# Towards a refined estimate for topological degree in one dimension

Felipe Hernández

###### Abstract

In this paper we prove an inequality inspired by a conjecture of Brezis, which
asks for a bound for the topological degree of a map from the circle to itself
in terms of a nonlocal integral.

##  1 Introduction

Thie note concerns a question posed by Haim Brezis regarding the topological
degree of continuous maps from the circle ${{\mathbb{S}}}^{1}$ to itself
[Bre06, Open Problem 3].

######  Conjecture 1.1.

There exists a constant $C>0$ and $\delta_{0}>0$ such that the following
holds: For any continuous function $f\in
C^{0}({{\mathbb{S}}}^{1};{{\mathbb{S}}}^{1})$ and $\delta<\delta_{0}$,

$|\deg f|\leq C\delta\iint_{|f(s)-f(t)|>\delta}\frac{dsdt}{|s-t|^{2}}.$ (1)

This conjecture was inspired by the work of Bourgain, Brezis, and Nguyen
[BBN05], which established (1) without the factor of $\delta$ in the right
hand side. More recently, the analogous statement in dimension $d\geq 2$ has
been proven by Hoai-Minh Nguyen [Ngu17]. Remarkably, the one-dimensional case
remains the most difficult to handle.

In this note we prove a closely related statement which still seems to capture
some of the difficulty of Conjecture 1.1. For any map $f\in
C^{0}({{\mathbb{S}}}^{1};{{\mathbb{S}}}^{1})$ we can find some continuous
phase function $\varphi:{\mathbf{R}}\to{\mathbf{R}}$ so that $f(e^{2\pi
is})=e^{2\pi i\varphi(s)}$ and $\deg f=\varphi(1)-\varphi(0)$. The main result
is phrased in terms of functions $\varphi$ defined in this way.

######  Theorem 1.2.

Let $\varphi:{\mathbf{R}}\to{\mathbf{R}}$ be a continuous function such that
there exists $D\in{\mathbf{R}}$ such that $\varphi(x+1)=\varphi(x)+D$ for any
$x\in{\mathbf{R}}$. Then for $\delta<0.01$,

$|D|\leq
10^{9}\delta\iint_{\\{s,t\in[0,2]\mid|\varphi(s)-\varphi(t)|>\delta\\}}\frac{dsdt}{|s-t|^{2}}.$
(2)

To prove Theorem 1.2, we investigate a closely related problem in which the
domain of integration is restricted to $[0,1]$ rather than $[0,2]$, and so
that no periodicity conditions are imposed on the function.

######  Question 1.

Is it the case that there exists $C,\delta_{0}>0$ such that for all $g\in
C^{0}([0,1];{\mathbf{R}}^{+})$ and $\delta<\delta_{0}$,

$|g(1)-g(0)|\leq
C\delta\cdot\iint_{\\{s,t\in[0,1]\mid|g(s)-g(t)|>\delta\\}}\frac{dsdt}{|s-t|^{2}}?$

The answer to Question 1 is negative: for each $\delta>0$ there exist examples
of functions for which $g(1)-g(0)$ grows as large as desired and for which the
integral on the right is as small as desired. In fact, the function
$g(s)=\alpha\log(s+\beta)$ works for sufficiently small $\beta$ and $\alpha$
depending on $\delta$.

The proof of Theorem 1.2 will proceed by showing that all such counterexamples
to Question 1 have such a logarithmic-type singularity, either at $0$ or at
$1$. This will allow us to recover the desired contribution from the
integration for $s,t$ on either side of the point $1$.

The main challenge is that tight logarithmic-type singularities appearing
within the interval $[0,1]$ do not contribute much to the nonlocal integral
(2), so there could be millions of such almost-singularities in $[0,1]$.
Therefore, although $\varphi$ is continuous, its value at any given point is
almost meaningless so instead the argument works more with the values of
$\varphi$ on finite intervals.

###  1.1 Acknowledgements

The author is supported by the John and Fannie Hertz Foundation.

##  2 The structure result

The main result is actually a structure theorem for positive functions $g\in
C^{0}([0,1];{\mathbf{R}})$ that are negative examples for Question 1. We fix
the function $g$ and the scale $\delta$. We suppose for simplicity that
$M\delta=g(1)$ for some $100<M\in{{\mathbb{N}}}$, and that $g(0)=0$. We also
assume without loss of generality that the level sets $\\{s\mid
g(s)=k\delta\\}$ for $k\in{{\mathbb{N}}}$ are finite. If not, one may make an
extremely small perturbation to $g$ to ensure that this is the case.

To state the structure theorem we will need several definitions. First we
define the discretized level sets

$E_{k}:=\\{s\in[0,1]\mid g(s)\in[k\delta,(k+1)\delta]\\}$ (3)

and their neighbors

$N_{k}:=E_{k-1}\cup E_{k}\cup E_{k+1}.$ (4)

Using $E_{k}$ and $N_{k}$ we may write a simple lower bound for our nonlocal
integral quantity.

$\sum_{k\in{{\mathbb{N}}}}\iint_{s\in E_{k};t\not\in
N_{k}}\frac{dsdt}{|s-t|^{2}}\leq\iint_{|g(s)-g(t)|>\delta}\frac{dsdt}{|s-t|^{2}}.$
(5)

To simplify the study of the left hand side, we make the following
observation. If there exists an interval $I\subset[0,1]$ such that $|E_{k}\cap
I|>0.01|I|$ and $|N_{k}^{c}\cap I|>0.01|I|$, then

$\iint_{s\in E_{k};t\not\in N_{k}}\frac{dsdt}{|s-t|^{2}}\geq 10^{-4}.$

This motivates us to define the set of satisfied indices,

$\mathcal{S}=\\{k\in\mathbb{N}\mid\exists I=(a,b)\subset[0,1]\text{ s.t.
}|E_{k}\cap I|>0.01|I|\text{ and }|N_{k}^{c}\cap I|>0.01|I|\\},$ (6)

for which we have the bound

$\\#{\mathcal{S}}\leq 10^{4}\iint_{|g(s)-g(t)|>\delta}\frac{dsdt}{|s-t|^{2}}.$
(7)

The definition of $\mathcal{S}$ can be rephrased in terms of the notions of
empty and full intervals, which we define now.

######  Definition 2.1 (Empty and full intervals).

Let $I\subset[0,1]$ be an interval and let $0<\alpha<1$. We say that $I$ is
$\alpha$-full for the index $k\in{\mathbf{Z}}$ if

$|E_{k}\cap I|>\alpha|I|.$

On the other hand, we say that $I$ is $\alpha$-empty for $k$ if

$|N_{k}^{c}\cap I|>\alpha|I|.$

Observe that for $\alpha<0.5$, it is possible for an index to be both
$\alpha$-full and $\alpha$-empty for an interval.

The idea is to prove that either $\mathcal{S}$ is a large set, or else $g$ has
a special behavior near the endpoints $0$ and $1$. There are several possible
behaviors, corresponding to the different counterexamples to Question 1.

######  Definition 2.2 (Logarithmic singularities).

We say that a function $g\in C^{0}([0,1];{\mathbf{R}}^{+})$ has a _logarithmic
singularity at the boundary_ if there exists an $m>0.1M$ and sequences of
indices

$k_{0}<k_{1}<\cdots<k_{m}$

and lengths

$\tau_{0}>3\tau_{1}>\cdots>3^{m}\tau_{m}$

such that one of the following holds:

  * •

Growth near 1: The interval $(1-\tau_{i},1)$ is an $0.03$-full interval for
the index $k_{i}$ and $(0,\tau_{i})$ is an $0.03$-empty interval for the
indices $k_{i}-M-1$, $k_{i}-M$, and $k_{i}-M+1$.

  * •

Growth near 0: The interval $(0,\tau_{m-i})$ is an $0.03$-full interval for
the index $k_{i}$ and $(1-\tau_{m-i},1)$ is an $0.03$-empty interval for the
indices $k_{i}+M-1$, $k_{i}+M$, and $k_{i}+M+1$.

A set $\\{(k_{i},\tau_{i})\\}_{i=1}^{m}$ satisfying one of these conditions is
called a _witness_ to the logarithmic growth of $g$.

We are now ready to state the structure result.

######  Theorem 2.3.

At least one of the following holds. Either

$\\#\mathcal{S}\geq 0.01M$

or else $g$ has a logarithmic singularity as defined above.

###### Proof of Theorem 1.2 using Theorem 2.3.

We apply Theorem 2.3 to the function $\varphi$ restricted to the interval
$[0,1]$. In the case that $\\#\mathcal{S}\geq 0.01M$, then (7) implies the
result (since $M=\varphi(1)-\varphi(0)$).

We can therefore suppose the existence of a logarithmic singularity, and
without loss of generality we will suppose that there is growth at $1$,
meaning that there exist an increasing sequence $\\{k_{j}\\}_{j=1}^{m}$,
$m>0.1M$, and a geometrically decreasing sequence $\\{\tau_{j}\\}_{j=1}^{m}$
such that $(1-\tau_{i},1)$ is a $0.03$-full interval for $k_{i}$ and
$(0,\tau_{i})$ is an $0.03$-empty interval for $k_{i}-M-1,k_{i}-M$,and
$k_{i}-M+1$.

Then, extending the notion of empty intervals to the function $\varphi$ on
$[0,2]$, this implies that $(1,1+\tau_{i})$ is a $0.03$-empty interval for
$k_{i}-1$,$k_{i}$, and $k_{i}+1$. Therefore

$\int_{s\in(1-\tau_{i},1)\cap E_{i}}\int_{t\in(1,1+\tau_{i})\cap
N_{i}^{c}}\frac{dsdt}{|s-t|^{2}}\geq 10^{-6}.$ (8)

Each $1\leq i\leq m$ contributes $10^{-6}$, so the result follows. ∎

##  3 Proof of Theorem 2.3

Throughout the proof we will suppose that $g\in C^{0}([0,1];{\mathbf{R}}^{+})$
is some fixed positive function with $g(0)=0$ and $M\delta=g(1)$, and that
$\\#{\mathcal{S}}\leq 0.01M$, where ${\mathcal{S}}$ is defined in (6).

We first define the set $\mathcal{P}$ of pairs of unsatisfied indices

$\mathcal{P}=\\{k\in{{\mathbb{N}}}\mid
k\not\in\mathcal{S},k-1\not\in\mathcal{S},\text{ and
}k+1\not\in\mathcal{S}\\}.$ (9)

Observe that $\\#\mathcal{P}^{c}\leq 3\\#\mathcal{S}$, so we will be able to
find plenty of indices in $\mathcal{P}$. This is useful because the condition
$k\in\mathcal{P}$ is actually rather strong.

######  Lemma 3.1.

If $k\in\mathcal{P}$ and $I$ is an interval with $0.01|I|\leq|E_{k}\cap I|\leq
0.1|I|$, then either $|E_{k-1}\cap I|>0.8|I|$ or $|E_{k+1}\cap I|>0.8|I|$.

###### Proof.

Since $k\not\in\mathcal{S}$, it follows that $|N_{k}\cap I|>0.99|I|$. Suppose
that $|E_{k-1}\cap I|>0.01|I|$. Then, using the fact that
$k-1\not\in\mathcal{S}$, we have $|E_{k+1}\cap I|<0.01|I|$. ∎

We will now pick a very special set of intervals $\mathcal{I}_{k}$ which we
call canonical intervals.

######  Definition 3.2 (Canonical intervals).

We say that the collection of intervals
$\mathcal{I}=\\{\\{(s_{i,k},t_{i,k})\\}_{i=1}^{n_{k}}\\}_{k=1}^{M}$ are
_canonical_ if for each $k\in{{\mathbb{N}}}$, the collection $\mathcal{I}$
satisfies the following conditions:

  * •

Covering:

$E_{k}\subset\bigcup_{i=1}^{n_{k}}(s_{i,k},t_{i,k}).$

  * •

Ordering: For each $k\in[M]$,

$s_{1,k}<t_{1,k}<s_{2,k}<t_{2,k}<\cdots<s_{n_{k},k}<t_{n_{k},k}.$

In addition we require $s_{1}>a$ when $k>1$.

  * •

Balance: For each $i\in[n_{k}]$

$|E_{k}\cap(s_{i,k},t_{i,k})|\geq 0.1(t_{i,k}-s_{i,k}).$

with equality holding unless $i=n_{k}$ and $t_{i}=1$.

We call the elements of $\mathcal{I}^{J}_{k}$ _canonical intervals_ for the
index $k$.

Notice that the balance condition enforces that either $n_{k}=1$ and
$E_{k}=(s,1)$, or else

$E_{k}\subsetneq\bigcup_{i=1}^{n_{k}}(s_{i,k},t_{i,k}).$

It is not difficult to see that a canonical collection always exists, and we
choose now a canonical collection $\mathcal{I}$. We write
${\mathcal{I}}_{k}=\\{(s_{i,k},t_{i,k})\\}_{i=1}^{n_{k}}$ for the collection
of canonical intervals covering $E_{k}$.

