Real-time and efficient driver distraction detection is of great importance for road traffic safety and assisted driving. The design of a real-time lightweight model is crucial for in-vehicle edge devices that have limited computational resources. However, most existing approaches focus on lighter and more efficient architectures, ignoring the cost of losing tiny target detection performance that comes with lightweighting. In this paper, we present MTNet, a lightweight detector for driver distraction detection scenarios. MTNet consists of a multidimensional adaptive feature extraction block, a lightweight feature fusion block and utilizes the IoU-NWD weighted loss function, all while considering the accuracy gain of tiny target detection. In the feature extraction component, a lightweight backbone network is employed in conjunction with four attention mechanisms strategically integrated across the kernel space. This approach enhances the performance limits of the lightweight network. The lightweight feature fusion module is designed to reduce computational complexity and memory access. The interaction of channel information is improved through the use of lightweight arithmetic techniques. Additionally, CFSM module and EPIEM module are employed to minimize redundant feature map computations and strike a better balance between model weights and accuracy. Finally, the IoU-NWD weighted loss function is formulated to enable more effective detection of tiny targets. We assess the performance of the proposed method on the LDDB benchmark. The experimental results demonstrate that our proposed method outperforms multiple advanced detection models.
Citation: Zhiqin Zhu, Shaowen Wang, Shuangshuang Gu, Yuanyuan Li, Jiahan Li, Linhong Shuai, Guanqiu Qi. Driver distraction detection based on lightweight networks and tiny object detection[J]. Mathematical Biosciences and Engineering, 2023, 20(10): 18248-18266. doi: 10.3934/mbe.2023811
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Real-time and efficient driver distraction detection is of great importance for road traffic safety and assisted driving. The design of a real-time lightweight model is crucial for in-vehicle edge devices that have limited computational resources. However, most existing approaches focus on lighter and more efficient architectures, ignoring the cost of losing tiny target detection performance that comes with lightweighting. In this paper, we present MTNet, a lightweight detector for driver distraction detection scenarios. MTNet consists of a multidimensional adaptive feature extraction block, a lightweight feature fusion block and utilizes the IoU-NWD weighted loss function, all while considering the accuracy gain of tiny target detection. In the feature extraction component, a lightweight backbone network is employed in conjunction with four attention mechanisms strategically integrated across the kernel space. This approach enhances the performance limits of the lightweight network. The lightweight feature fusion module is designed to reduce computational complexity and memory access. The interaction of channel information is improved through the use of lightweight arithmetic techniques. Additionally, CFSM module and EPIEM module are employed to minimize redundant feature map computations and strike a better balance between model weights and accuracy. Finally, the IoU-NWD weighted loss function is formulated to enable more effective detection of tiny targets. We assess the performance of the proposed method on the LDDB benchmark. The experimental results demonstrate that our proposed method outperforms multiple advanced detection models.
Recently, fractional calculus methods became of great interest, because it is a powerful tool for calculating the derivation of multiples systems. These methods study real world phenomena in many areas of natural sciences including biomedical, radiography, biology, chemistry, and physics [1,2,3,4,5,6,7]. Abundant publications focus on the Caputo fractional derivative (CFD) and the Caputo-Hadamard derivative. Additionally, other generalization of the previous derivatives, such as Ψ-Caputo, study the existence of solutions to some FDEs (see [8,9,10,11,12,13,14]).
In general, an m-point fractional boundary problem involves a fractional differential equation with fractional boundary conditions that are specified at m different points on the boundary of a domain. The fractional derivative is defined using the Riemann-Liouville fractional derivative or the Caputo fractional derivative. Solving these types of problems can be challenging due to the non-local nature of fractional derivatives. However, there are various numerical and analytical methods available for solving such problems, including the spectral method, the finite difference method, the finite element method, and the homotopy analysis method. The applications of m-point fractional boundary problems can be found in various fields, including physics, engineering, finance, and biology. These problems are useful in modeling and analyzing phenomena that exhibit non-local behavior or involve memory effects (see [15,16,17,18]).
