# Inequalities for the polar derivative of a polynomial

N. A. Rather and Suhail Gulzar  Department of Mathematics
University of Kashmir
Srinagar, Hazratbal 190006
India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $P(z)$ be a polynomial of degree $n$ and for any real or complex number
$\alpha,$ let $D_{\alpha}P(z)=nP(z)+(\alpha-z)P^{\prime}(z)$ denote the polar
derivative with respect to $\alpha.$ In this paper, we obtain generalizations
of some inequalities for the polar derivative of a polynomial.

###### Key words and phrases:

Polynomials; Inequalities in the complex domain; Polar derivative; Bernstein’s
inequality.

###### 2000 Mathematics Subject Classification:

30A10, 30C10, 30E10.

##  1\. Introduction and statement of results

Let $P(z)$ be a polynomial of degree $n,$ then

(1.1) $\underset{\left|z\right|=1}{Max}\left|P^{\prime}(z)\right|\leq
n\underset{\left|z\right|=1}{Max}\left|P(z)\right|.$

Inequality (1.1) is an immediate consequence of S. Bernstein’s Theorem on the
derivative of a trigonometric polynomial (for reference, see [9, p.531], [10,
p.508] or [11]) and the result is best possible with equality holding for the
polynomial $P(z)=az^{n},$ $a\neq 0.$
If we restrict ourselves to the class of polynomials having no zero in
$|z|<1$, then inequality (1.1) can be replaced by

(1.2)
$\underset{\left|z\right|=1}{Max}\left|P^{\prime}(z)\right|\leq\frac{n}{2}\underset{\left|z\right|=1}{Max}\left|P(z)\right|.$

Inequality (1.2) was conjectured by Erdös and later verified by Lax [6]. The
result is sharp and equality holds for $P(z)=az^{n}+b,$ $|a|=|b|.$
For the class polynomials having all zeros in $|z|\leq 1,$ it was proved by
Turán [12] that

(1.3)
$\underset{\left|z\right|=1}{Max}\left|P^{\prime}(z)\right|\geq\frac{n}{2}\underset{\left|z\right|=1}{Max}\left|P(z)\right|.$

The inequality (1.3) is best possible and become equality for polynomial
$P(z)=(z+1)^{n}.$ As an extension of (1.2) and (1.3) Malik [7] proved that if
$P(z)\neq 0$ in $|z|<k$ where $k\geq 1,$ then

(1.4)
$\underset{\left|z\right|=1}{Max}\left|P^{\prime}(z)\right|\leq\frac{n}{1+k}\underset{\left|z\right|=1}{Max}\left|P(z)\right|,$

where as if $P(z)$ has all its zeros in $|z|\leq k$ where $k\leq 1,$ then

(1.5)
$\underset{\left|z\right|=1}{Max}\left|P^{\prime}(z)\right|\geq\frac{n}{1+k}\underset{\left|z\right|=1}{Max}\left|P(z)\right|.$

Let $D_{\alpha}P(z)$ denotes the polar derivative of the polynomial $P(z)$ of
degree $n$ with respect to the point $\alpha,$ then

$D_{\alpha}P(z)=nP(z)+(\alpha-z)P^{\prime}(z).$

The polynomial $D_{\alpha}P(z)$ is a polynomial of degree at most $n-1$ and it
generalizes the ordinary derivative in the sense that

$\underset{\alpha\rightarrow\infty}{Lim}\left[\dfrac{D_{\alpha}P(z)}{\alpha}\right]=P^{\prime}(z).$

Now corresponding to a given $n^{th}$ degree polynomial $P(z),$ we construct a
sequence of polar derivatives

$D_{\alpha_{1}}P(z)=nP(z)+(\alpha_{1}-z)P^{\prime}(z)=P_{1}(z)$ $\displaystyle
D_{\alpha_{s}}D_{\alpha_{s-1}}\cdots D_{\alpha_{2}}D_{\alpha_{1}}P(z)=(n-s+1)$
$\displaystyle\left\\{D_{\alpha_{s-1}}\cdots
D_{\alpha_{2}}D_{\alpha_{1}}P(z)\right\\}$
$\displaystyle+(\alpha_{s}-z)\left\\{D_{\alpha_{s-1}}\cdots
D_{\alpha_{2}}D_{\alpha_{1}}P(z)\right\\}^{\prime}.$

The points $\alpha_{1},\alpha_{2},\cdots,\alpha_{s},$ $s=1,2,\cdots,n,$ may be
equal or unequal complex numbers. The $s^{th}$ polar derivative
$D_{\alpha_{s}}D_{\alpha_{s-1}}\cdots D_{\alpha_{2}}D_{\alpha_{1}}P(z)$ of
$P(z)$ is a polynomial of degree at most $n-s.$ For

$P_{j}(z)=D_{\alpha_{j}}D_{\alpha_{j-1}}\cdots
D_{\alpha_{2}}D_{\alpha_{1}}P(z),$

we have

$\displaystyle P_{j}(z)$
$\displaystyle=(n-j+1)P_{j-1}(z)+(\alpha_{j}-z)P^{\prime}_{j-1}(z),\,\,\,\,\,\,j=1,2,\cdots,s,$
$\displaystyle P_{0}(z)$ $\displaystyle=P(z).$

A. Aziz [2] extended the inequality (1.2) to the $s^{th}$ polar derivative by
proving if $P(z)$ is a polynomial of degree $n$ not vanishing in $|z|<1,$ then
for $|z|\geq 1$

(1.6) $\displaystyle|D_{\alpha_{s}}\cdots$ $\displaystyle
D_{\alpha_{1}}P(z)|\leq\dfrac{n(n-1)\cdots(n-s+1)}{2}\left\\{|\alpha_{1}\cdots\alpha_{s}z^{n-s}|+1\right\\}\underset{|z|=1}{Max}|P(z)|,$

where $|\alpha_{j}|\geq 1,$ for $j=1,2,\cdots,s.$ The result is best possible
and equality holds for the polynomial $P(z)=(z^{n}+1)/2.$

As a refinement of inequality (1.6), Aziz and Wali Mohammad [5] proved if
$P(z)$ is a polynomial of degree $n$ not vanishing in $|z|<1,$ then for
$|z|\geq 1$

$\displaystyle|D_{\alpha_{s}}\cdots
D_{\alpha_{1}}P(z)|\leq\dfrac{n(n-1)\cdots(n-s+1)}{2}$
$\displaystyle\Big{\\{}\left(|\alpha_{1}\cdots\alpha_{s}z^{n-s}|+1\right)\underset{|z|=1}{Max}|P(z)|,$
(1.7)
$\displaystyle-\left(|\alpha_{1}\cdots\alpha_{s}z^{n-s}|-1\right)\Big{\\}}\underset{|z|=1}{Min}|P(z)|$

where $|\alpha_{j}|\geq 1,$ for $j=1,2,\cdots,s.$ The result is best possible
and equality holds for the polynomial $P(z)=(z^{n}+1)/2.$

In this paper, we shall obtain several inequalities concerning the polar
derivative of a polynomial and thereby obtain compact generalizations of
inequalities (1.6) and (Inequalities for the polar derivative of a
polynomial).

We first prove following result from which certain interesting results follow
as special cases.

######  Theorem 1.1.

If $F(z)$ be a polynomial of degree $n$ having all its zeros in the disk
$\left|z\right|\leq k$ where $k\leq 1$, and $P(z)$ is a polynomial of degree n
such that

$\left|P(z)\right|\leq\left|F(z)\right|\,\,\,for\,\,\,|z|=k,$

then for $\alpha_{j},\beta\in\mathbb{C}$ with $\left|\alpha_{j}\right|\geq
k,\left|\beta\right|\leq 1$, $j=1,2,\cdots,s$ and $|z|\geq 1$,

(1.8)
$\displaystyle\left|z^{s}P_{s}(z)+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)\right|\leq\left|z^{s}F_{s}(z)+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}F(z)\right|,$

where

(1.9) $\displaystyle
n_{s}=n(n-1)\cdots(n-s+1)\,\,\&\,\,\Lambda_{s}=(|\alpha_{1}|-k)(|\alpha_{2}|-k)\cdots(|\alpha_{s}|-k).$

If we choose $F(z)=z^{n}M/k^{n}$, where $M=Max_{|z|=k}\left|P(z)\right|$ in
Theorem 1.1, we get the following result.

######  Corollary 1.2.

If $P(z)$ be a polynomial of degree $n$, then for
$\alpha_{j},\beta\in\mathbb{C}$ with $\left|\alpha_{j}\right|\geq
k,\left|\beta\right|\leq 1$, $j=1,2,\cdots,s$ and $|z|\geq 1$,

(1.10)
$\displaystyle\left|z^{s}P_{s}(z)+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)\right|\leq\dfrac{n_{s}|z|^{n}}{k^{n}}\left|\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\beta\dfrac{\Lambda_{s}}{(1+k)^{s}}\right|\underset{\left|z\right|=k}{Max}\left|P(z)\right|,$

where $n_{s}$ and $\Lambda_{s}$ are given by (1.9).

If $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{s}=\alpha,$ then dividing both sides
of (1.10) by $|\alpha|^{s}$ and letting $|\alpha|\rightarrow\infty,$ we obtain
the following result.

######  Corollary 1.3.

If $P(z)$ be a polynomial of degree $n$ then for $\beta\in\mathbb{C}$ with
$\left|\beta\right|\leq 1$, and $|z|\geq 1$,

(1.11)
$\displaystyle\left|z^{s}P^{(s)}(z)+\beta\dfrac{n_{s}}{(1+k)^{s}}P(z)\right|\leq\dfrac{n_{s}|z|^{n}}{k^{n}}\left|1+\dfrac{\beta}{(1+k)^{s}}\right|\underset{\left|z\right|=k}{Max}\left|P(z)\right|,$

where $n_{s}$ is given by (1.9).

Again, if we take $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{s}=\alpha,$ then
divide both sides of (1.8) by $|\alpha|^{s}$ and letting
$|\alpha|\rightarrow\infty,$ we obtain the following result.

######  Corollary 1.4.

If $F(z)$ be a polynomial of degree $n$ having all its zeros in the disk
$\left|z\right|\leq k$ where $k\leq 1$, and $P(z)$ is a polynomial of degree
$n$ such that

$\left|P(z)\right|\leq\left|F(z)\right|\,\,\,for\,\,\,|z|=k,$

then for $\beta\in\mathbb{C}$ with $\left|\beta\right|\leq 1$, and $|z|\geq
1$,

(1.12)
$\displaystyle\left|z^{s}P^{(s)}(z)+\beta\dfrac{n_{s}}{(1+k)^{s}}P(z)\right|\leq\left|z^{s}F^{(s)}(z)+\beta\dfrac{n_{s}}{(1+k)^{s}}F(z)\right|,$

where $n_{s}$ is given by (1.9).

If we choose $P(z)=mz^{n}/k,$ where $m=Min_{|z|=k}|F(z)|,$ in Theorem 1.1 we
get the following result.

######  Corollary 1.5.

If $F(z)$ be a polynomial of degree $n$ having all its zeros in the disk
$\left|z\right|\leq k$ where $k\leq 1$, then for
$\alpha_{j},\beta\in\mathbb{C}$ with $\left|\alpha_{j}\right|\geq
k,\left|\beta\right|\leq 1$, where $j=1,2,\cdots,s$ and $|z|\geq 1$,

(1.13)
$\displaystyle\left|z^{s}F_{s}(z)+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}F(z)\right|\geq\dfrac{n_{s}|z|^{n}}{k^{n}}\left|\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\right|\underset{|z|=k}{Min}|F(z)|,$

where $n_{s}$ and $\Lambda_{s}$ are given by (1.9).

######  Remark 1.6.

For $\beta=0$ and $k=1,$ we get the result due to Aziz and Wali Mohammad [5,
Theorem 1].

Again, if we take $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{s}=\alpha,$ then
divide both sides of (1.13) by $|\alpha|^{s}$ and letting
$|\alpha|\rightarrow\infty,$ we obtain the following result.

######  Corollary 1.7.

If $F(z)$ be a polynomial of degree $n$ having all its zeros in the disk
$\left|z\right|\leq k$ where $k\leq 1$, then for $\beta\in\mathbb{C}$ with
$\left|\beta\right|\leq 1$, and $|z|\geq 1$,

(1.14) $\displaystyle\left|z^{s}F^{(s)}(z)+\dfrac{\beta
n_{s}}{(1+k)^{s}}F(z)\right|\geq\dfrac{n_{s}|z|^{n}}{k^{n}}\left|1+\dfrac{\beta}{(1+k)^{s}}\right|\underset{|z|=k}{Min}|F(z)|,$

where $n_{s}$ is given by (1.9).

For $s=1,$ and $\alpha_{1}=\alpha$ in Theorem 1.1 we get the following result:

######  Corollary 1.8.

If $F(z)$ be a polynomial of degree $n$ having all its zeros in the disk
$\left|z\right|\leq k$ where $k\leq 1$, and $P(z)$ is a polynomial n such that

$\left|P(z)\right|\leq\left|F(z)\right|\,\,\,for\,\,\,|z|=k,$

then for $\alpha,\beta\in\mathbb{C}$ with $\left|\alpha\right|\geq
k,\left|\beta\right|\leq 1$, and $|z|\geq 1$,

$\left|zD_{\alpha}P(z)+n\beta\left(\dfrac{|\alpha|-k}{k+1}\right)P(z)\right|\leq\left|zD_{\alpha}F(z)+n\beta\left(\dfrac{|\alpha|-k}{k+1}\right)F(z)\right|.$

######  Theorem 1.9.

If $P(z)$ is a polynomial of degree $n,$ which does not vanish in the disk
$|z|<k$ where $k\leq 1,$ then for
$\alpha_{1},\alpha_{2},\cdots,\alpha_{s},\beta\in\mathbb{C}$ with
$|\alpha_{1}|\geq k,|\alpha_{2}|\geq k,\cdots,|\alpha_{s}|\geq k,|\beta|\leq
1$ and $|z|\geq 1$ then

$\displaystyle\Bigg{|}z^{s}P_{s}(z)$
$\displaystyle+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)\Bigg{|}$ (1.15)
$\displaystyle\leq\dfrac{n_{s}}{2}\left\\{\dfrac{|z^{n}|}{k^{n}}\Bigg{|}\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}+\Bigg{|}z^{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}\right\\}\underset{|z|=k}{Max}|P(z)|,$

where $n_{s}$ and $\Lambda_{s}$ are given by (1.9).

