# Cyclic and Abelian CLT groups

Khyati Sharma and A. Satyanarayana Reddy
Department of Mathematics
Shiv Nadar University, India - $201314$
E-mail<EMAIL_ADDRESS>&<EMAIL_ADDRESS>
###### Abstract

A group $G$ of order $n$ is said to be an ACLT (CCLT) group, if for every
divisor $d$ of $n,$ where $d<n,$ $G$ has an abelian (cyclic) subgroup of order
$d.$ A natural number $n$ is said to be an ACLT (CCLT) number if every group
of order $n$ is an ACLT (CCLT) group. In this paper we find all ACLT and CCLT
numbers and study various properties of ACLT (CCLT) groups.

Key Words: CLT groups, CLT numbers, Cyclic numbers, Abelian numbers
AMS(2020): 20D60,20D05

##  1 Introduction and preliminaries

One of the important theorems of group theory is Lagrange’s theorem which
states that order of a subgroup of a finite group divides the order of that
group. However the converse of Lagrange’s theorem (CLT) is not true in general
and over the years, different researchers have investigated this issue from a
variety of angles. A group is said to be a CLT group if it satisfies the
converse of Lagrange’s theorem. For further details about CLT groups refer
[5]. A natural number $n$ is said to be a cyclic (abelian, CLT) number if
every group of order $n$ is a cyclic (abelian, CLT) group. Berger in $1978$
and Nganou in $2017$ studied CLT numbers and gave a complete characterization
of CLT numbers (see [3],[13]). In fact, Berger showed that every squarefree
number and a number which is divisible by only one prime is a CLT number.

Throughout this paper all groups are finite. In this paper we generalize the
notions of CLT groups and CLT numbers to the notions of ACLT (CCLT) groups and
ACLT (CCLT) numbers respectively. A finite group of order $n$ is said to be an
Abelian (Cyclic) CLT group, if for each divisor $d$ of $n,$ where $d<n,$ there
exists an abelian (cyclic) subgroup of order $d.$ We call such abelian CLT and
cyclic CLT groups as ACLT and CCLT groups respectively. A natural number $n$
is said to be an ACLT (CCLT) number if every group of order $n$ is an ACLT
(CCLT) group.

This paper is organized as follows. In Section 2, we find all CCLT numbers and
CCLT groups. We also investigate important properties of CCLT groups. In
Section 3 we characterize ACLT numbers, study various properties of ACLT
groups and discuss some of their concrete examples. In section 4, we summarize
the relationship between cyclic, abelian, CLT, CCLT and ACLT numbers using
Venn diagram.

However in this work, all groups which arise out of semidirect products are
non-abelian. Furthermore, depending on the context, we advise the reader to
choose suitable semidirect product. We denote cyclic group of order $n$ by
$C_{n}.$ For the examples of groups of given order and for their properties
refer [10].

##  2 CCLT Groups

In this section we find all CCLT numbers. We characterize all CCLT groups,
discuss their properties and produce some related examples. We begin with the
following observations about CCLT numbers.

It is obvious that every cyclic group is a CCLT group, hence every cyclic
number is a CCLT number. The smallest number which is not a cyclic number but
CCLT number is $4$ and the smallest CCLT number which is not an abelian number
is $6.$ If $p$ is a prime number, then the group $C_{p}\times C_{p}\times
C_{p}$ is not a CCLT group as it has no cyclic subgroup of order $p^{2}.$
Hence $p^{3}$ is not a CCLT number. But it is interesting to see that every
other group of order $p^{3}$ is a CCLT group.

######  Theorem 1.

A natural number $n$ is a CCLT number if and only if either $n$ is a cyclic
number or $n=pq$, where $p$ and $q$ are prime numbers, need not be distinct.

###### Proof.

If $n$ is a cyclic number or $n=p^{2}$ or $n=pq$, where $p$ and $q$ are
distinct prime numbers, then it is easy to prove that $n$ is a CCLT number.
For the converse, we prove that if $n$ is not a cyclic number or $n$ is not of
the form $p^{2}$ or $pq,$ then we give an example of a non-CCLT group of that
order.

Let $n=p^{k},$ where $k\geq 3.$ Then the group $G=C_{p}\times C_{p}\times
C_{p^{k-2}},$ does not contain a cyclic subgroup of order $p^{k-1}.$ Suppose
$n$ is a square free number of the form $n=p_{1}\times p_{2}\times\cdots\times
p_{k},$ where $k\geq 3.$ Since $n$ is not a cyclic number, there exists
$p_{i},p_{j}\in\\{p_{1},p_{2},\ldots,p_{k}\\}$ such that $p_{i}|p_{j}-1.$ If
we consider $G=H\times C_{\frac{n}{p_{i}p_{j}}},$ where $H$ is a non-abelian
group of order $p_{i}p_{j},$ then it is easy to see that $G$ does not contain
an abelian subgroup of order $p_{i}p_{j}.$ Finally if $n$ is of the form
$p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots p_{k}^{a_{k}},$ where $a_{1}\cdot
a_{2}\cdots a_{k}\geq 2,$ then $G=\left(C_{p_{j}}\times
C_{p_{j}^{a_{j-1}}}\right)\times C_{\frac{n}{p_{j}^{a_{j}}}},$ where
$a_{j}\geq 2$ does not contain a cyclic subgroup of order $p_{j}^{a_{j}}.$ ∎

Now we have $n\in{\mathbb{N}}$ is a CCLT number if and only if either
$\gcd(n,\varphi(n))=1$ or $n=pq,$ where $p,q$ are primes need not be distinct.
For every natural number $n,$ there is a cyclic group of order $n,$ hence a
CCLT group of order $n.$ The following are few nontrivial examples of CCLT
groups of order $n,$ where $n$ need not be a CCLT number.

######  Example 2.

  1. 1.

Let $G$ be a dihedral group $D_{n}$ of order $2n$ or a Dicyclic group
$Dic_{n}$ of order $4n.$ Then $G$ is a CCLT group if and only if either $n$ is
a prime number or $n=2^{k},$ where $k\in{\mathbb{N}}.$

###### Proof.

Since $m=2p,$ where $p$ is prime is a CCLT number, every group of order $2p$
is a CCLT group. If $n=2^{k},$ then $D_{n}$ is a group of order $2^{k+1}$ and
it contains a cyclic subgroup of order $2^{k}$ hence $D_{n}$ is a CCLT group.

If $n=2p$ by definition, $Dic_{n}$ contains an element of order $4$ and and an
element of order $2p.$ Hence it is CCLT group. If $n=2^{k},$ then $Dic_{n}$
contains cyclic subgroup of order $2^{k+1}.$

Conversely suppose that if $n$ is not of the above forms, then $n$ is
divisible by an odd prime say $p$ and $n>p.$ Then it is easy to see that
$D_{n}$ ( or $Dic_{n}$) has no cyclic subgroup of order $2^{m+1}p^{k}$ (or
$2^{m+2}p^{k}$), where $2^{m}|n$ but $2^{m+1}\nmid n,k\in\\{0,1\\}.$ ∎

  2. 2.

The group $G=C_{p^{n-1}}\times C_{p}$ is a CCLT group, where $p$ is a prime
number and $n\in{\mathbb{N}}.$

It is well known that cyclic groups $\subset$ abelian groups $\subset$
nilpotent groups $\subset$ supersolvable groups $\subset$ CLT groups $\subset$
solvable groups. By definition CCLT and ACLT groups are CLT groups. We already
know that abelian group need not be a CCLT group and $S_{3}$ is a CCLT group
but not a nilpotent group. Also recall that $S_{4}$ is a CLT group, but its
subgroup $A_{4}$ is not a CLT group, whereas every subgroup of a supersolvable
group is supersolvable. A natural question: is every subgroup of a CCLT group
is CCLT? To answer this question, we first prove that every CCLT group is
supersolvable, in fact we prove more. Consequently, every subgroup of CCLT
group is CLT.

Recall that a group $G$ is a metacyclic if $G$ contains a cyclic normal
subgroup $H$ such that $G/H$ is cyclic. Every cyclic group is metacyclic. Let
$G$ be a CCLT group of order $n$, $p$ is the smallest prime dividing $n$ and
$H$ be a cyclic subgroup of order $\frac{n}{p}.$ Then it is clear that $H$ is
normal in $G$ and $G/H$ is cyclic. Hence the following result.

######  Proposition 3.

Every CCLT group is metacyclic hence supersolvable.

An interesting observation is that most of the CCLT groups have a property
that all its proper subgroups are cyclic. It is not true in general as groups
of prime power order provides a counterexample. In particular, if $n\geq 3$
then the group $C_{p}\times C_{p}^{n-1}$ is a CCLT group of order $p^{n}$ and
its subgroups $C_{p^{n-1}}$ and $C_{p}\times C_{p^{n-2}}$ of order $p^{n-1}$
are not isomorphic. In the literature non-cyclic groups all of whose proper
subgroups are cyclic are known as minimal non-cyclic groups. These groups are
completely characterized by Theorem $2.1$ of [18].

######  Theorem 4.

Let $G$ be a CCLT group of order $n=p_{1}^{a_{1}}\cdot p_{2}^{a_{2}}\cdots
p_{k}^{a_{k}},$ where $k\geq 2.$ Then every Sylow $p$-subgroup of $G,$ where
$p\in\\{p_{1},p_{2},\ldots,p_{k}\\}$ is cyclic.

###### Proof.

Let $G$ be a CCLT group of order $n=p_{1}^{a_{1}}\cdot p_{2}^{a_{2}}\cdots
p_{k}^{a_{k}},$ where $k\geq 2$ and $H_{i}$ be a cyclic Sylow $p_{i}$-subgroup
of $G.$ Then from Sylow’s theorem, every Sylow $p_{i}$-subgroup of $G$ is
conjugate to $H_{i}.$ Hence all Sylow $p$-subgroups of $G$ are cyclic. ∎

In the literature, the groups satisfying the property of hypothesis of Theorem
4 are called $Z$-groups and it is known that $Z$-groups are metacyclic. For
examples of $Z$-groups (see [10]). Dihedral group $D_{15}$ is an example of
$Z$-group but not a CCLT group. Hence if we exclude groups of prime power
order, then we have

$\mbox{cyclic groups}\subset\mbox{CCLT
groups}\subset\mbox{$Z$-groups}\subset\mbox{metacyclic
groups}\subset\mbox{supersolvable groups}.$

######  Theorem 5.

Any two subgroups of same order of a $Z$-group are isomorphic.

