Small object detection (SOD) is significant for many real-world applications, including criminal investigation, autonomous driving and remote sensing images. SOD has been one of the most challenging tasks in computer vision due to its low resolution and noise representation. With the development of deep learning, it has been introduced to boost the performance of SOD. In this paper, focusing on the difficulties of SOD, we analyze the deep learning-based SOD research papers from four perspectives, including boosting the resolution of input features, scale-aware training, incorporating contextual information and data augmentation. We also review the literature on crucial SOD tasks, including small face detection, small pedestrian detection and aerial image object detection. In addition, we conduct a thorough performance evaluation of generic SOD algorithms and methods for crucial SOD tasks on four well-known small object datasets. Our experimental results show that network configuring to boost the resolution of input features can enable significant performance gains on WIDER FACE and Tiny Person. Finally, several potential directions for future research in the area of SOD are provided.
Citation: Qihan Feng, Xinzheng Xu, Zhixiao Wang. Deep learning-based small object detection: A survey[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 6551-6590. doi: 10.3934/mbe.2023282
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Small object detection (SOD) is significant for many real-world applications, including criminal investigation, autonomous driving and remote sensing images. SOD has been one of the most challenging tasks in computer vision due to its low resolution and noise representation. With the development of deep learning, it has been introduced to boost the performance of SOD. In this paper, focusing on the difficulties of SOD, we analyze the deep learning-based SOD research papers from four perspectives, including boosting the resolution of input features, scale-aware training, incorporating contextual information and data augmentation. We also review the literature on crucial SOD tasks, including small face detection, small pedestrian detection and aerial image object detection. In addition, we conduct a thorough performance evaluation of generic SOD algorithms and methods for crucial SOD tasks on four well-known small object datasets. Our experimental results show that network configuring to boost the resolution of input features can enable significant performance gains on WIDER FACE and Tiny Person. Finally, several potential directions for future research in the area of SOD are provided.
We consider the following nonlinear parabolic system with damping and diffusion:
{ψt=−(σ−α)ψ−σθx+αψxx,θt=−(1−β)θ+νψx+2ψθx+βθxx,(x,t)∈QT, | (1.1) |
with the following initial and boundary conditions:
{(ψ,θ)(x,0)=(ψ0,θ0)(x),0≤x≤1,ψ(1,t)=ψ(0,t)=ξ(t),(θ,θx)(1,t)=(θ,θx)(0,t),0≤t≤T. | (1.2) |
Here σ,α,β and ν are constants with α>0, β>0, T>0, and QT=(0,1)×(0,T). The function ξ(t) is measurable in (0,T). System (1.1) was originally proposed by Hsieh in [1] as a substitution for the Rayleigh–Benard equation for the purpose of studying chaos, and we refer the reader to [1,2,3] for the physical background. It is worthy to point out that with a truncation similar to the mode truncation from the Rayleigh–Benard equations used by Lorenz in [4], system (1.1) also leads to the Lorenz system. In spite of it being a much simpler system, it is still as rich as the Lorenz system, and some different routes to chaos, including the break of the time-periodic solution and the break of the switching solution, have been discovered via numerical simulations [5].
Neglecting the damping and diffusion terms, system (1.1) is simplified as
(ψθ)t=(0−σν2ψ)(ψθ)x. |
It has two characteristic values: λ1=ψ+√ψ2−σν and λ2=ψ−√ψ2−σν. Obviously, the system is elliptic if ψ2−σν<0, and it is hyperbolic if ψ2−σν>0. In particular, it is always strictly hyperbolic if σν<0.
The mathematical theory of system (1.1) has been extensively investigated in a great number of papers; see [6,7,8,9,10,11,12,13,14,15,16] and the references therein. However, there is currently no global result with large initial data because it is very difficult to treat the nonlinear term of system (1.1) in analysis. Motivated by this fact, we will first study the well-posedness of global large solutions of the problem given by (1.1) and (1.2) for the case in which σν<0, and then we will discuss the limit problem as α→0+, as well as the problem on the estimation of the boundary layer thickness. For the case in which σν>0, we leave the investigation in the future.
The problem of a vanishing viscosity limit is an interesting and challenging problem in many settings, such as in the boundary layer theory (cf.[17]). Indeed, the presence of a boundary layer that accompanies the vanishing viscosity has been fundamental in fluid dynamics since the seminal work by Prandtl in 1904. In this direction, there have been extensive studies with a large number of references that we will not mention here. As important work on the mathematical basis for the laminar boundary layer theory, which is related to the problem considered in this paper, Frid and Shelukhin [18] studied the boundary layer effect of the compressible or incompressible Navier–Stokes equations with cylindrical symmetry and constant coefficients as the shear viscosity μ goes to zero; they proved the existence of a boundary layer of thickness O(μq) with any q∈(0,1/2). It should be pointed out that, for the incompressible case, the equations are reduced to
vt=μ(vx+vx)x,wt=μ(wxx+wxx),0<a<x<b,t>0, | (1.3) |
where μ is the shear viscosity coefficient and v and w represent the angular velocity and axial velocity, respectively. Recently, this result was investigated in more general settings; see for instance, [19,20,21] and the references therein. In the present paper, we will prove a similar result to that obtained in [18]. Note that every equation in (1.3) is linear. However, equation (1.1)2, with a nonconservative term 2ψθx, is nonlinear. It is the term that leads to new difficulties in analysis, causing all previous results on system (1.1) to be limited to small-sized initial data.
The boundary layer problem also arises in the theory of hyperbolic systems when parabolic equations with low viscosity are applied as perturbations (see [22,23,24,25,26,27,28]).
Formally, setting α=0, we obtain the following system:
{¯ψt=−σ¯ψ−σ¯θx,¯θt=−(1−β)¯θ+ν¯ψx+2¯ψ¯θx+β¯θxx,(x,t)∈QT, | (1.4) |
with the following initial and boundary conditions:
{(¯ψ,¯θ)(x,0)=(ψ0,θ0)(x),0≤x≤1,(¯θ,¯θx)(1,t)=(¯θ,¯θx)(0,t),0≤t≤T. | (1.5) |
Before stating our main result, we first list some notations.
Notations: For 1≤p,s≤∞, k∈N, and Ω=(0,1), we denote by Lp=Lp(Ω) the usual Lebesgue space on Ω with the norm ‖⋅‖Lp, and Hk=Wk,2(Ω) and H10=W1,20(Ω) as the usual Sobolev spaces on Ω with the norms ‖⋅‖Hk and ‖⋅‖H1, respectively. Ck(Ω) is the space consisting of all continuous derivatives up to order k on Ω, C(¯QT) is the set of all continuous functions on ¯QT, where QT=(0,1)×(0,T) with T>0, and Lp(0,T;B) is the space of all measurable functions from (0,T) to B with the norm ‖⋅‖Lp(0,T;B), where B=Ls or Hk. We also use the notations ‖(f1,f2,⋯)‖2B=‖f‖2B+‖g‖2B+⋯ for functions f1,f2,⋯ belonging to the function space B equipped with a norm ‖⋅‖B, and for L2(QT)=L2(0,T;L2).
The first result of this paper can be stated as follows.
Theorem 1.1. Let α,β,σ and ν be constants with α>0,β∈(0,1) and σν<0. Assume that (ψ0,θ0)∈H1([0,1]) and ξ∈C1([0,T]), and that they are compatible with the boundary conditions. Then, the following holds:
(i) For any given α>0, there exists a unique global solution (ψ,θ) for the problem given by (1.1) and (1.2) in the following sense:
(ψ,θ)∈C(¯QT)∩L∞(0,T;H1),(ψt,θt,ψxx,θxx)∈L2(QT). |
Moreover, for some constant C>0 independent of α∈(0,α0] with α0>0,
{‖(ψ,θ)‖L∞(QT)≤C,‖(√|ψx|,α1/4ψx,ω1/2ψx,θx)‖L∞(0,T;L2)≤C,‖(ψt,θt,α3/4ψxx,α1/4θxx,ω1/2θxx)‖L2(QT)≤C, | (1.6) |
where the function ω(x):[0,1]→[0,1] is defined by
ω(x)={x,0≤x<1/2,1−x,1/2≤x≤1. | (1.7) |
(ii) There exists a unique global solution (¯ψ,¯θ) for the problem given by (1.4) and (1.5) in the following sense:
{¯ψ∈L∞(QT)∩BV(QT),(¯ψx,ω¯ψ2x)∈L∞(0,T;L1),¯ψt∈L2(QT),¯θ∈C(¯QT)∩L∞(0,T;H1),¯θt∈L2(QT),¯θ(1,t)=¯θ(0,t),∀t∈[0,T], | (1.8) |
and
{∬Qt(¯ψζt−σ¯ψζ−ν¯θxζ)dxdτ=∫10ψ(x,t)ζ(x,t)dx−∫10ψ0(x)ζ(x,0)dx,∬QtL[¯ψ,¯θ;φ]dxdτ=∫10¯θ(x,t)φ(x,t)dx−∫10θ0(x)φ(x,0)dx,a.e.t∈(0,T),whereL[¯ψ,¯θ;φ]=¯θφt−(1−β)¯θφ−ν¯ψφx+2¯ψ¯θxφ−β¯θxφx | (1.9) |
for any functions ζ,φ∈C1(¯QT) with φ(1,t)=φ(0,t) for all t∈[0,T] such that, as α→0+,
{ψ→¯ψstronglyinLp(QT)foranyp≥2,ψ→¯ψweaklyinM(QT),ψt⇀¯ψtweaklyinL2(QT),αψ2x→0stronglyinL2(QT), | (1.10) |
where M(QT) is the set of the Radon measures on QT, and
{θ→¯θstronglyinC(¯QT),θx⇀θ0xweakly−∗inL∞(0,T;L2),θt⇀¯θtweaklyinL2(QT). | (1.11) |
(iii) For some constant C>0 independent of α∈(0,1),
‖(ψ−¯ψ,θ−¯θ)‖L∞(0,T;L2)+‖θx−¯θx‖L2(QT)≤Cα1/4. | (1.12) |
Remark 1.1. The L2 convergence rate for ψ is optimal whenever a boundary layer occurs as α→0+. The reason why this is optimal will be shown by an example in Section 3.3.
We next study the boundary layer effect of the problem given by (1.1) and (1.2). Before stating the main result, we first recall the concept of BL–thickness, defined as in [18].
Definition 1.2. A nonnegative function δ(α) is called the BL–thickness for the problem given by (1.1) and (1.2) with a vanishing α if δ(α)↓0 as α↓0, and if
{limα→0+‖(ψ−¯ψ,θ−¯θ)‖L∞(0,T;L∞(δ(α),1−δ(α))=0,liminfα→0+‖(ψ−¯ψ,θ−¯θ)‖L∞(0,T;L∞(0,1))>0, |
where (ψ,θ) and (¯ψ,¯θ) are the solutions for the problems given by (1.1), (1.2) and (1.4), (1.5), respectively.
The second result of this paper is stated as follows.
Theorem 1.3. Under the conditions of Theorem 1.1, any function δ(α) satisfying the condition that δ(α)↓0 and √α/δ(α)→0 as α↓0 is a BL–thickness whenever (¯ψ(1,t),¯ψ(0,t))≢(ξ(t),ξ(t)) in (0,T).
Remark 1.2. Here, the function θ satisfies the spatially periodic boundary condition that has been considered in some papers, e.g., [3,29,30]. One can see from the analysis that Theorems 1.1 and 1.3 are still valid when the boundary condition of θ is the homogeneous Neumann boundary condition or homogeneous Dirichlet boundary condition.