######  Remark 1.10.

If we take $\beta=0$ and $k=1,$ we get inequality (1.6).

We next prove the following refinement of Theorem 1.9.

######  Theorem 1.11.

If $P(z)$ is a polynomial of degree $n$ and does not vanish in the disk
$|z|<k$ where $k\leq 1,$ then for
$\alpha_{1},\alpha_{2},\cdots,\alpha_{s},\beta\in\mathbb{C}$ with
$|\alpha_{1}|\geq k,|\alpha_{2}|\geq k,\cdots,|\alpha_{s}|\geq k,|\beta|\leq
1$ and $|z|\geq 1,$ we have

$\displaystyle\Bigg{|}z^{s}$ $\displaystyle
P_{s}(z)+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)\Bigg{|}$
$\displaystyle\leq$
$\displaystyle\dfrac{n_{s}}{2}\Bigg{[}\left\\{\dfrac{|z|^{n}}{k^{n}}\Bigg{|}\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\beta\dfrac{\Lambda_{s}}{(1+k)^{s}}\Bigg{|}+\Bigg{|}z^{s}+\beta\dfrac{\Lambda_{s}}{(1+k)^{s}}\Bigg{|}\right\\}\underset{|z|=k}{Max}|P(z)|$
(1.16)
$\displaystyle-\left\\{\dfrac{|z|^{n}}{k^{n}}\Bigg{|}\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\beta\dfrac{\Lambda_{s}}{(1+k)^{s}}\Bigg{|}-\Bigg{|}z^{s}+\beta\dfrac{\Lambda_{s}}{(1+k)^{s}}\Bigg{|}\right\\}\underset{|z|=k}{Min}|P(z)|\Bigg{]},$

where $n_{s}$ and $\Lambda_{s}$ are given by (1.9).

If we take $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{s}=\alpha,$ then divide both
sides of (1.8) by $|\alpha|^{s}$ and letting $|\alpha|\rightarrow\infty,$ we
obtain the following result.

######  Corollary 1.12.

If $P(z)$ is a polynomial of degree $n,$ which does not vanish in the disk
$|z|<k$ where $k\leq 1,$ then for $\beta\in\mathbb{C}$ with $|\beta|\leq 1$
and $|z|\geq 1$ then

$\displaystyle\Bigg{|}z^{s}P^{(s)}(z)+\beta\dfrac{n_{s}}{(1+k)^{s}}P(z)\Bigg{|}\leq$
$\displaystyle\dfrac{n_{s}}{2}\Bigg{[}\left\\{\dfrac{|z|^{n}}{k^{n}}\Bigg{|}1+\dfrac{\beta}{(1+k)^{s}}\Bigg{|}+\Bigg{|}\dfrac{\beta}{(1+k)^{s}}\Bigg{|}\right\\}\underset{|z|=k}{Max}|P(z)|$
(1.17)
$\displaystyle-\left\\{\dfrac{|z|^{n}}{k^{n}}\Bigg{|}1+\dfrac{\beta}{(1+k)^{s}}\Bigg{|}-\Bigg{|}\dfrac{\beta}{(1+k)^{s}}\Bigg{|}\right\\}\underset{|z|=k}{Min}|P(z)|\Bigg{]},$

where $n_{s}$ is given by (1.9).

For $s=1$ in Corollary 1.12, we get the following result.

######  Corollary 1.13.

If $P(z)$ is a polynomial of degree $n,$ which does not vanish in the disk
$|z|<k$ where $k\leq 1,$ then for $\beta\in\mathbb{C}$ with $|\alpha|\geq
k,|\beta|\leq 1$ and $|z|\geq 1$ then

$\displaystyle\Bigg{|}zP^{\prime}(z)+\dfrac{n\beta}{1+k}P(z)\Bigg{|}\leq$
$\displaystyle\dfrac{n}{2}\Bigg{[}\left\\{\dfrac{|z|^{n}}{k^{n}}\Bigg{|}1+\dfrac{\beta}{1+k}\Bigg{|}+\Bigg{|}\dfrac{\beta}{1+k}\Bigg{|}\right\\}\underset{|z|=k}{Max}|P(z)|$
(1.18)
$\displaystyle-\left\\{\dfrac{|z|^{n}}{k^{n}}\Bigg{|}1+\dfrac{\beta}{1+k}\Bigg{|}-\Bigg{|}\dfrac{\beta}{1+k}\Bigg{|}\right\\}\underset{|z|=k}{Min}|P(z)|\Bigg{]},$

For $\beta=0,$ Theorem 1.11 reduces to the following result.

######  Corollary 1.14.

If $P(z)$ is a polynomial of degree $n,$ which does not vanish in the disk
$|z|<k$ where $k\leq 1,$ then for
$\alpha_{1},\alpha_{2},\cdots,\alpha_{s},\beta\in\mathbb{C}$ with
$|\alpha_{1}|\geq k,|\alpha_{2}|\geq k,\cdots,|\alpha_{s}|\geq k,|\beta|\leq
1$ and $|z|\geq 1$ then

$\displaystyle\big{|}P_{s}(z)\big{|}\leq\dfrac{n_{s}}{2}\Bigg{[}$
$\displaystyle\left\\{\dfrac{|z|^{n-s}}{k^{n}}|\alpha_{1}\alpha_{2}\cdots\alpha_{s}|+1\right\\}\underset{|z|=k}{Max}|P(z)|$
(1.19)
$\displaystyle-\left\\{\dfrac{|z|^{n-s}}{k^{n}}|\alpha_{1}\alpha_{2}\cdots\alpha_{s}|-1\right\\}\underset{|z|=k}{Min}|P(z)|\Bigg{]},$

where $n_{s}$ is given by (1.9).

######  Remark 1.15.

For $k=1,$ inequality (1.14) reduces to (Inequalities for the polar derivative
of a polynomial).

If we take $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{s}=\alpha,$ then divide both
sides of (1.8) by $|\alpha|^{s}$ and letting $|\alpha|\rightarrow\infty,$ we
obtain the following result.

######  Corollary 1.16.

If $P(z)$ is a polynomial of degree $n,$ which does not vanish in the disk
$|z|<k$ where $k\leq 1,$ then for
$\alpha_{1},\alpha_{2},\cdots,\alpha_{s},\beta\in\mathbb{C}$ with
$|\alpha_{1}|\geq k,|\alpha_{2}|\geq k,\cdots,|\alpha_{s}|\geq k,|\beta|\leq
1$ and $|z|\geq 1$ then

(1.20)
$\displaystyle\big{|}P^{(s)}(z)\big{|}\leq\dfrac{n(n-1)\cdots(n-t+1)|z|^{n-s}}{2k^{n}}\Bigg{[}\underset{|z|=k}{Max}|P(z)|-\underset{|z|=k}{Min}|P(z)|\Bigg{]}.$

If $s=1$ and $\alpha_{1}=\alpha$ then inequality (1.11) reduces to the
following result.

######  Corollary 1.17.

If $P(z)$ is a polynomial of degree $n,$ which does not vanish in the disk
$|z|<k$ where $k\leq 1,$ then for $\alpha,\beta\in\mathbb{C}$ with
$|\alpha|\geq k,|\beta|\leq 1$ and $|z|\geq 1$ then

$\displaystyle\Bigg{|}z$ $\displaystyle
D_{\alpha}P(z)+n\beta\left(\dfrac{|\alpha|-k}{1+k}\right)P(z)\Bigg{|}$
$\displaystyle\leq\dfrac{n}{2}\Bigg{[}\left\\{\dfrac{1}{k^{n}}\Bigg{|}\alpha+\beta\left(\dfrac{|\alpha|-k}{1+k}\right)\Bigg{|}+\Bigg{|}z^{s}+\beta\left(\dfrac{|\alpha|-k}{1+k}\right)\Bigg{|}\right\\}\underset{|z|=k}{Max}|P(z)|$
(1.21)
$\displaystyle-\left\\{\dfrac{1}{k^{n}}\Bigg{|}\alpha+\beta\left(\dfrac{|\alpha|-k}{1+k}\right)\Bigg{|}-\Bigg{|}z^{s}+\beta\left(\dfrac{|\alpha|-k}{1+k}\right)\Bigg{|}\right\\}\underset{|z|=k}{Min}|P(z)|\Bigg{]}.$

##  2\. Lemmas

For the proof of Theorems, we need the following Lemmas. The first Lemma
follows by repeated application of Laguerre’s theorem [1] or[8, p. 52].

######  Lemma 2.1.

If all the zeros of $nth$ degree polynomial lie in circular region $C$ and if
none of the points $\alpha_{1},\alpha_{2},\cdots,\alpha_{s}$ lie in circular
region $C$, then each of the polar derivatives

(2.1) $D_{\alpha_{s}}D_{\alpha_{s-1}}\cdots
D_{\alpha_{1}}P(z)=P_{s}(z),\,\,\,\,\,\,\,\,\,s=1,2,\cdots,n-1$

has all its zeros in $C.$

The next Lemma is due to Aziz and Rather [3].

######  Lemma 2.2.

If $P(z)$ is a polynomial of degree $n,$ having all zeros in the closed disk
$|z|\leq k,$ $k\leq 1,$ then for every real or complex number $\alpha$ with
$|\alpha|\geq k$ and $|z|=1$, we have

(2.2) $|D_{\alpha}P(z)|\geq n\left(\dfrac{|\alpha|-k}{1+k}\right)|P(z)|.$

######  Lemma 2.3.

If $P(z)=\sum_{j=0}^{n}a_{j}z^{j}$ is a polynomial of degree $n$ having all
its zeros in $|z|\leq k,$ $k\leq 1$ then

(2.3) $\dfrac{1}{n}\left|\dfrac{a_{n-1}}{a_{n}}\right|\leq k.$

The above lemma follows by taking $\mu=1$ in the result due to Aziz and Rather
[4].

######  Lemma 2.4.

If $P(z)$ be a polynomial of degree $n$ having all zeros in the disk $|z|\leq
k$ where $k\leq 1,$ then for
$\alpha_{1},\alpha_{2},\cdots,\alpha_{s}\in\mathbb{C}$ with $|\alpha_{1}|\geq
k,|\alpha_{2}|\geq k,\cdots,|\alpha_{s}|\geq k,$ $(1\leq s<n),$ and $|z|=1$

(2.4) $\displaystyle|P_{s}(z)|\geq\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}|P(z)|,$

where $n_{s}$ and $\Lambda_{s}$ are defined in (1.9).

###### Proof.

The result is trivial if $|\alpha_{j}|=k$ for at least one $j$ where
$j=1,2,\cdots,t.$ Therefore, we assume $|\alpha_{j}|>k$ for all
$j=1,2,\cdots,t.$ We shall prove Lemma by principle of mathematical induction.
For $t=1$ the result is follows by Lemma 2.2.
We assume that the result is true for $t=q,$ which means that for $|z|=1,$ we
have

(2.5) $|P_{q}(z)|\geq\dfrac{n_{q}A_{\alpha_{q}}}{(1+k)^{q}}|P(z)|\,\,\,q\geq
1,$

and we will prove that the Lemma is true for $t=q+1$ also.
Since
$D_{\alpha_{1}}P(z)=(na_{n}\alpha_{1}+a_{n-1})z^{n-1}+\cdots+(na_{0}+\alpha_{1}a_{1})$
and $|\alpha_{1}|>k,$ $D_{\alpha_{1}}P(z)$ is a polynomial of degree $n-1.$ If
this is not true, then

$na_{n}\alpha_{1}+a_{n-1}=0,$

which implies

$|\alpha_{1}|=\dfrac{1}{n}\left|\dfrac{a_{n-1}}{a_{n}}\right|.$

By Lemma 2.3, we have

$|\alpha_{1}|=\dfrac{1}{n}\left|\dfrac{a_{n-1}}{a_{n}}\right|\leq k.$

But this is the contradiction to the fact $|\alpha|>k.$ Hence,
$D_{\alpha_{1}}P(z)$ is a polynomial of degree $n-1$ and by Lemma 2.1,
$D_{\alpha_{1}}P(z)$ has all its zeros in $|z|\leq k.$ By the similar argument
as before, $D_{\alpha_{2}}D_{\alpha_{1}}P(z)$ must be a polynomial of degree
$n-2$ for $|\alpha_{1}|>k,$ $|\alpha_{2}|>k$ and all its zeros in $|z|\leq k.$
Continuing in this way, we conclude $D_{\alpha_{q}}D_{\alpha_{q-1}}\cdots
D_{\alpha_{1}}P(z)=P_{q}(z)$ is a polynomial of degree $n-q$ for all
$|\alpha_{j}|>k,$ $j=1,2,\dots,q$ and has all zeros in $|z|\leq k.$ Applying
Lemma 2.2 to $P_{q}(z),$ we get for $|\alpha_{q+1}|>k,$