Pantograph equations are a set of differential equations that describe the motion of a pantograph, which is a mechanism used for copying and scaling drawings or diagrams. The equations are based on the assumption that the pantograph arms are rigid and do not deform during operation, we can simply say that see [19]. One important application of the pantograph equations is in the field of drafting and technical drawing. Before the advent of computer-aided design (CAD) software, pantographs were commonly used to produce scaled copies of drawings and diagrams. By adjusting the lengths of the arms and the position of the stylus, a pantograph can produce copies that are larger or smaller than the original [20], electrodynamics [21] and electrical pantograph of locomotive [22].
Many authors studied a huge number of positive solutions for nonlinear fractional BVP using fixed point theorems (FPTs) such as SFPT, Leggett-Williams and Guo-Krasnosel'skii (see [23,24]). Some studies addressed the sign-changing of solution of BVPs [25,26,27,28,29].
In this work, we use Schauder's fixed point theorem (SFPT) to solve the semipostone multipoint Ψ-Caputo fractional pantograph problem
Dν;ψrϰ(ς)+F(ς,ϰ(ς),ϰ(r+λς))=0, ς in (r,ℑ) | (1.1) |
ϰ(r)=ϑ1, ϰ(ℑ)=m−2∑i=1ζiϰ(ηi)+ϑ2, ϑi∈R, i∈{1,2}, | (1.2) |
where λ∈(0,ℑ−rℑ),Dν;ψr is Ψ-Caputo fractional derivative (Ψ-CFD) of order ν, 1<ν≤2, ζi∈R+(1≤i≤m−2) such that 0<Σm−2i=1ζi<1, ηi∈(r,ℑ), and F:[r,ℑ]×R×R→R.
The most important aspect of this research is to prove the existence of a positive solution of the above m-point FBVP. Note that in [30], the author considered a two-point BVP using Liouville-Caputo derivative.
The article is organized as follows. In the next section, we provide some basic definitions and arguments pertinent to fractional calculus (FC). Section 3 is devoted to proving the the main result and an illustrative example is given in Section 4.
In the sequel, Ψ denotes an increasing map Ψ:[r1,r2]→R via Ψ′(ς)≠0, ∀ ς, and [α] indicates the integer part of the real number α.
Definition 2.1. [4,5] Suppose the continuous function ϰ:(0,∞)→R. We define (RLFD) the Riemann-Liouville fractional derivative of order α>0,n=[α]+1 by
RLDα0+ϰ(ς)=1Γ(n−α)(ddς)n∫ς0(ς−τ)n−α−1ϰ(τ)dτ, |
where n−1<α<n.
Definition 2.2. [4,5] The Ψ-Riemann-Liouville fractional integral (Ψ-RLFI) of order α>0 of a continuous function ϰ:[r,ℑ]→R is defined by
Iα;Ψrϰ(ς)=∫ςr(Ψ(ς)−Ψ(τ))α−1Γ(α)Ψ′(τ)ϰ(τ)dτ. |
Definition 2.3. [4,5] The CFD of order α>0 of a function ϰ:[0,+∞)→R is defined by
Dαϰ(ς)=1Γ(n−α)∫ς0(ς−τ)n−α−1ϰ(n)(τ)dτ, α∈(n−1,n),n∈N. |
Definition 2.4. [4,5] We define the Ψ-CFD of order α>0 of a continuous function ϰ:[r,ℑ]→R by
Dα;Ψrϰ(ς)=∫ςr(Ψ(ς)−Ψ(τ))n−α−1Γ(n−α)Ψ′(τ)∂nΨϰ(τ)dτ, ς>r, α∈(n−1,n), |
where ∂nΨ=(1Ψ′(ς)ddς)n,n∈N.
