IFKD: Implicit field knowledge distillation for single view reconstruction

Jianyuan Wang; Huanqiang Xu; Xinrui Hu; Biao Leng; Jianyuan Wang; Huanqiang Xu; Xinrui Hu; Biao Leng

doi:10.3934/mbe.2023617

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 8: 13864-13880. doi: 10.3934/mbe.2023617

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Research article Special Issues

IFKD: Implicit field knowledge distillation for single view reconstruction

1.
School of Intelligence Science and Technology, University of Science and Technology Beijing, Beijing 100083, China
2.
Key Laboratory of Intelligent Bionic Unmanned Systems, Ministry of Education, University of Science and Technology Beijing, Beijing 100083, China
3.
School of Computer Science and Engineering, Beihang University, Beijing 100191, China

Academic Editor: Jerry Chun-Wei Lin

Received: 30 November 2022 Revised: 14 May 2023 Accepted: 21 May 2023 Published: 19 June 2023

In 3D reconstruction tasks, camera parameter matrix estimation is usually used to present the single view of an object, which is not necessary when mapping the 3D point to 2D image. The single view reconstruction task should care more about the quality of reconstruction instead of the alignment. So in this paper, we propose an implicit field knowledge distillation model (IFKD) to reconstruct 3D objects from the single view. Transformations are performed on 3D points instead of the camera and keep the camera coordinate identified with the world coordinate, so that the extrinsic matrix can be omitted. Besides, a knowledge distillation structure from 3D voxel to the feature vector is established to further refine the feature description of 3D objects. Thus, the details of a 3D model can be better captured by the proposed model. This paper adopts ShapeNet Core dataset to verify the effectiveness of the IFKD model. Experiments show that IFKD has strong advantages in IOU and other core indicators compared with the camera matrix estimation methods, which verifies the feasibility of the new proposed mapping method.

Keywords:

Citation: Jianyuan Wang, Huanqiang Xu, Xinrui Hu, Biao Leng. IFKD: Implicit field knowledge distillation for single view reconstruction[J]. Mathematical Biosciences and Engineering, 2023, 20(8): 13864-13880. doi: 10.3934/mbe.2023617

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Abstract

1. Introduction

Middle East respiratory syndrome (MERS) is a viral respiratory disease caused by Middle East respiratory syndrome coronavirus (MERS-CoV). The intermediate host of MERS-CoV is probably the dromedary camel, a zoonotic virus ^[1]. Most MERS cases are acquired by human-to-human transmission. There is no vaccine or specific treatment available, and approximately $35\%$ of patients with MERS-CoV infection have died ^[2]. There has been extensive works on infectious disease models and viral infection models associated with MERS that can help in disease control and provide strategies for potential drug treatments ^{[3,4,5,6,7,8]}.

Dipeptidyl peptidase-4 (DPP4) plays an important role in viral infection ^[2]. Based on classic viral infection models developed in ^[9,10,11], a four-dimensional ordinary differential equation model is proposed and studied in ^[8]. The model in ^[8] describes the interaction mechanisms among uninfected cells, infected cells, DPP4 and MERS-CoV.

Recently, taking into account periodic factors such as diurnal temperature differences and periodic drug treatment, the model in ^[8] has been further extended a periodic case in ^[12], and then the existence of positive periodic solutions is studied by using the theorem in ^[13].

It is well-known that CTL immune responses play a very critical role in controlling viral load and the concentration of infected cells. Thus, many scholars have considered CTL immune responses in various viral infection models and have achieved many excellent research results ^{[14,15,16,17,18]}. CTL cells can kill virus-infected cells and are important for the control and clearance of MERS-CoV infections ^[19]. Inspired by the above research works, we consider the following periodic MERS-CoV infection model with CTL immune response:

$\begin{equation} \left\{ \begin{aligned} &\dot{T}(t) = \lambda(t)-\beta(t)D(t)v(t)T(t)-d(t)T(t), \\ &\dot{I}(t) = \beta(t)D(t)v(t)T(t)-d_{1}(t)I(t)-p(t)I(t)Z(t), \\ &\dot{v}(t) = d_{1}(t)M(t)I(t)-c(t)v(t), \\ &\dot{D}(t) = \lambda_{1}(t)-\beta_{1}(t)\beta(t)D(t)v(t)T(t)-\gamma(t)D(t), \\ &\dot{Z}(t) = q(t)I(t)Z(t)-b(t)Z(t). \end{aligned} \right. \end{equation}$

(1.1)

In model (1.1), $T(t)$ , $I(t)$ , $v(t)$ , $D(t)$ and $Z(t)$ represent the concentrations of uninfected cells, infected cells, free virus, DPP4 on the surface of uninfected cells and CTL cells at time $t$ , respectively. CTL cells increase at a rate bilinear rate $q(t)I(t)Z(t)$ by the viral antigen of the infected cells and decay at rate $b(t)Z(t)$ ; infected cells are killed by the CTL immune response at rate $p(t)I(t)Z(t)$ . Except for $p(t)$ , $q(t)$ and $b(t)$ , all the remaining parameters of model (1.1) have the same biological meanings as in ^[12].

Throughout the paper, it is assumed that the functions $\lambda(t)$ , $\beta(t)$ , $d(t)$ , $d_{1}(t)$ , $p(t)$ , $M(t)$ , $c(t)$ , $\lambda_{1}(t)$ , $\gamma(t)$ , $q(t)$ and $b(t)$ are positive, continuous and $\omega$ periodic ( $\omega > 0$ ); the function $\beta_{1}(t)$ is non-negative, continuous and $\omega$ periodic.

From point of view in both biology and mathematics, it is one of the most significant topics to study the existence of periodic oscillations of a system (see, for example, ^{[12,20,21,22,23,24,25,26]} and the references therein).

In the next section, some sufficient criteria are given for the existence of positive periodic oscillations of model (1.1). It should be mentioned here that, in the proofs of the main results in the following section, a new technique is developed to obtain a lower bound of the state variable $Z(t)$ characterizing CTL immune response in model (1.1).

