# A new construction of the degree of maximal monotone maps

Mohammad Niksirat  Department of Mathematics, University of Alberta, Edmonton,
Canada, T6G 2J5<EMAIL_ADDRESS>
###### Abstract.

The inclusion equations of the type $f\in T(x)$ where $T:X\to 2^{X^{\ast}}$ is
a maximal monotone map, are extensively studied in nonlinear analysis. In this
paper, we present a new construction of the degree of maximal monotone maps of
the form $T:Y\to 2^{X^{*}}$, where $Y$ is a locally uniformly convex and
separable Banach space continuously embedded in $X$. The advantage of the new
construction lies in the remarkable simplicity it offers for calculation of
degree in comparison with the classical one suggested by F. Browder. We prove
a few classical theorems in convex analysis through the suggested degree.

###### Key words and phrases:

degree theory, finite rank approximation, maximal monotone maps, multivalued
maps

###### 2010 Mathematics Subject Classification:

Primary 47H11, Secondary 47H07

##  1\. Introduction

Assume that $X$ is a uniformly convex Banach space, $Y$ is separable and
reflexive Banach spaces equipped with uniformly convex norms, and
$i:Y\rightarrow X$ is the continuous embedding. Furthermore, assume that
$T:Y\rightarrow 2^{X^{\ast}}$ is a maximal monotone map with the effective
domain $D(T)=Y$ in the following sense. A pair $(\tilde{y},\tilde{x}^{\ast})$
is in the graph of $T$ if the condition $\langle
x^{\ast}-\tilde{x}^{\ast},i(y-\tilde{y})\rangle\geq 0$ holds for all
$(y,x^{\ast})\in{\rm graph}(T)$ where $\langle,\rangle$ denotes the continuous
pairing between $X,X^{*}$. In this article, we propose a topological degree
$T$ possessing classical properties of topological degree in certain sense.
The construction generalizes the F. Browder’s classical degree of maximal
monotone maps [1]. The Browder’s degree is constructed as follows. Assume that
$X$ is a reflexive Banach space equipped with a uniformly convex norm and
$T:X\rightarrow 2^{X^{\ast}}$ a maximal monotone map. The map
$T_{\epsilon}=T+\epsilon J$, for $\epsilon>0$ where $J:X\rightarrow X^{\ast}$
is the duality mapping possesses the following properties:

  1. (1)

$T_{\epsilon}$ is a map satisfying the following condition: for any
$x_{n}\rightharpoonup x$ in $X$, if there is $x^{\ast}_{n}\in T(x_{n})$ such
that

$\limsup_{n\rightarrow\infty}\langle x^{\ast}_{n}+\epsilon
J(x_{n}),x_{n}-x\rangle\leq 0,$

then $x_{n}\rightarrow x$.

  2. (2)

The map $T_{\epsilon}$ is onto $X^{\ast}$,

  3. (3)

if $x_{1}\neq x_{2}$, the sets $T(x_{1}),T(x_{2})$ are disjoint, that is,

$T(x_{1})\cap T(x_{2})=\emptyset.$

  4. (4)

The map $T_{\epsilon}^{-1}:X^{\ast}\times(0,\infty)\rightarrow X$ is well
defined and continuous.

Notice that $J$ is single valued, bijective and bi-continuous if $X$ is
uniformly convex. It is shown that the map $(T_{\epsilon}^{-1}+\epsilon
J^{-1})^{-1}:X\rightarrow X^{\ast}$ is a single valued demi-continuous and
$(S)_{+}$ for which a degree theory has been developed by F. Browder. The
degree of $T$ at $0\in X^{*}$ in an open bounded set $D\subset X$ is defined
by the following relation

(1.1) $\deg(T,D,0)=\lim_{\epsilon\rightarrow
0}\deg((T_{\epsilon}^{-1}+\epsilon J^{-1})^{-1},D,0).$

The degree suggested in this article generalizes the Browder’s degree in the
sense that if $Y=X$, the two degrees are the same. The main advantage of the
suggested degree is the direct use of finite rank approximation we employed in
our previous work [2] for single valued mappings. It is seen that the
constructed degree makes calculations much simpler than (1.1). We take note
that that the suggested degree is different from the degree of the map
$i^{\ast}\circ T:Y\rightarrow 2^{Y^{\ast}}$. The difference between two
formulations is discussed in [2] for single valued maps.

######  Definition 1.1.

Assume that $X_{1}$ and $X_{2}$ are Banach spaces. A map $A:X_{1}\rightarrow
2^{X_{2}}$ is called upper semi-continuous at $x\in X_{1}$ if for every
neighborhood $V$ of $A(x)$, there exists an open neighborhood $U$ of $x$ such
that $T(U)\subset V$.

We have the following theorem for the upper semi-continuous multi-valued
mappings; see for example [3, 4].

######  Theorem 1.2.

($\epsilon$-continuous subgraph) Assume that $X_{1}$ and $X_{2}$ are Banach
spaces, and the map $A:X_{1}\rightarrow 2^{X_{2}}$ is upper semi-continuous.
If $A(x)$ is closed and convex for all $x\in X_{1}$, then for any
$\epsilon>0$, there exists a continuous single valued function
$A_{\epsilon}:X_{1}\rightarrow X_{2}$ such that for any $x\in X_{1}$, there
exists $z_{1}\in X_{1}$ and $\tilde{z}_{2}\in A(z)$ such that
$\|x-z_{1}\|<\epsilon$ and $\|A_{\epsilon}(x)-\tilde{z}_{2}\|<\epsilon$.

######  Proposition 1.3.

Let $X,Y$ be Banach spaces, $i:Y\rightarrow X$ a continuous embedding and
$T:Y\rightarrow 2^{X^{\ast}}$ a maximal monotone map with the effective domain
$Y$. Then $T(y)$ is closed and convex for all $y\in Y$, and $T$ is norm to
weak-star upper semi-continuous in the following sense. For arbitrary $y\in
Y$, and arbitrary sequence $(y_{n})$ converges to $y$ in norm, there is a
weakly limit point $x^{\ast}$ of $\cup_{n}T(y_{n})$ such that $x^{\ast}\in
T(y)$.

The proof is completely similar to one for the map $T:X\rightarrow
2^{X^{\ast}}$. For a proof of the standard version see for example [5].

If $Y$ is a separable and reflexive Banach space equipped with a uniformly
convex norm, a theorem by Browder and Ton [6] guarantees the existence of a
separable Hilbert space $H$ such that the embedding $j:H\hookrightarrow Y$ is
dense and compact. Choosing an orthogonal basis $\\{h_{k}\\}_{k=1}^{\infty}$
for $H$, we obtaine the basis
$\mathcal{Y}=\\{y^{1},y^{2},\cdots,y^{n},\cdots\\}$ for $Y$ where
$y^{k}=j(h_{k})$, and accordingly, the filtration $\mathbb{Y}=\\{Y_{n}\\}$,
where $Y_{n}={\rm span}\\{y^{1},\ldots,y^{n}\\}$. The following proposition is
simply verified.

######  Proposition 1.4.

For any $y\in Y$, there is a sequence $(y_{n}),y_{n}\in Y_{n}$ such that
$y_{n}\to y$.

The pairing in $Y_{n}$ is denoted by $(,)$ and is defined by the relation
$(y^{i},y^{j})=\delta_{ij}$ for all $y^{i},y^{j}\in\mathcal{Y}$. We define the
maximal monotone operato $T:Y\to 2^{X^{*}}$ in the following sense.

######  Definition 1.5.

Suppose $X$ and $Y$ are separable and reflexive Banach spaces equipped with
uniformly convex norm, and assume that $T:Y\rightarrow 2^{X^{\ast}}$ is a
maximal monotone map. The finite rank approximation of arbitrary $x^{\ast}\in
T(y)$ in $Y_{n}\in\mathbb{Y}$, is defined by
$\hat{x}_{n}=\sum_{k=1}^{n}\langle x^{\ast},i(y^{k})\rangle y^{k}$.
Accordingly, the finite rank map $T_{n}:Y\rightarrow 2^{Y_{n}}$ is defined by
the relation

(1.2) $T_{n}(y)=\bigcup_{x^{\ast}\in T(y)}\hat{x}_{n}.$

For any $x^{*}\in X^{*}$ and $y\in Y_{n}$, we have the property

$(\hat{x}_{n},y)=\langle x^{\ast},i(y)\rangle,$

where $\langle,\rangle$ is the pairing between $X^{\ast},X$. In fact, if
$x^{\ast}\in X^{*}$, then for $\hat{x}_{n}=\sum_{k=1}^{n}\langle
x^{\ast},y^{k}\rangle y^{k}$, we have

$\sum_{k=1}^{n}\langle x^{\ast},i(y^{k})\rangle(y^{k},y)=\sum_{k=1}^{n}\langle
x^{\ast},i((y^{k},y)y^{k})\rangle=\left\langle
x^{\ast},i\left(\sum_{k=1}^{n}(y^{k},y)y^{k}\right)\right\rangle,$

and thus $\langle x^{\ast},i(y)\rangle=(\hat{x}_{n},y)$.

######  Lemma 1.6.

The finite rank approximation $T_{n}$ is upper semi-continuous and for every
$x\in Y$, the set $T_{n}(x)$ is closed and convex.

