Survey on low-level controllable image synthesis with deep learning

Shixiong Zhang; Jiao Li; Lu Yang; Shixiong Zhang; Jiao Li; Lu Yang

doi:10.3934/era.2023374

Electronic Research Archive

2023, Volume 31, Issue 12: 7385-7426. doi: 10.3934/era.2023374

Previous Article Next Article

Review Special Issues

Survey on low-level controllable image synthesis with deep learning

1.
School of Automation Engineering, University of Electronic Science and Technology of China, Sichuan, China
2.
College of Information Engineering, Sichuan Agricultural University, Sichuan, China

Received: 03 September 2023 Revised: 29 October 2023 Accepted: 01 November 2023 Published: 21 November 2023

Deep learning, particularly generative models, has inspired controllable image synthesis methods and applications. These approaches aim to generate specific visual content using latent prompts. To explore low-level controllable image synthesis for precise rendering and editing tasks, we present a survey of recent works in this field using deep learning. We begin by discussing data sets and evaluation indicators for low-level controllable image synthesis. Then, we review the state-of-the-art research on geometrically controllable image synthesis, focusing on viewpoint/pose and structure/shape controllability. Additionally, we cover photometrically controllable image synthesis methods for 3D re-lighting studies. While our focus is on algorithms, we also provide a brief overview of related applications, products and resources for practitioners.

Keywords:

Citation: Shixiong Zhang, Jiao Li, Lu Yang. Survey on low-level controllable image synthesis with deep learning[J]. Electronic Research Archive, 2023, 31(12): 7385-7426. doi: 10.3934/era.2023374

Related Papers:

[1]	Ahmed M.A. El-Sayed, Eman M.A. Hamdallah, Hameda M. A. Alama . Multiple solutions of a Sturm-Liouville boundary value problem of nonlinear differential inclusion with nonlocal integral conditions. AIMS Mathematics, 2022, 7(6): 11150-11164. doi: 10.3934/math.2022624
[2]	Murugesan Manigandan, Kannan Manikandan, Hasanen A. Hammad, Manuel De la Sen . Applying fixed point techniques to solve fractional differential inclusions under new boundary conditions. AIMS Mathematics, 2024, 9(6): 15505-15542. doi: 10.3934/math.2024750
[3]	Mouffak Benchohra, John R. Graef, Nassim Guerraiche, Samira Hamani . Nonlinear boundary value problems for fractional differential inclusions with Caputo-Hadamard derivatives on the half line. AIMS Mathematics, 2021, 6(6): 6278-6292. doi: 10.3934/math.2021368
[4]	Bashir Ahmad, Badrah Alghamdi, Ahmed Alsaedi, Sotiris K. Ntouyas . Existence results for Riemann-Liouville fractional integro-differential inclusions with fractional nonlocal integral boundary conditions. AIMS Mathematics, 2021, 6(7): 7093-7110. doi: 10.3934/math.2021416
[5]	Madeaha Alghanmi, Shahad Alqurayqiri . Existence results for a coupled system of nonlinear fractional functional differential equations with infinite delay and nonlocal integral boundary conditions. AIMS Mathematics, 2024, 9(6): 15040-15059. doi: 10.3934/math.2024729
[6]	Abdissalam Sarsenbi, Abdizhahan Sarsenbi . Boundary value problems for a second-order differential equation with involution in the second derivative and their solvability. AIMS Mathematics, 2023, 8(11): 26275-26289. doi: 10.3934/math.20231340
[7]	Adel Lachouri, Mohammed S. Abdo, Abdelouaheb Ardjouni, Bahaaeldin Abdalla, Thabet Abdeljawad . On a class of differential inclusions in the frame of generalized Hilfer fractional derivative. AIMS Mathematics, 2022, 7(3): 3477-3493. doi: 10.3934/math.2022193
[8]	Djamila Chergui, Taki Eddine Oussaeif, Merad Ahcene . Existence and uniqueness of solutions for nonlinear fractional differential equations depending on lower-order derivative with non-separated type integral boundary conditions. AIMS Mathematics, 2019, 4(1): 112-133. doi: 10.3934/Math.2019.1.112
[9]	Nayyar Mehmood, Niaz Ahmad . Existence results for fractional order boundary value problem with nonlocal non-separated type multi-point integral boundary conditions. AIMS Mathematics, 2020, 5(1): 385-398. doi: 10.3934/math.2020026
[10]	Najla Alghamdi, Bashir Ahmad, Esraa Abed Alharbi, Wafa Shammakh . Investigation of multi-term delay fractional differential equations with integro-multipoint boundary conditions. AIMS Mathematics, 2024, 9(5): 12964-12981. doi: 10.3934/math.2024632

Abstract

1. Introduction

The models of the differential and integral equations have been appeared in different applications (see ^{[1,2,3,4,6,7,9,12,13,14,15,16,17,18,19,20]}).

Boundary value problems involving fractional differential equations arise in physical sciences and applied mathematics. In some of these problems, subsidiary conditions are imposed locally. In some other cases, nonlocal conditions are imposed. It is sometimes better to impose nonlocal conditions since the measurements needed by a nonlocal condition may be more precise than the measurement given by a local condition. Consequently, a variety of excellent results on fractional boundary value problems (abbreviated BVPs) with resonant conditions have been achieved. For instance, Bai ^[4] studied a type of fractional differential equations with m-points boundary conditions. The existence of nontrivial solutions was established by using coincidence degree theory. Applying the same method, Kosmatov ^[17] investigated the fractional order three points BVP with resonant case.