One way to think about $\mathcal{I}_{k}$ is that it describes a simplification
of the level set $E_{k}$ that is more robust to the kinds of rapid
oscillations that $g$ may have which ultimately do not affect the nonlocal
integral. A key concept that uses the collection $\mathcal{I}$ is that of a
compatible interval.

######  Definition 3.3.

An interval $I$ is _compatible_ with $\mathcal{I}_{k}$ if for every
$(s,t)\in\mathcal{I}_{k}$, either $(s,t)\subset I$ or $(s,t)$ is disjoint from
$I$.

Another important use of the idea of canonical intervals is that Proposition
3.1 allows us to define a dichotomy between bottom-heavy intervals and top-
heavy intervals.

######  Definition 3.4 (Bottom-heavy and top-heavy intervals).

Let $I\in\mathcal{I}_{k}$ be a canonical interval with $|E_{k}\cap I|=0.1|I|$.
We say that $I$ is a _bottom-heavy_ interval if $|I\cap E_{k+1}|>0.8|I|$. If
on the other hand $|I\cap E_{k-1}|>0.8|I|$, then we say that $I$ is a _top-
heavy_ interval.

If $k\in P$ and $I\in\mathcal{I}_{k}$, then $I$ must be either top-heavy,
bottom-heavy, or else $|I\cap E_{k}|>0.1|I|$. By the balance condition, the
latter case can only happen when the right-endpoint of $I$ is $1$. It will be
useful to introduce some terminology to handle this case.

######  Definition 3.5 (Terminal intervals).

A _terminal_ interval for the index $k$ is an interval $I=(a,b)\subset[0,1]$
such that $|I\cap E_{k}|>0.03|I|$ and either $a=0$ or $b=1$. If $a=0$, then
$I$ is a _left-terminal_ interval. Conversely if $b=1$ then $I$ is a _right-
terminal_ interval. A _maximal_ terminal interval $I$ is a terminal interval
with the property that any interval $I^{\prime}$ with $I\subset I^{\prime}$
satisfies $|I^{\prime}\cap E_{k}|<0.03|I|$.

The following simple lemma is one main reason we can hope to demonstrate the
existence of a logarithmic divergence.

######  Lemma 3.6.

Let $|k-k^{\prime}|>1$ and either $k\not\in\mathcal{S}$ or
$k^{\prime}\not\in\mathcal{S}$. Suppose $k$ and $k^{\prime}$ have maximal
left-terminal intervals of length $\tau$ and $\tau^{\prime}$, respectively.
Then either $\tau>3\tau^{\prime}$ or $\tau^{\prime}>3\tau$. The same statement
holds for maximal right-terminal intervals.

###### Proof.

Without loss of generality suppose that $\tau>\tau^{\prime}$ but
$\tau<3\tau^{\prime}$. Consider the interval $I=[0,3\tau^{\prime}]$. Observe
that $|I\cap E_{k}|,|I\cap E_{k^{\prime}}|>0.01|I|$, which is a contradiction.
∎

We now take another small step towards demonstrating logarithmic divergence,
which is to define a set of indices and lengths
$\mathcal{G}_{L}=\\{(k_{i},\tau_{i})\\}_{i=1}^{m}$ which we expect to serve as
a witness to logarithmic growth near the left endpoint $0$.

The set $\mathcal{G}_{L}$ is defined inductively as follows. We set $k_{0}=0$
and $\tau_{0}$ to be the length of the maximal left-terminal interval for the
index $0$.111Observe that $0$ must have a left-terminal interval because
$\theta(0)=0$. Given $k_{i}$ and $\tau_{i}$, let $k$ be the least index
$k\in\mathcal{P}$ such that $k>k_{i}$ and such that there is a maximal left-
terminal interval $(0,\tau)$ with $\tau>3\tau_{i}$. If such an index exists,
we set $k_{i+1}=k$ and $\tau_{i+1}=\tau$ and continue. If not, the process
halts and the definition of $\mathcal{G}_{L}$ is complete.

We also define $\mathcal{G}_{R}$, the set of indices that are separated from
$\mathcal{G}_{L}$ and are not in $\mathcal{G}_{L}$. More precisely, define

$\mathcal{G}_{R}=\\{k\in\mathcal{P}\setminus\mathcal{G}_{L}\mid
k-1\not\in\mathcal{G}_{L}\\}.$

We have used notation to suggest that $\mathcal{G}_{R}$ is a good candidate
for a witness of logarithmic growth near the right endpoint $1$. This is not
clear from the definition, and in fact the largest remaining task is to prove
that this is the case.

Our first main lemma sets up conditions under which we can control the size of
the discretized superlevel sets

$E_{>k}:=\\{t\in[0,1]\mid g(t)>(k+1)\delta\\}$

using only information about the index $k$. In particular we introduce the
notion of a grounded interval.

######  Definition 3.7 (Grounded interval).

Let $J=(a,b)\subset[0,1]$ be an interval. We say that $J$ is _grounded_ for
the index $k$ if $J$ is an interval compatible with $\mathcal{I}_{k}$, if
$g(a)<(k+1)\delta$, and if for every $I\in{\mathcal{I}}_{k}$ with $I\subset
J$, $I$ is a bottom-heavy interval.

######  Lemma 3.8.

Let $k\in\mathcal{P}$, and let $J=(a,b)\subset[0,1]$ be a grounded interval
for the index $k$. Then

$|E_{>k}\cap J|\leq\frac{1}{5}|E_{k}\cap J|$ (10)

and

$|E_{k}\cap J|<\frac{1}{5}|E_{k-1}\cap J|.$ (11)

###### Proof.

Let $I_{1},I_{2},\cdots,I_{n}$ with $I_{i}=(s_{i},t_{i})$ be an enumeration in
order of all bottom-heavy canonical intervals in $\mathcal{I}_{k}$ that are
contained in $J$. The proof goes by induction on $n$. In the case $n=0$, the
covering property implies $E_{k}\cap J$ is empty. Therefore $\theta(a)\leq
k\delta$ and the intermediate value theorem implies that $\theta(s)\leq
k\delta$ for all $s\in J$, and thus $|E_{>k}\cap J|=0$.

Now consider the case $n\geq 1$, and let $I_{1}=(s_{1},t_{1})$. Again by the
intermediate value theorem and the covering property of $\mathcal{I}_{k}$,
$\theta(s)\leq k\delta$ for $s\in(a,s_{1})$. Since $k\in\mathcal{P}$ and
$I_{1}$ is bottom-heavy, we have $|I_{1}\cap E_{k}|=0.1|I_{1}|$, $|I_{1}\cap
E_{>k}|<0.01|I_{1}|$, and $|I_{1}\cap E_{k-1}|>0.8|I_{1}|$. If
$\theta(t_{1})\leq(k+1)\delta$, then the interval $(t_{1},t)$ satisfies the
inductive hypothesis, having $n-1$ canonical intervals inside.

The remaining case is that $\theta(t_{1})>(k+1)\delta$. Let $t^{\prime}$ be
the first time $t^{\prime}>t_{1}$ for which $\theta(t^{\prime})=(k+1)\delta$.
Suppose that $t^{\prime}>(t_{1}-s_{1})$, and consider the interval
$I_{1}^{\prime}=(s_{1},2t_{1}-s_{1})$. In this case, $|N_{k-1}^{c}\cap
I_{1}^{\prime}|>0.5|I^{\prime}_{1}|$ and $|E_{k-1}\cap
I_{1}^{\prime}|>0.4|I_{1}^{\prime}|$, which contradicts the assumption that
$k-1\not\in\mathcal{S}$. Thus it must be that $t^{\prime}<(t_{1}-s_{1})$.

Define now the interval $I=(s_{1},t^{\prime})$. Then $|E_{k-1}\cap I|>0.4|I|$,
so it follows that $|N_{k-1}^{c}\cap I|<0.01|I|$, and in particular

$|E_{>k}\cap I|<0.01|I|<0.02|I_{1}|=\frac{1}{5}|E_{k}\cap I_{1}|.$

At this point one can now induct, using the smaller interval
$I^{\prime}=[t^{\prime},t]$ which has fewer bottom heavy canonical intervals.
∎

The following lemma is our main tool for finding grounded intervals.

######  Lemma 3.9.

Let $m\in\mathcal{G}_{R}$ and $k\in\mathcal{G}_{L}$ be the largest index in
$\mathcal{G}_{L}$ satisfying $k<m$. Let $\tau_{k}$ be the length of the
maximal left-terminal interval for $k$. Then there exists an interval
$J=(s,t)$ such that $s<\tau_{k}/2$, $J$ is grounded for the index $m$, and
$(t,1)$ is compatible with $\mathcal{I}_{m}$.

###### Proof.

Observe that, by the definition of $\mathcal{G}_{R}$, we have $k<m-1$.

We directly define

$s=\inf\\{s^{\prime}\in(\tau_{k}/3,1]\mid
g(s^{\prime})<(m+1)\delta,s^{\prime}\not\in\bigcup_{I\in{\mathcal{I}}_{m}\cup{\mathcal{I}}_{m+1}}I\\}.$

First, we claim that $s<\tau_{k}/2$. Indeed, for every point
$s^{\prime}\in(\tau_{k}/3,s)$, we either have $\theta(s)>(m+1)\delta$ or else
there exists an interval $I\in\mathcal{I}_{m}\cup\mathcal{I}_{m+1}$ such that
$I\subset[0,s)$ and $s^{\prime}\in I$. Thus

$|(0,s)\cap E_{\geq
m}|\geq\sum_{\begin{subarray}{c}I\in\mathcal{I}_{m}\cup\mathcal{I}_{m+1}\\\
I\subset(0,s)\end{subarray}}0.1|I|+\Big{|}(\tau_{k}/3,s)\setminus\bigcup_{I\in{\mathcal{I}}_{m}\cup{\mathcal{I}}_{m+1}}I\Big{|}\geq
0.1(s-\tau_{k}/3)$

If $s>\tau_{k}/2$, then this implies

$|(0,\tau_{k})\cap E_{\geq m}|\geq|(0,\tau_{k}/2)\cap E_{\geq
m}|\geq\frac{1}{60}\tau_{k},$

this would contradict $k\not\in\mathcal{S}$, since $|(0,\tau_{k})\cap
E_{k}|>0.03\tau_{k}$ and $k<m-1$.

It remains to show that every $I\in\mathcal{I}_{m}$ with
$|I\cap(s^{\prime},1)|>0$ is either bottom-heavy or a terminal interval.

On the contrary, suppose that $I=(s_{0},s_{1})$ is the first top-heavy
interval in $\mathcal{I}_{m}$ with $s_{0}>s$. Consider
$\sigma=s_{0}-2(s_{1}-s_{0})$. If $\sigma<0$, then

$|E_{m}\cap(0,s_{1})|\geq 0.1(s_{1}-s_{0})\geq 0.03(s_{1}-\sigma).$

In particular $(0,s_{1})$ is a left-terminal interval for $m$ with
$s_{1}>\tau_{k}/3$, which along with Proposition 3.6 contradicts
$m\not\in\mathcal{G}_{L}$.

Therefore we can assume that $\sigma>0$. The next sub-case to consider is
$\sigma<s$. In this case, split the interval $(\sigma,s_{0})$ into the
interval $(\sigma,s)$ and $(s,s_{0})$. Since $s\not\in I$ for any
$I\in\mathcal{I}_{m}\cup\mathcal{I}_{m+1}$, we have $|(\sigma,s)\cap
E_{m}|<0.1(s-\sigma)$ and $|(\sigma,s)\cap E_{m+1}|<0.1(s-\sigma)$. Moreover,
$|(s,s_{0})\cap E_{m}|<0.1(s_{0}-s^{\prime})$ because every canonical interval
that intersects $(s,s_{0})$ is contained in $(s,s_{0})$. Finally,
$|(s,s_{0})\cap E_{m+1}|<0.02(s_{0}-s)$ because the interval $(s,s_{0})$
satisfies the conditions of Lemma 11. In conclusion, we have

$|E_{m}\cap(\sigma,s_{0})|<0.1(s_{0}-\sigma)$

and

$|E_{m+1}\cap(\sigma,s_{0})|<0.1(s_{0}-\sigma).$

These inequalities are mutually incompatible with the hypotheses that
$m,m+1\not\in\mathcal{P}$, since
$|E_{m}\cap(\sigma,s_{1})|>0.03(s_{1}-\sigma)$ and
$|E_{m+1}\cap(\sigma,s_{1})|>0.1(s_{1}-\sigma)$ because $(s_{0},s_{1})$ is
top-heavy.

The final case to consider is $\sigma>s$. If $(\sigma,s)$ is compatible with
${\mathcal{I}}_{k}$, then the argument above applies to reach a contradiction.
The remaining case therefore is that
$\sigma\in(t_{0},t_{1})\in{\mathcal{I}}_{k}\cup{\mathcal{I}}_{k+1}$. If
$(t_{0},t_{1})\in{\mathcal{I}}_{k}$, then because $(t_{0},t_{1})$ is bottom-
heavy by hypothesis,

∎

One consequence of Lemma 3.9 and Lemma 11 is that the sets $\mathcal{G}_{L}$
and $\mathcal{G}_{R}$ cannot interlace much.