###### Proof.

Fix $n$ and $\epsilon>0$. If $T_{n}$ is not upper semi-continuous at $x\in Y$,
there is a sequence $(\delta_{m}),\delta_{m}\rightarrow 0$ and $x_{m}\in
B_{\delta_{m}}(x)$ such that for some $\hat{x}_{n,m}\in T_{n}(x_{m})$, we have
$\hat{x}_{n,m}\not\in V_{\epsilon}(T_{n}(x))$. $T$ is maximal monotone, and
thus locally bounded. Therefore, there is a subsequence (shown for the sake of
simplicity again by $\hat{x}_{n,m}$) such that $\hat{x}_{n,m}\to\hat{x}$. We
show $\hat{x}\in T_{n}(x)$. Since $\hat{x}_{n,m}\in T_{n}(x_{m})$, there is
$x^{\ast}_{m}\in T(x_{m})$ such that $\hat{x}_{n,m}$ are the finite rank
approximation in $Y_{n}$ of $x^{\ast}_{m}$, that is,

$\hat{x}_{n,m}=\sum_{k=1}^{n}\langle x^{\ast}_{m},i(y^{k})\rangle y^{k}.$

Since $T$ is norm to weak-star upper semi-continuous, and $x_{m}\rightarrow
x$, we have $x^{\ast}_{m}\rightharpoonup x^{\ast}$ for some $x^{\ast}\in
T(x)$. Thus $\langle x^{\ast}_{m},i(y^{k})\rangle\rightarrow\langle
x^{\ast},i(y^{k})\rangle$ for all $1\leq k\leq n$. Therefore

$\hat{x}_{n,m}=\sum_{k=1}^{n}\langle x^{\ast}_{m},i(y^{k})\rangle
y^{k}\rightarrow\sum_{k=1}^{n}\langle x^{\ast},i(y^{k})\rangle y^{k}\in
T_{n}(x),$

and therefore $\hat{x}\in T_{n}(x)$. Now we show that $T_{n}(x)$ is closed for
all $x\in Y$. Consider an arbitrary sequence $\hat{x}_{m}\in T_{n}(x)$, and
$\hat{x}_{m}\rightarrow\hat{x}$. Let $x^{\ast}_{m}\in T(y)$ be the sequence
such that $\hat{x}_{m}=\sum_{k=1}^{n}\langle x^{\ast}_{m},i(y^{k})\rangle
y^{k}$. Since $T(y)$ is bounded and convex, the sequence $x^{\ast}_{m}$
converges weakly (in a subsequence) to some $x^{\ast}\in T(x)$ and thus

$\hat{x}_{m}=\sum_{k=1}^{n}\langle x^{\ast}_{m},i(y^{k})\rangle
y^{k}\rightarrow\sum_{k=1}^{n}\langle x^{\ast},i(y^{k})\rangle y^{k},$

and thus

$\hat{x}=\sum_{k=1}^{n}\langle x^{\ast},i(y^{k})\rangle y^{k}\in T_{n}(x).$

That $T_{n}(x)$ is convex follows simply from the convexity of $T(x)$. ∎

By the Lemma (1.6) and Theorem (1.2), the $\epsilon$-continuous selection
$T_{n,\epsilon}$ of $T_{n}$ is well defined. The single valued map
$T_{n,\epsilon}$ is continuous and for any $x\in Y_{n}$, there is $z\in Y_{n}$
and $\hat{z}\in T_{n}(z)$ such that $\|z-x\|\in\epsilon$ and
$\|\hat{z}-T_{n,\epsilon}(x)\|<\epsilon$.

##  2\. Degree definition

Let $(\epsilon_{n})$ be a positive sequence such that $\epsilon_{n}\rightarrow
0$. Fix $\epsilon>0$. Consider the function
$\tilde{T}_{n,\epsilon_{n}}:Y_{n}\rightarrow Y_{n}$ defined by the relation

(2.1) $\tilde{T}_{n,\epsilon_{n}}=T_{n,\epsilon_{n}}+\epsilon J_{n},$

where $T_{n,\epsilon_{n}}$ is the $\epsilon_{n}$-continuous selection of
$T_{n}$ and $J_{n}$ is the finite rank approximation of $J\circ i:Y\rightarrow
X^{\ast}$ in $Y_{n}$ where $J:X\to X^{*}$ is the bi-continuous duality map.

######  Lemma 2.1.

Let $D\subset Y$ be an open bounded set and assume that for some $\epsilon>0$,
we have $0\not\in{\rm cl}T^{\epsilon}(\partial D)$, where
$T^{\epsilon}=T+\epsilon J\circ i$. Then there is $N>0$ such that
$0\not\in\tilde{T}_{n,\epsilon_{n}}(\partial D_{n})$ for all $n\geq N$ where
$D_{n}=D\cap Y_{n}$.

###### Proof.

Otherwise, there is a sequence $z_{n}\in\partial D_{n}$ such that
$\tilde{T}_{n,\epsilon_{n}}(z_{n})=0$ for all $n\geq 1$. Since $\partial D$ is
bounded, there is a subsequence (we show again by $z_{n}$) that weakly
converges to $z$. We first show that $z_{n}$ converges strongly to $z$. Choose
a sequence $\zeta_{n}\in Y_{n}$ that converges to $z$ in norm. Since
$\tilde{T}_{n,\epsilon_{n}}(z_{n})=0$ on $Y_{n}$, we have

$(T_{n,\epsilon_{n}}(z_{n}),z_{n}-\zeta_{n})+\epsilon(J_{n}(z_{n}),z_{n}-\zeta_{n})=0,$

because $z_{n}-\zeta_{n}\in Y_{n}$. By the relation

$(J_{n}(z_{n}),z_{n}-\zeta_{n})=\langle J\circ
i(z_{n}),i(z_{n}-\zeta_{n})\rangle,$

we can write

(2.2) $(T_{n,\epsilon_{n}}(z_{n}),z_{n}-\zeta_{n})+\epsilon\langle J\circ
i(z_{n}),i(z_{n}-\zeta_{n})\rangle=0.$

On the other hand, for each $z_{n}$, there is $x_{n}\in Y_{n}$ and
$\hat{x}_{n}\in T_{n}(x_{n})$ such that

$\|x_{n}-z_{n}\|<\epsilon_{n},\|\hat{x}_{n}-T_{n,\epsilon_{n}}(z_{n})\|<\epsilon_{n}.$

Therefore, we have

$(T_{n,\epsilon_{n}}(z_{n}),z_{n}-\zeta_{n})=(T_{n,\epsilon_{n}}(z_{n})-\hat{x}_{n},z_{n}-\zeta_{n})+(\hat{x}_{n},z_{n}-\zeta_{n}),$

and by the relation $\|T_{n,\epsilon_{n}}(z_{n})-\hat{x}_{n}\|<\epsilon_{n}$,
we obtain

$(T_{n,\epsilon_{n}}(z_{n}),z_{n}-\zeta_{n})\geq-\epsilon_{n}\|z_{n}-\zeta_{n}\|+(\hat{x}_{n},z_{n}-\zeta_{n})$

By the relation $\|x_{n}-z_{n}\|<\epsilon_{n}$, we have

$(\hat{x}_{n},z_{n}-\zeta_{n})\geq-\epsilon_{n}\|\hat{x}_{n}\|+(\hat{x}_{n},x_{n}-\zeta_{n}).$

Since $\hat{x}_{n}\in T_{n}(x)$, there are $x^{\ast}_{n}\in T(x_{n})$ such
that $\hat{x}_{n}$ are the finite rank approximations of $x^{\ast}_{n}$ in
$Y_{n}$. Thus, we can write

$(\hat{x}_{n},x_{n}-\zeta_{n})=\langle
x^{\ast}_{n},i(x_{n}-\zeta_{n})\rangle.$

Also, for some $C>0$, we can write

$\langle
x^{\ast}_{n},i(x_{n}-\zeta_{n})\rangle\geq-C\|x^{\ast}_{n}\|\|z-\zeta_{n}\|+\langle
x^{\ast}_{n},i(x_{n}-z)\rangle.$

Choose an arbitrary $z^{\ast}\in T(z)$. We have

$\langle x^{\ast}_{n},i(x_{n}-z)\rangle\geq\langle
x^{\ast}_{n}-z^{\ast},i(x_{n}-z)\rangle+\langle
z^{\ast},i(x_{n}-z)\rangle\geq\langle z^{\ast},i(x_{n}-z)\rangle.$

Since $z_{n}\rightharpoonup z$ and $\|x_{n}-z_{n}\|\to 0$, we conclude

$\lim_{n\rightarrow\infty}(T_{n,\epsilon_{n}}(z_{n}),z_{n}-\zeta_{n})\geq 0.$

Thus, the relation (2.2) implies

$\limsup_{n\rightarrow\infty}\langle J\circ
i(z_{n}),i(z_{n}-\zeta_{n})\rangle\leq 0.$

By the relation $\zeta_{n}\rightarrow z$, we conclude

$\limsup_{n\rightarrow\infty}\langle J\circ
i(z_{n}),i(z_{n}-\zeta_{n})\rangle\leq 0,$

and since $J$ is a map of class $(S)_{+}$, we obtain $z_{n}\rightarrow
z\in\partial D$. Now we show $0\in{\rm cl}T^{\epsilon}(z)$. Choose arbitrary
$y\in Y$ and sequence $y_{n}\in Y_{n}$, $y_{n}\rightarrow y$. We have

$(T_{n,\epsilon_{n}}(z_{n}),y_{n})+\epsilon\langle J\circ
i(z_{n}),i(y_{n})\rangle=0.$