The proofs of the above theorems are based on the uniform estimates given by (1.6). First, based on a key observation of the structure of system (1.1), we find two identities (see Lemma 2.1). In this step, an idea is to transform (1.1)2 into an equation with a conservative term (see (2.4)). And then, from the two identities, we deduce some basic energy-type estimates (see Lemma 2.2). The condition σν<0 plays an important role here. With the uniform estimates in hand, we derive the required uniform bound of (α1/4‖ψx‖L∞(0,t;L2)+α3/4‖ψxx‖L2(QT)) for the study of the boundary layer effect via standard analysis (see [18,19]). See Lemma 2.3 with proof. The uniform bound of ‖(ψ,θ)‖L∞(QT) is derived by a more delicate analysis (see Lemma 2.4). Finally, the boundary estimate ‖ω1/2ψx‖L∞(0,T;L2)≤C is established (see Lemma 2.6), through which we complete the proof for the estimation of the boundary layer thickness.
Before ending the section, let us introduce some of the previous works on system (1.1). It should be noted that most of those works focus on the case when σν>0. In this direction, Tang and Zhao [6] considered the Cauchy problem for the following system:
{ψt=−(1−α)ψ−θx+αψxx,θt=−(1−α)θ+νψx+2ψθx+αθxx, | (1.13) |
with the following initial condition: (ψ,θ)(x,0)=(ψ0,θ0)(x)→(0,0)asx→±∞, where 0<α<1 and 0<ν<4α(1−α). They established the global existence, nonlinear stability and optimal decay rate of solutions with small-sized initial data. Their result was extended in [7,8] to the case of the initial data with different end states, i.e.,
(ψ,θ)(x,0)=(ψ0,θ0)(x)→(ψ±,θ±)asx→±∞, | (1.14) |
where ψ±,θ± are constant states with (ψ+−ψ−,θ+−θ−)≢(0,0). For the initial-boundary value problem on quadrants, Duan et al. [9] obtained the global existence and the Lp decay rates of solutions for the problem given by (1.13) and (1.14) with small-sized initial data. For the Dirichlet boundary value problem, Ruan and Zhu [10] proved the global existence of system (1.1) with small-sized initial data and justified the limit as β→0+ under the following condition: ν=μβ for some constant μ>0. In addition, they established the existence of a boundary layer of thickness O(βδ) with any 0<δ<1/2. Following [10], some similar results on the Dirichlet–Neumann boundary value problem were obtained in [11]. For the case when σν<0, however, there are few results on system (1.1). Chen and Zhu [12] studied the problem given by (1.13) and (1.14) with 0<α<1; they proved global existence with small-sized initial data and justified the limit as α→0+. In their argument, the condition σν<0 plays a key role. Ruan and Yin [13] discussed two cases of system (1.1): α=β and α≢β, and they obtained some further results that include the C∞ convergence rate as β→0+.
We also mention that one can obtain a slightly modified system by replacing the nonlinear term 2ψθx in (1.1) with (ψθ)x. Jian and Chen [31] first obtained the global existence results for the Cauchy problem. Hsiao and Jian [29] proved the unique solvability of global smooth solutions for the spatially periodic Cauchy problem by applying the Leary–Schauder fixed-point theorem. Wang [32] discussed long time asymptotic behavior of solutions for the Cauchy problem. Some other results on this system is available in [33,34,35] and the references therein.
The rest of the paper is organized as follows. In Section 2, we will derive the uniform a priori estimates given by (1.6). The proofs of Theorem 1.1 and Theorem 1.3 will be given in Section 3 and Section 4, respectively.
In the section, we will derive the uniform a priori estimates by (1.6), and we suppose that the solution (ψ,θ) is smooth enough on ¯QT. From now on, we denote by C a positive generic constant that is independent of α∈(0,α0] for α0>0.
First of all, we observe that (˜ψ,˜θ):=(2ψ,2θ/ν) solves the following system:
{˜ψt=−(σ−α)˜ψ−σν˜θx+α˜ψxx,˜θt=−(1−β)˜θ+˜ψx+˜ψ˜θx+β˜θxx, | (2.1) |
with the following initial and boundary conditions:
{(˜ψ,˜θ)(x,0)=(2ψ0,2θ0/ν)(x),0≤x≤1,˜ψ(1,t)=˜ψ(0,t)=2ξ(t),(˜θ,˜θx)(1,t)=(˜θ,˜θx)(0,t),0≤t≤T. |
From the system, we obtain the following crucial identities for our analysis.
Lemma 2.1. (˜ψ,˜θ) satisfies the following for all t∈[0,T]:
ddt∫10(12˜ψ2+σν˜θ)dx+∫10[α˜ψ2x+σν(1−β)˜θ]dx=(α−σ)∫10˜ψ2dx+α[(˜ψ˜ψx)(1,t)−(˜ψ˜ψx)(0,t)] | (2.2) |
and
ddt∫10e˜θdx+β∫10e˜θ˜θ2xdx+(1−β)∫10˜θe˜θdx=0. | (2.3) |
Proof. Multiplying (2.1)1 and (2.1)2 by ˜ψ and σν, respectively, adding the equations and then integrating by parts over (0,1) with respect to x, we obtain (2.2) immediately.
We now multiply (2.1)2 by e˜θ to obtain
(e˜θ)t=−(1−β)˜θe˜θ+(˜ψe˜θ)x+βe˜θ˜θxx. | (2.4) |
Integrating it over (0,1) with respect to x, we get (2.3) and complete the proof.
Lemma 2.2. Let the assumptions of Theorem 1.1 hold. Then, we have the following for any t∈[0,T]:
∫10ψ2dx+α∬Qtψ2xdxdτ≤C | (2.5) |
and
|∫10θdx|≤C. | (2.6) |
Proof. Integrating (2.2) and (2.3) over (0,t), respectively, we obtain
12∫10˜ψ2dx+∬Qt[α˜ψ2x+(σ−α)˜ψ2]dxdτ=∫10(2ψ20+2σθ0)dx+α∫t0ξ(t)˜ψx|x=1x=0dτ−σν(∫10˜θdx+(1−β)∬Qt˜θdxdτ), | (2.7) |
and
∫10e˜θdx+β∬Qte˜θ˜θ2xdxdτ=∫10e2θ0νdx−(1−β)∬Qt˜θe˜θdxdτ. | (2.8) |
From the inequalities ex≥1+x and xex≥−1 for all x∈R, and given β∈(0,1), we derive from (2.8) that
∫10˜θdx≤C, |
so that
∫10˜θdx+(1−β)∬Qt˜θdxdτ≤C. | (2.9) |
We now treat the second term on the right-hand side of (2.7). Integrating (2.1)1 over (0,1) with respect to x, multiplying it by ξ(t) and then integrating over (0,t), we obtain
|α∫t0ξ(τ)˜ψx|x=1x=0dτ|=|ξ(t)∫10˜ψdx−2ξ(0)∫10ψ0dx−∫t0ξt(τ)∫10˜ψdxdτ+(σ−α)∫t0ξ(τ)∫10˜ψdxdτ|≤C+14∫10˜ψ2dx+C∬Qt˜ψ2dxdτ. | (2.10) |
Combining (2.9) and (2.10) with (2.7), and noticing that σν<0, we obtain (2.5) by using the Gronwall inequality.
Combining (2.5) and (2.10) with (2.7) yields
|∫10˜θdx+(1−β)∬Qt˜θdxdτ|≤C. | (2.11) |
Let φ=∬Qt˜θdxdτ. Then, (2.11) is equivalent to
|(e(1−β)tφ)t|≤Ce(1−β)t, |
which implies that |φ|≤C. This, together with (2.11), gives (2.6) and completes the proof.
Lemma 2.3. Let the assumptions of Theorem 1.1 hold. Then, for any t∈[0,T],
∫10θ2dx+∬Qt(θ2x+ψ2t)dxdτ≤C | (2.12) |
and
α∫10ψ2xdx+α2∬Qtψ2xxdxdτ≤C. | (2.13) |
Proof. Multiplying (1.1)2 by θ, integrating over (0,1), and using (2.5) and Young's inequality, we have
12ddt∫10θ2dx+β∫10θ2xdx=(β−1)∫10θ2dx+∫10(2ψθ−νψ)θxdx≤C+β4∫10θ2xdx+C‖θ‖2L∞. | (2.14) |
From the embedding theorem, W1,1↪L∞, and Young's inequality, we have
‖θ‖2L∞≤C∫10θ2dx+C∫10|θθx|dx≤Cε∫10θ2dx+ε∫10θ2xdx,∀ε∈(0,1). | (2.15) |
Plugging it into (2.14), taking ε sufficiently small and using the Gronwall inequality, we obtain
∫10θ2dx+∬Qtθ2xdxdτ≤C. | (2.16) |
Rewrite (1.1)1 as
−ψt+αψxx=(σ−α)ψ+σθx. |
Taking the square on the two sides, integrating over (0,1) and using (2.5), we obtain
αddt∫10ψ2xdx+∫10(ψ2t+α2ψ2xx)dx=∫10[(σ−α)ψ+σθx]2dx+αξtψx|x=1x=0≤C+C∫10θ2xdx+Cα‖ψx‖L∞. | (2.17) |
From the embedding theorem, W1,1↪L∞, and the Hölder inequality, we have
‖ψx‖L∞≤C∫10|ψx|dx+C∫10|ψxx|dx; |
hence, by Young's inequality,
α‖ψx‖L∞≤Cε+Cα∫10ψ2xdx+εα2∫10ψ2xxdx. |
Plugging it into (2.17), taking ε sufficiently small and using (2.16), we obtain the following by applying the Gronwall inequality:
α∫10ψ2xdx+∬Qt(ψ2t+α2ψ2xx)dxdτ≤C. |
This completes the proof of the lemma.
For η>0, let
Iη(s)=∫s0hη(τ)dτ,wherehη(s)={1,s≥η,sη,|s|<η,−1,s<−η. |
Obviously, hη∈C(R),Iη∈C1(R) and
h′η(s)≥0a.e.s∈R;|hη(s)|≤1,limη→0+Iη(s)=|s|,∀s∈R. | (2.18) |
With the help of (2.18), we obtain the following estimates, which are crucial for the study of the boundary layer effect.
Lemma 2.4. Let the assumptions of Theorem 1.1 hold. Then, for any t∈[0,T],
∫10(|ψx|+θ2x)dx+∬Qtθ2tdxdτ≤C. | (2.19) |
In particular, ‖(ψ,θ)‖L∞(QT)≤C.
Proof. Let W=ψx. Differentiating (1.1)1 with respect to x, we obtain
Wt=−(σ−α)W−σθxx+αWxx. |
Multiplying it by hη(W) and integrating over Qt, we get
∫10Iη(W)dx=∫10Iη(ψ0x)dx−∬Qt[(σ−α)W+σθxx]hη(W)dxdτ−α∬Qth′η(W)W2xdxdτ+α∫t0(Wxhη(W))|x=1x=0dτ=:∫10Iη(ψ0x)dx+3∑i=1Ii. | (2.20) |
By applying |hη(s)|≤1, we have
I1≤C+C∬Qt(|W|+|θxx|)dxdτ. | (2.21) |
By using h′η(s)≥0 for s∈R, we have
I2≤0. | (2.22) |
We now estimate I3. Since θ(1,t)=θ(0,t), it follows from the mean value theorem that, for any t∈[0,T], there exists some xt∈(0,1) such that θx(xt,t)=0; hence,
|θx(y,t)|=|∫yxtθxx(x,t)dx|≤∫10|θxx|dx,∀y∈[0,1]. |
It follows from (1.1)1 that
α∫t0|Wx(a,t)|dτ≤C+C∫t0|θx(a,τ)|dτ≤C+C∬Qt|θxx|dxdτ,wherea=0,1. |
Consequently, thanks to |hη(s)|≤1,
I3≤Cα∫t0[|Wx(1,τ)|+|Wx(0,τ)|]dτ≤C+C∬Qt|θxx|dxdτ. | (2.23) |
Plugging (2.21)–(2.23) into (2.20), and letting η→0+, we deduce the following by noticing that limη→0+Iη(s)=|s|:
∫10|ψx|dx≤C+C∬Qt|ψx|dxdτ+C∬Qt|θxx|dxdτ. | (2.24) |
Plugging (1.1)2 into the final term of the right-hand side on (2.24), and by using (2.5) and (2.16), we obtain the following by using the Gronwall inequality:
∫10|ψx|dx≤C+C∬Qt|θt|dxdτ. | (2.25) |
By using the embedding theorem, (2.5) and (2.25), we have
‖ψ‖2L∞≤(∫10|ψ|dx+∫10|ψx|dx)2≤C[1+P(t)], | (2.26) |
where P(t)=∬Qtθ2tdxdτ.