Lemma 2.1. [4,5] Suppose q,ℓ>0, and ϰinC([r,ℑ],R). Then ∀ς∈[r,ℑ] and by assuming Fr(ς)=Ψ(ς)−Ψ(r), we have
1) Iq;ΨrIℓ;Ψrϰ(ς)=Iq+ℓ;Ψrϰ(ς),
2) Dq;ΨrIq;Ψrϰ(ς)=ϰ(ς),
3) Iq;Ψr(Fr(ς))ℓ−1=Γ(ℓ)Γ(ℓ+q)(Fr(ς))ℓ+q−1,
4) Dq;Ψr(Fr(ς))ℓ−1=Γ(ℓ)Γ(ℓ−q)(Fr(ς))ℓ−q−1,
5) Dq;Ψr(Fr(ς))k=0, k=0,…,n−1, n∈N, qin(n−1,n].
Lemma 2.2. [4,5] Let n−1<α1≤n,α2>0, r>0, ϰ∈L(r,ℑ), Dα1;Ψrϰ∈L(r,ℑ). Then the differential equation
Dα1;Ψrϰ=0 |
has the unique solution
ϰ(ς)=W0+W1(Ψ(ς)−Ψ(r))+W2(Ψ(ς)−Ψ(r))2+⋯+Wn−1(Ψ(ς)−Ψ(r))n−1, |
and
Iα1;ΨrDα1;Ψrϰ(ς)=ϰ(ς)+W0+W1(Ψ(ς)−Ψ(r))+W2(Ψ(ς)−Ψ(r))2+⋯+Wn−1(Ψ(ς)−Ψ(r))n−1, |
with Wℓ∈R, ℓ∈{0,1,…,n−1}.
Furthermore,
Dα1;ΨrIα1;Ψrϰ(ς)=ϰ(ς), |
and
Iα1;ΨrIα2;Ψrϰ(ς)=Iα2;ΨrIα1;Ψrϰ(ς)=Iα1+α2;Ψrϰ(ς). |
Here we will deal with the FDE solution of (1.1) and (1.2), by considering the solution of
−Dν;ψrϰ(ς)=h(ς), | (2.1) |
bounded by the condition (1.2). We set
Δ:=Ψ(ℑ)−Ψ(r)−Σm−2i=1ζi(Ψ(ηi)−Ψ(r)). |
Lemma 2.3. Let ν∈(1,2] and ς∈[r,ℑ]. Then, the FBVP (2.1) and (1.2) have a solution ϰ of the form
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)h(τ)Ψ′(τ)dτ, |
where
ϖ(ς,τ)=1Γ(ν){[(Ψ(ℑ)−Ψ(r))ν−1−Σm−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1]Ψ(ς)−Ψ(r)Δ−(Ψ(ς)−Ψ(τ))ν−1,τ≤ς,ηi−1<τ≤ηi,[(Ψ(ℑ)−Ψ(τ))ν−1−Σm−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1]Ψ(ℑ)−Ψ(r)Δ,ς≤τ,ηi−1<τ≤ηi, | (2.2) |
i=1,2,...,m−2.
Proof. According to the Lemma 2.2 the solution of Dν;ψrϰ(ς)=−h(ς) is given by
ϰ(ς)=−1Γ(ν)∫ςr(Ψ(ς)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+c0+c1(Ψ(ς)−Ψ(r)), | (2.3) |
where c0,c1∈R. Since ϰ(r)=ϑ1 and ϰ(ℑ)=∑m−2i=1ζiϰ(ηi)+ϑ2, we get c0=ϑ1 and
c1=1Δ(−1Γ(ν)m−2∑i=1ζi∫ηjr(Ψ(ηi)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+1Γ(ν)∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+ϑ1[m−2∑i=1ζi−1]+ϑ2). |
By substituting c0,c1 into Eq (2.3) we find,
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2−1Γ(ν)(∫ςr(Ψ(ς)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+(Ψ(ς)−Ψ(r))Δm−2∑i=1ζi∫ηjr(Ψ(ηi)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ−Ψ(ς)−Ψ(r)Δ∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2+∫ℑrϖ(ς,τ)h(τ)Ψ′(τ)dτ, |
where ϖ(ς,τ) is given by (2.2). Hence the required result.