###### Proof.

Let $G$ be a $Z$-group of order $mn.$ Then from Theorem 11 of Chapter V of
[22] we have

$G=\langle a,b|a^{m}=b^{n}=e,bab^{-1}=a^{r},\gcd((r-1)n,m)=1,r^{n}\equiv
1\;\pmod{m}\rangle.$ (1)

Let $H$ and $K$ be subgroups of $G$ of same order $d.$ If $d|m$ or $d|n,$ then
we are done as both $H$ and $K$ are cyclic groups of order $d$. Suppose
$d=ts,$ where $t,s>1,$ $t|m$ and $s|n.$ Then

$H=\langle a^{k_{1}},b^{k_{2}}\rangle,K=\langle
a^{v_{1}},b^{v_{2}}\rangle,\;\;\mbox{where}\;\;o(a^{k_{1}})=o(a^{v_{1}})=t,o(b^{k_{2}})=o(b^{v_{2}})=s.$

We define a map $\psi:H\to K$ as $\psi(a^{k_{1}})=a^{v_{1}}$ and
$\psi(b^{k_{2}})=b^{v_{2}}.$ Then by using relations in Equation (1) we can
see that $\psi$ is an isomorphism. ∎

######  Corollary 6.

A CCLT group which is neither cyclic nor a $p$-group is minimal non-cyclic.

######  Theorem 7.

Every subgroup of a CCLT group is CCLT.

###### Proof.

Let $G$ be a CCLT group.
Let $G$ be a $p$-group of order $p^{n}.$ It is clear that the result is true
for $n\leq 3.$ Let $G$ be a CCLT group of order $p^{n},$ where $n\geq 4.$ We
prove the result by the method of contradiction. Suppose $k$ is the largest
such that there exists a non-CCLT subgroup $H$ of order $p^{k}.$ By Sylow’s
theorem $H$ is a normal subgroup of a CCLT subgroup $N$ of $G$ of order
$p^{k+1}.$ Let $K$ be a cyclic subgroup of $N$ of order $p^{k}.$ Then
$HK\subseteq N$ and

$|HK|=\frac{|H||K|}{|H\cap K|}=p^{2k-{\alpha}}\leq p^{k+1},$

where ${\alpha}=|H\cap K|.$ Also we can prove that ${\alpha}=k-1.$ Thus we get
a contradiction as $H$ contains a cyclic subgroup of order $p^{k-1}.$
G is not a $p$-group. In this case by Corollary 6 we can say that every proper
subgroup of CCLT group which is not a $p$-group is cyclic. Hence the result
follows. ∎

######  Remark 8.

Before proceeding by Proposition $2.5$ of [11] and Theorem 5 one can see that
order of every CCLT group which is neither cyclic nor a $p$-group is of the
form $p^{r}q,$ where $p,q$ are primes and $p|q-1.$ Further there exists a
unique non-cyclic CCLT group of that order. With these observations, we can
make the following conclusions. Let $G_{CCLT}(n)$ denotes the number of non-
isomorphic CCLT groups of order $n.$ If $n$ is not a prime power, then we have

$G_{CCLT}(n)=\begin{cases}2&\mbox{if $n=p^{r}q$ and $p|(q-1)$}\\\
1&\mbox{otherwise.}\end{cases}$

We know that from Theorem 1, $p$ and $p^{2}$ are CCLT number so every group of
order $p$ and $p^{2}$ are CCLT. We also observe that if $p$ is a prime number
and $k\in{\mathbb{N}}$ then $G_{CCLT}(2^{3})=4$ and

$G_{CCLT}(p^{k})\geq\begin{cases}6&\mbox{if $p=2$ and $k>3$}\\\ 3&\mbox{if
$p\not=2$ and $k>2$.}\end{cases}$

######  Theorem 9.

Let $G$ be a CCLT group of order $n$ and ${\mathcal{S}}(G)$ denotes the number
of subgroups of $G.$ Then

${\mathcal{S}}(G)=\begin{cases}\tau(n)&\mbox{if $G$ is cyclic}\\\
2+(p+1)(k-1)&\mbox{if $G$ is abelian and $n=p^{k}$}\\\ 2r+q+1&\mbox{if
$n=p^{r}q,p\not=q$ and $p|(q-1).$}\end{cases}$

###### Proof.

If $G$ is cyclic then it is a well known fact that $S(G)=\tau(n)$ (see [8]).
However, when $G$ is non-cyclic abelian group, then one can easily deduce that
$G$ is a $p$-group as well as it is isomorphic to $C_{p}\times C_{p}^{k-1},$
therefore by Theorem A of [1] it is immediate that $S(G)=2+(p+1)(k-1)$.

On the other hand if $G$ is non-abelian, then Corollary 6 and Remark 8 yields
that $o(G)=p^{r}q$ and all proper subgroups of $G$ are cyclic. According to a
result given in [19] it can be concluded that $G$ has a unique cyclic subgroup
$H$ of order $p^{r-1}q.$ Since any subgroup of $G$ of order $p^{r-1}q$ is
maximal, therefore all other proper subgroups of $G$ are contained in $H$
except the subgroups of order $p^{r}.$ Now it is easy to see that number of
subgroups of $H$ is $\tau(p^{r-1}q)=2r.$ Also, number of elements of order
$p^{r}$ in $G$ is $p^{r}q-p^{r-1}q,$ which implies that number of subgroups of
order $p^{r}$ in $G$ is $q.$ Hence, $S(G)=2r+q+1.$ ∎

######  Corollary 10.

Let $G$ be a CCLT group and $C(G)$ denotes the set of all cyclic subgroups of
$G.$ Then

$|C(G)|=\begin{cases}\tau(n)&\mbox{if $G$ is cyclic}\\\ (k-1)p+2&\mbox{if $G$
is abelian and $n=p^{k}$}\\\ 2r+q&\mbox{if $n=p^{r}q,p\not=q$ and
$p|(q-1).$}\end{cases}$

###### Proof.

The proof is immediate by using Theorem $1$ of [20] and Corollary 6. ∎

The following result answers, when the direct product of CCLT groups is a CCLT
group.

######  Theorem 11.

Let $H$ and $K$ be groups of order $m$ and $n$ respectively. If $H\times K$ is
a CCLT group, then both $H$ and $K$ are cyclic groups. Conversely if $H$ and
$K$ are cyclic groups, then $H\times K$ is a CCLT group if one of the
following conditions hold:

  1. 1.

If $\gcd(m,n)=1.$

  2. 2.

If $\gcd(m,n)\neq 1,$ then $H\times K\cong C_{p}\times C_{p^{k}},$ where $p$
is a prime number and $k\in{\mathbb{N}}.$

###### Proof.

Let $H\times K$ be a CCLT group and $H$ be not a cyclic group. If
$\gcd(m,n)=1$ then it is easy to see that there is no cyclic subgroup of order
$m.$ If $\gcd(m,n)\neq 1$ and $p|\gcd(m,n),$ where $p$ is a prime number. Then
$m=p^{r}x$ and $n=p^{s}y,$ where $\gcd(p,x)=\gcd(p,y)=1.$ We can show that
$H\times K$ has no cyclic subgroup of order $p^{\max\\{r,s\\}+1}.$ Conversely,
suppose that $H$ and $K$ are cyclic groups. If $\gcd(m,n)=1,$ then $H\times K$
is cyclic and hence CCLT. Let $p|\gcd(m,n),$ where $p$ is a prime number. Then
$m=p^{r}x$ and $n=p^{s}y,$ where $\gcd(p,x)=\gcd(p,y)=1.$ Now we claim that
$xy=1$ and either $r=1$ or $s=1.$ Suppose $xy\neq 1$ or both $r>1$ and $s>1,$
then $H\times K$ has no cyclic subgroup of order $p^{\max\\{r,s\\}+1}.$ This
contradicts $H\times K$ is a CCLT group. ∎

Now from Theorem 4 and Part 2 of Theorem 11, we can say that every abelian
CCLT group is cyclic except $C_{p^{n-1}}\times C_{p}.$

######  Theorem 12.

Quotient group of a CCLT group is CCLT.

###### Proof.

Let $G$ be a CCLT group and $N$ be a normal subgroup of $G.$

$G$ is a $p$-group. Let $G$ be a group of order $p^{n},$ where $p$ is a prime
number. It is sufficient to consider the case that $o(N)=p^{m},$ where $m\leq
n-2.$ Since $G$ is a CCLT group, there exists an element $g\in G$ of order
$p^{n-1}.$ Let $o(gN)=p^{k},$ where $k\leq{n-m}.$ Then $o({g^{p}}^{k})\leq
p^{m}.$ Consider,

$o({g^{p}}^{k})=\frac{o(g)}{(o(g),p^{k})}=\frac{p^{n-1}}{(p^{n-1},p^{k})}=\frac{p^{n-1}}{p^{k}}=p^{n-k-1}\leq
p^{m}.$

As a consequence $k\geq n-m-1.$ Which shows that $k=n-m-1$ or $k=n-m.$ In both
the cases $G/N$ is CCLT.

$G$ is not a $p$-group. Then by Corollary 6 all proper subgroups of $G$ are
cyclic. If $N$ is a normal subgroup of $G$ then by Correspondence theorem
every proper subgroup of $G/N$ is cyclic. Hence $G/N$ is CCLT.

∎

##  3 ACLT Groups

In this section we characterize all ACLT numbers. Later we find out important
properties of ACLT groups and discuss some related examples. We begin with the
necessary and sufficient condition for a natural number $n$ to be an ACLT
number.

######  Theorem 13.

A natural number $n$ is an ACLT number if and only if one of the following
conditions hold

  1. 1.