Multiplying (1.1)2 by θt and integrating over Qt, we have
P(t)+∫10(β2θ2x+1−β2θ2)dx≤C+∬Qt(νψx+2ψθx)θtdxdτ. | (2.27) |
By applying (2.5), (2.12), (2.16), (2.26) and Young's inequality, we deduce the following:
ν∬Qtψxθtdxdτ=−ν∬Qtψ(θx)tdxdτ=−ν∫10ψθxdx+ν∫10ψ0θ0xdx+ν∬Qtψtθxdxdτ≤C+β4∫10θ2xdx | (2.28) |
and
2∬Qtψθxθtdxdτ≤12P(t)+C∫t0(∫10θ2x(x,τ)dx)‖ψ‖2L∞dτ≤C+12P(t)+C∫t0(∫10θ2x(x,τ)dx)P(τ)dτ. | (2.29) |
Substituting (2.28) and (2.29) into (2.27), and noticing that ∬QTθ2xdxdt≤C, we obtain the following by using the Gronwall inequality:
∫10θ2xdx+∬Qtθ2tdxdτ≤C. |
This, together with (2.25), gives
∫10|ψx|dx≤C. |
And, the proof of the lemma is completed.
Lemma 2.5. Let the assumptions of Theorem 1.1 hold. Then, for any t∈[0,T],
α1/2∫10ψ2xdx+α3/2∬Qtψ2xxdxdτ≤C. | (2.30) |
Proof. Multiplying (1.1)2 by θxx, integrating over (0,1) and using Lemmas 2.2–2.4, we have
12ddt∫10θ2xdx+β∫10θ2xxdx=∫10[(1−β)θ−νψx−2ψθx]θxxdx≤C+C∫10ψ2xdx+β4∫10θ2xxdx+C‖θx‖2L∞≤C+C∫10ψ2xdx+β2∫10θ2xxdx, | (2.31) |
where we use the estimate with ε sufficiently small based on a proof similar to (2.15):
‖θx‖2L∞≤Cε∫10θ2xdx+ε∫10θ2xxdx,∀ε∈(0,1). |
It follows from (2.31) that
∫10θ2xdx+∬Qtθ2xxdxdτ≤C+C∬Qtψ2xdxdτ. | (2.32) |
Similarly, multiplying (1.1)1 by αψxx, and integrating over (0,1), we have
α2ddt∫10ψ2xdx+α2∫10ψ2xxdx+α(σ−α)∫10ψ2xdx=−σα∫10ψxθxxdx+[(σ−α)ξ+ξt][α˜ψx]|x=1x=0+σα(θxψx)|x=1x=0≤Cα∫10ψ2xdx+Cα∫10θ2xxdx+Cα[1+|θx(1,t)|+|θx(0,t)|]‖ψx‖L∞. | (2.33) |
Then, by integrating over (0,t) and using (2.32) and the Hölder inequality, we obtain
α∫10ψ2xdx+α2∬Qtψ2xxdxdτ≤Cα+Cα∬Qtψ2xdxdτ+CαA(t)(∫t0‖ψx‖2L∞dτ)1/2, | (2.34) |
where
A(t):=(∫t0[1+|θx(1,τ)|2+|θx(0,τ)|2]dτ)1/2. |
To estimate A(t), we first integrate (1.1)2 over (x,1) for x∈[0,1], and then we integrate the resulting equation over (0,1) to obtain
θx(1,t)=1β∫10∫1y[θt+(1−β)θ−2ψθx]dxdy−νξ(t)+ν∫10ψ(x,t)dx. |
By applying Lemmas 2.3 and 2.4, we obtain
∫T0θ2x(1,t)dt≤C. | (2.35) |
Similarly, we have
∫T0θ2x(0,t)dt≤C. | (2.36) |
Therefore,
A(t)≤C. | (2.37) |
From the embedding theorem, W1,1↪L∞, and the Hölder inequality, we have
‖ψx‖2L∞≤C∫10ψ2xdx+C∫10|ψxψxx|dx≤C∫10ψ2xdx+C(∫10ψ2xdx)1/2(∫10ψ2xxdx)1/2; |
therefore, by the Hölder inequality and Young's inequality, we obtain the following for any ε∈(0,1):
α(∫t0‖ψx‖2L∞dτ)1/2≤C√αε+Cα∬Qtψ2xdxdτ+εα2∬Qtψ2xxdxdτ. | (2.38) |
Combining (2.37) and (2.38) with (2.34), and taking ε sufficiently small, we obtain (2.30) by using the Gronwall inequality. The proof of the lemma is completed.
As a consequence of Lemma 2.5 and (2.32), we have that α1/2∬QTθ2xxdxdt≤C.
Lemma 2.6. Let the assumptions of Theorem 1.1 hold. Then, for any t∈[0,T],
∫10ψ2xω(x)dx+∬Qt(θ2xx+αψ2xx)ω(x)dxdτ≤C, |
where ω is the same as that in (1.7).
Proof. Multiplying (1.1)1 by ψxxω(x) and integrating over Qt, we have
12∫10ψ2xω(x)dx+α∬Qtψ2xxω(x)dxdτ≤C+C∬Qtψ2xω(x)dxdτ+3∑i=1Ii, | (2.39) |
where
{I1=(α−σ)∬Qtψψxω′(x)dxdτ,I2=−σ∬Qtψxθxxω(x)dxdτ,I3=−σ∬Qtψxθxω′(x)dxdτ. |
By applying Lemmas 2.3 and 2.4, we have
I1≤C∬Qt|ψ||ψx|xdτ≤C, | (2.40) |
and, by substituting (1.1)2 into I2, we get
I2=−σβ∬Qtψxω(x)[θt+(1−β)θ−νψx−2ψθx]dxdτ≤C+C∬Qtψ2xω(x)dxdτ. | (2.41) |
To estimate I3, we first multiply (1.1)2 by θxω′(x), and then we integrate over Qt to obtain
ν∬Qtψxθxω′(x)dxdτ=E−β2∫t0[2θ2x(1/2,τ)−θ2x(1,τ)−θ2x(0,τ)]dτ, |
where
E=∬Qtθxω′(x)[θt+(1−β)θ−2ψθx]dxdτ. |
Note that |E|≤C by Lemmas 2.3 and 2.4. Then, given σν<0, we have
I3=−σνE+σνβ2∫t0[2θ2x(1/2,t)−θ2x(1,t)−θ2x(0,t)]dt≤C+C∫t0[θ2x(1,τ)+θ2x(0,τ)]dτ. | (2.42) |
Combining (2.35) and (2.36) with (2.42), we obtain
I3≤C. | (2.43) |
Substituting (2.40), (2.41) and (2.43) into (2.39), we complete the proof of the lemma by applying the Gronwall inequality.
In summary, all uniform a priori estimates given by (1.6) have been obtained.
For any fixed α>0, the global existence of strong solutions for the problem given by (1.1) and (1.2) can be shown by a routine argument (see [29,36]). First, by a smooth approximation of the initial data satisfying the conditions of Theorem 1.1, we obtain a sequence of global smooth approximate solutions by combining the a priori estimates given by (1.6) with the Leray–Schauder fixed-point theorem (see [36, Theorem 3.1]). See [29] for details. And then, a global strong solution satisfying (1.6) is constructed by means of a standard compactness argument.
The uniqueness of solutions can be proved as follows. Let (ψ2,θ2) and (ψ1,θ1) be two strong solutions, and denote (Ψ,Θ)=(ψ2−ψ1,θ2−θ1). Then, (Ψ,Θ) satisfies
{Ψt=−(σ−α)Ψ−σΘx+αΨxx,Θt=−(1−β)Θ+νΨx+2ψ2Θx+2θ1xΨ+βΘxx, | (3.1) |
with the following initial and boundary conditions:
{(Ψ,Θ)|t=0=(0,0),0≤x≤1,(Ψ,Θ,Θx)|x=0,1=(0,0,0),0≤t≤T. |
Multiplying (3.1)1 and (3.1)2 by Ψ and Θ, respectively, integrating them over [0,1] and using Young's inequality, we obtain
12ddt∫10Ψ2dx+α∫10Ψ2xdx≤β4∫10Θ2xdx+C∫10Ψ2dx | (3.2) |
and
12ddt∫10Θ2dx+β∫10Θ2xdx=∫10[(β−1)Θ2−νΘxΨ+2ψ2ΘxΘ+2θ1xΨΘ]dx≤β4∫10Θ2xdx+C∫10(Ψ2+Θ2)dx, | (3.3) |
where we use ‖ψ2‖L∞(QT)+‖θ1x‖L∞(0,T;L2)≤C and
∬Qtθ21xΘ2dxdτ≤C∫t0‖Θ‖2L∞dτ≤∬QtΘ2dxdτ+β16∬QtΘ2xdxdτ. |
First, by adding (3.2) and (3.3), and then using the Gronwall inequality, we obtain that ∫10(Ψ2+Θ2)dx=0on[0,T] so that (Ψ,Θ)=0 on ¯QT. This completes the proof of Theorem 1.1(ⅰ).
The aim of this part is to prove (1.10) and (1.11). Given the uniform estimates given by (1.6) and Aubin–Lions lemma (see [37]), there exists a sequence {αn}∞n=1, tending to zero, and a pair of functions (¯ψ,¯θ) satisfying (1.8) such that, as αn→0+, the unique global solution of the problem given by (1.1) and (1.2) with α=αn, still denoted by (ψ,θ), converges to (¯ψ,¯θ) in the following sense:
{ψ→¯ψstronglyinLp(QT)foranyp≥2,ψ→¯ψweaklyinM(QT),θ→¯θstronglyinC(¯QT),θx⇀θ0xweakly−∗inL∞(0,T;L2),(ψt,θt)⇀(¯ψt,¯θt)weaklyinL2(QT), | (3.4) |
where M(QT) is the set of Radon measures on QT. From (3.4), one can directly check that (¯ψ,¯θ) is a global solution for the problem given by (1.4) and (1.5) in the sense of (1.8) and (1.9).
We now return to the proof of the uniqueness. Let (¯ψ2,¯θ2) and (¯ψ1,¯θ1) be two solutions, and denote (M,N)=(¯ψ2−¯ψ1,¯θ2−¯θ1). It follows from (1.9) that
12∫10M2dx+σ∬QtM2dxdτ=−ν∬QtMNxdxdτ, | (3.5) |
and
12∫10N2dx+∬Qt[βN2x+(1−β)N2]dxdτ=∬Qt(−νMNx+2ψ2NxN+2MNθ1x)dxdτ. | (3.6) |
By using Young's inequality, we immediately obtain
12∫10M2dx≤C∬QtM2dxdτ+β4∬QtN2xdxdτ,12∫10N2dx+β∬QtN2xdxdτ≤C∬Qt(M2+N2)dxdτ+β4∬QtN2xdxdτ, |
where we use ψ2∈L∞(QT),θ1x∈L∞(0,T;L2), and the estimate with ε sufficiently small:
∬QtN2θ21xdxdτ≤∫T0‖N‖2L∞∫10θ21xdxdτ≤Cε∬QtN2dxdτ+ε∬QtN2xdxdτ,∀ε∈(0,1). |
Then, the Gronwall inequality gives (M, N) = (0, 0) a.e. in Q_T . Hence, the uniqueness follows.