(2.6)
$|P_{q+1}(z)|=|D_{\alpha_{q+1}}P_{q}(z)|\geq\dfrac{(n-q)(|\alpha_{q+1}|-k)}{1+k}|P_{q}(z)|\,\,\,\textnormal{for}\,\,\,\,|z|=1.$

Inequality (2.6) in conjunction with (2.5) gives for $|z|=1,$

(2.7) $|P_{q+1}(z)|\geq\dfrac{n_{q+1}A_{\alpha_{q+1}}}{(1+k)^{q+1}}|P(z)|,$

where $n_{q+1}=n(n-1)\cdots(n-q)$ and
$A_{\alpha_{q+1}}=(|\alpha_{1}|-k)(|\alpha_{2}|-k)\cdots(|\alpha_{q+1}|-k).$
This shows that the result is true for $s=q+1$ also. This completes the proof
of Lemma 2.4. ∎

######  Lemma 2.5.

If $P(z)$ be a polynomial of degree $n,$ then for
$\alpha_{1},\alpha_{2},\cdots,\alpha_{s}\in\mathbb{C}$ with $|\alpha_{1}|\geq
k,|\alpha_{2}|\geq k,\cdots,|\alpha_{s}|\geq k,$ $(1\leq t<n),$ $|\beta|\leq
1$ and $|z|\geq 1,$

$\displaystyle\Bigg{|}z^{s}$ $\displaystyle
P_{s}(z)+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)\Bigg{|}+k^{n}\left|z^{s}Q_{s}(z/k^{2})+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}Q(z/k^{2})\right|$
(2.8) $\displaystyle\leq$ $\displaystyle
n_{s}\left\\{\dfrac{|z^{n}|}{k^{n}}\Bigg{|}\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}+\Bigg{|}z^{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}\right\\}\underset{|z|=k}{Max}|P(z)|,$

where $n_{s}$ and $\Lambda_{s}$ are defined in (1.9) and
$Q(z)=z^{n}\overline{P(1/\overline{z})}.$

###### Proof.

Let $M=\underset{|z|=k}{Max}|P(z)|.$ Therefore, for every $\lambda$ with
$|\lambda|>1,$ $|P(z)|<|\lambda Mz^{n}/k^{n}|$ on $|z|=k.$ By Rouche’s theorem
it follows that all the zeros of $F(z)=P(z)+\lambda Mz^{n}/k^{n}$ lie in
$|z|<k.$ If $G(z)=z^{n}\overline{F(1/\overline{z})}$ then
$|k^{n}G(z/k^{2})|=|F(z)|$ for $|z|=k$ and hence for any $\delta$ with
$|\delta|>1,$ the polynomial $H(z)=k^{n}G(z/k^{2})+\delta F(z)$ has all its
zeros in $|z|<k.$ By applying Lemma 2.4 to $H(z),$ we have for
$\alpha_{1},\alpha_{2},\cdots,\alpha_{s}\in\mathbb{C}$ with
$|\alpha_{1}|>k,|\alpha_{2}|>k,\cdots,|\alpha_{s}|>k,$ $(1\leq t<n),$

$|z^{s}H_{s}(z)|\geq\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}|H(z)|\,\,\,\,\,\textnormal{for}\,\,\,\,|z|=1.$

Therefore, for any $\beta$ with $|\beta|<1$ and $|z|=1,$ we have

$|z^{s}H_{s}(z)|>|\beta|\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}|H(z)|.$

Since $|\alpha_{j}|>k,$ $j=1,2,\cdots,s$ by Lemma 2.1 the polynomial
$z^{s}H_{s}(z)$ has all its zeros in $|z|<1$ and by Rouche’s theorem, the
polynomial

$T(z)=z^{s}H_{s}(z)+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}H(z)$

has all its zeros in $|z|<1.$ Replacing $H(z)$ by $k^{n}G(z/k^{2})+\delta
F(z),$ we conclude that the polynomial

$T(z)=k^{n}\left\\{z^{s}G_{s}(z/k^{2})+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}G(z/k^{2})\right\\}+\delta\left\\{z^{s}F_{s}(z)+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}F(z)\right\\}$

has all its zero in $|z|<1.$ This gives for $|\beta|<1,$ $|\alpha_{j}|\geq k,$
where $j=1,2,\cdots,t$ and $|z|\geq 1.$

(2.9) $k^{n}\left|z^{s}G_{s}(z/k^{2})+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}G(z/k^{2})\right|\leq\left|z^{s}F_{s}(z)+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}F(z)\right|$

If inequality (2.9) is not true, then there is a point $z_{0}$ with
$|z_{0}|\geq 1$ such that

$k^{n}\left|z_{0}^{s}G_{s}(z_{0}/k^{2})+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}G(z_{0}/k^{2})\right|>\left|z_{0}^{s}F_{s}(z_{0})+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}F(z_{0})\right|$

Since all the zeros of $F(z)$ lie in $|z|<k,$ proceeding similarly as in case
of $H(z),$ it follows that the polynomial $z^{s}F_{s}(z)+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}F(z)$ has all its zeros in $|z|<1,$ and hence
$z_{0}^{s}F_{s}(z_{0})+\dfrac{\beta n_{s}\Lambda_{s}}{(1+k)^{s}}F(z_{0})\neq
0.$ Now, choosing

$\delta=-\dfrac{k^{n}\left\\{z_{0}^{s}G_{s}(z_{0}/k^{2})+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}G(z_{0}/k^{2})\right\\}}{z_{0}^{s}F_{s}(z_{0})+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}F(z_{0})}.$

then $\delta$ is a well-defined real or complex number, with $|\delta|>1$ and
$T(z_{0})=0,$ which contradicts the fact that $T(z)$ has all zeros in $|z|<1.$
Thus, (2.9) holds. Now, replacing $F(z)$ by $P(z)+\lambda Mz^{n}/k^{n}$ and
$G(z)$ by $Q(z)+\overline{\lambda}M/k^{n}$ in (2.9), we have for $|z|\geq 1,$

$\displaystyle k^{n}\Bigg{|}z^{s}$ $\displaystyle Q_{s}(z/k^{2})+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}Q(z/k^{2})+\dfrac{\overline{\lambda}n_{s}}{k^{n}}\left\\{z^{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\right\\}M\Bigg{|}$
(2.10) $\displaystyle\leq\Bigg{|}z^{s}P_{s}(z)+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)+\dfrac{\lambda
n_{s}}{k^{n}}\left\\{\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\right\\}Mz^{n}\Bigg{|}.$

Choosing the argument of $\lambda$ in the right hand side of (2) such that

$\displaystyle\Bigg{|}z^{s}$ $\displaystyle P_{s}(z)+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)+\dfrac{\lambda
n_{s}}{k^{n}}\left\\{\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\right\\}Mz^{n}\Bigg{|}$
$\displaystyle=$ $\displaystyle\Bigg{|}\dfrac{\lambda
n_{s}}{k^{n}}\left\\{\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\right\\}Mz^{n}\Bigg{|}-\Bigg{|}z^{s}P_{s}(z)+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)\Bigg{|}.$

which is possible by Corollary 1.2, then for $|z|\geq 1,$ we have

$\displaystyle\Bigg{|}z^{s}$ $\displaystyle P_{s}(z)+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)\Bigg{|}+k^{n}\Bigg{|}z^{s}Q_{s}(z/k^{2})+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}Q(z/k^{2})\Bigg{|}$ (2.11) $\displaystyle\leq$
$\displaystyle|\lambda|n_{s}\Bigg{[}\dfrac{|z^{n}|}{k^{n}}\Bigg{|}\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}+\Bigg{|}z^{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}\Bigg{]}M$

Making $|\lambda|\rightarrow 1$ and using the continuity for $|\beta|=1$ and
$|\alpha_{j}|=k,$ where $j=1,2,\cdots,t,$ in (2), we get the inequality (2.5).
∎

##  3\. Proof of Theorems

######  Proof of Theorem 1.1.

By hypothesis $F(z)$ is a polynomial of degree $n$ having all in zeros in the
closed disk $|z|\leq k$ and $P(z)$ is a polynomial of degree $n$ such that

(3.1) $|P(z)|\leq|F(z)|\,\,\,\,\textrm{for}\,\,\,\,|z|=k,$

therefore, if $F(z)$ has a zero of multiplicity $s$ at $z=ke^{i\theta_{0}}$,
then $P(z)$ has a zero of multiplicity at least $s$ at $z=ke^{i\theta_{0}}$.
If $P(z)/F(z)$ is a constant, then inequality (3.1) is obvious. We now assume
that $P(z)/F(z)$ is not a constant, so that by the maximum modulus principle,
it follows that

$|P(z)|<|F(z)|\,\,\,\textrm{for}\,\,|z|>k\,\,.$

Suppose $F(z)$ has $m$ zeros on $|z|=k$ where $0\leq m<n$, so that we can
write

$F(z)=F_{1}(z)F_{2}(z)$

where $F_{1}(z)$ is a polynomial of degree $m$ whose all zeros lie on $|z|=k$
and $F_{2}(z)$ is a polynomial of degree exactly $n-m$ having all its zeros in
$|z|<k$. This implies with the help of inequality (3.1) that

$P(z)=P_{1}(z)F_{1}(z)$

where $P_{1}(z)$ is a polynomial of degree at most $n-m$. Again, from
inequality (3.1), we have

$|P_{1}(z)|\leq|F_{2}(z)|\,\,\,for\,\,|z|=k\,$

where $F_{2}(z)\neq 0\,\,for\,\,|z|=k$. Therefore for every real or complex
number $\lambda$ with $|\lambda|>1$, a direct application of Rouche’s theorem
shows that the zeros of the polynomial $P_{1}(z)-\lambda F_{2}(z)$ of degree
$n-m\geq 1$ lie in $|z|<k$ hence the polynomial

$G(z)=F_{1}(z)\left(P_{1}(z)-\lambda F_{2}(z)\right)=P(z)-\lambda F(z)$

has all its zeros in $|z|\leq k.$ Therefore for $r>1,$ all the zeros of
$G(rz)$ lie in $|z|\leq k/r<k$ By applying Lemma 2.4 to the polynomial
$G(rz),$ then for $|z|=1,$ we have

$|z^{s}G_{s}(rz)|\geq\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}|G(rz)|.$

Equivalently for $|z|=1,$ we have

(3.2) $|z^{s}P_{s}(rz)-\lambda
z^{s}F_{s}(rz)|\geq\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}|P(rz)-\lambda F(rz)|.$

Therefore, we have for any $\beta$ with $|\beta|<1$ and $|z|=1,$

(3.3) $|z^{s}P_{s}(rz)-\lambda
z^{s}F_{s}(rz)|>\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}|\beta||P(rz)-\lambda
F(rz)|.$

Since $|\alpha_{j}|>k,$ $j=1,2,\cdots,t$ by Rouche’s theorem, the polynomial

$\displaystyle T(rz)$ $\displaystyle=\left\\{z^{s}P_{s}(rz)-\lambda
z^{s}F_{s}(rz)\right\\}+\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}\beta\left\\{P(rz)-\lambda
F(rz)\right\\}$
$\displaystyle=z^{s}P_{s}(rz)+\dfrac{n_{s}\Lambda_{s}\beta}{(1+k)^{s}}P(rz)-\lambda\left\\{z^{s}F_{s}(rz)+\dfrac{n_{s}\Lambda_{s}\beta}{(1+k)^{s}}F(rz)\right\\}$

has all zeros in $|z|<1.$ This implies for $|z|\geq 1$

(3.4)
$|z^{s}P_{s}(rz)+\dfrac{n_{s}\Lambda_{s}\beta}{(1+k)^{s}}P(rz)|\leq|z^{s}F_{s}(rz)+\dfrac{n_{s}\Lambda_{s}\beta}{(1+k)^{s}}F(rz)|$

If it is true, then we have for some $z_{0}$ with $|z_{0}|\geq k,$

(3.5)
$|z_{0}^{s}P_{s}(rz_{0})+\dfrac{n_{s}\Lambda_{s}\beta}{(1+k)^{s}}P(rz_{0})|>|z_{0}^{s}F_{s}(rz_{0})+\dfrac{n_{s}\Lambda_{s}\beta}{(1+k)^{s}}F(rz_{0})|$

Since
$z_{0}^{s}P_{s}(rz_{0})+\dfrac{n_{s}\Lambda_{s}\beta}{(1+k)^{s}}P(rz_{0})\neq
0,$ we can choose

$\lambda=\dfrac{z_{0}^{s}F_{s}(rz_{0})+\dfrac{n_{s}\Lambda_{s}\beta}{(1+k)^{s}}F(rz_{0})}{z_{0}^{s}P_{s}(rz_{0})+\dfrac{n_{s}\Lambda_{s}\beta}{(1+k)^{s}}P(rz_{0})}$

$\lambda$ is well-defined real or complex number with $|\lambda|<1$ and with
this choice of $\lambda,$ and $T(rz_{0})=0$ for $|z_{0}|\geq k.$ But this is a
contradiction to the fact that $T(rz)\neq 0$ for $|z|\geq 1.$ Thus (3.4)
holds. Letting $r\rightarrow 1,$ in (3.4), we get the desired result. ∎

######  Proof of Theorem 1.9.

Let $Q(z)=z^{n}\overline{P(1/\overline{z})}.$ Since $P(z)$ does not vanish in
the disk $|z|<k,\,\,k\leq 1$, then the polynomial $Q(z/k^{2})$ has all zeros
in $|z|\leq k.$ Applying Theorem 1.1 to $k^{n}Q(z/k^{2})$ and noting that
$|P(z)|=|k^{n}Q(z/k^{2})|$ for $|z|=k,$ we have for all
$\alpha_{j},\beta\in\mathbb{C}$ with $|\alpha_{j}|\geq 1,$ $j=1,2,\cdots,t,$
$|\beta|\leq 1,$ and $|z|\geq 1,$