Since $J$ is continuous, we have $\langle J\circ
i(z_{n}),i(y_{n})\rangle\rightarrow\langle J\circ i(z),y\rangle$. Choose
$x_{n}\in Y_{n}$ and $\hat{x}_{n}\in T_{n}(x_{n})$ such that

$\|x_{n}-z_{n}\|<\epsilon_{n},\|\hat{x}_{n}-T_{n,\epsilon_{n}}(z_{n})\|<\epsilon_{n}.$

We have

$\lim|(T_{n,\epsilon_{n}}(z_{n}),y_{n})-(\hat{x}_{n},y_{n})|=0.$

For $x^{\ast}_{n}\in T(x_{n})$, and by the relation $y_{n}\rightarrow y$, we
obtain

$\lim|(T_{n,\epsilon_{n}}(z_{n}),y_{n})-\langle x^{\ast}_{n},i(y)\rangle|=0.$

Since $T$ is norm to weak-star upper semi-continuous, we conclude $\langle
x^{\ast}_{n},i(y)\rangle\rightarrow\langle x^{\ast},y\rangle$ for some
$x^{\ast}\in T(z)$. This implies that $x^{\ast}+\epsilon J\circ i(z)$=0 and
thus $0\in{\rm cl}T^{\epsilon}(z)$ that contradicts the condition
$0\not\in{\rm cl}T^{\epsilon}(\partial D)$. ∎

######  Proposition 2.2.

Assume that $0\not\in{\rm cl}T(\partial D)$. Then there is $\epsilon>0$ such
that $0\not\in{\rm cl}T^{\epsilon}(\partial D)$.

###### Proof.

By the assumption, there is $r>0$ such that ${\rm dist}(0,{\rm cl}T(\partial
D))=r$. Let $z\in\partial D$ is arbitrary. Take arbitrary $z^{\ast}\in T(z)$.
We have

$\|z^{\ast}+\epsilon J(z)\|\geq\|z^{\ast}\|-\epsilon\|z\|\geq
r-\epsilon\|z\|.$

Therefore $0\not\in{\rm cl}T^{\epsilon}(\partial D)$ if

(2.3) $0<\epsilon<\frac{r}{\max_{z\in\partial D}\|z\|}.$

The boundedness of $\partial D$ guarantees the existence of $\epsilon>0$. ∎

######  Definition 2.3.

Assume that $X$ and $Y$ are separable and reflexive Banach spaces equipped
with uniformly convex norms, $D\subset Y$ is an open bounded set and $T:Y\to
2^{X^{\ast}}$ is a maximal monotone map such that $0\not\in{\rm cl}T(\partial
D)$. Choose $\epsilon>0$ satisfying $\left(\ref{epsilon}\right)$ and consider
the map $\tilde{T}_{n,\epsilon_{n}}$ defined in $\left(\ref{tildeT}\right)$.
The degree of $T$ in $D$ with respect to $0$ is defined by the following
formula

(2.4)
$\deg(T,D,0)=\lim_{n\rightarrow\infty}\deg_{B}(\tilde{T}_{n,\epsilon_{n}},D_{n},0),$

where $\deg_{B}$ is the usual Brouwer’s degree of the map
$\tilde{T}_{n,\epsilon_{n}}$.

######  Proposition 2.4.

The degree defined in the relation $\left(\ref{mmdeg}\right)$ is stable with
respect to $n$.

###### Proof.

Consider the sequence of mappings $(\tilde{T}_{k,\epsilon_{k}})$ such that for
sufficiently large $n$ the condition $0\not\in{\rm
cl}\tilde{T}_{k,\epsilon_{k}}(\partial D_{k})$ is satisfied for $k\geq n-1$ .
First note that there is $\epsilon_{0}>0$ such that for
$0<\epsilon_{1},\epsilon_{2}<\epsilon_{0}$, the following relation holds

(2.5)
$\deg_{B}(\tilde{T}_{n,\epsilon_{1}},D_{n},0)=\deg_{B}(\tilde{T}_{n,\epsilon_{2}},D_{n},0),$

In fact, for any $x\in Y_{n}$, there is $z_{1},z_{2}\in Y_{n}$ and
$\hat{z}_{1}\in T_{n}(z_{1})$, $\hat{z}_{2}\in T_{n}(z_{2})$ such that

$\|z_{1}-x\|<\epsilon_{1},\|\hat{z}_{1}-T_{n,\epsilon_{1}}(x)\|<\epsilon_{1},\|z_{2}-x\|<\epsilon_{2},\|\hat{z}_{2}-T_{n,\epsilon_{2}}(x)\|<\epsilon_{2}.$

The continuity of $T_{n,\epsilon_{1}}$, $T_{n,\epsilon_{2}}$ implies that
$\|\tilde{T}_{n,\epsilon_{1}}-\tilde{T}_{n,\epsilon_{2}}\|$ can be controlled
and thus (2.5) holds. Let us write $\tilde{T}_{n,\epsilon_{n}}$ as

$\tilde{T}_{n,\epsilon_{n}}=\tilde{T}_{n,\epsilon_{n}}^{1}+\tilde{T}_{n,\epsilon_{n}}^{2},$

where $\tilde{T}_{n,\epsilon_{n}}^{1}$ is the projection of
$\tilde{T}_{n,\epsilon_{n}}$ into $Y_{n-1}$ and
$\tilde{T}_{n,\epsilon_{n}}^{2}$ is the projection into $\\{y^{n}\\}$. Define
the map $S_{n,\epsilon_{n}}:Y_{n}\rightarrow 2^{Y_{n}}$ as

$S_{n,\epsilon_{n}}(x)=\tilde{T}_{n,\epsilon_{n}}^{1}(x)+(x,y^{n})y^{n}$

Obviously, we have

(2.6)
$\deg_{B}(S_{n,\epsilon_{n}},D_{n},0)=\deg_{B}(\tilde{T}_{n,\epsilon_{n}}^{1},D_{n-1},0).$

First we show

(2.7)
$\deg_{B}(\tilde{T}_{n,\epsilon_{n}}^{1},D_{n-1},0)=\deg_{B}(\tilde{T}_{n-1,\epsilon_{n}},D_{n-1},0)=\deg_{B}(\tilde{T}_{n-1,\epsilon_{n-1}},D_{n-1},0).$

The last equality follows from (2.5). In order to prove the first equality, we
note that if $T_{n,\epsilon_{n}}$ is an $\epsilon_{n}$-continuous selection of
$T_{n}$, then $T_{n,\epsilon_{1}}^{1}$ is also an $\epsilon_{n}$-continuous
selection of $T_{{n-1},\epsilon_{n}}$. In fact, let $x\in Y_{n-1}$ be
arbitrary, then there is $z\in Y_{n}$ and $\hat{z}\in T_{n}(y)$ such that

$\|\hat{z}^{1}-T^{1}_{n,\epsilon_{n}}(x)+\hat{z}^{2}-T^{2}n,\epsilon_{n}\|^{2}<\epsilon_{n}^{2},$

where $\hat{z}^{1}\in Y_{n-1}$ and $\hat{z}^{2}\in\\{y^{n}\\}$. This implies

$\|\hat{z}^{1}-T^{1}_{n,\epsilon_{n}}(x)\|<\epsilon_{n},\|z^{1}-y\|<\epsilon_{n}.$

Again it follows that $\|T_{n-1,\epsilon_{n}}-T^{1}_{n,\epsilon_{n}}\|$ can be
controlled and thus the first equality in (2.7) is proved. Now, we show

(2.8)
$\deg_{B}(\tilde{T}_{n,\epsilon_{n}},D_{n},0)=\deg_{B}(S_{n,\epsilon_{n}},D_{n},0).$

Consider the convex homotopy

$h_{n}(t)=(1-t)\tilde{T}_{n,\epsilon_{n}}+tS_{n,\epsilon_{n}}.$

It is enough to show $0\not\in h_{n}(t)(\partial D_{n})$ for $t\in[0,1]$.
Clearly, $0\not\in h_{n}(t)(\partial D_{n})$ for $t=0,1$. For $t\in(0,1)$
assume that there exists a sequence $t_{n}\in(0,1)$ and $(z_{n})$,
$z_{n}\in\partial D_{n}$ such that $h_{n}(t_{n})(z_{n})=0$. According to the
construction of $h_{n}(t)$ we have

$0=h_{n}(t_{n})(z_{n})=\tilde{T}^{1}_{n,\epsilon_{n}}(z_{n})+(1-t_{n})\tilde{T}^{2}_{n,\epsilon_{n}}(z_{n})+t_{n}(z_{n},y^{n})y^{n}.$

The above relation implies $\tilde{T}_{n,\epsilon_{n}}^{1}(z_{n})=0$ and

$\tilde{T}^{2}_{n,\epsilon_{n}}(z_{n})=-\frac{t_{n}}{1-t_{n}}(z_{n},y^{n})y^{n}.$

Since $\partial D$ is bounded then $z_{n}\rightharpoonup z$ in a subsequence.
Choose the sequence $(\zeta_{n}),\zeta_{n}\in Y_{n}$ and $\zeta_{n}\rightarrow
z$ and obtain

$(\tilde{T}_{n,\epsilon_{n}}(z_{n}),z_{n}-\zeta_{n})=-\frac{t_{n}}{1-t_{n}}|(z_{n},y^{n})|^{2}+\frac{t_{n}}{1-t_{n}}(z_{n},y^{n})(\zeta_{n},y^{n}).$

