# Complex connections on conformal Kähler manifolds with Norden metric

Marta Teofilova

###### Abstract.

An eight-parametric family of complex connections on a class complex manifolds
with Norden metric is introduced. The form of the curvature tensor with
respect to each of these connections is obtained. The conformal group of the
considered connections is studied and some conformal invariants are obtained.

_2010 Mathematics Subject Classification_ : 53C15, 53C50.

_Keywords_ : complex connection, complex manifold, Norden metric.

##  1\. Introduction

Almost complex manifolds with Norden metric were first studied by A. P. Norden
[9] and are introduced in [4] as generalized $B$-manifolds. A classification
of these manifolds with respect to the covariant derivative of the almost
complex structure is obtained in [1] and two equivalent classifications are
given in [2, 3].

An important problem in the geometry of almost complex manifolds with Norden
metric is the study of linear connections preserving the almost complex
structure or preserving both, the structure and the metric. The first ones are
called almost complex connections, and the second ones are known as natural
connections. A special type of natural connection is the canonical one. In [2]
it is proved that on an almost complex manifold with Norden metric there
exists a unique canonical connection. The canonical connection (called also
the $B$-connection) and its conformal group on a conformal Kähler manifold
with Norden metric are studied in [3].

In [11] we have obtained a two-parametric family of complex connections on a
conformal Kähler manifold with Norden metric and have proved that the
curvature tensors corresponding to these connections coincide with the
curvature tensor of the canonical connection.

In the present work we continue our research on complex connections on complex
manifolds with Norden metric by focusing our attention on the class of the
conformal Kähler manifolds, i.e. manifolds which are conformally equivalent to
Käher manifolds with Norden metric. We introduce an eight-parametric family of
complex connections on such manifolds and consider their curvature properties.
We also study the conformal group of these connections and obtain some
conformal invariants. In the last section we give an example of a four-
dimensional conformal Kähler manifold with Norden metric, on which the
considered complex connections are flat.

##  2\. Preliminaries

Let $(M,J,g)$ be a $2n$-dimensional almost complex manifold with Norden
metric, i.e. $J$ is an almost complex structure and $g$ is a pseudo-Riemannian
metric on $M$ such that

$J^{2}x=-x,\qquad g(Jx,Jy)=-g(x,y)$ (2.1)

for all differentiable vector fields $x$, $y$ on $M$, i.e.
$x,y\in\mathfrak{X}(M)$.

The associated metric $\widetilde{g}$ of $g$ is given by
$\widetilde{g}(x,y)=g(x,Jy)$ and is a Norden metric, too. Both metrics are
necessarily neutral, i.e. of signature $(n,n)$.

If $\nabla$ is the Levi-Civita connection of the metric $g$, the fundamental
tensor field $F$ of type $(0,3)$ on $M$ is defined by

$F(x,y,z)=g\left((\nabla_{x}J)y,z\right)$ (2.2)

and has the following symmetries

$F(x,y,z)=F(x,z,y)=F(x,Jy,Jz).$ (2.3)

Let $\left\\{e_{i}\right\\}$ ($i=1,2,\ldots,2n$) be an arbitrary basis of
$T_{p}M$ at a point $p$ of $M$. The components of the inverse matrix of $g$
are denoted by $g^{ij}$ with respect to the basis $\left\\{e_{i}\right\\}$.
The Lie 1-forms $\theta$ and $\theta^{\ast}$ associated with $F$, and the Lie
vector $\Omega$, corresponding to $\theta$, are defined by, respectively

$\theta(x)=g^{ij}F(e_{i},e_{j},x),\qquad\theta^{\ast}=\theta\circ
J,\qquad\theta(x)=g(x,\Omega).$ (2.4)

The Nijenhuis tensor field $N$ for $J$ is given by [7]

$N(x,y)=[Jx,Jy]-[x,y]-J[Jx,y]-J[x,Jy].$

It is known [8] that the almost complex structure is complex if and only if it
is integrable, i.e. iff $N=0$.

A classification of the almost complex manifolds with Norden metric is
introduced in [1], where eight classes of these manifolds are characterized
according to the properties of $F$. The three basic classes $\mathcal{W}_{i}$
($i=1,2,3$) are given by

$\bullet$ the class $\mathcal{W}_{1}$:

$\begin{array}[]{l}F(x,y,z)=\frac{1}{2n}\left[g(x,y)\theta(z)+g(x,Jy)\theta(Jz)\right.\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\quad\qquad\qquad\quad\left.+g(x,z)\theta(y)+g(x,Jz)\theta(Jy)\right];\end{array}$
(2.5)

$\bullet$ the class $\mathcal{W}_{2}$ of the _special complex manifolds with
Norden metric_ :

$F(x,y,Jz)+F(y,z,Jx)+F(z,x,Jy)=0,\quad\theta=0\quad\Leftrightarrow\quad
N=0,\quad\theta=0;$ (2.6)

$\bullet$ the class $\mathcal{W}_{3}$ of the _quasi-Kähler manifolds with
Norden metric_ :

$F(x,y,z)+F(y,z,x)+F(z,x,y)=0.$ (2.7)

The special class $\mathcal{W}_{0}$ of the _Kähler manifolds with Norden
metric_ is characterized by the condition $F=0$ ($\nabla J=0$) and is
contained in each one of the other classes.

Let $R$ be the curvature tensor of $\nabla$, i.e.
$R(x,y)z=\nabla_{x}\nabla_{y}z-\nabla_{y}\nabla_{x}z-\nabla_{\left[x,y\right]}z$.
The corresponding (0,4)-type tensor is defined by
$R(x,y,z,u)=g\left(R(x,y)z,u\right)$.

A tensor $L$ of type (0,4) is called a _curvature-like_ tensor if it has the
properties of $R$, i.e.

$\begin{array}[]{l}L(x,y,z,u)=-L(y,x,z,u)=-L(x,y,u,z),\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ L(x,y,z,u)+L(y,z,x,u)+L(z,x,y,u)=0.\end{array}$ (2.8)

The Ricci tensor $\rho(L)$ and the scalar curvatures $\tau(L)$ and
$\tau^{\ast}(L)$ of $L$ are defined by:

$\begin{array}[]{c}\rho(L)(y,z)=g^{ij}L(e_{i},y,z,e_{j}),\vskip 6.0pt plus
2.0pt minus 2.0pt\\\
\tau(L)=g^{ij}\rho(L)(e_{i},e_{j}),\quad\tau^{\ast}(L)=g^{ij}\rho(L)(e_{i},Je_{j}).\end{array}$
(2.9)

A curvature-like tensor $L$ is called a _Kähler tensor_ if

$L(x,y,Jz,Ju)=-L(x,y,z,u).$ (2.10)

Let $S$ be a tensor of type (0,2). We consider the following tensors [3]:

$\begin{array}[]{l}\psi_{1}(S)(x,y,z,u)=g(y,z)S(x,u)-g(x,z)S(y,u)\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\
\phantom{\psi_{1}(S)(x,y,z,u)}+g(x,u)S(y,z)-g(y,u)S(x,z),\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ \psi_{2}(S)(x,y,z,u)=g(y,Jz)S(x,Ju)-g(x,Jz)S(y,Ju)\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\phantom{\psi_{1}(S)(x,y,z,u)}+g(x,Ju)S(y,Jz)-g(y,Ju)S(x,Jz),\vskip 6.0pt plus
2.0pt minus 2.0pt\\\
\pi_{1}=\frac{1}{2}\psi_{1}(g),\qquad\pi_{2}=\frac{1}{2}\psi_{2}(g),\qquad\pi_{3}=-\psi_{1}(\widetilde{g})=\psi_{2}(\widetilde{g}).\end{array}$
(2.11)

The tensor $\psi_{1}(S)$ is curvature-like if $S$ is symmetric, and the tensor
$\psi_{2}(S)$ is curvature-like if $S$ is symmetric and hybrid with respect to
$J$, i.e. $S(x,Jy)=S(y,Jx)$. In the last case the tensor
$\\{\psi_{1}-\psi_{2}\\}(S)$ is Kählerian. The tensors $\pi_{1}-\pi_{2}$ and
$\pi_{3}$ are also Kählerian.

The usual conformal transformation of the Norden metric $g$ (conformal
transformation of type I [3]) is defined by

$\overline{g}=e^{2u}g,$ (2.12)

where $u$ is a pluriharmonic function, i.e. the 1-form $du\circ J$ is closed.

A $\mathcal{W}_{1}$-manifold with closed 1-forms $\theta$ and $\theta^{\ast}$
(i.e. $\mathrm{d}\theta=\mathrm{d}\theta^{\ast}=0$) is called a _conformal
Kähler manifold with Norden metric_. Necessary and sufficient conditions for a
$\mathcal{W}_{1}$-manifold to be conformal Kählerian are:

$(\nabla_{x}\theta)y=(\nabla_{y}\theta)x,\qquad(\nabla_{x}\theta)Jy=(\nabla_{y}\theta)Jx.$
(2.13)

The subclass of these manifolds is denoted by $\mathcal{W}_{1}^{\hskip
0.72229pt0}$.

It is proved [3] that a $\mathcal{W}_{1}^{\hskip 0.72229pt0}$-manifold is
conformally equivalent to a Kähler manifold with Norden metric by the
transformation (2.12).

It is known that on a pseudo-Riemannian manifold $M$ ($\dim M=2n\geq 4$) the
conformal invariant Weyl tensor has the form

$W(R)=R-\frac{1}{2(n-1)}\big{\\{}\psi_{1}(\rho)-\frac{\tau}{2n-1}\pi_{1}\big{\\}}.$
(2.14)

Let $L$ be a Kähler curvature-like tensor on an almost complex manifold with
Norden metric $(M,J,g)$, $\dim M=2n\geq 6$. Then the Bochner tensor $B(L)$ for
$L$ is defined by [3]:

$\begin{array}[]{l}B(L)=L-\frac{1}{2(n-2)}\big{\\{}\psi_{1}-\psi_{2}\big{\\}}\big{(}\rho(L)\big{)}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\phantom{B(L)}+\frac{1}{4(n-1)(n-2)}\big{\\{}\tau(L)\big{(}\pi_{1}-\pi_{2}\big{)}+\tau^{\ast}(L)\pi_{3}\big{\\}}.\end{array}$
(2.15)

##  3\. Complex Connections on $\mathcal{W}_{1}$-manifolds

######  Definition 3.1 ([7]).

_A linear connection $\nabla^{\prime}$ on an almost complex manifold $(M,J)$
is said to be_ almost complex _if $\nabla^{\prime}J=0$._

We introduce an eight-parametric family of complex connections in the
following

######  Theorem 3.1.

On a $\mathcal{W}_{1}$-manifold with Norden metric there exists an eight-
parametric family of complex connections $\nabla^{\prime}$ defined by

$\begin{array}[]{l}\nabla_{x}^{\prime}y=\nabla_{x}y+Q(x,y),\end{array}$ (3.1)

where the deformation tensor $Q(x,y)$ is given by

$\begin{array}[]{l}Q(x,y)=\frac{1}{2n}\left[\theta(Jy)x-g(x,y)J\Omega\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \hskip
4.69772pt\phantom{Q(x,y)}+\frac{1}{n}\left\\{\lambda_{1}\theta(x)y+\lambda_{2}\theta(x)Jy+\lambda_{3}\theta(Jx)y+\lambda_{4}\theta(Jx)Jy\right.\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \phantom{Q(x,y)}\hskip
4.33601pt+\lambda_{5}\left[\theta(y)x-\theta(Jy)Jx\right]+\lambda_{6}\left[\theta(y)Jx+\theta(Jy)x\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \left.\hskip
4.33601pt\phantom{Q(x,y)}+\lambda_{7}\left[g(x,y)\Omega-g(x,Jy)J\Omega\right]+\lambda_{8}\left[g(x,Jy)\Omega+g(x,y)J\Omega\right]\right\\},\end{array}$
(3.2)

$\lambda_{i}\in\mathbb{R}$, $i=1,2,...,8$.