######  Lemma 3.10.

Let $b\in\mathcal{G}_{R}$ and suppose that there exists $k\in\mathcal{G}_{L}$
such that $k>b$. Then the length $\tau_{k}$ of the maximal left-terminal
interval for $k$ satisfies $\tau_{k}>\frac{1}{3}$. In particular, $k$ is the
largest element in $\mathcal{G}_{L}$.

###### Proof.

Let $a\in\mathcal{G}_{L}$ be the largest index in $\mathcal{G}_{L}$ with
$a<b$, and let $\tau_{a}$ be the length of the maximal left-terminal interval
for $a$.

We will first show that $\tau_{k}$ is in a terminal interval of
$\mathcal{I}_{b}$, by first showing that $\tau_{k}$ is in a canonical interval
of $\mathcal{I}_{b}$, and then by demonstrating that it is terminal.

Applying Lemma 3.9, we may find an interval $(s,t)$ with $s<\tau_{a}/2$ which
is grounded for the index $b$. If $\tau_{k}\not\in I$ for every
$I\in\mathcal{I}_{b}$, then the interval $(s,\tau_{k})$ is grounded for the
index $b$. Then by Lemma 11,

$|E_{k}\cap(s,\tau_{k})|\leq|E_{>b}\cap(s,\tau_{k})|<\frac{1}{5}|E_{b}\cap(s,\tau_{k})|=0.02(\tau_{k}-s).$

Moreover since $a\not\in\mathcal{S}$,

$|E_{k}\cap(0,s)|<0.01s.$

Combined, these give $|E_{k}\cap(0,\tau_{k})|<0.02\tau_{k}$, so $\tau_{k}$
cannot be a left-terminal interval.

Now we check that $\tau_{k}$ must be contained in a terminal interval. If not,
then $\tau_{k}\in(s_{b},t_{b})\in\mathcal{I}_{b}$ with $t_{b}<1$. The same
argument used above shows that $|E_{k}\cap(0,s_{b})|<0.02s_{b}$. We will now
prove that $|E_{k}\cap(s_{b},\tau_{k})|<0.02(\tau_{k}-s_{b})+0.01\tau_{k}$,
which will suffice to arrive at the contradiction
$|E_{k}\cap(0,\tau_{k})|<0.03\tau_{k}$.

First, using the fact that $t_{b}<1$ and $b\not\in\mathcal{G}_{L}$, we have

$0.1(t_{b}-s_{b})=|E_{b}\cap(s_{b},t_{b})|\leq|E_{b}\cap(0,t_{b})|<0.03t_{b}.$

Rearranging and using $s_{b}<\tau_{k}$, we can certainly achieve the bound
$t_{b}<2\tau_{k}$.

We combine this with the bound $|E_{k}\cap(s_{b},t_{b})|<0.01(t_{b}-s_{b})$
which comes from the fact that $J$ is bottom heavy, to conclude

$\displaystyle|E_{k}\cap(s_{b},\tau_{k})|\leq|E_{k}\cap(s_{b},t_{b})|$
$\displaystyle\leq 0.01(t_{b}-s_{b})$ $\displaystyle<0.01(2\tau_{k}-s_{b})\leq
0.02(\tau_{k}-s_{b})+0.01\tau_{k},$

as desired.

We have thus demonstrated that $\tau_{k}\in(s_{b},1)\in\mathcal{I}_{b}$. Since
$\tau_{k}<1$, it must follow that $\tau_{a}<\frac{1}{3}$. This allows us to
write

$0.1(1-s_{b})<|E_{b}\cap(s_{b},1)|<|E_{b}\cap(0,1)|<0.03,$

from which we conclude $s_{b}>0.7>\frac{1}{3}$. Since $\tau_{k}>s_{b}$ this
certainly implies the conclusion. ∎

Let $\tau_{L}$ be the length of the largest maximal left-interval in
$\mathcal{G}_{L}$, and $\tau_{R}$ be the largest maximal right-interval in
$\mathcal{G}_{R}$ ($\tau_{R}=0$ is possible). The thrust of the previous two
lemmas is that we achieve exponential decay for the maximal right-intervals of
the indices in $\mathcal{G}_{R}$.

######  Lemma 3.11.

Let $m\in\mathcal{G}_{R}$, and let $k>m+1$. Let $\tau_{m}$ be the length of
the maximal right-interval for $m$. Then the length $\tau_{k}$ of the maximal
right-interval for $k$ satisfies either $3\tau_{k}<\tau_{m}$ or
$\tau_{k}>\frac{1}{2}$.

###### Proof.

By Proposition 3.6, we only need to rule out the case
$3\tau_{m}<\tau_{k}<\frac{1}{2}$. By Lemma 3.9, there exists an interval $J$
containing $(\frac{1}{2},1-\tau_{m})$ which is grounded for index $m$. If
$\tau_{k}<\frac{1}{2}$, then $1-\tau_{k}\in(\frac{1}{2},1-\tau_{m})$. Let
$s\in J$ be maximal such that $(s,1-\tau_{m})$ is a grounded interval and
$s\leq 1-\tau_{k}$. Observe that either $s=1-\tau_{m}$ or else
$(s,t)\in\mathcal{I}_{m}$ for some $t>1-\tau_{k}$. In either case we have

$|E_{m}\cap(s,1)|>0.1(1-\tau_{k}-s).$

Combined with the inequality

$|E_{m}\cap(s,1)|<0.03(1-s),$

which follows from the maximality of $\tau_{m}$, we obtain

$1-s<\frac{10}{7}\tau_{k}.$

Next, because $(s,1-\tau_{m})$ is a grounded interval, it follows from Lemma
11 that

$|E_{k}\cap(1-\tau_{k},1-\tau_{m})|\leq|E_{k}\cap(s,1-\tau_{m})|\leq\frac{1}{5}|E_{m}\cap(s,1-\tau_{m})|<0.03\frac{2}{7}\tau_{k}.$

Now we compute, using also $\tau_{k}>3\tau_{m}$,

$\displaystyle|E_{k}\cap(1-\tau_{m},1)|$
$\displaystyle=|E_{k}\cap(1-\tau_{k},1)|-|E_{k}\cap(1-\tau_{k},1-\tau_{m})|$
$\displaystyle\geq 0.03\tau_{k}(1-\frac{2}{7})>0.01\tau_{m}.$

This violates the fact that $m\not\in\mathcal{S}$, so we have arrived at a
contradiction. ∎

We need one final lemma, which is useful in the case that
$\\#\mathcal{G}_{L}<0.99M$.

######  Lemma 3.12.

If $k\in\mathcal{G}_{R}$ and $k\leq M$, then there is a right-terminal
interval for $k$.

###### Proof.

By Lemma 3.9, there exists $s$ and $t$ such that $(s,t)$ is a grounded
interval for $k$, and $(t,1)$ is a terminal interval.

In particular, there exist canonical intervals $J\in\mathcal{I}_{k}$ contained
in $(s^{\prime},1)$. Consider the last such interval
$J=(s_{n_{k},k},t_{n_{k},k})=:(s_{f},t_{f})$. Our claim is that $(s_{f},1)$ is
a right-terminal interval. Indeed, if $t_{f}=1$, this means that $|J\cap
E_{k}|>0.1|J|$, and we are done.

Otherwise, we have that $|J\cap E_{k}|=0.1|J|$ and $|J\cap E_{k-1}|>0.8|J|$,
because $J$ must be bottom-heavy. In this case, it suffices to show that
$1-t_{f}<(1-s_{f})/2$.

The hypothesis that $J$ is the final canonical interval and $k<\delta^{-1}D$
together imply that $\theta(t)>(k+1)\delta$ for all $t>t_{f}$. Suppose that
$t_{f}<1-\tau/2$, and consider the interval
$J^{\prime}=(1-\tau,1-\tau/2)\subset[0,1]$. Observe that $|J^{\prime}\cap
E_{k-1}|>0.35(\tau/2)$ and $|J^{\prime}\cap L_{k-1}^{c}|>0.5(\tau/2)$. This
would contradict the hypothesis that $k-1\not\in\mathcal{S}$.

In particular, the interval $I=(s_{f},1+(t_{f}-s_{f}))\subset[0,2]$ satisfies
$|I\cap E_{k}|>0.01|I|$ and

$|I\cap E_{>N}|=0.5|I|.$

∎

We are now ready to complete the proof of the main result.

###### Proof of Theorem 2.3.

We can first assume that $\\#\mathcal{S}<0.01M$, in which case we need to
demonstrate a logarithmic singularity for $g$. As we observed, this assumption
implies that $\\#\mathcal{P}^{c}\leq 0.03M$.

We now split the analysis into two cases. In the first case, we suppose that
$\\#\mathcal{G}_{L}<0.9M$. In this case, one has
$\\#(\mathcal{G}_{R}\cap[M-1])>0.07M$. Consider the set

$\mathcal{W}=\\{k\in\mathcal{G}_{R}\cap[M]\mid\tau_{k}<\frac{1}{2}\text{ and
}k\text{ is even}\\},$

Replacing ‘even’ with ‘odd’ if necessary, we may assume that
$\\#\mathcal{W}>0.03M$ (provided that $M$ is sufficiently large, because at
most $100$ indices can have $\tau_{k}>\frac{1}{2}$). Using Lemma 3.11, we see
that $\tau_{k}>3\tau_{k^{\prime}}$ when $k^{\prime}>k$ and
$k,k^{\prime}\in\mathcal{W}$. Moreover, as $E_{k-M}$ is empty for each
$k\in\mathcal{W}$, we see that the indices in $\mathcal{W}$ serve as a witness
for a logarithmic singularity of $g$ at $1$.

In the second case, we have $\\#\mathcal{G}_{L}>0.9M$. Let $\ell$ be the
second-largest index in $\mathcal{G}_{L}$. By Lemma 3.10,
$\mathcal{G}_{R}\cap[\ell]$ is empty. Thus the set

$\mathcal{W}=\\{k\in\mathcal{G}_{L}\mid k+M\geq\ell+1\text{ and
}k,k+M\in\mathcal{P}\\}$

satisfies $\\#\mathcal{W}\geq 0.45M$. By discarding at most two elements of
$\mathcal{W}$ and then filtering to odd or even indices, we can arrange that
$\mathcal{W}>0.2M$ and that $k+M\in\mathcal{G}_{R}$ for every
$k\in\mathcal{W}$.

Now enumerate the elements of $\mathcal{W}$ in order, writing

$k_{0}<k_{1}<\cdots<k_{m}$

where $m=\\#\mathcal{W}$. Let $\tau^{L}_{i}$ be the length of the maximal
left-terminal interval for the index $k_{i}$, and let $\tau^{R}_{i}$ be the
length of the maximal right-terminal interval for the index $k_{i}+M$. Then by
the definition of $\mathcal{G}_{L}$,

$\tau_{0}^{L}<\frac{1}{3}\tau^{L}_{1}<\cdots<\frac{1}{3^{k}}\tau^{L}_{i}<\cdots<\frac{1}{3^{m}}\tau^{L}_{m}.$

Moreover, by Lemma 3.11,

$\tau_{0}^{R}>3\tau_{1}^{R}>\cdots>3^{k}\tau^{R}_{i}>\cdots>3^{m}\tau^{R}_{m}.$

Suppose that for some index $k_{i}$, $\tau_{i}^{L}>30\tau_{i}^{R}$. Then
$(0,\tau_{i}^{L})$ is $0.03$-full for $k_{i}$ and $(1-\tau_{i}^{L},1)$ is
$0.03$-empty for $k_{i}+M$. On the other hand if
$\tau_{i}^{L}<\frac{1}{30}\tau_{i}^{R}$ then $(1-\tau_{i}^{R},1)$ is
$0.03$-full for $k_{i}+M$ and $0.03$-empty for $k_{i}$.

There can only be at most, say, $20$ indices for which both of the above
hypotheses fail. Then splitting into the most common case, we either have a
witness for a singularity at $0$ or at $1$. ∎

## References

  * [BBN05] Jean Bourgain, Haïm Brezis, and Hoai-Minh Nguyen.  A new estimate for the topological degree.  Comptes Rendus Mathematique, 340(11):787–791, 2005.
  * [Bre06] Haïm Brezis.  New questions related to the topological degree.  In The unity of mathematics, pages 137–154. Springer, 2006.
  * [Ngu17] Hoai-Minh Nguyen.  A refined estimate for the topological degree.  Comptes Rendus Mathematique, 355(10):1046–1049, 2017.


# Symplectic Representation and Turbulent Global Solutions of Incompressible
Navier-Stokes Equations in ${\mathbb{R}}^{3}$

Yongqian Han  Institute of Applied Physics and Computational Mathematics,
Beijing 100088, China  National Key Laboratory of Computational Physics,
Beijing 100088, China<EMAIL_ADDRESS>
###### Abstract.

The incompressible Navier-Stokes equations are considered. We find that there
exist infinite non-trivial solutions of static Euler equations. Moreover there
exist random solutions of static Euler equations. Provided Reynolds number is
large enough and time variable $t$ goes to infinity, these random solutions of
static Euler equations are the path limits of corresponding Navier-Stokes
flows. But the double limit of these Navier-Stokes flows do not exist.
Therefore these solutions are called turbulent solutions.