$n$ is an abelian number.

  2. 2.

$n=pq$, where $p$ and $q$ are distinct primes.

  3. 3.

$n=p^{m}$, where $p$ is prime $m\in\\{0,1,2,3,4\\}$.

  4. 4.

$n=p^{2}q$, where $p$ and $q$ are distinct primes and

    1. (a)

$p=2,q\in\\{4k+3,:k\in\mathbb{N}\\}.$

    2. (b)

$p|q-1,\ p^{2}\nmid q-1,\ p\nmid q+1$ and $q\nmid p^{2}-1$.

###### Proof.

First we show that if $n$ is one of the given form, then $n$ is an ACLT
number. If $n$ is an abelian number or $n=pq$ or $n=p^{3},$ then it is easy to
see that $n$ is an ACLT number. If $n=p^{4},$ then in order to show that $n$
is an ACLT number it is sufficient to prove that it contains an abelian
subgroup of order $p^{3}.$ But it is known that a group of order $p^{m}$ has
an abelian subgroup of order $p^{k},$ where $k(k+1)\geq 2m$ (see [6] or [4]).
Thus $n=p^{4}$ is an ACLT number. If $n=p^{2}q,$ then every group of order $n$
contains subgroups of order $1,p,q,p^{2}$ and they are abelian. Hence in order
to show that $n$ is an ACLT number it is sufficient to show that it contains
an abelian subgroup of order $pq.$

If $n=4q,$ where $q\in\\{4k+3|k\in{\mathbb{N}}\\},$ then $G(n)=4,$ where
$G(k)$ denotes the number of groups of order $k$ refer [9]. Further, these
four distinct groups up to isomorphism are $C_{4q},C_{2}\times C_{2q},Dic_{q}$
or $D_{2q}.$ From Example 2 and Example 14 we know that the groups $Dic_{q}$
and $D_{2q}$ are ACLT respectively. If $n=p^{2}q,$ where $p|q-1,\ p^{2}\nmid
q-1,\ p\nmid q+1$ and $q\nmid p^{2}-1$, then $G(n)=4$ and if $G$ is a group of
order $n,$ then $G$ is isomorphic to $C_{p^{2}q}$ or $C_{p}\times C_{pq}$ or
$C_{q}\rtimes C_{p^{2}}$ or $C_{q}\rtimes(C_{p}\times C_{p}).$ It is easy to
verify that all these groups contain an abelian subgroup of order $pq.$ Hence
they are ACLT groups.

To prove the converse part, we show that whenever $n$ is not in any one of the
forms given in the hypotheses, then there exist a non-ACLT group of order $n$.

Suppose $n$ is a squarefree number of the form $p_{1}\times p_{2}\times
p_{3}\times\cdots\times p_{m},$ where $m\geq 3.$ Since $n$ is not an abelian
number there exists $p_{i},p_{j}\in\\{p_{1},p_{2},\ldots,p_{m}\\}$ such that
$p_{i}|p_{j}-1.$ In this case if we consider the group $G=(C_{p_{j}}\rtimes
C_{p_{i}})\times C_{\frac{n}{p_{i}p_{j}}},$ then $G$ does not contain an
abelian subgroup of order $p_{i}p_{j}.$

If $n=p^{m},$ where $p$ is prime and $m\geq 5,$

$G=\begin{cases}\left((C_{2})^{3}\rtimes C_{4}\right)\times
C_{2^{m-5}}&\mbox{if $p=2$}\\\ \left((C_{p^{3}})\rtimes C_{p^{2}}\right)\times
C_{p^{m-5}}&\mbox{if $p\neq 2$}\\\ \end{cases}$

Then $G$ has no abelian subgroup of order $p^{m-1}.$

Suppose $n=p_{1}^{a_{1}}p_{2}^{a_{2}}\dots p_{k}^{a_{k}},$ where $k\geq 2$ and
$a_{j}\geq 3$ for some $j.$ Then if we consider the group $G=H\times
C_{\frac{n}{p_{j}^{a_{j}}}},$ where $H$ is a non-abelian group of order
$p_{j}^{a_{j}},$ then $G$ does not contain an abelian subgroup of order
$p_{j}^{a_{j}}.$ Now onwards we suppose that
$n=p_{1}^{a_{1}}p_{2}^{a_{2}}\dots p_{k}^{a_{k}},$ where $k\geq 2$ and
$a_{j}\leq 2.$ Let us assume that $k\geq 3.$ Since $n$ is not an abelian
number, there exists $p_{i}$ and $p_{j}$ such that $p_{i}|p_{j}^{a_{j}}-1$ and
hence there exists a non-abelian group $H$ of order $p_{i}p_{j}^{a_{j}}.$ If
we consider $G=H\times C_{\frac{n}{p_{i}p_{j}^{a_{j}}}}$ then $G$ does not
contain an abelian subgroup of order $p_{i}^{a_{i}}p_{j}^{a_{j}}.$ Finally the
problem boils down to the cases $n=p^{2}q$ or $n=p^{2}q^{2},$ it is known that
some of these numbers are non-CLT numbers (see [2, 17, 15, 16]). Hence we
confine to CLT numbers of this form.

Suppose $n=4q,$ where $q\in\\{4k+1|k\in{\mathbb{N}}\\}.$ Consider the group
$G=C_{q}\rtimes C_{4}$ of order $n$ which has no abelian subgroup of order
$2q.$ The groups $G=C_{p^{2}}\rtimes C_{q}$ or $G=(C_{p}\times C_{p})\rtimes
C_{q},$ where $p>2$ have no abelian subgroups of order $pq.$ If
$n=p^{2}q^{2},$ where $p|q-1,p^{2}\nmid q-1,p\nmid q+1,q\nmid p^{2}-1,$ then
it is easy to see that the group $G=(C_{q}\times C_{q})\rtimes C_{p^{2}}$ has
no abelian subgroup of order $p^{2}q.$ For all other CLT groups of order
$p^{2}q^{2}$ there exists a non-ACLT group $H$ of order $p^{2}q.$ Therefore in
this case the group $G=H\times C_{q}$ has no abelian subgroup of order
$pq^{2}.$ Hence the result. ∎

The following are few nontrivial examples of ACLT groups of order $n,$ where
$n$ need not be an ACLT number.

######  Example 14.

  1. 1.

A finite group $G$ is said to be minimal non-abelian if $G$ is non-abelian but
every proper subgroup of $G$ is abelian. These groups are completely
classified in [12] and [14]. Every minimal non-abelian group is an ACLT group.

  2. 2.

Dihedral group $D_{n}$ of order $2n$ is an ACLT group if and only if $n$ is
one of the following

    1. (a)

$n$ is a prime number.

    2. (b)

$n=2^{k},k\in\mathbb{N}.$

    3. (c)

$n=2p,$ where $p$ is a prime number.

###### Proof.

From Part 1 of Example 2 we already have the equivalence of $(a),(b)$ and CCLT
of $D_{n}.$ If $n=2p,$ then $D_{n}$ has a cyclic subgroup of order $2p$ hence
it is an ACLT group. Note that $D_{n}$ does not have a cyclic subgroup of
order $4$ hence it is not a CCLT group. Conversely suppose that $n$ has two
odd prime divisors $p$ and $q$ or $n=2^{k}p,$ where $k>1.$ Then any subgroup
of order $2^{m+1}p^{\delta}$ is not abelian, where $2^{m}|n,2^{m+1}\nmid n$
and $\delta\in\\{0,1\\}.$ ∎

  3. 3.

The group $G=C_{p}^{3}\rtimes C_{p}^{2}$ is an ACLT group.

  4. 4.

The semi-dihedral group $S_{2^{n}}$ and generalised quaternion group
$Q_{2^{n}}$ of order $2^{n}$ are ACLT groups.

A group whose commutator subgroup is an abelian group is called metabelian
group. It is known that a group $G$ is metabelian if and only if there is an
abelian normal subgroup $A$ of $G$ such that the quotient group $G/A$ is
abelian. With this result in hand we have a necessary condition for a group to
be an ACLT group. Let $G$ be an ACLT group of order $n.$ Suppose that $p$ is
the smallest prime dividing $n$ and $H$ is an abelian subgroup of $G$ of order
$\frac{n}{p}.$ Then it is clear that $H$ is normal in $G$ and $G/H$ is
abelian. Hence the following result.

######  Proposition 15.

Every ACLT group is a metabelian group.

We know that direct product of abelian groups is abelian. The group
$D_{p}\times C_{p}$ is suggesting that $H\times K$ can be an ACLT group even
if one of $H$ and $K$ is not an abelian group. For example, if $H$ is an ACLT,
non-abelian group of order $m$ and $K$ is an abelian group of order $n$ with
$\gcd(m,n)=1,$ then $H\times K$ does not contain an abelian subgroup of order
$m.$ The group $D_{p}\times C_{pq}$ is not ACLT as it does not contain abelian
subgroup of order $2p^{2}$ even though $D_{p}$ is ACLT, $C_{pq}$ is abelian
and $(2p,pq)>1.$ We summarize these observations and state in the following
result.

######  Theorem 16.

Let $H$ and $K$ be groups with $m$ and $n$ elements respectively. If $H\times
K$ is an ACLT group, then both $H$ and $K$ are ACLT and one of them is
abelian. Conversely suppose $H$ is an ACLT group and $K$ is an abelian group.
Then $H\times K$ is an ACLT group if one of the following conditions hold.

  1. 1.

If $H$ is an abelian group.

  2. 2.