###### Proof.

By (2.5), (3.1) and (3.2) we verify that
$(\nabla^{\prime}_{x}J)y=\nabla^{\prime}_{x}Jy-J\nabla^{\prime}_{x}y=0$, and
hence the connections $\nabla^{\prime}$ are complex for any
$\lambda_{i}\in\mathbb{R}$, $i=1,2,...,8$. ∎

Let us remark that the two-parametric family of complex connections obtained
for $\lambda_{1}=\lambda_{4}$, $\lambda_{3}=-\lambda_{2}$,
$\lambda_{5}=\lambda_{7}=0$, $\lambda_{8}=-\lambda_{6}=\frac{1}{4}$, is
studied in [11].

Let $T^{\prime}$ be the torsion tensor of $\nabla^{\prime}$, i.e.
$T^{\prime}(x,y)=\nabla^{\prime}_{x}y-\nabla^{\prime}_{x}y-[x,y]$. Taking into
account that the Levi-Civita connection $\nabla$ is symmetric, we have
$T^{\prime}(x,y)=Q(x,y)-Q(y,x)$. Then by (3.2) we obtain

$\begin{array}[]{l}T^{\prime}(x,y)=\frac{1}{n}\left\\{\left(\lambda_{1}-\lambda_{5}\right)\left[\theta(x)y-\theta(y)x\right]+\left(\lambda_{2}-\lambda_{6}\right)\left[\theta(x)Jy-\theta(y)Jx\right]\right.\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \qquad\quad\hskip
1.4457pt\left.+\left(\lambda_{3}-\lambda_{6}-\frac{1}{2}\right)\left[\theta(Jx)y-\theta(Jy)x\right]+\left(\lambda_{4}+\lambda_{5}\right)\left[\theta(Jx)Jy-\theta(Jy)Jx\right]\right\\}.\end{array}$
(3.3)

It is easy to verify the following

$\underset{x,y,z}{\mathfrak{S}}T^{\prime}(x,y,z)=\underset{x,y,z}{\mathfrak{S}}T^{\prime}(Jx,Jy,z)=\underset{x,y,z}{\mathfrak{S}}T^{\prime}(x,y,Jz)=0,$
(3.4)

where $\mathfrak{S}$ is the cyclic sum by the arguments $x,y,z$.

Next, we obtain necessary and sufficient conditions for the complex
connections $\nabla^{\prime}$ to be symmetric (i.e. $T^{\prime}=0$).

######  Theorem 3.2.

The complex connections $\nabla^{\prime}$ defined by (3.1) and (3.2) are
symmetric on a $\mathcal{W}_{1}$-manifold with Norden metric if and only if
$\lambda_{1}=-\lambda_{4}=\lambda_{5}$,
$\lambda_{2}=\lambda_{3}-\frac{1}{2}=\lambda_{6}$.

Then, by putting $\lambda_{1}=-\lambda_{4}=\lambda_{5}=\mu_{1}$,
$\lambda_{2}=\lambda_{6}=\lambda_{3}-\frac{1}{2}=\mu_{2}$,
$\lambda_{7}=\mu_{3}$, $\lambda_{8}=\mu_{4}$ in (3.2) we obtain a four-
parametric family of complex symmetric connections $\nabla^{\prime\prime}$ on
a $\mathcal{W}_{1}$-manifold which are defined by

$\begin{array}[]{l}\nabla^{\prime\prime}_{x}y=\nabla_{x}y+\frac{1}{2n}\left[\theta(Jx)y+\theta(Jy)x-g(x,y)J\Omega\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\phantom{\nabla^{\prime}_{x}y}+\frac{1}{n}\left\\{\mu_{1}\left[\theta(x)y+\theta(y)x-\theta(Jx)Jy-\theta(Jy)Jx\right]\right.\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\phantom{\nabla^{\prime}_{x}y}+\mu_{2}\left[\theta(Jx)y+\theta(Jy)x+\theta(x)Jy+\theta(y)Jx\right]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\phantom{\nabla^{\prime}_{x}y}+\left.\mu_{3}\left[g(x,y)\Omega-g(x,Jy)J\Omega\right]+\mu_{4}\left[g(x,Jy)\Omega+g(x,y)J\Omega\right]\right\\}.\end{array}$
(3.5)

The well-known Yano connection [12, 13] on a $\mathcal{W}_{1}$-manifold is
obtained from (3.5) for $\mu_{1}=\mu_{3}=0$, $\mu_{4}=-\mu_{2}=\frac{1}{4}$.

######  Definition 3.2 ([2]).

_A linear connection $\nabla^{\prime}$ on an almost complex manifold with
Norden metric $(M,J,g)$ is said to be_ natural _if
$\nabla^{\prime}J=\nabla^{\prime}g=0$ (or equivalently,
$\nabla^{\prime}g=\nabla^{\prime}\widetilde{g}=0$)._

From (3.1) and (3.2) we compute the covariant derivatives of $g$ and
$\widetilde{g}$ with respect to the complex connections $\nabla^{\prime}$ as
follows

$\begin{array}[]{l}\left(\nabla^{\prime}_{x}g\right)(y,z)=-Q(x,y,z)-Q(x,z,y)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
=-\frac{1}{n}\left\\{2\left[\lambda_{1}\theta(x)g(y,z)+\lambda_{2}\theta(x)g(y,Jz)+\lambda_{3}\theta(Jx)g(y,z)\right.\right.\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\left.+\lambda_{4}\theta(Jx)g(y,Jz)\right]+(\lambda_{5}+\lambda_{7})\left[\theta(y)g(x,z)+\theta(z)g(x,y)\right.\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\left.-\theta(Jy)g(x,Jz)-\theta(Jz)g(x,Jy)\right]+(\lambda_{6}+\lambda_{8})\left[\theta(y)g(x,Jz)\right.\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\left.+\theta(z)g(x,Jy)+\theta(Jy)g(x,z)+\theta(Jz)g(x,y)\right]\left.\right\\},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\left(\nabla^{\prime}_{x}\widetilde{g}\right)(y,z)=-Q(x,y,Jz)-Q(x,Jz,y).\end{array}$
(3.6)

Then, by (3.6) we get the following

######  Theorem 3.3.

The complex connections $\nabla^{\prime}$ defined by (3.1) and (3.2) are
natural on a $\mathcal{W}_{1}$-manifold if and only if
$\lambda_{1}=\lambda_{2}=\lambda_{3}=\lambda_{4}=0$,
$\lambda_{7}=-\lambda_{5}$, $\lambda_{8}=-\lambda_{6}$.

If we put $\lambda_{8}=-\lambda_{6}=s$, $\lambda_{7}=-\lambda_{5}=t$,
$\lambda_{i}=0$, $i=1,2,3,4$, in (3.2), we obtain a two-parametric family of
natural connections $\nabla^{\prime\prime\prime}$ defined by

$\begin{array}[]{l}\nabla^{\prime\prime\prime}_{x}y=\nabla_{x}y+\frac{1-2s}{2n}\left[\theta(Jy)x-g(x,y)J\Omega\right]+\frac{1}{n}\left\\{s\left[g(x,Jy)\Omega-\theta(y)Jx\right]\right.\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\phantom{\nabla^{\prime}_{x}y}\left.+t\left[g(x,y)\Omega-g(x,Jy)J\Omega-\theta(y)x+\theta(Jy)Jx\right]\right\\}.\end{array}$
(3.7)

The well-known canonical connection [2] (or $B$-connection [3]) on a
$\mathcal{W}_{1}$-manifold with Norden metric is obtained from (3.7) for
$s=\frac{1}{4}$, $t=0$.

We give a summery of the obtained results in the following table

Connection type | Symbol | Parameters
---|---|---
$\begin{array}[]{l}\text{Complex}\end{array}$ | $\nabla^{\prime}$ |  $\lambda_{i}\in\mathbb{R}$, $i=1,2,...,8$.
$\begin{array}[]{l}\text{Complex}\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ \text{symmetric}\end{array}$ | $\nabla^{\prime\prime}$ | $\begin{array}[]{c}\mu_{i},\hskip 2.168pti=1,2,3,4,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ \mu_{1}=\lambda_{1}=-\lambda_{4}=\lambda_{5},\hskip 2.168pt\mu_{2}=\lambda_{2}=\lambda_{6}=\lambda_{3}-\frac{1}{2},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ \mu_{3}=\lambda_{7},\hskip 2.168pt\mu_{4}=\lambda_{8}\end{array}$
$\begin{array}[]{l}\text{Natural}\end{array}$ | $\nabla^{\prime\prime\prime}$ | $\begin{array}[]{c}s,t,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ s=\lambda_{8}=-\lambda_{6},\hskip 2.168ptt=\lambda_{7}=-\lambda_{5},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ \lambda_{i}=0,\hskip 2.168pti=1,2,3,4.\end{array}$

Our next aim is to study the curvature properties of the complex connections
$\nabla^{\prime}$. Let us first consider the natural connection $\nabla^{0}$
obtained from (3.7) for $s=t=0$, i.e.

$\nabla^{0}_{x}y=\nabla_{x}y+\frac{1}{2n}\left[\theta(Jy)x-g(x,y)J\Omega\right].$
(3.8)

This connection is a semi-symmetric metric connection, i.e. a connection of
the form $\nabla_{x}y+\omega(y)x-g(x,y)U$, where $\omega$ is a 1-form and $U$
is the corresponding vector of $\omega$, i.e. $\omega(x)=g(x,U)$. Semi-
symmetric metric connections are introduced in [5] and studied in [6, 14]. The
form of the curvature tensor of an arbitrary connection of this type is
obtained in [14]. The geometry of such connections on almost complex manifolds
with Norden metric is considered in [10].