###### Key words and phrases:

Incompressible Navier-Stokes Equation, Symplectic Representation, Randomness
and Turbulence of Incompressible Fluid.

###### 2000 Mathematics Subject Classification:

35Q30, 76D05, 76F02, 37L20

##  1\. Introduction

The Navier-Stokes equations in ${\mathbb{R}}^{3}$ are given by

(1.1) $u_{t}-\nu\Delta u+(u\cdot\nabla)u+\nabla P=0,$ (1.2) $\nabla\cdot u=0,$

where $u=u(t,x)=\big{(}u^{1}(t,x),u^{2}(t,x),u^{3}(t,x)\big{)}$ and $P=P(t,x)$
stand for the unknown velocity vector field of fluid and its pressure, $\nu>0$
is the coefficient of viscosity. Here $\partial_{x_{j}}$ denotes by
$\partial_{j}$ ($j=1,2,3$).

The Fourier transformation of $u(t,x)$ with respect to $x$ denotes by
$\hat{u}(t,\xi)$. Then equation (1.2) is equivalent to

(1.3)
$\xi_{j}\hat{u}^{j}=\xi_{1}\hat{u}^{1}+\xi_{2}\hat{u}^{2}+\xi_{3}\hat{u}^{3}=0.$

It is that $\hat{u}(t,\xi)$ is perpendicular to $\xi$. Denote by
$\xi\bot\hat{u}$. Equivalently $\hat{u}(t,\xi)\in T_{\xi}{\mathbb{S}}^{2}$.

Let $A=(a,b,c)\in{\mathbb{R}}^{3}-\\{0\\}$ and $\xi\in{\mathbb{S}}^{2}$. The
vector $\\{A\times\xi\\}\in T_{\xi}{\mathbb{S}}^{2}$ is called the 1st order
incompressible symplectic symmetry. Take $3\times 3$ matrix
$M=\big{(}m^{ij}\big{)}$. The vector $\\{\xi\times(M\xi)\\}\in
T_{\xi}{\mathbb{S}}^{2}$ is called the 2nd order incompressible symplectic
symmetry. Generally let $T=\big{(}T^{i_{1}\cdots i_{n}}\big{)}$ be $n$th order
tensor and $T(\xi\cdots\xi)=T^{i_{1}i_{2}\cdots
i_{n}}\xi_{i_{2}}\cdots\xi_{i_{n}}$. The vector $\\{\xi\times
T(\xi\cdots\xi)\\}\in T_{\xi}{\mathbb{S}}^{2}$ is called the $n$th order
incompressible symplectic symmetry.

Let vector $A\in{\mathbb{R}}^{3}-\\{0\\}$. Then $A\times\xi$ and
$(A\times\xi)\times\xi$ are perpendicular for any
$\xi\in{\mathbb{R}}^{3}-\\{0\\}$, and form the basis of space
$T_{\xi}{\mathbb{S}}^{2}$ which is so-called moving frame. There exist
$\hat{\phi}(\xi)$ and $\hat{\psi}(\xi)$ such that

$\hat{u}(\xi)=\hat{\phi}(\xi)\cdot\\{A\times\xi\\}+\hat{\psi}(\xi)\cdot\\{(A\times\xi)\times\xi\\},\;\;\forall\hat{u}(\xi)\in
T_{\xi}{\mathbb{S}}^{2}.$

Therefore for any velocity vector $u$ satisfying the equation (1.2), there
exist a vector $A\in{\mathbb{R}}^{3}-\\{0\\}$ and real scalar functions $\phi$
and $\psi$ such that

(1.4)
$u(t,x)=\\{A\times\nabla\\}\phi(t,x)+\\{(A\times\nabla)\times\nabla\\}\psi(t,x).$

We call that the formulation (1.4) is (1,2)-symplectic representation of
velocity vector $u$.

Let

(1.5) $\begin{split}\omega(t,x)=&\nabla\times u(t,x)\\\
=&-\\{(A\times\nabla)\times\nabla\\}\phi(t,x)+(A\times\nabla)\Delta\psi(t,x).\\\
\end{split}$

Taking curl with equation (1.1), we have

(1.6)
$\omega_{t}-\nu\Delta\omega+(u\cdot\nabla)\omega-(\omega\cdot\nabla)u=0.$

Thanks the following observations

(1.7) $\begin{split}(A\times\nabla)\cdot
u(t,x)&=(A\times\nabla)\cdot(A\times\nabla)\phi(t,x)\\\ &=\big{\\{}(A\cdot
A)\Delta-(A\cdot\nabla)^{2}\big{\\}}\phi(t,x),\\\ \end{split}$ (1.8)
$\begin{split}(A\times\nabla)\cdot\omega(t,x)&=(A\times\nabla)\cdot(A\times\nabla)\Delta\psi(t,x)\\\
&=\big{\\{}(A\cdot A)\Delta-(A\cdot\nabla)^{2}\big{\\}}\Delta\psi(t,x),\\\
\end{split}$

taking scalar product of equation (1.1) with $A\times\nabla$, we have

(1.9) $\big{\\{}(A\cdot
A)\Delta-(A\cdot\nabla)^{2}\big{\\}}\\{\phi_{t}-\nu\Delta\phi\\}+(A\times\nabla)\cdot\\{(u\cdot\nabla)u\\}=0.$

And taking scalar product of (1.6) with $A\times\nabla$, we derive

(1.10) $\big{\\{}(A\cdot
A)\Delta-(A\cdot\nabla)^{2}\big{\\}}\Delta\\{\psi_{t}-\nu\Delta\psi\\}+(A\times\nabla)\cdot\\{(u\cdot\nabla)\omega-(\omega\cdot\nabla)u\\}=0.$

There is a large literature (see [1, 2, 3] and therein references) studying
the incompressible Navier-Stokes equations. Here we construct some turbulent
global solutions of equations (1.1) (1.2) by using (1.4) (1.9) (1.10),
provided $(\phi,\psi)$ is radial symmetric and cylindrical symmetric
respectively.

##  2\. Radial Symplectic Representation

In this section, we assume that $\phi$ and $\psi$ are radial symmetric
functions with respect to space variable $x\in{\mathbb{R}}^{3}$. It is that
$\phi(t,x)=\phi(t,r)$, $\psi(t,x)=\psi(t,r)$ and
$r^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$. Inserting
$\big{(}\phi(t,r),\psi(t,r)\big{)}$ into (1.9) (1.10), by long long
straightforward calculation, we find that $u$ satisfies Navier-Stokes
equations (1.1) (1.2), provided

(2.1)
$u(t,x)=\\{A\times\nabla\\}e^{-\nu\lambda^{2}t}\Phi(r)+\\{(A\times\nabla)\times\nabla\\}e^{-\nu\lambda^{2}t}\Psi(r),$
(2.2) $\begin{split}\Phi=&\lambda\Psi,\\\ \end{split}$ (2.3)
$\begin{split}-\Delta\Psi=&\lambda^{2}\Psi.\\\ \end{split}$

Here $\Delta=\partial_{r}^{2}+\frac{2}{r}\partial_{r}$ and
$A\in{\mathbb{R}}^{3}-\\{0\\}$.

Moreover let

(2.4)
$u(x)=\\{A\times\nabla\\}\Phi(r)+\\{(A\times\nabla)\times\nabla\\}\Psi(r)$

and $(\Phi,\Psi)$ satisfy (2.2) (2.3), then $u$ is the solution of static
Euler equations

(2.5) $\begin{split}&(u\cdot\nabla)u+\nabla P=0,\\\ &\nabla\cdot u=0.\\\
\end{split}$

The equation (2.3) is equivalent to

(2.6) $\begin{split}-\partial_{r}^{2}(r\Psi)=&\lambda^{2}r\Psi.\\\
\end{split}$

There exist infinite solutions (see [2])

(2.7)
$\begin{split}&\Psi(r)=\Psi_{\lambda\alpha\beta}(r)=\alpha\frac{1}{r}\sin(\lambda
r)+\beta\frac{1}{r}\cos(\lambda r)\\\ \end{split}$

of equation (2.6) for any real constants $\lambda,\alpha$ and $\beta$.

Therefore there exist solution class ${\mathbb{X}}_{E}$ of static Euler
equations and solution class ${\mathbb{X}}_{NS}$ of Navier-Stokes equations.
Here

(2.8)
$\begin{split}&{\mathbb{X}}_{E}=\cup_{\lambda\in{\mathbb{R}}}X_{e\lambda},\\\
&X_{e\lambda}=\mbox{span}\big{\\{}\lambda(A\times\nabla)\big{(}\frac{1}{r}\sin(\lambda
r)\big{)}+\\{(A\times\nabla)\times\nabla\\}\big{(}\frac{1}{r}\sin(\lambda
r)\big{)},\\\ &\hskip
65.44133pt\lambda(A\times\nabla)\big{(}\frac{1}{r}\cos(\lambda
r)\big{)}+\\{(A\times\nabla)\times\nabla\\}\big{(}\frac{1}{r}\cos(\lambda
r)\big{)}\big{\\}},\\\ \end{split}$ (2.9)
$\begin{split}&{\mathbb{X}}_{NS}=\cup_{\lambda\in{\mathbb{R}}}X_{ns\lambda},\\\
&X_{ns\lambda}=\mbox{span}\big{\\{}\lambda
e^{-\nu\lambda^{2}t}(A\times\nabla)\big{(}\frac{1}{r}\sin(\lambda
r)\big{)}+e^{-\nu\lambda^{2}t}\\{(A\times\nabla)\times\nabla\\}\big{(}\frac{1}{r}\sin(\lambda
r)\big{)},\\\ &\hskip 65.44133pt\lambda
e^{-\nu\lambda^{2}t}(A\times\nabla)\big{(}\frac{1}{r}\cos(\lambda
r)\big{)}+e^{-\nu\lambda^{2}t}\\{(A\times\nabla)\times\nabla\\}\big{(}\frac{1}{r}\cos(\lambda
r)\big{)}\big{\\}},\\\ \end{split}$

where $X_{e\lambda}$ and $X_{ns\lambda}$ are linear spaces for any
$\lambda\in{\mathbb{R}}$.

There also exist non-trivial solutions of equation (2.3) with boundary
condition.

Let $R_{0}>0$ and ball $B_{0}=\\{r<R_{0}\\}\subset{\mathbb{R}}^{3}$. We
consider Dirichlet eigenvalue problem

(2.10)
$\begin{split}-(\partial_{1}^{2}+\partial_{2}^{2}+\partial_{3}^{2})\Psi=&\lambda^{2}\Psi,\\\
\end{split}$ (2.11) $\Psi|_{r=R_{0}}=0.$

The eigenvalue $\lambda$ of problem (2.10) (2.11) is so-called Dirichlet
eigenvalue. By spectrum theory of compact operator $(-\Delta)^{-1}$, there
exist countable infinite Dirichlet eigenvalues

$0<\lambda_{1}\leq\cdots\leq\lambda_{n}\leq\cdots$

Each Dirichlet eigenvalue is real, and may be geometrically simple or finite.

Let $\rho$ be orthogonal transformation in ${\mathbb{R}}^{3}$, and
$\Psi_{j}(x)$ be the eigenfunction corresponding to Dirichlet eigenvalue
$\lambda_{j}$. Then $\Psi_{j}(\rho x)$ is also eigenfunction, and
$\Psi_{j}(x)$ is radial function provided $\lambda_{j}$ is simple. Therefore
$\lambda_{1}=\frac{\pi}{R_{0}}$ and

(2.12)
$\begin{split}&\Psi_{1}(x)=\Psi_{1}(r)=\alpha\frac{1}{r}\sin(\lambda_{1}r),\end{split}$

since the first eigenvalue $\lambda_{1}$ is simple and $\Psi_{1}(x)\not=0$ for
any $x\in B_{0}$.