Lemma 2.4. If 0<∑m−2i=1ζi<1, then
i) Δ>0,
ii) (Ψ(ℑ)−Ψ(τ))ν−1−∑m−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1>0.
Proof. i) Since ηi<ℑ, we have
ζi(Ψ(ηi)−Ψ(r))<ζi(Ψ(ℑ)−Ψ(r)), |
−m−2∑i=1ζi(Ψ(ηi)−Ψ(r))>−m−2∑i=1ζi(Ψ(ℑ)−Ψ(r)), |
Ψ(ℑ)−Ψ(r)−m−2∑i=1ζi(Ψ(ηi)−Ψ(r))>Ψ(ℑ)−Ψ(r)−m−2∑i=1ζi(Ψ(ℑ)−Ψ(r))=(Ψ(ℑ)−Ψ(r))[1−m−2∑i=1ζi]. |
If 1−Σm−2i=1ζi>0, then (Ψ(ℑ)−Ψ(r))−Σm−2i=1ζi(Ψ(ηi)−Ψ(r))>0. So we have Δ>0.
ii) Since 0<ν−1≤1, we have (Ψ(ηi)−Ψ(τ))ν−1<(Ψ(ℑ)−Ψ(τ))ν−1. Then we obtain
m−2∑j=iζj(Ψ(ηj)−Ψ(τ))ν−1<m−2∑j=iζj(Ψ(ℑ)−Ψ(τ))ν−1≤(Ψ(ℑ)−Ψ(τ))ν−1m−2∑i=1ζi<(Ψ(ℑ)−Ψ(τ))ν−1, |
and so
(Ψ(ℑ)−Ψ(τ))ν−1−m−2∑j=iζj(Ψ(ηj)−Ψ(τ))ν−1>0. |
Remark 2.1. Note that ∫ℑrϖ(ς,τ)Ψ′(τ)dτ is bounded ∀ς∈[r,ℑ]. Indeed
∫ℑr|ϖ(ς,τ)|Ψ′(τ)dτ≤1Γ(ν)∫ςr(Ψ(ς)−Ψ(τ))ν−1Ψ′(τ)dτ+Ψ(ς)−Ψ(r)Γ(ν)Δm−2∑i=1ζi∫ηir(Ψ(ηj)−Ψ(τ))ν−1Ψ′(τ)dτ+Ψ(ς)−Ψ(r)ΔΓ(ν)∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1Ψ′(τ)dτ=(Ψ(ς)−Ψ(r))νΓ(ν+1)+Ψ(ς)−Ψ(r)ΔΓ(ν+1)m−2∑i=1ζi(Ψ(ηi)−Ψ(r))ν+Ψ(ς)−Ψ(r)ΔΓ(ν+1)(Ψ(ℑ)−Ψ(r))ν≤(Ψ(ℑ)−Ψ(r))νΓ(ν+1)+Ψ(ℑ)−Ψ(r)ΔΓ(ν+1)m−2∑i=1ζi(Ψ(ηi)−Ψ(r))ν+(Ψ(ℑ)−Ψ(r))ν+1ΔΓ(ν+1)=M. | (2.4) |
Remark 2.2. Suppose Υ(ς)∈L1[r,ℑ], and w(ς) verify
{Dν;ψrw(ς)+Υ(ς)=0,w(r)=0, w(ℑ)=Σm−2i=1ζiw(ηi), | (2.5) |
then w(ς)=∫ℑrϖ(ς,τ)Υ(τ)Ψ′(τ)dτ.
Next we recall the Schauder fixed point theorem.
Theorem 2.1. [23] [SFPT] Consider the Banach space Ω. Assume ℵ bounded, convex, closed subset in Ω. If ϝ:ℵ→ℵ is compact, then it has a fixed point in ℵ.
We start this section by listing two conditions which will be used in the sequel.
● (Σ1) There exists a nonnegative function Υ∈L1[r,ℑ] such that ∫ℑrΥ(ς)dς>0 and F(ς,ϰ,v)≥−Υ(ς) for all (ς,ϰ,v)∈[r,ℑ]×R×R.