Let us denote by $R^{0}$ the curvature tensor of $\nabla^{0}$, i.e.
$R^{0}(x,y)z=\nabla^{0}_{x}\nabla^{0}_{y}z-\nabla^{0}_{y}\nabla^{0}_{x}z-\nabla^{0}_{[x,y]}z$.
The corresponding tensor of type (0,4) is defined by
$R^{0}(x,y,z,u)=g(R^{0}(x,y,)z,u)$. According to [14] it is valid

######  Proposition 3.1.

On a $\mathcal{W}_{1}$-manifold with closed 1-form $\theta^{\ast}$ the Kähler
curvature tensor $R^{0}$ of $\nabla^{0}$ defined by (3.8) has the form

$R^{0}=R-\frac{1}{2n}\psi_{1}(P),$ (3.9)

where

$P(x,y)=\left(\nabla_{x}\theta\right)Jy+\frac{1}{2n}\theta(x)\theta(y)+\frac{\theta(\Omega)}{4n}g(x,y)+\frac{\theta(J\Omega)}{2n}g(x,Jy).$
(3.10)

Since the Weyl tensor $W(\psi_{1}(S))=0$ (where $S$ is a symmetric
(0,2)-tensor), from (3.9) and (3.10) we conclude that

$W(R^{0})=W(R).$ (3.11)

Thus, the last equality implies

######  Proposition 3.2.

Let $(M,J,g)$ be a $\mathcal{W}_{1}$-manifold with closed 1-form
$\theta^{\ast}$, and $\nabla^{0}$ be the natural connection defined by (3.8).
Then, the Weyl tensor is invariant by the transformation
$\nabla\rightarrow\nabla^{0}$.

Further in this section we study the curvature properties of the complex
connections $\nabla^{\prime}$ defined by (3.1) and (3.2). Let us denote by
$R^{\prime}$ the curvature tensors corresponding to these connections.

If a linear connection $\nabla^{\prime}$ and the Levi-Civita connection
$\nabla$ are related by an equation of the form (3.1), then, because of
$\nabla g=0$, their curvature tensors of type (0,4) satisfy

$\begin{array}[]{l}g(R^{\prime}(x,y)z,u)=R(x,y,z,u)+(\nabla_{x}Q)(y,z,u)-(\nabla_{y}Q)(x,z,u)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\phantom{g(R^{\prime}(x,y)z,u)}+Q(x,Q(y,z),u)-Q(y,Q(x,z),u),\end{array}$
(3.12)

where $Q(x,y,z)=g(Q(x,y),z)$.

Then, by (3.1), (3.2), (3.9), (3.10), (3.12) we obtain the relation between
$R^{\prime}$ and $R^{0}$ as follows

$\begin{array}[]{l}R^{\prime}(x,y,z,u)=R^{0}(x,y,z,u)+g(y,z)A_{1}(x,u)-g(x,z)A_{1}(y,u)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
+g(x,u)A_{2}(y,z)-g(y,u)A_{2}(x,z)-g(y,Jz)A_{1}(x,Ju)\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ +g(x,Jz)A_{1}(y,Ju)-g(x,Ju)A_{2}(y,Jz)+g(y,Ju)A_{2}(x,Jz)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
+\left[\frac{\lambda_{5}\lambda_{7}-\lambda_{6}\lambda_{8}}{n^{2}}\theta(\Omega)+\frac{\lambda_{7}-\lambda_{5}+2(\lambda_{5}\lambda_{8}+\lambda_{6}\lambda_{7})}{2n^{2}}\theta(J\Omega)\right]\\{\pi_{1}-\pi_{2}\\}(x,y,z,u)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
-\left[\frac{\lambda_{5}\lambda_{8}+\lambda_{6}\lambda_{7}}{n^{2}}\theta(\Omega)-\frac{\lambda_{6}-\lambda_{8}+2(\lambda_{5}\lambda_{7}-\lambda_{6}\lambda_{8})}{2n^{2}}\theta(J\Omega)\right]\pi_{3}(x,y,z,u),\end{array}$
(3.13)

where

$\begin{array}[]{l}A_{1}(x,y)=\frac{\lambda_{7}}{n}\left\\{\left(\nabla_{x}\theta\right)y+\frac{\lambda_{7}}{n}[\theta(x)\theta(y)-\theta(Jx)\theta(Jy)]\right\\}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\phantom{A_{1}(x,y)}+\frac{\lambda_{8}}{n}\left\\{\left(\nabla_{x}\theta\right)Jy+\frac{1-2\lambda_{8}}{2n}[\theta(x)\theta(y)-\theta(Jx)\theta(Jy)]\right\\}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\phantom{A_{1}(x,y)}+\frac{\lambda_{7}(4\lambda_{8}-1)}{2n^{2}}[\theta(x)\theta(Jy)+\theta(Jx)\theta(y)],\vskip
12.0pt plus 4.0pt minus 4.0pt\\\
A_{2}(x,y)=-\frac{\lambda_{5}}{n}\left\\{\left(\nabla_{x}\theta\right)y-\frac{\lambda_{5}}{n}[\theta(x)\theta(y)-\theta(Jx)\theta(Jy)]\right\\}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\phantom{A_{2}(x,y)}-\frac{\lambda_{6}}{n}\left\\{\left(\nabla_{x}\theta\right)Jy+\frac{1+2\lambda_{6}}{2n}[\theta(x)\theta(y)-\theta(Jx)\theta(Jy)]\right\\}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\phantom{A_{2}(x,y)}+\frac{\lambda_{5}(4\lambda_{6}+1)}{2n^{2}}[\theta(x)\theta(Jy)+\theta(Jx)\theta(y)].\end{array}$
(3.14)

We are interested in necessary and sufficient conditions for $R^{\prime}$ to
be a Kähler curvature-like tensor, i.e. to satisfy (2.8) and (2.10). From
(2.11), (3.13) and (3.14) it follows that $R^{\prime}$ is Kählerian if and
only if $A_{1}(x,y)=A_{2}(x,y)$. Hence we obtain

######  Theorem 3.4.

Let $(M,J,g)$ be a conformal Kähler manifold with Norden metric, and
$\nabla^{\prime}$ be the complex connection defined by (3.1) and (3.2). Then,
$R^{\prime}$ is a Kähler curvature-like tensor if and only if
$\lambda_{7}=-\lambda_{5}$ and $\lambda_{8}=-\lambda_{6}$. In this case, from
(3.1) and (3.2) we obtain a six-parametric family of complex connections
$\nabla^{\prime}$ whose curvature tensors $R^{\prime}$ have the form

$\begin{array}[]{l}R^{\prime}=R^{0}+\frac{\lambda_{7}}{n}\left\\{\psi_{1}-\psi_{2}\right\\}(S_{1})+\frac{\lambda_{8}}{n}\left\\{\psi_{1}-\psi_{2}\right\\}(S_{2})\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\phantom{R^{\prime}}+\frac{\lambda_{7}(4\lambda_{8}-1)}{2n^{2}}\left\\{\psi_{1}-\psi_{2}\right\\}(S_{3})+\frac{\lambda_{7}(1-2\lambda_{8})\theta(J\Omega)}{n^{2}}\left\\{\pi_{1}-\pi_{2}\right\\}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\phantom{R^{\prime}}+\frac{2\lambda_{7}\lambda_{8}\theta(\Omega)}{n^{2}}\pi_{3},\end{array}$
(3.15)

where $R^{0}$ is given by (3.9), (3.10), and

$\begin{array}[]{l}S_{1}(x,y)=\left(\nabla_{x}\theta\right)y+\frac{\lambda_{7}}{n}[\theta(x)\theta(y)-\theta(Jx)\theta(Jy)]-\frac{\lambda_{7}\theta(\Omega)}{2n}g(x,y)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\phantom{S_{1}(x,y)}+\frac{\lambda_{7}\theta(J\Omega)}{2n}g(x,Jy),\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\
S_{2}(x,y)=\left(\nabla_{x}\theta\right)Jy+\frac{1-2\lambda_{8}}{2n}[\theta(x)\theta(y)-\theta(Jx)\theta(Jy)]\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\phantom{S_{2}(x,y)}+\frac{\lambda_{8}\theta(\Omega)}{2n}g(x,y)+\frac{(1-\lambda_{8})\theta(J\Omega)}{2n}g(x,Jy),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
S_{3}(x,y)=\theta(x)\theta(Jy)+\theta(Jx)\theta(y).\end{array}$ (3.16)

From (3.15), (3.16) and Theorem 3.4 we get

######  Corollary 3.1.

Let $(M,J,g)$ be a conformal Kähler manifold with Norden metric and
$\nabla^{\prime}$ be the eight-parametric family of complex connections
defined by (3.1) and (3.2). Then $R^{\prime}=R^{0}$ if and only if
$\lambda_{i}=0$ for $i=5,6,7,8$.

Let us remark that by putting $\lambda_{i}=0$ for $i=1,2,5,6,7,8$ in (3.2) we
obtain a two-parametric family of complex connections whose Kähler curvature
tensors coincides with $R^{0}$ on a $\mathcal{W}_{1}$-manifold with closed
1-form $\theta^{\ast}$.

Theorem 3.2 and Corollary 3.1 imply

######  Corollary 3.2.

On a conformal Kähler manifold with Norden metric the Weyl tensor is invariant
by the transformation of the Levi-Civita connection in any of the complex
connection $\nabla^{\prime}$ defined by (3.1) and (3.2) for $\lambda_{i}=0$,
$i=5,6,7,8$.

Since for the Bochner tensor of $\\{\psi_{1}-\psi_{2}\\}(S)$ it is valid
$B\left(\\{\psi_{1}-\psi_{2}\\}(S)\right)=0$, where $S$ is symmetric and
hybrid with respect to $J$, from Theorem 3.4 and (2.11) it follows

$B(R^{\prime})=B(R^{0}).$ (3.17)

By this way we proved the truthfulness of the following

######  Theorem 3.5.

Let $(M,J,g)$ be a conformal Kähler manifold with Norden metric, $R^{\prime}$
be the curvature tensor of $\nabla^{\prime}$ defined by (3.1) and (3.2) for
$\lambda_{7}=-\lambda_{5}$, $\lambda_{8}=-\lambda_{6}$, and $R^{0}$ be the
curvature tensor of $\nabla^{0}$ given by (3.8). Then the Bochner tensor is
invariant by the transformations $\nabla^{0}\rightarrow\nabla^{\prime}$.

##  4\. Conformal transformations of complex connections

In this section we study usual conformal transformations of the complex
connections $\nabla^{\prime}$ defined in the previous section.