2. Main results

For some function $f(t)$ which is continuous and $\omega$ -periodic on $\mathbb{R}$ , let us define the following notations:

$f^{U} = \max\limits_{t\in[0,\omega]}f(t), \; \; \; f^{l} = \min\limits_{t \in[0,\omega]} f(t), \; \; \; \widehat{f} = \frac{1}{\omega}\int_{0}^{\omega}f(t)dt.$

Moreover, for convenience, let us give the following parameters:

$\begin{align*} & R^{*} = \frac{\widehat{\lambda}\beta^{l}\exp\{L_{3}+L_{4}\}} {\widehat{d_{1}}\exp\{M_{2}\}(\beta^{l}\exp\{L_{3}+L_{4}\}+d^{U})} > 1, \; \; \; \omega^{*} = \frac{\widehat{b}}{2\widehat{\lambda}\widehat{q}}, \; \; \; \delta^{*} = \frac{\widehat{d_{1}}}{2\widehat{p}}(R^{*}-1),\\ &M_1 = \ln(\frac{\lambda^{U}}{d^{l}}), \; \; \; M_2 = \ln (\frac{\widehat{b}}{\widehat{q}}+2 \widehat{\lambda}\omega), \; \; \; M_3 = \ln(\frac{\widehat{(d_{1}M)}}{\widehat{c}})+M_2+2\widehat{c}\omega,\\ &M_4 = \ln(\frac{\lambda_{1}^{U}}{\gamma^{l}}), \; \; \; M_5 = \ln(\frac{\widehat{\beta}\exp\{M_{1}+M_{3}+M_{4}\}} {\widehat{p}\exp \{L_{2}\}})+2\widehat{b}\omega,\\ &{L}_1 = \ln(\frac{\lambda^{l}}{\beta^{U}\exp \{M_3+M_4\}+d^{U}}), \; \; \; {L}_2 = \ln (\frac{\widehat{b}}{\widehat{q}}-2\widehat{\lambda}\omega), \; \; \; {L}_3 = \ln(\frac{(\widehat{{d}_{1}{M}})}{\widehat{c}}) +L_{2}-2\widehat{c}\omega,\\ &{L}_4 = \ln(\frac{\lambda_{1}^{l}}{({\beta_1\beta})^{U} \exp\{M_1+M_3\}+\gamma^{U}}), \; \; \; {L}_5 = \ln(\delta^{*})-2\widehat{b}\omega. \end{align*}$

The following theorem is the main result of this paper.

Theorem 2.1. If $R^{*} > 1$ and $\omega < \omega^{*}$ , then model (1.1) has at least one positive $\omega$ -periodic solution.

Proof. Making the change of variables $T(t) = \exp\{u_1(t)\}$ , $I(t) = \exp\{u_2(t)\}$ , $v(t) = \exp\{u_3(t)\}$ , $D(t) = \exp\{u_4(t)\}$ , $Z(t) = \exp\{u_5(t)\}$ , then model (1.1) can be rewritten as

$\begin{equation} \left\{\begin{aligned} \dot{u}_1(t) = &\frac{\lambda(t)}{\exp\{u_1(t)\}}-\beta(t)\exp\{u_3(t)+u_4(t)\}-d(t),\\ \dot{u}_2(t) = &\beta(t)\frac{\exp \{u_1(t)+u_3(t)+u_4(t)\}}{\exp \{u_2(t)\}}-d_1(t)-p(t)\exp \{u_5(t)\}, \\ \dot{u}_3(t) = &d_1(t)M(t)\frac{\exp\{u_2(t)\}}{\exp\{u_3(t)\}}-c(t), \\ \dot{u}_4(t) = &\frac{\lambda_1(t)}{\exp\{u_4(t)\}}-\beta_1(t) \beta(t)\exp\{u_1(t)+u_3(t)\}-\gamma(t), \\ \dot{u}_5(t) = &q(t)\exp\{u_2(t)\}-b(t). \end{aligned} \right. \end{equation}$

(2.1)

Thus, we only need to consider model (2.1).

Let us set

$X = Y = \left\{ u = (u_1(t), u_2(t), u_3(t), u_4(t), u_5(t))^{T} \in C(\mathbb{R}, \mathbb{R}^{5})\; |\; u(t) = u(t+\omega) \right\}$

with the norm

$||u|| = \max\limits_{t\in[0,\omega]}|u_1(t)|+ \max\limits_{t\in[0,\omega]}|u_2(t)|+ \max\limits_{t\in[0,\omega]}|u_3(t)|+ \max\limits_{t\in[0,\omega]}|u_4(t)|+ \max\limits_{t\in[0,\omega]}|u_5(t)|.$

It can be shown that $X$ and $Y$ are Banach spaces. Define

$\begin{align*} &N u = \left[ \begin{array}{c} \frac{\lambda(t)}{\exp\{u_1(t)\}}-\beta(t)\exp\{u_3(t)+u_4(t)\}-d(t) \\ \beta(t)\frac{\exp \{u_1(t)+u_3(t)+u_4(t)\}}{\exp \{u_2(t)\}}-d_1(t)-p(t) \exp \{u_5(t)\}\\ d_1(t)M(t)\frac{\exp\{u_2(t)\}}{\exp \{u_3(t)\}}-c(t) \\ \frac{\lambda_1(t)}{\exp \{u_4(t)\}}-\beta_1(t) \beta(t) \exp \{u_1(t)+u_3(t)\}-\gamma(t) \\ q(t)\exp\{u_2(t)\}-b(t) \end{array} \right]: = \left[\begin{array}{c} N_1(t) \\ N_2(t) \\ N_3(t) \\ N_4(t) \\ N_5(t) \end{array} \right]\; (u\in X), \end{align*}$

$Lu = \dot{u}\; (u\in {\rm{Dom}}\; L),\; \; \; P u = \frac{1}{\omega}\int_{0}^{\omega}u(t)dt\; (u\in X),\; \; \; Q u = \frac{1}{\omega}\int_{0}^{\omega}u(t)dt\; (u\in Y),$

here ${\rm{Dom}}\; L = \{u\in X, \; \dot{u}\in X\}$ . It easily has that ${\rm{Ker}}\; L = \{u\in X\; |\; u\; \in \mathbb{R}^5\}$ and ${\rm{Im}}\; L = \{u\in Y\; |\; \int_{0}^{\omega}u(t)dt = 0\}$ . Further, it is clear that ${\rm{Im}}\; L$ is closed in $Y$ and ${\rm{dim\; Ker}}\; L = {\rm{codim\; Im}}\; L = 5$ . Hence, $L$ is a Fredholm mapping with index zero.

For $\mu\in(0, 1)$ , let us consider the equation $Lu = \mu Nu$ , i.e.,

$\begin{equation} \left\{\begin{aligned} \dot{u}_1(t) = &\mu \left[\frac{\lambda(t)}{\exp\{u_1(t)\}}-\beta(t)\exp \{u_3(t)+u_4(t)\}-d(t)\right],\\ \dot{u}_2(t) = &\mu \left[\beta(t)\frac{\exp \{u_1(t)+u_3(t)+u_4(t)\}}{\exp\{u_2(t)\}}-d_1(t)-p(t)\exp \{u_5(t)\}\right],\\ \dot{u}_3(t) = &\mu \left[d_1(t)M(t)\frac{\exp\{u_{2}(t)\}}{\exp\{u_3(t)\}}-c(t)\right], \\ \dot{u}_4(t) = &\mu \left[\frac{\lambda_1(t)}{\exp\{u_4(t)\}}-\beta_1(t)\beta(t)\exp \{u_1(t)+u_3(t)\}-\gamma(t)\right],\\ \dot{u}_5(t) = &\mu\left[q(t)\exp\{u_2(t)\}-b(t)\right]. \end{aligned} \right. \end{equation}$