(3.6)
$\displaystyle\left|z^{s}P_{s}(z)+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)\right|\leq
k^{n}\left|z^{s}Q_{s}(z/k^{2})+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}Q(z/k^{2})\right|.$

Inequality (3.6) in conjunction with Lemma 2.5 gives for all
$\alpha_{j},\beta\in\mathbb{C}$ with $|\alpha_{j}|\geq k,$ $j=1,2,\cdots,t$
$|\beta|\leq 1,$ and $|z|\geq 1,$

$\displaystyle 2\Bigg{|}$ $\displaystyle
z^{s}P_{s}(z)+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)\Bigg{|}$
$\displaystyle\leq\left|z^{s}P_{s}(z)+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)\right|+k^{n}\left|z^{s}Q_{s}(z/k^{2})+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}Q(z/k^{2})\right|$
$\displaystyle\leq
n_{s}\left\\{\dfrac{|z^{n}|}{k^{n}}\Bigg{|}\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}+\Bigg{|}z^{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}\right\\}\underset{|z|=k}{Max}|P(z)|,$

which is equivalent to (1.9). ∎

######  Proof of Theorem 1.11.

Let $m=Min_{|z|=k}|P(z)|.$ If $P(z)$ has a zero on $|z|=k,$ then $m=0$ and
result follows from Theorem 1.9. Therefore, We assume that $P(z)$ has all its
zeros in $|z|>k$ where $k\leq 1$ so that $m>0$. Now for every $\lambda$ with
$|\lambda|<1$, it follows by Rouche’s theorem that $h(z)=P(z)-\lambda m$ does
not vanish in $|z|<k$. Let
$g(z)=z^{n}\overline{h(1/\overline{z})}=z^{n}\overline{P(1/\overline{z})}-\overline{\lambda}mz^{n}=Q(z)-\overline{\lambda}mz^{n}$
then, the polynomial $g(z/k^{2})$ has all its zeros in $|z|\leq k.$ As
$|k^{n}g(z/k^{2})|=|h(z)|$ for $|z|=k,$ applying Theorem 1.1 to
$k^{n}g(z/k^{2}),$ we get for
$\alpha_{1},\alpha_{2},\cdots,\alpha_{s},\beta\in\mathbb{C},$ with
$|\alpha_{j}|\geq k,$ $j=1,2,\cdots,t$ $|\beta|\leq 1$ and $|z|\geq 1,$

(3.7)
$\displaystyle\left|z^{s}h_{s}(z)+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}h(z)\right|\leq
k^{n}\left|z^{s}g_{s}(z/k^{2})+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}g(z/k^{2})\right|.$

Equivalently for $|z|\geq 1,$ we have

$\displaystyle\Bigg{|}z^{s}$ $\displaystyle P_{s}(z)+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)-\lambda
n_{s}\left\\{z^{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\right\\}m\Bigg{|}$
(3.8) $\displaystyle\leq k^{n}\Bigg{|}z^{s}Q_{s}(z/k^{2})+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}Q(z/k^{2})-\dfrac{\overline{\lambda}n_{s}}{k^{2n}}\left\\{\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\right\\}mz^{n}\Bigg{|}.$

Since $Q(z/k^{2})$ has all its zeros in $|z|\leq k$ and
$k^{n}\underset{|z|=k}{Min}|Q(z/k^{2})|=\underset{|z|=k}{Min}|P(z)|,$ by
Corollary 1.5 applied to $Q(z/k^{2}),$ we have for $|z|\geq 1,$

$\displaystyle\Bigg{|}z^{s}Q_{s}(z/k^{2})+\beta\dfrac{n_{s}\Lambda_{s}}{(1+k)^{s}}Q(z/k^{2})\Bigg{|}$
$\displaystyle\geq\dfrac{n_{s}}{k^{n}}\left|\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\right|\underset{|z|=k}{Min}|Q(z)|$
$\displaystyle=\dfrac{n_{s}}{k^{2n}}\left|\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\right|\underset{|z|=k}{Min}|P(z)|$
(3.9)
$\displaystyle=\dfrac{n_{s}}{k^{2n}}\left|\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\right|m.$

Now, choosing the argument of $\lambda$ on the right hand side of inequality
(3) such that

$\displaystyle k^{n}$ $\displaystyle\Bigg{|}z^{s}Q_{s}(z/k^{2})+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}Q(z/k^{2})-\dfrac{\overline{\lambda}n_{s}}{k^{2n}}\left\\{\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\right\\}mz^{n}\Bigg{|}$
$\displaystyle=k^{n}\Bigg{|}z^{s}Q_{s}(z/k^{2})+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}Q(z/k^{2})\Bigg{|}-\dfrac{|\overline{\lambda}|n_{s}}{k^{n}}\Bigg{|}\left\\{\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\right\\}mz^{n}\Bigg{|},$

which is possible by inequality (3), we get for $|z|\geq 1,$

$\displaystyle\Bigg{|}z^{s}$ $\displaystyle P_{s}(z)+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)\Bigg{|}-|\lambda|n_{s}\Bigg{|}z^{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}m$
$\displaystyle=k^{n}\Bigg{|}z^{s}Q_{s}(z/k^{2})+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}Q(z/k^{2})\Bigg{|}-\dfrac{|\lambda||z|^{n}n_{s}}{k^{n}}\Bigg{|}\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}m,$

Letting $\lambda\rightarrow 1,$ we have for $|z|\geq 1,$

$\displaystyle\Bigg{|}z^{s}$ $\displaystyle P_{s}(z)+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)\Bigg{|}-k^{n}\Bigg{|}z^{s}Q_{s}(z/k^{2})+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}Q(z/k^{2})\Bigg{|}$ (3.10) $\displaystyle\leq
n_{s}\left\\{\Bigg{|}z^{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}-\dfrac{|z|^{n}}{k^{n}}\Bigg{|}\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}\right\\}m$

Adding (2.5) and (3), we get for $|z|\geq 1$

$\displaystyle 2\Bigg{|}z^{s}$ $\displaystyle P_{s}(z)+\dfrac{\beta
n_{s}\Lambda_{s}}{(1+k)^{s}}P(z)\Bigg{|}$ $\displaystyle\leq
n_{s}\Bigg{[}\left\\{\dfrac{|z^{n}|}{k^{n}}\Bigg{|}\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}+\Bigg{|}z^{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}\right\\}\underset{|z|=k}{Max}|P(z)|$
$\displaystyle+\left\\{\Bigg{|}z^{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}\Bigg{|}-\dfrac{|z|^{n}}{k^{n}}\Bigg{|}\alpha_{1}\alpha_{2}\cdots\alpha_{s}+\dfrac{\beta\Lambda_{s}}{(1+k)^{s}}mz^{n}\Bigg{|}\right\\}m\Bigg{]},$

which is equivalent to (1.11). This completes the proof of Theorem 1.11. ∎

## References

  * [1] A. Aziz, A new proof of Laguerre’s theorem about the zeros of polynomials, Bull. Austral. Math. Soc. 33 (1986), 131-138.
  * [2] A. Aziz, Inequalities for the polar derivatives of a polynomial, J. Approx. Theory, 55 (1988) 183-193
  * [3] A. Aziz, N. A. Rather, A refinement of a theorem of Paul Turán concerning polynomials. Math Ineq Appl. 1, 231–238 (1998).
  * [4] A. Aziz and N. A. Rather, Some Zygmund type $L^{q}$ inequalities for polynomials, J. Math. Anal. Appl. 289 (2004) 14-29.
  * [5] A. Aziz and Wali Mohammad Shah, Inequalities for polar derivatives of polynomial, Proc. Indian Acad. Sci. (Math. Sci.), 107, 263-270.
  * [6] P. D. Lax, Proof of a conjecture of P.Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc., 50 (1944), 509-513
  * [7] M. A. Malik, On the derivative of a polynomial, J. Lond. Math. Soc., Second Series 1 (1969), 57-60.
  * [8] M. Marden, Geometry of Polynomials, Math. Surveys, Amer. Math. Soc., Providence, 1966.
  * [9] G. V. Milovanovic, D. S. Mitrinovic and Th. M. Rassias, Topics in Polynomials: Extremal Properties, Inequalities, Zeros, World scientific Publishing Co., Singapore, (1994)
  * [10] Q. I. Rahman and G. Schmessier, Analytic theory of polynomials, Claredon Press, Oxford, 2002.
  * [11] A. C. Schaffer, Inequalities of A. Markoff and S. Bernstein for polynomials and related functions, Bull. Amer. Math. Soc., 47(1941), 565-579.
  * [12] P. Turán, U¨ber die Ableitung von Polynomen, Compositio Mathematica 7 (1939), 89-95 (German).


# Annotation Cost-Efficient Active Learning for Deep Metric Learning Driven
Remote Sensing Image Retrieval

Genc Hoxha, Gencer Sumbul, Julia Henkel, Lars Möllenbrok, and Begüm Demir,
Genc Hoxha, Julia Henkel, Lars Möllenbrok and Begüm Demir are with the Faculty
of Electrical Engineering and Computer Science, Technische Universität Berlin,
10623 Berlin, Germany and also with the Berlin Institute for the Foundations
of Learning and Data (BIFOLD), 10623 Berlin, Germany (emails: genc.hoxha@tu-
berlin.de<EMAIL_ADDRESS><EMAIL_ADDRESS>demir@tu-
berlin.de). Gencer Sumbul is with the Environmental Computational Science and
Earth Observation Laboratory (ECEO), École Polytechnique Fédérale de Lausanne
(EPFL), 1950 Sion, Switzerland (e-mail: gencer.sumbul@epfl.ch).

###### Abstract

Deep metric learning (DML) has shown to be very effective for content-based
image retrieval (CBIR) in remote sensing (RS). Most of DML methods for CBIR
rely on a high number of annotated images to accurately learn model parameters
of deep neural networks (DNNs). However, the process of gathering a large
number of image annotations is time consuming and costly. To address this
problem, we propose an annotation cost-efficient active learning (ANNEAL)
method specifically designed for DML driven CBIR in RS. ANNEAL aims to create
a small but informative training set made up of similar and dissimilar image
pairs to be utilized for accurately learning a deep metric space. The
informativeness of the image pairs is evaluated based on the combination of
uncertainty and diversity criteria. To assess the uncertainty of image pairs,
we introduce two different algorithms: 1) metric-guided uncertainty estimation
(MGUE); and 2) binary classifier guided uncertainty estimation (BCGUE). MGUE
algorithm automatically estimates a threshold value that acts as a ”boundary”
between similar and dissimilar image pairs based on the distances in the
metric space. The closer the similarity between image pairs is to the
estimated threshold value the higher is their uncertainty. BCGUE algorithm
automatically estimates the uncertainty of the image pairs based on the
confidence of the classifier in assigning the correct similarity label. The
diversity criterion is assessed through a clustering-based strategy. ANNEAL
selects the most informative image pairs by combining either MGUE algorithm or
BCGUE algorithm with clustering-based strategy. The selected most informative
image pairs are forwarded to expert annotators to be labeled as similar or
dissimilar. This way of annotating images significantly reduces the annotation
cost compared to the cost of annotating images with land-use land-cover class
labels. Experimental results carried out on two RS benchmark datasets
demonstrate the effectiveness of our method. The code of the proposed method
will be publicly available upon the acceptance of the paper.

###### Index Terms:

Active learning, content-based image retrieval, deep metric learning, remote
sensing.

##  I Introduction

With the rapid development of remote sensing (RS) technology, we have
witnessed an unprecedented growth in the volume of RS image archives.
Accordingly, one of the most important research topics in RS is the
development of fast and accurate content based image retrieval (CBIR) methods.
CBIR methods aim at searching and retrieving semantically similar images to a
user-defined query image from massive archives [1, 2, 3, 4]. This is usually
achieved based on two main steps. The first step is devoted to the
characterization of the complex content of a RS image scene with a set of
discriminitave features, whereas the second step is devoted to the retrieval
of the most similar images by evaluating the similarity between the query
image and those of an archive in the feature space. In particular, the
accurate characterization of images is crucial to reach an accurate search
capability within huge data archives. In this context, deep metric learning
(DML) has recently shown to be very effective for CBIR in RS. DML aims at
learning a feature (i.e., metric) space, where similar images are mapped close
to each other and dissimilar samples are mapped apart from each other. The
amount and quality of the available training samples (i.e., images) are
important to learn an accurate metric space. However, in operational scenarios
gathering a sufficient number of labeled training images is not realistic due
to the high cost and the related time consuming process of this task. To deal
with this problem, active learning (AL) methods have been presented in the
literature in the context of CBIR [5, 6]. AL aims to expand an initial and
sub-optimal training set in an iterative manner by finding the most
informative images from an archive that when annotated and included in the
training set can significantly improve the retrieval performance[6]. The
informativeness of the images is usually assessed based on two criteria: 1)
uncertainty; and 2) diversity. Given a query image, the uncertainty criterion
aims at finding images for which the confidence of a supervised algorithm in
correctly classifying them as relevant or irrelevant to the given query image
is low. The diversity criterion aims at finding diverse images to reduce the
redundancy of the selected images as much as possible. As an example,
Feratcatu et al. [5] present an AL method in the context of binary support
vector machines (SVMs) [7, 8] that combines the two aforementioned criteria to
select the most informative images based on two steps. In the first step, the
most uncertain images are selected from the archive based on the margin
sampling (MS) strategy [9, 10]. In the second step, the most diverse images
among the uncertain ones are defined by selecting the most distant samples in
the feature space. In [6] a triple criteria AL method is proposed that
includes a density criterion in addition to the uncertainty and the diversity
criteria. The three criteria are jointly evaluated in two steps. In the first
step, similarly as in[5], the most uncertain images from the archive are
selected based on the MS strategy. In the second step the most diverse images
among the uncertain ones are selected from the highest density regions in the
feature space. This is achieved using kernel k-means clustering technique to
cluster the uncertain images into different clusters in the feature space.
Then, the most representative image from each cluster is selected as the one
situated in the highest density region of the respective cluster.