Although the study of fractional BVPs at resonance has acquired fruitful achievements, it should be noted that such problems with Riemann-Stieltjes integrals are very scarce, so it is worthy of further explorations. Riemann-Stieltjes integral has been considered as both multipoint and integral in a single framework, which is more common, see the relevant works due to Ahmad et al. ^[1].

The boundary value problems with nonlocal, integral and infinite points boundary conditions have been studied by some authors (see, for example ^[8,10,11,12]).

Here, we discuss the boundary value problem of the nonlinear differential inclusions of arbitrary (fractional) orders

$\begin{equation} \frac{dx}{dt}\in F_{1}(t,x(t),I^{\gamma}f_{2}(t,D^{\alpha}x(t))),\; \; \alpha,\gamma\in(0,1] \; \; \; \; t\in(0,1) \end{equation}$

(1.1)

with the nonlocal boundary condition

$\begin{equation} {{\sum\limits^{m}_{k = 1} a_{k}}} x(\tau_{k} ) = x_{0},\; a_{k} > 0\; \; \tau_{k}\in[0,1], \end{equation}$

(1.2)

the integral condition

$\begin{equation} \int_{0}^{1}x(s)dg(s) = x_{0} \end{equation}$

(1.3)

and the infinite point boundary condition

$\begin{equation} {{\sum\limits^{\infty}_{k = 1} a_{k}}} x(\tau_{k} ) = x_{0},\; a_{k} > 0\; and\; \tau_{k}\in[0,1]. \end{equation}$

(1.4)

We study the existence of solutions $x\in C[0, 1]$ of the problems (1.1) and (1.2), and deduce the existence of solutions of the problem of (1.1) with the conditions (1.3) and (1.4). Then the existence of the maximal and minimal solutions will be proved. The sufficient condition for the uniqueness and continuous dependence of the solution will be studied.

This paper is organised as: In Section 2, we prove the existence of continuous solutions of the problems (1.1) and (1.2), and deduce the existence of solutions of the problem of (1.1) with the conditions (1.3) and (1.4). In Section 3, the existence of the maximal and minimal solutions is proved. In Section 4, the sufficient condition for the uniqueness and continuous dependence of the solution are studied. Next, in Section 5, we extend our results to the nonlocal problems (1.3) and (2.1). Finally, some existence results is proved for the nonlocal problems (1.4) and (2.1) in Section 6.

2. Main results

Consider the following assumptions:

(I) (i) The set $F_{1}(t, x, y)$ is nonempty, closed and convex for all $(t, x, y)\in [0, 1]\times R\times R$ .

(ii) $F_{1}(t, x, y)$ is measurable in $t\in [0, 1]$ for every $x, y\in R$ .

(iii) $F_{1}(t, x, y)$ is upper semicontinuous in $x$ and $y$ for every $t\in [0, 1]$ .

(iv) There exist a bounded measurable function $a_{1}: [0, 1]\longrightarrow R$ and a positive constant $K_{1}$ , such that

$\begin{eqnarray*} \|F_{1}(t,x,y)\|& = &\sup\{|f_{1}|:f_{1}\in F_{1}(t,x,y)\}\\ &\leq& |a_{1}(t)|+K_{1}(|x|+|y|). \end{eqnarray*}$

Remark 2.1. From the assumptions (i)–(iv) we can deduce that (see ^[3,6,7,13]) there exists $f_{1}\in F_{1}(t, x, y)$ , such that

(v) $f_{1}:[0, 1]\times R\times R\longrightarrow R$ is measurable in $t$ for every $x, y\in R$ and continuous in $x, y$ for $t\in [0, 1]$ , and there exist a bounded measurable function $a_{1}:[0, 1]\rightarrow R$ and a positive constant $K_{1} > 0$ such that

$|f_{1}(t,x,y)|\leq |a_{1}(t)|+K_{1}(|x|+|y|),$

and the functional $f_{1}$ satisfies the differential equation

$\begin{equation} \frac{dx}{dt} = f_{1}(t,x(t),I^{\gamma}f_{2}(t,D^{\alpha}x(t))),\; \; \alpha,\gamma\in(0,1]\; and\; t\in(0,1]. \end{equation}$

(2.1)

(II) $f_{2}:[0, 1]\times R\longrightarrow R$ is measurable in $t$ for any $x\in R$ and continuous in $x$ for $t\in [0, 1]$ , and there exist a bounded measurable function $a_{2}:[0, 1]\rightarrow R$ and a positive constant $K_{2} > 0$ such that

$|f_{2}(t,x)|\leq |a_{2}(t)|+K_{2}|x|,\; \forall t\in [0,1]\; \; \mbox {and}\; x\in R$

and

$\sup\limits_{t\in [0,1]}|a_{i}(t)|\leq a_{i}, i = 1,2.$

(III) $2K_{1}\gamma+K_{1}K_{2}\alpha < \alpha\gamma\Gamma(2-\alpha), \; \alpha, \gamma \in (0, 1].$

Remark 2.2. From (I) and (v) we can deduce that every solution of (1.1) is a solution of (2.1). Now, we shall prove the following lemma.