(2.2)

For any solution $u = (u_1(t), u_2(t), u_3(t), u_4(t), u_5(t))^{T}\in X$ of (2.2), it has

$\begin{equation} \left\{ \begin{aligned} &\int_{0}^{\omega}\left[\frac{\lambda(t)}{\exp \{u_1(t)\}}-\beta(t)\exp \{u_3(t)+u_4(t)\}-d(t)\right]dt = 0, \\ &\int_{0}^{\omega}\left[\beta(t)\frac{\exp \{u_1(t)+u_3(t)+u_4(t)\}}{\exp \{u_2(t)\}}-d_1(t)-p(t)\exp\{u_5(t)\}\right]dt = 0, \\ &\int_{0}^{\omega}\left[d_1(t) M(t)\frac{\exp\{u_2(t)\}}{\exp \{u_3(t)\}}-c(t)\right]dt = 0, \\ &\int_{0}^{\omega}\left[\frac{\lambda_1(t)}{\exp\{u_4(t)\}}-\beta_1(t) \beta(t)\exp\{u_1(t)+u_3(t)\}-\gamma(t)\right]dt = 0, \\ &\int_{0}^{\omega}\left[q(t)\exp\{u_2(t)\}-b(t)\right]dt = 0 . \end{aligned} \right. \end{equation}$

(2.3)

From the first two equations in (2.2), it has

$\begin{equation*} \dot{u}_1(t)\exp \{u_1(t)\} = \mu \left[\lambda(t)-\beta(t)\exp \{u_1(t)+u_3(t)+u_4(t)\}-d(t)\exp \{u_1(t)\}\right], \end{equation*}$

and

$\begin{equation*} \dot{u}_2(t)\exp \{u_2(t)\} = \mu \left[\beta(t)\exp \{u_1(t)+u_3(t)+u_4(t)\}-d_{1}(t)\exp\{u_{2}(t)\}-p(t)\exp\{u_{2}(t)+u_5(t)\}\right]. \end{equation*}$

Hence, by integrating the above two equations on $[0, \omega]$ , it has

$\begin{equation} \int_{0}^{\omega} \left[\lambda(t)-\beta(t)\exp \{u_1(t)+u_3(t)+u_4(t)\}-d(t)\exp \{u_1(t)\}\right]dt = 0 \end{equation}$

(2.4)

and

$\begin{equation} \int_{0}^{\omega} \left[\beta(t)\exp \{u_1(t)+u_3(t)+u_4(t)\}-d_{1}(t)\exp\{u_{2}(t)\}-p(t)\exp\{u_{2}(t)+u_5(t)\}\right]dt = 0. \end{equation}$

(2.5)

Note that $I(t): = \exp\{u_{2}(t)\}$ satisfies

$\dot{I}(t) = \dot{u}_{2}(t)\exp\{u_{2}(t)\} = \mu\left[\beta(t)\exp \{u_1(t)+u_3(t)+u_4(t)\}-d_{1}(t)-p(t)\exp\{u_{2}(t)+u_5(t)\}\right].$

Then, from (2.4) and (2.5), it has

$\begin{equation} \begin{aligned} \int_{0}^{\omega}|\dot{I}(t)| dt \leq & \mu\int_{0}^{\omega}[\beta(t)\exp \{u_1(t)+u_3(t)+u_4(t)\}+d_{1}(t)+p(t)\exp\{u_{2}(t)+u_5(t)\}]dt\\ \leq & 2\int_{0}^{\omega}\beta(t)\exp \{u_1(t)+u_3(t)+u_4(t)\}dt\\ \leq & 2 \widehat{\lambda}\omega. \end{aligned} \end{equation}$

(2.6)

From the third and the fifth equations of (2.2), it has

$\begin{equation} \begin{aligned} &\int_{0}^{\omega}|\dot u_{3}(t)|dt \leq\mu \left[\int_{0}^{\omega}d_1(t)M(t)\frac{\exp \{u_2(t)\}}{\exp \{u_{3}(t)\}}dt+\int_{0}^{\omega} c(t)dt\right] < 2\widehat{c}\omega,\\ &\int_{0}^{\omega}|\dot u_5(t)|dt \leq \mu \left[\int_{0}^{\omega} q(t)\exp \{u_2(t)\}dt+\int_{0}^{\omega}b(t)dt\right] < 2\widehat{b}\omega. \end{aligned} \end{equation}$

(2.7)

Note that $u\in X$ , there exist $\xi_{i}, \; \eta_{i}\in[0, \omega]$ $(i = 1, 2, 3, 4, 5)$ , such that

$u_{i}(\xi_{i}) = \min\limits_{t\in[0,\omega]}u_i(t), \; \; u_i(\eta_{i}) = \max\limits_{t\in[0,\omega]}u_i(t)\; (i = 1,2,3,4,5).$

From (2.2), $\dot{u}_{1}(\eta_{1}) = 0$ and $\dot{u}_{4}(\eta_{4}) = 0$ , it has

$\begin{align*} &\frac{\lambda(\eta_1)}{\exp\{u_1(\eta_1)\}}-\beta(\eta_1) \exp \{u_3(\eta_1)+u_4(\eta_1)\}-d(\eta_1) = 0,\\ &\frac{\lambda_1(\eta_4)}{\exp\{u_4(\eta_4)\}} -\beta_1(\eta_4)\beta(\eta_4) \exp \{u_1(\eta_4)+u_3(\eta_4)\}-\gamma(\eta_4) = 0, \end{align*}$

which imply that

$\begin{equation} \begin{aligned} & u_1(t)\leq u_{1}(\eta_{1})\leq \ln\left(\frac{\lambda(\eta_{1})}{d(\eta_{1})}\right) \leq\ln \left(\frac{\lambda^{U}}{d^l}\right) = {M}_1,\\ & u_4(t)\leq u_{4}(\eta_{4})\leq \ln\left(\frac{\lambda_{1}(\eta_{4})}{\gamma(\eta_{4})}\right) \leq \ln\left(\frac{\lambda_{1}^{U}}{\gamma^{l}}\right) = M_{4}. \end{aligned} \end{equation}$

(2.8)

From the last equation of (2.3), it has

$\int_{0}^{\omega}q(t)\exp\{u_{2}(\xi_{2})\}dt\leq\widehat{b}\omega \leq \int_{0}^{\omega}q(t)\exp\{u_{2}(\eta_{2})\}dt,$

which implies that

$I(\xi_{2}) = \exp\{u_{2}(\xi_{2})\}\leq \frac{\widehat{b}}{\widehat{q}} \leq \exp\{u_{2}(\eta_{2})\} = I(\eta_{2}).$

