# Fixed point results for contractions of polynomial type

Mohamed Jleli111Department of Mathematics, College of Science, King Saud
University, Riyadh 11451, Saudi Arabia; E-mail address<EMAIL_ADDRESS>Cristina Maria Păcurar222Faculty of Mathematics and Computer Science,
Transilvania University of Braşov, 50 Iuliu Maniu Blvd., Braşov, Romania;
E-mail address<EMAIL_ADDRESS>Bessem Samet333Department of
Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi
Arabia; E-mail address<EMAIL_ADDRESS>
###### Abstract

We introduce two new classes of single-valued contractions of polynomial type
defined on a metric space. For the first one, called the class of polynomial
contractions, we establish two fixed point theorems. Namely, we first consider
the case when the mapping is continuous. Next, we weaken the continuity
condition. In particular, we recover Banach’s fixed point theorem. The second
class, called the class of almost polynomial contractions, includes the class
of almost contractions introduced by Berinde [Nonlinear Analysis Forum. 9(1)
(2004) 43–53]. A fixed point theorem is established for almost polynomial
contractions. The obtained result generalizes that derived by Berinde in the
above reference. Several examples showing that our generalizations are
significant, are provided.

2020 Mathematics Subject Classification: 47H10, 54H25.
Key words and phrases: Polynomial contractions, almost polynomial
contractions, Picard-continuous mappings, weakly Picard operators, fixed
point.

##  1 Introduction

The most used techniques for studying the existence an uniqueness of solutions
to nonlinear problems (for instance, integral equations, differential
equations, fractional differential equations, evolution equations) are based
on the reduction of the problem to an equation of the form $Tu=u$, where $T$
is a self-mapping defined on a certain set $X$ (usually equipped with a
certain topology) and $u\in X$ is the unknown solution. Any solution $u$ to
the previous equation is called a fixed point of $T$. So, the study of fixed
point problems for different classes of mappings $T$ and different topological
structures on $X$ is of great importance.

One of the most celebrated results in fixed point theory is the Banach fixed
point theorem [2], which states that, if $(X,d)$ is a complete metric space
and $T:X\to X$ is a mapping satisfying

$d(Tw,Tz)\leq\lambda d(z,w)$ (1.1)

for all $w,z\in X$, where $\lambda\in(0,1)$ is a constant, then

  * ($B_{1}$)

$T$ possesses one and only one fixed point;

  * ($B_{2}$)

For all $z_{0}\in X$, the sequence $\\{z_{n}\\}$ defined by $z_{n+1}=Tz_{n}$,
converges to this fixed point.

Any mapping $T:X\to X$ satisfying (1.1) is called a contraction. The
literature includes several generalizations and extensions of Banach’s fixed
point theorem. Some of them were focused on weakening the right-side of
inequality (1.1), see e.g. [3, 8, 11, 16, 24, 25, 27, 30]. In other results,
the underlying space is equipped with a generalized distance, see e.g. [9, 12,
13, 14, 17]. We also cite the paper [18] of Nadler, who initiated the study of
fixed points for multi-valued mappings. More recent fixed point results
extending Banach’s fixed point theorem can be found in [1, 5, 7, 20, 21, 22,
23, 19].

On the other hand, despite the importance of Banach’s fixed point theorem,
this result is only concerned with continuous mappings. Namely, any mapping
$T:X\to X$ satisfying (1.1) is continuous on $(X,d)$. So, it is natural to ask
whether it is possible to extend Banach’s fixed point theorem to mappings that
are not necessarily continuous. The first work in this direction is due to
Kannan [15], where he introduced the class of mappings $T:X\to X$ satisfying
the condition

$d(Tx,Ty)\leq\lambda\left[d(x,Tx)+d(y,Ty)\right]$

for all $x,y\in X$, where $\lambda\in\left(0,\frac{1}{2}\right)$ is a
constant. Namely, it was shown that ($B_{1}$) and ($B_{2}$) hold also for the
above class of mappings (when $(X,d)$ is a complete metric space). Following
Kannan’s result, several fixed point theorems have been obtained without the
requirement of the continuity of the mapping, see e.g. [3, 4, 6, 10, 11, 25].
In particular, Berinde [3] introduced an interesting class of mappings, called
the class of almost contractions (or weak contractions), which includes
Kannan’s mappings and many other classes of mappings. We recall below the
definition of almost contractions.

######  Definition 1.1.

Let $(X,d)$ be a metric space. A mapping $T:X\to X$ is called an almost
contraction, if there exists $\lambda\in(0,1)$ and $\ell>0$ such that

$d(Tx,Ty)\leq\lambda d(x,y)+\ell d(y,Tx)$ (1.2)

for every $x,y\in X$.

Berinde [3] proved the following fixed point theorem for the above class of
mappings.

######  Theorem 1.2.

Let $(X,d)$ be a complete metric space and $T:X\to X$ be an almost
contraction. Then

  * (i)

$T$ admits at least one fixed point;

  * (ii)

For all $z_{0}\in X$, the sequence $\\{z_{n}\\}$ defined by $z_{n+1}=Tz_{n}$,
converges to a fixed point of $T$.

We point out that almost contractions can have more than one fixed point (see
[3, Example 1]).

In this paper, we first introduce the class of polynomial contractions. Two
fixed point results are obtained for such mappings. Namely, we first consider
the case when $T$ is continuous. Next, we weaken the continuity condition. Our
obtained results recover Banach’s fixed point theorem. Next, we introduce the
class of almost polynomial contractions and establish a fixed point theorem
for this class of mappings. Our obtained result generalizes Theorem 1.2.
Several examples are provided to illustrate our results.

##  2 The class of polynomial contractions

We introduce below the class of polynomial contractions.

######  Definition 2.1.

Let $(X,d)$ be a metric space and $T:X\to X$ be a given mapping. We say that
$T$ is a polynomial contraction, if there exists $\lambda\in(0,1)$, a natural
number $k\geq 1$ and a family of mappings $a_{i}:X\times X\to[0,\infty)$,
$i=0,\cdots,k$, such that

$\sum_{i=0}^{k}a_{i}(Tx,Ty)d^{i}(Tx,Ty)\leq\lambda\sum_{i=0}^{k}a_{i}(x,y)d^{i}(x,y)$
(2.1)

for every $x,y\in X$.

In this section, we are concerned with the study of fixed points for the above
class of mappings.

We first consider the case when $T$ is a continuous mapping.

######  Theorem 2.2.

Let $(X,d)$ be a complete metric space and $T:X\to X$ be a polynomial
contraction. Assume that the following conditions hold:

  * (i)

$T$ is continuous;

  * (ii)

There exist $j\in\\{1,\cdots,k\\}$ and $A_{j}>0$ such that

$a_{j}(x,y)\geq A_{j},\quad x,y\in X.$

Then $T$ admits a unique fixed point $z^{*}\in X$. Moreover, for every
$z_{0}\in X$, the Picard sequence $\\{z_{n}\\}\subset X$ defined by
$z_{n+1}=Tz_{n}$ for all $n\geq 0$, converges to $z^{*}$.

###### Proof.

We first prove that the set of fixed points of $T$ is nonempty. Let $z_{0}\in
X$ be fixed and $\\{z_{n}\\}\subset X$ be the Picard sequence defined by

$z_{n+1}=Tz_{n},\quad n\geq 0.$

Making use of (2.1) with $(x,y)=(z_{0},z_{1})$, we obtain

$\sum_{i=0}^{k}a_{i}(Tz_{0},Tz_{1})d^{i}(Tz_{0},Tz_{1})\leq\lambda\sum_{i=0}^{k}a_{i}(z_{0},z_{1})d^{i}(z_{0},z_{1}),$

that is,

$\sum_{i=0}^{k}a_{i}(z_{1},z_{2})d^{i}(z_{1},z_{2})\leq\lambda\sum_{i=0}^{k}a_{i}(z_{0},z_{1})d^{i}(z_{0},z_{1}).$
(2.2)

Using again (2.1) with $(x,y)=(z_{1},z_{2})$, we obtain

$\sum_{i=0}^{k}a_{i}(Tz_{1},Tz_{2})d^{i}(Tz_{1},Tz_{2})\leq\lambda\sum_{i=0}^{k}a_{i}(z_{1},z_{2})d^{i}(z_{1},z_{2}),$

that is,

$\sum_{i=0}^{k}a_{i}(z_{2},z_{3})d^{i}(z_{2},z_{3})\leq\lambda\sum_{i=0}^{k}a_{i}(z_{1},z_{2})d^{i}(z_{1},z_{2}),$

which implies by (2.2) that

$\sum_{i=0}^{k}a_{i}(z_{2},z_{3})d^{i}(z_{2},z_{3})\leq\lambda^{2}\sum_{i=0}^{k}a_{i}(z_{0},z_{1})d^{i}(z_{0},z_{1}).$

Continuing in the same way, we obtain by induction that

$\sum_{i=0}^{k}a_{i}(z_{n},z_{n+1})d^{i}(z_{n},z_{n+1})\leq\lambda^{n}\sum_{i=0}^{k}a_{i}(z_{0},z_{1})d^{i}(z_{0},z_{1}),\quad
n\geq 0.$ (2.3)

Since

$a_{j}(z_{n},z_{n+1})d^{j}(z_{n},z_{n+1})\leq\sum_{i=0}^{k}a_{i}(z_{n},z_{n+1})d^{i}(z_{n},z_{n+1}),$

we obtain by (ii) that

$A_{j}d^{j}(z_{n},z_{n+1})\leq\sum_{i=0}^{k}a_{i}(z_{n},z_{n+1})d^{i}(z_{n},z_{n+1}),$

which implies by (2.3) that

$d^{j}(z_{n},z_{n+1})\leq\lambda^{n}\sigma_{j,0},\quad n\geq 0,$ (2.4)

where

$\sigma_{j,0}=A_{j}^{-1}\sum_{i=0}^{k}a_{i}(z_{0},z_{1})d^{i}(z_{0},z_{1}).$
(2.5)

Then, making use of (2.4) and the triangle inequality, we obtain that for all
$n\geq 0$ and $m\geq 1$,

$\displaystyle d^{j}(z_{n},z_{n+m})$ $\displaystyle\leq
d^{j}(z_{n},z_{n+1})+d^{j}(z_{n+1},z_{n+2})+\cdots+d^{j}(z_{n+m-1},z_{n+m})$
$\displaystyle\leq\sigma_{j,0}\left(\lambda^{n}+\lambda^{n+1}+\cdots+\lambda^{n+m-1}\right)$
$\displaystyle=\sigma_{j,0}\lambda^{n}\frac{1-\lambda^{m}}{1-\lambda}$
$\displaystyle\leq\sigma_{j,0}\frac{\lambda^{n}}{1-\lambda},$

which yields

$d(z_{n},z_{n+m})\leq\left(\frac{\sigma_{j,0}}{1-\lambda}\right)^{\frac{1}{j}}(\lambda^{\frac{1}{j}})^{n}\to
0\mbox{ as }n,m\to\infty.$

This shows that $\\{z_{n}\\}$ is a Cauchy sequence. Since $(X,d)$ is complete,
there exists $z^{*}\in X$ such that

$\lim_{n\to\infty}d(z_{n},z^{*})=0,$

which implies by the continuity of $T$ that

$\lim_{n\to\infty}d(z_{n+1},Tz^{*})=\lim_{n\to\infty}d(Tz_{n},Tz^{*})=0.$

Then, from the uniqueness of the limit, we deduce that $Tz^{*}=z^{*}$, that
is, $z^{*}$ is a fixed point of $T$.