Let $(M,J,g)$ and $(M,J,\bar{g})$ be conformally equivalent almost complex
manifolds with Norden metric by the transformation (2.12). It is known that
the Levi-Civita connections $\nabla$ and $\overline{\nabla}$ of $g$ and
$\overline{g}$, respectively, are related as follows

$\overline{\nabla}_{x}y=\nabla_{x}y+\sigma(x)y+\sigma(y)x-g(x,y)\Theta,$ (4.1)

where $\sigma(x)=du(x)$ and $\Theta=\textrm{grad}\hskip 1.4457pt\sigma$, i.e.
$\sigma(x)=g(x,\Theta)$. Let $\overline{\theta}$ be the Lie 1-form of
$(M,J,\overline{g})$. Then by (2.4) and (4.1) we obtain

$\overline{\theta}=\theta+2n\big{(}\sigma\circ
J\big{)},\qquad\quad\overline{\Omega}=e^{-2u}\big{(}\Omega+2nJ\Theta\big{)}.$
(4.2)

It is valid the following

######  Lemma 4.1.

Let $(M,J,g)$ be an almost complex manifold with Norden metric and
$(M,J,\overline{g})$ be its conformally equivalent manifold by the
transformation (2.12). Then, the curvature tensors $R$ and $\overline{R}$ of
$\nabla$ and $\overline{\nabla}$, respectively, are related as follows

$\overline{R}=e^{2u}\big{\\{}R-\psi_{1}\big{(}V\big{)}-\pi_{1}\sigma\big{(}\Theta\big{)}\big{\\}},$
(4.3)

where $V(x,y)=\big{(}\nabla_{x}\sigma\big{)}y-\sigma(x)\sigma(y)$.

Let us first study the conformal group of the natural connection $\nabla^{0}$
given by (3.8). Equalities (3.8) and (4.1) imply that its conformal group is
defined analytically by

$\overline{\nabla}^{\hskip 1.4457pt0}_{x}y=\nabla^{0}_{x}y+\sigma(x)y.$ (4.4)

It is known that if two linear connections are related by an equation of the
form (4.4), where $\sigma$ is a 1-form, then the curvature tensors of these
connections coincide iff $\sigma$ is closed. Hence, it is valid

######  Theorem 4.1.

Let $(M,J,g)$ be a $\mathcal{W}_{1}$-manifold with closed 1-form
$\theta^{\ast}$. Then the curvature tensor $R^{0}$ of $\nabla^{0}$ is
conformally invariant, i.e.

$\overline{R}^{0}=e^{2u}R^{0}.$ (4.5)

Further in this section let $(M,J,g)$ be a conformal Kähler manifold with
Norden metric. Then $(M,J,\overline{g})$ is a Kähler manifold and thus
$\overline{\theta}=0$. From (4.2) it follows $\sigma=\frac{1}{2n}(\theta\circ
J)$. Then, from (3.1) and (3.2) we get $\overline{\nabla}^{\hskip
1.4457pt\prime}=\overline{\nabla}$ and hence
$\overline{R}^{\prime}=\overline{R}$ for all $\lambda_{i}\in\mathbb{R}$,
$i=1,2,...,8$. In particular, $\overline{R}^{\prime}=\overline{R}^{0}$. Then,
Theorems 3.5 and (3.17) imply

######  Theorem 4.2.

On a conformal Kähler manifold with Norden metric the Bochner curvature tensor
of the complex connections $\nabla^{\prime}$ defined by (3.1) and (3.2) with
the conditions $\lambda_{7}=-\lambda_{5}$ and $\lambda_{8}=-\lambda_{6}$ is
conformally invariant by the transformation (2.12), i.e.

$B(\overline{R}^{\prime})=e^{2u}B(R^{\prime}).$ (4.6)

Let us remark that the conformal invariancy of the Bochner tensor of the
canonical connection on a conformal Kähler manifold with Norden metric is
proved in [3].

From Theorem 4.1 and Corollary 3.1 we obtain

######  Corollary 4.1.

Let $(M,J,g)$ be a conformal Kähler manifold with Norden metric and
$\nabla^{\prime}$ be a complex connection defined by (3.1) and (3.2). If
$\lambda_{i}=0$ for $i=5,6,7,8$, then the curvature tensor of
$\nabla^{\prime}$ is conformally invariant by the transformation (2.12).

##  5\. An Example

Let $G$ be a real connected four-dimensional Lie group, and $\mathfrak{g}$ be
its corresponding Lie algebra. If $\\{e_{1},e_{2},e_{3},e_{4}\\}$ is a basis
of $\mathfrak{g}$, we equip $G$ with a left-invariant almost complex structure
$J$ by

$Je_{1}=e_{3},\qquad Je_{2}=e_{4},\qquad Je_{3}=-e_{1},\qquad Je_{4}=-e_{2}.$
(5.1)

We also define a left-invariant pseudo-Riemannian metric $g$ on $G$ by

$\begin{array}[]{l}g(e_{1},e_{1})=g(e_{2},e_{2})=-g(e_{3},e_{3})=-g(e_{4},e_{4})=1,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ g(e_{i},e_{j})=0,\quad i\neq j,\quad
i,j=1,2,3,4.\end{array}$ (5.2)

Then, because of (2.1), (5.1) and (5.2), $(G,J,g)$ is an almost complex
manifold with Norden metric.

Further, let the Lie algebra $\mathfrak{g}$ be defined by the following
commutator relations

$\begin{array}[]{l}[e_{1},e_{2}]=[e_{3},e_{4}]=0,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\
[e_{1},e_{4}]=[e_{2},e_{3}]=\lambda(e_{1}+e_{4})+\mu(e_{2}-e_{3}),\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\
[e_{1},e_{3}]=-[e_{2},e_{4}]=\mu(e_{1}+e_{4})-\lambda(e_{2}-e_{3}),\end{array}$
(5.3)

where $\lambda,\mu\in\mathbb{R}$.

The well-known Koszul’s formula for the Levi-Civita connection of $g$ on $G$,
i.e. the equality

$2g(\nabla_{e_{i}}e_{j},e_{k})=g([e_{i},e_{j}],e_{k})+g([e_{k},e_{i}],e_{j})+g([e_{k},e_{j}],e_{i}),$
(5.4)

and (5.2) imply the following essential non-zero components of the Levi-Civita
connection:

$\begin{array}[]{ll}\nabla_{e_{1}}e_{1}=\nabla_{e_{2}}e_{2}=\mu e_{3}+\lambda
e_{4},&\nabla_{e_{3}}e_{3}=\nabla_{e_{4}}e_{4}=-\lambda e_{1}+\mu e_{2},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\nabla_{e_{1}}e_{3}=\mu(e_{1}+e_{4}),&\nabla_{e_{1}}e_{4}=\lambda e_{1}-\mu
e_{3},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \nabla_{e_{2}}e_{3}=\mu
e_{1}+\lambda e_{4},&\nabla_{e_{2}}e_{4}=\lambda(e_{2}-e_{3}).\end{array}$
(5.5)

Then, by (2.2), (2.3) and (5.5) we compute the following essential non-zero
components $F_{ijk}=F(e_{i},e_{j},e_{k})$ of $F$:

$\begin{array}[]{l}F_{111}=F_{422}=2\mu,\quad F_{222}=-F_{311}=2\lambda,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
F_{112}=-F_{214}=F_{314}=-F_{412}=\lambda,\quad
F_{212}=-F_{114}=F_{312}=-F_{414}=\mu.\end{array}$ (5.6)

Having in mind (2.4) and (5.6), the components $\theta_{i}=\theta(e_{i})$ and
$\theta_{i}^{\ast}=\theta^{\ast}(e_{i})$ of the 1-forms $\theta$ and
$\theta^{\ast}$, respectively, are:

$\theta_{2}=\theta_{3}=\theta^{\ast}_{1}=-\theta^{\ast}_{4}=4\lambda,\qquad\theta_{1}=-\theta_{4}=-\theta^{\ast}_{2}=-\theta^{\ast}_{3}=4\mu.$
(5.7)

By (2.4) and (5.7) we compute

$\begin{array}[]{c}\Omega=4\mu(e_{1}+e_{4})+4\lambda(e_{2}-e_{3}),\qquad
J\Omega=4\lambda(e_{1}+e_{4})-4\mu(e_{2}-e_{3}),\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ \theta(\Omega)=\theta(J\Omega)=0.\end{array}$ (5.8)

By the characteristic condition (2.5) and equalities (5.6), (5.7) we prove
that the manifold $(G,J,g)$ with Lie algebra $\mathfrak{g}$ defined by (5.3)
belongs to the basic class $\mathcal{W}_{1}$. Moreover, by (5.5) and (5.7) it
follows that the conditions (2.13) hold and thus

######  Proposition 5.1.

The manifold $(G,J,g)$ defined by (5.1), (5.2) and (5.3) is a conformal Kähler
manifold with Norden metric.

According to (3.8), (5.2), (5.5) and (5.7) the components of the natural
connection $\nabla^{0}$ are given by

$\begin{array}[]{ll}\nabla_{e_{1}}^{0}e_{1}=-\nabla_{e_{4}}^{0}e_{1}=\mu
e_{2},&\nabla_{e_{2}}^{0}e_{1}=\nabla_{e_{3}}^{0}e_{1}=\lambda e_{2},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\nabla_{e_{1}}^{0}e_{2}=-\nabla_{e_{4}}^{0}e_{2}=-\mu
e_{1},&\nabla_{e_{2}}^{0}e_{2}=\nabla_{e_{3}}^{0}e_{2}=-\lambda e_{1},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\nabla_{e_{1}}^{0}e_{3}=-\nabla_{e_{4}}^{0}e_{3}=\mu
e_{4},&\nabla_{e_{2}}^{0}e_{3}=\nabla_{e_{3}}^{0}e_{3}=\lambda e_{4},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\nabla_{e_{1}}^{0}e_{4}=-\nabla_{e_{4}}^{0}e_{4}=-\mu
e_{3},&\nabla_{e_{2}}^{0}e_{4}=\nabla_{e_{4}}^{0}e_{4}=-\lambda
e_{3}.\end{array}$ (5.9)

By (5.9) we obtain $R^{0}=0$. Then, by (3.9) and (3.10) the curvature tensor
$R$ of $(G,J,g)$ has the form

$R=\frac{1}{4}\psi_{1}(A),\qquad
A(x,y)=(\nabla_{x}\theta)Jy+\frac{1}{4}\theta(x)\theta(y).$ (5.10)

Moreover, having in mind (3.16), (5.5), (5.7) and (5.8), we compute
$S_{1}=S_{2}=S_{3}=0$. Hence, for the tensors $R^{\prime}$ of the complex
connections $\nabla^{\prime}$ given by (3.15), it is valid $R^{\prime}=0$.