Then, from (2.6) and $\omega < \omega^{*}$ , it has

$\begin{align*} & I(t)\leq I(\xi_2)+\int_{0}^{\omega}|\dot{I}(t)|dt \leq\frac{\widehat{b}}{\widehat{q}}+2\widehat{\lambda}\omega,\\ & I(t)\geq I(\eta_2)-\int_{0}^{\omega}|\dot{I}(t)|dt\geq \frac{\widehat{b}}{\widehat{q}}-2\widehat{\lambda}\omega = 2\widehat{\lambda}\left(\omega^{*}-\omega\right) > 0. \end{align*}$

Thus, it has

$\begin{equation} \begin{aligned} &u_2(t)\leq\ln\left(\frac{\widehat{b}}{\widehat{q}}+2\widehat{\lambda}\omega\right) = {M}_{2}, \; \; \; u_2(t)\geq\ln\left(\frac{\widehat{b}}{\widehat{q}}-2\widehat{\lambda}\omega\right) = {L}_{2}. \end{aligned} \end{equation}$

(2.9)

From the third equation of (2.3), it has

$\int_{0}^{\omega}d_1(t)M(t)\frac{\exp\{M_2\}} {\exp\{u_3(\xi_3)\}}dt \geq \widehat{c}\omega \geq \int_{0}^{\omega}d_1(t)M(t)\frac{\exp\{L_2\}} {\exp\{u_3(\eta_3)\}}dt,$

which implies that

$u_3(\xi_3)\leq\ln\left(\frac{\widehat{(d_{1}M)}}{\widehat{c}}\right)+M_{2}, \; \; \; u_3(\eta_3)\geq \ln\left(\frac{\widehat{(d_{1}M)}}{\widehat{c}}\right)+L_{2}.$

Then, from (2.7), it has

$\begin{equation} \begin{aligned} &u_3(t)\leq u_3(\xi_3)+\int_{0}^{\omega}|\dot{u}_3(t)| dt \leq\ln\left(\frac{\widehat{(d_1{M})}}{\widehat{c}}\right)+M_{2}+2\widehat{c}\omega = {M}_{3},\\ &u_3(t)\geq u_3(\eta_3)-\int_{0}^{\omega}|\dot{u}_3(t)| dt \geq\ln\left(\frac{\widehat{(d_1{M})}}{\widehat{c}}\right)+L_{2} -2\widehat{c}\omega = {L}_{3}. \end{aligned} \end{equation}$

(2.10)

From the second equation of (2.3), it has

$\widehat{p}\exp\{u_5(\xi_{5})\}\omega\leq \int_{0}^{\omega}\left[\beta(t)\frac{\exp\{M_{1}+M_{3}+M_{4}\}} {\exp \{L_{2}\}}-d_1(t)\right]dt\leq \frac{\exp\{M_{1}+M_{3}+M_{4}\}} {\exp \{L_{2}\}}\widehat{\beta}\omega,$

which implies that

$u_5(\xi_{5})\leq \ln\left( \frac{\widehat{\beta}\exp\{M_{1}+M_{3}+M_{4}\}} {\widehat{p}\exp \{L_{2}\}}\right): = l_{5}.$

Then, from (2.7), it has

$u_{5}(t)\leq u_{5}(\xi_{5})+\int_{0}^{\omega}|\dot{u}_{5}(t)| dt \leq l_{5}+2 \widehat{b}\omega = {M}_5.$

From $\dot{u}_1(\xi_1) = 0$ , $\dot{u}_4(\xi_4) = 0$ , (2.8) and (2.10), it has

$\begin{align*} &\exp\{u_1(\xi_1)\} = \frac{\lambda(\xi_1)}{\beta(\xi_1) \exp\{u_3(\xi_1)+u_4(\xi_1)\}+d(\xi_1)} \geq\frac{\lambda^l}{\beta^U\exp\{M_3+M_4\}+d^U},\\ &\exp\{u_4(\xi_4)\} = \frac{\lambda_1(\xi_4)}{\beta_1(\xi_4) \beta(\xi_4)\exp\{u_1(\xi_4)+u_3(\xi_4)\}+\gamma(\xi_4)} \geq \frac{\lambda_1^l}{(\beta_{1}\beta)^U\exp\{M_1+M_3\}+\gamma^U}. \end{align*}$

Thus, it has

$\begin{equation} \begin{aligned} &u_{1}(t)\geq u_{1}\left(\xi_{1}\right)\geq \operatorname{ln}\left(\frac{\lambda^{l}}{\beta^{U} \exp \left\{M_{3}+M_{4}\right\}+d^{U}}\right) = L_{1},\\ &u_4(t)\geq u_4(\xi_4) = \ln\left(\frac{\lambda_{1}^l}{(\beta_1\beta)^U \exp \{M_1+M_3\}+\gamma^U}\right) = L_4. \end{aligned} \end{equation}$

(2.11)

Let us give an estimate of the lower bound of the state variable $u_5(t)$ related to CTL immune response. It should be mentioned here that a completely different method from that in ^[12] has been used.

Claim A If $R^{*} > 1$ and $\omega < \omega^{*}$ , then

$\exp\{u_5(\eta_{5})\}\geq \delta^{*}.$

If Claim A is not true, then it has that, for any $t$ , $\exp\{u_5(t)\} \leq \exp\{u_5(\eta_{5})\} < \delta^{*}$ . Hence, it has from (2.3), (2.9)–(2.11) that

$\begin{align*} 0 = &\int_{0}^{\omega}\left[\beta(t)\frac{\exp \{u_1(t)+u_3(t)+u_4(t)\}}{\exp \{u_2(t)\}}-d_1(t)-p(t)\exp\{u_5(t)\}\right]dt\\ \geq&\int_{0}^{\omega}\left[\beta(t)\frac{\exp \{u_1(t)+L_{3}+L_4\}} {\exp \{M_{2}\}}-d_1(t)-p(t)\exp \{u_5(\eta_{5})\}\right]dt\\ \geq&\frac{\beta^{l}\exp\{L_3+L_{4}\}}{\exp \{M_2\}}\int_{0}^{\omega}\exp\{u_{1}(t)\}dt -(\widehat{d}_1+\widehat{p} \delta^{*})\omega, \end{align*}$

which implies that

$\begin{equation} \int_{0}^{\omega} d(t)\exp\{u_{1}(t)\}dt \leq d^{U}\int_{0}^{\omega}\exp\{u_{1}(t)\}dt \leq \frac{ d^{U}(\widehat{d}_1+\widehat{p} \delta^{*}) \exp \{M_2\}}{\beta^{l}\exp\{L_3+L_{4}\}}\omega : = \Psi\omega. \end{equation}$

(2.12)