● (Σ2) G(ς,ϰ,v)≠0, for (ς,ϰ,v)∈[r,ℑ]×R×R.
Let ℵ=C([r,ℑ],R) the Banach space of CFs (continuous functions) with the following norm
‖ϰ‖=sup{|ϰ(ς)|:ς∈[r,ℑ]}. |
First of all, it seems that the FDE below is valid
Dν;ψrϰ(ς)+G(ς,ϰ∗(ς),ϰ∗(r+λς))=0, ς∈[r,ℑ]. | (3.1) |
Here the existence of solution satisfying the condition (1.2), such that G:[r,ℑ]×R×R→R
G(ς,z1,z2)={F(ς,z1,z2)+Υ(ς), z1,z2≥0,F(ς,0,0)+Υ(ς), z1≤0 or z2≤0, | (3.2) |
and ϰ∗(ς)=max{(ϰ−w)(ς),0}, hence the problem (2.5) has w as unique solution. The mapping Q:ℵ→ℵ accompanied with the (3.1) and (1.2) defined as
(Qϰ)(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)G(ς,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ, | (3.3) |
where the relation (2.2) define ϖ(ς,τ). The existence of solution of the problems (3.1) and (1.2) give the existence of a fixed point for Q.
Theorem 3.1. Suppose the conditions (Σ1) and (Σ2) hold. If there exists ρ>0 such that
[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≤ρ, |
where L≥max{|G(ς,ϰ,v)|:ς∈[r,ℑ], |ϰ|,|v|≤ρ} and M is defined in (2.4), then, the problems (3.1) and (3.2) have a solution ϰ(ς).
Proof. Since P:={ϰ∈ℵ:‖ϰ‖≤ρ} is a convex, closed and bounded subset of B described in the Eq (3.3), the SFPT is applicable to P. Define Q:P→ℵ by (3.3). Clearly Q is continuous mapping. We claim that range of Q is subset of P. Suppose ϰ∈P and let ϰ∗(ς)≤ϰ(ς)≤ρ, ∀ς∈[r,ℑ]. So
|Qϰ(ς)|=|[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)G(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ|≤[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≤ρ, |
for all ς∈[r,ℑ]. This indicates that ‖Qϰ‖≤ρ, which proves our claim. Thus, by using the Arzela-Ascoli theorem, Q:ℵ→ℵ is compact. As a result of SFPT, Q has a fixed point ϰ in P. Hence, the problems (3.1) and (1.2) has ϰ as solution.
Lemma 3.1. ϰ∗(ς) is a solution of the FBVP (1.1), (1.2) and ϰ(ς)>w(ς) for every ς∈[r,ℑ] iff the positive solution of FBVP (3.1) and (1.2) is ϰ=ϰ∗+w.
Proof. Let ϰ(ς) be a solution of FBVP (3.1) and (1.2). Then
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)G(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)(F(τ,ϰ∗(τ),ϰ∗(r+λτ))+p(τ))Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ+1Γ(ν)∫ℑrϖ(ς,τ)p(τ)Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)G(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ+w(ς). |
So,
ϰ(ς)−w(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ. |
Then we get the existence of the solution with the condition
ϰ∗(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ. |
For the converse, if ϰ∗ is a solution of the FBVP (1.1) and (1.2), we get
Dν;ψr(ϰ∗(ς)+w(ς))=Dν;ψrϰ∗(ς)+Dν;ψrw(ς)=−F(ς,ϰ∗(ς),ϰ∗(r+λς))−p(ς)=−[F(ς,ϰ∗(ς),ϰ∗(r+λς))+p(ς)]=−G(ς,ϰ∗(ς),ϰ∗(r+λς)), |
which leads to
Dν;ψrϰ(ς)=−G(ς,ϰ∗(ς),ϰ∗(r+λς)). |
We easily see that
ϰ∗(r)=ϰ(r)−w(r)=ϰ(r)−0=ϑ1, |
i.e., ϰ(r)=ϑ1 and
ϰ∗(ℑ)=m−2∑i=1ζiϰ∗(ηi)+ϑ2, |
ϰ(ℑ)−w(ℑ)=m−2∑i=1ζiϰ(ηi)−m−2∑i=1ζjw(ηi)+ϑ2=m−2∑i=1ζi(ϰ(ηi)−w(ηi))+ϑ2. |
So,
ϰ(ℑ)=m−2∑i=1ζiϰ(ηi)+ϑ2. |
Thus ϰ(ς) is solution of the problem FBVP (3.1) and (3.2).