On the other hand since $\zeta_{n}\rightarrow z$, we have
$(\zeta_{n},y^{n})\rightarrow 0$. Since there exists sequence $\hat{z}_{n}\in
T_{n}(z_{n})$ such that
$\|\hat{z}_{n}-T_{n,\epsilon_{n}}(z_{n})\|<\epsilon_{n}$ we can write for some
$z^{\ast}_{n}\in T(z_{n})$

$\limsup_{n\rightarrow\infty}\langle z^{\ast}_{n}+\epsilon J\circ
i(z_{n}),i(z_{n}-z)\rangle=\limsup_{n\rightarrow\infty}(\tilde{T}_{n,\epsilon_{n}}(z_{n}),z_{n}-\zeta_{n}).$

Therefore we obtain

$\limsup_{n\rightarrow\infty}\langle z^{\ast}_{n}+\epsilon J\circ
i(z_{n}),i(z_{n}-z)\rangle\leq 0.$

Since

$\lim_{n\rightarrow\infty}\langle z^{\ast}_{n},i(z_{n}-z)\rangle\geq 0,$

we obtain

$\limsup_{n\rightarrow\infty}\langle J\circ i(z_{n}),i(z_{n}-z)\rangle\leq 0,$

and thus $z_{n}\rightarrow z$. This is impossible because $0\not\in{\rm
cl}T^{\epsilon}(\partial D)$. ∎

Now, we show that the definition (2.4) satisfies the classical properties of a
topological degree including the solvability and the homotopy invariance.

######  Theorem 2.5.

Let $D\subset Y$ be an open bounded set and assume that $T:Y\rightarrow
2^{X^{\ast}}$ is maximal monotone and $0\not\in{\rm cl}T(\partial D)$. If

$\deg(T,D,0)\neq 0,$

then there is $y\in D$ such that $0\in T(y)$.

###### Proof.

Assume $\deg(T,D,0)\neq 0$, then there exists a sequence $z_{n}\in D$ such
that $\tilde{T}_{n,\epsilon_{n}}(z_{n})=0$ for sufficiently small
$\epsilon_{n}>0$. This implies that there is the sequence $\hat{z}_{n}\in
T_{n}(z_{n})$ such that $\|\hat{z}_{n}+\epsilon J_{n}(z_{n})\|<\epsilon_{n}$.
Since $D$ is bounded then $z_{n}$ converges weakly (in a subsequence) to some
$z$. Since $T$ is monotone, we conclude

$\limsup_{n\rightarrow\infty}\langle J\circ i(z_{n}),i(z_{n}-z)\rangle\leq 0,$

and thus $z_{n}$ converges strongly to $z\in{\rm cl}(D)$. Let $\hat{z}_{n}$ be
the $n$-approximation of $z^{\ast}_{n}\in T(z_{n})$. Since $T$ is norm to
weak-star upper semi-continuous, the sequence $z^{*}_{n}$ converges weakly in
a subsequence to some $z^{\ast}\in T(z)$. Let $v\in Y$ be arbitrary. Consider
the sequence $(v_{n}),v_{n}\in Y_{n}$ and $v_{n}\rightarrow y$. Then we have

$\langle z^{\ast}_{n},i(v_{n})\rangle=-\epsilon\langle J\circ
i(z_{n}),i(v_{n})\rangle-\epsilon(T_{n,\epsilon_{n}}(z_{n})-\hat{z}_{n},v_{n}\rangle\rightarrow
0,$

On the other hand, $\langle z^{\ast}_{n},i(v_{n})\rangle\rightarrow\langle
z^{\ast},i(y)\rangle$, and thus $z^{\ast}=0$ or equivalently $0\in T(z)$.
Since $0\not\in{\rm cl}T(\partial D)$, we conclude $z\in D$. ∎

######  Definition 2.6.

Let $D\subset X$ be an open bounded set, and assume $h:[0,1]\times
Y\rightarrow 2^{X^{\ast}}$ be a continuous homotopy with respect to $t$ such
that for any $t\in[0,1]$, the map $h(t):Y\rightarrow 2^{X^{\ast}}$ is maximal
monotone. Furthermore assume that $0\not\in{\rm cl}h([0,1]\times\partial D)$.
The map $h$ is called an admissible homotopy for maximal monotone maps.

######  Proposition 2.7.

The degree defined in $\left(\ref{mmdeg}\right)$ is stable under the
admissible homotopy of maximal monotone maps.

###### Proof.

According to the definition of the admissible homotopy, the degree

$\deg_{B}(\tilde{h}_{n,\epsilon_{n}}(t),D_{n},0)$

is independent of $t$ due to the fact
$0\not\in\tilde{h}_{n,\epsilon_{n}}(t)(\partial D_{n})$ for $t\in[0,1]$ and
the homotopy invariance of the Brouwer’s degree. Now, the stability of the
defined degree $\left(\ref{mmdeg}\right)$ with respect to $n$ implies that the
degree $\deg(h(t),D,0)$ is independent of $t$. ∎

##  3\. Degree theoretic proofs

We give degree theoretic proofs of some theorems in convex analysis. The first
theorem is due to D. DeFigueirdo [7].

######  Theorem 3.1.

Assume that $X$ is a separable uniformly convex Banach space, $T:X\rightarrow
2^{X^{\ast}}$ is a maximal monotone map such that $0\not\in(T+\lambda
J)(S_{r})$, where $S_{r}$ is the sphere of radius $r$ and $\lambda>0$ is
arbitrary. Then there exists $u\in{\rm cl}(B_{r})$ such that $0\in T(u)$,
where $B_{r}$ is the ball of radius $r$ in $X$.

###### Proof.

Assume that $0\not\in T({\rm cl}(B_{r}))$. We show first that $0\not\in{\rm
cl}T(S_{r})$. Otherwise there exist a sequence $u_{n}\in S_{r}$ and
$u^{\ast}_{n}\in T(u_{n})$ such that $u^{\ast}_{n}\rightarrow 0$. The sequence
$(u_{n})$ converges weakly in a subsequence (that we show again by $u_{n}$) to
some $u\in{\rm cl}(B_{r}).$ Claim: $[u,0]\in{\rm graph}(T)$. For any
$[x,x^{\ast}]\in{\rm graph}(T)$ we have the inequality

$\langle x^{\ast},x-u\rangle=\lim\langle
x^{\ast}-u^{\ast}_{n},x-u_{n}\rangle\geq 0.$

Since $T$ is maximal monotone, the above inequality implies $[x,0]\in{\rm
graph}(T)$ or equivalently $0\in T(u)$. This contradicts the assumption
$0\not\in T({\rm cl}(B_{r}))$. It is also apparent that $0\not\in{\rm
cl}(J(S_{r}))$. Next, we show $0\not\in{\rm cl}((1-t)T+tJ)(S_{r}))$ for
$t\in(0,1)$. Otherwise there exist $t_{n}\in(0,1)$, $u_{n}\in S_{r}$ and
$u^{\ast}_{n}\in T(u_{n})$ such that

$(1-t_{n})u^{\ast}_{n}+t_{n}J(u_{n})\rightarrow 0.$

Again for $u_{n}\rightharpoonup u$ and $t_{n}\rightarrow t$ we obtain by the
monotonicity property of $T$ the following inequality

$\limsup_{n\rightarrow\infty}\langle J(u_{n}),u_{n}-u\rangle\leq 0,$

that implies $u_{n}\rightarrow u\in S_{r}$. Claim:
$[u,\frac{-t}{1-t}J(u)]\in{\rm graph}(T)$. For any $[x,x^{\ast}]\in{\rm
graph}(T)$ we obtain by the fact $\frac{-t}{1-t}J(u_{n})\in T(u_{n})$ the
following relation

$\langle x^{\ast}+\frac{t}{1-t}J(u),x-u\rangle=\lim\langle
x^{\ast}+\frac{t}{1-t}J(u_{n}),x-u_{n}\rangle\geq 0,$

that proves the claim. Now by degree theoretic argument we have

$\deg(T,B_{r},0)=\deg((1-t)T+tJ,B_{r},0)=\deg(J,B_{r},0)=1.$

The above calculation guarantees the existence of $u\in B_{r}$ such that $0\in
T(u)$ and this contradicts the assumption $0\not\in T({\rm cl}(B_{r}))$.
Therefore the assumption $0\not\in T({\rm cl}(B_{r}))$ is wrong and thus $0\in
T({\rm cl}(B_{r}))$. ∎

The next theorem is again from DeFigueirdo [7].

######  Proposition 3.2.

Let $X$ be a separable uniformly convex Banach space and assume that
$f:X\rightarrow X^{\ast}$ is a pseudo-monotone map. Then ${\rm Rang}(\partial
N_{r}+f)=X^{\ast}$ where $N_{r}$ is the map

$N_{r}(x)=\left\\{\begin{array}[]{l}0\hskip 28.45274pt{\rm if\ }x\in B_{r}\\\
1\hskip 28.45274pt{\rm if\ }x\in S_{r}\end{array}\right.,$

and $\partial N_{r}$ is the set of the sub-gradients of $N_{r}$.