Adding (2.4) and (2.5) together, it has

$\begin{align*} \int_{0}^{\omega} [\lambda(t)-d(t)\exp\{u_{1}(t)\}]dt = &\int_{0}^{\omega}[d_{1}(t)\exp\{u_{2}(t)\}+p(t)\exp\{u_{2}(t)+u_{5}(t)\}]dt\\ \leq &\int_{0}^{\omega}\exp\{M_{2}\}[d_{1}(t)+p(t)\exp\{u_{5}(\eta_{5})\}]dt\\ \leq &\exp\{M_{2}\}(\widehat{d_{1}}+\widehat{p}\delta^{*})\omega, \end{align*}$

which implies that

$\begin{align*} \int_{0}^{\omega} d(t)\exp\{u_{1}(t)\}dt \geq & \left[\widehat{\lambda}-\exp\{M_{2}\}(\widehat{d_{1}}+\widehat{p}\delta^{*})\right]\omega\\ = & \Psi \omega+ \left[\widehat{\lambda}-\Psi-\exp\{M_{2}\}(\widehat{d_{1}}+\widehat{p}\delta^{*})\right]\omega\\ = &\Psi \omega+ \left[\widehat{\lambda}-\exp\{M_{2}\} \left( 1+\frac{d^{U}}{\beta^{l}\exp\{L_{3}+L_{4}\}} \right) (\widehat{d_{1}}+\widehat{p}\delta^{*})\right]\omega\\ = &\Psi \omega+\widehat{d_{1}}\exp\{M_{2}\} \left( 1+\frac{d^{U}}{\beta^{l}\exp\{L_{3}+L_{4}\}} \right) \left(R^{*}-1 -\frac{\widehat{p}}{\widehat{d_{1}}}\delta^{*}\right)\omega\\ = & \Psi \omega+\frac{\widehat{d_{1}}}{2}\exp\{M_{2}\} \left( 1+\frac{d^{U}}{\beta^{l}\exp\{L_{3}+L_{4}\}} \right) (R^{*}-1)\omega\\ > &\Psi \omega, \end{align*}$

which is a contradiction to (2.12). Thus, the claim holds.

From Claim A and (2.7), it has

$\begin{equation} u_5(t)\geq u_5(\eta_5)-\int_{0}^{\omega}|\dot{u}_5(t)|dt \geq\ln(\delta^{*})-2\widehat{b}\omega = L_5. \end{equation}$

(2.13)

Now, for convenience, let us define

$\overline{R^{*}} = \left(\widehat{\lambda}-\frac{\widehat{d_{1}}\widehat{b}}{\widehat{q}}\right) \frac{\widehat{\beta}\widehat{(d_{1}M)}}{\widehat{d}\widehat{d_{1}}\widehat{c}} \frac{\widehat{\lambda_{1}}}{\widehat{({\beta_1\beta})} \frac{\widehat{\lambda}\widehat{(d_{1}M)} \widehat{b}}{\widehat{d}\widehat{c}\widehat{q}}+\widehat{\gamma}}, \; \; \; Z_{max} = \frac{\widehat{q}}{\widehat{p}\widehat{b}} \left(\widehat{\lambda}-\frac{\widehat{d_{1}}\widehat{b}}{\widehat{q}}\right).$

Note that if $R^{*} > 1$ , then it has

$\widetilde{R^{*}}: = \left(\widehat{\lambda}-\widehat{d_{1}}\exp\{M_{2}\}\right) \frac{\beta^{l}\exp\{L_{3}+L_{4}\}}{d^{U}\widehat{d_{1}}\exp\{M_{2}\}} > 1,$

which implies that

$\begin{align*} Z_{max} > &\frac{\widehat{q}}{\widehat{p}\widehat{b}} \left[\widehat{\lambda}-\widehat{d_{1}} \left(\frac{\widehat{b}}{\widehat{q}}+2\widehat{\lambda}\omega\right)\right] = \frac{\widehat{q}}{\widehat{p}\widehat{b}} \left(\widehat{\lambda}-\widehat{d_{1}}\exp\{M_{2}\}\right) > 0,\\ \overline{R^{*}}\geq& \left(\widehat{\lambda}-\frac{\widehat{d_{1}}\widehat{b}}{\widehat{q}}\right) \frac{\widehat{\beta}\widehat{(d_{1}M)}}{\widehat{d}\widehat{d_{1}}\widehat{c}} \frac{\lambda_{1}^{l}}{({\beta_1\beta})^{U} \exp\{M_3+M_1\}+\gamma^{U}}\geq \widetilde{R^{*}} > 1. \end{align*}$

Let $(u_{1}, u_{2}, u_{3}, u_{4}, u_{5})^{T}\in \mathbb{R}^{5}$ be the solution of the following equations:

$\begin{equation} \left\{ \begin{aligned} &\frac{\widehat{\lambda}}{\exp\{u_1\}}-\widehat{\beta}\exp \{u_3+u_4\}-\widehat{d} = 0,\\ &\frac{\widehat{\beta}\exp \{u_1+u_3+u_4\}}{\exp\{u_2\}}-\widehat{d}_1-\widehat{p}\exp\{u_5\} = 0,\\ &\widehat{(d_{1}M)}\frac{\exp\{u_2\}}{\exp\{u_3\}}-\widehat{c} = 0, \\ &\frac{\widehat{\lambda}_1}{\exp\{u_4\}}-\widehat{({\beta}_{1}{\beta})} \exp \{u_1+u_3\}-\widehat{\gamma} = 0, \\ &\widehat{q}\exp\{u_2\}-\widehat{b} = 0. \end{aligned}. \right. \end{equation}$

(2.14)

Define $\Gamma : [0, Z_{max}] \rightarrow \mathbb{R}$ , via

$\Gamma(x) = \frac{\widehat{\beta}\widehat{(d_{1}M)}}{\widehat{c}} \frac{\widehat{\lambda_{1}}\Gamma_{1}(x)} {\widehat{(\beta_{1}\beta)}\Gamma_{1}(x)\frac{\widehat{(d_{1}M)}\widehat{b}}{\widehat{q}\widehat{c}} +\widehat{\gamma}} -\widehat{d}_1-\widehat{p}x,$

where

$\Gamma_{1}(x) = \frac{\widehat{\lambda}}{\widehat{d}} -\frac{\widehat{d_{1}}\widehat{b}}{\widehat{d}\widehat{q}} -\frac{\widehat{p}\widehat{b}}{\widehat{d}\widehat{q}}x = \frac{\widehat{p}\widehat{b}}{\widehat{d}\widehat{q}}(Z_{max}-x).$