Thanks to the uniqueness, the convergence results of (3.4) hold for \alpha\rightarrow 0^+ .
Finally, by using Lemma 2.3, we immediately obtain
\begin{equation*} \begin{aligned} \alpha^2\iint_{Q_t}\psi_x^4dxd\tau\leq C\sqrt{\alpha}. \end{aligned} \end{equation*} |
So, \alpha\psi_x^2\rightarrow 0 strongly in L^2(Q_T) as \alpha\rightarrow 0^+ . Thus, the proof of Theorem 1.1(ⅱ) is completed.
In addition, the following local convergence results follows from (1.6) and (3.4):
\begin{equation*} \label{convergence4} \left\{\begin{split} &\psi\rightarrow \overline\psi\, \, \, strongly \, \, in\, \, C([\epsilon, 1-\epsilon]\times[0, T]), \\ &\psi_{x}\rightharpoonup\psi_{x}^0\, \, \, weakly-*\, \, in\, \, L^{\infty}(0, T;L^2(\epsilon, 1-\epsilon)), \\ & \theta_{xx} \rightharpoonup \overline\theta_{xx} \, \, \, weakly\, \, in\, \, L^{2}((\epsilon, 1-\epsilon)\times(0, T)) \end{split}\right. \end{equation*} |
for any \epsilon\in (0, 1/4) . Consequently, the equations comprising (1.1) hold a.e in Q_T .
This purpose of this part is to prove (1.12). Let (\psi_i, \theta_i) be the solution of the problem given by (1.1) and (1.2) with \alpha = \alpha_i \in (0, 1) for i = 1, 2 . Denote U = \psi_2-\psi_1 and V = \theta_2-\theta_1 . Then it satisfies
\begin{equation} \left\{\begin{aligned} &U_t = - \sigma U+\alpha_2\psi_2-\alpha_1\psi_1-\sigma V_x+\alpha_2\psi_{2xx}-\alpha_1\psi_{1xx}, \\ &V_t = -(1-\beta)V+\nu U_x+2\psi_2 V_x+2U \theta_{1x}+\beta V_{xx}. \end{aligned}\right. \end{equation} | (3.7) |
Multiplying (3.7) _1 and (3.7) _2 by U and V , respectively, integrating them over Q_t and using Lemmas 2.3–2.5, we have
\begin{equation*} \begin{aligned} \frac12\int_0^1 U^2dx \leq C(\sqrt{\alpha_2}+\sqrt{\alpha_1})+ C \iint_{Q_t}U^2dxd\tau+\frac{\beta}{4}\iint_{Q_t}V_x^2dxd\tau \end{aligned} \end{equation*} |
and
\begin{equation*} \begin{aligned} \frac12\int_0^1 V^2dx +\beta\iint_{Q_t}V_x^2dxd\tau \leq& C\iint_{Q_t}(U^2+V^2)dxd\tau +\frac{\beta}{4}\iint_{Q_t}V_x^2dxd\tau, \end{aligned} \end{equation*} |
where we use \|\psi_2\|_{L^\infty(Q_T)}+\|\theta_{1x}\|_{L^\infty(0, T; L^2)}\leq C and the estimate with \varepsilon sufficiently small:
\begin{equation*} \begin{aligned} \iint_{Q_t}V^2( \theta_{1x})^2dxd\tau\leq& C\int_0^t\|V^2\|_{L^\infty}d\tau\\ \leq& \frac{C}{\varepsilon}\iint_{Q_t}V^2dxd\tau+\varepsilon\iint_{Q_t}V_x^2dxd\tau, \quad\forall \varepsilon\in(0, 1). \end{aligned} \end{equation*} |
Then, the Gronwall inequality gives
\begin{equation*} \begin{aligned} \int_0^1 (U^2+V^2)dx+\iint_{Q_t}V_x^2dxd\tau \leq C(\sqrt{\alpha_2}+\sqrt{\alpha_1}). \end{aligned} \end{equation*} |
We now fix \alpha = \alpha_2 , and then we let \alpha_1\rightarrow 0^+ to obtain the desired result by using (1.10) and (1.11).
In summary, we have completed the proof of Theorem 1.1.
We next show the optimality of the L^2 convergence rate O(\alpha^{1/4}) for \psi , as stated in Remark 1.1. For this purpose, we consider the following example:
\begin{equation*} \label{example} \begin{aligned} (\psi_0, \theta_0)\equiv (0, 1)\; \; \mbox{on}\; \; [0, 1]\; \; and\; \; \xi(t)\equiv t^3\; \; \mbox{in}\; \; [0, T]. \end{aligned} \end{equation*} |
It is easy to check that (\overline\psi, \overline\theta) = (0, e^{(\beta-1)t}) is the unique solution of the problem given by (1.4) and (1.5). According to Theorem 1.3 of Section 4, a boundary layer exists as \alpha\rightarrow 0^+ . To achieve our aim, it suffices to prove the following:
\begin{equation} \begin{split} \mathop{\lim\inf}_{\alpha\rightarrow 0^+}\left(\alpha^{-1/4}\|\psi\|_{L^\infty(0, T;L^2)}\right) > 0. \end{split} \end{equation} | (3.8) |
Suppose, on the contrary, that there exists a subsequence \{\alpha_n\} satisfying \alpha_n\rightarrow 0^+ such that the solution of the problem given by (1.1) and (1.2) with \alpha = \alpha_n , still denoted by (\psi, \theta) , satisfies
\begin{equation} \begin{split} \sup\limits_{0 < t < T}\int_0^1 \psi^2 dx = o(1)\alpha^{1/2}_n, \end{split} \end{equation} | (3.9) |
where o(1) indicates a quantity that uniformly approaches to zero as \alpha_n\rightarrow 0^+ . Then, by using the embedding theorem, we obtain
\begin{equation*} \begin{split} \|\psi\|_{L^\infty}^2&\leq C\int_0^1 \psi^2 dx +C\int_0^1 |\psi||\psi_x| dx\\ &\leq C\alpha_n^{1/2}+C\left(\int_0^1 \psi^2 dx \int_0^1 \psi_x^2 dx\right)^{1/2}. \end{split} \end{equation*} |
Hence, we get that \|\psi\|_{L^\infty(0, T; L^\infty)}\rightarrow 0\; \; \hbox{as}\; \; \alpha_n\rightarrow 0^+ by using (3.9) and \alpha_n^{1/2}\|\psi_x\|_{L^2}^2 \leq C . On the other hand, it is obvious that \|\theta-\overline\theta\|_{L^\infty(0, T; L^\infty)}\rightarrow 0\; \; \hbox{as}\; \; \alpha_n\rightarrow 0^+ by using (1.11). This shows that a boundary layer does not occur as \alpha_n\rightarrow 0^+ , and this leads to a contradiction. Thus, (3.8) follows. This proof is complete.
Thanks to \sup_{0 < t < T}\int_0^1(\psi_x^2+\overline\psi_x^2) \omega(x)dx\leq C , we have
\begin{equation*} \begin{split} \sup\limits_{0 < t < T}\int_{\delta}^{1-\delta}\psi_x^2 dx\leq \frac{C}{\delta}, \quad\forall \delta \in (0, 1/4). \end{split} \end{equation*} |
By the embedding theorem, and from Theorem 1.1(ⅲ), we obtain the following for any \delta \in (0, 1/4) :
\begin{equation*} \begin{split} \|\psi-\overline \psi\|_{L^\infty(\delta, 1-\delta)}^2&\leq C\int_0^1(\psi-\overline \psi)^2 dx +C\int_{\delta}^{1-\delta} \big|(\psi-\overline \psi)(\psi_x-\overline \psi_x)\big|dx\\ &\leq C\sqrt{\alpha}+C\left(\int_{\delta}^{1-\delta} (\psi-\overline \psi)^2 dx \int_{\delta}^{1-\delta} (\psi_x-\overline \psi_x)^2 dx\right)^{1/2}\\ &\leq C\sqrt{\alpha}+C\left(\frac{\sqrt{\alpha}}{\delta}\right)^{1/2}. \end{split} \end{equation*} |
Hence, for any function \delta(\alpha) satisfying that \delta(\alpha)\downarrow 0 and \sqrt{\alpha}/\delta(\alpha)\rightarrow 0 as \alpha\downarrow 0 , it holds that
\begin{equation*} \begin{split} \|\psi-\overline \psi\|_{L^\infty(0, T;L^\infty(\delta(\alpha), 1-\delta(\alpha)))} \rightarrow 0\quad \hbox{as}\; \; \alpha\rightarrow 0^+. \end{split} \end{equation*} |
On the other hand, it follows from (1.11) that \|\theta-\overline\theta\|_{L^\infty(0, T; L^\infty)}\rightarrow 0\; \; \hbox{as}\; \; \alpha \rightarrow 0^+ . Consequently, for any function \delta(\alpha) satisfying that \delta(\alpha)\downarrow 0 and \sqrt{\alpha}/\delta(\alpha)\rightarrow 0 as \alpha\downarrow 0 , we have
\begin{equation*} \begin{split} \|(\psi-\overline \psi, \theta-\overline\theta)\|_{L^\infty(0, T;L^\infty(\delta(\alpha), 1-\delta(\alpha)))} \rightarrow 0\quad \hbox{as}\; \; \alpha\rightarrow 0^+. \end{split} \end{equation*} |
Finally, we observe that
\begin{equation*} \begin{aligned} \mathop{\lim\inf}\limits_{\alpha\rightarrow 0^+}\|\psi-\overline \psi\|_{L^\infty(0, T;L^\infty)} > 0 \end{aligned} \end{equation*} |
whenever (\overline\psi(1, t), \overline\psi(0, t)) \not\equiv(\xi(t), \xi(t)) on [0, T] . This ends the proof of Theorem 1.3.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
We would like to express deep thanks to the referees for their important comments. The research was supported in part by the National Natural Science Foundation of China (grants 12071058, 11971496) and the research project of the Liaoning Education Department (grant 2020jy002).
The authors declare that there is no conflict of interest.