######  Proposition 5.2.

The complex connections $\nabla^{\prime}$ defined by (3.1) and (3.2) are flat
on $(G,J,g)$.

## References

  * [1] G. Ganchev, A. Borisov, _Note on the almost complex manifolds with a Norden metric_ , Compt. Rend. Acad. Bulg. Sci. 39(5) (1986), 31–34.
  * [2] G. Ganchev, V. Mihova, _Canonical connection and the canonical conformal group on an almost complex manifold with $B$-metric_, Ann. Univ. Sofia Fac. Math. Inform., 81(1) (1987), 195–206.
  * [3] G. Ganchev, K. Gribachev, V. Mihova, $B$_-connections and their conformal invariants on conformally Kähler manifolds with $B$-metric_, Publ. Inst. Math. (Beograd) (N.S.) 42(56) (1987), 107–121.
  * [4] K. Gribachev, D. Mekerov, G. Djelepov, _Generalized_ B _-manifolds_ , Compt. Rend. Acad. Bulg. Sci. 38(3) (1985), 299–302.
  * [5] A. Hayden, _Subspaces of a space with torsion_ , Proc. London Math. Soc. 34 (1932), 27–50.
  * [6] T. Imai, _Notes on semi-symmetric metric connections_ , Tensor N.S. 24 (1972), 293–296.
  * [7] S. Kobayshi, K. Nomizu, _Foundations of differential geometry_ vol. 1, 2, Intersc. Publ., New York, 1963, 1969.
  * [8] A. Newlander, L. Nirenberg, _Complex analytic coordinates in almost complex manifolds_ , Ann. Math. 65 (1957), 391–404.
  * [9] A. P. Norden, _On a class of four-dimensional A-spaces_ , Russian Math. (Izv VUZ) 17(4) (1960), 145–157.
  * [10] S. D. Singh, A. K. Pandey, _Semi-symmetric metric connections in an almost Norden metric manifold_ , Acta Cienc. Indica Math. 27(1) (2001), 43–54.
  * [11] M. Teofilova, _Almost complex connections on almost complex manifolds with Norden metric_ , In: Trends in Differential Geometry, Complex Analysis and Mathematical Physics, eds. K. Sekigawa, V. Gerdjikov and S. Dimiev, World Sci. Publ., Singapore (2009), 231–240.
  * [12] K. Yano, _Affine connections in an almost product space_ , Kodai Math. Semin. Rep. 11(1) (1959), 1–24.
  * [13] K. Yano, _Differential geometry on complex and almost complex spaces_ , Pure and Applied Math. vol. 49, Pergamon Press Book, New York, 1965.
  * [14] K. Yano, _On semi-symmetric metric connection_ , Rev. Roumanie Math. Pure Appl. 15 (1970), 1579–1586.

_Marta Teofilova
Department of Geometry
Faculty of Mathematics and Informatics
University of Plovdiv
236 Bulgaria Blvd.
4003 Plovdiv, Bulgaria
e-mail:_<EMAIL_ADDRESS>


∎

11institutetext: Narinder Singh Punn 22institutetext: IIIT Allahabad,
Prayagraj, India, 211015
Tel.: +91-7018466740
22email<EMAIL_ADDRESS>33institutetext: Sonali Agarwal
44institutetext: IIIT Allahabad, Prayagraj, India, 211015
Tel.: +91-9415647042
44email<EMAIL_ADDRESS>
# Automated diagnosis of COVID-19 with limited posteroanterior chest X-ray
images using fine-tuned deep neural networks

Narinder Singh Punn  Sonali Agarwal

###### Abstract

The novel coronavirus 2019 (COVID-19) is a respiratory syndrome that resembles
pneumonia. The current diagnostic procedure of COVID-19 follows reverse-
transcriptase polymerase chain reaction (RT-PCR) based approach which however
is less sensitive to identify the virus at the initial stage. Hence, a more
robust and alternate diagnosis technique is desirable. Recently, with the
release of publicly available datasets of corona positive patients comprising
of computed tomography (CT) and chest X-ray (CXR) imaging; scientists,
researchers and healthcare experts are contributing for faster and automated
diagnosis of COVID-19 by identifying pulmonary infections using deep learning
approaches to achieve better cure and treatment. These datasets have limited
samples concerned with the positive COVID-19 cases, which raise the challenge
for unbiased learning. Following from this context, this article presents the
random oversampling and weighted class loss function approach for unbiased
fine-tuned learning (transfer learning) in various state-of-the-art deep
learning approaches such as baseline ResNet, Inception-v3, Inception
ResNet-v2, DenseNet169, and NASNetLarge to perform binary classification (as
normal and COVID-19 cases) and also multi-class classification (as COVID-19,
pneumonia, and normal case) of posteroanterior CXR images. Accuracy,
precision, recall, loss, and area under the curve (AUC) are utilized to
evaluate the performance of the models. Considering the experimental results,
the performance of each model is scenario dependent; however, NASNetLarge
displayed better scores in contrast to other architectures, which is further
compared with other recently proposed approaches. This article also added the
visual explanation to illustrate the basis of model classification and
perception of COVID-19 in CXR images.

###### Keywords:

COVID-19 Classification Deep learning Transfer learning Pneumonia Chest X-ray
(CXR) Imbalanced learning

††journal: Applied Intelligence

##  1 Introduction

Coronaviruses are a large family of viruses that can cause severe illness to
the human being. The first known severe epidemic is severe acute respiratory
syndrome (SARS) occurred in 2003, whereas the second outbreak began in 2012 in
Saudi Arabia with the middle east respiratory syndrome (MERS). The novel
coronavirus disease 2019 (COVID-19), began as an outbreak from epicentre
Wuhan, People’s Republic of China in late December 2019, and till April 15,
2020, it caused 1,996,681 infections and 127,590 deaths worldwide WHO1 . The
coronavirus (COVID-19) outbreak was declared a public health emergency of
international concern by WHO on January 30, 2020 WHO2 . On March 11, as the
number of COVID-19 cases has increased thirteen times apart from China with
more than 118,000 cases in 114 countries and over 4,000 deaths, WHO declared
this a pandemic punn2020covid . Globally, many researchers of medicine,
clinical and artificial intelligence areas are trying hard to mobilize
preventive action plans for COVID-19 with identified research priorities.
Since this disease is highly contagious, the most desirable preventive measure
is to identify the infected people to control the spread. Unfortunately, there
is no well-known treatment available to cure COVID-19, therefore the
identified infected person must be kept in isolation to break the transmission
chain as this patient may become the source of community transfer. Till now,
the testing kit is the only available option for diagnosis of COVID-19.
Unavailability of testing kits due to excessive demand all over the world is a
severe problem in the mission against this pandemic. Though several healthcare
organizations are claiming for successful development of testing kits, there
is a huge gap in demand and supply. The healthcare agencies have accelerated
the rate of development of low-cost testing kits, but the inability to
diagnose at early-stage and due to exponential growth of COVID-19 cases,
medical professionals are bound to rely on other diagnostic measures.

Clinical studies have shown that most COVID-19 patients suffer from lung
infection guo2020origin . Although chest CT is a more effective imaging
technique for lung-related disease diagnosis; CXR is preferred because it is
widely available, faster and cheaper than CT. Since COVID-19 infection attacks
the epithelial cells that line our respiratory tract, hence X-rays images can
be used to analyse the lungs to diagnose pneumonia, lung inflammation,
abscesses, and/or enlarged lymph nodes allen1994eosinophilic . Due to its easy
transmission, developing techniques to accurately and easily identify the
presence of COVID-19 and distinguish it from other forms of flu and pneumonia
is crucial.

Biomedical image analysis (segmentation and classification) is an admired area
of research to make the healthcare system more promising fourcade2019deep . In
this area, advancement in computing infrastructure makes it possible to deploy
the deep learning techniques for complex medical image analysis tasks. Recent
works have shown that the chest X-rays of patients suffering from COVID-19
depicts certain abnormalities in the radiography rajinikanth2020harmony . For
medical image analysis; deep learning techniques, specifically, convolutional
neural networks (CNN) are very effective and efficient in feature extraction
and learning, hence becoming the most popular choice among researchers
krizhevsky2012imagenet . CNNs have been successfully deployed in the analysis
of video endoscopy gomez2019low  and CT images, and also used for the
diagnosis of pediatric pneumonia via chest X‐ray images choe2019deep ;
kermany2018identifying . Chouhan et al. chouhan2020novel  proposed a transfer
learning based deep network approach pre-trained on ImageNet 5206848  for
pneumonia detection. Wang et al. wang2017hospital  proposed a customized VGG16
model for lung regions identification to classify different types of
pneumonia. Later, Ronneburger et al. ronneberger2015u  demonstrated the
effectiveness of image augmentation with CNN in the presence of a small set of
images. In the area of biomedical image classification, Rajpurkar et al.
rajpurkar2018deep  proposed a dense CNN with 121‐layers to detect several
pathologies including pneumonia using chest X‐rays. Lakhani et al.
lakhani2017deep  obtained an area under the curve (AUC) of 0.95 in pneumonia
detection using AlexNet and GoogLeNet along with image augmentation. Recently,
several deep‐learning based COVID‐19 detection techniques have been proposed
wang2020covid ; wang2020deep ; WHO3 . Linda et al. wang2020covid  introduced a
deep CNN, named COVID‐Net for the detection of COVID‐19 cases from the chest
X‐ray images. Shuai et al. wang2020deep  achieved accuracy, specificity and
sensitivity of 89.5%, 88% and 87% respectively for COVID‐19 identification
using CT images.

There are many datasets available about chest X-rays for the detection of
pneumonia WHO4 ; irvin2019chexpert ; cohen2020covid ; WHO5 ; but in present
research work, COVID-19 X-ray chest images cohen2020covid  and Radiological
Society of North America (RSNA) images WHO5  are utilized to generate all
possible samples of chest infection and also to make the study comparable with
other research works. The number of COVID-19 infected samples present in this
dataset is very limited that may lead to biased outcome, hence the objective
of this research is to maximize the learning ability in presence of a small
set of positive class samples. For early diagnosis of COVID-19, this article
presents the effectiveness of random oversampling and weighted class loss
function approaches for unbiased fine-tuned learning (transfer learning) in
various state-of-the-art deep learning techniques.

Rest of the manuscript is organized as follows: recent research articles are
discussed in section II, and section III briefs the dataset. Section IV
contains the proposed methodology followed by the evaluation metrics in
section V. Results are discussed in section VI whereas the last section
contains concluding remarks.