We propose the given FBVP as follows
D75ϰ(ς)+F(ς,ϰ(ς),ϰ(1+0.5ς))=0, ς∈(1,e), | (4.1) |
ϰ(1)=1, ϰ(e)=17ϰ(52)+15ϰ(74)+19ϰ(115)−1. | (4.2) |
Let Ψ(ς)=logς, where F(ς,ϰ(ς),ϰ(1+12ς))=ς1+ςarctan(ϰ(ς)+ϰ(1+12ς)).
Taking Υ(ς)=ς we get ∫e1ςdς=e2−12>0, then the hypotheses (Σ1) and (Σ2) hold. Evaluate Δ≅0.366, M≅3.25 we also get |G(ς,ϰ,v)|<π+e=L such that |ϰ|≤ρ, ρ=17, we could just confirm that
[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≅16.35≤17. | (4.3) |
By applying the Theorem 3.1 there exit a solution ϰ(ς) of the problem (4.1) and (4.2).
In this paper, we have provided the proof of BVP solutions to a nonlinear Ψ-Caputo fractional pantograph problem or for a semi-positone multi-point of (1.1) and(1.2). What's new here is that even using the generalized Ψ-Caputo fractional derivative, we were able to explicitly prove that there is one solution to this problem, and that in our findings, we utilize the SFPT. The results obtained in our work are significantly generalized and the exclusive result concern the semi-positone multi-point Ψ-Caputo fractional differential pantograph problem (1.1) and (1.2).
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Small Groups (RGP.1/350/43).
The authors declare no conflict of interest.
[1] |
A. Krizhevsky, I. Sutskever, G. E. Hinton, Imagenet classification with deep convolutional neural networks, Adv. Neural Inform. Process. Syst., 6 (2017), 84–90. https://doi.org/10.1145/3065386 doi: 10.1145/3065386
![]() |
[2] | K. He, X. Zhang, S. Ren, J. Sun, Deep residual learning for image recognition, in Proceedings of the IEEE conference on computer vision and pattern recognition, (2016), 770–778. https://doi.org/10.1109/CVPR.2016.90 |
[3] | J. Redmon, S. Divvala, R. Girshick, A. Farhadi, You only look once: Unified, real-time object detection, preprint, arXiv: 1506.02640. |
[4] | J. Redmon, A. Farhadi, Yolov3: An incremental improvement, preprint, arXiv: 1804.02767. |
[5] | Ultralytics, Yolov5, 2021. Available from: https://github.com/ultralytics/yolov5. |
[6] |
A. Misra, S. Samuel, S. Cao, K. Shariatmadari, Detection of driver cognitive distraction using machine learning methods, IEEE Access, 11 (2023), 18000–18012. https://doi.org/10.1109/ACCESS.2023.3245122 doi: 10.1109/ACCESS.2023.3245122
![]() |
[7] |
S. M. Iranmanesh, H. N. Mahjoub, H. Kazemi, Y. P. Fallah, An adaptive forward collision warning framework design based on driver distraction, IEEE Trans. Intell. Trans. Syst., 19 (2018), 3925–3934. https://doi.org/10.1109/TITS.2018.2791437 doi: 10.1109/TITS.2018.2791437
![]() |
[8] | A. Jamsheed V., B. Janet, U. S. Reddy, Real time detection of driver distraction using cnn, in 2020 Third International Conference on Smart Systems and Inventive Technology (ICSSIT), (2020), 185–191. https://doi.org/10.1109/ICSSIT48917.2020.9214233 |
[9] |
C. Huang, X. Wang, J. Cao, S. Wang, Y. Zhang, Hcf: A hybrid cnn framework for behavior detection of distracted drivers, IEEE access, 8 (2020), 109335–109349. https://doi.org/10.1109/ACCESS.2020.3001159 doi: 10.1109/ACCESS.2020.3001159
![]() |
[10] | C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, Z. Wojna, Rethinking the inception architecture for computer vision, in Proceedings of the IEEE conference on computer vision and pattern recognition, (2016), 2818–2826. https://doi.org/10.1109/CVPR.2016.308 |