[1] | S. Agarwal, J. O. D. Terrail, F. Jurie, Recent advances in object detection in the age of deep convolutional neural networks, preprint, arXiv: 1809.03193. |
[2] | R. Girshick, J. Donahue, T. Darrell, J. Malik, Rich feature hierarchies for accurate object detection and semantic segmentation, in 2014 IEEE Conference on Computer Vision and Pattern Recognition, (2014), 580–587. https://doi.org/10.1109/CVPR.2014.81 |
[3] | R. Girshick, Fast R-CNN, in 2015 IEEE International Conference on Computer Vision (ICCV), (2015), 1440–1448. https://doi.org/10.1109/ICCV.2015.169 |
[4] |
S. Ren, K. He, R. Girshick, J. Sun, Faster R-CNN: towards real-time object detection with region proposal networks, IEEE Trans. Pattern Anal. Mach. Intell., 39 (2016), 1137–1149. https://doi.org/10.1109/TPAMI.2016.2577031 doi: 10.1109/TPAMI.2016.2577031
![]() |
[5] |
K. He, G. Gkioxari, P. Dollár, R. Girshick, Mask R-CNN, IEEE Trans. Pattern Anal. Mach. Intell., 42 (2020), 386–397. https://doi.org/10.1109/TPAMI.2018.2844175 doi: 10.1109/TPAMI.2018.2844175
![]() |
[6] | J. Redmon, S. Divvala, R. Girshick, A. Farhadi, You Only Look Once: Unified, real-time object detection, in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2016), 779–88. https://doi.org/10.1109/CVPR.2016.91 |
[7] | J. Redmon, A. Farhadi, YOLOv3: An incremental improvement, preprint, arXiv: 1804.02767. |
[8] | J. C. Y. Wang, A. Bochkovskiy, H. Y. M. Liao, YOLOv7: Trainable bag-of-freebies sets new state-of-the-art for real-time object detectors, preprint, arXiv: 2207.02696. |
[9] |
K. Kang, H. Li, J. Yan, X. Zeng, B. Yang, T. Xiao, et al., T-CNN: tubelets with convolutional neural networks for object detection from videos, IEEE Trans. Circuits Syst. Video Technol., (2017), 2896–2907. https://doi.org/10.1109/TCSVT.2017.2736553 doi: 10.1109/TCSVT.2017.2736553
![]() |
[10] | T. Yin, X. Zhou, P. Krahenbuhl, Center-based 3d object detection and tracking, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2021), 11784–11793. https://doi.org/10.1109/CVPR46437.2021.01161 |
[11] | J. Dai, K. He, J. Sun, Instance-aware semantic segmentation via multi-task network cascades, in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2016), 3150–3158. https://doi.org/10.1109/CVPR.2016.343 |
[12] | B. Hariharan, P. Arbeláez, R. Girshick, J. Malik, Hypercolumns for object segmentation and fine-grained localization, in 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2015), 447–456. https://doi.org/10.1109/CVPR.2015.7298642 |
[13] | B. Hariharan, P. Arbeláez, R. Girshick, J. Malik, Simultaneous detection and segmentation, in Computer Vision–ECCV 2014: 13th European Conference, Zurich, Switzerland, September 6-12, 2014, Proceedings, Part VII 13, (2014), 297–312. https://doi.org/10.1007/978-3-319-10584-0_20 |
[14] | C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, et al., Going deeper with convolutions, in 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2015), 1–9. https://doi.org/10.1109/CVPR.2015.7298594 |
[15] | H. Wang, F. He, Z. Peng, T. Shao, Y. L. Yang, K. Zhou, et al., Understanding the robustness of skeleton-based action recognition under adversarial attack, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2021), 14656–14665. https://doi.org/10.1109/CVPR46437.2021.01442 |
[16] | L. Wang, Z. Tong, B. Ji, G. Wu, TDN: Temporal difference networks for efficient action recognition, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2021), 1895–1904. https://doi.org/10.48550/arXiv.2012.10071 |
[17] | D. Li, Z. Qiu, Y. Pan, T. Yao, H. Li, T. Mei, Representing videos as discriminative sub-graphs for action recognition, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2021), 3310–3319. https://doi.org/10.48550/arXiv.2201.04027 |
[18] | C. F. R. Chen, R. Panda, K. Ramakrishnan, R. Feris, J. Cohn, A. Oliva, et al., Deep analysis of cnn-based spatio-temporal representations for action recognition, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2021), 6165–6175. https://doi.org/10.1109/CVPR46437.2021.00610 |
[19] |
S. Jha, C. Seo, E. Yang, G. P. Joshi, Real time object detection and trackingsystem for video surveillance system, Multimed. Tools Appl., 80 (2021), 3981–3996. https://doi.org/10.1007/s11042-020-09749-x doi: 10.1007/s11042-020-09749-x
![]() |
[20] | M. A. Farooq, A. A. Khan, A. Ahmad, R. H. Raza, Effectiveness of state-of-the-art super resolution algorithms in surveillance environment, in Conference on Multimedia, Interaction, Design and Innovation, 1376 (2021), 79–88. https://doi.org/10.48550/arXiv.2107.04133 |
[21] | X. Zheng, X. Li, K. Xu, X. Jiang, T. Sun, Gait identification under surveillance environment based on human skeleton, preprint, arXiv: 2111.11720. |
[22] | F. Wu, Q. Wang, J. Bian, H. Xiong, N. Ding, F. Lu, et al., A survey on video action recognition in sports: datasets, methods and applications, preprint, arXiv: 2206.01038. |
[23] | C. J. Roros, A. C. Kak, maskGRU: Tracking small objects in the presence of large background motions, preprint, arXiv: 2201.00467. |
[24] | Y. B. Can, A. Liniger, D. P. Paudel, L. Van Gool, Structured bird's-eye-view traffic scene understanding from onboard images, in 2021 IEEE/CVF International Conference on Computer Vision (ICCV), (2021), 15641–15650. https://doi.org/10.1109/ICCV48922.2021.01537 |
[25] | S. Hampali, S. Stekovic, S. D. Sarkar, C. S. Kumar, F. Fraundorfer, V. Lepetit, Monte carlo scene search for 3d scene understanding, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2021), 13804–13813. https://doi.org/10.1109/CVPR46437.2021.01359 |
[26] | J. Hou, B. Graham, M. Niessner, S. Xie, Exploring data-efficient 3d scene understanding with contrastive scene contexts, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2021), 15587–15597. https://doi.org/10.1109/CVPR46437.2021.01533 |
[27] | Y. Liu, R. Wang, S. Shan, X. Chen, Structure inference net: object detection using scene-level context and instance-level relationships, in 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2018), 6985–6994. https://doi.org/10.1109/CVPR.2018.00730 |
[28] | M. Schön, M. Buchholz, K. Dietmayer, MGNet: monocular geometric scene understanding for autonomous driving, in 2021 IEEE/CVF International Conference on Computer Vision (ICCV), (2021), 15784–15795. https://doi.org/10.1109/ICCV48922.2021.01551 |
[29] | K. He, X. Zhang, S. Ren, J. Sun, Deep residual learning for image recognition, in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2016), 770–778. https://doi.org/10.1109/CVPR.2016.90 |
[30] | S. H. Gao, M. M. Cheng, K. Zhao, X. Y. Zhang, M. H. Yang, P. Torr, Res2Net: a new multi-scale backbone architecture, in IEEE Trans. Pattern Anal. Mach. Intell., 43 (2021), 652–662. https://doi.org/10.1109/TPAMI.2019.2938758 |
[31] | K. Simonyan, A. Zisserman, Very deep convolutional networks for large-scale image recognition, preprint, arXiv: 1409.1556. |
[32] | 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. |
[33] | M. Sandler, A. Howard, M. Zhu, A. Zhmoginov, L. C. Chen, MobileNetV2: inverted residuals and linear bottlenecks, in 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2018), 4510–4520. https://doi.org/10.48550/arXiv.1801.04381 |
[34] |
K. He, X. Zhang, S. Ren, J. Sun, Spatial pyramid pooling in deep convolutional networks for visual recognition, IEEE Trans. Pattern Anal. Mach. Intell., 37 (2015), 1904–1916. https://doi.org/10.1109/TPAMI.2015.2389824 doi: 10.1109/TPAMI.2015.2389824
![]() |
[35] | T. Y. Lin, P. Dollár, R. Girshick, K. He, B. Hariharan, S. Belongie, Feature pyramid networks for object detection, in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2017), 936–944. https://doi.org/10.1109/CVPR.2017.106 |
[36] | W. Liu, D. Anguelov, D. Erhan, C. Szegedy, S. Reed, C. Y. Fu, et al., SSD: single shot multibox detector, in European Conference on Computer Vision, (2016), 21–37. https://doi.org/10.1007/978-3-319-46448-0_2 |
[37] | C. Zhu, Y. He, M. Savvides, Feature selective anchor-free module for single-shot object detection, in 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), (2019), 840–849. |
[38] | H. Law, J. Deng, CornerNet: Detecting objects as paired keypoints, in European Conference on Computer Vision, (2018), 765–781. https://doi.org/10.1007/978-3-030-01264-9_45 |
[39] | Z. Tian, C. Shen, H. Chen, T. He, FCOS: fully convolutional one-stage object detection, in 2019 IEEE/CVF International Conference on Computer Vision (ICCV), (2019), 9626–9635. https://doi.org/10.1109/ICCV.2019.00972 |
[40] | X. Zhou, D. Wang, P. Krähenbühl, Objects as points, preprint, arXiv: 1904.07850. |
[41] | C. Eggert, S. Brehm, A. Winschel, D. Zecha, R. Lienhart, A closer look: small object detection in faster R-CNN, in 2017 IEEE International Conference on Multimedia and Expo (ICME), (2017), 421–426. https://doi.org/10.1109/ICME.2017.8019550 |
[42] | C. Chen, M. Y. Liu, O. Tuzel, J. Xiao, R-CNN for small object detection, in Asian Conference on Computer Vision, 10115 (2017), 214–230. https://doi.org/10.1007/978-3-319-54193-8_14 |
[43] | T. Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, et al., Microsoft COCO: common objects in context, in European Conference on Computer Vision, (2014), 740–755. https://doi.org/10.48550/arXiv.1405.0312 |
[44] | J. Deng, W. Dong, R. Socher, L. J. Li, K. Li, F. Li, ImageNet: a large-scale hierarchical image database, in 2009 IEEE Conference on Computer Vision and Pattern Recognition, (2009), 248–255. https://doi.org/10.1109/CVPR.2009.5206848 |
[45] |
M. Everingham, L. Van Gool, C. K. I. Williams, J. Winn, A. Zisserman, The pascal visual object classes (voc) challenge, Int. J. Comput. Vis., 88 (2010), 303–338. https://doi.org/10.1007/s11263-009-0275-4 doi: 10.1007/s11263-009-0275-4
![]() |
[46] | Z. Zong, G. Song, Y. Liu, DETRs with collaborative hybrid assignments training, preprint, arXiv: 2211.12860. |
[47] | S. Yang, P. Luo, C. C. Loy, X. Tang, WIDER FACE: a face detection benchmark, in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2016), 5525–5533. https://doi.org/10.1109/CVPR.2016.596 |
[48] | A. B. Chan, Z. S. J. Liang, N. Vasconcelos, Privacy preserving crowd monitoring: counting people without people models or tracking, in 2008 IEEE Conference on Computer Vision and Pattern Recognition, (2008), 1–7. https://doi.org/10.1109/CVPR.2008.4587569 |