We now show that $z^{*}$ is the unique fixed point of $T$. Indeed, if
$z^{**}\in X$ is another fixed point of $T$, i.e., $Tz^{**}=z^{**}$ and
$d(z^{*},z^{**})>0$, then making use of (2.1) with $(x,y)=(z^{*},z^{**})$, we
get

$\sum_{i=0}^{k}a_{i}(Tz^{*},Tz^{**})d^{i}(Tz^{*},Tz^{**})\leq\lambda\sum_{i=0}^{k}a_{i}(z^{*},z^{**})d^{i}(z^{*},z^{**}),$

that is,

$\sum_{i=0}^{k}a_{i}(z^{*},z^{**})d^{i}(z^{*},z^{**})\leq\lambda\sum_{i=0}^{k}a_{i}(z^{*},z^{**})d^{i}(z^{*},z^{**}).$
(2.6)

On the other hand, from (ii), we have

$\displaystyle\sum_{i=0}^{k}a_{i}(z^{*},z^{**})d^{i}(z^{*},z^{**})$
$\displaystyle\geq a_{j}(z^{*},z^{**})d^{j}(z^{*},z^{**})$ $\displaystyle\geq
A_{j}d^{j}(z^{*},z^{**}).$

Since $A_{j}>0$ and $d(z^{*},z^{**})>0$, we deduce that

$\sum_{i=0}^{k}a_{i}(z^{*},z^{**})d^{i}(z^{*},z^{**})>0.$

Then, dividing (2.6) by
$\sum_{i=0}^{k}a_{i}(z^{*},z^{**})d^{i}(z^{*},z^{**})$, we reach a
contradiction with $\lambda\in(0,1)$. Consequently, $z^{*}$ is the unique
fixed point of $T$. This completes the proof of Theorem 2.2. ∎

We now study some particular cases of Theorem 2.2.

######  Proposition 2.3.

Let $(X,d)$ be a metric space and $T:X\to X$ be a polynomial contraction (in
the sense of Definition 2.1). Assume that the following conditions hold:

  * (i)

$a_{0}\equiv 0$, i.e., $a_{0}(x,y)=0$ for all $x,y\in X$;

  * (ii)

For all $i\in\\{1,\cdots,k\\}$, there exists $B_{i}>0$ such that

$a_{i}(x,y)\leq B_{i},\quad x,y\in X;$

  * (iii)

There exist $j\in\\{1,\cdots,k\\}$ and $A_{j}>0$ such that

$a_{j}(x,y)\geq A_{j},\quad x,y\in X.$

Then $T$ is continuous.

###### Proof.

Let $\\{u_{n}\\}\subset X$ be a sequence such that

$\lim_{n\to\infty}d(u_{n},u)=0,$ (2.7)

for some $u\in X$. Using (i) and making use of (2.1) with $(x,y)=(u_{n},u)$,
we get

$\sum_{i=1}^{k}a_{i}(Tu_{n},Tu)d^{i}(Tu_{n},Tu)\leq\lambda\sum_{i=1}^{k}a_{i}(u_{n},u)d^{i}(u_{n},u),\quad
n\geq 0,$

which implies by (ii) and (iii) that

$A_{j}d^{j}(Tu_{n},Tu)\leq\lambda\sum_{i=1}^{k}B_{i}d^{i}(u_{n},u),\quad n\geq
0.$ (2.8)

Then, making use of (2.7) and passing to the limit as $n\to\infty$ in (2.8),
we get

$\lim_{n\to\infty}d^{j}(Tu_{n},Tu)=0,$

which is equivalent to

$\lim_{n\to\infty}d(Tu_{n},Tu)=0.$

This shows that $T$ is a continuous mapping. ∎

From Theorem 2.2 and Proposition 2.3, we deduce the following result.

######  Corollary 2.4.

Let $(X,d)$ be a complete metric space and $T:X\to X$ be a polynomial
contraction. Assume that the following conditions hold:

  * (i)

$a_{0}\equiv 0$;

  * (ii)

For all $i\in\\{1,\cdots,k\\}$, there exists $B_{i}>0$ such that

$a_{i}(x,y)\leq B_{i},\quad x,y\in X;$

  * (iii)

There exist $j\in\\{1,\cdots,k\\}$ and $A_{j}>0$ such that

$a_{j}(x,y)\geq A_{j},\quad x,y\in X.$

Then $T$ admits a unique fixed point $z^{*}\in X$. Moreover, for every
$z_{0}\in X$, the Picard sequence $\\{z_{n}\\}\subset X$ defined by
$z_{n+1}=Tz_{n}$ for all $n\geq 0$, converges to $z^{*}$.

The following result is an immediate consequence of Corollary 2.4.

######  Corollary 2.5.

Let $(X,d)$ be a complete metric space and $T:X\to X$ be a given mapping.
Assume that there exist $\lambda\in(0,1)$, a natural number $k\geq 1$ and a
finite sequence $\\{a_{i}\\}_{i=1}^{k}\subset(0,\infty)$ such that

$\sum_{i=1}^{k}a_{i}d^{i}(Tx,Ty)\leq\lambda\sum_{i=1}^{k}a_{i}d^{i}(x,y)$
(2.9)

for every $x,y\in X$. Then $T$ admits a unique fixed point $z^{*}\in X$.
Moreover, for every $z_{0}\in X$, the Picard sequence $\\{z_{n}\\}\subset X$
defined by $z_{n+1}=Tz_{n}$ for all $n\geq 0$, converges to $z^{*}$.

######  Remark 2.6.

Observe that Corollary 2.5 recovers Banach’s fixed point theorem. Indeed,
taking $k=1$ and $a_{1}=1$, (2.9) reduces to

$d(Tx,Ty)\leq\lambda d(x,y),\quad x,y\in X.$

We provide below an example to illustrate Theorem 2.2.

######  Example 2.7.

Let $X=\\{x_{1},x_{2},x_{3},x_{4}\\}$ and $T:X\to X$ be the mapping defined by

$Tx_{1}=x_{1},\,\,Tx_{2}=x_{3},\,\,Tx_{3}=x_{4},\,\,Tx_{4}=x_{1}.$

Let $d$ be the discrete metric on $X$, i.e.,

$d(x_{i},x_{j})=\left\\{\begin{array}[]{llll}1&\mbox{if}&i\neq j,\\\\[4.0pt]
0&\mbox{if}&i=j.\end{array}\right.$

Consider the mapping $a_{0}:X\times X\to[0,\infty)$ defined by

$\displaystyle a_{0}(x_{i},x_{j})=a_{0}(x_{j},x_{i}),$ $\displaystyle
a_{0}(x_{i},x_{i})=0,$ $\displaystyle
a_{0}(x_{1},x_{2})=a_{0}(x_{2},x_{3})=3,$ $\displaystyle
a_{0}(x_{1},x_{3})=a_{0}(x_{3},x_{4})=2,$ $\displaystyle
a_{0}(x_{1},x_{4})=1,$ $\displaystyle a_{0}(x_{2},x_{4})=6.$

We claim that

$a_{0}(Tx,Ty)+d(Tx,Ty)\leq\frac{3}{4}\left(a_{0}(x,y)+d(x,y)\right)$ (2.10)

for every $x,y\in X$, that is, $T$ is a polynomial contraction in the sense of
Definition 2.1 with $k=1$, $a_{1}\equiv 1$ and $\lambda=\frac{3}{4}$. If $x=y$
or $(x,y)=(x_{1},x_{4})$, then (2.10) is obvious. Then, by symmetry, we have
just to show that (2.10) holds for all $x_{i},x_{j}\in X$ with $1\leq i<j\leq
4$ and $(i,j)\neq(1,4)$. Table 1 provides the different values of
$a_{0}(Tx_{i},Tx_{j})+d(Tx_{i},Tx_{j})$ and
$a_{0}(x_{i},x_{j})+d(x_{i},x_{j})$ for all $1\leq i<j\leq 4$, which confirm
(2.10).

$(i,j)$ | $a_{0}(Tx_{i},Tx_{j})+d(Tx_{i},Tx_{j})$ | $a_{0}(x_{i},x_{j})+d(x_{i},x_{j})$
---|---|---
$(1,2)$ | 3 | 4
$(1,3)$ | 2 | 3
$(2,3)$ | 3 | 4
$(2,4)$ | 3 | 7
$(3,4)$ | 2 | 3
Table 1: The values of $a_{0}(Tx_{i},Tx_{j})+d(Tx_{i},Tx_{j})$ &
$a_{0}(x_{i},x_{j})+d(x_{i},x_{j})$

Then, all the conditions of Theorem 2.2 are satisfied ((ii) is satisfied with
$A_{1}=1$). On the other hand, $T$ admits a unique fixed point $z^{*}=x_{1}$,
which confirms our obtained result.