[11] | F. Chollet, Xception: Deep learning with depthwise separable convolutions, in Proceedings of the IEEE conference on computer vision and pattern recognition, (2017), 1251–1258. https://doi.org/10.1109/CVPR.2017.195 |
[12] |
F. Sajid, A. R. Javed, A. Basharat, N. Kryvinska, A. Afzal, M. Rizwan, An efficient deep learning framework for distracted driver detection, IEEE Access, 9 (2021), 169270–169280. https://doi.org/10.1109/ACCESS.2021.3138137 doi: 10.1109/ACCESS.2021.3138137
![]() |
[13] |
D. L. Nguyen, M. D. Putro, K. H. Jo, Driver behaviors recognizer based on light-weight convolutional neural network architecture and attention mechanism, IEEE Access, 10 (2022), 71019–71029. https://doi.org/10.1109/ACCESS.2022.3187185 doi: 10.1109/ACCESS.2022.3187185
![]() |
[14] | F. N. Iandola, S. Han, M. W. Moskewicz, K. Ashraf, W. J. Dally, K. Keutzer, Squeezenet: Alexnet-level accuracy with 50x fewer parameters and¡ 0.5 mb model size, preprint, arXiv: 1602.07360. |
[15] | A. G. Howard, M. Zhu, B. Chen, D. Kalenichenko, W. Wang, T. Weyand, et al., Mobilenets: Efficient convolutional neural networks for mobile vision applications, preprint, arXiv: 1704.04861. |
[16] | M. Sandler, A. Howard, M. Zhu, A. Zhmoginov, L. C. Chen, Mobilenetv2: Inverted residuals and linear bottlenecks, in Proceedings of the IEEE conference on computer vision and pattern recognition, (2018), 4510–4520. https://doi.org/10.1109/CVPR.2018.00474 |
[17] | A. Howard, M. Sandler, G. Chu, L. C. Chen, B. Chen, M. Tan, et al., Searching for mobilenetv3, in Proceedings of the IEEE/CVF international conference on computer vision, (2019), 1314–1324. https://doi.org/10.1109/ICCV.2019.00140 |
[18] | X. Zhang, X. Zhou, M. Lin, J. Sun, Shufflenet: An extremely efficient convolutional neural network for mobile devices, in Proceedings of the IEEE conference on computer vision and pattern recognition, (2018), 6848–6856. https://doi.org/10.1109/CVPR.2018.00716 |
[19] | N. Ma, X. Zhang, H. T. Zheng, J. Sun, Shufflenet v2: Practical guidelines for efficient cnn architecture design, in Proceedings of the European conference on computer vision (ECCV), (2018), 116–131. https://doi.org/10.1007/978-3-030-01264-9 |
[20] | C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, et al., Going deeper with convolutions, in Proceedings of the IEEE conference on computer vision and pattern recognition, (2015), 1–9. https://doi.org/10.1109/CVPR.2015.7298594 |
[21] | S. Ioffe, C. Szegedy, Batch normalization: Accelerating deep network training by reducing internal covariate shift, in International conference on machine learning, (2015), 448–456. |
[22] | C. Szegedy, S. Ioffe, V. Vanhoucke, A. Alemi, Inception-v4, inception-resnet and the impact of residual connections on learning, preprint, arXiv: 1602.07261. |
[23] | M. Tan, Q. Le, Efficientnet: Rethinking model scaling for convolutional neural networks, in International conference on machine learning, (2019), 6105–6114. |
[24] | M. Tan, Q. Le, Efficientnetv2: Smaller models and faster training, in International conference on machine learning, (2021), 10096–10106. |