[49] | L. Wang, J. Shi, G. Song, Object detection combining recognition and segmentation, in Asian Conference on Computer Vision, 4843 (2007), 189. |
[50] | E. Bondi, R. Jain, P. Aggrawal, S. Anand, R. Hannaford, A. Kapoor, et al., BIRDSAI: a dataset for detection and tracking in aerial thermal infrared videos, in 2020 IEEE Winter Conference on Applications of Computer Vision (WACV), (2020), 1736–1745. https://doi.org/10.1109/WACV45572.2020.9093284 |
[51] | L. Neumann, M. Karg, S. Zhang, C. Scharfenberger, E. Piegert, S. Mistr, et al., NightOwls: a pedestrians at night dataset, in Asian Conference on Computer Vision, (2019), 691–705. https://doi.org/10.1007/978-3-030-20887-5_43 |
[52] | K. Behrendt, L. Novak, R. Botros, A deep learning approach to traffic lights: Detection, tracking, and classification, in 2017 IEEE International Conference on Robotics and Automation (ICRA), (2017), 1370–1377. https://doi.org/10.1109/ICRA.2017.7989163 |
[53] | C. Ertler, J. Mislej, T. Ollmann, L. Porzi, G. Neuhold, Y. Kuang, The Mapillary Traffic sign dataset for detection and classification on a global scale, in European Conference on Computer Vision, (2020), 68–84. https://doi.org/10.48550/arXiv.1909.04422 |
[54] |
J. Zhang, M. Huang, X. Jin, X. Li, A real-time chinese traffic sign detection algorithm based on modified yolov2, Algorithms, 10 (2017), 127. https://doi.org/10.3390/a10040127 doi: 10.3390/a10040127
![]() |
[55] | D. Tabernik, D. Skočaj, Deep learning for large-scale traffic-sign detection and recognition, preprint, arXiv: 1904.00649. |
[56] | Z. Zhu, D. Liang, S. Zhang, X. Huang, B. Li, S. Hu, Traffic-sign detection and classification in the wild, in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2016), 2110–2118. https://doi.org/10.1109/CVPR.2016.232 |
[57] |
Z. Zhao, P. Zheng, S. T. Xu, X. Wu, Object detection with deep learning: a review, IEEE Trans. Neural Networks Learn. Syst., 30 (2019), 3212–3232. https://doi.org/10.1109/TNNLS.2018.2876865 doi: 10.1109/TNNLS.2018.2876865
![]() |
[58] |
K. Li, G. Wan, G. Cheng, L. Meng, J. Han, Object detection in optical remote sensing images: A survey and a new benchmark, ISPRS J. Photogramm. Remote Sens., 159 (2020), 296–307. https://doi.org/10.1016/j.isprsjprs.2019.11.023 doi: 10.1016/j.isprsjprs.2019.11.023
![]() |
[59] | K. Oksuz, B. C. Cam, S. Kalkan, E. Akbas, Imbalance problems in object detection: a review, preprint, arXiv: 1909.00169. |
[60] | A. G. Menezes, G. de Moura, C. Alves, A. C. P. L. F. de Carvalho, Continual object detection: a review of definitions, strategies, and challenges, preprint, arXiv: 2205.15445. |
[61] |
L. Jiao, R. Zhang, F. Liu, S. Yang, B. Hou, L. Li, et al., New generation deep learning for video object detection: a survey, IEEE Trans. Neural Networks Learn. Syst., 33 (2022), 3195–3215. https://doi.org/10.1109/TNNLS.2021.3053249 doi: 10.1109/TNNLS.2021.3053249
![]() |
[62] |
L. Jiao, F. Zhang, F. Liu, S. Yang, L. Li, Z. Feng, et al., A survey of deep learning-based object detection, IEEE Access, 7 (2019), 128837–128868. https://doi.org/10.1109/ACCESS.2019.2939201 doi: 10.1109/ACCESS.2019.2939201
![]() |
[63] |
G. Chen, H. Wang, K. Chen, Z. Li, Z. Song, Y. Liu, et al., A survey of the four pillars for small object detection: multiscale representation, contextual information, super-resolution, and region proposal, IEEE Trans. Syst. Man Cybern, Syst., 52 (2022), 936–953. https://doi.org/10.1109/TSMC.2020.3005231 doi: 10.1109/TSMC.2020.3005231
![]() |
[64] | K. Chen, J. Wang, J. Pang, Y. Cao, Y. Xiong, X. Li, et al., MMDetection: open mmlab detection toolbox and benchmark, preprint, arXiv: 1906.07155. |
[65] |
K. Tong, Y. Wu, F. Zhou, Recent advances in small object detection based on deep learning: A review, Image Vis. Comput., 97 (2020), 103910. https://doi.org/10.1016/j.imavis.2020.103910 doi: 10.1016/j.imavis.2020.103910
![]() |
[66] |
Y. Liu, P. Sun, N. Wergeles, Y. Shang, A survey and performance evaluation of deep learning methods for small object detection, Expert Syst. Appl., 172 (2021), 114602. https://doi.org/10.1016/j.eswa.2021.114602 doi: 10.1016/j.eswa.2021.114602
![]() |
[67] |
K. Tong, Y. Wu, Deep learning-based detection from the perspective of small or tiny objects: A survey, Image Vis. Comput., 123 (2022), 104471. https://doi.org/10.1016/j.imavis.2022.104471 doi: 10.1016/j.imavis.2022.104471
![]() |
[68] | A. M. Rekavandi, L. Xu, F. Boussaid, A. K. Seghouane, S. Hoefs, M. Bennamoun, A guide to image and video based small object detection using deep learning: case study of maritime surveillance, preprint, arXiv: 2207.12926. |
[69] | G. Cheng, X. Yuan, X. Yao, K. Yan, Q. Zeng, J. Han, Towards large-scale small object detection: survey and benchmarks, preprint, arXiv: 2207.14096. |
[70] | S. Liu, L. Qi, H. Qin, J. Shi, J. Jia, Path aggregation network for instance segmentation, in 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2018), 8759–8768. https://doi.org/10.1109/CVPR.2018.00913 |
[71] | M. Tan, R. Pang, Q. V. Le, EfficientDet: scalable and efficient object detection, preprint, arXiv: 1911.09070. |
[72] | S. Liu, D. Huang, Y. Wang, Learning spatial fusion for single-shot object detection, preprint, arXiv: 1911.09516. |
[73] | G. Ghiasi, T. Y. Lin, R. Pang, Q. V. Le, NAS-FPN: learning scalable feature pyramid architecture for object detection, preprint, arXiv: 1904.07392. |
[74] | T. Y. Lin, P. Goyal, R. Girshick, K. He, P. Dollár, Focal loss for dense object detection, in 2017 IEEE International Conference on Computer Vision (ICCV), (2017), 2999–3007. https://doi.org/10.1109/ICCV.2017.324 |
[75] | Z. Li, F. Zhou, FSSD: feature fusion single shot multibox detector, preprint, arXiv: 1712.00960. |
[76] | L. Cui, R. Ma, P. Lv, X. Jiang, Z. Gao, B. Zhou, et al., MDSSD: multi-scale deconvolutional single shot detector for small objects, preprint, arXiv: 1805.07009. |
[77] | Y. Gong, X. Yu, Y. Ding, X. Peng, J. Zhao, Z. Han, Effective fusion factor in fpn for tiny object detection, preprint, arXiv: 2011.02298. |
[78] | Z. Liu, G. Gao, L. Sun, Z. Fang, HRDNet: High-resolution detection network for small objects, preprint, arXiv: 2006.07607. |
[79] | Z. Liu, G. Gao, L. Sun, L. Fang, IPG-Net: image pyramid guidance network for small object detection, in 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), (2020), 4422–4430. https://doi.org/10.1109/CVPRW50498.2020.00521 |
[80] | P. Y. Chen, J. W. Hsieh, C. Y. Wang, H. Y. M. Liao, Recursive hybrid fusion pyramid network for real-time small object detection on embedded devices, in 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), (2020), 1612–1621. https://doi.org/10.1109/CVPRW50498.2020.00209 |
[81] | C. Yang, Z. Huang, N. Wang, QueryDet: cascaded sparse query for accelerating high-resolution small object detection, preprint, arXiv: 2103.09136. |
[82] |
C. Deng, M. Wang, L. Liu, Y. Liu, Y. Jiang, Extended feature pyramid network for small object detection, IEEE Trans. Multimedia, 24 (2022), 1968–1979. https://doi.org/10.1109/TMM.2021.3074273 doi: 10.1109/TMM.2021.3074273
![]() |
[83] | J. Li, X. Liang, Y. Wei, T. Xu, J. Feng, S. Yan, Perceptual generative adversarial networks for small object detection, in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2017), 1951–1959. https://doi.org/10.1109/CVPR.2017.211 |
[84] | Y. Bai, Y. Zhang, M. Ding, B. Ghanem, SOD-MTGAN: small object detection via multi-task generative adversarial network, in European Conference on Computer Vision, 11217 (2018), 210–226. https://doi.org/10.1007/978-3-030-01261-8_13 |
[85] | J. Noh, W. Bae, W. Lee, J. Seo, G. Kim, Better to follow, follow to be better: towards precise supervision of feature super-resolution for small object detection, in 2019 IEEE/CVF International Conference on Computer Vision (ICCV), (2019), 9724–9733. https://doi.org/10.1109/ICCV.2019.00982 |
[86] | F. Zhang, L. Jiao, L. Li, F. Liu, X. Liu, MultiResolution attention extractor for small object detection, preprint, arXiv: 2006.05941. |
[87] | J. Rabbi, N. Ray, M. Schubert, S. Chowdhury, D. Chao, Small-object detection in remote sensing images with end-to-end edge-enhanced gan and object detector network, preprint, arXiv: 2003.09085. |
[88] |
K. Jiang, Z. Wang, P. Yi, G. Wang, T. Lu, J. Jiang, Edge-enhanced GAN for remote sensing image super-resolution, IEEE Trans. Geosci. Remote Sens., 57 (2019), 5799–5812. https://doi.org/10.1109/TGRS.2019.2902431 doi: 10.1109/TGRS.2019.2902431
![]() |
[89] | X. Wang, K. Yu, S. Wu, J. Gu, Y. Liu, C. Dong, et al., ESRGAN: enhanced super-resolution generative adversarial networks, in Proceedings of the European conference on computer vision (ECCV), (2018). https://doi.org/10.1007/978-3-030-11021-5_5 |
[90] | A. Jolicoeur-Martineau, The relativistic discriminator: a key element missing from standard gan, preprint, arXiv: 1807.00734. |
[91] |
I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, et al., Generative adversarial nets, Adv. Neural Inf. Process Syst., 27 (2014). https://doi.org/10.48550/arXiv.1406.2661 doi: 10.48550/arXiv.1406.2661
![]() |
[92] |
J. Cao, Y. Pang, S. Zhao, X. Li, High-level semantic networks for multi-scale object detection, IEEE Trans. Circuits Syst. Video Technol., 30 (2020), 3372–3386. https://doi.org/10.1109/TCSVT.2019.2950526 doi: 10.1109/TCSVT.2019.2950526
![]() |
[93] |
K. Zhang, Z. Zhang, Z. Li, Y. Qiao, Joint face detection and alignment using multitask cascaded convolutional networks, IEEE Signal Process. Lett., 23 (2016), 1499–1503. https://doi.org/10.1109/LSP.2016.2603342 doi: 10.1109/LSP.2016.2603342
![]() |
[94] | Z. Hao, Y. Liu, H. Qin, J. Yan, X. Li, X. Hu, Scale-aware face detection, in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2017), 1913–1922. https://doi.org/10.1109/CVPR.2017.207 |
[95] | B. Singh, L. S. Davis, An analysis of scale invariance in object detection - snip, in 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2018), 3578–3587. https://doi.org/10.1109/CVPR.2018.00377 |
[96] |
B. Singh, M. Najibi, L. S. Davis, SNIPER: efficient multi-scale training, Adv. Neural Inf. Process Syst., 31 (2018). https://doi.org/10.48550/arXiv.1805.09300 doi: 10.48550/arXiv.1805.09300
![]() |
[97] | Y. Kim, B. N. Kang, D. Kim, SAN: learning relationship between convolutional features for multi-scale object detection, in European Conference on Computer Vision, 11209 (2018), 328–343. https://doi.org/10.1007/978-3-030-01228-1_20 |