Lemma 2.1. If the solution of the problems (1.2)–(2.1) exists then it can be expressed by the integral equation

$\begin{eqnarray} y(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\limits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}} {\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}} {\Gamma(\gamma)}f_{2}(\phi,y(\theta))d\theta)ds. \end{eqnarray}$

(2.2)

Proof. Consider the boundary value problems (1.2)–(2.1) be satisfied. Operating by $I^{1-\alpha}$ on both sides of (2.1) we can obtain

$\begin{equation} D^{\alpha}x(t) = I^{1-\alpha}\frac{dx}{dt} = I^{1-\alpha} f_{1}(t,x(t),I^{\gamma}f_{2}(t,D^{\alpha}x(t))). \end{equation}$

(2.3)

Taking

$\begin{equation} D^{\alpha}x(t) = y(t), \end{equation}$

(2.4)

then we obtain

$\begin{equation} x(t) = x(0)+I^{\alpha}y(t). \end{equation}$

(2.5)

Putting $t = \tau$ and multiplying (2.5) by $\Sigma_{k = 1}^{m}a_{k}$ , then we get

$\begin{equation} \Sigma_{k = 1}^{m}a_{k}x(\tau_{k}) = \Sigma_{k = 1}^{m}a_{k}x(0)+\Sigma_{k = 1}^{m}a_{k}I^{\alpha}y(\tau_{k}), \end{equation}$

(2.6)

$\begin{equation} x_{0} = \Sigma_{k = 1}^{m}a_{k}x(0)+\Sigma_{k = 1}^{m}a_{k}I^{\alpha}y(\tau_{k}) \end{equation}$

(2.7)

and

$\begin{equation} x(0) = \frac{1}{\Sigma_{k = 1}^{m}a_{k}}[x_{0}-\Sigma_{k = 1}^{m}a_{k}I^{\alpha}y(\tau_{k})]. \end{equation}$

(2.8)

Then

$\begin{equation} x(t) = \frac{1}{\Sigma_{k = 1}^{m}a_{k}}[x_{0}-\Sigma_{k = 1}^{m}a_{k}I^{\alpha}y(\tau_{k})]+I^{\alpha}y(t). \end{equation}$

(2.9)

Substituting (2.8) and (2.9) in (2.5), which completes the proof.

Theorem 2.1. Let assumptions (I)–(III) be satisfied. Then the integral equation (2.2) has at least one continuous solution.

Proof. Define a set $Q_{r}$ as

$Q_{r} = \{y\in C[0,1]:\|y\|\leq r\},\; \;$

$\begin{eqnarray*} r = \frac{\Gamma(\gamma+1)[a_{1}+K_{1}A|x_{0}|]+k_{1}a_{2}}{\alpha\gamma\Gamma(2-\alpha)-[2K_{1}\alpha+K_{1}K_{2}\gamma]},\; \; \; (\sum\limits^{m}_{k = 1}a_{k})^{-1} = A, \end{eqnarray*}$

and the operator $F$ by

$\begin{eqnarray*} Fy(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\limits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,y(\theta))d\theta)ds. \end{eqnarray*}$

For $y\in Q_{r}$ , then

$\begin{eqnarray*} |Fy(t)|& = &|\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\limits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}} {\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}} {\Gamma(\gamma)}f_{2}(\theta,y(\theta))d\theta)ds|\\ &\leq& K_{1}\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)}[\frac{|x_{0}|}{\sum\limits^{m}_{k = 1}a_{k}}+\frac{\sum\limits^{m}_{k = 1} a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta}{\sum\limits^{m}_{k = 1}a_{k}}\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta+\int_{0}^{s}\frac{(s-\phi)^{\gamma-1}} {\Gamma(\gamma)}(K_{2}|y(\theta)|+|a_{2}(t)|)d\theta]+|a_{1}(t)|)ds\\ &\leq& \frac{1}{\Gamma(2-\alpha)}[K_{1}A|x_{0}|+\frac{K_{1}\|y\|}{\Gamma(\alpha+1)}+K_{1}\frac{\|y\|}{\Gamma(\alpha+1)} +\frac{K_{1}K_{2}\|y\|}{\Gamma(\gamma+1)}+\frac{K_{1}a_{2}}{\Gamma(\gamma+1)}+a_{1}]\\ &\leq& \frac{1}{\Gamma(2-\alpha)}[K_{1}A|x_{0}|+\frac{K_{1}r}{\Gamma(\alpha+1)}+K_{1}\frac{r}{\Gamma(\alpha+1)}+ \frac{K_{1}K_{2}r}{\Gamma(\gamma+1)}+\frac{K_{1}a_{2}}{\Gamma(\gamma+1)}+a_{1}]\\ &\leq& \frac{1}{\Gamma(2-\alpha)}[K_{1}A|x_{0}|+\frac{2K_{1}r}{\Gamma(\alpha+1)}+\frac{K_{1}K_{2}r}{\Gamma(\gamma+1)}+ \frac{K_{1}a_{2}}{\Gamma(\gamma+1)}+a_{1}]\; \leq r. \end{eqnarray*}$

Thus, the class of functions $\{Fy\}$ is uniformly bounded on $Q_{r}$ and $F:Q_{r}\rightarrow Q_{r}$ . Let $y\in Q_{r}$ and $t_{1}, t_{2}\in [0, 1]$ such that $|t_{2}-t_{1}| < \delta$ , then