Remark that Banach’s fixed point theorem is not applicable in this example.
Indeed, we have

$\frac{d(Tx_{1},Tx_{2})}{d(x_{1},x_{2})}=\frac{d(x_{1},x_{3})}{d(x_{1},x_{2})}=1.$

We also notice that, if we consider the mapping

$\mathcal{D}(x,y)=d(x,y)+a_{0}(x,y),\quad x,y\in X,$

then (2.10) reduces to

$\mathcal{D}(Tx,Ty)\leq\frac{3}{4}\mathcal{D}(x,y),\quad x,y\in X.$

However, $\mathcal{D}$ is not a metric on $X$. This can be easily seen
observing that (see Table 1)

$\mathcal{D}(x_{2},x_{4})=7>6=\mathcal{D}(x_{2},x_{1})+\mathcal{D}(x_{1},x_{4}).$

We now weaken the continuity condition imposed on $T$ in Theorem 2.2.

######  Definition 2.8.

Let $(X,d)$ be a metric space. A mapping $T:X\to X$ is called Picard-
continuous, if for all $z,w\in X$, we have

$\lim_{n\to\infty}d(T^{n}z,w)=0\implies\lim_{n\to\infty}d(T(T^{n}z),Tw)=0,$

where $T^{0}z=z$ and $T^{n+1}z=T(T^{n}z)$ for all $n\geq 0$.

Remark that, if $T:X\to X$ is continuous, then $T$ is Picard-continuous.
However, the converse is not true. The following example shows this fact.

######  Example 2.9.

Let $X=[a,b]$, where $a,b\in\mathbb{R}$ and $a<b$. Consider the mapping
$T:X\to X$ defined by

$Tx=\left\\{\begin{array}[]{llll}a&\mbox{if}&a\leq x<b,\\\\[4.0pt]
\frac{a+b}{2}&\mbox{if}&x=b.\end{array}\right.$

Let $d$ be the standard metric on $X$, that is, $d(x,y)=|x-y|$ for all $x,y\in
X$. Clearly, the mapping $T$ is not continuous at $b$. However, $T$ is Picard-
continuous in the sense of Definition 2.8. Indeed, observe that for all $z\in
X$, we have

$T^{n}z=a\quad\mbox{ for all }n\geq 2.$

So, if for some $z,w\in X$, we have

$\lim_{n\to\infty}d(T^{n}z,w)=0,$

then $w=a$ and

$\lim_{n\to\infty}d(T(T^{n}z),w)=\lim_{n\to\infty}d(Ta,w)=d(Ta,w)=0,$

which shows that $T$ is Picard-continuous.

######  Theorem 2.10.

Let $(X,d)$ be a complete metric space and $T:X\to X$ be a polynomial
contraction. Assume that the following conditions hold:

  * (i)

$T$ is Picard-continuous;

  * (ii)

There exist $j\in\\{1,\cdots,k\\}$ and $A_{j}>0$ such that

$a_{j}(x,y)\geq A_{j},\quad x,y\in X.$

Then $T$ admits a unique fixed point $z^{*}\in X$. Moreover, for every
$z_{0}\in X$, the Picard sequence $\\{z_{n}\\}\subset X$ defined by
$z_{n+1}=Tz_{n}$ for all $n\geq 0$, converges to $z^{*}$.

###### Proof.

We first prove that the set of fixed points of $T$ is nonempty. Let $z_{0}\in
X$ be fixed and $\\{z_{n}\\}\subset X$ be the Picard sequence defined by

$z_{n+1}=Tz_{n},\quad n\geq 0,$

that is,

$z_{n}=T^{n}z_{0},\quad n\geq 0.$

From the proof of Theorem 2.2, we know that $\\{z_{n}\\}$ is a Cauchy
sequence, which implies by the completeness of $(X,d)$ that there exists
$z^{*}\in X$ such that

$\lim_{n\to\infty}d(T^{n}z_{0},z^{*})=0.$

Then, by the Picard continuity of $T$, it holds that

$\lim_{n\to\infty}d(T^{n+1}z_{0},Tz^{*})=\lim_{n\to\infty}d(T(T^{n}z_{0}),Tz^{*})=0,$

which implies by the uniqueness of the limit that $z^{*}$ is a fixed point of
$T$. The rest of the proof is similar to that of Theorem 2.2. ∎

We now provide an example to illustrate Theorem 2.10.

######  Example 2.11.

Let $X=[0,1]$ and $T:X\to X$ be the mapping defined by

$Tx=\left\\{\begin{array}[]{llll}\frac{1}{4}&\mbox{if}&0\leq x<1,\\\\[4.0pt]
0&\mbox{if}&x=1.\end{array}\right.$

Let $d$ be the standard metric on $X$, i.e.,

$d(x,y)=|x-y|,\quad x,y\in X.$

Remark that $T$ is not a continuous mapping, but it is Picard-continuous (see
Example 2.9).

Consider now the mapping $a_{0}:X\times X\to[0,\infty)$ defined by

$a_{0}(x,y)=\frac{5}{6}\left(x\left|x-\frac{1}{4}\right|+y\left|y-\frac{1}{4}\right|\right),\quad
x,y\in X.$

We claim that

$a_{0}(Tx,Ty)+d(Tx,Ty)\leq\frac{1}{2}\left(a_{0}(x,y)+d(x,y)\right)$ (2.11)

for all $x,y\in X$, that is, $T$ is a polynomial contraction in the sense of
Definition 2.1 with $k=1$, $a_{1}\equiv 1$ and $\lambda=\frac{1}{2}$. We
discuss three possible cases (due to the symmetry of $a_{0}$).
Case 1: $x,y\in[0,1)$. In this case, we have

$\displaystyle a_{0}(Tx,Ty)+d(Tx,Ty)$
$\displaystyle=a_{0}\left(\frac{1}{4},\frac{1}{4}\right)+d\left(\frac{1}{4},\frac{1}{4}\right)$
$\displaystyle=0,$

which yields (2.11).
Case 2: $x\in[0,1)$ and $y=1$. In this case, we have

$\displaystyle a_{0}(Tx,Ty)+d(Tx,Ty)$
$\displaystyle=a_{0}\left(\frac{1}{4},0\right)+d\left(\frac{1}{4},0\right)$
$\displaystyle=\frac{1}{4}$

and

$\displaystyle\frac{a_{0}(x,y)+d(x,y)}{2}$
$\displaystyle=\frac{a_{0}\left(x,1\right)+d\left(x,1\right)}{2}$
$\displaystyle=\frac{5}{12}x\left|x-\frac{1}{4}\right|+\frac{5}{16}+\frac{|x-1|}{2}$
$\displaystyle\geq\frac{1}{4},$

which yields (2.11).
Case 3: $x=y=1$. In this case, we have

$\displaystyle a_{0}(Tx,Ty)+d(Tx,Ty)$
$\displaystyle=a_{0}(0,0)+d\left(0,0\right)$ $\displaystyle=0,$

which yields (2.11).

Therefore, (2.11) holds. Consequently, Theorem 2.10 applies. On the other
hand, $z^{*}=\frac{1}{4}$ is the unique fixed point of $T$, which confirms the
obtained result given by Theorem 2.10.

Notice that in this example, Theorem 2.2 is inapplicable since $T$ is not
continuous.

##  3 The class of almost polynomial contractions

Motivated by Berinde [3], we introduce below the class of almost polynomial
contractions.

######  Definition 3.1.

Let $(X,d)$ be a metric space and $T:X\to X$ be a given mapping. We say that
$T$ is an almost polynomial contraction, if there exists $\lambda\in(0,1)$, a
natural number $k\geq 1$, a finite sequence
$\\{L_{i}\\}_{i=0}^{k}\subset(0,\infty)$ and a family of mappings
$a_{i}:X\times X\to[0,\infty)$, $i=0,\cdots,k$, such that

$\sum_{i=0}^{k}a_{i}(Tx,Ty)d^{i}(Tx,Ty)\leq\lambda\sum_{i=0}^{k}a_{i}(x,y)\left[d^{i}(x,y)+L_{i}d^{i}(y,Tx)\right]$
(3.1)

for every $x,y\in X$.

We recall below the concept of weakly Picard operators, which was introduced
by Rus (see e.g. [26, 28, 29]).

######  Definition 3.2.

Let $(X,d)$ be a metric space and $T:X\to X$ be a given mapping. We say that
$T$ is a weakly Picard operator, if

  * (i)

The set of fixed points of $T$ is nonempty;

  * (ii)

For all $z_{0}\in X$, the Picard sequence $\\{T^{n}z_{0}\\}$ is convergent and
its limit belongs to the set of fixed points of $T$.

Our main result in this section is the following fixed point theorem.

######  Theorem 3.3.

Let $(X,d)$ be a complete metric space and $T:X\to X$ be an almost polynomial
contraction. Assume that the following conditions hold:

  * (i)

$T$ is Picard-continuous;

  * (ii)

There exist $j\in\\{1,\cdots,k\\}$ and $A_{j}>0$ such that

$a_{j}(x,y)\geq A_{j},\quad x,y\in X.$

Then $T$ is a weakly Picard operator.