###### Proof.

Apparently, we have

(3.1) $\partial N_{r}(x)=\left\\{\begin{array}[]{ll}0&{\rm if\ }x\in B_{r}\\\
\\{\lambda J(x),\lambda\geq 0\\}&{\rm if\ }x\in S_{R}\end{array}\right..$

Claim: for every $f_{0}\in X^{{}^{\ast}}$, we have

(3.2) $\deg(\partial N_{r}+f-f_{0},B_{r},0)\neq 0.$

First we show if $0\not\in(\partial N_{r}+f-f_{0})({\rm cl}(B_{r}))$ then

(3.3) $0\not\in{\rm cl}(\partial N_{r}+f-f_{0})(S_{r}).$

Otherwise, there is a sequence $u_{n}\in S_{r}$ and $u^{\ast}_{n}\in\partial
N_{r}(u_{n})$ such that $u^{\ast}_{n}+f(u_{n})-f_{0}\rightarrow 0$. But
$u_{n}\rightharpoonup u\in{\rm cl}(B_{r})$ in a sub-sequence. We prove that
$[u,f_{0}-f(u)]\in{\rm graph}(\partial N_{r})$. Let
$f_{0}=u^{\ast}_{n}+f(u_{n})+\epsilon(n)$ where $\epsilon(n)\in X^{\ast}$ and
$\epsilon(n)\rightarrow 0$. For any arbitrary $[x,x^{\ast}]\in{\rm
graph}(\partial N_{r})$, we have

$\displaystyle\langle x^{\ast}-f_{0}+f(u),x-u\rangle=\lim\langle
x^{\ast}+f(u)-u^{\ast}_{n}-f(u_{n}),x-u_{n}\rangle\geq$
$\displaystyle\lim\langle f(u)-f(u_{n}),x-u\rangle.$

But

(3.4) $0=\lim\langle
u^{\ast}_{n}+f(u_{n})-f_{0},u_{n}-u\rangle\geq\limsup\langle
f(u_{n}),u_{n}-u\rangle$

Since $f$ is pseudo-monotone we obtain $f(u_{n})\rightharpoonup f(u)$ and
therefore

$\langle x^{\ast}-f_{0}+f(u),x-u\rangle\geq 0.$

This implies that $[u,f_{0}-f(u)]\in{\rm graph}(\partial N_{r})$ and thus
$0\in(\partial N_{r}+f-f_{0})({\rm cl}(B_{r}))$ which is impossible by the
assumption. Now consider the affine homotopy

(3.5) $h(t)=(1-t)(\partial N_{r}+f-f_{0})+tJ.$

for $t\in(0,1]$. We show

$0\not\in{\rm cl}((1-t)(\partial N_{r}+f-f_{0})+tJ)(S_{r}).$

Otherwise, there is a sequence $u_{n}\in S_{r}$, $u^{\ast}_{n}\in\partial
N_{r}(u_{n})$ and $t_{n}\rightarrow t$ such that

$(1-t_{n})(u^{\ast}_{n}+f(u_{n})-w)+t_{n}Ju_{n}\rightarrow 0.$

But $u_{n}\rightharpoonup u\in{\rm cl}(B_{r})$ in a subsequence. W show

$[u,f_{0}-f(u)-\frac{t}{1-t}J(u)]\in{\rm graph}(\partial N_{r}).$

For any $[x,x^{\ast}]\in{\rm graph}(\partial N_{r})$ we have

$\displaystyle\langle x^{\ast}-f_{0}+f(u)+\frac{t}{1-t}J(u),x-u\rangle=$
$\displaystyle\lim\langle
x^{\ast}+f(u)+\frac{t}{1-t}J(u)-u^{\ast}_{n}-f(u_{n})-\frac{t_{n}}{1-t_{n}}J(u_{n}),x-u\rangle\geq$
$\displaystyle\geq\limsup\langle
f(u)-f(u_{n}),x-u\rangle+\liminf\langle\frac{t}{1-t}J(u)-\frac{t_{n}}{1-t_{n}}J(u_{n}),x-u\rangle$

We conclude $u_{n}\rightarrow u\in S_{r}$ and $f(u_{n})\rightharpoonup f(u)$
because $f$ is pseudo-monotone. Therefore we obtain again

$0\in(\partial N_{r}+f+\frac{t}{1-t}J-f_{0})({\rm cl}(B_{r})).$

But $(\partial N_{r}+f+\frac{t}{1-t}J-f_{0})({\rm cl}(B_{r}))=(\partial
N_{r}+f-f_{0})({\rm cl}(B_{r}))$ and then $0\in(\partial N_{R}+f-f_{0})({\rm
cl}(B_{r}))$ that is impossible. Finally we use the homotopy invariance
property of degree and write

$\deg(\partial N_{R}+f-f_{0},B_{r},0)=\deg(h(t),B_{R},0)=\deg(J,B_{r},0)=1.$

Therefore there exist $u\in{\rm cl}(B_{r})$ such that $f_{0}\in\partial
N_{r}(u)+f(u)$. ∎

The following theorem is due to F. Browder [8] for the surjectivity of the
monotone maps with locally bounded inverse.

######  Theorem 3.3.

Assume $A:X\rightarrow X^{\ast}$ is a demi-continuous monotone map such that
$A^{-1}$ is locally bounded, that is, for every $f\in X^{\ast}$ there is a
bounded $V_{f}\ni f$ such that $A^{-1}(V_{f})$ is bounded. Then $A$ is onto.

###### Proof.

For any $f\in X^{\ast}$, we show that there is sufficiently large $r=r(f)$
such that:

$\deg(A,B_{r},f)\neq 0.$

Choose $r>0$ such that for a neighborhood $V_{f}\ni f$ the following condition
is satisfied

$S_{r}\cap A^{-1}(V_{f})=\emptyset,$

or equivalently $f\not\in{\rm cl}A(S_{r})$. Since there is $\epsilon>0$ such
that

$\deg(A,B_{r},f)=\deg(A+\epsilon J,B_{r},f),$

it is enough to show

(3.6) $\deg(A+\epsilon J,B_{r},f)\neq 0,$

for sufficiently large $r$ and sufficiently small $\epsilon>0$. First, we show

$\deg(A+\epsilon J,B_{r},0)\neq 0.$

In fact, if $(A+\epsilon J)(z)=0$ for $z\in\partial B_{r}$, then

$\langle A(z)-A(0),z\rangle+\epsilon\|z\|^{2}+\langle A(0),z\rangle=0.$

Since $A$ is monotone, the inequality $\epsilon\|z\|^{2}+\langle
A(0),z\rangle\leq 0$ implies $\epsilon\|z\|\leq\|A(0)\|$, that is impossible
for sufficiently large $r$. Since $A+\epsilon J$ is a map of class $(S)_{+}$,
define the homotopy $h(t)=tA+\epsilon J$. It is simply seen that $0\not\in
h(t)(\partial B_{r})$ and then

$\deg(A+\epsilon J,B_{r},0)=\deg(h(t),B_{r},0)=\deg(J,B_{r},0)\neq 0.$

The proof of (3.6) is completely similar to one presented above. ∎

## References

  * [1] F. Browder. The theory of degree of mapping for nonlinear mappings of monotone type, Nonlinear partial differential equations and their applications, 6:165–177, 1982.
  * [2] M. Niksirat, A new generalization of the Browder’s degree of (S)+ maps, appear soon.
  * [3] A. Cellina, Approximation of set valued functions and fixed point theorems, Ann. Mat. Pura Appl., 82:17–24, 1969.
  * [4] , N. G. Lloyd, Degree Theory, Cambrige University Press, 1978.
  * [5] V. P. Barbu, Nonlinear differential equations of monotone types in Banach spaces, Springer, 2010.
  * [6] F. Browder and B. A. Ton, Nonlinear functional equations in Banach spaces and elliptic super regularization, Math Z., 105:177–195, 1968.
  * [7] D. de Figuerido, An existence theorem for pseudo-monotone operator equation in Banach spaces, J. Math. Anal. Appl., 34:151–156, 1971.
  * [8] F. Browder. Nonlinear operators and nonlinear equations of evolution in Banach spaces. Americam Math. Soc., 1976.


# DeepSegmenter: Temporal Action Localization for Detecting Anomalies in
Untrimmed Naturalistic Driving Videos

Armstrong Aboah & Ulas Bagci
Department of Radiology
Northwestern University
armstrong.aboah<EMAIL_ADDRESS>Corresponding and primary author
Abdul Rashid Mussah & Neema Jakisa Owor
Department of Civil Engineering
University of Missouri-Columbia
<EMAIL_ADDRESS>equal contributionequal contribution Yaw Adu-Gyamfi
Department of Civil Engineering
University of Missouri-Columbia
<EMAIL_ADDRESS>
###### Abstract

Identifying unusual driving behaviors exhibited by drivers during driving is
essential for understanding driver behavior and the underlying causes of
crashes. Previous studies have primarily approached this problem as a
classification task, assuming that naturalistic driving videos come
discretized. However, both activity segmentation and classification are
required for this task due to the continuous nature of naturalistic driving
videos. The current study therefore departs from conventional approaches and
introduces a novel methodological framework, DeepSegmenter, that
simultaneously performs activity segmentation and classification in a single
framework. The proposed framework consists of four major modules namely Data
Module, Activity Segmentation Module, Classification Module and Postprocessing
Module. Our proposed method won 8th place in the 2023 AI City Challenge, Track
3, with an activity overlap score of 0.5426 on experimental validation data.
The experimental results demonstrate the effectiveness, efficiency, and
robustness of the proposed system. The code is available at
https://github.com/aboah1994/DeepSegment.git.