$\begin{eqnarray*} &&|Fy(t_{2})-Fy(t_{1})|\\ & = &|\int_{0}^{t_{2}}\frac{(t_{2}-s)^{-\alpha}}{\Gamma(1-\alpha)} (f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))ds-\int_{0}^{t_{1}}\frac{(t_{1}-s)^{-\alpha}}{\Gamma(1-\alpha)} (f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))ds|\\ &\leq&|\int_{0}^{t_{2}}\frac{(t_{2}-s)^{-\alpha}}{\Gamma(1-\alpha)} (f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))ds-\int_{0}^{t_{1}}\frac{(t_{2}-s)^{-\alpha}}{\Gamma(1-\alpha)} (f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))ds\\ &&+\int_{0}^{t_{1}}\frac{(t_{2}-s)^{-\alpha}}{\Gamma(1-\alpha)} (f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))ds-\int_{0}^{t_{1}}\frac{(t_{1}-s)^{-\alpha}}{\Gamma(1-\alpha)} (f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))ds|\\ &\leq&\int_{t_{1}}^{t_{2}}\frac{(t_{2}-s)^{-\alpha}}{\Gamma(1-\alpha)} |(f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))|ds\\ &&+\int_{0}^{t_{1}}\frac{(t_{2}-s)^{-\alpha}-(t_{1}-s)^{-\alpha}}{\Gamma(1-\alpha)} |(f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))ds|\\ &\leq&\int_{t_{1}}^{t_{2}}\frac{(t_{2}-s)^{-\alpha}}{\Gamma(1-\alpha)} |(f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))|ds\\ &&+\int_{0}^{t_{1}}\frac{(t_{2}-s)^{\alpha}-(t_{1}-s)^{\alpha}}{\Gamma(1-\alpha)(t_{2}-s)^{\alpha}(t_{1}-s)^{\alpha}}. |(f_{1}(s,x(s),I^{\gamma}f_{2}(s,y(s)))|ds. \end{eqnarray*}$

Thus, the class of functions $\{Fy\}$ is equicontinuous on $Q_{r}$ and $\{Fy\}$ is compact operator by the Arzela-Ascoli Theorem ^[5].

Now we prove that $F$ is continuous operator. Let ${y_{n}}\subset Q_{r}$ be convergent sequence such that $y_{n}\rightarrow y$ , then

$\begin{eqnarray*} Fy_{n}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{n}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{n}(\theta)d\theta, \int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,y_{n}(\theta)))ds. \end{eqnarray*}$

Using Lebesgue dominated convergence Theorem ^[5] and assumptions (iv)–(II) we have

$\begin{eqnarray*} \lim\limits_{n\to \infty} Fy_{n}(t)& = &\lim\limits_{n\to \infty}\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{n}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{n}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,y_{n}(\theta)))ds\\ & = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}\lim\limits_{n\to \infty}y_{n}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}\lim\limits_{n\to \infty}y_{n}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,\lim\limits_{n\to \infty}y_{n}(\theta)))ds\\ & = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2} (\theta,y(\theta)))ds\; = \; Fy(t). \end{eqnarray*}$

Then $F:Q_{r}\rightarrow Q_{r}$ is continuous, and by Schauder Fixed Point Theorem^[5] there exists at least one solution $y\in C[0, 1]$ of (2.2). Now

$\begin{eqnarray*} \frac{dx}{dt}& = &\frac{d}{dt}[x(0)+I^{\alpha}y(t)]\\ & = &\frac{d}{dt}I^{\alpha}I^{1-\alpha}f_{1}(t,x(t),I^{\gamma}f_{2}(t,y(t)))\\ & = &\frac{d}{dt}If_{1}(t,x(t),I^{\gamma}f_{2}(t,y(t)))\\ & = &f_{1}(t,x(t),I^{\gamma}f_{2}(t,y(t))). \end{eqnarray*}$

Putting $t = \tau$ and using (2.9), we obtain

$\begin{eqnarray*} &&x(t) = \frac{1}{\Sigma_{k = 1}^{m}a_{k}}[x_{0}-\Sigma_{k = 1}^{m}a_{k}I^{\alpha}y(\tau_{k})]+I^{\alpha}y(t),\\ &&\Sigma_{k = 1}^{m}a_{k}x(\tau_{k}) = \Sigma_{k = 1}^{m}a_{k}\frac{1}{\Sigma_{k = 1}^{m}a_{k}}[x_{0}-\Sigma_{k = 1}^{m}a_{k}I^{\alpha}y(\tau_{k})]+\Sigma_{k = 1}^{m}a_{k}I^{\alpha}y(\tau_{k}), \end{eqnarray*}$

then

$\begin{eqnarray*} &&\Sigma_{k = 1}^{m}a_{k}x(\tau_{k}) = x_{0}, \; a_{k} > 0\; and\; \tau_{k}\in[0,1]. \end{eqnarray*}$

This proves the equivalence between the problems (1.2)–(2.1) and the integral equation (2.2). Then there exists at least one solution $y\in C[0, 1]$ of the problems (1.2)–(2.1).

3. Maximal and minimal solutions

Here, we shall study the maximal and minimal solutions for the problems (1.2) and (2.1). Let $y(t)$ be any solution of (2.2), let $u(t)$ be a solution of (2.2), then $u(t)$ is said to be a maximal solution of (2.2) if it satisfies the inequality

$y(t)\leq u(t),\; \; \; t\in[0,1].$

A minimal solution can be defined by similar way by reversing the above inequality.

Lemma 3.1. Let the assumptions of Theorem 2.1 be satisfied. Assume that $x(t)$ and $y(t)$ are two continuous functions on $[0, 1]$ satisfying

$\begin{eqnarray*} y(t)&\leq&\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,y(\theta)))ds\; \; t\in[0,1],\\ x(t)&\geq&\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}x(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}x(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2} (\theta,x(\theta)))ds\; \; t\in[0,1], \end{eqnarray*}$

where one of them is strict.