[98] | Y. Li, Y. Chen, N. Wang, Z. Zhang, Scale-aware trident networks for object detection, preprint, arXiv: 1901.01892. |
[99] | J. Peng, M. Sun, Z. X. Zhang, T. Tan, J. Yan, POD: practical object detection with scale-sensitive network, in 2019 IEEE/CVF International Conference on Computer Vision (ICCV), (2019), 9606–9615. https://doi.org/10.1109/ICCV.2019.00970 |
[100] |
A. Oliva, A. Torralba, The role of context in object recognition, Trends Cogn. Sci., 11 (2007), 520–527. https://doi.org/10.1016/j.tics.2007.09.009 doi: 10.1016/j.tics.2007.09.009
![]() |
[101] | S. Bell, C. L. Zitnick, K. Bala, R. Girshick, Inside-outside net: detecting objects in context with skip pooling and recurrent neural networks, in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2016), 2874–2883. https://doi.org/10.1109/CVPR.2016.314 |
[102] | C. Y. Fu, W. Liu, A. Ranga, A. Tyagi, A. C. Berg, DSSD: deconvolutional single shot detector, preprint, arXiv: 1701.06659. |
[103] | W. Xiang, D. Q. Zhang, H. Yu, V. Athitsos, Context-aware single-shot detector, in 2018 IEEE Winter Conference on Applications of Computer Vision (WACV), (2018), 1784–1793. https://doi.org/10.1109/WACV.2018.00198 |
[104] | X. Chen, A. Gupta, Spatial memory for context reasoning in object detection, in 2017 IEEE International Conference on Computer Vision (ICCV), (2017), 4106–4116. https://doi.org/10.1109/ICCV.2017.440 |
[105] | K. Fu, J. Li, L. Ma, K. Mu, Y. Tian, Intrinsic relationship reasoning for small object detection, preprint, arXiv: 2009.00833. |
[106] | J. S. Lim, M. Astrid, H. J. Yoon, S. I. Lee, Small object detection using context and attention, in 2021 International Conference on Artificial Intelligence in Information and Communication (ICAIIC), (2021), 181–186. https://doi.org/10.1109/ICAIIC51459.2021.9415217 |
[107] | A. Bochkovskiy, C. Y. Wang, H. Y. M. Liao, YOLOv4: optimal speed and accuracy of object detection, preprint, arXiv: 2004.10934. |
[108] | H. Zhang, M. Cisse, Y. N. Dauphin, D. Lopez-Paz, Mixup: beyond empirical risk minimization, preprint, arXiv: 1710.09412. |
[109] | S. Yun, D. Han, S. J. Oh, S. Chun, J. Choe, Y. Yoo, CutMix: regularization strategy to train strong classifiers with localizable features, in Proceedings of the IEEE International Conference on Computer Vision, (2019), 6023–6032. https://doi.org/10.1109/ICCV.2019.00612 |
[110] | M. Kisantal, Z. Wojna, J. Murawski, J. Naruniec, K. Cho, Augmentation for small object detection, preprint, arXiv: 1902.07296. |
[111] | C. Chen, Y. Zhang, Q. Lv, S. Wei, X. Wang, X. Sun, et al., RRNet: a hybrid detector for object detection in drone-captured images, in 2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW), (2019), 100–108. https://doi.org/10.1109/ICCVW.2019.00018 |
[112] | F. O. Unel, B. O. Ozkalayci, C. Cigla, The power of tiling for small object detection, in 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), (2019), 582–591. https://doi.org/10.1109/CVPRW.2019.00084 |
[113] | Y. Chen, P. Zhang, Z. Li, Y. Li, X. Zhang, L. Qi, et al., Dynamic scale training for object detection, preprint, arXiv: 2004.12432. |
[114] | B. Zoph, E. D. Cubuk, G. Ghiasi, T. Y. Lin, J. Shlens, Q. V. Le, Learning data augmentation strategies for object detection, in European Conference on Computer Vision, (2020), 566–583. https://doi.org/10.1007/978-3-030-58583-9_34 |
[115] | E. D. Cubuk, B. Zoph, D. Mane, V. Vasudevan, Q. V. Le, AutoAugment: learning augmentation policies from data, preprint, arXiv: 1805.09501. |
[116] | Y. Chen, Y. Li, T. Kong, L. Qi, R. Chu, L. Li, et al., Scale-aware automatic augmentation for object detection, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), (2021), 9563–9572. https://doi.org/10.1109/CVPR46437.2021.00944 |
[117] | N. Samet, S. Hicsonmez, E. Akbas, Reducing label noise in anchor-free object detection, preprint, arXiv: 2008.01167. |
[118] | K. Duan, S. Bai, L. Xie, H. Qi, Q. Huang, Q. Tian, CenterNet++ for object detection, preprint, arXiv: 2204.08394. |
[119] | J. Wang, C. Xu, W. Yang, L. Yu, A normalized gaussian wasserstein distance for tiny object detection, preprint, arXiv: 2110.13389. |
[120] | C. Xu, J. Wang, W. Yang, H. Yu, L. Yu, G. Xia, RFLA: Gaussian receptive field based label assignment for tiny object detection, in Proceedings of the European conference on computer vision (ECCV), (2022). https://doi.org/10.1007/978-3-031-20077-9_31 |
[121] | C. Lee, S. Park, H. Song, J. Ryu, S. Kim, H. Kim, et al., Interactive multi-class tiny-object detection, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2022), 14136–14145. https://doi.org/10.1109/CVPR52688.2022.01374 |
[122] | F. C. Akyon, S. Altinuc, A. Temi̇zel, Slicing aided hyper inference and fine-tuning for small object detection, preprint, arXiv: 2202.06934. |
[123] | P. Hu, D. Ramanan, Finding tiny faces, in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2017), 1522–1530. https://doi.org/10.1109/CVPR.2017.166 |
[124] | S. Zhang, X. Zhu, Z. Lei, H. Shi, X. Wang, S. Z. Li, S.3FD: single shot scale-invariant face detector, in 2017 IEEE International Conference on Computer Vision (ICCV), (2017), 192–201. https://doi.org/10.1109/ICCV.2017.30 |
[125] | Y. Bai, Y. Zhang, M. Ding, B. Ghanem, Finding tiny faces in the wild with generative adversarial network, in 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2018), 21–30. https://doi.org/10.1109/CVPR.2018.00010 |
[126] | P. Samangouei, M. Najibi, L. Davis, R. Chellappa, Face-magnet: magnifying feature maps to detect small faces, preprint, arXiv: 1803.05258. |
[127] | C. Zhu, R. Tao, K. Luu, M. Savvides, Seeing small faces from robust anchor's perspective, preprint, arXiv: 1802.09058. |
[128] | Y. Zhu, H. Cai, S. Zhang, C. Wang, Y. Xiong, TinaFace: strong but simple baseline for face detection, preprint, arXiv: 2011.13183. |
[129] | J. Dai, H. Qi, Y. Xiong, Y. Li, G. Zhang, H. Hu, et al., Deformable convolutional networks, in 2017 IEEE International Conference on Computer Vision (ICCV), (2017), 764–773. https://doi.org/10.1109/ICCV.2017.89 |
[130] | Z. Zheng, P. Wang, W. Liu, J. Li, R. Ye, D. Ren, Distance-IoU loss: faster and better learning for bounding box regression, in Proceedings of the AAAI conference on artificial intelligence, 34 (2019), 12993–13000. https://doi.org/10.1609/aaai.v34i07.6999 |
[131] | A. Shrivastava, A. Gupta, R. Girshick, Training region-based object detectors with online hard example mining, in Proceedings of the IEEE conference on computer vision and pattern recognition, (2016), 761–769. https://doi.org/10.1109/CVPR.2016.89 |
[132] | Z. Zhang, W. Shen, S. Qiao, Y. Wang, B. Wang, A. Yuille, Robust face detection via learning small faces on hard images, in Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, (2020), 1361–1370. https://doi.org/10.48550/arXiv.1811.11662 |
[133] | T. Song, L. Sun, D. Xie, H. Sun, S. Pu, Small-scale pedestrian detection based on somatic topology localization and temporal feature aggregation, preprint, arXiv: 1807.01438. |
[134] | S. Das, P. S. Mukherjee, U. Bhattacharya, Seek and you will find: a new optimized framework for efficient detection of pedestrian, preprint, arXiv: 1912.10241. |
[135] | W. Liu, S. Liao, W. Ren, W. Hu, Y. Yu, High-level semantic feature detection: a new perspective for pedestrian detection, in 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), (2019), 5182–5191. https://doi.org/10.1109/CVPR.2019.00533 |
[136] | X. Yu, Y. Gong, N. Jiang, Q. Ye, Z. Han, Scale match for tiny person detection, in 2020 IEEE Winter Conference on Applications of Computer Vision (WACV), (2020), 1246–1254. https://doi.org/10.1109/WACV45572.2020.9093394 |
[137] |
D. Božić-Štulić, Ž. Marušić, S. Gotovac, Deep learning approach in aerial imagery for supporting land search and rescue missions, Int. J. Comput Vis., 127 (2019), 1256–1278. https://doi.org/10.1007/s11263-019-01177-1 doi: 10.1007/s11263-019-01177-1
![]() |
[138] | G. Adaimi, S. Kreiss, A. Alahi, Perceiving traffic from aerial images, preprint, arXiv: 2009.07611. |
[139] | C. Gheorghe, N. Filip, Road traffic analysis using unmanned aerial vehicle and image processing algorithms, in 2022 IEEE International Conference on Automation, Quality and Testing, Robotics (AQTR), (2022), 1–5. https://doi.org/10.1109/AQTR55203.2022.9802058 |
[140] |
J. Han, J. Ding, J. Li, G. S. Xia, Align deep features for oriented object detection, IEEE Trans. Geosci. Remote Sens., 60 (2022), 5602511. https://doi.org/10.1109/TGRS.2021.3062048 doi: 10.1109/TGRS.2021.3062048
![]() |
[141] | 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 2019 IEEE/CVF International Conference on Computer Vision (ICCV), (2019), 8231–8240. https://doi.org/10.1109/ICCV.2019.00832 |
[142] | X. Xie, G. Cheng, J. Wang, X. Yao, J. Han, Oriented r-cnn for object detection, in 2021 IEEE/CVF International Conference on Computer Vision (ICCV), (2021), 3500–3509. https://doi.org/10.1109/ICCV48922.2021.00350 |
[143] | R. Qin, Q. Liu, G. Gao, D. Huang, Y. Wang, MRDet: a multi-head network for accurate oriented object detection in aerial images, preprint, arXiv: 2012.13135. |
[144] | X. Zhang, E. Izquierdo, K. Chandramouli, Dense and small object detection in uav vision based on cascade network, in 2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW), (2019), 118–126. https://doi.org/10.1109/ICCVW.2019.00020 |
[145] | J. Yi, P. Wu, B. Liu, Q. Huang, H. Qu, D. Metaxas, Oriented object detection in aerial images with box boundary-aware vectors, in Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, (2021), 2150–2159. https://doi.org/10.1109/WACV48630.2021.00220 |
[146] | O. Ronneberger, P. Fischer, T. Brox, U-Net: convolutional networks for biomedical image segmentation, in Medical Image Computing and Computer-Assisted Intervention, (2015), 234–241. https://doi.org/10.1007/978-3-319-24574-4_28 |
[147] | J. Han, J. Ding, N. Xue, G. S. Xia, ReDet: a rotation-equivariant detector for aerial object detection, preprint, arXiv: 2103.07733. |
[148] | J. Ding, N. Xue, Y. Long, G. S. Xia, Q. Lu, Learning ROI transformer for oriented object detection in aerial images, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), (2019), 2849–2858. https://doi.org/10.1109/CVPR.2019.00296 |
[149] |
M. Zand, A. Etemad, M. Greenspan, Oriented bounding boxes for small and freely rotated objects, IEEE Trans. Geosci. Remote Sensing, 60 (2022), 1–15. https://doi.org/10.1109/TGRS.2021.3076050 doi: 10.1109/TGRS.2021.3076050