It is obvious that $\Psi_{1}\in H^{1}_{0}(B_{0})\cap H^{2}(B_{0})$. By the
regular theory of elliptic equations, we have $\Psi_{1}\in
H^{\infty}_{loc}({\mathbb{R}}^{3})$. Since the regularity of $\Psi_{1}$ is
independent of $\lambda_{1}$, we derive

$\Psi_{\lambda\alpha
0}(r)=\Psi_{\lambda\alpha\beta}{\big{|}}_{\beta=0}(r)=\alpha\frac{1}{r}\sin(\lambda
r)\in H^{\infty}_{loc}({\mathbb{R}}^{3}).$

Therefore there exist solution class ${\mathbb{X}}^{s}_{E}$ of static Euler
equations and solution class ${\mathbb{X}}^{s}_{NS}$ of Navier-Stokes
equations. Here

(2.13)
$\begin{split}&{\mathbb{X}}^{s}_{E}=\cup_{\lambda\in{\mathbb{R}}}X^{s}_{e\lambda}\subset
H^{\infty}_{loc}({\mathbb{R}}^{3}),\\\
&X^{s}_{e\lambda}=\mbox{span}\big{\\{}\lambda(A\times\nabla)\big{(}\frac{1}{r}\sin(\lambda
r)\big{)}+\\{(A\times\nabla)\times\nabla\\}\big{(}\frac{1}{r}\sin(\lambda
r)\big{)}\big{\\}},\\\ \end{split}$ (2.14)
$\begin{split}&{\mathbb{X}}^{s}_{NS}=\cup_{\lambda\in{\mathbb{R}}}X^{s}_{ns\lambda}\subset
H^{\infty}_{loc}({\mathbb{R}}^{3}),\\\
&X^{s}_{ns\lambda}=\mbox{span}\big{\\{}\lambda
e^{-\nu\lambda^{2}t}(A\times\nabla)\big{(}\frac{1}{r}\sin(\lambda
r)\big{)}+e^{-\nu\lambda^{2}t}\\{(A\times\nabla)\times\nabla\\}\big{(}\frac{1}{r}\sin(\lambda
r)\big{)}\big{\\}},\\\ \end{split}$

where $X^{s}_{e\lambda}\subset H^{\infty}_{loc}({\mathbb{R}}^{3})$ and
$X^{s}_{ns\lambda}\subset H^{\infty}_{loc}({\mathbb{R}}^{3})$ are linear
spaces for any $\lambda\in{\mathbb{R}}$.

##  3\. Cylindrical Symplectic Representation

In this section, we assume that $A=ae_{3}=(0,0,a)$, $\phi$ and $\psi$ are
cylindrical symmetric functions with respect to space variable
$x\in{\mathbb{R}}^{3}$. It is that $\phi(t,x)=\phi(t,r,x_{3})$,
$\psi(t,x)=\psi(t,r,x_{3})$ and $r^{2}=x_{1}^{2}+x_{2}^{2}$.

Assume that symplectic representation $(\phi,\psi)$ is independent of $x_{3}$.
Inserting $\big{(}\phi(t,r),$ $\psi(t,r)\big{)}$ into (1.9) (1.10), by
straightforward calculation, we find that there exist solution classes
${\mathbb{X}}^{2R}_{E}$ of static Euler equations and ${\mathbb{X}}^{2R}_{NS}$
of Navier-Stokes equations. Here

(3.1)
$\begin{split}{\mathbb{X}}^{2R}_{E}=\big{\\{}&(A\times\nabla)\Phi(r)+\\{(A\times\nabla)\times\nabla\\}\Psi(r)\\\
&\big{|}\mbox{$\Phi(r)$ and $\Psi(r)$ are any functions}\big{\\}},\end{split}$
(3.2)
$\begin{split}{\mathbb{X}}^{2R}_{NS}=\big{\\{}&(A\times\nabla)e^{-t\nu\Delta}\Phi(r)+\\{(A\times\nabla)\times\nabla\\}e^{-t\nu\Delta}\Psi(r)\\\
&\big{|}\mbox{$\Phi(r)$ and $\Psi(r)$ are any functions}\big{\\}}.\end{split}$

Assume that symplectic representation $(\phi,\psi)$ depends on $x_{3}$.
Inserting $\big{(}\phi(t,r,x_{3}),$ $\psi(t,r,x_{3})\big{)}$ into (1.9)
(1.10), by long long straightforward calculation, we find that $u$ satisfies
Navier-Stokes equations (1.1) (1.2), provided

(3.3)
$u(t,x)=\\{A\times\nabla\\}e^{-\nu\lambda^{2}t}\Phi(r,x_{3})+\\{(A\times\nabla)\times\nabla\\}e^{-\nu\lambda^{2}t}\Psi(r,x_{3}),$
(3.4) $\begin{split}\Phi=&\lambda\Psi,\\\ \end{split}$ (3.5)
$\begin{split}-\Delta\Psi=&\lambda^{2}\Psi.\\\ \end{split}$

Here $\Delta=\partial_{r}^{2}+\frac{1}{r}\partial_{r}+\partial_{3}^{2}$.

Moreover let

(3.6)
$u(x)=\\{A\times\nabla\\}\Phi(r,x_{3})+\\{(A\times\nabla)\times\nabla\\}\Psi(r,x_{3})$

and $(\Phi,\Psi)$ satisfy (3.4) (3.5), then $u$ is the solution of static
Euler equations

(3.7) $\begin{split}&(u\cdot\nabla)u+\nabla P=0,\\\ &\nabla\cdot u=0.\\\
\end{split}$

There exist non-trivial solutions of equation (3.5) with boundary conditions.
Now we construct some solutions of (3.5).

Let $R_{0}>0$ and disc $D_{0}=\\{r<R_{0}\\}\subset{\mathbb{R}}^{2}$. We
consider Dirichlet eigenvalue problem

(3.8) $\begin{split}-(\partial_{1}^{2}+\partial_{2}^{2})W=&\xi^{2}W,\\\
\end{split}$ (3.9) $W|_{r=R_{0}}=0.$

By spectrum theory of compact operator $(-\Delta)^{-1}$, there exist countable
infinite Dirichlet eigenvalues

$0<\xi_{1}\leq\cdots\leq\xi_{n}\leq\cdots$

Each Dirichlet eigenvalue is real, and may be geometrically simple or finite.

Let $\rho$ be orthogonal transformation in ${\mathbb{R}}^{2}$, and $W_{j}(x)$
be the eigenfunction corresponding to Dirichlet eigenvalue $\xi_{j}$. Then
$W_{j}(\rho x)$ is also eigenfunction, and $W_{j}(x)$ is radial function
provided $\xi_{j}$ is simple. Therefore $W_{1}(x)$ is radial since the first
eigenvalue $\xi_{1}$ is simple.

It is obvious that $W_{j}\in H^{1}_{0}(D_{0})\cap H^{2}(D_{0})$. By the
regular theory of elliptic equations, we have $W_{j}\in H^{\infty}(D_{0})$ for
any $j=1,2,\cdots$.

Therefore there exist solutions

(3.10) $\begin{split}&\lambda_{j}^{2}=\xi_{j}^{2}+\eta^{2},\;\;j=1,2,\cdots\\\
&\Psi_{j}(r,x_{3})=\alpha W_{j}(r)\sin(\eta x_{3})+\beta W_{j}(r)\cos(\eta
x_{3})\\\ \end{split}$

of problem (3.5) provided $W_{j}(x)$ is radial. Here $\alpha$, $\beta$ and
$\eta$ are any real constants.

Thus there exist solution class ${\mathbb{Y}}_{E}$ of static Euler equations
and solution class ${\mathbb{Y}}_{NS}$ of Navier-Stokes equations. Here

(3.11)
$\begin{split}&{\mathbb{Y}}_{E}=\cup^{\infty}_{j=1}\cup_{\eta\in{\mathbb{R}}}Y^{e}_{j\eta},\\\
&Y^{e}_{j\eta}=\mbox{span}\big{\\{}\lambda_{j}(A\times\nabla)\big{(}W_{j}(r)\sin(\eta
x_{3})\big{)}+\\{(A\times\nabla)\times\nabla\\}\big{(}W_{j}(r)\sin(\eta
x_{3})\big{)},\\\ &\hskip
62.59605pt\lambda_{j}(A\times\nabla)\big{(}W_{j}(r)\cos(\eta
x_{3})\big{)}+\\{(A\times\nabla)\times\nabla\\}\big{(}W_{j}(r)\cos(\eta
x_{3})\big{)}\big{\\}},\\\ \end{split}$ (3.12)
$\begin{split}{\mathbb{Y}}_{NS}=&\cup^{\infty}_{j=1}\cup_{\eta\in{\mathbb{R}}}Y^{ns}_{j\eta},\\\
Y^{ns}_{j\eta}=&\mbox{span}\big{\\{}u^{s}_{j\eta},u^{c}_{j\eta}\big{\\}},\\\
u^{s}_{j\eta}=&\lambda_{j}e^{-\nu\lambda^{2}_{j}t}(A\times\nabla)\big{(}W_{j}(r)\sin(\eta
x_{3})\big{)}\\\
&+e^{-\nu\lambda^{2}_{j}t}\\{(A\times\nabla)\times\nabla\\}\big{(}W_{j}(r)\sin(\eta
x_{3})\big{)},\\\
u^{c}_{j\eta}=&\lambda_{j}e^{-\nu\lambda^{2}_{j}t}(A\times\nabla)\big{(}W_{j}(r)\cos(\eta
x_{3})\big{)}\\\
&+e^{-\nu\lambda^{2}_{j}t}\\{(A\times\nabla)\times\nabla\\}\big{(}W_{j}(r)\cos(\eta
x_{3})\big{)},\\\ \end{split}$

provided $W_{j}(x)$ is radial. It is obvious that $Y^{e}_{j\eta}$ and
$Y^{ns}_{j\eta}$ are linear spaces for any $j=1,2,\cdots$ and
$\eta\in{\mathbb{R}}$.

We observe that

$\Delta_{r}V=(\partial_{r}^{2}+\frac{1}{r}\partial_{r})V=\frac{1}{r}\partial_{r}(r\partial_{r}V).$

Let $0<R_{1}<R_{2}$ and ring ${\cal
R}=\\{R_{1}<r<R_{2}\\}\subset{\mathbb{R}}^{2}$. We consider equation

(3.13) $\begin{split}-\partial_{r}(r\partial_{r}V)=\zeta^{2}rV\\\ \end{split}$

with separated or coupled boundary condition.

The separated boundary condition is defined by

(3.14)
$\begin{split}&m_{11}V(R_{1})+m_{12}\\{r\partial_{r}V(r)\\}\big{|}_{r=R_{1}}=0,\\\
&m_{21}V(R_{2})+m_{22}\\{r\partial_{r}V(r)\\}\big{|}_{r=R_{2}}=0,\\\
\end{split}$

where $m_{ij}\in{\mathbb{R}}$, $(m_{11},m_{12})\not=(0,0)$ and
$(m_{21},m_{22})\not=(0,0)$.

The coupled boundary condition is defined by

(3.15) $\begin{split}&Y(r)=\left(\begin{array}[]{c}V(r)\\\
r\partial_{r}V(r)\end{array}\right),\\\ &Y(R_{1})=KY(R_{2}),\\\ \end{split}$

where $2\times 2$ matrix $K\in SL_{2}({\mathbb{R}})$ and $\det K$=1.

By Sturm-Liouville theory (see [4], Page 72, Theorem 4.3.1), there exist
countable infinite eigenvalues

$0<\zeta_{1}\leq\cdots\leq\zeta_{n}\leq\cdots$

and eigenfunction $V_{j}(r)$ corresponding to eigenvalue $\zeta_{j}$ of
equation (3.13) with separated boundary condition (3.14) or coupled boundary
condition (3.15). Each eigenvalue is real, and may be geometrically simple or
double.

For the eigenfunctions $V_{j}$ of problem (3.13) (3.14), by the regular theory
of elliptic equations, we have $V_{j}\in H^{1}_{0}({\cal R})\cap
H^{\infty}({\cal R})$ ($j=1,2,\cdots$) provided $(m_{11},m_{12})=(1,0)$ and
$(m_{21},m_{22})=(1,0)$.

Therefore there exist solutions

(3.16)
$\begin{split}&\lambda_{j}^{2}=\zeta_{j}^{2}+\eta^{2},\;\;j=1,2,\cdots\\\
&\Psi_{j}(r,x_{3})=\alpha V_{j}(r)\sin(\eta x_{3})+\beta V_{j}(r)\cos(\eta
x_{3})\\\ \end{split}$

of problem (3.5). Here $\alpha$, $\beta$ and $\eta$ are any real constants.

Thus there exist solution class ${\mathbb{Z}}_{E}$ of static Euler equations
and solution class ${\mathbb{Z}}_{NS}$ of Navier-Stokes equations. Here

(3.17)
$\begin{split}&{\mathbb{Z}}_{E}=\cup^{\infty}_{j=1}\cup_{\eta\in{\mathbb{R}}}Z^{e}_{j\eta},\\\
&Z^{e}_{j\eta}=\mbox{span}\big{\\{}\lambda_{j}(A\times\nabla)\big{(}V_{j}(r)\sin(\eta
x_{3})\big{)}+\\{(A\times\nabla)\times\nabla\\}\big{(}V_{j}(r)\sin(\eta
x_{3})\big{)},\\\ &\hskip
62.59605pt\lambda_{j}(A\times\nabla)\big{(}V_{j}(r)\cos(\eta
x_{3})\big{)}+\\{(A\times\nabla)\times\nabla\\}\big{(}V_{j}(r)\cos(\eta
x_{3})\big{)}\big{\\}},\\\ \end{split}$ (3.18)
$\begin{split}{\mathbb{Z}}_{NS}=&\cup^{\infty}_{j=1}\cup_{\eta\in{\mathbb{R}}}Z^{ns}_{j\eta},\\\
Z^{ns}_{j\eta}=&\mbox{span}\big{\\{}u^{s}_{j\eta},u^{c}_{j\eta}\big{\\}},\\\
u^{s}_{j\eta}=&\lambda_{j}e^{-\nu\lambda^{2}_{j}t}(A\times\nabla)\big{(}V_{j}(r)\sin(\eta
x_{3})\big{)}\\\
&+e^{-\nu\lambda^{2}_{j}t}\\{(A\times\nabla)\times\nabla\\}\big{(}V_{j}(r)\sin(\eta
x_{3})\big{)},\\\
u^{c}_{j\eta}=&\lambda_{j}e^{-\nu\lambda^{2}_{j}t}(A\times\nabla)\big{(}V_{j}(r)\cos(\eta
x_{3})\big{)}\\\
&+e^{-\nu\lambda^{2}_{j}t}\\{(A\times\nabla)\times\nabla\\}\big{(}V_{j}(r)\cos(\eta
x_{3})\big{)},\\\ \end{split}$

where $Z^{e}_{j\eta}$ and $Z^{ns}_{j\eta}$ are linear spaces for any
$j=1,2,\cdots$ and $\eta\in{\mathbb{R}}$.