Let functions $f_{1}$ and $f_{2}$ be monotonic nondecreasing in $y$ , then

$\begin{equation} y(t) < x(t),\; \; \; t > 0. \end{equation}$

(3.1)

Proof. Let the conclusion (3.1) be not true, then there exists $t_{1}$ with

$y(t_{1}) < x(t_{1}),\; \; \; t_{1} > 0\; \; \; \mbox{and}\; \; \; y(t) < x(t),\; \; \; 0 < t < t_{1}.$

Since $f_{1}$ and $f_{2}$ are monotonic functions in $y,$ then we have

$\begin{eqnarray*} y(t_{1})&\leq&\int_{0}^{t_{1}}\frac{(t_{1}-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,y(\theta)))ds\\ & < &\int_{0}^{t_{1}}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}x(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}x(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2} (\theta,x(\theta)))ds\\ & < &x(t_{1}),\; t_{1}\in[0,1]. \end{eqnarray*}$

This contradicts the fact that $y(t_{1}) = x(t_{1})$ , then $y(t) < x(t)$ . This completes the proof.

For the existence of the continuous maximal and minimal solutions for (2.1), we have the following theorem.

Theorem 3.1. Let the assumptions of Theorem 2.1 be hold. Moreover, if $f_{1}$ and $f_{2}$ are monotonic nondecreasing functions in $y$ for each $t\in [0, 1]$ , then Eq (2.1) has maximal and minimal solutions.

Proof. First, we should demonstrate the existence of the maximal solution of (2.1). Let $\epsilon > 0$ be given. Now consider the integral equation

$\begin{eqnarray} y_{\epsilon}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1,\epsilon}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}} {\Gamma(\alpha)}y_{\epsilon}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_\epsilon(\theta)d\theta,\int_{0}^{s} \frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2,\epsilon}(\theta,y_{\epsilon}(\theta))d\theta \bigg)ds,\; \; t\in[0,1], \end{eqnarray}$

where

$f_{1,\epsilon}(s,x_{\epsilon}(s),y_{\epsilon}(s) ) = f_{1}(s,x_{\epsilon}(s),y_{\epsilon}(s))+\epsilon,$

$f_{2,\epsilon}(s,x_{\epsilon}(s)) = f_{2}(s,x_{\epsilon}(s))+\epsilon.$

For $\epsilon_{2} > \epsilon_{1},$ we have

$\begin{eqnarray} y_{\epsilon_{2}}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1,\epsilon_2}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{2}}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{2}}(\theta)d\theta,\int_{0}^{s} \frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2,\epsilon_{2}}(\theta,y_{\epsilon_{2}}(\theta))d\theta\bigg )ds,\; \; t\in[0,1], \end{eqnarray}$

$\begin{eqnarray} y_{\epsilon_{2}}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} \bigg[f_{1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{2}}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{2}}(\theta)d\theta,\int_{0}^{s} \frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}(f_{2}(\theta,y_{\epsilon_{2}}(\theta))+\epsilon_{2})d\theta \bigg)+\epsilon_2\bigg]ds,\; t\in[0,1]. \end{eqnarray}$

Also

$\begin{eqnarray*} y_{\epsilon_{1}}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1,\epsilon_1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{1}}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{1}}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^ {\gamma-1}}{\Gamma(\gamma)}f_{2,\epsilon_{1}}(\theta,y_{\epsilon_{1}}(\theta))d\theta\bigg)ds,\\ y_{\epsilon_{1}}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} \bigg[f_{1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\phi)^{\alpha-1}} {\Gamma(\alpha)}y_{\epsilon_{1}}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{1}}(\theta)d\theta,\int_{0}^{s} \frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}(f_{2}(\theta,y_{\epsilon_{1}}(\theta))+\epsilon_{1})d\theta \bigg)+\epsilon_1\bigg] ds \\ & > &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} \bigg[f_{1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}} {\Gamma(\alpha)}y_{\epsilon_{2}}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{2}}(\theta)d\theta,\int_{0}^{s} \frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}(f_{2}(\theta,y_{\epsilon_{2}}(\tau))+\epsilon_{2})d\theta \bigg)+\epsilon_2\bigg]ds. \end{eqnarray*}$

Applying Lemma 3.1, we obtain

$y_{\epsilon_{2}} < y_{\epsilon_{1}},\; t\in[0,1].$

As shown before, the family of function $y_{\epsilon}(t)$ is equi-continuous and uniformly bounded, then by Arzela Theorem, there exist decreasing sequence $\epsilon_{n}$ , such that $\epsilon_{n}\rightarrow 0$ as $n\rightarrow \infty$ , and $u(t) = {\lim_{n\rightarrow \infty}}y_{\epsilon_{n}}(t)$ exists uniformly in $[0, 1]$ and denote this limit by $u(t)$ . From the continuity of the functions, $f_{2, \epsilon_n}(t, y_{\epsilon_n}(t))$ , we get $f_{2, \epsilon_n}(t, y_{\epsilon_n}(t))\longrightarrow f_{2}(t, y(t))$ as $n\rightarrow \infty$ and

$\begin{eqnarray*} &&u(t) = \lim\limits_{n\rightarrow \infty}y_{\epsilon_{n}}(t) = \int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} \bigg[f_{1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{n}}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon_{n}}(\theta)d\theta, \int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}(f_{2}(\theta,y_{\epsilon_{n}}(\theta))+\epsilon_{n})\bigg)+\epsilon_n \bigg]ds,\; \; t\in[0,1]. \end{eqnarray*}$

Now we prove that $u(t)$ is the maximal solution of (2.1). To do this, let $y(t)$ be any solution of (2.1), then

$\begin{eqnarray*} y(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,y(\theta))d\theta \bigg)ds,\\ y_{\epsilon}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} \bigg[f_{1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)} y_{\epsilon}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}(f_{2} (\theta,y_{\epsilon}(\theta))+\epsilon)d\theta \bigg)+\epsilon \bigg]ds \end{eqnarray*}$

and

$\begin{eqnarray*} y_{\epsilon}(t)& > &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}\bigg(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y_{\epsilon}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)} f_{2}(\theta,y_{\epsilon}(\theta))\bigg)ds. \end{eqnarray*}$

Applying Lemma 3.1, we obtain

$y(t) < y_{\epsilon}(t), t\in [0,1].$

From the uniqueness of the maximal solution it clear that $y_{\epsilon}(t)$ tends to $u(t)$ uniformly in $[0, 1]$ as $\epsilon\rightarrow 0.$

By a similar way as done above, we can prove the existence of the minimal solution.