##  1 Introduction

Driving is a complex activity that necessitates a high level of concentration
and coordination. Even with the best intentions, drivers may engage in
behaviors that can result in accidents, such as distracted driving, aggressive
driving, or driving while impaired. Our ability to identify and characterize
these behaviors is essential for enhancing road safety and preventing traffic
accidents. To do the aforementioned, a comprehensive dataset detailing
different anomalous driving behaviors and differentiating them from safe
driving actions is required.

Figure 1: Cameras mounted on vehicle’s dashboards to monitor drivers’ behavior
in a naturalistic environment.

The naturalistic driving dataset is one such dataset that has been used
extensively to study unsafe driving behaviors. As shown in Fig. 1,
naturalistic driving data refers to information collected from sensors or
cameras mounted on vehicles as they are driven in real-world environments.
This type of data contains a wealth of information about the driving task,
including the actions of the driver and the context in which they are
performed. This makes it possible to identify patterns and trends in
naturalistic driving videos that may indicate the presence of anomalous or
unsafe driving behaviors.

Driving activity recognition models usually capitalize on the richness of the
visual feed provided by driver-facing camera data, to detect anomalous driving
behaviors[1, 2, 3, 4, 5]. The usual approach uses rule-based pose estimation
and posture tracking, through feature detection and tracking algorithms[1, 2,
6]. Other studies have focused on gaze mapping[7], as well as the fusion of
driver visual data with vehicle state characteristics[8]. This approach,
whilst widely used and successful, only solves one part of the problem.
Another limitation to this approach is the one-size-fits-all nature, which
brings limited flexibility and adaptability to the defined rules governing the
separation of normal and anomalous driving.

Given these limitations, advancements in anomalous driving behavior
recognition and classification have tapped into the potential of more robust
deep learning (DL) approaches[8, 3]. Researchers favor deep learning
approaches because they produce better predictions and outcomes than
conventional rule based and machine learning algorithms. Deep learning models
are mostly utilized to enable automatic feature extraction through the
training of complicated features with little external assistance to provide
meaningful representations of data through deep neural networks[9].

Whilst DL approaches come with many advantages, the initial hurdle for getting
them off the ground involves providing large amounts of pre-labeled data for
their training[10, 11, 12]. Although this limitation is easily overcome with
the availability of naturalistic driving data, the DL models are trained via
sequence of event frames from video data. As such, these models expect pre-
segmented video data clips in order to accurately classify the type of event
taking place[13].

Video data streamed from cameras in a naturalistic driving environment are
continuous and untrimmed in nature. In order to actively deploy activity
recognition and classification models to such a situation, the modeling
framework should have the ability to identify when anomalous driving activity
is taking place whilst accurately classifying what kind of activity that is.
This involves a summarized three step process of:

  1. 1.

Identifying and extracting visual features from video frames

  2. 2.

Defining the state of the driving behavior and observational period based on
instances related to the extracted features

  3. 3.

Classifying the type of driving behavior observed in the localized time
sequence where the action is observed

As earlier highlighted, the more popular rule based models do a stellar job
with steps 1 and 2, but are limited in their ability to accurately classify
the type of observed action (step 3), especially when there are multiple
classes of observed behaviors with little variations between groups of
them[3]. DL methods also have no trouble with steps 1 and 3 but usually
require sequential or batch processing of the video frames[13]. Whilst it’s
possible to localize actions from temporal sequence by introducing techniques
from rule-based algorithms, the research in this area is lacking.

The purpose of this study is to introduce a hybrid approach to continuous
video activity recognition by capitalizing on the advantages of the two
aforementioned approaches. The NVIDIA AI City 2023 challenge presented an
opportunity to tackle this problem. In this study, we introduce DeepSegmenter,
a temporal action localization algorithm which combines a DL feature detection
algorithm and rule based feature tracking algorithm to first localize
instances of anomalous driving behaviors (steps 1 and 2), before passing the
clipped sequence of frames into a DL based activity classifier (step 3) to
label the observed action as either normal driving or one of fifteen other
anomalous driving behaviors.

DeepSegmenter performs incredibly well at localizing the instances of
anomalous driving behaviors regardless of the length of the activity. The
benefit of this approach is realized in its capacity to improve upon the
current capabilities of Advanced Driver Assistance Systems (ADAS).

The rest of the paper is organized as follows. The second section discusses
related research work on methods used in characterizing anomalous driving
behaviour in naturalistic driving videos. The third section presents our
proposed approach, while the fourth section details the experiment procedure.
We then present our experimental results, which demonstrate the effectiveness
of our proposed method in segmenting and classifying anomaly detection.
Lastly, we discuss the implications of our findings and propose future
directions for research in this field.

##  2 Related works

Driver action recognition has been thoroughly explored in recent years, with
research into it still ongoing. By creating models to recognize or forecast
driving behaviors, and implementing remedial actions for anomalous and risky
driving, it will be possible to improve the safety of vehicle driving and
reduce the number of driver-caused road traffic accidents. Several research
publications have employed various strategies to present solutions to this
idea, with the most modern methodologies relying on supervised learning
strategies.

###  2.1 Traditional Methods

Earlier research in human activity recognition relied on empirical rule
inference and quantitative statistical analysis, such as Hidden Markov Model
(HMM), Gaussian Mixture Model (GMM), Random Forest (RF), Support Vector
Machine (SVM), Fuzzy Neural Network (FNN), and k-Nearest Neighbor (kNN) [14,
15, 16, 17, 18]. Using a random forest classification algorithm, Ahnagari et
al. [19] identified six prevalent distracted driving behaviors with a 0.765
accuracy rate. Yao et al. [20] also developed a random forest model to
identify distracted driving behavior with 0.9 accuracy. Whilst these
accuracies suggest incredible strides, they are limited to binary outcomes of
anomalous and normal driving, and fail in multi-classification tasks of
identifying the types of anomalous driving activity observed. Traditional
methods have provided us with a reasonable degree of accuracy, but their
drawbacks, such as dependence on specialist experience on artificial
extraction of characteristics, and inability to consider driving time sequence
and correlation, can lead to driving behavior identification errors [14, 21].
More recent advancements in the area of driver action recognition have looked
into bridging this limitation by utilizing more advanced deep learning based
models.

###  2.2 Deep learning Methods

The deep spatiotemporal features of driving behavior data can be automatically
extracted using deep learning techniques like Convolutional Neural Networks
(CNN), Recurrent Neural Networks (RNN), as well as Transformers. These
techniques also incorporate feature extraction and recognition prediction into
a model for end-to-end learning with high recognition accuracy. Current
temporal action detection models can be classified into either single-stage
detectors or two-stage detectors.

Single-Stage Detector. Single-stage detectors predict action proposals
directly from video features, without the need for intermediate processes like
region proposals or sliding windows [22, 23, 24, 25]. Anchor Free single-stage
detector employed by Tang et al. [22] predicts the action boundaries and
scores directly from the video features using an anchor free regression head
and a focal loss function. Decouple-SSAD method was proposed by Rahman and
Laganiere [24], this method decouples the localization and classification
tasks by using parallel branches and a soft-NMS module to refine the
proposals. Chen et al. [26] offers a single-stage approach that makes use of
Timeception (temporal CNN), which collects multi-scale temporal information
and produces action proposals and classification. Yan et al. [27] pre-trained
a CNN model using unsupervised feature learning via sparse filtering, followed
by fine-tuning with classification. Their system monitors the position of the
driver’s hands and uses extracted information to predict whether the posture
is safe or unsafe. Baheti et al. [28] also used a MobileVGG network to detect
and classify driver distraction. With the rise of the Transformer’s attention
technique that has achieved state-of-the-art results, many recent research
solutions have implemented transformers in temporal action detection [23, 29,
30].

Two-Stage Detector. A two-stage detector has two stages: one for extracting
frames and one for classifying proposals/redefining temporal boundaries.
Inspired by Faster-RCNN, Chao et al. [31], as well as Gao et al. [32], used a
multi-scale architecture to improve receptive field alignment and exploits the
temporal context of actions for proposal generation and action classification.
ActionFormer, initially developed by Zhang et al.[30], was employed by Nguyen
[33] to predict the temporal location of an event and do classification
simultaneously, a second-stage classifier is then employed to improve
prediction precision. Stragazer is also another efficient multi-scale vision
transformer that learns hierarchical robust representations and then uses a
sliding window for temporal localization[34]. Ding et al.[35] also proposed a
Coarse-to-Fine Boundary Localization Method in which the features of the video
are extracted first, and then a sliding window is used to generate coarse
boundaries. Following that, the boundaries are refined to obtain the fine
boundaries.

##  3 Approach

The overall structure of our DeepSegmenter system is illustrated in Fig 2. Our
design is simple yet effective. The proposed system consists of four primary
modules: Data Module, Activity Segmentation Module, Classification Module and
Postprocessing Module.

Figure 2: Overall Structure of DeepSegmenter.

###  3.1 Data Module

The data module is tasked with organizing and preprocessing the naturalistic
driving videos. To accomplish this, the data module executes a series of
preprocessing operations, including resolution normalization and pixel-value
normalization. Resolution normalization entails resizing the video to a
standard resolution, which ensures that the video can be analyzed uniformly
across all samples as shown in Equation 1. Pixel-value normalization involves
rescaling the pixel values to a standard range of 0 and 1, which eliminates
differences in brightness and contrast between videos (Equation 2). By
performing these preprocessing steps, the data module ensures that the
collected data is in a format that subsequent modules can easily process and
analyze.

$X_{resized}=\text{resize}(X_{original},(W,H))$ (1)

where $X_{original}$ is the original video, $X_{resized}$ is the resized
video, and $\text{resize}()$ is a function that resizes the video to the
specified dimensions.