###### Proof.

Let $z_{0}\in X$ be fixed and $\\{z_{n}\\}\subset X$ be the Picard sequence
defined by

$z_{n+1}=Tz_{n},\quad n\geq 0.$

Making use of (3.1) with $(x,y)=(z_{0},z_{1})$, we obtain

$\sum_{i=0}^{k}a_{i}(Tz_{0},Tz_{1})d^{i}(Tz_{0},Tz_{1})\leq\lambda\sum_{i=0}^{k}a_{i}(z_{0},z_{1})\left[d^{i}(z_{0},z_{1})+L_{i}d^{i}(z_{1},Tz_{0})\right],$

that is,

$\sum_{i=0}^{k}a_{i}(z_{1},z_{2})d^{i}(z_{1},z_{2})\leq\lambda\sum_{i=0}^{k}a_{i}(z_{0},z_{1})d^{i}(z_{0},z_{1}).$
(3.2)

Again, making use of (3.1) with $(x,y)=(z_{1},z_{2})$, we obtain

$\sum_{i=0}^{k}a_{i}(Tz_{1},Tz_{2})d^{i}(Tz_{1},Tz_{2})\leq\lambda\sum_{i=0}^{k}a_{i}(z_{1},z_{2})\left[d^{i}(z_{1},z_{2})+L_{i}d^{i}(z_{2},Tz_{1})\right],$

that is,

$\sum_{i=0}^{k}a_{i}(z_{2},z_{3})d^{i}(z_{2},z_{3})\leq\lambda\sum_{i=0}^{k}a_{i}(z_{1},z_{2})d^{i}(z_{1},z_{2}),$

which gives us thanks to (3.2) that

$\sum_{i=0}^{k}a_{i}(z_{2},z_{3})d^{i}(z_{2},z_{3})\leq\lambda^{2}\sum_{i=0}^{k}a_{i}(z_{0},z_{1})d^{i}(z_{0},z_{1}).$

Continuing this process, we get by induction that

$\sum_{i=0}^{k}a_{i}(z_{n},z_{n+1})d^{i}(z_{n},z_{n+1})\leq\lambda^{n}\sum_{i=0}^{k}a_{i}(z_{0},z_{1})d^{i}(z_{0},z_{1}),\quad
n\geq 0,$

which implies from (ii) that

$d^{j}(z_{n},z_{n+1})\leq\lambda^{n}\sigma_{j,0},\quad n\geq 0,$

where $\sigma_{j,0}$ is given by (2.5). Next, proceeding as in the proof of
Theorem 2.2, we obtain that $\\{z_{n}\\}$ is a Cauchy sequence, which implies
by the completeness of $(X,d)$ the existence of $z^{*}\in X$ such that

$\lim_{n\to\infty}d(z_{n},z^{*})=0.$

Finally, taking into consideration that $T$ is Picard-continuous, we get

$\lim_{n\to\infty}d(z_{n+1},Tz^{*})=0,$

which implies by the uniqueness of the limit that $z^{*}=Tz^{*}$. The proof of
Theorem 3.3 is then completed. ∎

We now investigate some special cases of Theorem 3.3.

######  Proposition 3.4.

Let $(X,d)$ be a metric space and $T:X\to X$ be a given mapping. Assume that
there exist $\lambda\in(0,1)$, a natural number $k\geq 1$ and two finite
sequence $\\{a_{i}\\}_{i=1}^{k},\\{L_{i}\\}_{i=1}^{k}\subset(0,\infty)$ such
that

$\sum_{i=1}^{k}a_{i}d^{i}(Tx,Ty)\leq\lambda\sum_{i=1}^{k}a_{i}\left[d^{i}(x,y)+L_{i}d^{i}(y,Tx)\right]$
(3.3)

for every $x,y\in X$. Then $T$ is Picard-continuous

###### Proof.

Let $z,u\in X$ be such that

$\lim_{q\to\infty}d(T^{q}z,u)=0.$ (3.4)

Making use of (3.3) with $(x,y)=(T^{q}z,u)$, we obtain

$\sum_{i=1}^{k}a_{i}d^{i}(T(T^{q}z),Tu)\leq\lambda\sum_{i=1}^{k}a_{i}\left[d^{i}(T^{q}z,u)+L_{i}d^{i}(u,T(T^{q}z))\right],$

that is,

$\sum_{i=1}^{k}a_{i}d^{i}(T^{q+1}z,Tu)\leq\lambda\sum_{i=1}^{k}a_{i}\left[d^{i}(T^{q}z,u)+L_{i}d^{i}(u,T^{q+1}z)\right],$

which implies that

$d(T(T^{q}z),Tu)\leq\frac{\lambda}{a_{1}}\sum_{i=1}^{k}a_{i}\left[d^{i}(T^{q}z,u)+L_{i}d^{i}(u,T^{q+1}z)\right].$

Then, passing to the limit as $q\to\infty$ in the above inequality and making
use of (3.4), we obtain

$\lim_{q\to\infty}d(T(T^{q}z),Tu)=0,$

which proves that $T$ is Picard-continuous. ∎

From Theorem 3.3 and Proposition 3.4, we deduce the following result.

######  Corollary 3.5.

Let $(X,d)$ be a complete metric space and $T:X\to X$ be a given mapping.
Assume that there exist $\lambda\in(0,1)$, a natural number $k\geq 1$ and two
finite sequence $\\{a_{i}\\}_{i=1}^{k},\\{L_{i}\\}_{i=1}^{k}\subset(0,\infty)$
such that (3.3) holds for every $x,y\in X$. Then $T$ is a weakly Picard
operator.

###### Proof.

Remark that (3.3) is a special case of (3.1) with $a_{0}\equiv 0$ and $a_{i}$
is constant for all $i\in\\{1,\cdots,k\\}$. Then, by Proposition 3.4, Theorem
3.3 applies. ∎

######  Remark 3.6.

Taking $k=1$, $a_{1}=1$ and $L_{1}=\frac{\ell}{\lambda}$, where $\ell>0$,
(3.3) reduces to (1.2). Then, by Corollary 3.5, we recover Berinde’s fixed
point theorem (Theorem 1.2).

We now give some examples to illustrate the above obtained results. The
following example shows that under the conditions of Theorem 3.3, we may have
more than one fixed point.

######  Example 3.7.

Let $X=\left\\{x_{1},x_{2},x_{3}\right\\}$ and $T:X\to X$ be the mapping
defined by

$Tx_{1}=x_{1},\,\,Tx_{2}=x_{2},\,\,Tx_{3}=x_{1}.$

The set $X$ is equipped with the discrete metric

$d(x_{i},x_{j})=\left\\{\begin{array}[]{llll}1&\mbox{if}&i\neq j,\\\\[4.0pt]
0&\mbox{if}&i=j.\end{array}\right.$

We claim that the mapping $T$ satisfies (3.3) with $k=2$, $a_{1}=a_{2}=1$,
$\lambda=\frac{2}{3}$ and $L_{1}=L_{2}=\frac{1}{2}$, that is,

$d(Tx,Ty)+d^{2}(Tx,Ty)\leq\frac{2}{3}\left[d(x,y)+\frac{1}{2}d(y,Tx)+d^{2}(x,y)+\frac{1}{2}d^{2}(y,Tx)\right]$
(3.5)

for all $x,y\in X$.

$(i,j)$ | $d(Tx_{i},Tx_{j})+d^{2}(Tx_{i},Tx_{j})$ | $d(x_{i},x_{j})+\frac{1}{2}d(x_{j},Tx_{i})+d^{2}(x_{i},x_{j})+\frac{1}{2}d^{2}(x_{j},Tx_{i})$
---|---|---
$(1,2)$ | 2 | 3
$(2,1)$ | 2 | 3
$(2,3)$ | 2 | 3
$(3,2)$ | 2 | 3
Table 2: The values of $d(Tx_{i},Tx_{j})+d^{2}(Tx_{i},Tx_{j})$ &
$d(x_{i},x_{j})+\frac{1}{2}d(x_{j},Tx_{i})+d^{2}(x_{i},x_{j})+\frac{1}{2}d^{2}(x_{j},Tx_{i})$

If $x=y$ or $(x,y)\in\\{(x_{1},x_{3}),(x_{3},x_{1})\\}$, then (3.5) is
obvious. Table 2 gives the different values of
$d(Tx_{i},Tx_{j})+d^{2}(Tx_{i},Tx_{j})$ and
$d(x_{i},x_{j})+\frac{1}{2}d(x_{j},Tx_{i})+d^{2}(x_{i},x_{j})+\frac{1}{2}d^{2}(x_{j},Tx_{i})$
for all $i,j\in\\{1,2,3\\}$ with $i\neq j$ and
$(i,j)\not\in\\{(1,3),(3,1)\\}$, which confirm (3.5). Then Corollary 3.5
applies. On the other hand, the set of fixed points of $T$ is nonempty, which
confirms the obtained result provided by Corollary 3.5. Remark that the set of
fixed points of $T$ is equal to $\\{x_{1},x_{2}\\}$.

An other example that illustrates Theorem 3.3 is given below.

######  Example 3.8.

Let $X=[0,1]$ and $T:X\to X$ be the mapping defined by

$Tx=\left\\{\begin{array}[]{llll}\frac{1}{4}&\mbox{if}&0\leq x<1,\\\\[4.0pt]
0&\mbox{if}&x=1.\end{array}\right.$

Let $d$ be the standard metric on $X$, i.e.,

$d(x,y)=|x-y|,\quad x,y\in X.$

We recall that $T$ is Picard-continuous (see Example 2.9).

Consider the mapping $a_{0}:X\times X\to[0,\infty)$ defined by

$a_{0}(x,y)=\left|4x^{2}-3x+\frac{1}{2}\right|+\left|4y^{2}-3y+\frac{1}{2}\right|,\quad
x,y\in X.$

We claim that

$a_{0}(Tx,Ty)+d(Tx,Ty)\leq\frac{1}{2}\left(2a_{0}(x,y)+d(x,y)+d(y,Tx)\right)$
(3.6)

for all $x,y\in X$, that is, $T$ is an almost polynomial contraction in the
sense of Definition 3.1 with $k=1$, $a_{1}\equiv 1$, $L_{0}=L_{1}=1$ and
$\lambda=\frac{1}{2}$. We discuss four possible cases.
Case 1: $0\leq x,y<1$. In this case, we have

$\displaystyle a_{0}(Tx,Ty)+d(Tx,Ty)$
$\displaystyle=a_{0}\left(\frac{1}{4},\frac{1}{4}\right)+d\left(\frac{1}{4},\frac{1}{4}\right)$
$\displaystyle=0.$

Then (3.6) holds.
Case 2: $0\leq x<1$, $y=1$. In this case, we have

$\displaystyle a_{0}(Tx,Ty)+d(Tx,Ty)$
$\displaystyle=a_{0}\left(\frac{1}{4},0\right)+d\left(\frac{1}{4},0\right)$
$\displaystyle=\frac{1}{2}+\frac{1}{4}$ $\displaystyle=\frac{3}{4}$
$\displaystyle=d(y,Tx)$
$\displaystyle\leq\frac{1}{2}\left(2a_{0}(x,y)+d(x,y)+d(y,Tx)\right).$

Then (3.6) holds.
Case 3: $x=1$, $0\leq y<1$. In this case, we have

$\displaystyle a_{0}(Tx,Ty)+d(Tx,Ty)$
$\displaystyle=a_{0}\left(0,\frac{1}{4}\right)+d\left(0,\frac{1}{4}\right)$
$\displaystyle=\frac{1}{2}+\frac{1}{4}$ $\displaystyle=\frac{3}{4}$

and

$\displaystyle a_{0}(x,y)$ $\displaystyle=a_{0}(1,y)$
$\displaystyle=\frac{3}{2}+\left|4y^{2}-3y+\frac{1}{2}\right|$
$\displaystyle\geq\frac{3}{2}.$

Therefore, it holds that

$\displaystyle a_{0}(Tx,Ty)+d(Tx,Ty)$ $\displaystyle=\frac{3}{4}$
$\displaystyle\leq\frac{3}{2}$
$\displaystyle\leq\frac{1}{2}\left[2a_{0}(x,y)\right]$
$\displaystyle\leq\frac{1}{2}\left(2a_{0}(x,y)+d(x,y)+d(y,Tx)\right).$

Then (3.6) holds.
Case 4: $x=y=1$. In this case, we have

$\displaystyle a_{0}(Tx,Ty)+d(Tx,Ty)$ $\displaystyle=a_{0}(0,0)+d(1,1)$
$\displaystyle=1$ $\displaystyle=d(y,Tx)$
$\displaystyle\leq\frac{1}{2}\left(2a_{0}(x,y)+d(x,y)+d(y,Tx)\right).$

Then (3.6) holds.