(a)

(b)

Figure 1: Examples of image pairs from AID dataset [11]: a) similar pairs (in
blue frame) and b) dissimilar pairs (in red frame).

The aforementioned AL methods have shown their effectiveness in reducing the
need of annotating a high number of images. However, they result in limited
retrieval performance as they are defined in the context of traditional
classifiers such as SVMs that rely on hand-crafted features. Moreover, the
CBIR systems with these AL methods require a new AL process when a new query
image is selected. This involves: i) asking the human expert to annotate the
selected images as relevant or irrelevant with respect to the considered query
image; ii) training a separate (ad-hoc) binary classifier using the annotated
images with two class labels (i.e., relevant/irrelevant) with respect to the
query image; and iii) repeating the process until the user is satisfied with
the results or a predefined labeling budged is reached. This procedure is
intractable in operational scenarios as the number of query images and their
diversity can be high. As a result, AL methods that allow the creation of a
training set with informative images independent from the selected query
images are needed. To this end, AL methods proposed for classification
problems with DNNs [12, 13, 14] could be used to simultaneously construct
informative training sets and learn image representations. However, in this
way image representations are encoded to discriminate land use land cover
(LULC) classes rather than modeling semantic similarities. Furthermore, the
collection of LULC class labels is complex and time-consuming, especially when
dealing with large number of LULC class labels [15, 16].

To overcome these issues, in this paper we introduce an annotation cost-
efficient AL method (ANNEAL) for DML driven CBIR in RS. ANNEAL aims to
construct a small but informative training set made up of similar and
dissimilar RS image pairs to be utilized for accurately and efficiently
learning a deep metric space. To this end, it is defined based on a two-step
procedure: 1) the selection of the most uncertain image pairs; and 2) the
selection of the most diverse image pairs among the most uncertain ones. For
the first step, we propose two different algorithms: 1) a metric-guided
uncertainty estimation (MGUE); and 2) a binary classifier guided uncertainty
estimation algorithm (BCGUE). The first algorithm employs uncertainty
assessment of image pairs directly in a deep metric space as the first time in
AL literature. This is achieved based on the automatic estimation of an
adequate threshold value to distinguish between similar and dissimilar image
pairs. The unlabeled image pairs that have a similarity value closest to the
threshold value are the ones characterized by the highest degree of
uncertainty. The estimation of the threshold value is based on the distance in
the feature (i.e., metric) space between the images that compose the similar
and dissimilar pairs in the current training set. In the second algorithm, the
uncertainty of the image pairs is assessed based on the confidence of the
classifier in assigning the correct similarity label that requires the
incorporation of a binary classifier in the considered CBIR system. As the
confidence of the classifier becomes lower, the uncertainty of the considered
image pair becomes higher. In the second step, we select the most diverse
image pairs among the most uncertain ones through a clustering based strategy.
In detail, the $k$-means clustering algorithm is utilized to first cluster the
most uncertain pairs into different clusters, and then we select one image
pair per cluster. ANNEAL is defined as the combination of one of the proposed
uncertainty with the clustering-based strategy. We call ANNEAL with MGUE
algorithm as ANNEAL-MGUE, while that with BCGUE algorithm as ANNEAL-BCGUE. The
selected most informative pairs either by ANNEAL-MGUE or ANNEAL-BCGUE are
labeled by a human annotator as similar/dissimilar with respect to each other.
Defining the similarity of two RS images based on their contents naturally
aligns with the goal of CBIR. Fig. 1 depicts some examples of similar and
dissimilar RS image pairs. Based on the visual inspection of such pairs, a
human expert is required to define their similarity. Annotating images as
similar/dissimilar significantly reduces the annotation cost compared to
annotating images with LULC class labels. Referring to the information theory,
it requires only one bit of information to label a pair of images (i.e., one
sample) as similar/dissimilar. In contrast, $\text{log}_{2}{C}$ bits of
information are required to label an image with one of the predefined $C$ LULC
class labels. Moreover, annotating images as similar/dissimilar allows to
obtain further training pairs with zero cost due to transitive property of
similarity [17].

We would like to note that BCGUE has been briefly presented in [18] with
limited experimental analysis. The contribution of this paper, which
significantly extends our previous work presented in [18], consists in: 1) the
detailed description of ANNEAL-BCGUE with enriched experiments; 2) introducing
ANNEAL-MGUE (which does not require the use of a binary classifier in the
considered CBIR system); and 3) comparing theoretically and experimentally
ANNEAL-BCGUE and ANNEAL-MGUE with an extended experimental analysis on two RS
benchmark datasets. Experimental results show the effectiveness of ANNEAL in
general. Among the proposed uncertainty evaluation algorithms, we observe that
MGUE leads to more accurate uncertainty estimation of the image pairs directly
on a metric space, while reducing the model complexity in terms of the number
of parameters.

The remaining part of this paper is organized as follows. Section II presents
the related work on CBIR in RS. Section III introduces the proposed method.
Section IV describes the considered RS image archives and the experimental
setup, while the experimental results are presented in Section V. Finally, in
Section VI, the conclusion of the work is drawn.

##  II Related Work on DML for CBIR in RS

The development of DML methods, which learn a feature (i.e., metric) space in
which similar images are close to each other and vice versa, has attracted
attention in RS due to their capabilities to model image similarities for
accurate CBIR [1]. As an example, Chaundhuri et al. [19] introduce a DML-based
CBIR system for very high resolution (VHR) images that employs a Siamese graph
CNN to learn image similarity from region adjacency graphs (RAGs) using
contrastive loss function [20]. The authors create mini-batches of tuples of
similar and dissimilar images based on the class labels (i.e., if the images
share the same class label they are considered as similar and vice versa).
Subsequently, RAG representations are fed into two graph CNNs with shared
weights that are trained through contrastive loss to model image similarity
[20]. A triplet CNN network with shared weights is proposed in [21], where
image triplets are used instead of tuples to learn image similarities. The
triplets consist of: a) an anchor image; b) a positive image that is similar
(i.e., share the same class label) to the anchor; and c) a negative image that
is dissimilar (i.e., different class label) to the anchor. The triplet loss
[22] is used to train the network, aiming to minimize the distance in the
feature space between the anchor and the positive pair, while maximizing that
of the anchor and the negative pair. Imbriacco et al. [23] present a CBIR
system based on a multi-branch CNN to extract global and local features
combining the triplet loss [22] with the cross-entropy loss. A global optimal
structured loss function that uses image triplets is proposed in [24] to
produce an embedding space in which the positive pairs are forced to stay in a
compact form within a hyper-sphere while the negative ones far away from its
boundaries. To this end, the authors in [24] combine the global lifted
structure loss [25] and the cross-entropy loss to learn global discriminative
features with the hardest positive and negative mining strategy [26] that
increases intra-class compactness and inter-class sparsity. Similarly, Fan et
al. [27] propose a distribution consistency loss function that uses multiple
positive and negative images for each anchor (unlike the triplet loss
function) considering the equal distribution of the mined easy and hard
samples within each class. The distribution consistency loss function is a
combination of sample balance loss and the ranking consistency loss [28]. The
former assigns dynamic weights to the selected hard samples based on the ratio
of hard and easy samples in a given class, while the latter ranks the negative
samples according to their category distribution. Sumbul et al [29] present a
triplet selection strategy to select informative and representative triplets
based on relevancy, hardness and diversity of the selected images. DML has
been also successfully applied in the context hashing-based CBIR systems [30,
31, 32, 33]. As an example, Roy et al. [31] present a DML based hashing method
that combines the triplet loss with bit-balancing and push loss [34] to
generate discriminative hash codes for CBIR. Similarly, to obtain
discriminative hash code representations, Li et al. [30] combine contrastive
and bit-balance loss functions. For a comprehensive and an extensive survey on
the recent advancement of hashing-based CBIR systems in RS we refer the reader
to the recent review paper proposed by Zhou et al. [4].

##  III Proposed Annotation Cost-Efficient Active Learning (ANNEAL) Method

Let $\mathcal{I}$ be an RS image archive and
$\mathcal{T}=\\{\boldsymbol{X}_{i}^{\mathcal{T}}\\}^{M}_{i=1}$ be an initial
training set made up of $M$ RS image pairs, where
$\boldsymbol{X}_{i}^{\mathcal{T}}$ is the i-th pair composed of images
$\boldsymbol{I}^{i,\mathcal{T}}_{1}$ and $\boldsymbol{I}^{i,\mathcal{T}}_{2}$
from $\mathcal{I}$, i.e., $\forall\
\boldsymbol{X}_{i}^{\mathcal{T}}=(\boldsymbol{I}^{i,\mathcal{T}}_{1},\boldsymbol{I}^{i,\mathcal{T}}_{2})\in\mathcal{T}\\!\\!\\!\quad\exists\\!\\!\\!\quad\boldsymbol{I}^{i,\mathcal{T}}_{1},\boldsymbol{I}^{i,\mathcal{T}}_{2}\in\mathcal{I}$.
Each image pair $\boldsymbol{X}_{i}^{\mathcal{T}}$ is associated with a
similarity label $\boldsymbol{y}_{i}^{\mathcal{T}}$, where
$\boldsymbol{y}_{i}^{\mathcal{T}}=1$ if $\boldsymbol{I}^{i,{\mathcal{T}}}_{1}$
and $\boldsymbol{I}^{i,\mathcal{T}}_{2}$ are similar to each other and
$\boldsymbol{y}_{i}^{\mathcal{T}}=0$ otherwise. Let
$\boldsymbol{X}_{k}^{\mathcal{U}}$ be an unlabeled image pair and
$\mathcal{U}=\\{\boldsymbol{X}_{k}^{\mathcal{U}}\\}^{N}_{k=1}$ be a set of $N$
unlabeled pairs, where $N\gg M$. We assume that the number $|\mathcal{T}|$ of
labeled training image pairs is not sufficient for accurately learning a deep
metric space. To construct a small but informative training set of similar and
dissimilar RS image pairs, we introduce an annotation cost-efficient AL
(ANNEAL) method for DML driven CBIR in RS. To this end, ANNEAL iteratively
selects a batch
$\mathcal{S}=\\{\boldsymbol{X}_{1}^{\mathcal{U}},\boldsymbol{X}_{2}^{\mathcal{U}},\ldots,\boldsymbol{X}_{h}^{\mathcal{U}}\\}$
of $h$ most informative image pairs from $\mathcal{U}$ that are uncertain
(i.e., ambiguous) and as much diverse as possible to each other. This is
achieved based on a two-step procedure. In the first step, ANNEAL selects the
set
$\mathcal{S}^{Unc}=\\{\boldsymbol{X}_{1},\boldsymbol{X}_{2},\ldots,\boldsymbol{X}_{p}\\}$
of $p$ most uncertain image pairs with $p>h$. In the second step, the $h$ most
diverse samples are selected from the most uncertain set $\mathcal{S}^{Unc}$.
The selected most informative image pairs are annotated by human experts as
similar or dissimiliar and added to the current training set. The iterative
process is terminated based on CBIR performance or labeling budget. We would
like to note that this way of annotating images reduces the annotation cost
for a given sample (i.e., image pair) to only one bit of information
($log_{2}{2}=1$) compared to associating each image with one of $C$ LULC class
labels (which require $log_{2}{C}$ bits of information). Before explaining in
detail the proposed ANNEAL method (which is illustrated in 2), we first
present the considered DML based CBIR system in the following.

Figure 2: Block diagram of the proposed ANNEAL method.

###  III-A The Considered DML Driven CBIR System

The CBIR system considered in this work is based on DML. As in any CBIR
system, one of the two fundamental steps is the image characterization with
discriminative features. To this end, we consider two DNNs with shared weights
(i.e., Siamese neural network [SNN]) that are trained using the contrastive
loss [20] for learning a metric space. To this end, each image pair
$\boldsymbol{X}_{i}^{\mathcal{T}}=(\boldsymbol{I}^{i,\mathcal{T}}_{1},\boldsymbol{I}^{i,\mathcal{T}}_{2})\in\mathcal{T}$
is given as input to the SNN to obtain corresponding image features
$\boldsymbol{F}_{i}^{\mathcal{T}}=(\boldsymbol{f}^{i,\mathcal{T}}_{1},\boldsymbol{f}^{i,\mathcal{T}}_{2})$,
where $\boldsymbol{f}^{i,\mathcal{T}}_{1}$ and
$\boldsymbol{f}^{i,\mathcal{T}}_{2}$ denote the image features of
$\boldsymbol{I}^{i,\mathcal{T}}_{1}$ and $\boldsymbol{I}^{i,\mathcal{T}}_{2}$,
respectively. Then, the considered contrastive loss function
$\mathcal{L}_{CL}$ is formulated on image features as follows:

$\displaystyle\\!\\!\mathcal{L}_{CL}\\!$
$\displaystyle=\\!\left\\{\\!\\!\\!\begin{array}[]{ll}1-s(\boldsymbol{f}^{i,\mathcal{T}}_{1},\boldsymbol{f}^{i,\mathcal{T}}_{2})),\\!\\!\\!&\boldsymbol{y}_{i}^{\mathcal{T}}=1\\\
\max{(0,s(\boldsymbol{f}^{i,\mathcal{T}}_{1},\boldsymbol{f}^{i,\mathcal{T}}_{2})\\!-\\!m}),\\!\\!\\!&\boldsymbol{y}_{i}^{\mathcal{T}}=0\end{array}\right.$
(3)

where $m$ is the margin parameter and $s(\cdot,\cdot)$ measures cosine
similarity.