If $H$ is non-abelian then if $p|n,$ where $p$ is a prime number, then $p|m.$

###### Proof.

If $H$ is non-ACLT group then there exists $d\in{\mathbb{N}}$ such that $d|m$
and $H$ has no abelian subgroup of order $d.$ Then we can show that $H\times
K$ has no abelian subgroup of order $d^{k+1},$ where $d^{k}|n$ but
$d^{k+1}\not|n.$ Let $H$ and $K$ be non-abelian groups. Then it is obvious
that $H\times K$ has no abelian subgoup of order $\frac{mn}{p},$ where $p$ is
the smallest prime dividing $m.$ Therefore $H\times K$ is ACLT then both $H$
and $K$ are ACLT and one of them is abelian. Conversly, if $H$ and $K$ both
are abelian then $H\times K$ is abelian hence ACLT. If $H$ is non-abelian and
$p|n$ but $p\not|m$ then there is no subgroup of order $\frac{mn}{p}.$ Hence
the result holds. ∎

Let $\pi(n)$ denotes the set of all primes dividing $n.$ Our objective is to
prove that if $G$ is a non-abelian ACLT group, then $|\pi(|G|)|\leq 2.$ By
using this result we can show that every ACLT group is supersolvable and every
subgroup of an ACLT group is CLT.

######  Theorem 17.

If $G$ is a non-abelian, ACLT group then order of $G$ has at most two prime
divisors.

Before proving the above theorem we see few more properties of ACLT groups.
Similar to Theorem 4 we have every Sylow $p$-subgroup of an ACLT group is
abelian. In the literature, groups of this nature are called $A$-groups. Thus
if we exclude $p$-groups, every ACLT group is an $A$-group. For examples of
various $A$-groups refer[10]. We need following result on $A$-groups which is
a collection of several results or observations of $A$-groups given by
D.R.Taunt [19].

######  Theorem 18 (D.R.Taunt [19]).

  1. 1.

Any subgroup or homomorphic image of an $A$-group is $A$-group.

  2. 2.

Every solvable $A$-group contains unique maximal abelian normal subgroup.

  3. 3.

Let $Z[G]$ and $G^{\prime}$ denote the center and commutator subgroups of $G$
respectively. Then $Z[G]\cap G^{\prime}=\\{e\\},$ whenever $G$ is an
$A$-group.

A subgroup $H$ of a group $G$ is Carter, if $H$ is both self-normalizing and
nilpotent. Carter [7] proved that every finite solvable group has a Carter
subgroup and all its Carter subgroups are conjugate.

######  Lemma 19.

Let $G$ be a finite, non-abelian, CLT and $A$-group of order $n$, where
$|\pi(n)|\geq 3.$ Then for some $s\in\pi(n)$ all the subgroups of index $s$ in
$G$ are non-abelian.

###### Proof.

Let $S$ be the set of all non-isomorphic prime index abelian subgroups of $G.$
Since every element in $S$ is a maximal subgroup of $G,$ hence they are either
normal or self-normalizing. Suppose the hypothesis is not true. Then for every
$s\in\pi(n)$ there exists an abelian subgroup of index $s$ in G. Therefore
$|S|\geq 3.$ Further, from Part 2 of Theorem 18, $S$ contains exactly one
normal subgroup. Also if any two elements of $S$ are self-normalizing, then
they are Carter subgroups, thus they are conjugate to each other. Which shows
that $|S|\leq 2.$ Hence we got a contradiction. ∎

Now the proof of Theorem 17 is immediate. Our next goal is to show that every
ACLT group is supersolvable. We prove the following lemma to prove the same.

######  Lemma 20.

Let $G$ be an ACLT group and $q$ be the largest prime divisor of $o(G),$ then

  1. 1.

Sylow $q$-subgroup is normal in $G.$

  2. 2.

If $K$ is the Sylow $q$-subgroup of $G,$ then $G^{\prime}\leq K.$

  3. 3.

Further, if $G$ is a non-abelian group, then $C_{G}(K),$ the centralizer of
$K$ in $G$, is the index $p$ abelian subgroup of $G.$

###### Proof.

Proof of Part 1. If $G$ is abelian or $p=q$, then we are done. Let $G$ be a
non-abelian group and $o(G)=p^{a}q^{b},$ where $p<q.$ Since $G$ is an ACLT
group, it has an abelian subgroup $H$ of index $p.$ If $K$ is a subgroup of
$H$ of order $q^{b},$ then $K$ is also a Sylow $q$-subgroup of $G.$ Further,
$H\leq N_{G}(K)$ follows from the fact that $H$ is abelian. Which shows that
either $N_{G}(K)=H$ or $N_{G}(K)=G.$ If $N_{G}(K)=H$, then number of Sylow-$q$
subgroups of $G$ is $p$ which is not possible from third Sylow theorem. Hence
$K$ is normal in $G.$
Proof of Part 2. We observe from Part 1 that $G$ is isomorphic to $K\rtimes
L,$ where $L$ is a Sylow $p$-subgroup of $G.$ We can check from the definition
of commutator subgroup that $G^{\prime}\leq K.$ Hence the result follows.
Proof of Part 3. Since $H$ is abelian so we can see that $H\leq C_{G}(K).$
Consequently, either $C_{G}(K)=H$ or $C_{G}(K)=G.$ If $C_{G}(K)=G$, then
$K\leq Z(G).$ But from Part 2 we have $G^{\prime}\leq K.$ Hence we got
contradiction from part 3 of Theorem 18 and the fact that $G$ is non-abelian.
∎

######  Theorem 21.

Every ACLT group is supersolvable.

###### Proof.

If $G$ is abelian, then we are done. Let $G$ be a non-abelian ACLT group of
order $p^{a}q^{b},$ where $p$ and $q$ are prime numbers. If $p=q,$ then $G$ is
supersolvable. If $p<q$ then by Part 1 of Lemma 20, $G$ has a normal Sylow
$q$-subgroup say $K$. If $L$ is a Sylow $p$-subgroup of $G,$ then $KL=G$.
Which shows that $K$ and $KL$ are normal in $G.$ Therefore $G$ has a Sylow-
tower $\\{e\\}\leq K\leq KL.$ Finally from Part 3 of Lemma 20 we have
$N_{G}(K)/C_{G}(K)$ is a cyclic group of order $p,$ thus $N_{G}(K)/C_{G}(K)$
is strictly $p$-closed. Also if $L$ be any Sylow $p$-subgroup of $G$ and $N$
be an abelian subgroup of index $q$ in G. Then we can show that
$N_{G}(L)=C_{G}(L)=N.$ Therefore $N_{G}(L)/C_{G}(L)$ is also strictly
$p$-closed. Hence the result follows from Theorem $1.12$ of [21]. ∎

######  Theorem 22.

Every subgroup of an ACLT group is ACLT.

We need few lemmas before proving Theorem 22.

######  Lemma 23.

Every subgroup of an ACLT group of order $p^{n},$ where $p$ is a prime number
and $n\in{\mathbb{N}}$ is ACLT.

###### Proof.

The proof of this is same as the proof of Lemma 7. ∎

######  Lemma 24.

Let $G$ be a finite group of order $n=p^{a}q^{b},$ where $p$ and $q$ are prime
numbers. If $G$ has abelian subgroups of order $n_{1}=p^{a-1}q^{b}$ and
$n_{2}=p^{a}q^{b-1}$ then $G$ is ACLT.

###### Proof.

Let $H$ and $K$ be abelian subgroups of $G$ order $n_{1}$ and $n_{2}$
respectively. We can easily show that if $d|n$ and $d<n$ then either $d|n_{1}$
or $d|n_{2}.$ Thus $G$ has an abelian subgroup of order $d.$ Hence $G$ is
ACLT. ∎

######  Lemma 25.

Let $G$ be an ACLT group of order $p^{a}q^{b}$, where $p$ and $q$ are prime
numbers and $p<q.$ Then all maximal subgroups of $G$ are ACLT.

###### Proof.

From Theorem 21 $G$ is supersolvable, hence it follows from Theorem 1.7 of
[21] that all maximal subgroups of $G$ are of prime index. Thus order of every
maximal subgroup of $G$ is either $p^{a-1}q^{b}$ or $p^{a}q^{b-1}.$ Let $M$
and $N$ be any two non-abelian subgroups of $G$ of order $p^{a-1}q^{b}$ and
$p^{a}q^{b-1}.$ Since $G$ is ACLT, it has abelian subgroups of order
$p^{a-1}q^{b}$ and $p^{a}q^{b-1}$ say $H$ and $K$ respectively. Also
$HM\subseteq G$ and $KM\subseteq G$ and

$|HM|=\frac{o(H)\times o(M)}{o(H\cap M)}=\frac{p^{2a-2}q^{2b}}{o(H\cap
M)}\leq{p^{a}q^{b}}.$

Consequently, $o(H\cap M)=p^{a-2}q^{b},$ similarly one can see that $o(K\cap
M)=p^{a-1}q^{b-1}.$ With the similar approach one can show that $o(H\cap
N)=p^{a-1}q^{b-1}$ and $o(K\cap N)=p^{a}q^{b-2}.$ Therefore by Lemma 24, $M$
and $N$ are ACLT. ∎

Now we will prove Theorem 22 by induction on the row index of the hasse
diagram of $G,$ starting from the top most row. The result is trivially true
for the first row as $G$ is ACLT. Let us assume that the result is true for
the $k^{th}$ row that is all the subgroups of $G$ in the $k^{th}$ row of the
hasse diagram are ACLT. Let $H$ be a subgroup of $G$ in $(k+1)^{th}$ row. Then
$H$ is a maximal subgroup of some subgroup of $G$ in the $k^{th}$ row say $N.$
By induction hypothesis $N$ is ACLT. Then by Lemma 25 $H$ is ACLT. Hence all
subgroups of $G$ are ACLT.

##  4 Conclusion

This section is devoted to summarize the relationship among cyclic, abelian
CLT, ACLT and CCLT numbers using diagrammatic visualization.

  1. 1.

Figure $1$ summarize, how ACLT and CCLT numbers interact with other known
numbers.

CLT numbersACLT numberscyclic numbersabeliannumbersCCLTnumbers Figure 1:

  2. 2.