Consequently, (3.6) is satisfied for all $x,y\in X$. Notice also that
condition (ii) of Theorem 3.3 holds with $j=1$ and $A_{1}=1$. Then Theorem 3.3
applies. Observe that $z^{*}=\frac{1}{4}$ is a fixed point of $T$, which
confirms the result given by Theorem 3.3.

### Data availability statement

No new data were created or analyzed in this study.

### Conflict of interest

The authors declare that they have no competing interests.

### Funding

The third author is supported by Researchers Supporting Project number
(RSP2024R4), King Saud University, Riyadh, Saudi Arabia.

### Authors’ contributions

All authors contributed equally and significantly in writing this article. All
authors read and approved the final manuscript.

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# Object-QA: Towards High Reliable Object Quality Assessment

Jing Lu1∗, Baorui Zou2∗, Zhanzhan Cheng31∗, Shiliang Pu1, Shuigeng Zhou2, Yi
Niu1, Fei Wu3 1Hikvision Research Institute, China; 2Fudan University,
Shanghai, China; 3Zhejiang University, Hangzhou, China

(2020)

###### Abstract.

In object recognition applications, object images usually appear with
different quality levels. Practically, it is very important to indicate object
image qualities for better application performance, e.g. filtering out low-
quality object image frames to maintain robust video object recognition
results and speed up inference. However, no previous works are explicitly
proposed for addressing the problem. In this paper, we define the problem of
object quality assessment for the first time and propose an effective approach
named Object-QA to assess high-reliable quality scores for object images.
Concretely, Object-QA first employs a well-designed relative quality assessing
module that learns the intra-class-level quality scores by referring to the
difference between object images and their estimated templates. Then an
absolute quality assessing module is designed to generate the final quality
scores by aligning the quality score distributions in inter-class. Besides,
Object-QA can be implemented with only object-level annotations, and is also
easily deployed to a variety of object recognition tasks. To our best
knowledge this is the first work to put forward the definition of this problem
and conduct quantitative evaluations. Validations on 5 different datasets show
that Object-QA can not only assess high-reliable quality scores according with
human cognition, but also improve application performance.

*Authors contributed equally to this research.

††copyright: acmcopyright††journalyear: 2020††doi:
10.1145/xxxxxxx.1122456††conference: Xxxx ’20: ACM Symposium on XXX; June
xx–xx, 2020; Xxxx, XX††booktitle: xxx ’20: ACM Symposium on XXX, June xx–xx,
2020, Xxxx, XX††price: 15.00††isbn: 978-1-4503-XXXX-X/18/06

##  1\. Introduction

Figure 1. The comparison of assessing target between IQA and OQA. (a)$\sim$(d)
are examples of IQA: (a) high-quality image containing no object, (b) low-
quality image with gaussian noise containing no object, (c) low-quality image
containing 1 clear object but with blurred background, (d) low-quality image
with 2 distorted objects. (e)$\sim$(h) are examples of OQA: (e) high-quality
image with 1 vehicle, (f) low-quality image with 1 clear but occluded vehicle
(IQA would regard it as a high-quality image since no physical noise occurs,
which is a failed case of IQA methods applied on object images), (g) low-
quality image with 1 blurred vehicle, (h) low-quality image with 1 vehicle
under poor illumination.

Problem | IQA | OQA
---|---|---
assessing target | whole image | cropped object image
object number | none/multiple | one
background quality | consider | ignore
noise type | constrained physical noise (e.g. | unconstrained noise (e.g.
gaussian blur/noise, transmission errors | occlusion, motion blur, poor illumination
application | image processing task (e.g. | object recognition task (e.g.
image restoration, image super-resolution | person/vehicle re-identification, text recognition

Table 1. Differences between OQA and IQA.  Figure 2. Illustration of object
quality assessment.

In many real-life object recognition systems (e.g. vehicle (Wang et al., 2019)
or person (Li et al., 2018) re-identification, license plate recognition (Xu
et al., 2018) and scene text recognition (Bai et al., 2018)), object images
usually contain one centering object and appear with different qualities (as
(e)$\sim$(h) in Figure 1) caused by complicated environmental inteferences
(e.g. occlusion, motion, illumination change). Practically, high-quality (e.g.
clear, undistorted) object images are beneficial to recognition, while low-
quality (e.g. blurred, distorted) object images may result in poor recognition
performance.

Generally, high reliable quality scores of object images can be used to
enhance application performance. For instance, (1) object quality assessment
can be used as a pre-processing technique: filtering out low-quality images to
avoid wrong results, boosting applications’ efficiency and even saving storage
space; (2) It can also be applied in post-processing: removing unreliable
recognitions caused by low quality to promote application experience. Thus,
_identifying object image quality_ can be a powerful strategy to maintain
robust application performance.

Problem definition. Here, we first explicitly present the definition of _o
bject quality assessment_ (_abbr._ OQA) problem: given a specific
recognition/classification task in which each image normally only contains one
object for recognition ((e)$\sim$(h) in Figure 1), the quality level of each
object image requires to be assessed with reasonable quality scores by
considering how well the object image is beneficial for recognition (i.e.
higher scores represent better quality and vice versa), in this way the
quality relations among any object images in this task can be determined.

An intuitive application of OQA is to filter out relative low-quality images
tending to result in wrong recognitions, so that recognition/classification
tasks can be enhanced. Emphatically, the object quality scores are domain-
specific. For example, the quality scores can reasonably express the quality
relations among different vehicle images, but can not distinguish the quality
between a vehicle image and a person image.

The OQA task is a different research topic compared with the well-know image
quality assessment (_abbr._ IQA) in the following aspects: (1) assessing
target: IQA attempts to assess the quality of the entire image in which none
or multiple objects may exist, and jointly considers the quality of background
area and foreground objects (see (a)$\sim$(d) in Figure 1), while OQA simply
evaluates the quality of the only foreground object in the cropped image (see
(e)$\sim$(h)); (2) goal: IQA predicts the distortion extent of the entire
image by considering constrained types of physical noises (e.g. gaussian
blur/noise, JPEG2000 compression and fast fading channel distortion) (Sheikh
et al., 2006), thus it can be applied in image restoration (Katsaggelos,
2012), image super-resolution (Ouwerkerk, 2006) and even image retrieval (Yan
et al., 2014) applications. But the goal of OQA is to achieve object-level
quality estimation indicating whether object images are suitable for
recognition, by considering unconstrained noises (e.g. occlusion, blur and
poor illumination) in real-life scenarios, so it can be used for boosting the
performance and speed of various object recognition systems. Figure 1(f) is an
object image with no physical noise but containing occluded object, IQA may
generate high scores while OQA should consider it a low-quality image since
its recognition result is very possibly incorrect. We summarize the
differences between OQA and IQA in Table 1.

Proposed idea. In this paper, our focus is the OQA task. Firstly we consider
an intriguing question that how exactly does human identify object’s quality
level?

As illustrated in many works from IQA, human visual system needs a reference
image to quantify the perceptual discrepancy by comparing the distorted image
either directly with the original undistorted image (reference based methods
(Seo et al., 2019; Prashnani et al., 2018)) or implicitly with a hallucinated
image in mind (no-reference based methods (Lin and Wang, 2018; Ren et al.,
2018)). We transfer this common knowledge to OQA, that is,

given an object image, human may firstly assume the high-quality morphology of
this object class as its quality benchmark, like the benchmark (T1) for object
images (1)$\sim$(3) in Figure 2, then estimate the difference between the
object image and its benchmark as its quality. Smaller difference represents
better quality and vice versa. We describe the quality benchmark as ‘template’
and denote this class-aware template comparison strategy as _relative quality
assessment_. As shown in (R1)/(R2) of part (i), the predicted quality scores
can reasonably express the relative intra-class quality relations.

However, inter-class quality score distribution is still unreasonable
(‘relative quality distribution’ in part (i) leads to the wrong quality
ranking of image (c) and (d) in part (ii)). It’s because the hypothetic
‘template’ is subjective and varies from person to person, making the
estimated templates of different classes correspond to inconsistent quality
levels, as template (T1) and (T2) in part (i) imply, thus the quality scores
between different classes are incomparable directly. Fortunately, we discover
that the misordered ranking can be re-aligned by judging whether object images
are beneficial to correct recognition (similar idea appeared in (Liu et al.,
2017)). For instance in part (ii), image (d) has a better chance for producing
correct recognition result compared with (c) since the contour of vehicle in
(d) can still be identified, then the wrong ranking in between can be fixed.
Based on this observation, we can re-align the quality relations among
different classes while maintaining the quality distribution learned by
_relative quality assessment_ , and obtain rational ‘absolute quality
distribution’ (see part (ii)). Such sorting strategy is denoted as _absolute
quality assessment_ in inter-class.

