\right)\right\\}\right)}}$,
where $a$ is a decaying factor, and $b$ is the threshold. Finally,
$\boldsymbol{\omega}_{b}$ can be obtained by traversing all pixels and re-
ordering $\omega_{p}$ into a vector.

##  IV Experiments

Extensive experiments are conducted on both public and self-developed LF
datasets. The 3$\times$3 sub-images of the Stanford LF dataset [24] were first
down-sampled by a factor of 2 and then used to test different methods. We
compared our method with a refocusing method [9], an LF reconstruction method
[14], and a bokeh rendering method [20]. For method [14], the 3$\times$3 LF
input was angularly rendered to 5$\times$5 views and spatially super-resolved
to the original resolution. To achieve fair comparison between method [20] and
our method, we used the disparity estimation approach proposed in [23]. We
consider the HR center-view sub-image as the groundtruth and quantitatively
evaluate the SR performance by calculating PSNRs of focused regions between
the results of each method and the groundtruth. Since there is no groundtruth
for the bokeh, we follow the existing bokeh rendering methods [16, 17, 18, 19,
20], and use visual inspection for evaluation.

For the self-developed LF dataset, we adopted a scanning scheme since view-by-
view scanning in static occasion is equivalent to a single shot by a camera
array [12]. We installed an iPhone 6S camera (with a $4.8mm\times 3.6mm$ image
sensor and an $F=2.2$, $f=29mm$ lens) on a gantry. Similar to the approach
proposed in [25], we shifted the camera to 9 intersections of a $2\times 2$
grid (the size of each grid is $5\times 5mm^{2}$), and scanned scenes in our
laboratory. The captured images are calibrated by using the method in [26]. We
then used a Canon EOS 5D Mark IV DSLR (with a $36mm\times 24mm$ image sensor
and an $F=2.2$, $f=50mm$ lens) to provide “groundtruth” images.

The parameter setting of our method is listed in Table I. All the listed
parameters are tuned through experiments to achieve good performance.
Specifically, the performance of our method can be improved gradually as NoI
increases. To achieve a compromise between visual performance and
computational cost, we set NoI to 10 in our implementation. Additionally, we
adopted the same settings of $J_{\textit{BTV}}\left(\boldsymbol{x}\right)$ as
in [21]. We observed through experiments that the final result was insensitive
to these parameters except $b$, since $b$ has a dominant influence on the
classification of focus and bokeh.

All algorithms were implemented in MATLAB on a PC with a 2.40 GHz CPU (Intel
Core i7-5500U) and a 12 GB RAM.

###  IV-A Visual Performance Comparison

(a)

(b)

(c)

(d)

Figure 3: Results achieved on scenes Knights (top), Cards (middle), and Chess (bottom) in the Stanford LF dataset [24]. (a) Vaish et al. [9], (b) Farrugia et al. [14], (c) Liu et al. [20], (d) ours ($K$=2). Note that, sharpness of the focused region (images in red boxes) is evaluated quantitatively using PSNR. Table I: Parameter Setting Parameters |  $\lambda_{b}$ |  $\lambda_{BTV}$ |  Step Size |  NoI |  $a$ |  $b$
---|---|---|---|---|---|---
Values |  5 |  0.2 |  0.1 |  10 |  15 |  0.3

(a) Center view

(b) Vaish et al. [9]

(c) Farrugia et al. [14]

(d) Liu et al. [20]

(e) Canon DSLR

(f) Ours ($K$=3)

Figure 4:  Results achieved on scene Dolls in the self-developed LF dataset.
Note that, the photograph taken by a Canon DSLR is used as the “groundtruth”.

From Figs. 3 and 4, we can observe that obvious aliasing artifacts are
produced by [9] although shallow DoF is achieved. Farrugia et al. [14] improve
the image resolution and approximate the real bokeh to a certain extent. That
is, the aliasing artifact is alleviated but not eliminated. Liu et al. [20]
address the aliasing problem but do not improve image resolution, leading to
blurring in focused region. Compared to [9], [14], and [20], our method
achieves the highest PSNR in focused regions and promising performance in the
bokeh. Note that, the bokeh rendered by our method (Fig. 4(f)) is similar to
that generated by the DSLR (Fig. 4(e)). Although the focused region in Fig.
4(f) is not as sharp as that in Fig. 4(e), the natural gap caused by unideal
low-cost cameras is partially filled by using our method.