Given a natural number $n$ we define a graph $X_{n}=(V_{n},E_{n}),$ where the
vertex set $V_{n}={\mathcal{G}}_{n},$ the set of all non-isomorphic groups of
order $n$ and there is an edge between $G_{i},G_{j}\in{\mathcal{G}}_{n}$ if
$G_{i}\times G_{j}$ is an ACLT group. It is easy to check that $X_{n}$ is a
complete graph if and only if $n$ is an abelian number or a CCLT number. For
example,

${\mathcal{G}}_{28}=\\{C_{28},C_{2}\times C_{14},D_{14},Dic_{7}\\}.$

$C_{28}$$C_{2}\times C_{14}$$D_{14}$$Dic_{7}$ Figure 2: Corresponding Graph of
${\mathcal{G}}_{28}.$

Note $n=28$ is an ACLT number, but one can see that corresponding graph is not
a complete graph. It is also easy to see that the graph $X_{n}$ is a connected
graph if and only if $n$ is an ACLT number. It is interesting to find various
properties of these graphs.

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# UIEC^2-Net: CNN-based Underwater Image Enhancement Using Two Color Space

Yudong Wang  School of Electrical and Information Engineering, Tianjin
University, Tianjin, 300072, China  Jichang Guo The work described in this
paper was fully supported by a grant from the National Natural Science
Foundation of China (No.61771334) (_Corresponding author: Jichang Guo_) School
of Electrical and Information Engineering, Tianjin University, Tianjin,
300072, China  Huan Gao  School of Electrical and Information Engineering,
Tianjin University, Tianjin, 300072, China  Huihui Yue  School of Electrical
and Information Engineering, Tianjin University, Tianjin, 300072, China

###### Abstract

Underwater image enhancement has attracted much attention due to the rise of
marine resource development in recent years. Benefit from the powerful
representation capabilities of Convolution Neural Networks(CNNs), multiple
underwater image enhancement algorithms based on CNNs have been proposed in
the last few years. However, almost all of these algorithms employ RGB color
space setting, which is insensitive to image properties such as luminance and
saturation. To address this problem, we proposed Underwater Image Enhancement
Convolution Neural Network using 2 Color Space (UICE^2-Net) that efficiently
and effectively integrate both RGB Color Space and HSV Color Space in one
single CNN. To our best knowledge, this method is the first to use HSV color
space for underwater image enhancement based on deep learning. UIEC^2-Net is
an end-to-end trainable network, consisting of three blocks as follow: a RGB
pixel-level block implements fundamental operations such as denoising and
removing color cast, a HSV global-adjust block for globally adjusting
underwater image luminance, color and saturation by adopting a novel neural
curve layer, and an attention map block for combining the advantages of RGB
and HSV block output images by distributing weight to each pixel. Experimental
results on synthetic and real-world underwater images show the good
performance of our proposed method in both subjective comparisons and
objective metrics. The code are available at
https://github.com/BIGWangYuDong/UWEnhancement

###### Index Terms:

underwater image enhancement, HSV color space, deep learning.

##  I Introduction

Nowadays, exploration and utilization of marine resources has become the
strategy center in international community. It is very important to make use
of these images because underwater images carry plenty of information.
However, due to the impact of backscatter in far distances, light selective
absorption and scattering in water, the raw underwater images usually suffer
from low-quality, including low contrast and brightness, color deviations,
blurry details and uneven bright speck. The purpose of underwater image
enhancement is to obtain higher quality and clearer underwater images, so as
to make more effective use of image information. It can make full use of
images information and has been widely used to promote numerous engineering
and high-level research tasks such as underwater fish detection, shipwreck
detection, underwater archaeology, etc.

Figure 1: Samples of real-world underwater image enhancement by UIEC^2-Net.
Top row: raw underwater images from DUIE datasets [1]; Bottom row: the
corresponding results of our model.

Early underwater image enhancement methods often directly use traditional
techniques(e.g. histogram equalization based methods,and physical-based
methods[2, 3]). However, these algorithms have certain limitations: the
classic in-air image enhancement methods[4, 5] are often not outstanding
enough in processing underwater image. Most physical model-based methods[6, 7]
are based on a simplified underwater image formation model[8], which is
inaccurate because it assume many parameters. This leads to get not good
enough results for some types of underwater images and is only suitable for
underwater scenes under certain circumstances. In the past few decades, deep
learning technology has developed rapidly, which has brought a series of
breakthroughs to various of computer vision and image processing tasks[9, 10,
11, 12, 13, 14, 15, 16]. Affected by deep learning, many researchers began to
use CNN-based and GAN-based algorithms to enhance underwater image. The goal
of CNN-based algorithms is to be faithful to the original underwater image,
while the core of GAN-based algorithms is to improve the perceptual quality of
the entire underwater image. However, the current underwater image enhancement
still fails to observe attractive performance. This might be attributable to
the fact that most deep learning-based underwater image enhancement algorithms
almost all used RGB Color Space. The RGB color space can deal with the
scattering problem and improve the underwater image issues such as color
deviations, some issues of quality degradation such as low contrast,
saturation and brightness cannot get further improvement due to the RGB color
space cannot directly reflect some important parameters of the image.

6.8in

Figure 2: An overview of the proposed UIEC^2-Net architecture. UIEC^2-Net
consists of three blocks trained end-to-end: a RGB pixel-level block for
simple and basic processing, a HSV global-adjust block that leverages neural
curve layers for globally refine image property(saturation and luminance), and
a attention map block for getting better underwater enhanced image through
attention mechanism.

In this paper, to make up for the above shortcomings of RGB color space based
enhancement methods, we propose UIEC^2-Net, a novel CNN-based underwater image
enhancement methods, using both RGB Color Space and HSV Color Space.
UIEC^2-Net contains three blocks as follow: a RGB pixel-level block for simple
and basic operations such as denoising and removing color cast, a HSV global-
adjust block for globally adjust and fine-tune underwater image such as
saturation and brightness, and an attention map block for combining the
advantages of RGB pixel-level block and HSV global-adjust block output images.
Similar to [17], UIEC^2-Net includes global image transformations to adjust
colors, brightness, saturation, etc. Same as [1], the RGB pixel-level block is
a plain fully CNN, which we do not use skip connection or downpooling.
Although most of the processing results of the RGB block is already doing very
well, there are still exists some problem such as insufficient texture details
and the background color cast cannot restored well. Inspired by [18], we
applied a HSV global-adjust block to enhance the properties of underwater
images through the learned global adjustment curves in HSV color space, which
more expressively adjust underwater image properties and make the underwater
image enhancement effect better, no matter on visual effect or scores of image
quality evaluation standard. We simplified the model of [18], but it still
shows good performance in underwater image enhancement. However, because the
H-channel of HSV color space is very sensitive, when HSV color space is used
to process some underwater images, this may cause color distortion in some
areas of the image. To solve the problem and further improve the quality of
enhanced image, attention map block is used to combine the advantages of the
output results of the above two blocks and avoid the related disadvantages at
the same time. Therefore, attention map block has been proposed to learn the
weight of each pixel for the output of each block. This is similar to [19],
but we are pixel-level instead of channel-level for better results. Our
contributions in this paper are three-fold:

  * •

An end-to-end convolution neural network for underwater image enhancement: The
convolution neural network architecture incorporating a RGB pixel-level block
for simple and basic operations such as denoising and removing color cast, a
HSV global-adjust block for adjusting global color cast and fine-tuning
underwater image such as saturation and brightness, and a attention map block
for getting better underwater enhanced image. UIEC^2-Net parameters are much
less than deep CNN models, meanwhile, our model has better generalization
ability and gets better results on real-world underwater image datasets.

  * •

HSV global-adjust block: We introduce a simple neural network curve layer,
that learns a piece-wise linear scaling curve to adjust image properties in
HSV color space, especially the saturation and brightness of the underwater
image.

  * •

Two color space: We apply a differentiable transformations of the image in HSV
color spaces. To our best knowledge, this method is the first to use HSV color
space for underwater image enhancement based on deep learning.

##  II Relate Work

Underwater image enhancement based on deep learning: Exploring the underwater
world has become a popular direction in recent years [20, 21]. Underwater
image enhancement as an important pretreatment process aim to improve the
visual quality of low-quality images. A variety of methods based on deep
learning have been proposed and can be organized into two main categories,
which are CNN-based algorithms and GAN-based algorithms. Moreover, the deep
learning-based underwater image enhancement can be divided into 5 methods[22]
based on their architectural difference: encoder-decoder models [23, 24, 25],
which is a famous architecture in many low-level tasks to advance image
quality; designed a modular or block has the same structure and are repeatedly
applied in the network to improve the feature extraction capability of the
network[26, 27]; designed multiple branch aims to learn different features
through separate branches[28, 1, 29]; improving the performance of enhancement
and restoration by predicted the depth map or transmission map of the
underwater image[30, 31, 32]; employed multiple generators to inference the
improved image[33, 34, 35].

Underwater image dataset: Deep learning-based methods always need to be driven
by large datasets. Unlike other low-level vision tasks such as image super-
resolution[36], image denoising[37], image deblurring[38] and image
dehazing[39], which can use plenty of synthetic degraded images and high-
quality counterparts for training, it is difficult to synthesize realistic
underwater images for deep learning training, due to the complex underwater
image formation models are affected by many factors(e.g. turbidity and
lighting conditions). Therefore, it is necessary to synthesize underwater
images, using GAN to synthesize low-quality underwater images is a common way
to obtain paired image, Li et al.[32] proposed a GAN-based method to
synthesize underwater images from RGB-D in-air images and Fabbri et al. [25]
used Cyclegan[40, 27] to degrade the underwater image quality. Meanwhile,
using underwater image formation model has gradually attracted attention
recently. Blasinski [41] provided a three-parameter underwater image formation
model [42] for underwater image simulation. Anwar et al.[26] incorporate a new
underwater image synthesis method that simulates 10 different categories
underwater images using NYU-v2 indoor dataset[43], which is used in related
research[24]. Although the synthetic image is similar to the real-world
underwater image, there still exists a gap between them. Other methods include
Duarte et al.[44] simulated underwater image degradation using milk,
chlorophyll, or green tea in a tank to get paired-dataset. More recently, Li
et al. constructed a real-world underwater image enhancement dataset,
including 950 underwater images, 890 of which have the correspond reference
images. These potential reference images are produced by 12 enhancement
methods, and voted by 50 volunteers to select the final references.