##  2 Related work

Due to the ample availability of X-ray machines, disease diagnosis using CXR
images are widely used by healthcare experts. In case of any suspect of
COVID-19; instead of using test kits, an alternate way to detect pneumonia
from the CXR images is required, so that further investigation can be narrowed
down for COVID-19 identification. Many studies have been performed on similar
ground with several CXR datasets for diagnosis of pneumonia and other
complications WHO4 ; irvin2019chexpert ; cohen2020covid ; WHO5 . These studies
also advocate the need of an automated system for quick diagnosis because the
manual methods of X-ray analysis are time consuming and unable to serve the
purpose due to limited availability of X-ray machine operators or
radiologists.

Amid the COVID-19 outbreak, many companies at the global level around the
world embraced a flurry of Artificial Intelligence (AI) based solutions to
detect COVID-19 on chest X-ray scans. It is evident that deep learning tools
are effectively used to screen mild cases, triage new infections, and monitor
disease advancements. This way of diagnosis can reduce the growing burden on
radiologists, and also supplant standard nucleic acid tests as the primary
diagnostic tool for coronavirus infection. It is also reported that a swab
test needs isolation for testing procedure whereas chest X-ray based detection
can be easily manageable. Kermany et al. kermany2018identifying  proposed CXR
image-based deep learning model to detect pneumonia and classify other
diseases using different medical datasets with testing accuracy of 92.80%. In
another similar research, Stephen et al. stephen2019efficient  illustrated an
efficient deep learning approach for pneumonia classification by using four
convolutional layers and two dense layers in addition to classical image
augmentation and achieved 93.73% testing accuracy. Later, Saraiva et al.
saraiva2019classification  experimented convolutional neural networks to
classify images of childhood pneumonia by using a deep learning model with
seven convolutional layers along with three dense layers while achieving
95.30% testing accuracy. Liang and Zheng liang2019transfer  demonstrated a
transfer learning method with a deep residual network for pediatric pneumonia
diagnosis with 49 convolutional layers and two dense layers and achieved
96.70% testing accuracy. In similar research, Wu et al. wu2020predict  focused
on convolutional deep neural learning networks and random forest to propose a
pneumonia prediction using CXR images and achieved 97% testing accuracy.

Afterwards, Narin et al. narin2020automatic  proposed a deep convolutional
neural network based automatic prediction model of COVID-19 with the help of
pre-trained transfer models using CXR images. In this research, authors used
ResNet50, InceptionV3 and Inception-ResNetV2 pre-trained models to obtain a
higher prediction accuracy for a subset of X-ray dataset. Apostolopoulos et.
al. apostolopoulos2020covid  in their study, utilised state-of-the-art
convolutional neural network architectures for classifying the CXR images.
Transfer Learning was adopted to handle various abnormalities present in the
dataset. Two datasets from different repositories have been used to study
images of three classes: COVID-19, bacterial/viral pneumonia and normal
condition. The article establishes the suitability of the deep learning model
with the help of accuracy, sensitivity, and specificity parameters.

In another research, generative adversarial networks (GAN) are used by Khalifa
et al. khalifa2020detection  to detect pneumonia from CXR images. The authors
addressed the overfitting problem and claimed its robustness by generating
more images through GAN. The dataset containing 5863 CXR images of two
categories: normal and pneumonia, has been used with typical deep learning
models such as AlexNet, GoogLeNet, Squeeznet and Resnet18 to detect pneumonia.
This research highlights that the Resnet18 outperformed among other deep
transfer models in combination with GAN. Further, Sethy et al.
sethy2020detection  proposed a deep learning based model to identify
coronavirus infections using CXR images. Deep features from CXR images have
been extracted and support vect or machine (SVM) classifier is used to measure
accuracy, false positive rate, F1 score, Matthew’s correlation coefficient
(MCC) and kappa. It is found that ResNet50 in combination with SVM is
statistically superior when compared to other models. Later, Bukhari et al.
bukhari2020diagnostic  also used ResNet-50 CNN architectures on 278 CXR
images, partitioned under 3 groups as normal, pneumonia and COVID-19. This
approach gave promising results and indicated substantial differentiation of
pulmonary changes caused by COVID-19 from the other types of pneumonia.

Recently, in another research work, an improved ResNet-50 CNN architecture
named COVIDResNet has been proposed farooq2020covid , where conventional
ResNet-50 model is applied with different training techniques including
progressive resizing, cyclical learning rate finding and discriminative
learning rates to gain fast and accurate training. The experiment is performed
through progressively re-sizing of input images to 128x128x3, 224x224x3 and
229x229x3 pixels, and automatic learning rate selection for fine-tuning the
network at each stage. This work claimed to be computationally efficient and
highly accurate for multi-class classification.

A new deep anomaly detection model is developed by Zhang et. al.
zhang2020covid  for fast and more reliable screening. To evaluate the model
performance, CXR image data of COVID-19 cases and other pneumonia has been
collected from two different sources. To eliminate the data imbalance problem
in the collected samples, authors proposed a CXR based COVID-19 screening
model through anomaly detection task pang2019deep .

Following this context, this article proposes to contribute for early
diagnosis of COVID-19 using the state-of-the-art deep learning architectures,
assisted with transfer learning and class imbalance learning approaches.

##  3 Dataset description

In this research three datasets are utilized for experiments: COVID-19 image
cohen2020covid , Radiological Society of North America (RSNA) WHO5  and U.S.
national library of medicine (USNLM) collected Montgomery country - NLM(MC)
jaeger2014two . COVID-19 image dataset is a public database of pneumonia cases
with CXR images related to COVID-19, MERS, SARS, and ARDS collected by Cohen
et al. cohen2020covid  from multiple resources available at public domains
without infringing patient’s confidentiality (Fig. 1(b)). It is claimed that
this dataset can help to identify characteristics of COVID-19 in contrast to
other types of pneumonia; therefore it can play a major role in predicting
survival rate. The dataset includes the statistics up to March 25, 2020
consisting of 5 types of pneumonia such as SARSr-CoV-2 or COVID-19, SARSr-
CoV-1 or SARS, Streptococcus spp., Pneumocystis spp. and ARDS with following
attributes: patient ID, offset, sex, age, finding, survival, view, modality,
date, location, filename, doi, url, license, clinical notes, and other notes.

Figure 1: Sample chest radiographs. Table 1: Fused dataset disease class summary details. Dataset | Findings | PCXR images | Total samples
---|---|---|---
COVID-19 | COVID-19 | 108 | 153
Other pneumonia | 45
RSNA | Normal | 453 | 923
Other pneumonia | 470
NLM(MC) | Tuberculosis | 58 | 138
Normal | 80

Another dataset utilized in this study is published under RSNA pneumonia
detection challenge is a subset of 30,000 examinations taken from the NIH
CXR14 dataset WHO5 . Out of 30,000 selected images, 15,000 examinations had
positive cases of pneumonia and from the remaining 15000 cases, 7500 cases had
no findings and other 7500 cases had symptoms other than pneumonia. All these
images are annotated by a group of experts including radiologists in two
stages. A sample image is shown in Fig. 1(c). This dataset has been published
in two stages. In Stage one, 25,684 training images were considered to test
1,000 images. Later in stage two 1000 testing samples were added to the
training set to form the dataset of 26,684 training images and a new set of
3,000 radiographs were introduced for the test. For robust testing and
comprehensive coverage of the comparative analysis, NLM(MC) jaeger2014two
dataset is also utilized that consists of 138 chest posterior-anterior x-rays
samples of tuberculosis and normal cases. A sample image is represented in
Fig. 1(d). Table 1 presents the class summary details of the fused dataset
resulting from the above discussed datasets which is utilized for training,
testing and validation of the proposed approach. The fused dataset is composed
of 1214 posteroanterior chest x-ray samples with classes labeled as COVID-19
(108), other pneumonia (515), tuberculosis (58) and normal (533). The
generated fused dataset is publicly available CXR1 .

##  4 Proposed contribution

The era of artificial intelligence has brought significant improvements in the
living society makridakis2017forthcoming . The recent advancements in deep
learning have extended its domain in various applications such as healthcare,
pixel restoration, visual recognition, signal processing and a lot more
liu2017survey . In healthcare domain, the deep learning based image processing
approaches for classification and segmentation are applied for faster,
efficient, and early diagnosis of the deadly diseases e.g. breast cancer,
brain tumor, etc. by using different imaging modalities such as X-ray, CT,
MRI, shen2017deep  and fused modalities punn2020inception  along with its
future possibilities. The success of these approaches is dependent on the
large amount of data availability, which however is not in the case of
automated COVID-19 detection.

Figure 2: Schematic representation of the proposed components for COVID-19
identification.

The main contribution of the work is divided into the components as shown in
Fig. 2. It has two concrete components: data preprocessing and classification.
COVID-19 image, RSNA and NLM(MC) datasets are used to generate the final
working set. The newly generated dataset contains CXR images of the following
classes: coronavirus caused diseases, pneumonia, other diseases and normal
cases. Further, binary classification (COVID-19 vs others) and multi-class
classification (COVID-19, other types of pneumonia, tuberculosis and normal)
are achieved using random oversampling and weighted class loss function
approaches for unbiased fine-tuned learning (transfer learning) in various
state-of-the-art deep learning approaches such as baseline ResNet,
Inception-v3, Inception ResNet-v2, DenseNet169, and NASNetLarge
szegedy2016rethinking ; szegedy2017inception ; huang2017densely ;
zoph2018learning . The trained models are utilized for identification and
classification of COVID-19 in novel samples. Later, visualization techniques
are utilized to understand and elaborate the basis of the classification
results.

###  4.1 Imbalanced Learning Approach

Class balancing techniques are necessary when minority classes are more
important. The dataset used in this research is highly imbalanced which may
lead to biased learning of the model. Number of coronavirus infected CXR
images are very less compared to other classes, hence class balancing
techniques must be insured to smoothen the learning process. This section
discusses two approaches to handle the class imbalance problem: weight class
approach and random oversampling johnson2019survey .

####  4.1.1 Weighted class approach

In this approach, the intention is to balance the data by altering the weights
that each training sample class carries when computing the loss. Normally,
each class carries equal weights, but sometimes certain classes with minority
samples are required to hold more weights if they are more important because
training examples within that class should have a significant effect on the
loss function. In the used dataset the coronavirus infected image class
samples must be given more weights as they are more significant. In this
article, the weights for each class is generated based on the Eq. 1.

${w(c)}=C_{c}.{\frac{\sum\limits_{c=0}^{N}{n_{c}}}{N.n_{c}}}$ (1)

where $C_{c}$ is the class constant for a class c, N is the number of classes,
and $n_{c}$ is the number of samples in a class c. The computed class weights
are later fused with the objective function (loss function) of the deep
learning model in order to heavily penalize the false predictions concerned
with the minority samples, which in this case is coronavirus.