##  4\. Conclusion

Some interesting and significant phenomena are founded by observing the
solutions in Section 2 and Section 3.

For Navier-Stokes equations, there exist some special solution classes, for
instance ${\mathbb{X}}_{NS}^{2R}$, ${\mathbb{X}}_{NS}$,
${\mathbb{X}}^{s}_{NS}$, ${\mathbb{Y}}_{NS}$ and ${\mathbb{Z}}_{NS}$, which
will be called turbulent solution classes.

Let $\omega$ be random number and $\Gamma_{\omega}$ be a path. In the path
$\Gamma_{\omega}$, we have

(4.1) $\begin{split}t\nu=\omega\\\ \end{split}$

for any $(\nu,t)\in\Gamma_{\omega}$.

Assume that $u_{ns}(t,x)\in{\mathbb{X}}_{NS}^{2R}$ is the solution of Navier-
Stokes equation. Then there exist $\Phi(r)$ and $\Psi(r)$ such that

$\begin{split}u_{ns}(t,x)=&(A\times\nabla)\Big{\\{}\frac{1}{4\pi
t\nu}\int_{{\mathbb{R}}^{2}}\exp\\{-\frac{|y|^{2}}{4t\nu}\\}\Phi(|x-y|)dy\Big{\\}}\\\
&+\\{(A\times\nabla)\times\nabla\\}\Big{\\{}\frac{1}{4\pi
t\nu}\int_{{\mathbb{R}}^{2}}\exp\\{-\frac{|y|^{2}}{4t\nu}\\}\Psi(|x-y|)dy\Big{\\}}.\\\
\end{split}$

Letting $(\nu,t)\in\Gamma_{\omega}$ and $(\nu,1/t)\rightarrow(0,0)$, we have
path limit

(4.2)
$\begin{split}u_{e}(r)=&\lim_{(\nu,t)\in\Gamma_{\omega},(\nu,1/t)\rightarrow(0,0)}u_{ns}(t,x)\\\
=&(A\times\nabla)\Phi_{l}(r)+\\{(A\times\nabla)\times\nabla\\}\Psi_{l}(r),\\\
\Phi_{l}(r)=&\Phi_{l}(x)=\frac{1}{4\pi\omega}\int_{{\mathbb{R}}^{2}}\exp\\{-\frac{|y|^{2}}{4\omega}\\}\Phi(|x-y|)dy,\\\
\Psi_{l}(r)=&\Psi_{l}(x)=\frac{1}{4\pi\omega}\int_{{\mathbb{R}}^{2}}\exp\\{-\frac{|y|^{2}}{4\omega}\\}\Psi(|x-y|)dy.\\\
\end{split}$

It is obvious that random $u_{e}$ is the solution of static Euler equations
and $u_{e}\in{\mathbb{X}}_{E}^{2R}$. Moreover double limit
$\lim_{(\nu,1/t)\rightarrow(0,0)}u_{ns}(t,x)$ does not exist.

Assume that $u_{ns}(t,x)\in{\mathbb{X}}_{NS}$ is the solution of Navier-Stokes
equation. Then there exist linear space $X_{ns\lambda}$ and real constants
$\alpha$, $\beta$ such that $u_{ns}(t,x)\in X_{ns\lambda}$ and

$\begin{split}u_{ns}(t,x)=&\alpha\big{\\{}\lambda
e^{-\nu\lambda^{2}t}(A\times\nabla)\big{(}\frac{1}{r}\sin(\lambda
r)\big{)}+e^{-\nu\lambda^{2}t}\\{(A\times\nabla)\times\nabla\\}\big{(}\frac{1}{r}\sin(\lambda
r)\big{)}\big{\\}}\\\ &+\beta\big{\\{}\lambda
e^{-\nu\lambda^{2}t}(A\times\nabla)\big{(}\frac{1}{r}\cos(\lambda
r)\big{)}+e^{-\nu\lambda^{2}t}\\{(A\times\nabla)\times\nabla\\}\big{(}\frac{1}{r}\cos(\lambda
r)\big{)}\big{\\}}.\\\ \end{split}$

Letting $(\nu,t)\in\Gamma_{\omega}$ and $(\nu,1/t)\rightarrow(0,0)$, we have
path limit

(4.3)
$\begin{split}u_{e}(r)=&\lim_{(\nu,t)\in\Gamma_{\omega},(\nu,1/t)\rightarrow(0,0)}u_{ns}(t,x)\\\
=&\alpha\big{\\{}\lambda
e^{-\omega\lambda^{2}}(A\times\nabla)\big{(}\frac{1}{r}\sin(\lambda
r)\big{)}+e^{-\omega\lambda^{2}}\\{(A\times\nabla)\times\nabla\\}\big{(}\frac{1}{r}\sin(\lambda
r)\big{)}\big{\\}}\\\ &+\beta\big{\\{}\lambda
e^{-\omega\lambda^{2}}(A\times\nabla)\big{(}\frac{1}{r}\cos(\lambda
r)\big{)}+e^{-\omega\lambda^{2}}\\{(A\times\nabla)\times\nabla\\}\big{(}\frac{1}{r}\cos(\lambda
r)\big{)}\big{\\}}.\\\ \end{split}$

It is obvious that random $u_{e}$ is the solution of static Euler equations
and $u_{e}\in X_{e\lambda}$ since $X_{e\lambda}$ is linear space. Moreover
double limit $\lim_{(\nu,1/t)\rightarrow(0,0)}u_{ns}(t,x)$ does not exist.

Similarly assume that $u_{ns}(t,x)\in{\mathbb{X}}^{s}_{NS}$ (
${\mathbb{Y}}_{NS}$ or ${\mathbb{Z}}_{NS}$ respectively) is the solution of
Navier-Stokes equation. Then there exists linear space $X^{s}_{ns\lambda}$
($Y^{ns}_{j\lambda}$ or $Z^{ns}_{j\lambda}$ resp.) such that $u_{ns}(t,x)\in
X^{s}_{ns\lambda}$ ($Y^{ns}_{j\lambda}$ or $Z^{ns}_{j\lambda}$ resp.). Since
$X^{s}_{e\lambda}$ ($Y^{e}_{j\lambda}$ or $Z^{e}_{j\lambda}$ resp.) is linear
space, we can prove that there exists the path limit

$u_{e}(r)=\lim_{(\nu,t)\in\Gamma_{\omega},(\nu,1/t)\rightarrow(0,0)}u_{ns}(t,x)$

such that $u_{e}(r)\in X^{s}_{e\lambda}$ ($Y^{e}_{j\lambda}$ or
$Z^{e}_{j\lambda}$ resp.). But the double limit
$\lim_{(\nu,1/t)\rightarrow(0,0)}u_{ns}(t,x)$ does not exist.

In experiment or numerical simulation of fluid motion ruled by Navier-Stokes
equations, large Reynolds number $\frac{1}{\nu}\gg 1$ is firstly given. Then
after a long long time, we observe the fluid motion at some time $1\ll
t_{1}<t_{2}<\cdots<t_{n}<\cdots$. $(\nu,t_{n})$ is located in the path
$\Gamma_{\omega_{n}}$ where $\omega_{n}=t_{n}\nu$ ($n=1,2,\cdots$). We can see
random of fluid based on the existence of path limit of velocity along
$\Gamma_{\omega_{n}}$ provided the initial velocity of fluid is in one class
of ${\mathbb{X}}_{E}^{2R}$, ${\mathbb{X}}_{E}$, ${\mathbb{X}}^{s}_{E}$,
${\mathbb{Y}}_{E}$ and ${\mathbb{Z}}_{E}$.

## Acknowledgments

This work is supported by National Natural Science Foundation of China–NSF,
Grant No.11971068 and No.11971077.

## References

  * [1] P. Constantin.  Some open problems and research dierections in the mathematical study of fluid dynamics. In Mathematics Unlimited-2001 and Beyond, Springer Verlag, Berlin, 2002, 353–360.
  * [2] Yongqian Han. Symplectic symmetry and rarial symmetry either persistence or breaking of incompressible fluid.  arXiv 2305.13737.
  * [3] R. Temam.  Navier-Stokes Equations. Theory and numerical analysis. (Reprint of the 1984 edition). AMS Chelsea Publishing, Providence, RI, 2001.
  * [4] A. Zettl.  Sturm-Liouville Theory. Mathematical Surveys and Monographs Vol. 121. American Mathematical Society, 2005.


# Supervised learning for crop/weed classification based on color and texture
features

Faiza Mekhalfa  Division Productique et Robotique
Centre de Developpement de Technologies Avancees
Algiers, Algeria
<EMAIL_ADDRESS>Fouad Yacef  Division Productique et Robotique
Centre de Developpement de Technologies Avancees
Algiers, Algeria
<EMAIL_ADDRESS>
###### Abstract

Computer vision techniques have attracted a great interest in precision
agriculture, recently. The common goal of all computer vision-based precision
agriculture tasks is to detect the objects of interest (e.g., crop, weed) and
discriminating them from the background. The Weeds are unwanted plants growing
among crops competing for nutrients, water, and sunlight, causing losses to
crop yields. Weed detection and mapping is critical for site-specific weed
management to reduce the cost of labor and impact of herbicides. This paper
investigates the use of color and texture features for discrimination of
Soybean crops and weeds. Feature extraction methods including two color spaces
(RGB, HSV), gray level Co-occurrence matrix (GLCM), and Local Binary Pattern
(LBP) are used to train the Support Vector Machine (SVM) classifier. The
experiment was carried out on image dataset of soybean crop, obtained from an
unmanned aerial vehicle (UAV), which is publicly available. The results from
the experiment showed that the highest accuracy (above $96$%) was obtained
from the combination of color and LBP features.

###### Index Terms:

Crop/weed classification, Color features, Gray Level Co-occurrence matrix
(GLCM), Local Binary Pattern (LBP), Support Vector Machine (SVM)

##  I Introduction

In recent years, there has been a strong activity in precision agriculture
(PA), particularly the monitoring aspect. Precision agriculture employs data
from multiple sources for the purpose of improving crop yields and increasing
the cost-effectiveness of crop management strategies including fertilizer
inputs, irrigation management, and pesticide application [1]. PA offers the
opportunity for a farmer to apply the right amount of treatment at the right
time and at the right place [2]. Nowadays, (UAVs) can be exploited in a
variety of applications related to crops management, by capturing high spatial
and temporal resolution images of the entire agricultural field. UAVs have
been considered more efficient, compared to the ground robot or satellite
acquisitions, since they allow a fast acquisition of the field with very high
spatial resolution and at a low cost [3].

Among the most popular application of UAVs in Precision Agriculture is weed
mapping [4][5]. Weeds are undesirable plants, which grow in agricultural crops
and can cause several problems. They are competing for available resources
such as water or even space, causing losses to crop yields [3]. The knowledge
of weed infestation is an essential procedure for the use of preventive
measures in their control. The challenge of crop/weed classification was
addressed by considering various machine learning techniques. The most popular
classification techniques are the Artificial Neural Networks (ANNs) family [6]
and the Random Forest algorithm [7].

Based on the principle of structure minimum according to the statistical
learning theory developed by Vapnik and Chervonenkis [8], Support Vector
Machines (SVMs) can solve practical problems encountered with traditional
classifiers in the aspects of small training samples, nonlinearity, high
dimension and local extreme values [9]. The aim of this work is to use Support
Vector Machine (SVM) classifier to perform the identification of weeds in
relation to soybean and soil and classification of them in grass and
broadleaf, aiming to apply the specific herbicide to weed detected. The
critical component of any classification challenge is the available data. The
publicly available dataset is used to train and evaluate machine learning
algorithm. The image database collected by Dos Santos Ferreira and al. [5]
contains over fifteen thousand images of the soil, soybean, broadleaf and
grass weeds. As a classical pattern recognition problem, crop/weed
classification primarily consists of two critical subproblems: feature
extraction and classifier designation. Feature extraction is crucial step to
find the suitable descriptors that can provide good discrimination between
different classes. Principally, there are three main approaches for weed
detection: based on color, shape and texture analysis. Various texture
analysis techniques exist, texture features derived from gray-level co-
occurrence matrix (GLCM) [10] and Local Binary Pattern (LBP) [11] are the most
popular because of their simplicity and adaptability. In this study, color, as
a primary input feature, was combined with the texture features for training
SVM classifier in order to discriminate between crop/weed plants and soil.