On the basis of above observations, we propose an effective object-level
quality assessment approach named Object-QA to assess high-reliable quality
scores for object images. Specifically, Object-QA consists of two modules: the
Relative Quality Assessing module (_abbr._ RQA) for obtaining intra-class-
level quality scores, and the Absolute Quality Assessing module (_abbr._ AQA)
for obtaining inter-class-level quality scores. RQA first synthesizes
hypothetical templates for each class of object images, then computes the
distances between selected images and their templates as their relative
quality scores. AQA is responsible for learning the quality distribution
alignment from different classes to obtain the final absolute quality scores
in inter-class. Main contributions are:

(1) We give the explicit definition of _object quality assessment_ problem for
the first time.

(2) We propose a general object quality assessment approach named Object-QA
for assessing high-reliable quality scores with only recognition-based
annotations. Inspired by human cognition (Lin and Wang, 2018; Liu et al.,
2017), we implement Object-QA with two modules: the relative quality assessing
module for obtaining intra-class quality scores and the absolute quality
assessing module for generating inter-class quality scores. The proposed
network is quite light-weighted and can be easily equipped on various object
recognition applications for improving their performance and inference speed,
yet brings extremely low computational cost.

(3) Experiments on 5 tasks, including 1 synthetic MNIST dataset and 4 real-
world scenarios (person (Wang et al., 2014) and vehicle (Zapletal and Herout,
2016) re-identification, video text spotting (Karatzas et al., 2013), license
plate recognition (Xu et al., 2018)) demonstrate the effectiveness of Object-
QA.

##  2\. Related work

In fact, no previous work can be found for specially study the problem of
object quality assessment, which is a stark contrast to its potential power
for improving the robustness and time efficiency of object recognition
systems. In the few related works, object quality assessment strategies are
all existed as by-products in video-based recognition tasks, and these methods
didn’t report any quantitative results regarding quality assessment.

Here we categorize them as follows: (1) attention-based strategy. Most
previous methods fall into this category by adopting temporal attention
mechanisms to indicate the attending weight of different images, the attending
weights are treated as quality scores. Liu et al. (Liu et al., 2017) and Yang
et al. (Yang et al., 2017) adopted a temporal attention module to aggregate
discriminative features from multiple frames for video person re-
identification and face recognition respectively, then attention weight of
each frame was treated as the quality score. Song et al. (Song et al., 2018)
also followed this quality assessment routine for person re-identification
task. (2) feature clustering strategy. Recently Cheng et al. (Cheng et al.,
2019) proposed a quality scoring method for selecting the highest-quality text
regions from tracked text streams, in which quality scores were generated by
computing the feature distance between testing images and their templates
obtained by K-means clustering.

All the above strategies only concentrate on depicting the intra-class quality
relations between consecutive object frames in video-based tasks, but are
limited to assessing quality in more general single-image scenario, in which
quality relations between images from different classes needs identifying.

Overcoming the limitation of previous work, in this paper we explicitly
explore the quality correlations among object images from different classes,
and propose a robust object quality assessment method to generate absolute
quality scores for object images with only recognition-based annotations.

##  3\. Methodology

Figure 3.  The workflow of Object-QA, which consists of 3 steps: (a)
Extracting features for RQA and AQA with a pre-trained recognition model; (b)
Generating relative quality scores by RQA network; (c) Generating the final
absolute quality scores by AQA network. Details of neural networks are
described in _Experiment_ section.

###  3.1. Overview

The workflow of our method is illustrated in Figure 3, which consists of three
steps: (1) extracting features for the following parts by a pre-trained
recognition model, (2) learning the relative quality ranking between intra-
class images by Relative Quality Assessment (_abbr_. RQA) network, and (3)
adjusting the learned relative quality scores by Absolute Quality Assessment
(_abbr_. AQA) network to obtain final absolute quality scores.

Specifically, RQA first synthesizes a template (representing the relative
quality benchmark) by exploring the quality relations between intra-class
object images, then generates relative quality scores by calculating the
differences between testing images and their templates. AQA is responsible for
re-aligning relative quality scores among different classes by considering
correlations between relative quality distributions, and generating absolute
quality scores.

During inference only AQA is needed, which can be easily deployed in existing
recognition network as a quality generation branch.

###  3.2. Pre-feature Extracting

Given a set of intra-class object images $I=\\{I_{1},I_{2},...,I_{N}\\}$, we
extract their features as $F=[F_{1},F_{2},...,F_{N}]\in\mathbb{R}^{C\times N}$
with a pre-trained recognition model. The recognition models adopted in
different scenarios are detailed in _Experiment_.

###  3.3. Relative Quality Assessment

RQA is designed to generate relative quality scores, which includes 2 major
steps: 1) synthesizing the hypothetical template represented in feature space,
and 2) producing the quality scores by calculating the distance between
testing images and their templates.

_Synthesizing template._ Empirically, the features of different quality images
are actually localised at different positions in feature space, which implies
the quality correlations are contained in feature distributions intrinsically.
Naturally, the represented template should be closer to those high-quality
images with correct recognitions while be farther from the low-quality ones
easily leading to wrong recognitions. Here, we adopt the multi-head attention
structure (Vaswani et al., 2017) to explore the quality correlations among
different quality images in intra-class, then further synthesize the feature
representation of template.

Specifically, $F\in\mathbb{R}^{C\times N}$ is firstly sliced into $M$ pieces
$(F^{1},F^{2},...,F^{M})$ along the channel dimension ($C$), and then they are
separately fed into M heads. Concretely, in the $m$-th head, $F^{m}$ is
firstly fed into the linear layer to obtain the triple of query, key and
value: $(Q^{m},K^{m},V^{m})$. Then we conduct the scaled dot-product attention
(Vaswani et al., 2017) on each head to obtain head outputs,

(1) ${T}^{m}\text{=}softmax(\frac{Q^{m}*V^{m}}{\sqrt{d}})*V^{m},$

where ‘*’ is dot-product and $\sqrt{d}$ is the scale factor according to
(Vaswani et al., 2017). Finally, the corresponding template of $F$ is
synthesized by directly concatenating all head outputs along the channel
dimension, i.e.,

(2) ${T}\text{=}Concat(T^{1},T^{2},...,T^{M}),$

where $T\in\mathbb{R}^{C\times N}$. Then the template feature representation
$\bar{T}\in\mathbb{R}^{C}$ of each class is generated by conducting average
pooling on $T$.

Theoretically, the optimal template feature should be the one that is most
beneficial to recognition. So in order to learn the templates, we use object
recognition loss (denoted by $\mathcal{L}_{recog}$) during optimization, so
that the estimated template feature can approximate the optimal one. Besides,
the intra-class quality scores should be computed with respect to the same
quality benchmark, thus a template consistency loss (denoted by
$\mathcal{L}_{consis}$) is introduced to constrain the template feature of
each image in the same class being consistent. The loss function of RQA is
formalized as:

(3)
$\mathcal{L}_{RQA}\text{=}\mathcal{L}_{recog}+\lambda_{c}\mathcal{L}_{consis}$

where $\lambda_{c}$ is a tunable parameter and the consistency loss is:

(4)
$\mathcal{L}_{consis}\text{=}\frac{2}{N*(N-1)}\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}MSE(T_{i},T_{j})$

in which $MSE$ is the average mean square error. $T_{i}\in\mathbb{R}^{C}$ and
$T_{j}\in\mathbb{R}^{C}$ mean the learned templates of image $F_{i}$ and
$F_{j}$, respectively.

_Generating quality scores._ Relative quality score $q_{i}$ can be generated
by computing the feature distance between testing image and its class’s
template, i.e.,

(5) ${q}_{i}\text{=}\frac{\bar{T}*F_{i}}{||\bar{T}||*||F_{i}||},$

###  3.4. Absolute Quality Assessment

Since the learned templates of different classes usually correspond to
different quality levels, RQA is incapable of assigning reasonable quality
scores in inter-class. To address this, AQA network is designed to re-adjust
the relative quality scores and produce absolute quality scores, which
contains three progressive parts as follows.

_Binary inter-class quality alignment._ We first align the quality
distributions of different classes by referring to a learnable quality anchor
$th$. That is, those images with quality score larger than the quality anchor
always can be recognized correctly, vice versa. Concretely, we assume that all
scores fall into the range (0,1). Then all images should be mapped into 2
absolute quality score ranges: the correctly recognized ones are mapped into
the higher score range $[th,1)$ while the wrongly recognized ones are mapped
into the lower score range $(0,th)$. This alignment is ensured by:

(6)
$\mathcal{L}_{align}\text{=}\frac{1}{M}\sum_{i=1}^{M}relu[\delta(I_{i})*(q_{i}-th-\epsilon)]$

where indicator function $\delta(I_{i})$ equals -1 when $I_{i}$ can be
correctly recognized and 1 otherwise, and $\epsilon$ is a regularization
parameter for avoiding regressing the same quality scores. Quality anchor $th$
is jointly optimized with $q_{i}$.

_Intra-class quality distribution maintaining._ Obviously the learned relative
quality ranking in intra-class should be kept. Given the relative quality
score triplet $(q_{1},q_{2},q_{3})$ in a class, the corresponding absolute
score triplet generated by AQA are $(Q_{1},Q_{2},Q_{3})$. The absolute quality
ranking of each image pair should remain the same as relative scores, which is
constrained in a loss function:

(7)
$\mathcal{L}_{rank}\text{=}\frac{1}{3}\sum_{i=1}^{2}\sum_{j=i+1}^{3}relu[\Delta(q_{i},q_{j})*(Q_{i}-Q_{j}-\epsilon)]$

where $\Delta(\cdot,\cdot)$ equals -1 when $q_{i}>q_{j}$ and 1 otherwise.

Additionally, the correlation between quality differences
$d_{1}=|q_{1}-q_{2}|$ and $d_{2}=|q_{2}-q_{3}|$ ($|\cdot|$ denotes absolute
value) should also be taken into consideration for better characterizing the
intra-class quality distribution, which is denoted as:

(8)
$\mathcal{L}_{d-rank}\text{=}relu[\Delta(d_{1},d_{2})*(d_{1}-d_{2}-\epsilon)].$

Then total loss function for keeping intra-class quality ranking is:

(9)
$\mathcal{L}_{intra}\text{=}\mathcal{L}_{rank}+\lambda_{intra}\mathcal{L}_{d-rank},$

where $\lambda_{intra}$ is a tunable parameter.