####  4.1.2 Random oversampling approach

In this approach, the objective is to increase the number of minority samples
by utilizing the existing samples belonging to the minority class. The
minority samples are increased until the samples associated with every class
become equal. Hence the procedure follows by identifying the difference
between the number of samples in majority and minority class. To fill this
void of difference, the samples are generated from the randomly selected
sample belonging to the minority class by applying certain statistical
operations. In this work, the samples of CXR image of COVID-19 positive cases
are less as compared to other classes, therefore, these minority class images
are randomly oversampled by means of rotation, scaling, and displacement with
the objective to achieve equal distribution of classes and accommodate
unbiased learning among the deep learning models.

###  4.2 Classification Strategy

Based on the type of data samples availability of CXR images the COVID-19
classification is divided into two following schemes:

  * •

Binary Classification - In this classification scheme, the coronavirus
positive samples labelled as “1” (COVID-19) are identified against the rest of
the samples labelled as “0” (non COVID-19 case) which involves other cases
e.g. chlamydophila, SARS, streptococcus, tuberculosis, etc., along with the
normal cases.

  * •

Multi-class Classification - In this classification scheme, the aim is to
distinguish and identify the COVID-19 samples from the other pneumonia cases
along with the presence of tuberculosis and normal case findings. The multi-
class classification is performed with three and four classes. The three
classes are provided with labels as “0” being a normal case, “1” being a
COVID-19 case, and “2” being other pneumonia and tuberculosis cases, whereas
four classes are labeled as “0” being a normal case, “1” being a COVID-19
case, and “2” being other pneumonia case and “3” as tuberculosis case.

In both the classification strategies, the deep learning models are trained
with the above discussed imbalanced learning approaches using the weighted
categorical cross entropy (WCE) loss function as given by Eq. 2 and 3
koidl2013loss :

$f{(s)}_{i}=\frac{e^{s_{i}}}{\sum\limits_{C}^{j}{e^{s_{i}}}}$ (2)
$WCE=-{\sum\limits_{i}^{C}{w(i).t_{i}.\log(f{(s)}_{i})}}$ (3)

In categorical cross entropy, the distribution of the predictions (the
activations in the output layer, one for each class) is compared with the true
distribution only, to ensure the clear representation of the true class as
one-hot encoded vector; here, closer the model’s outputs are to that vector,
the lower the loss.

###  4.3 Data Preprocessing

In this article, due to the limited samples of posteroanterior chest X-ray
images concerned with positive COVID-19 cohen2020covid  cases, the data
samples are mixed with the other randomly selected CXR images selected from
other datasets-, RSNA WHO5  and NLM(MC) jaeger2014two . The RSNA and NLM(MC)
datasets consists of posteroanterior CXR images covering sample cases labelled
as pneumonia and tuberculosis respectively along with normal samples. Table 2
describes the distribution of training, testing, and validation sets using the
fused dataset for binary and multi-class classification along with different
class imbalance strategies i.e. class weighted loss function that penalizes
the model for any false negative prediction and random oversampling
yap2014application  of minority classes which in this case is COVID-19.

Table 2: Posteroanterior CXR images distribution into training, validation,
and test sets from the fused datasets for different problem definitions.

Findings | Class imbalanced learning strategy
---|---
Classification with class weighted loss function | Classification with random oversampling
| Binary
---
(2 classes)
| Multi-class
---
(3 classes)
| Multi-class
---
(4 classes)
| Binary
---
(2 classes)
| Multi-class
---
(3 classes)
| Multi-class
---
(4 classes)
Tr | Val | Tst | Tr | Val | Tst | Tr | Val | Tst | Tr | Val | Tst | Tr | Val | Tst | Tr | Val | Tst
Normal | 906 | 90 | 110 | 437 | 44 | 52 | 437 | 44 | 52 | 960 | 90 | 110 | 469 | 44 | 52 | 437 | 44 | 52
Tuberculosis | 469 | 46 | 58 | 47 | 5 | 6 | 469 | 46 | 58 | 437 | 5 | 6
| Other
---
pneumonia
422 | 41 | 52 | 437 | 41 | 52
COVID-19 | 88 | 9 | 11 | 88 | 9 | 11 | 88 | 9 | 11 | 960 | 9 | 11 | 469 | 9 | 11 | 437 | 9 | 11
Total | 994 | 99 | 121 | 994 | 99 | 121 | 994 | 99 | 121 | 1920 | 99 | 121 | 1407 | 99 | 121 | 1748 | 99 | 121

The CXR images in the aggregated dataset also consists of unwanted artifacts
such as bright texts, symbols, varying resolutions and pixel level noise,
which necessitates its preprocessing. In order to suppress the highlighted
textual and symbolic noise, the images are inpainted with the image mask
generated using binary thresholding WHO6  as given by equation 4, followed by
resizing the images to a fixed size resolution of 331x331x3.

$M(x,y)=\begin{cases}{max}\\_{th},&i(x,y)\geq{{min}\\_{th}}.\\\
0,&\text{otherwise}.\end{cases}$ (4)

where i(x,y) is an input image, $max_{th}$ and $min_{th}$ are max and min
thresholds to design the mask. Despite filtering the unwanted information,
there is still the possibility of uncertainty at the deep pixel level
representation hasinoff2014photon . The denoising or removal of such
uncertainty is carried through the adaptive total variation method
szegedy2015going  while preserving the original distribution of pixel values.

Let for a given grayscale image f, on a bounded set $\Omega$ over
${\mathbb{R}}^{2}$, where $\Omega\subset{\mathbb{R}}^{2}$, denoising image u
that closely matches to observed image $x=(x_{1},x_{2})$ $\epsilon$ $\Omega$
\- pixels, given as

$u=\operatorname*{arg\,min}_{u}{\left(\int_{\Omega}{}(u-f.\ln{u})dx+\int_{\Omega}{}(\omega(x)|\bigtriangledown
u|dx)\right)}$ (5)

where $\omega(x)=\frac{1}{1+k{\mod{G_{\sigma}*\bigtriangledown u}}^{\prime}}$,
$G_{\rho}$ \- the Gaussian kernel for smoothing with $\sigma$ variance, k $>$
0 is contrast parameter and * is convolution operator.

Fig. 3 illustrates the data preprocessing stages by considering an instance of
COVID-19 case consisting of textual and symbolic artifacts from the generated
dataset. The resulting distributed pixels histograms at each stage of
preprocessing shown in Fig. 3, illustrates that the preprocessing approach
tends to preserve the original nature of distribution of the pixels while
removing the irregular intensities. The preprocessed images are then divided
into training, testing, and validation set for training and evaluation of the
state-of-the-art deep learning classification models.

Figure 3: Data preprocessing stages of raw posteroanterior CXR image. Table 3: Recent deep learning architectures that are reported with top 5 error rate per year. Architecture | Year | Contribution | Param. | Dataset | E. rate
---|---|---|---|---|---
| Inception-v3
---
szegedy2016rethinking
2015 |  | Avoided the bottleneck
---
representations,
Dimension reduction
promotes faster learning.
23.6M | ILSVRC | 5.6
| Inception
---
ResNet-v2
szegedy2017inception
2016 |  | Focused on residual
---
connection rather
than filter connection
via split and merge
approach.
56M | ILSVRC | 4.9
| DenseNet169
---
huang2017densely
2017 |  | Dense connection blocks
---
for activation flow across
the layers.
17.8M |  | CIFAR-10
---
CIFAR-100
CIFAR-10+
CIFAR-100+
| 5.19
---
19.64
3.46
17.18
| NASNetLarge
---
zoph2018learning
2018 |  | Search and transfer the
---
architecture block from
small dataset to large
dataset,
Scheduled drop path
regularization.
22.6M | CIFAR-10 | 2.4

###  4.4 Deep learning models

This section incorporates the state-of-the-art deep learning models utilized
in the present research work as shown in Table 3 along with their respective
contribution, parameters, and performance on the standard benchmark datasets.
The inception deep convolutional architectures proposed by GoogLeNet are
considered as the state-of-the-art deep learning architectures for image
analysis and object identification with the basic model as inception-v1
chen2010adaptive . Later, this base model was refined by introducing the batch
normalization and established as the inception-v2 szegedy2016rethinking . In
further iterations, additional factorisation was introduced and released as
the inception-v3. It is one of the pre-trained models to perform two types of
specific tasks: dimensionality reduction using CNN and classification using
fully-connected and softmax layers. Since it is originally trained on over a
million images consisting of 1,000 classes of ImageNet, its head layers can be
retrained for the generated dataset using historical knowledge to reduce the
extensive training and computational power. Later, Inception-ResNet-v2 was
proposed by Szegedy et al. szegedy2017inception . This hybrid model is a
combination of residual connections and a recent version of Inception
architecture. It is intended to train very deep convolutional models by the
additive merging of signals, both for image recognition and object detection.
This network is more robust and learns rich feature representations.
Afterwards, DenseNet was proposed by Huang et al. huang2017densely . It works
on the concept of reuse, in which each layer receives inputs from all previous
layers and yields a condensed model to pass its own feature-maps to all
subsequent layers. This makes the network thinner and compact with the fewer
number of channels, while improving variation in the input of subsequent
layers, and becomes easy to train and highly parameter efficient. Google Brain
research team proposed NASNet model based on reinforcement learning search
methods zoph2018learning . It creates search space by factoring the network
into cells and further dividing it into a number of blocks. Each block is
supported by a set of popular operations in CNN models with various kernel
size e.g: convolutions, max pooling, average pooling, dilated convolution,
depth-wise separable convolutions etc.

###  4.5 Fine tuning

Training deep learning models require (Inception-v3, InceptionResNet-v2, etc.)
exhaustive amount of resources and time. While these networks, attain
relatively excellent performance on ImageNet 5206848 , training them on a CPU
is an exercise in futility. These CNNs are often trained for a couple of weeks
or more using arrays of GPUs to get good results on the complex and
complicated datasets. In most deep CNNs the first few convolution layers learn
low-level features (edges, curves, blobs) and with progress, through the
network, it learns more mid/high-level features or patterns associated with
the on-going task. In fine tuning, the aim is to keep or freeze these trained
low-level features, and only train the high-level features needed for our new
image classification problem.

Figure 4: Schematic representation of training framework for deep learning
architectures via transfer learning.

In this article, the trials of training the deep learning classification
models initiates with the baseline residual network that is composed of five
residual blocks he2016deep , defined by two convolutions whose rectified
linear unit (ReLU) activation is concatenated with the input of the first
convolution layer, followed by max-pooling and instance normalization, except
the final output layer which uses softmax activation function. Later, transfer
learning is utilized on the state-of-the-art architectures discussed in Table
3 to utilize the pre-trained models while fine tuning the head layers to serve
the purpose of classifying the COVID-19 samples, as illustrated in Fig. 4.
This follows by enabling the four head layers of the network to adjust its
trainable parameters along with the addition of fully connected layers with
128 neurons and 2 or 3 neurons in the output layer depending on binary or
multi-class classification, accompanied with the softmax activation function.