The remainder of the paper is organized as follows. Section II describes the
methodology to develop the weed detection and classification system. In
section III we present the experimental results to classify weeds in soybean
crop images where performance comparison between different features is carried
out, and we conclude our work in section IV.

##  II METHODOLOGY

###  II-A Image dataset

The database used in this study is available online and can be downloaded from
https://www.kaggle.com/fpeccia/weed-detection-in-soybean-crops. It was built
by Dos Santos Ferreira and al. [5], using 400 images of soybean crop captured
by the UAV. The Simple Linear Iterative Clustering (SLIC) algorithm [12] was
used to segment the UAV images. The segments of each image, that identified
one of the four classes used in this experiment, were annotated manually. The
image dataset contained 15,336 segments, being 3249 of soil, 7376 of soybean,
3520 grass and 1191 of broadleaf weeds.

###  II-B Feature extraction

Feature extraction is one of the most important stages in pattern recognition.
It generates an input vector called descriptor for each image which is then
used as input to multiclass classifiers. Although color attributes make sense
in distinguishing between vegetation and Soil, they become less effective when
applied to classify plant species. Sometimes, the color of weeds and crop
leaves look almost the same. In this study, color, as a primary input feature,
was combined with the texture features (GLCM, LBP) for discriminating
soybean/weed plants and soil.

1) Color features: The color features are means and standard deviations of the
three RGB and HSV image bands.

2) Gray-Level Cooccurrence Matrix (GLCM): Textural analysis is a very useful
tool for discrimination of weeds from the main crop [13]. One of the earliest
methods used for texture feature extraction was proposed by Haralick et al.
[10], known as Gray-Level Cooccurrence Matrix (GLCM) and since then it has
been widely used in many texture analysis applications.

GLCM is a second-order statistical texture analysis method. It examines the
spatial relationship among pixels and defines how frequently a combination of
pixels are present in an image in a given direction $\theta$ and distance
d.Various research studies show d values ranging from 1, 2 to 10. Applying
large displacement value to a fine texture would yield a GLCM that does not
capture detailed textural information. GLCM directions $\theta$ of analysis
are: Horizontal (0°or 180°),Vertical (90°or 270°), Right Diagonal (45°or 225
°) and Left diagonal (135°or 315°) (See Fig 1).

Figure 1: The direction angles for GLCM .

Let I be a given grey scale image. Let $\textit{N}_{g}$ be the total number of
grey levels in the image. The Grey Level Co-occurrence Matrix defined by
Haralick [14] is a square matrix p, where the $(\textit{i},\textit{j})^{th}$
entry of p represents the number of occasions a pixel with intensity i is
adjacent to a pixel with intensity j. The normalized co-occurrence matrix
$\textit{p}_{d}$ is obtained by dividing each element of p by the total number
of co-occurrence pairs in p.

The fourteen textural features proposed by Haralick et al [10] contain
information about image texture characteristics such as homogeneity, gray-tone
linear dependencies, contrast, number and nature of boundaries present and the
complexity of the image. We used nine textural features in our study. The
following equations define these features [14].

Energy: This statistic is also called Uniformity or Angular second moment. It
measures the textural uniformity that is pixel pair repetitions. Energy
reaches a maximum value equal to one for a constant image.

$Energy=\sum_{i}\sum_{j}\textit{p}_{\textit{d}}(\textit{i},\textit{j})^{2}$
(1)

Contrast is a measure of intensity or gray level variations between a pixel
and its neighbor over the whole image. Large contrast reflects large intensity
differences in GLCM. Contrast is 0 for a constant image.

$Contrast=\sum_{i}\sum_{j}(\textit{i}-\textit{j})^{2}\textit{p}_{\textit{d}}(\textit{i},\textit{j})$
(2)

Entropy: This feature measures the disorder or complexity of an image. Complex
textures tend to have high entropy.

$Entropy=\sum_{i}\sum_{j}\textit{p}_{\textit{d}}(\textit{i},\textit{j})\log(\textit{p}_{d}(\textit{i},\textit{j}))$
(3)

Homogeneity: This feature is also called as Inverse Difference Moment. It
measures image homogeneity as it assumes larger values for smaller gray tone
differences in pair elements. It is more sensitive to the presence of near
diagonal elements in the GLCM. Homogeneity is 1 for a diagonal GLCM.

$Homogeneity=\sum_{i}\sum_{j}\frac{\textit{p}_{\textit{d}}(\textit{i},\textit{j})}{1+(\textit{i}-\textit{j})^{2}}$
(4)

Correlation: The correlation feature is a measure of gray tone linear
dependencies in the image. Correlation is 1 or -1 for a perfectly positively
or negatively correlated image.

$Correlation=\sum_{i}\sum_{j}\textit{p}_{\textit{d}}(\textit{i},\textit{j})\frac{(\textit{i}-\mu_{\textit{x}})(\textit{j}-\mu_{\textit{y}})}{\sigma_{\textit{x}}^{2}\sigma_{\textit{y}}^{2}}$
(5)

where
$\mu_{\textit{x}},\mu_{\textit{y}}and\sigma_{\textit{x}},\sigma_{\textit{y}}$
are the means and standard deviations and are expressed as:

$\begin{split}\mu_{\textit{x}}=\sum_{i}\sum_{j}\textit{i}\textit{p}_{\textit{d}}(\textit{i},\textit{j})\\\
\mu_{\textit{y}}=\sum_{i}\sum_{j}\textit{j}\textit{p}_{\textit{d}}(\textit{i},\textit{j})\\\
\sigma_{\textit{x}}=\sqrt{\sum_{i}\sum_{j}(\textit{i}-\mu_{\textit{x}})^{2}\textit{p}_{\textit{d}}(\textit{i},\textit{j})}\\\
\sigma_{\textit{y}}=\sqrt{\sum_{i}\sum_{j}(\textit{j}-\mu_{\textit{y}})^{2}\textit{p}_{\textit{d}}(\textit{i},\textit{j})}\end{split}$
(6)

The moments are the statistical expectation of certain power functions of a
random variable and are characterized as follows. Moment 1 is the mean which
is the average of pixel values in an image and it is represented as

$Mean=\sum_{i}\sum_{j}(\textit{i}-\textit{j})\textit{p}_{\textit{d}}(\textit{i},\textit{j})$
(7)

Moment 2 is the standard deviation that can be denoted as

$Standarddeviation=\sum_{i}\sum_{j}(\textit{i}-\textit{j})^{2}\textit{p}_{\textit{d}}(\textit{i},\textit{j})$
(8)

Moment 3 measures the degree of asymmetry in the distribution and it is
defined as skewness

$Skewness=\sum_{i}\sum_{j}(\textit{i}-\textit{j})^{3}\textit{p}_{\textit{d}}(\textit{i},\textit{j})$
(9)

Moment 4 measures the relative peak or flatness of a distribution and is also
known as kurtosis:

$Kurtosis=\sum_{i}\sum_{j}(\textit{i}-\textit{j})^{4}\textit{p}_{d}(\textit{i},\textit{j})$
(10)

3) Local binary patterns (LBP):

Local Binary Patterns (LBP) is a kind of gray-scale texture operator that is
used for describing the spatial structure of an image texture [11]. Due to its
discriminative power and computational simplicity, LBP texture extractor has
become a popular approach in various applications [15].

The original LBP operator [11] forms labels for the image pixels by
thresholding the 3 × 3 neighborhood of each pixel with the center value and
considering the result as a binary number. The histogram of these $2^{8}=256$
different labels can then be used as a texture descriptor.

The LBP operator was extended to use neighborhoods of different sizes [16].
Using a circular neighborhood and bilinearly interpolating values at non-
integer pixel coordinates allow any radius and number of pixels in the
neighborhood. Fig. 2 illustrates three neighbor-sets, where the notation (M,
R) denotes a neighborhood of M sampling points on a circle of radius of R.

Given a pixel at $(\textit{x}_{\textit{c}},\textit{y}_{\textit{c}})$, the
resulting LBP can be expressed in decimal form as:

$\textit{LBP}_{\textit{M,R}}(\textit{x}_{\textit{c}},\textit{y}_{\textit{c}})=\sum_{i=0}^{M-1}\textit{s}(\textit{i}_{m}-\textit{i}_{c})2^{m}$
(11)

where $\textit{i}_{m},\textit{i}_{c}$ are respectively gray-level values of
the central pixel and M surrounding pixels in the circle neighborhood with a
radius R, and function s(x) is defined as:

$s(x)=\left\\{\begin{array}[]{l l}1&\quad\text{if $x\geq 0$}\\\
0&\quad\text{if $x<0$}\\\ \end{array}\right.$ (12)

After the LBP extraction, each pixel in an image is replaced by a binary
pattern, except at the borders of the image where all of the neighbor values
do not exist. The feature vector of an image consists of a histogram of the
pixel LBPs. The length of the histogram is $2^{M}$ since each possible LBP is
assigned a separate bin. In order to remove rotation effect, a rotation-
invariant LBP is proposed in [16]

$LBP_{M,R}^{ri}=\min\\{ROR(LBP_{M,R},i),i=0,1,...,M-1\\}$ (13)

where $ROR(x,i)$ performs an $i$ -step circular bit-wise right shift on $x$.

Uniform Local Binary Patterns are patterns with at most two circular 0-1 and
1-0 transitions. For example, 00000000 (0 transitions) and 01110000 (2
transitions) are both uniform whereas 11001001 (4 transitions) and 01010011 (6
transitions) are not. Selecting only uniform patterns contributes to both
reducing the feature dimensionality and improving the performance of
classifiers using the LBP features.

Figure 2: LBP Neighboring Pixels System.

###  II-C Support vector machine classifier

Once a feature descriptor is calculated, the next step deals with crop/weeds
classification. Support vector machine (SVM) classifier is one of the most
successful machine learning methods, because it is robust, accurate and is
effective even when using a small training sample.

Support vector machines are originally developed for binary classification.
But, they can be adopted to handle the multiclass classification tasks. The
basic theory of SVM consists to draw an optimal hyperplane separating data
points of different classes. Both separable and non-separable problems are
handled by SVM in the linear and nonlinear cases. The idea behind SVM is to
map the original data points from the input space to a high dimensional one,
called feature space. The mapping is done by a suitable choice of Kernel
function [9].

To implement SVM on image classification we are given a certain number n of
training data, each data has two parts: the d-dimensional vector of image
features and the corresponding labels of classes (either +1 or -1) [17]:

$E=\\{(x_{i},y_{i})/x_{i}\in\mathbb{R}^{d},y_{i}\in\\{-1,1\\},i=1,...n\\}$
(14)

SVM maps the d-dimensional input vector $x$ from the input space to the
$d_{h}$-dimensional feature space using a nonlinear function
$\varphi(.):\mathbb{R}^{d}\longrightarrow\mathbb{R}^{d_{h}}$. The separating
hyperplane in the feature space is then defined as

$w.\varphi(x)+b=0\ /\ w\in\mathbb{R}^{d_{h}},b\in\mathbb{R}$ (15)

The classifier should satisfy the condition of existence of $w$ and $b$ such
that:

$y_{i}(w.\varphi(x_{i})+b)\geq 1$ (16)

However, in practical applications, data of both classes are overlapping,
which makes a perfect linear separation impossible. Therefore, a restricted
number of misclassifications should be tolerated around the margin. The
resulting optimization problem for SVM where the violation of the constraints
is penalized is given as:

$\left\\{\begin{array}[]{l l
l}\min\frac{1}{2}\|w\|^{2}+C\sum_{i=1}^{n}\xi_{i}&\quad\text{such that}\\\
y_{i}(w.\varphi(x_{i})+b)\geq 1-\xi_{i}\\\ \xi_{i}\geq 0\\\
\end{array}\right.$ (17)

where $\xi_{i}$ is the relaxation factor considering classification error and
$C$ is the cost parameter that controls the tradeoff between allowing training
errors and forcing strict margins (i.e. empirical risk minimization [9].

Typically, the constrained optimization problem is referred as the primal
optimization problem, which can be written in the dual space by Lagrange
multipliers $\alpha_{i}\geq 0$. The solution should maximize the following
expression:

$\begin{split}L(w,b,\xi_{i},\alpha_{i})=\frac{1}{2}\|w\|^{2}+C\sum_{i=1}^{n}\xi_{i}\\\
-\sum_{i=0}^{n}\alpha_{i}(y_{i}(w.\varphi(x_{i})+b)-1)\end{split}$ (18)

and the dual problem is given as

$\begin{array}[]{l l}\max
L(\alpha)=\sum_{i=1}^{n}\alpha_{i}-\frac{1}{2}\sum_{i,j=1}^{n}\alpha_{i}\alpha_{j}y_{i}y{j}K(x_{i},x_{j})\\\
\text{subject to}\>0\leq\alpha_{i}\leq C\text{and}\sum\alpha_{i}y_{i}=0\\\
\end{array}$ (19)

where the Kernel function $K(x_{i},x_{j})$ corresponds to the inner product
belonging to transformation space:

$K(x_{i},x_{j})=\varphi(x_{i})\varphi(x_{j})$ (20)

There are three kind of commonly used Kernel functions: linear kernel
function, polynomial kernel function and radial basis function (RBF) [18].
Finally, the SVM classifier function can be written as

$f(x)=sign(\sum_{i=1}^{n}\alpha_{i}y_{i}K(x_{i},x_{j})+b)$ (21)

Numerous approaches have been suggested to use SVM for multiclass
classification, but there are two most popular approaches, one versus all
(1.vs. all) and one versus one (1.vs.1) [19].