Equation (2.14) can be rewritten as

$\begin{align*} &\exp\{u_{2}\} = \frac{\widehat{b}}{\widehat{q}}, \; \; \; \exp\{u_{3}\} = \frac{\widehat{(d_{1}M)}}{\widehat{c}}\exp\{u_{2}\} = \frac{\widehat{(d_{1}M)}\widehat{b}}{\widehat{q}\widehat{c}},\\ &\exp\{u_{1}\} = \frac{\widehat{\lambda}}{\widehat{d}} -\frac{\widehat{d_{1}}\exp\{u_{2}\}}{\widehat{d}} -\frac{\widehat{p}\exp\{u_{2}\}}{\widehat{d}}\exp\{u_{5}\} = \Gamma_{1}(\exp\{u_{5}\}),\\ &\exp\{u_{4}\} = \frac{\widehat{\lambda_{1}}} {\widehat{(\beta_{1}\beta)}\exp\{u_{1}+u_{3}\}+\widehat{\gamma}} = \frac{\widehat{\lambda_{1}}} {\widehat{(\beta_{1}\beta)}\Gamma_{1}(\exp\{u_{5}\})\frac{\widehat{(d_{1}M)}\widehat{b}}{\widehat{q}\widehat{c}} +\widehat{\gamma}},\\ &\frac{\widehat{\beta}\widehat{(d_{1}M)}}{\widehat{c}} \exp \{u_1+u_4\} -\widehat{d}_1-\widehat{p}\exp\{u_5\} = \Gamma(\exp\{u_5\}) = 0. \end{align*}$

It is obvious that if there is a solution $(u_{1}, u_{2}, u_{3}, u_{4}, u_{5})^{T}\in \mathbb{R}^{5}$ for (2.14), it must have $0 < \exp\{u_5\} < Z_{max}$ . In addition, note that $\Gamma(x)$ is monotonically decreasing with respect to $x$ on $[0, Z_{max}]$ . It has from $\Gamma\left(Z_{max}\right) = -\widehat{d_{1}}-\widehat{p}Z_{max} < 0$ and

$\Gamma(0) = \frac{\widehat{\beta}\widehat{(d_{1}M)}}{\widehat{c}} \frac{\widehat{\lambda_{1}}\Gamma_{1}(0)} {\widehat{(\beta_{1}\beta)}\Gamma_{1}(0) \frac{\widehat{(d_{1}M)}\;\widehat{b}}{\widehat{q}\widehat{c}} +\widehat{\gamma}}-\widehat{d_1} > \frac{\widehat{\beta}\widehat{(d_{1}M)}}{\widehat{c}} \frac{\widehat{\lambda_{1}}(\frac{\widehat{\lambda}}{\widehat{d}} -\frac{\widehat{d_{1}}\widehat{b}}{\widehat{d}\widehat{q}})} {\widehat{(\beta_{1}\beta)}\frac{\widehat{\lambda}}{\widehat{d}} \frac{\widehat{(d_{1}M)}\;\widehat{b}}{\widehat{q}\widehat{c}} +\widehat{\gamma}}-\widehat{d_1} = \widehat{d_{1}}(\overline{R^{*}}-1) > 0$

that there exists a unique positive constant $x = Z^{*}\in(0, Z_{max})$ such that $\Gamma(Z^{*}) = 0$ .

The above discussions show that, if $R^{*} > 1$ , (2.14) has a unique solution $(u_{1}^{*}, u_{2}^{*}, u_{3}^{*}, u_{4}^{*}, u_{5}^{*})^{T}$ , here $u_i^* = \ln(e_{i})$ $(i = 1, 2, 3, 4, 5)$ ,

$e_{1} = \Gamma_{1}(Z^{*}) > 0,\; \; e_{2} = \frac{\widehat{b}}{\widehat{q}} > 0,\; \; e_{3} = \frac{\widehat{(d_{1}M)}\widehat{b}}{\widehat{q}\widehat{c}} > 0,\; \; e_{4} = \frac{\widehat{\lambda_{1}}} {\widehat{(\beta_{1}\beta)}\Gamma_{1}(Z^{*}) \frac{\widehat{(d_{1}M)}\;\widehat{b}}{\widehat{q}\widehat{c}} +\widehat{\gamma}} > 0,\; \; e_{5} = Z^{*} > 0.$

Let us define the following set

$\Omega = \{u\in X\; |\; ||u|| < U_{1} = 1+\sum\limits_{i = 1}^{5} (\max\{|M_{i}|,|L_{i}|\}+|u_{i}^{*}|)\}\subset X.$

Moreover, by similar arguments as in ^[12], it has that $N$ is $L$ -compact on $\overline{\Omega}$ .

Now, let us compute the Leray-Schauder degree ${\rm{deg}}\left\{QN, \partial\Omega\cap Ker\; L, (0, 0, 0, 0, 0)^{T}\right\} : = \Delta$ as follows,

$\begin{align*} \Delta = &{\rm{sign}}\left|\begin{array}{ccccc} -\frac{\widehat{\lambda}}{e_{1}}&0&-\widehat{\beta} e_{3}e_{4} &-\widehat{\beta} e_{3}e_{4}&0\\ \widehat{\beta}\frac{e_{1}e_{3}e_{4}}{e_{2}} &-\widehat{\beta} \frac{e_{1}e_{3}e_{4}}{e_{2}} &\widehat{\beta}\frac{e_{1}e_{3}e_{4}}{e_{2}}& \widehat{\beta}\frac{e_{1}e_{3}e_{4}}{e_{2}}&-\widehat{p}e_{5}\\ 0& \widehat{(d_{1}M)}\frac{e_{2}}{e_{3}}&-\widehat{(d_{1}M)}\frac{e_{2}}{e_{3}}&0&0\\ -\widehat{(\beta_{1}\beta)} e_{1}e_{3} &0 &-\widehat{(\beta_{1}\beta)} e_{1}e_{3}&-\frac{\widehat{\lambda_{1}}}{e_{4}}&0 \\ 0&\widehat{q}e_{2}&0&0&0 \end{array}\right|\\ = &{\rm{sign}}\left\{ -\widehat{(d_{1}M)}\widehat{p}\widehat{q}\frac{e_{2}^{2}e_{5}}{e_{3}} \left(\frac{\widehat{\lambda}\widehat{\lambda_{1}}}{e_{1}e_{4}} -\widehat{\beta}\widehat{(\beta_{1}\beta)}e_{1}e_{3}^{2}e_{4} \right)\right\}\\ = &{\rm{sign}}\left\{ -\widehat{(d_{1}M)}\widehat{p}\widehat{q}\frac{e_{2}^{2}e_{5}}{e_{3}e_{1}e_{4}} \left[\widehat{d}e_{1} \left(\widehat{(\beta_{1}\beta)}e_{1}e_{3}e_{4}+\widehat{\gamma}e_{4}\right) +\widehat{\beta}\widehat{\gamma}e_{1}e_{3}e_{4}^{2} \right]\right\}\\ = &-1\neq0, \end{align*}$

where $\widehat{\lambda} = \widehat{\beta}e_{1}e_{3}e_{4}+\widehat{d}e_{1}$ and $\widehat{\lambda_{1}} = \widehat{(\beta_{1}\beta)}e_{1}e_{3}e_{4}+\widehat{\gamma}e_{4}$ are used.

Finally, it has those all the conditions of the continuation theorem in ^[13] (also see, for example, Lemma 2.1 in ^[12]) are satisfied. This proves that, if $\omega < \omega^{*}$ and $R^{*} > 1$ , model (2.1) has at least one $\omega$ -periodic solution.