_Inter-class quality distribution maintaining._ We firstly define the quality
distribution entropy ($e$) to depict the quality distribution in each quality
score triplet $(s_{1},s_{2},s_{3})$:

(10)
${e}\text{=}-\frac{min(D_{1},D_{2})}{max(D_{1},D_{2})}\log(\frac{min(D_{1},D_{2})}{max(D_{1},D_{2})})$

where $D1=|s_{1}-s_{2}|$ and $D2=|s_{2}-s_{3}|$. Larger value of $e$ implies
more disordered quality distribution while smaller value represents more
uniform distribution.

Given two relative quality score triplets $(q^{A}_{1},q^{A}_{2},q^{A}_{3})$
and $(q^{B}_{1},q^{B}_{2},q^{B}_{3})$ from two different classes $A$ and $B$,
we can obtain their relative $e$ as $e^{A}_{r}$ and $e^{B}_{r}$.
Correspondingly, the absolute $e$ are represented as $e^{A}_{a}$ and
$e^{B}_{a}$. Then we can align quality distributions between class A and class
B by:

(11)
$\mathcal{L}_{inter}\text{=}relu[\Delta(e^{A}_{r},e^{B}_{r})*(e^{A}_{a}-e^{B}_{a}-\zeta)]$

where $\zeta$ is a regularization parameter.

Finally, AQA is optimized as:

(12)
$\mathcal{L}_{AQA}\text{=}\mathcal{L}_{align}+\lambda_{a1}\mathcal{L}_{intra}+\lambda_{a2}\mathcal{L}_{inter}$

where $\lambda_{a1}$ and $\lambda_{a2}$ are tunable parameters.

##  4\. Experiments

Since all the previous works didn’t report any quantitative results regarding
object quality assessment, we can only re-implement two representative state-
of-art strategies ((Liu et al., 2017), (Cheng et al., 2019)) for comparison.
We first explore the effectiveness of our method on a synthetic MNIST (LeCun
et al., 1998), then conduct extensive experiments on 4 different real-life
scenarios to demonstrate its robustness and potential power for enhancing
recognition performance.

Unfortunately we are unable to compare our method with previous literature in
IQA for two reasons: (1) training of IQA methods requires the availability of
annotated quality labels or reference images, which is absent in existing
object classification/recognition datasets, making the re-implementation
impossible; (2) to say the least, even if we re-implement IQA methods, since
IQA only considers constrained physical noise, the performance of IQA methods
will degenerate in OQA-oriented scenarios where unconstrained types of
interferences exist (e.g. occlusion, pose changes), a failed case is shown in
Figure 1(f).

###  4.1. Implementation Details

Our work is built on the ‘PyTorch’ framework.

The pre-trained recognition model. In five evaluation scenarios, the pre-
trained recognition models mentioned in Section 3.2 are firstly pre-trained,
applying the same training strategies as used in (LeCun et al., 1998), (Gao
and Nevatia, 2018), (Liu et al., 2018), (Cheng et al., 2017) and (Shi et al.,
2016). Implementation details can be found in our supplementary meterial.

Relative quality assessment network. In each training batch, RQA takes the
features of 3 randomly selected images from the same class as input, followed
by a multi-head attention structure. The num of heads ($M$) is set to 4 in all
of our experiments. In the scaled dot-product attention module, 3 fully
connected layers are employed for embedding the input feature and outputting
$Q$, $K$, $V$, and the output channel number is the same as input feature.
During training, $\lambda_{c}$ is set to 1 and batch size is set to 9600, and
‘Adam’ is used for optimization with learning rate = 0.0005 and decay rate =
0.94 for every 200 epochs.

Absolute quality assessment network. The regularization term $\epsilon$ and
$\zeta$ are set to 0.02 and 0.01 respectively. $\lambda_{intra}$,
$\lambda_{a1}$ and $\lambda{{}_{a2}}$ in Equation 12 are all set to 1. In
training stage, batch size is set to 9600, and ‘Adam’ is used to optimize the
model with fixed learning rate=$10^{-3}$ and weight decay=0.0005.

###  4.2. Evaluation protocol

The fact is that existing classification/recognition datasets all lack the
ground truth of quality scores. Besides, the annotation of object quality is
tricky since the accuracy of annotated scores can not be ensured due to the
subjective perceptual errors of different human annotators. Thus we attempt to
design some strategies to obtain the ground truth of quality scores for
quantitative evaluations (detailed strategies for ground truth generation are
illustrated in corresponding validation scenario).

Two quality evaluation metrics are adopted to evaluate our method: 1)
Spearman’s Rank Order Correlation Coefficient (SROCC) (Lin and Wang, 2018) for
evaluating monotonic relations between ground truth and predicted scores, 2)
Linear Correlation Coefficient (LCC) (Lin and Wang, 2018) for assessing linear
correlations between ground truth and predictions. Additionally, we also
validate the improvements on recognition performance as an auxiliary evidence
to prove the effectiveness of our method.

###  4.3. Ablation Study

Dataset. Arising from the lack of object quality assessment benchmarks, we
generate a MNIST (LeCun et al., 1998)-based quality assessment dataset in
which images appear in different qualities. Correspondingly, quality score
labels are provided for evaluation.

The synthetic MNIST dataset is generated by introducing two quality factors:
1) Blur: gaussian blur with evenly spaced kernel sizes of
[3,5,7,9,11,13,15,17,19] for synthesizing images with different blur degrees.
Thereby, a total of 9 images are synthesized respecting to 1 original image.
2) Illumination deterioration: contrast and intensity modification for
generating images with various illumination conditions, in which contrast
changes from 0.4 to 0.9 and intensity changes from -160 to 250. 3) Mixed:
considering both blur and illumination.

The object quality scores should be able to represent the extent of how an
object image is suitable for recognition. Consider that the original MNIST
image is noise-free, so it can be regarded as the ‘ideal’ template which is
most beneficial for recognition. Then it’s reasonable to generate the ground
truth of quality score by calculating feature distances between the original
MNIST images and their polluted images. Note that, the ground truth is not
used in training but only used to evaluate the method’s performance.

We randomly select half of the images in each class as the training set, i.e.
600 thousand images with blur and 400 thousand images with illumination, while
the rest of them are used for testing. In order to evaluate our method, we
design two test sets: the ‘Intra’ and ‘Inter’ set. ‘Intra’ set contains 10000
groups of images, in which images in a group are from the same class. ‘Inter’
set contains 10000 groups of images, in which images in a group are from
different classes. The group size varies from 3 to 10. The pre-trained model
is the same to (LeCun et al., 1998), and the outputs before the final
classification layer are used as the extracted features.

Evaluation protocol. We first compute SROCC and LCC in each image group, then
report the average value of all groups.

Metrics | Blur | Illumination | Mixed
---|---|---|---
RQA | AQA | RQA | AQA | RQA | AQA
SROCC | Intra | 0.874 | 0.857 | 0.933 | 0.929 | 0.920 | 0.893
Inter | 0.641 | 0.776 | 0.765 | 0.805 | 0.767 | 0.814
LCC | Intra | 0.880 | 0.861 | 0.966 | 0.950 | 0.957 | 0.936
Inter | 0.680 | 0.866 | 0.850 | 0.886 | 0.845 | 0.895

Table 2. Quality assessment evaluation of RQA and AQA under different
conditions. ‘Blur’ and ‘Illumination’ mean images are only polluted by blur
and lighting, respectively. ‘Intra’ and ‘Inter’ denote evaluations on ‘Intra’
and ‘Inter’ sets respectively.

(a) MNIST

(b) CCPD

Figure 4. The quality distribution curves on different scenarios. (a) The
predicted quality score curves under different interferences (‘Blur’,
‘Illumination’ and ‘Mixed’) on synthetic MNIST. (b) The predicted quality
score curve on CCPD.

Overall performance. Table 2 shows that our method achieves promising quality
assessment results under all conditions. And Figure 4(a) implies the predicted
scores also appear in approximate distribution with their ground truth under
different interferences, which verifies the robustness of our method.

Figure 5. The visualization of quality scores by RQA and AQA.

Effects of RQA and AQA. In Table 2, we observed that both RQA and AQA succeed
in assessing quality scores in intra-class, and AQA significantly outperforms
RQA on ‘Inter’ sets under all conditions (See ‘Inter’ rows). An example is
illustrated in Figure 5, the intra-class quality ranking of RQA seems rational
but it fails on assessing quality scores in inter-class, while AQA can
generate satisfying quality ranking in inter-class.

$\mathcal{L}_{align}$ | ✓ | ✓ | ✓ | ✓
---|---|---|---|---
$\mathcal{L}_{intra}$ |  | ✓ |  | ✓
$\mathcal{L}_{inter}$ |  |  | ✓ | ✓
SROCC | 0.774 | 0.792 | 0.802 | 0.814
LCC | 0.832 | 0.863 | 0.869 | 0.895

Table 3.  The evaluation of AQA’s constraints on the ‘Mixed’ set.

Exploration on AQA. We further validate the effect of each constraint in AQA,
as shown in Table 3. Since $\mathcal{L}_{align}$ is used to provide the
quality anchor for aligning scores between different classes, it is
indispensable in our method. We find that both $\mathcal{L}_{intra}$ and
$\mathcal{L}_{inter}$ can significantly improve the results of SROCC and LCC,
and the jointly learning further improves the performance.

Inference speed. AQA is very light-weighted and its inference speed achieves
1820 frames per second as we test on 12GB Titan-X GPU with ‘PyTorch’.

Figure 6. The quality scores (under each image) generated by AQA on 4 tasks.
(a), (b), (c) and (d) show the predicted quality scores in vehicle re-
identification, video text spotting, license plate recognition and person re-
identification scenario, respectively.

###  4.4. Evaluation on Vehicle Re-identification

Dataset. We choose VehicleReId dataset (Zapletal and Herout, 2016) for our
evaluation on vehicle re-identification task. Half of the dataset is used as
training set while the rest as testing set. We adopt the model used in (Liu et
al., 2018) as pre-trained recognition model, and the outputs before final
classification layer are used as the extracted features.