Once the network is trained on a given training set, CBIR is performed by
comparing the feature of the given query with that of each image in the
archive $\mathcal{I}$. We would like to note that the proposed ANNEAL is
independent from the considered metric learning loss function. Thus, instead
of (1) different pairwise loss functions such as NT-Xent [35] can be also
utilized.

###  III-B ANNEAL: Uncertainty Criterion

The first step of the proposed ANNEAL method is devoted to the selection of
the most uncertain image pairs. To assess the uncertainty of image pairs, we
introduce two different algorithms: 1) metric guided uncertainty estimation
(MGUE); and 2) binary classifier guided uncertainty estimation (BCGUE). Each
algorithm is combined with the considered diversity criterion. ANNEAL with
MGUE algorithm is denoted as ANNEAL-MGUE, whereas ANNEAL with BCGUE algorithm
is denoted as ANNEAL-BCGUE.

####  III-B1 Metric Guided Uncertainty Estimation (MGUE)

The proposed MGUE algorithm aims at assessing uncertainty of the image pairs
directly in the metric space. To this end, we estimate a threshold value to
distinguish between similar and dissimilar images. This is automatically
achieved based on the distances between similar and dissimilar images in the
feature space. If the similarity in the feature space between any two images
exceeds the threshold value, then the images are considered as similar, vice
versa. The image pairs that have a cosine similarity value, which is closest
to the estimated threshold value, are the most uncertain ones. Let
$\alpha_{t}$ be the estimated threshold value. Then, the similarity of an
image pair
$\boldsymbol{X}_{i}=(\boldsymbol{I}^{i}_{1},\boldsymbol{I}^{i}_{2})$ can be
determined as follows:

$\displaystyle\\!\\!{X}_{i}=(\boldsymbol{I}^{i}_{1},\boldsymbol{I}^{i}_{2})\\!$
$\displaystyle=\\!\left\\{\\!\\!\\!\begin{array}[]{ll}\text{similar},\\!\\!\\!&\text{if}\
\ s(\boldsymbol{f}^{i}_{1},\boldsymbol{f}^{i}_{2})\geq\alpha_{t}\\\
\text{dissimilar},\\!\\!\\!&\text{if}\ \
s(\boldsymbol{f}^{i}_{1},\boldsymbol{f}^{i}_{2})<\alpha_{t}\end{array}\right.$
(6)

To estimate the threshold value $\alpha_{t}$, we calculate the mean and the
standard deviation of similar and dissimilar image pairs in the current
training set as follows:

$\displaystyle\mu_{sim}=\frac{1}{S}\sum_{i=1}^{S}s(\boldsymbol{f}^{i,\mathcal{T}}_{1},\boldsymbol{f}^{i,\mathcal{T}}_{2}),$
$\displaystyle\boldsymbol{y}_{i}^{\mathcal{T}}=1$ (7)

$\displaystyle\sigma_{sim}=\sqrt{\frac{1}{S}\sum_{i=1}^{S}(s(\boldsymbol{f}^{i,\mathcal{T}}_{1},\boldsymbol{f}^{i,\mathcal{T}}_{2})-\mu_{sim})^{2}},$
$\displaystyle\boldsymbol{y}_{i}^{\mathcal{T}}=1$ (8)

$\displaystyle\mu_{dsim}=\frac{1}{D}\sum_{i=1}^{D}s(\boldsymbol{f}^{i,\mathcal{T}}_{1},\boldsymbol{f}^{i,\mathcal{T}}_{2}),$
$\displaystyle\boldsymbol{y}_{i}^{\mathcal{T}}=0$ (9)

$\displaystyle\sigma_{dsim}=\sqrt{\frac{1}{D}\sum_{i=1}^{D}(s(\boldsymbol{f}^{i,\mathcal{T}}_{1},\boldsymbol{f}^{i,\mathcal{T}}_{2})-\mu_{dsim})^{2}},$
$\displaystyle\boldsymbol{y}_{i}^{\mathcal{T}}=0$ (10)

where $\mu_{sim}$,$\sigma_{sim}$ and $\mu_{dsim}$,$\sigma_{dsim}$ are the mean
and the standard deviation of the cosine similarity between the images
composing the similar and dissimilar pairs, respectively. $S$ and $D$ are the
number of total similar and dissimilar pairs in the training set,
respectively. Based on these statistics, the threshold value $\alpha_{t}$ is
defined as follows:

$\displaystyle\alpha_{t}=\frac{\mu_{sim}+\mu_{dsim}-\lambda(\sigma_{sim}-\sigma_{dsim})}{2},$
(11)

where $\lambda$ is a hyper-parameter that defines the width of the standard
deviation. The closer the cosine similarity between the images that compose
the unlabeled pairs are to the customized threshold $\alpha_{t}$ the more
uncertain the image pairs are. We define the uncertainty of a given unlabeled
pair
$\boldsymbol{X}_{i}^{\mathcal{U}}=(\boldsymbol{I}^{i,\mathcal{U}}_{1},\boldsymbol{I}^{i,\mathcal{U}}_{2})\in\mathcal{U}$
as the absolute difference between the cosine similarity of the image pair in
the feature space and the threshold value $\alpha_{t}$, as follows:

$\displaystyle
Unc(\boldsymbol{X}_{i}^{\mathcal{U}})=|s(\boldsymbol{f}^{i,\mathcal{U}}_{1},\boldsymbol{f}^{i,\mathcal{U}}_{2})-\alpha_{t}|.$
(12)

We calculate the uncertainty of each pair in the unlabeled set $\mathcal{U}$,
and then sort the pairs in ascending order and we select the set
$\mathcal{S}^{Unc}=\\{\boldsymbol{X}_{1},\boldsymbol{X}_{2},\ldots,\boldsymbol{X}_{p}\\}$
of $p$ most uncertain image pairs.

####  III-B2 Binary Classifier Guided Uncertainty Estimation (BCGUE)

The proposed BCGUE algorithm aims at assessing the uncertainty of the image
pairs based on the confidence of a binary classifier used to classify image
pairs as similar/dissimilar with respect to each other. The lower the
confidence of the classifier in assigning the similarity label to an image
pair, the higher is the uncertainty of the pair. To this end, we employ a
binary classifier after the considered SNN to classify images as similar or
dissimilar. The binary classifier consists of three fully connected layers,
while the last layer is made up of a single neuron. We obtain the class
posterior probability of $\boldsymbol{y}_{i}^{\mathcal{T}}$, for a given image
pair $\boldsymbol{X}_{i}^{\mathcal{T}}$, activating the single neuron of the
last layer with sigmoid activation function as follows:

$\displaystyle
P(\boldsymbol{y}_{i}^{\mathcal{T}}|\boldsymbol{X}_{i}^{\mathcal{T}})=\frac{1}{1+e^{-\phi(g(\boldsymbol{X}_{i}^{\mathcal{T}}))}},$
(13)

where $g$ and $\phi$ represent the SNN and the binary classifier,
respectively. To optimize the model parameters, in addition to
$\mathcal{L}_{CL}$, we employ the binary cross entropy loss function
$\mathcal{L}_{BCE}$ as follows:

$\displaystyle\mathcal{L}_{BCE}$
$\displaystyle=\boldsymbol{y}_{i}^{\mathcal{T}}\log(P(\boldsymbol{y}_{i}^{\mathcal{T}}|\boldsymbol{X}_{i}^{\mathcal{T}}))$
$\displaystyle\quad+(1-\boldsymbol{y}_{i}^{\mathcal{T}})\log(1-P(\boldsymbol{y}_{i}^{\mathcal{T}}|\boldsymbol{X}_{i}^{\mathcal{T}})).$
(14)

We train the whole system end-to-end combining $\mathcal{L}_{CL}$ and
$\mathcal{L}_{BCE}$ as follows:

$\displaystyle\mathcal{L}=(1-\gamma)\mathcal{L}_{CL}+\gamma\mathcal{L}_{BCE},$
(15)

where $\gamma$ is the balancing factor.

Once the model is trained, we use the posterior class probabilities given by
(13) to evaluate the confidence of the image pairs. Specifically, for a binary
classification problem the most uncertain image pairs are the ones
characterized by the posterior class probability
$P(\boldsymbol{y}_{i}^{\mathcal{U}}|\boldsymbol{X}_{i}^{\mathcal{U}})\approx
0.5$. We calculate the uncertainty of each pair in the unlabeled set
$\mathcal{U}$ using (12), where we substitute the cosine similarity between
image pairs
$s(\boldsymbol{f}^{i,\mathcal{U}}_{1},\boldsymbol{f}^{i,\mathcal{U}}_{2})$
with the class posterior probabilities
$P(\boldsymbol{y}_{i}^{\mathcal{U}}|\boldsymbol{X}_{i}^{\mathcal{U}})$ and set
the value of $\alpha_{t}=0.5$. Then, we sort the image pairs in ascending
order and we select the set
$\mathcal{S}^{Unc}=\\{\boldsymbol{X}_{1},\boldsymbol{X}_{2},\ldots,\boldsymbol{X}_{p}\\}$
of $p$ most uncertain image pairs.

###  III-C ANNEAL: Diversity Criterion

The second step of the proposed ANNEAL method is devoted to the selection of
the most diverse image pairs among the uncertain ones. To this end, we exploit
a clustering-based strategy to first partition the uncertain pairs into $h$
sets (i.e., clusters). We select the most uncertain pair from each cluster to
obtain a set
$\mathcal{S}=\\{\boldsymbol{X}_{1},\boldsymbol{X}_{2},\ldots,\boldsymbol{X}_{h}\\}$
of $h$ image pairs that are at the same time the most uncertain and diverse
ones. As a clustering strategy we use k-means clustering algorithm presented
in [36]. However, any clustering approach can be used in the proposed method.

After the selection of the most informative image pairs, a human expert
annotates each pair as similar or dissimilar and the labeled pairs are added
to the training set. We further enrich the training set with zero annotation
cost exploiting the transitivity property of similarity [17]. Let
$\boldsymbol{X}_{i}^{\mathcal{T}}$ and $\boldsymbol{X}_{j}^{\mathcal{T}}$ be
two image pairs that share a common image (i.e., same image is present in both
pairs). Let
$\boldsymbol{I}^{i,\mathcal{T}}_{2}=\boldsymbol{I}^{j,\mathcal{T}}_{2}$ be the
shared image between the pairs $\boldsymbol{X}_{i}^{\mathcal{T}}$ and
$\boldsymbol{X}_{j}^{\mathcal{T}}$. If $\boldsymbol{X}_{i}^{\mathcal{T}}$ and
$\boldsymbol{X}_{j}^{\mathcal{T}}$ are both similar pairs (i.e.,
$\boldsymbol{y}_{i}^{\mathcal{T}}=1$ and $\boldsymbol{y}_{j}^{\mathcal{T}}=1$)
then from transitivity property we form a new similar pair
$\boldsymbol{X}_{k}^{\mathcal{T}}=(\boldsymbol{I}^{i,\mathcal{T}}_{1},\boldsymbol{I}^{j,\mathcal{T}}_{1})$
with $\boldsymbol{y}_{k}^{\mathcal{T}}=1$. Similarly, if
$\boldsymbol{X}_{i}^{\mathcal{T}}$ is a similar pair and
$\boldsymbol{X}_{j}^{\mathcal{T}}$ is a dissimilar pair (i.e.,
$\boldsymbol{y}_{i}^{\mathcal{T}}=1$ and $\boldsymbol{y}_{j}^{\mathcal{T}}=0$)
then from transitivity property we can infer that
$\boldsymbol{X}_{k}^{\mathcal{T}}=(\boldsymbol{I}^{i,\mathcal{T}}_{1},\boldsymbol{I}^{j,\mathcal{T}}_{1})$
is a dissimilar pair and can be labeled with
$\boldsymbol{y}_{k}^{\mathcal{T}}=0$. If instead
$\boldsymbol{X}_{i}^{\mathcal{T}}$ and $\boldsymbol{X}_{j}^{\mathcal{T}}$ are
both dissimilar pairs (i.e., $\boldsymbol{y}_{i}^{\mathcal{T}}=0$ and
$\boldsymbol{y}_{j}^{\mathcal{T}}=0$) we cannot infer any information about
the pair
$\boldsymbol{X}_{k}^{\mathcal{T}}=(\boldsymbol{I}^{i,\mathcal{T}}_{1},\boldsymbol{I}^{j,\mathcal{T}}_{1})$.
As a result, no new pair will be formed in this case. It is worth noting that
we only perform a single transitive step and we do not proceed another
transitive step with the newly generated pairs.

##  IV Dataset Description and Experimental Design

###  IV-A Datasets

To asses the performance of the proposed cost-efficient AL method, we carried
out different experiments on two different archives that consists of VHR
images.

The first archive, namely UC-Merced, consists of 2100 images selected from
aerial ortho-imagery [37] and has in total 21 different categories:
agricultural, airplane, baseball diamond, beach, buildings, chaparral, dense
residential, forest, freeway, golf course, harbor, intersection, medium
density residential, mobile home park, overpass, parking lot, river, runway,
sparse residential, storage tanks, and tennis courts. The images are of size
$256$x$256$ pixels with a spatial resolution of 30 cm and are downloaded from
United States Geological Survey (USGS) National Map of the following U.S.
regions: Birmingham, Boston, Buffalo, Columbus, Dallas, Harrisburg, Houston,
Jacksonville, Las Vegas, Los Angeles, Miami, Napa, New York, Reno, San Diego,
Santa Barbara, Seattle, Tampa, Tucson, and Ventura.