Let us consider the following classical viral infection dynamic model ^[9] with CTL immune response:

$\left\{ \begin{aligned} &\dot{T}(t) = \lambda(t)-\beta(t)v(t)T(t)-d(t)T(t), \\ &\dot{I}(t) = \beta(t)v(t)T(t)-d_{1}(t)I(t)-p(t)I(t)Z(t), \\ &\dot{v}(t) = d_{1}(t)M(t)I(t)-c(t)v(t), \\ &\dot{Z}(t) = q(t)I(t)Z(t)-b(t)Z(t), \end{aligned} \right. \;\;\;\;\;\; ({\rm{A}})$

where, all the coefficients are the same with that in model (1.1).

Define $R_{1} : [0, \omega^{*}] \rightarrow \mathbb{R}$ , via

$R_{1}(x) = \frac{\widehat{\lambda}\beta^{l} \left[\frac{\widehat{(d_{1}M)}}{\widehat{c}}\exp\{-2\widehat{c}x\} \left(\frac{\widehat{b}}{\widehat{q}}-2\widehat{\lambda}x\right)\right] }{\widehat{d_{1}}\left(\frac{\widehat{b}}{\widehat{q}}+2\widehat{\lambda}x\right) \left\{d^{U}+\beta^{l} \left[\frac{\widehat{(d_{1}M)}}{\widehat{c}}\exp\{-2\widehat{c}x\} \left(\frac{\widehat{b}}{\widehat{q}}-2\widehat{\lambda}x\right)\right]\right\}}.$

Obviously, $R_1(x)$ is monotonically decreasing on $[0, \omega^{*}]$ and

$R_{1}(0) = \frac{\widehat{\lambda}\beta^{l}\widehat{(d_{1}M)}\widehat{q}} { \widehat{d_{1}}(d^{U}\widehat{c}\widehat{q}+ \beta^{l}\widehat{(d_{1}M)}\widehat{b}) },\; \; \; R_{1}(\omega^{*}) = 0.$

Therefore, if $R_{1}(0) > 1$ , then there exists a unique constant $\omega^{**}\in(0, \omega^{*})$ such that $R_{1}(\omega^{**}) = 1$ , $R_{1}(x) > 1$ for $0\leq x < \omega^{**}$ and $R_{1}(x) < 1$ for $\omega^{**} < x\leq\omega^{*}$ .

For model (A), it is not difficult to derive the following result.

Theorem 2.2. If $R_{1}(\omega) > 1$ and $\omega < \omega^{*}$ (i.e. $R_{1}(0) > 1$ and $\omega < \omega^{**} < \omega^{*}$ ), then model (A) has at least one positive $\omega$ -periodic solution.

Remark 2.1. If all the coefficients in model (A) take constants values, i.e., $\lambda(t)\equiv\lambda > 0$ , $\beta(t)\equiv\beta > 0$ , $d(t)\equiv d > 0$ , $d_{1}(t)\equiv d_{1} > 0$ , $p(t)\equiv p > 0$ , $M(t)\equiv M > 0$ , $c(t)\equiv c > 0$ , $q(t)\equiv q > 0$ and $b(t)\equiv b > 0$ , then model (A) becomes the classical model which is first proposed by Nowak and Bangham in ^[9]. the condition $\omega < \omega^{**}$ in Theorem 2.2 is naturally satisfied. Furthermore, it has $R_{1}(0) = (\lambda\beta M q)/(dcq+\beta d_{1}Mb): = R_{1}$ . From \emph{^[9]}, it has that the condition $R_{1} > 1$ implies the existence of a unique positive equilibrium. This shows that the conditions and conclusion in Theorem 2.2 are reasonable.

3. Conclusions and simulations

In summary, Theorem 2.1 in the paper successfully extends the main result in ^[12]) to a MERS-CoV viral infection model with CTL immune response. In the proof of Theorem 2.1, we use a very different method from that in ^[9] to obtain the lower bound $(\ln(\delta^{*})-2\widehat{b}\omega)$ of the state variable $u_5(t)$ . Furthermore, as a special case, Theorem 2.2 gives sufficient conditions for the existence of positive periodic solution of model (A). Model (A) is a natural extension of the classical model in ^[9]. As the end of the paper, let us give a example to summarize the applications of Theorem 2.1. Let us choose the coefficients in model (1.1) as follows (for the values of some parameters, please refer to ^[7,27] for the case of some autonomous models), $\lambda(t) = 45(1+0.1\sin(4\pi t))$ , $\beta(t) = 1.4\times 10^{-8}(1+0.1\cos(4\pi t))$ , $d(t) = 0.001(1+0.5\cos(4\pi t))$ , $d_{1}(t) = 0.056(1+0.5\cos(4\pi t))$ , $p(t) = 0.00092(1+0.5\cos(4\pi t))$ , $M(t) = 100000$ , $c(t) = 2.1(1+0.3\cos(4\pi t))$ , $\lambda_{1}(t) = 10(1+0.1\sin(4\pi t))$ , $\beta_{1}(t) = 0.001$ , $\gamma(t) = 0.01(1+0.1\cos(4\pi t))$ , $q(t) = 0.005(1+0.5\sin(4\pi t))$ , $b(t) = 0.5(1+0.4\cos(4\pi t))$ . Then, with the help of Maple mathematical software, it has $\omega = 0.5 < \omega^{*}\approx1.111111$ , $M_{1}\approx11.502875$ , $M_2\approx4.976734$ , $M_3\approx14.965318$ , $M_{4}\approx7.108426$ , $M_{5}\approx18.976215$ , $L_{1}\approx-0.383569$ , $L_{2} = 4.007333$ , $L_{3}\approx9.795917$ , $L_{4}\approx0.623402$ , $L_{5}\approx1.387881$ , $R^{*}\approx1.2170332 > 1$ . From Theorem 2.1, it follows that model (1.1) has at least one positive $\omega$ ( $\omega = 0.5$ )-periodic solution. gives the corresponding numerical simulation, and the initial value is chosen as $(T(0), I(0), v(0), D(0), Z(0))^T = (12.5,100, 265000,995.4,423)^{T}$ .

Figure 1. With the increasing of the time

$t$ , the evolution form of the solution of model (1.1).

DownLoad: Full-Size Img PowerPoint

Acknowledgments

This paper is supported by National Natural Science Foundation of China (No.11971055) and Beijing Natural Science Foundation (No.1202019).

Conflict of interest

The authors declare there is no conflict of interest.