Dataset | VehicleReId | IC13 | CCPD
---|---|---|---
Metric | $SRC_{a}$ | $SRC_{e}$ | $SRC_{e}$ | $QSHR$ | $SRC_{a}$ | $SRC_{e}$
PCW (Cheng et al., 2019) | 0.736 | 0.641 | 0.517 | 0.745 | 0.565 | 0.693
QAN (Liu et al., 2017) | 0.783 | 0.675 | 0.716 | 0.809 | - | -
YORO (Cheng et al., 2019) | 0.765 | 0.709 | 0.733 | 0.817 | - | -
RQA | 0.846 | 0.748 | 0.748 | 0.821 | 0.697 | 0.758
AQA | 0.837 | 0.853 | 0.761 | 0.827 | 0.651 | 0.855

Table 4. Quantitative evaluations of quality metrics on VehicleReId, IC13 and
CCPD referring to human annotations, including comparison with related
existing approaches. $SRC_{a}$ and $SRC_{e}$ are ‘intra-class’ and ‘inter-
class’ SROCC respectively, all the existing approaches are re-implemented with
our backbone.

Quality assessment result. We visualize quality scores in Figure 6 (a). which
shows that vehicle images of different qualities from different classes can be
well distinguished. For quantitative evaluations on quality scores, we
randomly select 48 groups of images from the testing set (each group contains
3 different quality images), in which 24 groups contain images from the same
class while the rest contain images from different classes. Then we equally
divide them into 6 subsets (each subset contains 4 groups of intra-class
images and 4 groups of inter-class images). Each subset is annotated by 1
person to decide the quality ranking in intra-class and inter-class, which is
regarded as ground truth. Finally, we compute average SROCC on these subsets,
shown in Table 4, $SRC_{a}$ and $SRC_{e}$ both indicate that our method
outperforms all the existing strategies (Liu et al., 2017; Cheng et al., 2019)
and achieves more consistent quality assessment with human perception,
especially the $SRC_{e}$ of AQA is significantly higher (0.15) than state-of-
art strategies.

Dataset | VehicleReId | iLIDS-VID
---|---|---
Metric | R1 | R5 | R10 | R1 | R5 | R10
BASE | 75.1% | 92.9% | 95.8% | 69.3% | 90.7% | 94.0%
QAN (Liu et al., 2017) | 75.5% | 92.1% | 95.8% | 71.2% | 89.6% | 94.0%
YORO (Cheng et al., 2019) | 75.8% | 92.5% | 95.8% | 71.5% | 90.2% | 94.0%
BASE+AQA | 77.1% | 92.9% | 95.8% | 73.3% | 90.7% | 94.0%

Table 5. The cumulative matching characteristics on VehicleReId and iLIDS-VID.
‘R1’ denotes rank-1 result.

Performance improvement. Table 5 shows the performance gain. Following (Gao
and Nevatia, 2018), we use temporal average pooled feature of each vehicle
sequence as its representation, and report its results as baseline (denoted as
‘BASE’). Then we discard images with scores lower than the learned threshold
(i.e. the learnable quality anchor $th$ (0.839) in Equation 6) in each
sequence to obtain its new feature representation, and report the results as
‘BASE+AQA’. By filtering out low-quality images, ‘BASE+AQA’ achieves 2% gain
at rank-1 (R1).

###  4.5. Evaluation on Video Text Spotting

Dataset. The ICDAR 2013 ‘Text in Video’ dataset (_abbr._ IC13) (Karatzas et
al., 2013) is used for our evaluation on video text spotting. IC13 provides
coarse quality annotations, i.e. each image is labeled with ‘HIGH’, ‘MODERATE’
or ‘LOW’ quality level. The model used in (Cheng et al., 2017) is treated as
our pre-trained recognition model, and the ‘glimpse’ features in its attention
decoder is used as the extracted features.

Quality assessment result. We first visualize the quality scoring results in
Figure 6 (b), in which the predicted quality scores are also satisfying. For
quantitative evaluations, we calculate the average quality score of ‘HIGH’,
‘MODERATE’ and ‘LOW’ images in each text sequence, denoted by
$(\overline{s_{h}},\overline{s_{m}},\overline{s_{l}})$. Apparently, the ground
truth quality ranking should be
$\overline{s_{h}}>\overline{s_{m}}>\overline{s_{l}}$, then SROCC metric can be
computed by comparing the predicted ranking with ground truth. Besides, we
also adopt ‘QSHR’ used in (Cheng et al., 2019) for evaluating the accuracy of
selecting the highest-quality image in each sequence. The results are shown in
Table 4, compared with PCW (Cheng et al., 2019) (using recognition confidence
as quality score) , QAN (Liu et al., 2017) and YORO (Cheng et al., 2019), our
method outperforms them by at least 1%.

Additionally, we separately compute the average score of all the ‘HIGH’,
‘MODERATE’ and ‘LOW’ images on the entire testing set, which are 0.661, 0.606
and 0.394. Notice that the estimated average score difference (0.055) between
‘HIGH’ and ‘MODERATE’ is far less than the difference (0.212) between ‘LOW’
and ‘MODERATE’. Thus we randomly selected 15 high-quality image pairs (1
‘HIGH’ image and 1 ‘MODERATE’ image in each pair) and 15 low-quality image
pairs (1 ‘MODERATE’ image and 1 ‘LOW’ image in each pair), then ask 2 persons
to annotate the quality ranking in each pair. Consequently, their decisions
are consistent on all the low-quality image pairs, but inconsistent on 4 high-
quality image pairs, which refects that the quality difference between ‘HIGH’
and ‘MODERATE’ images is harder to distinguish compared to that between
‘MODERATE’ and ‘LOW’ images.

Dataset | IC13 | CCPD
---|---|---
Metric | SRA | SRA
BASE | 60.72% | 82.30%
QAN (Liu et al., 2017) | 60.99% | -
YORO (Cheng et al., 2019) | 61.45% | -
BASE+AQA | 62.99% | 88.61%

Table 6. The sequence-level recognition accuracy (SRA) improvements on IC13
and CCPD.

Performance improvement. Table 6 shows the recognition accuracy gain by
adopting our quality scores. The baseline (denote by ‘BASE’) selects the most
frequently occurred recognition results as the final result for each sequence,
which is similar to majority voting (Wang et al., 2017). While equipped with
our method (denoted by ‘BASE+AQA’), we discard images with scores lower than
the learned threshold (0.711) by AQA, then proceed the voting strategy to get
final recognition results. We find that our quality assessment can help
recognition model improve accuracy by 2.27%, largely surpassing the 0.73% of
YORO (Cheng et al., 2019).

###  4.6. Evaluation on License Plate Recognition

Dataset. The large-scale license plate dataset ‘CCPD’ (Xu et al., 2018) is
selected for quality assessment evaluation. It contains over 300 thousands of
plates, in which many images are polluted by various noises and detailed
annotations of brightness and blurriness are available. Here, 66% images are
randomly selected as our training set and the rest as testing set. The
recognition model architecture is the same as that on IC13, and pre-trained on
CCPD.

Quality assessment result. We visualize the generated quality scores in Figure
6 (c), and the predictions are also satisfying. For quantifying the quality
assessment performance, we construct a sub test set (containing 508 images) by
selecting images with various brightness and blurriness. Then we generate
their ground truth of quality scores by two steps: 1) since all plates follow
the same character layout (e.g. the position of each character in a standard
license plate is fixed), we synthesize the standard plate images with no
interference as their standard templates; 2) following the same strategy on
MNIST, cosine similarity between the feature of testing image and its standard
template is calculated as the ground truth, which is further normalized to the
range [0,1]. SROCC on this sub set are shown in Table 4, our method
outperforms the naive strategy PCW (Cheng et al., 2019) by a large margin
(0.13/0.16 on $SRC_{a}$/$SRC_{e}$), and the $SRC_{e}$ of AQA is significantly
higher than that of RQA, which indicates AQA can produce much more reliable
absolute quality scores in inter-class. We haven’t re-implement other existing
strategies (QAN (Liu et al., 2017) and YORO (Cheng et al., 2019)) on this
dataset since they all require video frames as input, which can’t be satisfied
in this scenario.

Furthermore, we divide score range $[0,1]$ into 10 equal intervals $[0,0.1]$,
$...$, $[0.9,1]$, then generate 10 subsets according to the ground truth score
of each image. We compute the predicted average score of each subset, and
analyze its distribution in Figure 4(b), which implies that our predicted
scores remain consistent with ground truth whether images are polluted by blur
or brightness.

Performance improvement. We select 729 groups of license plates from the total
test set for evaluating the recognition performance gain, each group contains
at least 3 images, and experiment setting is the same as that on IC13. The
performance improvement is illustrated in Table 6, which shows the recognition
accuracy is significantly improved by 6.3% by discarding low-quality images.

###  4.7. Evaluation on Person Re-identification

Dataset. We choose iLIDS-VID dataset (Wang et al., 2014) for evaluations on
person re-identification, because it contains more diversified quality images
compared with other person re-identification datasets. We obtain the pre-train
model by following (Gao and Nevatia, 2018), then the outputs before the final
classification layer are used as the extracted features.

Quality assessment result. Due to lack of ground truth quality scores, we only
visualize and observe whether predicted scores are consistent with human
perception in this scenario, shown in Figure 6 (d). As a whole, predicted
scores can reasonably express the absolute quality relationships among
different class person images.

Performance improvement. We further evaluate the power of quality assessment
on improving person re-identification performance. As shown in Table 5, ‘BASE’
and ‘BASE+AQA’ share the same evaluation settings as that on vehicle re-
identification, our method significantly improves the top-1 matching (R1) by
4%, outperforming the improvements brought by existing strategies (Liu et al.,
2017; Cheng et al., 2019), which proves the advantage of our method compared
with state-of-art.

##  5\. Conclusion

In this paper, we define the new problem of _object quality assessment_ and
propose an object quality assessment approach named Object-QA to assess high-
reliable quality scores, which can help improve the performance and time
efficiency of various recognition applications. Inspired by human cognition,
Object-QA is implemented by two progressive modules: RQA for obtaining intra-
class-level quality scores and AQA for generating inter-class-level quality
scores. Extensive experiments on four real-life tasks demonstrate that Object-
QA can provide high-reliable quality scores which not only accords with human
cognition and also improves recognition performance. In future, we hope
quantitative benchmarks can be established and we’ll further explore the
object quality assessment in more diversified scenarios.

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