4. Uniqueness of the solution

Here, we study the sufficient condition for the uniqueness of the solution $y\in C[0, 1]$ of problems (1.2) and (2.1). Consider the following assumptions:

( $I^{*}$ ) (i) The set $F_{1}(t, x, y)$ is nonempty, closed and convex for all $(t, x, y)\in [0, 1]\times R\times R$ .

(ii) $F_{1}(t, x, y)$ is measurable in $t\in [0, 1]$ for every $x, y\in R$ .

(iii) $F_{1}$ satisfies the Lipschitz condition with a positive constant $K_{1}$ such that

$H(F_{1}(t,x_{1},y_{1}),F_{1}(t,x_{2},y_{2})|\leq K_{1}(|x_{1}-x_{_{2}}|+|y_{1}-y_{2}|),$

where $H(A, B)$ is the Hausdorff metric between the two subsets $A, B\in [0, 1]\times E.\;$

Remark 4.1. From this assumptions we can deduce that there exists a function $f_{1}\in F_{1}(t, x, y)$ , such that

(iv) $f_{1}:[0, 1]\times R\times R \rightarrow R$ is measurable in $t\in [0, 1]$ for every $x, y\in R$ and satisfies Lipschitz condition with a positive constant $K_{1}$ such that (see ^[3,7])

$|f_{1}(t,x_{1},y_{1})-f_{1}(t,x_{2},y_{2})|\leq K_{1}(|x_{1}-x_{2}|+|y_{1}-y_{2}|).$

$(II^{*}) \; f_{2}:[0, 1]\times R\longrightarrow R$ is measurable in $t \in [0, T]$ and satisfies Lipschitz condition with positive constant $K_{2}$ , such that

$|f_{2}(t,x)-f_{2}(t,y)|\leq K_{2}|x-y|.$

From the assumption ( $I^{*}$ ), we have

$|f_{1}(t,x,y)|-|f_{1}(t,0,0)|\leq |f_{1}(t,x,y)-f_{1}(t,0,0)|\leq K_{1}(|x|+|y|).$

Then

$\begin{eqnarray*} |f_{1}(t,x,y)|&\leq& K_{1}(|x|+|y|)+|f_{1}(t,0,0)|\\ &\leq& K_{1}(|x|+|y|)+\mid a_{1}(t)\mid, \end{eqnarray*}$

where $|a_{1}(t)| = \sup\limits_{t\in I}|f_{1}(t, 0, 0)|.$

From the assumption ( $II^{*}$ ), we have

$|f_{2}(t,y)|-|f_{2}(t,0)|\leq|f_{2}(t,y)-f_{2}(t,0)|\leq K_{2}|y|.$

Then

$\begin{eqnarray*} |f_{2}(t,x)|&\leq& K_{2}|x|+|f_{2}(t,0)|\\ &\leq& K_{2}|x|+\mid a_{2}(t)\mid, \end{eqnarray*}$

where $|a_{2}(t)| = \sup\limits_{t\in I}|f_{1}(t, 0)|$ .

Theorem 4.1. Let the assumptions $(I^{*})$ and $(II^{*})$ be satisfied. Then the solution of the problems (1.2) and (2.1) is unique.

Proof. Let $y_{1}(t)$ and $y_{2}(t)$ be solutions of the problems (1.2) and (2.1), then

Then

$\|y_{1}-y_{2}\|[\alpha\gamma\Gamma(2-\alpha)-(2K_{1}\alpha+K_{1}K_{2}\gamma)] < 0.$

Since $(\alpha\gamma\Gamma(2-\alpha)-(2K_{1}\alpha+K_{1}K_{2}\gamma)) < 1$ , then $y_{1}(t) = y_{2}(t)$ and the solution of (1.2) and (2.1) is unique.

4.1. Continuous dependence of the solution

Definition 4.1. The unique solution of the problems (1.2) and (2.1) depends continuously on initial data $x_{0}$ , if $\epsilon > 0$ , $\exists \delta > 0$ , such that

$|x_{0}-x^{*}_{0}|\leq \delta\; \; \; \; \Rightarrow \|y-y^{*}\|\leq \epsilon,$

where $y^{*}$ is the unique solution of the integral equation

$\begin{eqnarray*} y^{*}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x^{*}_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y^{*}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y^{*}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}} {\Gamma(\gamma)}f_{2}(\theta,y^{*}(\theta))d\theta)ds|. \end{eqnarray*}$

Theorem 4.2. Let the assumptions $(I^{*})$ and $(II^{*})$ be satisfied, then the unique solution of (1.2) and (2.1) depends continuously on $x_{0}.$

Proof. Let $y(t)$ and $y^{*}(t)$ be the solutions of problems (1.2) and (2.1), then

then we obtain

$\begin{eqnarray*} &&|y(t)-y^{*}(t)|\leq(AK_{1}\delta)(\alpha\gamma\Gamma(2-\alpha)-(2K_{1}\alpha+K_{1}K_{2}\gamma))^{-1}\leq\epsilon \end{eqnarray*}$

and

$\begin{eqnarray*} &&\|y(t)-y^{*}(t)\|\leq\epsilon. \end{eqnarray*}$

Definition 4.2. The unique solution of the problems (1.2) and (2.1) depends continuously on initial data $a_{k}$ , if $\epsilon > 0$ , $\exists \delta > 0$ , such that