Underwater image enhancement using multiple color spaces: Using multiple color
spaces for image enhancement is a popular research direction in recent years.
However, underwater image enhancement methods using multiple color spaces are
all focus on conventional algorithms. Iqbal et al.[45, 46] used histogram
stretching process on two different color spaces. They first use RGB color
space to correct underwater image contrast and then use HSI color space to
further improve the image quality. Hitam et al.[4] used CLAHE on RGB and HSV
color spaces and then combined Euclidean Norm to obtain enhanced image, the
results shows that the image quality can be improved by enhancing the
contrast. Ghani et al.[47] combined global and local contrast correction by
using RGB and HSV two color spaces to enhance underwater image quality. In
2015, Ghani et al.[48] used RGB and HSV color spaces during histogram
stretching process, and in 2017, Ghani et al.[49] proposed a method called
Recursive Adaptive Histogram Modification (RAHM) using both RGB and HSV color
space to improve the visual quality of the underwater images. In view of above
introduction, using multiple color space to enhance underwater image shows
better performance. Meanwhile, CNN has better enhancement effect than
traditional methods. Therefore, in this paper, we are the first using two
color spaces(RGB and HSV color space) in deep learning-based underwater image
enhancement to get higher quality images. As far as we know, using multiple
color spaces is not currently applied to deep learning-based underwater image
enhancement methods.

##  III Proposed Model

In this section, we first discuss the details of the proposed CNN-based
underwater image enhancement using 2 color spaces(RGB and HSV color space).
Then we introduce the loss function used in UIEC^2-Net. Finally, we present
the RGB color space and the HSV color space conversions to permit end-to-end
learning via stochastic gradient descent(SGD) and backpropagation.

###  III-A Network Architecture

UIEC^2-Net is an end-to-end trainable neural network that consists of three
blocks as shown in Figure 2. The RGB pixel-level block is a CNN-based network
for simple and basic processing such as denoising and removing color cast, the
HSV global-adjust block employing a novel neural curve layer that globally
refine image properties, especially the saturation and luminance, and the
attention map block distribute weight to result of RGB and HSV block at pixel-
level through the attention mechanism to obtain better enhanced underwater
images.

RGB pixel-level block: The architecture of the RGB pixel-level block and
parameter settings are shown at top of Figure 2. Because, downsampling may
cause some problems such as missing image information, especially, for pixel-
level computer vision tasks, it needs to be used in conjunction with
upsampling, and needs various of convolution to offset the impact of
downsampling and upsampling. We designed the RGB pixel-level block as a plain
fully CNN without downsampling. It consists with eight $3\times 3$
convolutional layers with stride 1, each of them using batch normalization and
the first 7 layers followed by a Leaky ReLU activation, the last layer using
sigmoid as the activation function, its purpose is to make the output in
[0,1]. RGB pixel-level block produces a result with output shape $H\times
W\times 3$, where $H,W$ is the height and width of feature maps, the number of
channels is 3, and the output is passed over to the HSV global-adjust block
for further processing. The performance of the RGB pixel-level block
architecture we proposed in this paper has reached a high level, however, we
believe that the widely used backbones for pixel-level tasks such as the U-Net
architecture [50] and the residual network architecture [51] can further
improve the performance, we will try in the future work.

HSV global-adjust block: The output of RGB pixel-level block is transformed to
HSV color space through differentiable RGB$\rightarrow$HSV and is used as
input of HSV global-adjust block. Our proposed HSV global-adjust block and
parameter settings are shown at the bottom of Figure 2, which consists of five
$3\times 3$ convolutional layers with stride 1, each of them followed by a
Leaky ReLU activation. There is a $2\times 2$ maxpooling after the first four
layers and the remaining one is not. Then we place a global average pooling
layer processes the shape of the feature maps to $1\times 1\times C$ and
followed by a fully connected layer that regress the knot points of a piece-
wise linear curve. The curve adjusts the predicted
image(${{\hat{I}}_{i}}\in[0,1]$) by scaling pixels with the formula presented
in Equation 1:

$S(\hat{I}_{i}^{jl})={k_{0}}+\sum\limits_{m=0}^{M-1}{({k_{m+1}}-{k_{m}})\delta(M\hat{I}_{i}^{jl}-m)}$
(1)

where

$\delta\left(x\right)=\left\\{\begin{array}[]{l}0{,\qquad}x<0\\\
x{,\qquad}0\leq x\leq 1\\\ 1{,\qquad}x>1\end{array}\right.$

where $M$ is the number of predicted knot points, $\hat{I}_{i}^{jl}$ is the
$j$-th pixel value in the $l$-th color channel of the $i$-th image, $k_{m}$ is
the value of the knot point $m$. The HSV global-adjust block can be seen as a
simple matter of multiplication of a pixel’s value: the pixel value is
multiplied by the value corresponding to the piece-wise linear curve to get
the globally refined images. Examples of the piece-wise linear curve that
learnt by proposed model are shown in Figure 3.

Figure 3: Examples of learnt neural global adjustment curves in HSV color
space.

HSV is composed of three channels(Hue, Saturation, Value), we using four
curves to adjust and refine image properties. We adjust luminance by using
value scaled based on value, adjust saturation by using saturation based on
saturation and saturation based on hue, and refine color by using hue based on
hue. The input of HSV global-adjust block is the HSV image converted from the
RGB pixel-level block’s output. The HSV global-adjust block produces a HSV
result with output shape $H\times W\times 3$ and the HSV image is converted
back to RGB space via a differentiable HSV to RGB conversion.

Attention map block: The raw input image is concatenated with the output of
RGB pixel-level block and HSV global-adjust block as the input of this block,
which is shown at the middle of Figure 2. The architecture of attention-map
block is similar to RGB pixel-level block, which consists with eight $3\times
3$ convolutional layers, each of them using batch normalization and the first
7 layers followed by a Leaky ReLU activation, the last layer uses Sigmoid as
activation function, which can set the output between 0 and 1. The output
shape of attention map block is $H\times W\times 6$, where the first three
channels are weights of RGB block output and the remaining three channels
belong to HSV block output. Finally, multiply the attention map with the
output of RGB image converted form HSV image and output of RGB block
respectively, and add the two results above to get high quality enhanced
underwater image, which is the UIEC^2-Net final output.

###  III-B Loss Function

The end-to-end training of UIEC^2-Net is supervised by four loss components,
which consists of two color space-specific terms. We defined ${{I_{i}}}$ is
the groundtruth image and ${{{\hat{I}}_{i}}}$ is the predicted image. Our
total loss function is presented in Equation 2:

$\begin{split}{\cal L}=&({\lambda_{pixel}}+{\lambda_{whole}})({\omega_{\ell
1}}{{\cal L}_{\ell 1}}+{\omega_{SSIM}}{{\cal L}_{SSIM}})+\\\
&{\omega_{hsv}}{{\cal L}_{hsv}}+{\omega_{perc}}{\cal L}_{j}^{\phi}\end{split}$
(2)

where ${{\cal L}_{hsv}}$ is the HSV color space loss function, and ${{\cal
L}_{\ell 1}}$, ${{\cal L}_{SSIM}}$, ${\cal L}_{j}^{\phi}$ are RGB color space
loss functions. ${\lambda_{pixel}}$ represent the weight applied in RGB pixel-
level block and ${\lambda_{whole}}$ is the weight applied in whole newtwork.
These functions and terms are defined in more detail below.

HSV loss(${{\cal L}_{hsv}}$): HSV image ${{I_{i}}}$ is divided into three
channels: hue ${H_{i}}\in[0,2\pi)$, saturation ${S_{i}}\in[0,1]$, value
${V_{i}}\in[0,1]$.Inspired by [18], we compute ${{\cal L}_{hsv}}$ in the
conical HSV color space:

${{\cal
L}_{hsv}}={\left\|{{{\hat{S}}_{i}}{{\hat{V}}_{i}}\cos({{{\hat{H}}_{i}}})-{S_{i}}{V_{i}}\cos\left({{H_{i}}}\right)}\right\|_{1}}$
(3)

Compared to RGB, HSV has more advantages because it separates color into
useful components(hue, saturation, value). It can globally adjust luminance
through value-channel, improve saturation through saturation-channel and
refine color through hue-channel.

L1 loss(${{\cal L}_{\ell 1}}$): We use $L_{1}$ loss on RGB pixels between the
predicted and groundtruth images. This loss function is first applied in RGB
pixel-level block, which computed $L_{1}$ distance between the predicted RGB
image and groundtruth image. Then it applied to compute the distance between
the UIEC^2-Net output and groundtruth image. The L1 losss presented in
Equation 4:

${{\cal L}_{\ell 1}}={\left\|{{{\hat{I}}_{i}}-{I_{i}}}\right\|_{1}}$ (4)

SSIM loss(${{\cal L}_{SSIM}}$): We use the SSIM loss [52] in our objective
function to impose the structure and texture similarity on the predicted
image. In this paper, we use gray images, which convert from RGB images, to
compute SSIM score, and for each pixel $x$, the SSIM value is computed within
a $11\times 11$ image patch around the pixel. The formula is as follows:

$\small
SSIM(x)=\frac{{2{\mu_{I}}(x){\mu_{\hat{I}}}(x)+{c_{1}}}}{{\mu_{I}^{2}(x)+\mu_{{}_{\hat{I}}}^{2}(x)+{c_{1}}}}\cdot\frac{{2{\sigma_{I\hat{I}}}(x)+{c_{2}}}}{{\sigma_{I}^{2}(x)+\sigma_{{}_{\hat{I}}}^{2}(x)+{c_{2}}}}$
(5)

where ${{\mu_{\hat{I}}}(x)}$ and ${\sigma_{{}_{\hat{I}}}(x)}$ correspond to
the mean and standard deviation of the RGB patch from the predicted image,
similarly, ${{\mu_{I}}(x)}$ and ${\sigma_{I}(x)}$ are the same corresponds to
groundtruth image. And ${{\sigma_{I\hat{I}}}(x)}$ is cross-covariance. We set
$c_{1}=0.02$ and $c_{2}=0.03$. The SSIM loss is expressed as:

${{\cal
L}_{SSIM}}=1-\frac{1}{N}\sum\limits_{i=1}^{N}{SSIM\left({{x_{i}}}\right)}$ (6)

The SSIM loss is similar to L1 loss, we applied it twice during training.