[25] | K. Han, Y. Wang, Q. Tian, J. Guo, C. Xu, C. Xu, Ghostnet: More features from cheap operations, in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, (2020), 1580–1589. https://doi.org/10.1109/CVPR42600.2020.00165 |
[26] | J. He, S. Erfani, X. Ma, J. Bailey, Y. Chi, X. Hua, Alpha-iou: A family of power intersection over union losses for bounding box regression, preprint, arXiv: 2110.13675. |
[27] |
C. Deng, M. Wang, L. Liu, Y. Liu, Y. Jiang, Extended feature pyramid network for small object detection, IEEE Trans. Multimedia, 24 (2021), 1968–1979. https://doi.org/10.1109/TMM.2021.3074273 doi: 10.1109/TMM.2021.3074273
![]() |
[28] | X. Yang, J. Yang, J. Yan, Y. Zhang, T. Zhang, Z. Guo, et al., Scrdet: Towards more robust detection for small, cluttered and rotated objects, in Proceedings of the IEEE/CVF international conference on computer vision, (2019), 8232–8241. https://doi.org/10.1109/ICCV.2019.00832 |
[29] | H. Li, J. Li, H. Wei, Z. Liu, Z. Zhan, Q. Ren, Slim-neck by gsconv: A better design paradigm of detector architectures for autonomous vehicles, preprint, arXiv: 2206.02424. |
[30] | J. Chen, S. h. Kao, H. He, W. Zhuo, S. Wen, C. H. Lee, et al., Run, don't walk: Chasing higher flops for faster neural networks, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2023), 12021–12031. https://doi.org/10.1109/CVPR52729.2023.01157 |
[31] |
V. M. Panaretos, Y. Zemel, Statistical aspects of wasserstein distances, Annual Rev. Stat. Appl., 6 (2019), 405–431. https://doi.org/10.1146/annurev-statistics-030718-104938 doi: 10.1146/annurev-statistics-030718-104938
![]() |
[32] | J. Wang, C. Xu, W. Yang, L. Yu, A normalized gaussian wasserstein distance for tiny object detection, preprint, arXiv: 2110.13389. |
[33] | S. Farm, State farm distracted driver detection, Technical report, 2016. Available from: : https://www.kaggle.com//state-farm-distracted-driver-detection. |
[34] | Z. Zhu, D. Liang, S. Zhang, X. Huang, B. Li, S. Hu, Traffic-sign detection and classification in the wild, in Proceedings of the IEEE conference on computer vision and pattern recognition, (2016), 2110–2118. https://doi.org/10.1109/CVPR.2016.232 |
[35] | Y. Li, P. Xu, Z. Zhu, X. Huang, G. Qi, Real-time driver distraction detection using lightweight convolution neural network with cheap multi-scale features fusion block, in Proceedings of 2021 Chinese Intelligent Systems Conference: Volume II, Springer, (2022), 232–240. |
[36] | M. Tan, Q. V. Le, Mixconv: Mixed depthwise convolutional kernels, preprint, arXiv: 1907.09595. |
[37] | A. Howard, C. Zhu, J. Chen, X. Wang, W. Wu, Y. He, et al., Mobilenext: Rethinking bottleneck structure for efficient mobile network design, preprint, arXiv: 2003.10888. |
[38] | Z. Zhu, Z. Yao, G. Qi, N. Mazur, P. Yang, B. Cong, Associative learning mechanism for drug-target interaction prediction, CAAI Trans. Intell. Technol., (2023). https://doi.org/10.1049/cit2.12194 |
[39] |
Z. Zhu, X. He, G. Qi, Y. Li, B. Cong, Y. Liu, Brain tumor segmentation based on the fusion of deep semantics and edge information in multimodal mri, Inform. Fusion, 91 (2023), 376–387. https://doi.org/10.1016/j.inffus.2022.10.022 doi: 10.1016/j.inffus.2022.10.022
![]() |