$\sum\limits^{m}_{k = 1}|a_{k}-a_{k}^{*}|\leq\delta\; \; \; \; \Rightarrow \|y-y^{*}\|\leq \epsilon,$

where $y^{*}$ is the unique solution of the integral equation

$\begin{eqnarray*} y^{*}(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}^{*}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y^{*}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y^{*}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)} f_{2}(\theta,y^{*}(\theta))d \theta)ds|. \end{eqnarray*}$

Theorem 4.3. Let the assumptions $(I^{*})$ and $(II^{*})$ be satisfied, then the unique solution of problems (1.2) and (2.1) depends continuously on $a_{k}$ .

Proof. Let $y(t)$ and $y^{*}(t)$ be the solutions of problems (1.2) and (2.1) and $(\sum\nolimits^{m}_{k = 1}a^{*}_{k})^{-1} = A^{*}$ ,

$\begin{eqnarray*} |y(t)-y^{*}(t)|& = &|\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma (\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^ {\gamma-1}}{\Gamma(\gamma)}f_{2}(\theta,y(\theta))d \theta)ds \\ &&-\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{m}_{k = 1}a^{*}_{k}}[x_{0}-\sum\limits^{m}_{k = 1}a^{*}_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y^{*}(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y^{*}(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}} {\Gamma(\gamma)}f_{2}(\theta,y^{*}(\theta))d \theta)ds|\\ &\leq&\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)}[\frac{|x_{0}|(\sum\nolimits^{m}_{k = 1}a^{*}_{k}-\sum\nolimits^{m}_{k = 1}a_{k})}{\sum\nolimits^{m}_{k = 1}a_{k}\sum\nolimits^{m}_{k = 1}a^{*}_{k}}\\ &&+\frac{\sum\nolimits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta}{\sum\nolimits^{m}_{k = 1}a_{k}} -\frac{\sum\nolimits^{m}_{k = 1}a^{*}_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y^{*}(\theta)|d\theta}{\sum\nolimits^{m}_{k = 1}a^{*}_{k}}\\ &&+K_{1}\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)-y^{*}(\theta)|d\theta+K_{1}K_{2} \int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}|y(\phi)-y^{*}(\theta)|d\theta]ds\\ &\leq&\frac{1}{\Gamma(2-\alpha)}[\frac{K_{1}|x_{0}|\sum\nolimits^{m}_{k = 1}|a^{*}_{k}-a_{k}|}{\sum\nolimits^{m}_{k = 1}a_{k}\sum\nolimits^{m}_{k = 1}a^{*}_{k}}\\ &&+\frac{K_{1}\sum\nolimits^{m}_{k = 1}a_{k}(\sum\nolimits^{m}_{k = 1}a^{*}_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta- \sum\nolimits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta)}{\sum\nolimits^{m}_{k = 1}a_{k}\sum\nolimits^{m}_{k = 1}a^{*}_{k}}\\ &&+\frac{K_{1}\sum\nolimits^{m}_{k = 1}a^{*}_{k}(\sum\nolimits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta- \sum\nolimits^{m}_{k = 1}a^{*}_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta)}{\sum\nolimits^{m}_{k = 1}a_{k}\sum\nolimits^{m}_{k = 1}a^{*}_{k}}\\ &&+K_{1}\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)-y^{*}(\theta)|d\theta+K_{1}K_{2} \int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}|y(\phi)-y^{*}(\theta)|d\theta]ds\\ &\leq&\frac{1}{\Gamma(2-\alpha)}[\frac{K_{1}|x_{0}|\delta}{\sum\nolimits^{m}_{k = 1}a_{k}\sum\nolimits^{m}_{k = 1}a^{*}_{k}}+ \frac{K_{1}(\sum\nolimits^{m}_{k = 1}a_{k}\delta\frac{r}{\Gamma(\alpha+1)}+\sum\nolimits^{m}_{k = 1}a^{*}_{k}\delta\frac{r}{\Gamma(\alpha+1)})}{\sum\nolimits^{m}_{k = 1}a_{k}\sum\nolimits^{m}_{k = 1}a^{*}_{k}}\\ &&+\frac{K_{1}\|y-y^{*}\|}{\Gamma(\alpha+1)}+\frac{K_{1}K_{2}\|y-y^{*}\|}{\Gamma(\gamma+1)}\\ &\leq&\frac{K_{1}\delta|x_{0}|}{\Gamma(2-\alpha)\sum\nolimits^{m}_{k = 1}a_{k}\sum\nolimits^{m}_{k = 1}a^{*}_{k}}+ \frac{K_{1}(\sum\nolimits^{m}_{k = 1}a_{k}\delta\frac{r}{\Gamma(\alpha+1)}+\sum\nolimits^{m}_{k = 1}a^{*}_{k}\delta\frac{r}{\Gamma(\alpha+1)})}{\Gamma(2-\alpha)\Gamma(\alpha+1)\sum\nolimits^{m}_{k = 1}a_{k} \sum\nolimits^{m}_{k = 1}a^{*}_{k}}\\ &&+\frac{K_{1}\|y-y^{*}\|}{\Gamma(2-\alpha)\Gamma(\alpha+1)}+\frac{K_{1}K_{2}\|y-y^{*}\|}{\Gamma(\gamma+1)\Gamma(2-\alpha)}, \end{eqnarray*}$