###  IV-B Adjustable DoF

(a) $K$=1

(b) $K$=2

(c) $K$=3

Figure 5: Adjustable DoF achieved by our method on scene Knights. Note that,
the middle knights are in focus in the top three images and the side knights
are in focus in the bottom three images

We use our method to refocus images to different depths with different
settings of $K$. As shown in Fig. 5, we can change both the depth in focus and
DoF in a post-processing mode. Note that, DoF post-adjustment is unavailable
for DSLRs, traditional refocusing methods [9, 8, 13], and LF reconstruction
methods [14, 2, 15].

###  IV-C Influence of Errors in Disparity Estimation

(a)

(b)

(c)

(d)

Figure 6: Influence of inaccurate disparity estimation on scene Box (top) and
Sideboard (bottom) in the HCI LF dataset [27]. (a) Groundtruth disparity, (b)
disparity estimated by the method in [23], (c) results achieved by our method
using the groundtruth disparity, (d) results achieved by our method using the
estimated disparity in (b).

We compared the visual performance of our method using the disparity estimated
by [23] to that using the groundtruth disparity (provided by the HCI LF
dataset [27]). Results show that the visual performance of our method is
partially influenced by the errors in disparity estimation. As shown in Fig.
6, disparity errors on the boundary of the focused region lead to some
artifacts. In contrast, our method is robust to minor disparity errors both in
bokeh and in focus.

###  IV-D Execution Time

Table II: Comparison of Execution Time |  Vaish et al. [9] |  Farrugia et al. [14] |  Liu et al. [20] |  Ours ($K$=2)
---|---|---|---|---
Knights ($512\times 512$) |  1.469 s |  2518 s |  42.59 s |  165.5 s
Cards ($512\times 512$) |  1.446 s |  1732 s |  38.27 s |  174.0 s
Chess ($700\times 400$) |  1.902 s |  2238 s |  48.74 s |  184.0 s
Dolls ($1024\times 1024$) |  6.218 s |  9548 s |  153.5 s |  655.8 s
Average of dataset [24] |  1.789 s |  2381 s |  46.83 s |  185.2 s
Figure 7: Execution time of our method achieved on scene Knights with
different settings of $K$ and NoI.

It can be observed from Table II that the shortest running time is achieved by
[9] since no depth estimation, pixel mapping or filtering is required. In
contrast, [14] is the most time-consuming method, and its running time
increases dramatically as image resolution increases. Method in [20] also
achieve a relative short execution time because it does not require view
rendering or SR, but the resolution is consequently limited. Since an
efficient bokeh rendering scheme is adopted in our method, the execution time
of our method is only $5\%$ to $10\%$ of that of [14].

As the execution time of our method may depend on $K$ (which determines the
size of the filter kernel) and NoI, we further investigate the execution time
of our method with different settings of $K$ and NoI. As shown in Fig. 7, NoI
has much more influence on the overall execution time than $K$ because the
adopted bokeh rendering approach is computationally efficient. Since a too
large bokeh (i.e., too shallow DoF) will destroy the esthetic value of a
photograph, we need to determine the DoF empirically based on the scenario.
Generally, a shallow DoF (e.g., $K$=3) is preferred for close-shot photographs
(e.g., scene Knights), and a deep DoF (e.g., $K$=1) is suggested for distant
scenes (e.g., landscape photograph).

##  V Conclusion

In this paper, we propose a method to refocus images captured by a camera
array. We formulate an SR model and design an anisotropic filter to increase
the image resolution and render the bokeh. The superiority of our method is
demonstrated by experiments on both public and self-developed datasets. Our
method enables camera arrays to produce aesthetical photographs with
acceptable computational cost.

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