References

[1]	M. Li, H. Zhang, D2im-net: Learning detail disentangled implicit fields from single images, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2021), 10246–10255. https://doi.org/10.1109/CVPR46437.2021.01011
[2]	S. Saito, Z. Huang, R. Natsume, S. Morishima, A. Kanazawa, H. Li, Pifu: Pixel-aligned implicit function for high-resolution clothed human digitization, in Proceedings of the IEEE/CVF International Conference on Computer Vision, (2019), 2304–2314. https://doi.org/10.1109/ICCV.2019.00239
[3]	T. Groueix, M. Fisher, V. G. Kim, B. C. Russell, M. Aubry, A papier-mâché approach to learning 3d surface generation, in Proceedings of the IEEE conference on computer vision and pattern recognition, (2018), 216–224. https://doi.org/10.1109/CVPR.2018.00030
[4]	C. B. Choy, D. Xu, J. Gwak, K. Chen, S. Savarese, 3d-r2n2: A unified approach for single and multi-view 3d object reconstruction, Eur. Conf. Comput. Vision, 9912 (2016), 628–644. https://doi.org/10.1007/978-3-319-46484-8_38 doi: 10.1007/978-3-319-46484-8_38
[5]	G. Yang, X. Huang, Z. Hao, M. Liu, S. Belongie, B. Hariharan, Pointflow: 3d point cloud generation with continuous normalizing flows, in Proceedings of the IEEE/CVF International Conference on Computer Vision, (2019), 4541–4550. https://doi.org/10.1109/ICCV.2019.00464
[6]	G. Chen, W. Choi, X. Yu, T. Han, M. Chandraker, Learning efficient object detection models with knowledge distillation, Adv. Neural Inform. Proc. Syst., (2017), 30.
[7]	J. J. Park, P. Florence, J. Straub, R. Newcombe, S. Lovegrove, Deepsdf: Learning continuous signed distance functions for shape representation, in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, (2019), 165–174. https://doi.org/10.1109/CVPR.2019.00025
[8]	G. Hinton, O. Vinyals, J. Dean, Distilling the knowledge in a neural network, preprint, arXiv: 1503.02531.
[9]	N. Wang, Y. Zhang, Z. Li, Y. Fu, W. Liu, Y. Jiang, Pixel2mesh: Generating 3d mesh models from single rgb images. in Proceedings of the European conference on computer vision (ECCV), (2018), 52–67. https://doi.org/10.1007/978-3-030-01252-6_4
[10]	H. Xie, H. Yao, X. Sun, S. Zhou, S. Zhang, Pix2vox: Context-aware 3d reconstruction from single and multi-view images, preprint, arXiv: 1901.11153.
[11]	H. Xie, H. Yao, S. Zhang, S. Zhou, W. Sun, Pix2vox++: Multi-scale context-aware 3d object reconstruction from single and multiple images, Int. J. Comput. Vision, 128 (2020), 2919–2935. https://doi.org/10.1007/s11263-020-01347-6 doi: 10.1007/s11263-020-01347-6
[12]	L. P. Tchapmi, V. Kosaraju, H. Rezatofighi, I. Reid, S. Savarese, Topnet: Structural point cloud decoder, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2019), 383–392. https://doi.org/10.1109/CVPR.2019.00047
[13]	L. Mescheder, M. Oechsle, M. Niemeyer, S. Nowozin, A. Geiger, Occupancy networks: Learning 3d reconstruction in function space. in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, 4460–4470. https://doi.org/10.1109/CVPR.2019.00459
[14]	Z. Chen, A. Tagliasacchi, H. Zhang, Bsp-net: Generating compact meshes via binary space partitioning, preprint, arXiv: 1911.06971.
[15]	Q. Xu, W. Wang, D. Ceylan, R. Mech, U. Neumann, Disn: Deep implicit surface network for high-quality single-view 3d reconstruction, Adv. Neural Inform. Process. Syst., (2019), 32.
[16]	J. Bechtold, M. Tatarchenko, V. Fischer, T. Brox, Fostering generalization in single-view 3d reconstruction by learning a hierarchy of local and global shape priors, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2021), 15880–15889. https://doi.org/10.1109/CVPR46437.2021.01562
[17]	Y. Duan, H. Zhu, H. Wang, L. Yi, R. Nevatia, L. J. Guibas, Curriculum deepsdf, Eur. Conf. Comput. Vision, (2020), 51–67. https://doi.org/10.1007/978-3-030-58598-3_4 doi: 10.1007/978-3-030-58598-3_4
[18]	A. Romero, N. Ballas, S. E. Kahou, A. Chassang, C. Gatta, Y. Bengio, Fitnets: Hints for thin deep nets, preprint, arXiv: 1412.6550.
[19]	Q. Li, S. Jin, J. Yan, Mimicking very efficient network for object detection, in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2017), 7341–7349. https://doi.org/10.1109/CVPR.2017.776
[20]	T. Wang, L. Yuan, X. Zhang, J. Feng, Distilling object detectors with fine-grained feature imitation, preprint, arXiv: 1906.03609.
[21]	W. E. Lorensen, H. E. Cline, Marching cubes: A high resolution 3d surface construction algorithm, ACM SIGGRAPH Comput. Graphics, 21 (1987), 163–169. https://doi.org/10.1145/37402.37422 doi: 10.1145/37402.37422
[22]	O. Ronneberger, P. Fischer, T. Brox, U-net: Convolutional networks for biomedical image segmentation, Medical Image Computing and Computer-Assisted Intervention, Springer, 9351 (2015), 234–241. https://doi.org/10.1007/978-3-319-24574-4_28
[23]	T. Hastie, R. Tibshirani, J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer, 2009.
[24]	Z. Chen, H. Zhang, Learning implicit fields for generative shape modeling, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2019), 5932-5941. https://doi.org/10.1109/CVPR.2019.00609
[25]	C. Häne, S. Tulsiani, J. Malik, Hierarchical surface prediction for 3d object reconstruction, preprint, arXiv: 1704.00710.
[26]	B. Xu, N. Wang, T. Chen, M. Li, Empirical evaluation of rectified activations in convolutional network, preprint, arXiv: 1505.00853.
[27]	J. Chibane, T. Alldieck, G. Pons-Moll, Implicit functions in feature space for 3d shape reconstruction and completion, in 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2020), 6968–6979. https://doi.org/10.1109/CVPR42600.2020.00700
[28]	J. Jin, A. G. Patil, Z. Xiong, H. Zhang, Dr-kfs: A differentiable visual similarity metric for 3d shape reconstruction, Eur. Conf. Comput. Vision, (2020), 295–311. https://doi.org/10.1007/978-3-030-58589-1_18 doi: 10.1007/978-3-030-58589-1_18
[29]	B. Yang, S. Wang, A. Markham, N. Trigoni, Robust attentional aggregation of deep feature sets for multi-view 3d reconstruction, Int. J. Comput. Vision, 128 (2020), 53–73. https://doi.org/10.1007/s11263-019-01217-w doi: 10.1007/s11263-019-01217-w
[30]	M. Jaderberg, K. Simonyan, A. Zisserman, K. Kavukcuoglu, Spatial transformer networks, in Proceedings of the 28th International Conference on Neural Information Processing Systems, 2 (2015), 2017–2025. https://doi.org/10.5555/2969442.2969465

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