$X_{normalized}=\frac{X_{resized}-X_{min}}{X_{max}-X_{min}}$ (2)

where $X_{resized}$ is the resized video, $X_{min}$ is the minimum pixel value
in $X_{resized}$, $X_{max}$ is the maximum pixel value in $X_{resized}$, and
$X_{normalized}$ is the normalized video.

###  3.2 Activity Segmentation Module

The activity segmentation module breaks down the continuous stream of video
data into discrete segments that can be analyzed and later classified. This
module consists of two submodules:

Keypoint Detection. The keypoint detection submodule identifies key points in
each frame of the video, such as the face and hands of the driver, which are
required by the activity segmentation step. This submodule employs a
pretrained yolov7[36] keypoint detections model to detect and track the
movement of these keypoints across multiple frames.

Activity Segmentation. The activity segmentation submodule uses detected key
points to identify and classify driver activities as either event (anomaly) or
non-event (normal driving). This submodule utilizes heuristic-based algorithm
for it categorization as illustrated in Algorithm 1. Anomalies are triggered
by either hand or head movements of the driver. In the case of a head anomaly,
the submodule detects when the angle between the eyes and the nose surpasses a
predefined threshold, if so, it classifies the frame as an anomaly, otherwise,
it is considered as normal driving. Similarly, for a hand anomaly, when the
angle between the hand exceeds a predefined threshold, an anomaly is
triggered, else it is classified as normal driving.

Algorithm 1 Activity Segmentation

Require: $\theta_{h},\theta_{hand}$ (threshold angles for head and hand
movements)

Ensure: classification result $c$ for each frame

  1. 1.

Function classifyActivity(frame):

    1. (a)

keyPoints $\leftarrow$ extractKeyPoints(frame)

    2. (b)

headAngle $\leftarrow$ calculateHeadAngle(keyPoints)

    3. (c)

handAngle $\leftarrow$ calculateHandAngle(keyPoints)

    4. (d)

if headAngle $>$ $\theta_{h}$ then

    5. (e)

return Anomaly

    6. (f)

else if handAngle $>$ $\theta_{hand}$ then

    7. (g)

return Anomaly

    8. (h)

else

    9. (i)

return Normal Driving

  2. 2.

Function Main():

    1. (a)

for each frame $f$ do

    2. (b)

classificationResult $\leftarrow$ classifyActivity($f$)

    3. (c)

storeResult(classificationResult)

###  3.3 Activity Classification Module

The activity classification module is in charge of classifying the segmented
event into 1 of 15 different categories of driving anomalies as shown in Table
1. This module uses a 3D CNN architecture known as X3D[37], developed for
video analysis.

X3D Architecture. The X3D structure is designed to efficiently process video
data using a combination of 2D and 3D convolutions. The central idea behind
X3D is to expand the network in both spatial and temporal dimensions to allow
for greater expressiveness while preserving efficiency. The three components
of the X3D network are the entry flow, the middle flow, and the exit flow. The
entry flow initially processes incoming video frames using a series of 2D
convolutional layers, followed by a 3D convolutional layer containing temporal
information. The middle flow adds additional 3D convolutional layers to the
network. The exit flow then reduces the output’s spatial dimensions using a
combination of 2D and 3D convolutions making it easier for classification by
fully connected layer as demonstrated in Fig 3.

Figure 3: X3D model architecture.

X3D networks enlarge a 2D network along multiple axes, such as duration, frame
rate, spatial resolution, width, bottleneck width, and depth. The application
of channel-wise spatio-temporal convolutions that enable the efficient
processing of video data is one of the most significant developments of X3D.
These convolutions make use of shared weights across neighboring channels in a
feature map, thereby reducing the number of required parameters and increasing
efficiency. The ”factorized” design of X3D, in which 3D convolutions are
broken down into a series of 2D convolutions, reduces the computational cost
of 3D convolutions while preserving their ability to capture temporal
information.

###  3.4 Postprocessing Module

The postprocessing module removes false positive detections, which consist
primarily of events extracted for less than one second. It employs a rule-
based algorithm that estimates the duration of classified events and either
ignores or adds to the final submission if the duration exceeds 1 second.
Finally, this module prepares the final results in the format required by the
evaluation system.

##  4 Experiment

In this study, we assess the effectiveness of the DeepSegmenter system on the
AI City Challenge dataset for Naturalistic Driver Action Recognition. Our
findings reveal that our model performs well compared to other systems when
tackling this difficult task.

Data. The data set consists of 210 video clips that amount to approximately 34
hours of footage, which were taken from 35 different drivers[38, 39]. Each
driver performed 16 different tasks, including actions like talking on the
phone, eating, and reaching back, once and in a random order. Each vehicle was
equipped with three cameras that recorded from different angles in
synchronization. To collect the data, each driver completed the tasks twice:
once without any appearance block and once while wearing an appearance block
such as sunglasses or a hat. This resulted in a total of 6 videos per each
driver, with 3 videos recorded without an appearance block and 3 videos
recorded with an appearance block. The summary of the activities performed by
the drivers are summarized in Table 1.

Table 1: Driver Activity Summary Activity Class type | Activity Label
---|---
1 | Drinking
2 | Phone Call(right)
3 | Phone Call(left)
4 | Eating
5 | Text (Right)
6 | Text (Left)
7 | Reaching behind
8 | Adjust control panel
9 | Pick up from floor (Driver)
10 | Pick up from floor (Passenger)
11 | Talk to passenger at the right
12 | Talk to passenger at backseat
13 | yawning
14 | Hand on head
15 | Singing or dancing with music

Training. The experiments were carried out on an NVIDIA GTX 1080ti GPU
graphics card. The activity classification model was built on the PyTorch
Lightning framework. The dataset was partitioned into ratios of 0.7:0.3,
corresponding to the training and validation datasets, respectively. We used
the Adam optimizer with a starting learning rate of 0.001 and a weight decay
of 0.001. We used the CosineAnnealingLR scheduler to adjust the learning rate
during training. The model was trained for 300 epochs with a batch size of 8.

Evaluation metrics. The average activity overlap score was used as the
evaluation metric in this study as shown in Equation 3. This score is
determined by comparing the predicted activity with the ground-truth activity
based on their overlap. The closest match will be considered as the predicted
activity with the highest overlap score (os), provided that it belongs to the
same class as the ground-truth activity. However, this match will only be
considered if the predicted activity’s start time (ps) and end time (pe) are
within a range of 10 seconds before or after the ground-truth activity’s start
time (gs) and end time (ge), respectively. The overlap score is computed by
finding the ratio between the time intersection and time union of the two
activities.

$os(p,g)=\frac{\max(\min(ge,pe)-\max(gs,ps),0)}{\max(ge,pe)-\min(gs,ps)}$ (3)

##  5 Results and Discussion

The 2023 NVIDIA AI City Challenge Track 3 test videos consists of 30 untrimmed
videos from 5 different drivers at 3 different camera positions. A submission
to the competition is a text file that follows the format: Video ID, Activity
ID, Start time, and End time. The Video ID is a numeric identifier for the
video, starting with 1 and indicating its position in the alphabetically
ordered list of all Track 3, Test Set videos. The Activity ID is a numeric
identifier beginning with 1 for the classified class of the anomaly.The Start
and End times are integer values representing the beginning and end of the
anomalous activity, respectively.

On the experimental test dataset, our proposed methodology achieved an overall
overlap score of 0.5426, ranking 8th on the public leader board, as shown in
Table 2.

Table 2: Top 10 Leader Board Ranking Rank | Team ID | Team Name | Score
---|---|---|---
1 | 209 | MeituanIoTCV | 0.7416
2 | 60 | JNU_boat | 0.7041
3 | 49 | ctcAI | 0.6723
4 | 118 | RW | 0.6245
5 | 8 | Purdue Digital Twin | 0.5921
6 | 48 | BUPTMCPRL | 0.5907
7 | 83 | DiveDeeper | 0.5881
8 | 217 | INTELLI_LAB (Ours) | 0.5426
9 | 152 | AI_LAB | 0.5424
10 | 11 | AIMIZ | 0.5409

##  6 Conclusion

This study presents a solution for Track 3 of the 2023 AI City Challenge that
focuses on performing both activity segmentation and classification in a
single framework called DeepSegmenter. The proposed framework is composed of
four modules: Data Module, Activity Segmentation Module, Classification
Module, and Postprocessing Module. According to the experimental results on
the test dataset, the proposed framework ranks 8th in the challenge with an
overlap score of 0.5426. On this challenge, we demonstrated the effectiveness
of our proposed framework.