The second archive for which we carried out our experiments is the Aerial
Image Dataset (AID) [11] which consists of 10000 images collected from Google
Earth imagery and has in total 30 different categories: airport, bare land,
baseball field, beach, bridge, center, church, commercial, dense residential,
desert, farmland, forest, industrial, meadow, medium residential, mountain,
park, parking, playground, pond, port, railway station, resort, river, school,
sparse residential, square, stadium, storage tanks and viaduct. [11]. The
number of images per category varies from 220 up to 420 images that are chosen
from different regions around the world, mainly from the United States,
England, France, Italy, Japan, Germany, etc, under different times, seasons
and imaging conditions . The images are of size $600$x$600$ pixels and have
different spatial resolutions in the range between 50 to 80 cm.

###  IV-B Experimental Setup

In all our experiments we have randomly divided the two archives into three
sets: training (80%), validation (10%) and test (10%). To derive the initial
training set $\mathcal{T}$ we randomly select $5\%$ and $1\%$ of the images
for UC-Merced and AID datasets, respectively and then create the pairs by
randomly choosing four similar and four dissimilar images for each labeled
images based on their categories (i.e., class labels). The unlabeled set
$\mathcal{U}$ consists of all training pairs that are not part of
$\mathcal{T}$. At each AL iteration we extend the initial training set
$\mathcal{T}$ by adding 336 and 392 most informative image pairs from
$\mathcal{U}$ of UC-Merced and AID, respectively. This means that 336 and 392
bits of information are included at each AL iteration of UC-Merced and AID,
respectively. This corresponds to the annotation of 5% and 1% of the images
with LULC class labels for UC-Merced and AID archives, respectively,
considering the datasets sizes (AID dataset is almost five times bigger than
UC-Merced). ResNet18 [38] is chosen as a backbone of the image
characterization module (i.e., SNN) with initial weights obtained from the
pre-trained model on ImageNet [39]. Subsequently the model’s output is
projected through a non-linear projection head composed of two fully connected
layers with hidden dimension of 512 and 256 units and RelU activation for both
the algorithms. The weights of the projection head are randomly initialized.
In the case of the BCGUE algorithm, the output of the ResNet18 is concatenated
and fed into the binary classifier. The model is trained for 15 epochs using a
batch size of 128 and Adam Optimizer [40] with a learning rate of $10^{-4}$.
For the retrieval, the projection head is discarded and the feature
representations obtained from the backbone (i.e., ResNet18) are used to
perform CBIR. The retrieval is performed on the test set, while the query
images are taken from the validation set. The retrieval performance is
measured using the widely used mean Average Precision (mAP) metric denoted as
follows:

$\displaystyle
mAP=\frac{1}{Q}\sum_{i=1}^{Q}\frac{1}{r_{i}}\sum_{j=1}^{k}Prec_{i}(j)\cdot\delta_{i}(j),$
(16)

where $Q$ is the size of the query set; $r_{i}$ is the number of items that
are related to the $i^{th}$ query image; $k$ is the number of images in the
retrieval set; $Prec_{i}(j)$ is the precision at $j^{th}$ position for the
$i^{th}$ query image and $\delta_{i}(j)$ is a binary relevance function that
returns 1 if the item at $j^{th}$ position is relevant to the $i^{th}$ query
image and 0 otherwise. In our experiments, the retrieval performance is
measured using mAP on 5 retrieved images (mAP@5).

All the experimental results refer to the average retrieval performances
obtained in three trials, while an initial training set was randomly
reconstructed in each trial. As suggested in [6] we set $p=4h$, whereas the
margin $m$ parameter of the loss function is chosen to be 0.5 as default.
While constructing the image pairs, there are more dissimilar pairs compared
to similar ones. Approximately, $5\%$ of the pairs are similar and $95\%$ are
dissimilar. To stabilize the training, we apply oversampling strategy on the
minority class presented in [41].

We train the CBIR system on the entire training set to have an upper bound on
the retrieval performance. We compare the proposed method with the following:
1) randomly selecting the image pairs at each AL iteration (i.e. random
selection) and 2) classification based AL (CAL) [42] for CBIR, where uncertain
images are selected to be labeled with LULC class labels. In CAL, the
uncertainty of an image is assessed based on the confidence of the supervised
classifier in correctly classifying it with one of the LULC class labels. To
make the methods comparable we use the same initial training set, the same
backbone, projection head and incorporate the same number of bits of
information to the labeled set at each AL iteration. For CAL, a fully-
connected layer is added on top of the projection head with softmax
activation, where the number of neurons is equal to the number of classes and
cross-entropy loss function is used to train the model. Since CAL only
evaluates uncertainty, we added a diversity criterion based on a clustering
strategy to allow for a fairer comparison.

##  V Experimental Results

We carried out different kinds of experiments to: 1) perform a sensitivity
analysis of the proposed MGUE algorithm with respect to the selection of the
hyparameter $\lambda$; 2) analyse the effect of the diversity criterion on the
proposed ANNEAL method; and 3) compare the proposed ANNEAL method with CAL
method and random selection.

###  V-A Analysis of the Effect of the Hyparameter $\lambda$ on the Proposed
MGUE Algorithm

To assess the uncertainty of the image pairs through the proposed MGUE
algorithm the threshold value needs to be estimated. The threshold value is
used by the proposed MGUE algorithm to select the most uncertain image pairs,
specifically those that have a cosine similarity value closest to the
estimated threshold value. The threshold value is estimated based on the mean
and the standard deviation of the cosine similarity values between the images
that compose the similar and dissimilar pairs and contains one hyperparameter,
$\lambda$ (see (11))]. This hyperparameter defines the width of the standard
deviation (dispersion of the cosine similarity values between the images of
similar and dissimilar pairs) and should be carefully selected.

We evaluated the effect of the hyperparameter $\lambda$ used in proposed MGUE
algorithm, while it is varied as $\lambda\in\\{1,2,3,4,5,6\\}$. Fig. 3 and 4
show the retrieval performance in terms of mAP@5 versus the bits of
information used and the hyperparameter $\lambda$ variation when the proposed
MGUE algorithm is applied on UC-Merced and AID datasets, respectively. From
the figures, one can observe that for both datasets when $\lambda=3$ the
proposed MGUE algorithm achieves the best trade-off between the number of bits
used for the training and the retrieval performance. When $\lambda$ increases
($\lambda>3$) or decreases ($\lambda<3$) the retrieval performance starts to
drop. This is due to the fact that very high threshold values result in
selecting easy and non informative dissimilar pairs whereas very low threshold
values result in selecting easy and non informative similar pairs. Based on
these results for all the subsequent experiments we fix the hyperparameter
$\lambda=3$ for the ANNEAL-MGUE.

Figure 3: The average retrieval performance in terms of mAP@5 versus the
number of bits of information obtained by the proposed MGUE for different
values of $\lambda$ for the UC-Merced dataset. Figure 4: The average retrieval
performance in terms of mAP@5 versus the number of bits of information
obtained by the proposed MGUE for different values of $\lambda$ for the AID
dataset.

###  V-B Analysis of the Effect of Diversity Criterion

In this subsection, we analyse the retrieval performance obtained by using
only the uncertainty criterion and the combination of uncertainty criterion
with diversity criterion for ANNEAL. Fig. 5 and 6 show the average retrieval
performance in terms of mAP@5 versus the number of bits of information
obtained by the ANNEAL-MGUE, the ANNEAL-BCGUE and also direct use of MGUE and
BCGUE without diversity criterion for the UC-Merced dataset and AID dataset,
respectively. One can observe that for both datasets the combination of
uncertainty and diversity criteria achieves the best results under different
bits of information used for training. In particular, this combination is much
important and effective for ANNEAL-BCGUE. As it can be seen in Fig. 5, the
proposed ANNEAL-BCGUE, not only converges faster to the upper bound compared
to BCGUE algorithm but also shows higher retrieval results under different
bits of information. This is particularly visible where the number of bits of
information used for training ranges between 1107 to 2583, when the
convergence to the upper bound is reached. The same behavior with a smaller
difference can also be seen with ANNEAL-MGUE.

Figure 5: The average retrieval performance in terms of mAP@5 versus the
number of bits of information obtained by the ANNEAL-MGUE, the ANNEAL-BCGUE
and also direct use of the MGUE and the BCGUE without diversity criterion for
the UC-Merced dataset. Figure 6: The average retrieval performance in terms of
mAP@5 versus the number of bits of information obtained by the ANNEAL-MGUE,
the ANNEAL-BCGUE and also direct use of the MGUE and the BCGUE without
diversity criterion for the the AID dataset.

###  V-C Comparison among the Proposed ANNEAL and Literature Methods

In this subsection, we compare the proposed ANNEAL method with CAL [42] and
random selection. Fig.  7 and Fig.  8 depict the retrieval performance versus
the number of bits of information used for training obtained by the proposed
ANNEAL-MGUE and ANNEAL-BCGUE methods, CAL [42] and random selection for UC-
Merced and AID datasets, respectively. By analyzing the figures, one can see
that ANNEAL-MGUE and ANNEAL-BCGUE lead to the highest retrieval performance
for all the iterations and outperform the CAL and random selection methods for
both datasets. As an example, for the UC-Merced dataset (see Fig.  7), when
1845 bits of information are used for training the ANNEAL-MGUE and ANNEAL-
BCGUE provide an average retrieval performance of 93.56%, 93.55% in terms of
mAP@5, respectively, whereas those obtained by CAL and random selection under
the same bits of information are of 87.82% and 71.96%, respectively. Moreover,
to reach to the same retrieval performance of ANNEAL-MGUE and ANNEAL-BCGUE,
the CAL method requires 4059 bits of information for training (i.e., final
training size). This corresponds to 6 more AL iterations compared to the
proposed ANNEAL-MGUE and ANNEAL-BCGUE methods. Another important result is
that ANNEAL-MGUE and ANNEAL-BCGUE reach convergence two times faster than the
CAL method. In particular, the convergence for both ANNEAL-MGUE and ANNEAL-
BCGUE is reached when using only 2214 bits of information. For random
selection, the retrieval performance obtained with the final training set does
not reach convergence.

By analyzing Fig.  8 one can observe that the retrieval performances of the
ANNEAL-MGUE and ANNEAL-BCGUE methods are in general better than those of CAL
and significantly better than random selection under different bits of
information used for training. For example, the retrieval performances for
ANNEAL-MGUE and ANNEAL-BCGUE obtained with the final size of the training set
(7440 bits) is of 92.85% and of 93.13% respectively, whereas the ones obtained
by CAL method and random selection are of 87.70% and 77.16% respectively. One
can observe that ANNEAL-MGUE can reach the final training size performance of
the CAL method using less than half bits (i.e., 3524) of information for
training whereas ANNEAL-BCGUE requires 4307 (slightly more than half) bits of
information. Another important observation is that, while ANNEAL-MGUE and
ANNEAL-BCGUE reach the convergence using 5482 and 6266 bits of information for
training, respectively, the retrieval performance obtained using the final
training set by the CAL and random selection methods does not reach
convergence. All these results prove that ANNEAL-MGUE and ANNEAL-BCGUE are
more effective than the CAL and random selection methods in creating
informative training set with a lower cost.

By analyzing Fig.  7 and Fig.  8, one can observe that for both datasets
ANNEAL-MGUE shows generally higher retrieval performances compared to ANNEAL-
BCGUE under each number of bits of information used for training. This is
particularly noticeable during the first AL iterations for UC-Merced dataset
(see Fig.  7), where ANNEAL-MGUE retrieval performances are generally much
better than ANNEAL-BCGUE. As an example, the retrieval results obtained by
ANNEAL-MGUE where the number of bits of information ranges from 738 to 1476
for UC-Merced dataset (see Fig.  7) are on average 5% higher compared to those
obtained by ANNEAL-BCGUE. One can observe that this trend is confirmed also in
the AID dataset for which the results obtained by ANNEAL-MGUE are on average
2% higher compared to those obtained by ANNEAL-BCGUE. This is due to the fact
that ANNEAL-MGUE is more effective in selecting uncertain pairs compared to
ANNEAL-BCGUE due to the MGUE algorithm that directly operates in the metric
space.

Figure 7: Comparison results between the proposed ANNEAL, CAL and random
selection methods on UC-Merced dataset. Figure 8: Comparison results between
the proposed ANNEAL, CAL and random selection methods on AID dataset.

Fig. 9 shows an example of images retrieved by the CAL method and the proposed
ANNEAL-MGUE and ANNEAL-BCGUE, where the query image is selected from medium
density residential category of the UC-Merced dataset when 1467 bits of
information are used for training. The images that are different from the
category of the query image are represented within red frames. As it can be
seen in Fig. 9 (a), the CAL method retrieves the highest number of images that
are different from the query category. In particular, we can see that the
first two and the ninth retrieved images are from the dense residential
category of the UC-Merced dataset. It is worth mentioning that the two
categories, medium density residential area and dense residential area, are
very similar to each other. We can see that among the images retrieved by
ANNEAL-BCGUE [see Fig. 9 (c)] two images are from the dense residential area.
However, different from CAL method, those images are positioned later in the
retrieval order (in the third and fourth position). In the case of ANNEAL-MGUE
method [see Fig. 9 (d)] almost all the retrieved images correspond to the
category of the medium density residential. The only image that does not share
the same category with the query image is the fourth retrieved one that
belongs to the dense residential area.