![]() |
[150] | Z. Yang, S. Liu, H. Hu, L. Wang, S. Lin, RepPoints: point set representation for object detection, in Proceedings of the IEEE/CVF International Conference on Computer Vision, (2019), 9657–9666. https://doi.org/10.1109/ICCV.2019.00975 |
[151] | W. Li, Y. Chen, K. Hu, J. Zhu, Oriented reppoints for aerial object detection, preprint, arXiv: 2105.11111. |
[152] | C. Xu, J. Wang, W. Yang, L. Yu, Dot distance for tiny object detection in aerial images, in IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), (2021), 1192–1201, https://doi.org/10.1109/CVPRW53098.2021.00130 |
[153] |
X. Fang, F. Hu, M. Yang, T. Zhu, R. Bi, Z. Zhang, Z. Gao, Small object detection in remote sensing images based on super-resolution, Pattern Recognit. Lett., 153 (2022), 107–112. https://doi.org/10.1016/j.patrec.2021.11.027.5 doi: 10.1016/j.patrec.2021.11.027.5
![]() |
[154] |
Y. Li, Q. Huang, X. Pei, Y. Chen, L. Jiao, R. Shang, Cross-layer attention network for small object detection in remote sensing imagery, IEEE J. Sel. Top Appl. Earth Obs. Remote Sens., 14 (2021), 2148–2161. https://doi.org/10.1109/JSTARS.2020.3046482 doi: 10.1109/JSTARS.2020.3046482
![]() |
[155] |
O. C. Koyun, R. K. Keser, İ. B. Akkaya, B. U. Töreyin, Focus-and-detect:a small object detection framework for aerial images, Signal Process. Image Commun., 104 (2022), 116675. https://doi.org/10.1016/j.image.2022.116675 doi: 10.1016/j.image.2022.116675
![]() |
[156] | B. F. Klare, B. Klein, E. Taborsky, A. Blanton, J. Cheney, K. Allen, et al., Pushing the frontiers of unconstrained face detection and recognition: IARPA Janus Benchmark A, in 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2015), 1931–1939. https://doi.org/10.1109/CVPR.2015.7298803 |
[157] | Y. Yuan, W. Yang, W. Ren, J. Liu, W. J. Scheirer, Z. Wang, UG2+: a collective benchmark effort for evaluating and advancing image understanding in poor visibility environments, preprint, arXiv: 1904.04474. |
[158] | H. Nada, V. A. Sindagi, H. Zhang, V. M. Patel, Pushing the limits of unconstrained face detection: a challenge ataset and baseline results, in 2018 IEEE 9th International Conference on Biometrics Theory, Applications and Systems (BTAS), (2018), 1–10. https://doi.org/10.1109/BTAS.2018.8698561 |
[159] | M. K. Yucel, Y. C. Bilge, O. Oguz, N. Ikizler-Cinbis, P. Duygulu, R. G. Cinbis, Wildest faces: face detection and recognition in violent settings, preprint, arXiv: 1805.07566. |
[160] |
S. Zhang, Y. Xie, J. Wan, H. Xia, S. Z. Li, G. Guo, WiderPerson: A diverse dataset for dense pedestrian detection in the wild, IEEE Trans. Multimedia, 22 (2020), 380–393. https://doi.org/10.1109/TMM.2019.2929005 doi: 10.1109/TMM.2019.2929005
![]() |
[161] | M. Braun, S. Krebs, F. Flohr, D. M. Gavrila, The eurocity persons dataset: a novel benchmark for object detection, preprint, arXiv: 1805.07193. |
[162] | S. Zhang, R. Benenson, B. Schiele, CityPersons: a diverse dataset for pedestrian detection, in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2017), 4457–4465. https://doi.org/10.1109/CVPR.2017.474 |
[163] | P. Dollar, C. Wojek, B. Schiele, P. Perona, Pedestrian detection: a benchmark, in 2009 IEEE Conference on Computer Vision and Pattern Recognition, (2009), 304–311. https://doi.org/10.1109/CVPR.2009.5206631 |
[164] |
P. Zhu, L. Wen, D. Du, X. Bian, H. Fan, Q. Hu, et al., Detection and tracking meet drones challenge, IEEE Trans. Pattern Anal. Mach. Intell., 44 (2022), 7380–7399. https://doi.org/10.1109/TPAMI.2021.3119563 doi: 10.1109/TPAMI.2021.3119563
![]() |
[165] | D. Du, Y. Qi, H. Yu, Y. Yang, K. Duan, G. Li, et al., The unmanned aerial vehicle benchmark: object detection and tracking, in Proceedings of the European Conference on Computer Vision (ECCV), (2018), 370–386. https://doi.org/10.1007/s11263-019-01266-1 |
[166] | G. S. Xia, X. Bai, J. Ding, Z. Zhu, S. Belongie, J. Luo, et al., DOTA: a large-scale dataset for object detection in aerial images, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2018), 3974–3983. https://doi.org/10.1109/CVPR.2018.00418 |
[167] |
G. Cheng, J. Han, P. Zhou, L. Guo, Multi-class geospatial object detection and geographic image classification based on collection of part detectors, ISPRS J. Photogramm. Remote Sens., 98 (2014), 119–132. https://doi.org/10.1016/j.isprsjprs.2014.10.002 doi: 10.1016/j.isprsjprs.2014.10.002
![]() |
[168] | H. Zhu, X. Chen, W. Dai, K. Fu, Q. Ye, J. Jiao, Orientation robust object detection in aerial images using deep convolutional neural network, in 2015 IEEE International Conference on Image Processing (ICIP), (2015), 3735–3739. https://doi.org/10.1109/ICIP.2015.7351502 |
[169] | L. Tuggener, I. Elezi, J. Schmidhuber, M. Pelillo, T. Stadelmann, DeepScores-a dataset for segmentation, detection and classification of tiny objects, in 2018 24th International Conference on Pattern Recognition (ICPR), (2018), 3704–3709. https://doi.org/10.1109/ICPR.2018.8545307 |
[170] | A. Geiger, P. Lenz, R. Urtasun, Are we ready for autonomous driving? The KITTI vision benchmark suite, in 2012 IEEE Conference on Computer Vision and Pattern Recognition, (2012), 3354–3361. https://doi.org/10.1109/CVPR.2012.6248074 |
[171] | S. Song, S. P. Lichtenberg, J. Xiao, SUN RGB-D: a rgb-d scene understanding benchmark suite, in 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2015), 567–576. https://doi.org/10.1109/CVPR.2015.7298655 |
[172] | S. Zhang, L. Wen, X. Bian, Z. Lei, S. Z. Li, Single-shot refinement neural network for object detection, in 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2018), 4203–4212. https://doi.org/10.1109/CVPR.2018.00442 |
[173] | J. Cao, H. Cholakkal, R. M. Anwer, F. S. Khan, Y. Pang, L. Shao, D2Det: towards high quality object detection and instance segmentation, in 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), (2020), 11482–11491. |
[174] |
Y. Chen, J. Li, H. Xiao, X. Jin, S. Yan, J. Feng, Dual path networks, Adv. Neural Inf. Process Syst., 30 (2017). https://doi.org/10.48550/arXiv.1707.01629 doi: 10.48550/arXiv.1707.01629
![]() |
[175] | Y. Zhu, C. Zhao, J. Wang, X. Zhao, Y. Wu, H. Lu, CoupleNet: coupling global structure with local parts for object detection, in 2017 IEEE International Conference on Computer Vision (ICCV), (2017), 4146–4154. https://doi.org/10.1109/ICCV.2017.444 |
[176] | H. Hu, J. Gu, Z. Zhang, J. Dai, Y. Wei, Relation networks for object detection, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2018), 3588–3597. https://doi.org/10.1109/CVPR.2018.00378 |
[177] | L. Tychsen-Smith, L. Petersson, Improving object localization with fitness nms and bounded iou loss, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2018), 6877–6885. https://doi.org/10.1109/CVPR.2018.00719 |
[178] | S. Xu, X. Wang, W. Lv, Q. Chang, C. Cui, K. Deng, et al., PP-YOLOE: an evolved version of YOLO, preprint, arXiv: 2203.16250. |
[179] |
J. Leng, Y. Ren, W. Jiang, X. Sun, Y. Wang, Realize your surroundings: exploiting context information for small object detection, Neurocomputing, 433 (2021). https://doi.org/10.1016/j.neucom.2020.12.093 doi: 10.1016/j.neucom.2020.12.093
![]() |
[180] | C. L. Zitnick, P. Dollár, Edge Boxes: locating object proposals from edges, in European Conference on Computer Vision, (2014), 391–405. https://doi.org/10.1007/978-3-319-10602-1_26 |
[181] | 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 |
[182] | X. Tang, D. K. Du, Z. He, J. Liu, PyramidBox: a context-assisted single shot face detector, in Proceedings of the European Conference on Computer Vision (ECCV), (2018), 797–813. https://doi.org/10.1007/978-3-030-01240-3_49 |
[183] | J. Deng, J. Guo, Y. Zhou, J. Yu, I. Kotsia, S. Zafeiriou, RetinaFace: single-stage dense face localisation in the wild, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2019), 5203–5212. https://doi.org/10.1109/CVPR42600.2020.00525 |
[184] |
Z. Liu, J. Du, F. Tian, J. Wen, MR-CNN: a multi-scale region-based convolutional neural network for small traffic sign recognition, IEEE Access, 7 (2019), 57120–57128. https://doi.org/10.1109/ACCESS.2019.2913882 doi: 10.1109/ACCESS.2019.2913882
![]() |
[185] | X. Lu, B. Li, Y. Yue, Q. Li, J. Yan, Grid R-CNN, in 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), (2019), 7355–7364, https://doi.org/10.1109/CVPR.2019.00754.(2018). |
[186] | J. Li, Y. Wang, C. Wang, Y. Tai, J. Qian, J. Yang, et al., DSFD: dual shot face detector, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2019), 5060–5069. https://doi.org/10.1109/CVPR.2019.00520 |
[187] |
X. Zhang, F. Wan, C. Liu, R. Ji, Q. Ye, FreeAnchor: learning to match anchors for visual object detection, IEEE Trans. Pattern Anal. Mach. Intell., 44 (2022), 3096–3109. https://doi.org/10.48550/arXiv.1909.02466 doi: 10.48550/arXiv.1909.02466
![]() |
[188] | J. Pang, K. Chen, J. Shi, H. Feng, W. Ouyang, D. Lin, Libra R-CNN: towards balanced learning for object detection, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2019), 821–830. https://doi.org/10.1109/CVPR.2019.00091 |
[189] |
G. Zhang, S. Lu, W. Zhang, CAD-Net: a context-aware detection network for objects in remote sensing imagery, IEEE Trans. Geosci. Remote Sens., 57 (2019), 10015–10024. https://doi.org/10.1109/TGRS.2019.2930982 doi: 10.1109/TGRS.2019.2930982
![]() |
[190] | N. Carion, F. Massa, G. Synnaeve, N. Usunier, A. Kirillov, S. Zagoruyko, End-to-end object detection with transformers, in European Conference on Computer Vision, 12346 (2020), 213–229. https://doi.org/10.1007/978-3-030-58452-8_13 |
[191] |
S. Li, F. Liu, L. Jiao, X. Liu, P. Chen, Learning salient feature for salient object detection without labels, IEEE Trans. Cybern., 53 (2022), 1012–1025. https://doi.org/10.1109/TCYB.2022.3209978 doi: 10.1109/TCYB.2022.3209978
![]() |
[192] |
F. Liu, X. Qian, L. Jiao, X. Zhang, L. Li, Y. Cui, Contrastive learning-based dual dynamic gcn for sar image scene classification, IEEE Trans. Neural Networks Learn Syst., (2022), 1–15. https://doi.org/10.1109/TNNLS.2022.3174873 doi: 10.1109/TNNLS.2022.3174873
![]() |
[193] |
Y. Du, F. Liu, L. Jiao, Z. Hao, S. Li, X. Liu, et al., Augmentative contrastive learning for one-shot object detection, Neurocomputing, 513 (2022), 13–24. https://doi.org/10.1016/j.neucom.2022.09.125 doi: 10.1016/j.neucom.2022.09.125
![]() |