## References

  * [1] Khaled Shaaban, Osama Halabi, and Eman Almughani.  A posture recognition system to track drivers’ activities while driving.  Journal of Traffic and Transportation Management, 3(2):25–31, 2021\.
  * [2] Christian Braunagel, Enkelejda Kasneci, Wolfgang Stolzmann, and Wolfgang Rosenstiel.  Driver-activity recognition in the context of conditionally autonomous driving.  In 2015 IEEE 18th International Conference on Intelligent Transportation Systems, pages 1652–1657. IEEE, 2015.
  * [3] Lichao Yang, Kuo Dong, Yan Ding, James Brighton, Zhenfei Zhan, and Yifan Zhao.  Recognition of visual-related non-driving activities using a dual-camera monitoring system.  Pattern Recognition, 116:107955, 2021.
  * [4] Armstrong Aboah, Yaw Adu-Gyamfi, Senem Velipasalar Gursoy, Jennifer Merickel, Matt Rizzo, and Anuj Sharma.  Driver maneuver detection and analysis using time series segmentation and classification.  Journal of Transportation Engineering, Part A: Systems, 149(3):04022157, 2023.
  * [5] Armstrong Aboah, Bin Wang, Bagci Ulas, and Yaw Adu-Gyamfi.  Real-time multi-class helmet violation detection using few-shot data sampling technique and yolov8.  In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, June 2023.
  * [6] Manuel Martin, Johannes Popp, Mathias Anneken, Michael Voit, and Rainer Stiefelhagen.  Body pose and context information for driver secondary task detection.  In 2018 IEEE Intelligent Vehicles Symposium (IV), pages 2015–2021. IEEE, 2018.
  * [7] Lichao Yang, Kuo Dong, Arkadiusz Jan Dmitruk, James Brighton, and Yifan Zhao.  A dual-cameras-based driver gaze mapping system with an application on non-driving activities monitoring.  IEEE Transactions on Intelligent Transportation Systems, 21(10):4318–4327, 2019.
  * [8] Lichao Yang, Mahdi Babayi Semiromi, Yang Xing, Chen Lv, James Brighton, and Yifan Zhao.  The identification of non-driving activities with associated implication on the take-over process.  Sensors, 22(1):42, 2021.
  * [9] Senem Tanberk, Zeynep Hilal Kilimci, Dilek Bilgin Tükel, Mitat Uysal, and Selim Akyokuş.  A hybrid deep model using deep learning and dense optical flow approaches for human activity recognition.  IEEE Access, 8:19799–19809, 2020.
  * [10] Chunhui Gu, Chen Sun, David A. Ross, Carl Vondrick, Caroline Pantofaru, Yeqing Li, Sudheendra Vijayanarasimhan, George Toderici, Susanna Ricco, Rahul Sukthankar, Cordelia Schmid, and Jitendra Malik.  Ava: A video dataset of spatio-temporally localized atomic visual actions, 2018.
  * [11] Hieu H. Pham, Louahdi Khoudour, Alain Crouzil, Pablo Zegers, and Sergio A. Velastin.  Video-based human action recognition using deep learning: A review, 2022\.
  * [12] M. Naphade, S. Wang, D. C. Anastasiu, Z. Tang, M. Chang, Y. Yao, L. Zheng, M. Shaiqur Rahman, A. Venkatachalapathy, A. Sharma, Q. Feng, V. Ablavsky, S. Sclaroff, P. Chakraborty, A. Li, S. Li, and R. Chellappa.  The 6th ai city challenge.  In 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pages 3346–3355. IEEE Computer Society, June 2022\.
  * [13] Yu-Wei Chao, Sudheendra Vijayanarasimhan, Bryan Seybold, David A. Ross, Jia Deng, and Rahul Sukthankar.  Rethinking the faster r-cnn architecture for temporal action localization.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2018.
  * [14] Dengfeng Zhao, Yudong Zhong, Zhijun Fu, Junjian Hou, Mingyuan Zhao, et al.  A review for the driving behavior recognition methods based on vehicle multisensor information.  Journal of Advanced Transportation, 2022, 2022.
  * [15] Armstrong Aboah and Yaw Adu-Gyamfi.  Smartphone-based pavement roughness estimation using deep learning with entity embedding.  Advances in Data Science and Adaptive Analysis, 12(03n04):2050007, 2020.
  * [16] Armstrong Aboah.  A vision-based system for traffic anomaly detection using deep learning and decision trees.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4207–4212, 2021.
  * [17] Armstrong Aboah, Michael Boeding, and Yaw Adu-Gyamfi.  Mobile sensing for multipurpose applications in transportation.  arXiv preprint arXiv:2106.10733, 2021.
  * [18] Armstrong Aboah and Elizabeth Arthur.  Comparative analysis of machine learning models for predicting travel time.  arXiv preprint arXiv:2111.08226, 2021.
  * [19] Samira Ahangari, Mansoureh Jeihani, Anam Ardeshiri, Md Mahmudur Rahman, and Abdollah Dehzangi.  Enhancing the performance of a model to predict driving distraction with the random forest classifier.  Transportation research record, 2675(11):612–622, 2021.
  * [20] Ying Yao, Xiaohua Zhao, Hongji Du, Yunlong Zhang, and Jian Rong.  Classification of distracted driving based on visual features and behavior data using a random forest method.  Transportation research record, 2672(45):210–221, 2018.
  * [21] Armstrong Aboah, Lydia Johnson, and Setul Shah.  Identifying the factors that influence urban public transit demand.  arXiv preprint arXiv:2111.09126, 2021.
  * [22] Yiping Tang, Chuang Niu, Minghao Dong, Shenghan Ren, and Jimin Liang.  Afo-tad: Anchor-free one-stage detector for temporal action detection.  arXiv preprint arXiv:1910.08250, 2019.
  * [23] Xiaolong Liu, Qimeng Wang, Yao Hu, Xu Tang, Shiwei Zhang, Song Bai, and Xiang Bai.  End-to-end temporal action detection with transformer.  IEEE Transactions on Image Processing, 31:5427–5441, 2022.
  * [24] Md Atiqur Rahman and Robert Laganière.  Single-stage end-to-end temporal activity detection in untrimmed videos.  In 2020 17th Conference on Computer and Robot Vision (CRV), pages 206–213. IEEE, 2020.
  * [25] Rao Cheng, Xiaowei He, Zhonglong Zheng, and Zhentao Wang.  Multi-scale safety helmet detection based on sas-yolov3-tiny.  Applied Sciences, 11(8):3652, 2021.
  * [26] Xiaoqiu Chen, Miao Ma, Zhuoyu Tian, and Jie Ren.  Timeception single shot action detector: A single-stage method for temporal action detection.  In Image and Graphics: 11th International Conference, ICIG 2021, Haikou, China, August 6–8, 2021, Proceedings, Part I 11, pages 340–354. Springer, 2021.
  * [27] Chao Yan, Frans Coenen, and Bailing Zhang.  Driving posture recognition by convolutional neural networks.  IET Computer Vision, 10(2):103–114, 2016.
  * [28] Bhakti Baheti, Sanjay Talbar, and Suhas Gajre.  Towards computationally efficient and realtime distracted driver detection with mobilevgg network.  IEEE Transactions on Intelligent Vehicles, 5(4):565–574, 2020.
  * [29] Wei Li, Shimin Chen, Jianyang Gu, Ning Wang, Chen Chen, and Yandong Guo.  Mv-tal: Mulit-view temporal action localization in naturalistic driving.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3242–3248, 2022.
  * [30] Chen-Lin Zhang, Jianxin Wu, and Yin Li.  Actionformer: Localizing moments of actions with transformers.  In Computer Vision–ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23–27, 2022, Proceedings, Part IV, pages 492–510. Springer, 2022.
  * [31] Yu-Wei Chao, Sudheendra Vijayanarasimhan, Bryan Seybold, David A Ross, Jia Deng, and Rahul Sukthankar.  Rethinking the faster r-cnn architecture for temporal action localization.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1130–1139, 2018.
  * [32] Jiyang Gao, Zhenheng Yang, Kan Chen, Chen Sun, and Ram Nevatia.  Turn tap: Temporal unit regression network for temporal action proposals.  In Proceedings of the IEEE international conference on computer vision, pages 3628–3636, 2017.
  * [33] Chuong Nguyen, Ngoc Nguyen, Su Huynh, Vinh Nguyen, and Son Nguyen.  Learning generalized feature for temporal action detection: Application for natural driving action recognition challenge.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3249–3256, 2022.
  * [34] Junwei Liang, He Zhu, Enwei Zhang, and Jun Zhang.  Stargazer: A transformer-based driver action detection system for intelligent transportation.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3160–3167, 2022.
  * [35] Guanchen Ding, Wenwei Han, Chenglong Wang, Mingpeng Cui, Lin Zhou, Dianbo Pan, Jiayi Wang, Junxi Zhang, and Zhenzhong Chen.  A coarse-to-fine boundary localization method for naturalistic driving action recognition.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3234–3241, 2022.
  * [36] Chien-Yao Wang, Alexey Bochkovskiy, and Hong-Yuan Mark Liao.  Yolov7: Trainable bag-of-freebies sets new state-of-the-art for real-time object detectors.  arXiv preprint arXiv:2207.02696, 2022.
  * [37] Christoph Feichtenhofer.  X3d: Expanding architectures for efficient video recognition.  In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 203–213, 2020.
  * [38] Milind Naphade, Shuo Wang, David C. Anastasiu, Zheng Tang, Ming-Ching Chang, Yue Yao, Liang Zheng, Mohammed Shaiqur Rahman, Meenakshi S. Arya, Anuj Sharma, Qi Feng, Vitaly Ablavsky, Stan Sclaroff, Pranamesh Chakraborty, Sanjita Prajapati, Alice Li, Shangru Li, Krishna Kunadharaju, Shenxin Jiang, and Rama Chellappa.  The 7th AI City Challenge.  In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, June 2023.
  * [39] Mohammed Shaiqur Rahman, Jiyang Wang, Senem Velipasalar Gursoy, David Anastasiu, Shuo Wang, and Anuj Sharma.  Synthetic Distracted Driving (SynDD2) dataset for analyzing distracted behaviors and various gaze zones of a driver, 2022.  arXiv:2204.08096.