##  5 Evaluation metrics

The deep learning models are trained using the training and validation set
under consideration of each possible combination of the discussed approaches
of class imbalance learning and classification strategy, thereby making four
possible scenarios for training a model. Later, the trained models are
evaluated on the test set using the standard benchmark performance metrics
such as accuracy, precision, recall (selectivity), area under curve (AUC),
specificity and F1 score (dice coefficient) as shown in Fig 5.

Figure 5: Confusion matrix and performance evaluation metrics.

##  6 Training and testing

The proposed approach is trained and tested on the above discussed fused
dataset using mini-batch gradient descent with learning rate optimizer as adam
ruder2016overview  via N-Vidia Titan GPU. The training, testing, and
validation sets are highlighted in Table 2. The training phase is assisted
with the batch size of 10, 4-fold cross validation, switch normalization
luo2018differentiable  to adapt to instance or batch or layer normalization,
and earlystopping technique that aims to stop the training process if the
validation error stops decreasing. For each epoch the above discussed
evaluation metrics are computed on the training and validation set to analyse
the model’s performance and improve the classification results. Later, the
test set is utilized to evaluate the results of the proposed approach.

##  7 Results and discussion

With extensive trials, it is observed that there is no individual model that
displayed best performance for all the scenarios in terms of accuracy,
precision, recall, specificity, AUC, and F1-score. However, the NASNetLarge
mostly claimed the best scores, thus making it best fit for the classification
of COVID-19 samples. It is also observed that the results of binary
classification (COVID-19 vs non-COVID-19) are better than the multi-class
classification (COVID-19 vs other classes). With this it is evident that by
grouping other diseases together as non-COVID-19 samples, models can
efficiently learn features and patterns belonging to the COVID-19. Fig. 6
shows the average performance curves of the NASNetLarge model evaluated and
monitored on the test set for each iteration during the training phase for
classification of COVID-19 samples, generated using the tensorboard. Since the
models are initialized with the trained weights on the ImageNet and head
layers are fine-tuned for classification, the training of the models initiates
with descent metrics values. The best scores of the models corresponding to
each scenario are highlighted in Table 4 along with the detailed comparative
analysis of the results for classifying the COVID-19 samples in terms of
discussed evaluation metrics for different scenarios CB, CM3, CM4 RB, RM3 and
RM4 where C and R indicates class imbalanced learning approaches, namely
weighted class loss functions and random oversampling of the minority class,
whereas B indicates the binary classification and M3 (3 classes), M4 (4
classes) indicate multi-class classification schemes. Furthermore, the results
obtained are compared with the other recently proposed approaches to detect
COVID-19 positive symptoms via X-ray images as shown in Table 5. It is
observed that the proposed approach outperforms other approaches, whereas
Apostolopoulos et al. apostolopoulos2020covid  achieved similar results with
the help of VGG19 model simonyan2014very  but with approximately double
trainable parameters (144M) as compared to the proposed approach (83M).

Figure 6: Monitoring every epoch of training with performance curves for the NASNetLarge model with average (a) accuracy, (b) AUC, (c) specificity, (d) F1-score, (e) precision, and (f) recall. The curves are smoothed using moving exponential average. Table 4: Comparative analysis of the classification results of the deep learning models for each scenario. Model | Scenario | Accuracy | Precision | Recall | AUC | Specifcity | F1-score
---|---|---|---|---|---|---|---
Baseline ResNet | CB | 0.84 | 0.67 | 0.89 | 0.90 | 0.85 | 0.76
CM3 | 0.77 | 0.62 | 0.89 | 0.90 | 0.77 | 0.73
CM4 | 0.77 | 0.61 | 0.83 | 0.90 | 0.77 | 0.70
RB | 0.89 | 0.56 | 0.56 | 0.89 | 0.60 | 0.56
RM3 | 0.76 | 0.63 | 0.42 | 0.83 | 0.77 | 0.50
RM4 | 0.76 | 0.65 | 0.43 | 0.82 | 0.73 | 0.52
Inception v2 | CB | 0.88 | 0.77 | 0.79 | 0.93 | 0.89 | 0.78
CM3 | 0.87 | 0.73 | 0.88 | 0.90 | 0.88 | 0.80
CM4 | 0.87 | 0.73 | 0.85 | 0.89 | 0.85 | 0.79
RB | 0.88 | 0.78 | 0.70 | 0.88 | 0.91 | 0.74
RM3 | 0.87 | 0.90 | 0.81 | 0.91 | 0.90 | 0.85
RM4 | 0.84 | 0.89 | 0.81 | 0.91 | 0.88 | 0.85
Inception ResNet v2 | CB | 0.95 | 0.86 | 0.66 | 0.98 | 0.87 | 0.75
CM3 | 0.92 | 0.85 | 0.92 | 0.99 | 0.89 | 0.88
CM4 | 0.91 | 0.83 | 0.91 | 0.98 | 0.89 | 0.87
RB | 0.90 | 0.85 | 0.75 | 0.93 | 0.88 | 0.80
RM3 | 0.92 | 0.97 | 0.96 | 0.99 | 0.94 | 0.96
RM4 | 0.91 | 0.95 | 0.93 | 0.98 | 0.93 | 0.94
DenseNet 169 | CB | 0.90 | 0.82 | 0.78 | 0.91 | 0.91 | 0.80
CM3 | 0.87 | 0.87 | 0.91 | 0.95 | 0.89 | 0.89
CM4 | 0.86 | 0.87 | 0.89 | 0.93 | 0.89 | 0.88
RB | 0.95 | 0.94 | 0.96 | 0.97 | 0.95 | 0.95
RM3 | 0.91 | 0.93 | 0.95 | 0.93 | 0.95 | 0.94
RM4 | 0.92 | 0.94 | 0.95 | 0.94 | 0.95 | 0.94
NASNet Large | CB | 0.97 | 0.82 | 0.91 | 0.99 | 0.98 | 0.86
CM3 | 0.94 | 0.89 | 0.91 | 0.94 | 0.89 | 0.90
CM4 | 0.95 | 0.88 | 0.89 | 0.92 | 0.89 | 0.88
RB | 0.98 | 0.87 | 0.90 | 0.99 | 0.94 | 0.98
RM3 | 0.96 | 0.93 | 0.91 | 0.96 | 0.94 | 0.92
RM4 | 0.95 | 0.95 | 0.90 | 0.95 | 0.94 | 0.92
Table 5: Comparative analysis of the proposed approach with recently proposed
strategies.

Authors |  Technique |  Network | Classes | Accuracy | Precision | Recall | AUC | Specificity
---|---|---|---|---|---|---|---|---
Yujin et al. oh2020deep  |  Fine tuning |  Res- Net18 | 4 | 0.88 | 0.83 | 0.86 | - | 0.96
Afshar et al. afshar2020covid  |  Fine tuning |  Capsule | 0.95 | - | 0.90 |  | 0.95
Proposed |  Fine tuning |  NASNet Large | 0.95 | 0.95 | 0.90 | 0.94 | 0.92
Wang et al. wang2020covid  |  Full training |  COVID-Net | 3 | 0.92 | 0.91 | 0.88 | - | -
Apostolopoulos et al. apostolopoulos2020covid  |  Fine tuning |  VGG19 | 0.87 | - | 0.92 | - | 0.98
Proposed |  Fine tuning |  NASNet Large | 0.96 | 0.93 | 0.91 | 0.96 | 0.94
Hall et al. hall2020finding  |  Fine tuning |  Ensemble | 2 | 0.91 | - | 0.78 | 0.94 | 0.93
Apostolopoulos et al. apostolopoulos2020covid  |  Fine tuning |  VGG19 | 0.98 | - | 0.92 | - | 0.98
Proposed |  Fine tuning |  NASNet Large | 0.98 | 0.87 | 0.90 | 0.99 | 0.98

Figure 7: LIME explanation of three distinct class samples using NASNetLarge
along with the prediction probabilities of sample being normal (N), COVID-19
(C), and other pneumonia (OP).

##  8 Visualization

Training the model and getting the results is not sufficient unless it is
understood that what is triggering the concerned output. To handle this
blackbox, visualization techniques assist in illustrating the basis of
prediction of the model. There are many visualization techniques for example
class activation maps (CAM) selvaraju2017grad , saliency maps (SM)
simonyan2013deep , local interpretable model-agnostic explanations (LIME)
ribeiro2016should , and a lot more. In this article, activation maps and LIME
techniques are utilized to present the model perception of identifying and
classifying the COVID-19 samples from CXR images. CAM aims at understanding
the feature space of an input image that influences the prediction, whereas
LIME is an innovative explanation technique to represent the model prediction
with local fidelity, interpretability and model agnostic. For instance, fine-
tuned NASNetLarge architecture is considered to generate LIME explanations for
some samples taken from the test set, whereas class activation maps tend to
present the patterns learned by the model for classification of samples as
shown in Fig. 8. Fig. 7 presents the LIME technique applied to four samples
belonging to four distinct classes as COVID-19, other types of pneumonia,
tuberculosis and normal cases. The red and green areas in the LIME generated
explanation (Fig. 7) correspond to the regions that contributed against the
predicted class and towards the predicted class respectively.

Figure 8: CAM of NASNetLarge model for (a) Normal, (b) COVID-19, and (c) Other
pneumonia.

##  9 Conclusion

This article proposes to leverage the state-of-the-art deep learning models to
aid in early identification and diagnosis of COVID-19 virus by using the
limited posteroanterior chest X-ray images. Each trained model was evaluated
using benchmark performance metrics e.g. accuracy, precision, recall, area
under curve, specificity, and F1 score under four different scenarios
concerned with imbalanced learning and classification strategy. With extensive
trials, it was observed that models achieve different scores in different
scenarios, among which NASNetLarge displayed better performance specially in
binary classification of COVID-19 samples. The visual representation based on
local interpretable model agnostic explanations is utilized to understand the
basis of prediction of the model. As an extension to this work more deep
learning models and preprocessing techniques can be explored to achieve better
results.

## Compliance with Ethical Standards

### Conflict of Interest

The authors have no conflict of interest to declare.

### Ethical approval

This article does not contain any studies with human participants or animals
performed by any of the authors.

###### Acknowledgements.

We are very much obliged to our institute, Indian Institute of Information
Technology Allahabad (IIITA), India and Big Data Analytics (BDA) lab for
ensuring the needed resources and support. We also would like to extend our
thanks to the colleagues for their valuable guidance and suggestions. The
manuscript is also available in the pre-print arXiv with the eprint-2004.11676
under the license Attribution 4.0 International (CC BY 4.0).

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