The one versus all approach consists of constructing one SVM per class, which
is trained to distinguish the samples of one class from the samples of all
remaining classes. Usually, classification of an unknown pattern is done
according to the maximum output among all SVMs. This multiclass method has an
advantage that the number of binary classifiers to construct equals number of
classes. However, there are some drawbacks: first, for large training data
sets and increasing class number, the memory requirement is very high; second,
when the number of training samples in each class is equal, the training
samples sizes will be unbalanced.

The one versus one approach consists in constructing one SVM for each pair of
classes. Thus, for a problem with Q classes, Q(Q-1)/2 SVMs are trained to
distinguish the samples of one class from the samples of another class.
Usually, classification of an unknown pattern is done according to the maximum
voting, where each SVM votes for one class. The number of training data
vectors required for each classifier is reduced than the previous method.
Hence, this method is considered more symmetric. Moreover, the memory required
is much smaller. However the main drawback of this method is the increase in
the number of classifiers as the number of class increases.

Figure 3: Illustration of SVM.

##  III Experimental results

To evaluate the performance of the classification system, we conducted
experiments on a publicly available dataset provided by Dos Santos Ferreira
and al. [5]. Totally 400 images covering four classes (100 images for each
class) were extracted from the database. They include Soil, Soybean (crop),
Broadleaf and Grass (weeds). A sample of the segmented images of each class
used in the experiments is shown in Fig. 4.

As the primary input feature in the SVM classifier, we used the mean and
standard deviation attributes of each channel of RGB and HSV color spaces.
Then the color features were combined with the texture features (GLCM and LBP)
for discriminating soybean/weed plants and soil. The horizontal direction 0°
with a range of 1 (nearest neighbor) was used to calculate the GLCM. Then, the
nine feature values mentioned in section II (contrast, correlation, energy,
entropy, homogeneity, mean, standard deviation, skewness and kurtosis) were
calculated as the texture features of each image. The LBP method uses the
uniform-rotation invariant LBP operator to extract texture features. The
number of the neighboring pixel M is set at 8 and the radius R is set at 1.
The LBP algorithm generates a feature matrix with 10 image features. The
obtained feature matrices and label values were divided randomly into two
sets: the training set, and testing set, in order to train the SVM classifiers
(one.vs.all and one.vs.one). The SVM kernel type is set to be Linear Basis
Function. The training stage was carried out using sequential minimal
optimization (SMO) algorithm [20].

All the algorithms were developed in Matlab environment and tested on Intel
Core i3 computer with a 2.53 GHz processor and 3 Gigabytes of RAM. The
performance was evaluated by means of the classification accuracy, which is
refer to the ability of the algorithm to predict the correct class label for
instances of unknown class labels (testing set), and the confusion matrices
which present the correctness of each class and the percentage of confusion of
a class with the others. Because the trained data are generated randomly, the
fore-coming presented scores are averaged on 10 iterations, for all
experiments

Firstly, we conducted a comparison experiment to investigate the performances
of the input features on SVM-based classifier. TABLE I displays the confusion
matrices of the SVM classifier (1.vs.1) using different features (COLOR,
COLOR+GLCM, COLOR+LBP), where the total data was divided in 70

Figure 4: A sample of segmented images, from top to down: broadleaf, grass, soil and soybean. TABLE I: CONFUSION MATRICES FOR SVM (1.VS.1) BY USING DIFFERENT ATTRIBUTES Classified /real | Broadleaf | Grass | Soil | Sybean
---|---|---|---|---
COLOR |  |  |  |
Broadleaf | 67.00 | 17.67 | 0 | 15.33
Grass | 14.33 | 70.33 | 1.00 | 14.33
Soil | 0 | 0 | 100 | 0
Soybean | 7.67 | 16.33 | 0 | 76.00
COLOR+GLCM |  |  |  |
Broadleaf | 72.33 | 22.00 | 0.33 | 5.33
Grass | 15.33 | 74.33 | 0.67 | 9.67
Soil | 0 | 1.00 | 99.00 | 0
Soybean | 4.00 | 10.33 | 0 | 85.67
COLOR+LBP |  |  |  |
Broadleaf | 95.33 | 1.00 | 0 | 3.67
Grass | 2.33 | 95.00 | 0 | 2.67
Soil | 0 | 0 | 100 | 0
Soybean | 3.00 | 2.67 | 0 | 94.33

The confusion matrices given in Table I show that, the color attributes could
perfectly discriminate between green vegetation (crops and weeds) and soil due
to the large difference in the color information. Meanwhile, the greatest
errors occurred in the classification between crops and weeds. The accuracy
for the class Soil obtained by using color attributes is equal to $100$%, i.e.
is ideal. However, the classes Broadleaf, Grass and Soybean presented a large
confusion. For example, $67.00$% of data for Broadleaf class are correctly
classified, $17.67$% are mismatched with Grass class and $15.33$% are confused
with Soybean class. This confusion was slightly reduced by adding GLCM Texture
information to color spaces ($72.33$% of Broadleaf are correctly classified).
The classification was improved by the combination of color attributes with
LBP features ($95.33$% of broadleaf are correctly classified). Examining the
results more closely, we can say that the LBP method produces more consistent
classification accuracy performance, in comparison with the GLCM features. The
COLOR+LBP was the best combination of input features for the SVM models for
the classification. Therefore, the COLOR+LBP features were used as the final
model for further analysis.

On the other hand, a performance comparison between one versus all and one
versus one approaches using COLOR+LBP features was established. The results of
the overall classification accuracies and the prediction times for the two
approaches with different numbers of input images in training and testing
phases are shown in Tables II and III.

TABLE II: TOTAL ACCURACY FOR SVM CLASSIFIERS USING COLOR+LBP FEATURES Method | Training set percentage
---|---
| 30% | 50 % | 70 %
SVM one.vs.all | 89.43% | 91.05 % | 93.92 %
SVM one.vs.one | 92.89% | 94.35 % | 96.17 %
TABLE III: COMPUTATIONAL TIME FOR SVM CLASSIFIERS USING COLOR+LBP FEATURES Method | Training set percentage
---|---
| 30% | 50 % | 70 %
SVM one.vs.all | 1.2966 s | 1.5382 s | 1.8442 s
SVM one.vs.one | 0.6242 s | 0.8580 s | 1.2206 s

From Tables II and III, We remark that the performance ratio increases with
the number of of input images. The SVM classifier with one.vs.one approach is
promising in terms of overall accuracy and algorithm speed where, it presents
the most outstanding performance compared to the one.vs. all method. The
overall classification accuracy, using one .vs. one approach reaches $96.17$%
yields over $2$% higher accuracy than the one .vs. all approach (i.e.,
$93.92$%). As shown in execution times, the SVM one.vs.one strategy is much
faster than one.vs.all approach. Indeed, the training time of SVM classifier
increases significantly with the number of training samples. Thus, since the
number of samples which are needed to train the SVM classifier of one .vs. one
strategy become smaller, it is generally faster to train the $6$ SVMs of the
one.vs.one method than the $4$ SVMs of the one .vs. all approach.

##  IV Conclusion

In this work, we have applied SVM classifier on UAV images to discriminate
crops, weeds and soil. We have evaluated the following input features: color
attributes, GLCM texture features and LBP extractor. The confusion matrices
have been obtained for different combinations. We notice that the
misclassifications occurred mainly between crop and weed classes, and these
were improved by adding Texture information to color space. Soil could be
accurately discriminated using only color features since soil has a strong
color difference with green vegetation. The LBP extractor combined with color
features produce more consistent classification accuracy performance. The use
of SVM classifier with one versus one approach achieved excellent results,
with accuracy higher than $96$% in the classification of all classes, and is
computationally effective.

In future, we can explore other features for training the classifiers and
analyze the effects of other machine learning algorithms for classifying crop
images. Particularly, we investigate the use of deep learning to reduce the
confusion between crops and weeds.

## References

  * [1] M.A. Friedl, ”Remote Sensing of Croplands,” in Comprehensive Remote Sensing, Vol. 6, Elsevier, 2018, pp.78-95.
  * [2] C. Zhang, and J. M. Kovacs, ”The application of small unmanned aerial systems for precision agriculture: a review,” Precision agriculture vol. 13, pp. 693-712, 2012.
  * [3] D. C.Tsouros, S. Bibi, P. G. Sarigiannidis, ”A Review on UAV-Based Applications for Precision Agriculture,” Information. vol. 10, pp. 349– 375, 2019.
  * [4] M. D. Bah, A. Hafiane, R. Canals, ”Weeds detection in UAV imagery using SLIC and the hough transform,” In Proceedings of the Seventh International Conference on Image Processing Theory, Tools and Applications (IPTA), Montreal, QC, Canada, pp. 1-6, 28 November-1 December 2017.
  * [5] A. Dos Santos Ferreira, D. M. Freitas, G. G. da Silva, H. Pistori, and M. T. Folhes, ”Weed detection in soybean crops using ConvNets,” Comput. Electron. Agric. vol. 143, pp. 314-324, 2017.
  * [6] H. Huang, J. Deng, Y. Lan, A. Yang, X. Deng, and L. Zhang, ”A fully convolutional network for weed mapping of unmanned aerial vehicle (UAV) imagery,” PLoS ONE, vol. 13, pp. 1-19,2018.
  * [7] De Castro, Ana I and Torres-Sánchez, Jorge and Peña, Jose M and Jiménez-Brenes, Francisco M and Csillik, Ovidiu and López-Granados, Francisca ” An Automatic Random Forest-OBIA Algorithm for Early Weed Mapping between and within Crop Rows Using UAV Imagery”. Remote Sensing.vol. 10 , pp. 285-306 ,2018.
  * [8] V. Vapnik and A. Chervonenkis, ”A note on one class of perceptrons”, Automation and Remote Control, vol. 25. 1964.
  * [9] N. Cristianin and J. Shawe-Taylor, Support vector machines, Cambridge University Press, 2000.
  * [10] R. Haralick, K. Shanmugam, and I. Dinstein, (1973) ”Textural Features for Image Classification”, IEEE Trans. on Systems, Man and Cybernetics, SMC- vol. 3, pp. 610-621, 1973.
  * [11] T. Ojala, M. Pietikainen, D. A. Harwood, ”comparative study of texture measures with classification based on feature distributions,” J.Pattern Recognition,vol.29,pp. 51-59,1996.
  * [12] R. Achanta, A. Shaji, K. Smith, A. Lucchi, P. Fua and S. Süsstrunk, ”Slic superpixels compared to state-of-the-art superpixel methods,” IEEE Trans. Pattern Anal. Mach. Intell. vol. 34, pp. 2274-2282, 2012.
  * [13] K. Kawamura, H. Asai, T. Yasuda, P. Soisouvanhc and S. Phongchanmixay, ”Discriminating crops/weeds in an upland rice field from UAV images with the SLIC-RF algorithm,” PLANT PRODUCTION SCIENCE, Taylor and Francis 2020,pp. 1-18.
  * [14] N. Zayed and H.A. Elnemr, ”Statistical Analysis of Haralick Texture Features to Discriminate Lung Abnormalities”. Int J Biomed Imaging. 2015.
  * [15] D. Huang, C. Shan, M. Ardabilian, Y. Wang and L. Chen, ”Local Binary Patterns and Its Application to Facial Image Analysis: A Survey,” in IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), vol. 41, no. 6, pp. 765-781, Nov. 2011.
  * [16] Ojala, Timo and Pietikainen, Matti and Maenpaa, Topi ”Multiresolution Gray-scale and Rotation Invariant Texture Classification with Local Binary Patterns,” IEEE Trans. Pattern Analysis and Machine Intelligence , vol. 24,no.7,pp. 971-987,2002 .
  * [17] F. Mekhalfa and N. Nacereddine, ”Multiclass Classification of Weld Defects in Radiographic Images based on Support Vector Machines”, 10th International Conference on Signal-Image Technology and Internet-Based Systems (SITIS), 23-27 November 2014, Marrakech, Morocco, pp. 1-6.
  * [18] C. J. C Burges, ”A tutorial on support vector machines for pattern recognition”, Data Min. Knowl. Discov., vol.2. 1998, pp.121-167.
  * [19] J. Milgram, M. Cheriet and R. Sabourin, One Against One or One Against All: Which One is Better for Handwriting Recognition with SVMs, Proceedings of 10th International Workshop on Frontiers in Handwriting Recognition, 2006.
  * [20] J. C. Patt, ”Fast training of support vector machines using sequential minimal optimization,” http://research. microsoft.com/pubs/68391/smo-book.pdf.