Perceptual loss(${\cal L}_{j}^{\phi}$): Inspired by [53], we defined the
perceptual loss based on VGG network [54], which pre-trained on the ImageNet
dataset[55].Due to the deep layer has a better understanding of semantic
information and can fully preserve the image content and overall spatial
structure, and the shallow layer is more sensitive to color and texture, we
select layer 4$\\_$3 from VGG-19 to make it sensitive to both color and
semantics. The perceptual loss expressed as the distance between the feature
representations of predicted RGB image and groundtruth image:

${\cal
L}_{j}^{\phi}=\frac{1}{{{C_{j}}{H_{j}}{W_{j}}}}\sum\limits_{i=1}^{N}{\left\|{{\phi_{j}}({{\hat{I}}_{i}})-{\phi_{j}}({I_{i}})}\right\|}$
(7)

where ${{\phi_{j}}}$ is the $j$-th convolutional layer of pre-trained VGG-19;
$N$ is the number of each batch in the training procedure;
${{C_{j}}{H_{j}}{W_{j}}}$ represents the $j$-th layer dimension of the feature
maps of the VGG-19 network, which ${{C_{j}},{H_{j}},{W_{j}}}$ represent
number, height and width of the feature map, respectively. The perceptual loss
computed the distance between the UIEC^2-Net output RGB image and groundtruth
image.

Loss term weights: Each loss term has a weight hyperparameter: ${\omega_{\ell
1}}$,${\omega_{SSIM}}$,${\omega_{hsv}}$,${\omega_{perc}}$, meanwhile, the L1
and SSIM loss has two hyperparameters:
${\lambda_{pixel}}$,${\lambda_{whole}}$. We set ${\omega_{\ell
1}=1}$,${\omega_{SSIM}=1}$,${\omega_{hsv}}=1$,${\omega_{perc}}=0.5$, and at
the first 20 epoches we set
${\lambda_{pixel}=0.5}$,${\lambda_{whole}=0.5}$,and set
${\lambda_{pixel}=0.1}$,${\lambda_{whole}=0.9}$ for the last epoches.

###  III-C Color Space Transformations

UIEC^2-Net relies on differentiable RGB$\rightarrow$HSV and
HSV$\rightarrow$RGB color space conversions to permit the end-to-end learning
via backpropagation. The related color space conversion methods can be found
on the OpenCV website
111https://docs.opencv.org/3.3.0/de/d25/imgproc_color_conversions.html.

RGB$\rightarrow$HSV: The color space transformation contains minimum and
maximum operations, each of them can differentiable. The RGB$\rightarrow$HSV
conversion functions shown as Equations 8-10:

$V=\max(R,G,B)$ (8)

$S=\left\\{\begin{array}[]{l}\frac{{V-\min({R,G,B})}}{V},\qquad if\ V\neq 0\\\
0,\qquad\qquad\qquad\ \ \ otherwise\end{array}\right.$ (9)

$H=\left\\{\begin{array}[]{l}\frac{{60({G-B})}}{{({V-\min({R,G,B})})}},\qquad
if\ V=R\\\ \frac{{120+60({B-R})}}{{({V-\min({R,G,B})})}},\qquad if\ V=G\\\
\frac{{240+60({R-G})}}{{({V-\min({R,G,B})})}},\qquad if\
V=B\end{array}\right.$ (10)

if $H<0$ then $H=H+360$, $0\leq V\leq 1,0\leq S\leq 1,0\leq H\leq 360$. The
operation involves conditional statements, which conditioned on the values of
R, G and B. However, these can be processed under the
pytorch222https://pytorch.org/ framework, and the operations such as
backpropagation can be performed.

HSV$\rightarrow$RGB: The conversion rarely applied in relevant deep learning-
based methods, due to the related formulas are hard to differentiable.
Inspired by [18], we replaced complex formulas with piecewise linear functions
based on Figure 4. The corresponding piecewise linear curves are defined by
linear segments and knot points, which makes them have a gradient between the
points. The formulas shown as:

$\displaystyle\small R({\hat{I}_{i}^{j}})=$
$\displaystyle{v_{j}}-({({{v_{s}}{v_{j}}})/60})\delta({360{h_{j}}-60})$ (11)
$\displaystyle+({({{v_{s}}{v_{j}}})/60})\delta({360{h_{j}}-240})$
$\displaystyle B({\hat{I}_{i}^{j}})=$
$\displaystyle{v_{j}}({1-{s_{j}}})+({({{v_{s}}{v_{j}}})/60})\delta({360{h_{j}}-120})$
(12) $\displaystyle-({({{v_{s}}{v_{j}}})/60})\delta({360{h_{j}}-300})$
$\displaystyle B({\hat{I}_{i}^{j}})=$
$\displaystyle{v_{j}}({1-{s_{j}}})+({({{v_{s}}{v_{j}}})/60})\delta({360{h_{j}}-120})$
(13) $\displaystyle-({({{v_{s}}{v_{j}}})/60})\delta({360{h_{j}}-300})$

where we define $\delta(x)$ as in Equation 14

$\delta(x)=\left\\{\begin{array}[]{l}0{,\qquad\ }x<0\\\ x{,\qquad\ }0\leq
x\leq 40\\\ 60{,\qquad}x>60\end{array}\right.$ (14)

where ${h_{j}},{s_{j}},{v_{j}}$ are the hue, saturation and value of pixel $j$
of image $I_{i}$ ,respectively. and both of them belongs $[0,1]$.

Figure 4: Piecewise linear functions used to convert HSV to RGB color space.
Figure source: https://commons.wikimedia.org/w/index.php?curid=1116423.

##  IV Experimental Results

To evaluate our method, we perform qualitative (subjective comparisons) and
quantitative (subjective comparisons) comparisons with traditional methods and
the recent state-of-the-art deep learning-based underwater image enhancement
methods on both synthetic and real-world underwater images. These methods
include Histogram Equalization(HE), White Balance(WB), UDCP[56], ULAP[57],
UWGAN[58], UGAN[25] ,UWCNN[26], DUIENet[1]. For deep learning-based methods,
we run the source codes with the pre-trained model parameters shared by
provided by corresponding authors to produce the best results for an objective
evaluation. In this section, we will first supplement the training details,
and then analyzed the experimental results of synthetic and real-world
underwater images.

###  IV-A Implementation Details

For training, the input of our network are both synthetic and real-world
underwater images. A random set of 800 pairs of real-world images extracted
from the UIEBD dataset [1] and 1200 pairs of synthetic images with ten types
are generated from the NYU-v2 RGB-D dataset based on [26], both of them (2000
images in total) are used to train our network. We resize the input images to
size $350\times 350$ and randomly crop them to size $320\times 320$. For
testing, 90 real-world and 900 synthetic images are treated as the testing
set. Compared to training, we do not resize or randomly crop the input image.
We trained our model using ADAM and set the learning rate to 0.0001. The batch
size is set to 8 and the epoch is set to 50. We use Pytorch as the deep
learning framework on an Inter(R) i7-8700k CPU, 32GB RAM, and a Nvidia GTX
1080Ti GPU.

###  IV-B Experiment on Synthetic Datasets

It is very common to train the network through synthetic underwater images
because there is no GT for underwater images, thus, We first evaluate the
capacity of our method on synthetic testing set. In Figure 5, we present the
results of underwater image enhancement on the synthetic underwater images
from our testing set.

6.5in

Figure 5: Subjective comparisons on synthetic underwater images. From left to
right are raw underwater images, the results of Histogram Equalization(HE),
White Balance(WB), UDCP[56], ULAP[57], UWGAN[58], UGAN[25] ,UWCNN[26],
DUIENet[1], the proposed UIEC^2-Net and reference images.

6.5in

Figure 6: Subjective comparisons on bluish underwater images. From left to
right are raw underwater images, the results of Histogram Equalization(HE),
White Balance(WB), UDCP[56], ULAP[57], UWGAN[58], UGAN[25] ,UWCNN[26],
DUIENet[1], the proposed UIEC^2-Net and reference images.

6.5in

Figure 7: Subjective comparisons on greenish underwater images. From left to
right are raw underwater images, the results of Histogram Equalization(HE),
White Balance(WB), UDCP[56], ULAP[57], UWGAN[58], UGAN[25] ,UWCNN[26],
DUIENet[1], the proposed UIEC^2-Net and reference images.

6.5in

Figure 8: Subjective comparisons on low-illuminated underwater images. From
left to right are raw underwater images, the results of Histogram
Equalization(HE), White Balance(WB), UDCP[56], ULAP[57], UWGAN[58], UGAN[25]
,UWCNN[26], DUIENet[1], the proposed UIEC^2-Net and reference images.

6.5in

Figure 9: Subjective comparisons on yellowish underwater images. From left to
right are raw underwater images, the results of Histogram Equalization(HE),
White Balance(WB), UDCP[56], ULAP[57], UWGAN[58], UGAN[25] ,UWCNN[26],
DUIENet[1], the proposed UIEC^2-Net and reference images.

6.5in

Figure 10: Subjective comparisons on shallow water images. From left to right
are raw underwater images, the results of Histogram Equalization(HE), White
Balance(WB), UDCP[56], ULAP[57], UWGAN[58], UGAN[25] ,UWCNN[26], DUIENet[1],
the proposed UIEC^2-Net and reference images. TABLE I:

Full-reference Image Quality Assessment in Terms of MSE,
PSNR and SSIM on Synthetic Images.