Figure 9: medium density residential image retrieval example. (a) Query image
and images retrieved when 1476 bits of information are used for training the
CBIR system when b) CAL; c) the proposed ANNEAL-BCGUE; and d) the proposed
ANNEAL-MGUE are applied on UC-Merced dataset. Red frames represent images that
are dissimilar to the query image based on the class label. Figure 10: Airport
image retrieval example. (a) Query image and images retrieved when 5090 bits
of information are used for training the CBIR system when b) CAL; c) the
proposed ANNEAL-BCGUE; and d) the proposed ANNEAL-MGUE are applied on the AID
dataset. Red frames represent images that are dissimilar to the query image
based on the class label.

Fig. 10 shows an example of images retrieved by CAL method and the proposed
ANNEAL-MGUE and ANNEAL-BCGUE methods, where the query image is selected from
airport category of the AID dataset when 5090 bits of information are used for
training. One can observe that CAL method retrieves four images that belong to
different categories with respect to the query one [see Fig. 10 (b)]. We can
see that both ANNEAL-MGUE and ANNEAL-BCGUE are capable of retrieving the
images that share the same category as the query one. This is also reflected
in the quantitative results of the AID dataset (see Fig. 8) where the proposed
ANNEAL method achieve better performances with respect to CAL.

##  VI Conclusion

In this paper, we have presented ANNEAL that is an annotation cost-efficient
AL method specifically designed for DML driven CBIR systems. The aim of the
proposed AL method is to construct a small but informative training set
composed of similar and dissimilar image pairs while maintaining high
retrieval performance. To this end, the proposed ANNEAL method selects the
most informative image pairs combining the uncertainty and diversity criteria
in two consecutive steps. In the first step, to select the most uncertain
image pairs we introduce two algorithms: 1) metric guided uncertainty
estimation (MCGUE); and 2) binary classifier guided uncertainty estimation
(BCGUE). MCGUE algorithm assesses the uncertainty of the image pairs directly
in the metric space by estimating a threshold value to distinguish between
similar and dissimilar image pairs based on their cosine similarity in the
metric space. The closer the cosine similarity between any two images (i.e.,
image pairs) the higher their uncertainty. BCGUE algorithm assesses the
uncertainty of the image pairs based on the confidence of a binary classifier
in predicting the correct similarity label. The lower the confidence of the
classifier, the higher the uncertainty of the image pair. In the second step,
the proposed ANNEAL method selects the most diverse image pairs among the most
uncertain ones based on k-means clustering algorithm. ANNEAL combines one of
the proposed uncertainty methods with clustering strategy to select the most
informative samples. ANNEAL with MGUE algorithm is denoted as ANNEAL-MGUE,
while that with BCGUE algorithm as ANNEAL-BCGUE. The selected most uncertain
and diverse image pairs are sent to a human expert to be annotated as being
similar or dissimilar.

Differently from previous AL methods in CBIR problems, ANNEAL is independent
from the selected query image. As a result, it does not require training an
ad-hoc binary classifier for each query image. This makes it more suitable for
operational scenarios compared to previous AL methods designed for CBIR
problems. Compared to a literature work (e.g., classification based active
learning [42]), for which an ad-hoc classifier is also not necessary,
experimental results show the effectiveness of ANNEAL in creating a small but
informative training set with a lower annotation cost, while achieving a
higher CBIR accuracy. This is due to the capability of the proposed
uncertainty estimation algorithms MGUE and BCGUE to assess the uncertainty of
image pairs rather than single images. This allows to: i) model image
similarities in a metric space (which is of great importance for CBIR
problems); and ii) reduce the annotation cost of a training sample from
$log_{2}{C}$ for $C$ classes to 1 ($log_{2}{2}$) in terms of bits of
information. In detail, among MGUE and BCGUE algorithms experimental results
show that MCGUE algorithm achieves higher performance than BCGUE algorithm.
This is due to the fact that MCGUE algorithm employs uncertainty estimation
directly in a deep metric space that further enhances the selection of
informative image pairs, and thus increases the CBIR performance compared to
BCGUE algorithm. We would like to point out that the way of annotating
training samples with binary labels in ANNEAL can significantly decrease the
annotation cost also for other RS image analysis problems that require a
training set composed of image pairs. As an example, ANNEAL can be integrated
for change detection problems when the label of a training sample becomes
change/no-change instead of similar/dissimilar. As a future work, we plan to
extend the proposed method to different RS image analysis tasks based on image
pairs. Moreover, using ternary labels while annotating training samples (i.e.,
image triplets) can further improve the effectiveness of ANNEAL to employ DML
for CBIR problems at the cost of a slight increase in annotation cost
($log_{2}{2}\to log_{2}{3}$). Accordingly, as a future development of this
work, we plan to extend the proposed method to DML based CBIR systems trained
with image triplets.

##  VII Acknowledgement

This work is supported by the European Research Council (ERC) through the
ERC-2017-STG BigEarth Project under Grant 759764 and by the European Space
Agency through the DA4DTE (Demonstrator precursor Digital Assistant interface
for Digital Twin Earth) project. Julia Henkel’s scholarship is funded by
Google Research.

## References

  * [1] G. Sumbul, J. Kang, and B. Demir, “Deep learning for image search and retrieval in large remote sensing archives,” in Deep Learning for the Earth Sciences: A Comprehensive Approach to Remote Sensing, Climate Science and Geosciences, ch. 11, pp. 150–160, Hoboken, NJ, USA: John Wiley & Sons, 2021.
  * [2] Y. Li, J. Ma, and Y. Zhang, “Image retrieval from remote sensing big data: A survey,” Information Fusion, vol. 67, pp. 94–115, 2021.
  * [3] X.-Y. Tong, G.-S. Xia, F. Hu, Y. Zhong, M. Datcu, and L. Zhang, “Exploiting deep features for remote sensing image retrieval: A systematic investigation,” IEEE Transactions on Big Data, vol. 6, no. 3, pp. 507–521, 2020.
  * [4] W. Zhou, H. Guan, Z. Li, Z. Shao, and M. R. Delavar, “Remote sensing image retrieval in the past decade: Achievements, challenges, and future directions,” IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 16, pp. 1447–1473, 2023.
  * [5] M. Ferecatu and N. Boujemaa, “Interactive remote-sensing image retrieval using active relevance feedback,” IEEE Transactions on Geoscience and Remote Sensing, vol. 45, no. 4, pp. 818–826, 2007.
  * [6] B. Demir and L. Bruzzone, “A novel active learning method in relevance feedback for content-based remote sensing image retrieval,” IEEE Transactions on Geoscience and Remote Sensing, vol. 53, no. 5, pp. 2323–2334, 2014.
  * [7] V. Vapnik, The nature of statistical learning theory.  Springer science & business media, 1999.
  * [8] F. Melgani and L. Bruzzone, “Classification of hyperspectral remote sensing images with support vector machines,” IEEE Transactions on Geoscience and Remote Sensing, vol. 42, no. 8, pp. 1778–1790, 2004.
  * [9] G. Schohn and D. Cohn, “Less is more: Active learning with support vector machines,” in ICML, vol. 2, p. 6, Citeseer, 2000.
  * [10] S. Tong and D. Koller, “Support vector machine active learning with applications to text classification,” Journal of machine learning research, vol. 2, no. Nov, pp. 45–66, 2001.
  * [11] G.-S. Xia, J. Hu, F. Hu, B. Shi, X. Bai, Y. Zhong, L. Zhang, and X. Lu, “Aid: A benchmark data set for performance evaluation of aerial scene classification,” IEEE Transactions on Geoscience and Remote Sensing, vol. 55, no. 7, pp. 3965–3981, 2017.
  * [12] O. Sener and S. Savarese, “Active learning for convolutional neural networks: A core-set approach,” in International Conference on Learning Representations, 2018.
  * [13] L. Möllenbrok, G. Sumbul, and B. Demir, “Deep active learning for multi-label classification of remote sensing images,” IEEE Geoscience and Remote Sensing Letters, vol. 20, pp. 1–5, 2023.
  * [14] L. Möllenbrok and B. Demir, “Active learning guided fine-tuning for enhancing self-supervised based multi-label classification of remote sensing images,” in IGARSS 2023 - 2023 IEEE International Geoscience and Remote Sensing Symposium, pp. 4986–4989, 2023.
  * [15] H. Hu, L. Xie, Z. Du, R. Hong, and Q. Tian, “One-bit supervision for image classification,” Advances in Neural Information Processing Systems, vol. 33, pp. 501–511, 2020.
  * [16] Y. Zhang, X. Zhang, L. Xie, J. Li, R. C. Qiu, H. Hu, and Q. Tian, “One-bit active query with contrastive pairs,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9697–9705, 2022.
  * [17] S. Roy, S. Paul, N. E. Young, and A. K. Roy-Chowdhury, “Exploiting transitivity for learning person re-identification models on a budget,” in Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 7064–7072, 2018.
  * [18] J. Henkel, G. Hoxha, G. Sumbul, L. Möllenbrok, and B. Demir, “Annotation cost efficient active learning for content based image retrieval,” in IGARSS 2023 - 2023 IEEE International Geoscience and Remote Sensing Symposium, pp. 4994–4997, 2023.
  * [19] U. Chaudhuri, B. Banerjee, and A. Bhattacharya, “Siamese graph convolutional network for content based remote sensing image retrieval,” Computer vision and image understanding, vol. 184, pp. 22–30, 2019.
  * [20] R. Hadsell, S. Chopra, and Y. LeCun, “Dimensionality reduction by learning an invariant mapping,” in 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06), vol. 2, pp. 1735–1742, 2006.
  * [21] R. Cao, Q. Zhang, J. Zhu, Q. Li, Q. Li, B. Liu, and G. Qiu, “Enhancing remote sensing image retrieval using a triplet deep metric learning network,” International Journal of Remote Sensing, vol. 41, no. 2, pp. 740–751, 2020.
  * [22] F. Schroff, D. Kalenichenko, and J. Philbin, “Facenet: A unified embedding for face recognition and clustering,” in Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 815–823, 2015.
  * [23] R. Imbriaco., T. Alkanat., E. Bondarev., and P. H. N. de With., “Multi-branch convolutional descriptors for content-based remote sensing image retrieval,” in Proceedings of the 15th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2020) - Volume 5: VISAPP, pp. 242–249, INSTICC, SciTePress, 2020.
  * [24] P. Liu, G. Gou, X. Shan, D. Tao, and Q. Zhou, “Global optimal structured embedding learning for remote sensing image retrieval,” Sensors, vol. 20, no. 1, p. 291, 2020.
  * [25] A. Hermans, L. Beyer, and B. Leibe, “In defense of the triplet loss for person re-identification,” arXiv preprint arXiv:1703.07737, 2017.
  * [26] X. Wang, X. Han, W. Huang, D. Dong, and M. R. Scott, “Multi-similarity loss with general pair weighting for deep metric learning,” in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 5022–5030, 2019.
  * [27] L. Fan, H. Zhao, and H. Zhao, “Distribution consistency loss for large-scale remote sensing image retrieval,” Remote Sensing, vol. 12, no. 1, p. 175, 2020.
  * [28] L. Fan, H. Zhao, H. Zhao, P. Liu, and H. Hu, “Distribution structure learning loss (dsll) based on deep metric learning for image retrieval,” Entropy, vol. 21, no. 11, 2019.
  * [29] G. Sumbul, M. Ravanbakhsh, and B. Demir, “Informative and representative triplet selection for multilabel remote sensing image retrieval,” IEEE Transactions on Geoscience and Remote Sensing, vol. 60, pp. 1–11, 2022.
  * [30] P. Li and P. Ren, “Partial randomness hashing for large-scale remote sensing image retrieval,” IEEE Geoscience and Remote Sensing Letters, vol. 14, no. 3, pp. 464–468, 2017.
  * [31] S. Roy, E. Sangineto, B. Demir, and N. Sebe, “Metric-learning-based deep hashing network for content-based retrieval of remote sensing images,” IEEE Geoscience and Remote Sensing Letters, vol. 18, no. 2, pp. 226–230, 2021.
  * [32] X. Tan, Y. Zou, Z. Guo, K. Zhou, and Q. Yuan, “Deep contrastive self-supervised hashing for remote sensing image retrieval,” Remote Sensing, vol. 14, no. 15, 2022.
  * [33] M. Huang, L. Dong, W. Dong, and G. Shi, “Supervised contrastive learning based on fusion of global and local features for remote sensing image retrieval,” IEEE Transactions on Geoscience and Remote Sensing, vol. 61, pp. 1–13, 2023.
  * [34] H.-F. Yang, K. Lin, and C.-S. Chen, “Supervised learning of semantics-preserving hash via deep convolutional neural networks,” IEEE transactions on pattern analysis and machine intelligence, vol. 40, no. 2, pp. 437–451, 2017.
  * [35] K. Sohn, “Improved deep metric learning with multi-class n-pair loss objective,” Advances in neural information processing systems, vol. 29, 2016.
  * [36] A. K. Jain and R. C. Dubes, Algorithms for clustering data.  Prentice-Hall, Inc., 1988.
  * [37] Y. Yang and S. Newsam, “Bag-of-visual-words and spatial extensions for land-use classification,” in Proceedings of the 18th SIGSPATIAL international conference on advances in geographic information systems, pp. 270–279, 2010.
  * [38] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016.
  * [39] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, “Imagenet: A large-scale hierarchical image database,” in 2009 IEEE Conference on Computer Vision and Pattern Recognition, pp. 248–255, 2009.
  * [40] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv preprint arXiv:1412.6980, 2014.
  * [41] P. Branco, L. Torgo, and R. Ribeiro, “A survey of predictive modeling on imbalanced domains,” ACM computing surveys (CSUR), vol. 49, no. 2, pp. 1–50, 2016.
  * [42] D. Wang and Y. Shang, “A new active labeling method for deep learning,” in 2014 International Joint Conference on Neural Networks (IJCNN), pp. 112–119, 2014.