Method | MSE | PSNR(dB) | SSIM
---|---|---|---
raw | 4142.5386 | 14.4978 | 0.6948
HE | 3869.0180 | 14.2563 | 0.6779
WB | 4246.7734 | 14.5174 | 0.6862
UDCP[56] | 4809.4566 | 13.6778 | 0.6373
ULAP[57] | 3725.4596 | 14.3739 | 0.6989
UWGAN[58] | 4028.8004 | 13.3007 | 0.7083
UGAN[25] | 2266.4678 | 16.2636 | 0.6625
UWCNN[26] | 3770.7036 | 14.8638 | 0.7419
DUIENet[1] | 2323.3417 | 16.1073 | 0.7734
UIEC^2-Net | 1126.3743 | 20.5442 | 0.8486

As shown in Figure 5(a),we simply give examples of synthetic underwater
images, generated from RGB-D in-air images. Figure 5(b)-(k) shows the
reference image of Histogram Equalization(HE), White Balance(WB), UDCP[56],
ULAP[57], UWGAN[58], UGAN[25] ,UWCNN[26], DUIENet[1], the proposed UIEC^2-Net
and the corresponding reference image respectively. It is noted that UWCNN[26]
has 10 different pre-train models, which are trained separately by 10 types.
We only choose type-1 model as a comparative experiment, so when the
underwater images category is rich, its generalization performance is not
good. Similarly, for other methods, they have better processing results in one
or two categories. However, in most cases, they are not good at processing
color cast and image details. The methods we proposed has good generalization
performance and can restore color cast better, which Figure 5 can
demonstrating its effectiveness and robustness.

We choose Mean Square Error (MSE), Peak Signal to Noise Ratio (PSNR) and
Structural SIMilarity index (SSIM) as full-reference image quality assessment
to assess the processed synthetic underwater images for all of the compared
methods. For MSE and PSNR metrics, the lower MSE (higher PSNR) denotes the
result is more close to the GT in terms of image content. For SSIM, the higher
SSIM scores mean the result is more similar to the GT in terms of image
structure and texture. Table I reports the quantitative results of different
methods on the synthetic testing images. We highlight the top 1 performance in
red, whereas the second best is in blue. Regarding the SSIM response,
UIEC^2-Net is higher than the second-best performer. Similarly, our PSNR is
obviously higher (less erroneous as indicated by the MSE scores) than the
compared methods. This can demonstrate that our proposed method achieves the
best performance in terms of full-reference image quality assessment on
synthetic underwater images.

###  IV-C Experiment on Real Datasets

From test dataset, we extracted five categories underwater images (greenish
underwater images, bluish underwater images, yellowish underwater images,
shallow water images, and underwater images image with limited illumination)
to compare the generalization capability of underwater enhancement methods.
Note that our test dataset and corresponding reference images are provided by
UIEBD[1]. Due to the nature of light propagation, the red light first
disappears in water, followed by green light and then the blue light. Most
underwater images are bluish and greenish, such as the raw underwater images
in (a) of Figure 6 and 7. Underwater images also include low-brightness
situations(shown in Figure 8), yellowish styles (shown in Figure 9) and
shallow water images(shown in Figure 10). As shown in Figure 6-10, color
deviation seriously affects the visual quality of underwater images.
Traditional methods usually consider only one type of underwater images. HE
effectively improves the contrast but cannot remove the color cast well. WB
can improve the greenish underwater images by supplementing red light, but the
results are not good in other cases. ULAP[57] can better enhance the greenish
underwater images due to a good estimate of the underwater optical attenuation
but the bluish underwater image cannot be improved well. UDCP[56] aggravate
the effect of the color cast. Deep learning-based methods have poor
generalization and low sensitivity to brightness and saturation of underwater
images, and tend to introduce artifacts, over-enhancement and color casts. By
contrast, (k) of Figure 6-10 shows that our proposed UIEC^2-Net effectively
removes the haze and color casts (especially the background color cast) of the
underwater images, adjusts underwater image properties (e.g. brightness and
saturation) and has good generalization capability in dealing with underwater
images. In addition, our results even achieve better visual quality than the
corresponding reference images(e.g. less noise and better details).

TABLE II:

Full-reference Image Quality Assessment in Terms of MSE,
PSNR and SSIM on Real-World Images.

Method | MSE | PSNR(dB) | SSIM
---|---|---|---
raw | 1322.1355 | 18.2701 | 0.8151
HE | 1078.9476 | 19.5854 | 0.8509
WB | 1455.7350 | 17.9261 | 0.8041
UDCP[56] | 5829.6013 | 11.1646 | 0.5405
ULAP[57] | 1517.6039 | 18.6789 | 0.8194
UWGAN[58] | 1256.0906 | 18.6209 | 0.8454
UGAN[25] | 558.2965 | 21.3031 | 0.7691
UWCNN[26] | 1342.7639 | 18.2851 | 0.8150
DUIENet[1] | 2023.3417 | 16.2906 | 0.7884
UIEC^2-Net | 365.5963 | 24.5663 | 0.9346

TABLE III:

No-reference Image Quality Evaluation in Terms of UCIQE,
UICM, UISM, UIConM and UIQM on Real-World Images.

Method | UCIQE | UICM | UISM | UIConM | UIQM
---|---|---|---|---|---
raw | 0.5044 | 2.5656 | 2.7255 | 0.5492 | 2.8407
HE | 0.5828 | 4.3170 | 3.6235 | 0.7017 | 3.7006
WB | 0.5429 | 5.1675 | 2.6102 | 0.5472 | 2.8731
UDCP[56] | 0.5747 | 4.0233 | 2.7689 | 0.6169 | 3.1367
ULAP[57] | 0.6014 | 4.9015 | 2.8597 | 0.5145 | 2.8224
UWGAN[58] | 0.5352 | 3.0011 | 3.1388 | 0.6169 | 3.2174
UGAN[25] | 0.6162 | 4.2476 | 4.2527 | 0.7384 | 4.0157
UWCNN[26] | 0.5044 | 2.55656 | 2.7255 | 0.5492 | 2.8407
DUIENet[1] | 0.6051 | 4.0727 | 3.8559 | 0.6940 | 3.7988
UIEC^2-Net | 0.6193 | 4.9046 | 4.7558 | 0.7094 | 4.0790
Reference | 0.6451 | 5.2137 | 3.9190 | 0.7116 | 3.8483

Similarly to Section IV-B, we choose MSE, PSNR and SSIM to assess the
recovered results on real-world underwater images. We calculate the results of
each method and the corresponding reference image, the quantitative results of
different methods on real-world underwater images are shown in Table II. Our
method achieves the best performance in terms of full-reference image quality
assessment, which can prove our proposed method is good at handling details.

Meanwhile, we choose underwater color image quality evaluation (UCIQE)[59] and
underwater image quality measure (UIQM)[60] as No-reference Image Quality
Evaluation. UCIQE evaluate underwater image quality by color density,
saturation and contrast. UIQM is a comprehensive underwater image evaluation
index, which is the weighted sum of underwater image colorfulness
measure(UICM), underwater image sharpness measure(UISM) and underwater image
conreast measure(UIConM):$UIQM={c_{1}}\times UICM+{c_{2}}\times
UISM+{c_{3}}\times UIQM$. We set $c_{1}=0.0282$,$c_{2}=0.2953$,$c_{3}=3.5753$
according to [60]. As shown in Table III, compared with other methods, our
proposed method achieve the best score in UCIQE, UISM and UIQM, and even get
higher score than reference image in UIQM and UISM. This also demonstrates
that our algorithm has more advantages in processing details.

6.5in

Figure 11: The enhanced result of only using RGB pixel-level block,
without(w/o) using HSV global-adjust block and UIEC^2-Net. Top row: the result
of only use RGB pixel-level block; Middle row: the result of UIEC^2-Net not
using HSV global-adjust block; Bottom row: the result of UIEC^2-Net. TABLE IV:

Image Quality Assessment of with/without HSV Rtouching Block and Attention Map
Block.

Method | MSE | PSNR(dB) | SSIM | UCIQE | UIQM
---|---|---|---|---|---
only RGB pixel-level block | 397.3553 | 23.8228 | 0.92849 | 0.6351 | 4.0732
w/o attention map block | 393.9875 | 24.1794 | 0.92651 | 0.60183 | 4.0979
UIEC^2-Net | 365.5963 | 24.5663 | 0.9346 | 0.6193 | 4.0790

###  IV-D Ablation Study

To demonstrate the effect of HSV global-adjust block and attention map block
in our network, we compared the proposed UIEC^2-Net with only using RGB pixel-
level block and without(w/o) attention map block as an ablation study. As show
in Table IV, although the use of HSV block and attention map block decreased
performance of UCIQE and UIQM, such a sacrifice is necessary to improved whole
network preference. In addition, it can achieve good results in subjective
perception of underwater image.

Such an example is presented in Figure 11, (a),(b) demonstrate that after
adding HSV global-adjust block, the background color cast of underwater image
is processed better and the saturation has been improved to make the processed
images more realistic. Figure 11(c) shows that although adopting global-adjust
block can effectively avoid the luminance problem such as overexposure, HSV
block may also over-processing the image to reduce the contrast and saturation
of the entire image. Meanwhile, due to the H-channel of HSV color space is
very sensitive, HSV block may causes color distortion in some reddish areas
when processing the underwater images. These shows in the second row of Figure
11(d). Attention map block can solve the problem by extracting the feature
information of the Raw images, results of RGB and HSV, then distributing
weight of each pixel to combine the advantages of results from RGB and HSV
block. As shown in (c),(d), some problems with HSV blocks can be avoided when
UIEC^2-Net using attention map block can. In addition, attention map block can
also avoid the appearance of noise blocks, which shown in (e)

##  V Conclusion

In this paper, we proposed a novel CNN-based underwater image enhancement
method, using both RGB Color Space and HSV Color Space. It is the first
attempt that using HSV color spaces in deep learning-based underwater
enhancement research. We first use pixel-level block based on RGB color space
for simple and basic enhancement operations such as removing color cast and
denosing, then we use a global-adjust block based on HSV color space for
globally refining underwater image properties such as luminance and
saturation. Our method has a good effect on removing color cast, especially
for the restoration and enhancement of background color, and also an greatly
retain the detailed information of the underwater image. Furthermore, our
method can be use as a guide for subsequent research of underwater image color
correction. Experiments on synthetic and real-world underwater images
including qualitative and quantitative assessment demonstrated the
effectiveness of our method.

## References

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