then we obtain

$\begin{eqnarray*} |y(t)-y^{*}(t)|\leq(AA^{*}K_{1}\delta [|x_{0}|+ r(\sum\limits^{m}_{k = 1}a_{k}+\sum\limits^{m}_{k = 1}a^{*}_{k})]).\; (\alpha\gamma\Gamma(2-\alpha)-(2K_{1}\alpha+K_{1}K_{2}\gamma))^{-1}\leq\epsilon \end{eqnarray*}$

and

$\begin{eqnarray*} &&\|y(t)-y^{*}(t)\|\leq\epsilon. \end{eqnarray*}$

5. Riemann-Stieltjes integral condition

Let $y\in C[0, 1]$ be the solution of the nonlocal boundary value problems (1.2) and (2.1). Let $a_{k} = (g(t_{k})-g(t_{k-1})$ , $g$ is increasing function, $\tau_{k}\in(t_{k-1}-t_{k})$ , $0 = t_{0} < t_{1} < t_{2}, ... < t_{m} = 1$ , then, as $m\longrightarrow \infty$ the nonlocal condition (1.2) will be

$\sum\limits^{m}_{k = 1}(g(t_{k})-g(t_{k-1})y(\tau_{k}) = x_{0}.$

As the limit $m\longrightarrow \infty$ , we obtain

$\lim\limits_{m\rightarrow \infty}\sum\limits^{m}_{k = 1}(g(t_{k})-g(t_{k-1})y(\tau_{k}) = \int_{0}^{1}y(s)dg(s) = x_{0}.$

Theorem 5.1. Let the assumptions (I)–(III) be satisfied. If $\sum\nolimits^{m}_{k = 1}a_{k}$ be convergent, then the nonlocal boundary value problems of (1.3) and (2.1) have at least one solution given by

$\begin{eqnarray*} y(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{g(1)-g(0)}[x_{0}-\int_{0}^{1}\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta dg(s)]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2} (\phi,y(\theta))d\theta)ds. \end{eqnarray*}$

Proof. As $m\rightarrow \infty$ , the solution of the nonlocal boundary value problem (2.1) will be

$\begin{eqnarray*} \lim\limits_{m\rightarrow \infty}y(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,(g(1)-g(0))^{-1}x_{0}\\ &&-(g(1)-g(0))^{-1}.\lim\limits_{m\rightarrow \infty}\sum\limits^{m}_{k = 1}[g(t_{k})-g(t_{k-1}]\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)} f_{2}(\phi,y(\theta))d\theta)ds\\ & = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{g(1)-g(0)}[x_{0}-\int_{0}^{1}\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta dg(s)]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)} f_{2}(\phi,y(\theta))d\theta)ds. \end{eqnarray*}$

6. Infinite-point boundary condition

Theorem 6.1. Let the assumptions (I)–(III) be satisfied, then the nonlocal boundary value problems of (1.4) and (2.1) have at least one solution given by

$\begin{eqnarray} y(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{\infty}_{k = 1}a_{k}}[x_{0}-\sum\limits^{\infty}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}} {\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)} f_{2}(\phi,y(\theta))d\theta)ds. \end{eqnarray}$

Proof. Let the assumptions of Theorem 2.1 be satisfied. Let $\sum\nolimits^{m}_{k = 1} a_{k}$ be convergent, then take the limit to (1.4), we have

$\begin{eqnarray} \lim\limits_{m\rightarrow \infty}y(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\lim\limits_{m\rightarrow m}\frac{1}{\sum\nolimits^{m}_{k = 1}a_{k}}[x_{0}-\lim\limits_{m\rightarrow \infty}\sum\limits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta]\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\phi,y(\theta))d\theta)ds. \end{eqnarray}$

Now

$\begin{eqnarray*} |a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta| \leq|a_{k}|\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}|y(\theta)|d\theta\leq\frac{|a_{k}|\|y\|}{\Gamma(\alpha+1)} \leq\frac{|a_{k}|r}{\Gamma(\alpha+1)} \end{eqnarray*}$

and by the comparison test $(\sum\nolimits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta)$ is convergent,

$\begin{eqnarray*} y(t)& = &\int_{0}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)} f_{1}(s,\frac{1}{\sum\nolimits^{\infty}_{k = 1}a_{k}}[x_{0}-\sum\limits^{\infty}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta]\nonumber\\ &&+\int_{0}^{s}\frac{(s-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta,\int_{0}^{s}\frac{(s-\theta)^{\gamma-1}}{\Gamma(\gamma)}f_{2}(\phi,y(\theta))d\theta)ds.\nonumber \end{eqnarray*}$

Furthermore, from (2.9) we have

$\begin{eqnarray*} \sum\limits^{\infty}_{k = 1}a_{k}x(\tau_{k})& = &\sum\limits^{\infty}_{k = 1}a_{k}[\frac{1}{\sum\nolimits^{\infty}_{k = 1}a_{k}}(x_{0}-\sum\limits^{\infty}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-s)^{\alpha-1}}{\Gamma(\alpha)}y(s)ds)]\\ &&+\int_{0}^{\tau_{k}}\frac{(\tau_{k}-s)^{\alpha-1}}{\Gamma(\alpha)}y(s)ds)]\\ & = &x_{0}. \end{eqnarray*}$

Example 6.1. Consider the following nonlinear integro-differential equation

$\begin{equation} \frac{dx}{dt} = t^{4}e^{-t}+\frac{x(t)}{\sqrt{t+2}}+\frac{1}{3}I^{\gamma}(\cos(5t+1)+\frac{1}{9}[t^{5}\sin D^{\frac{1}{3}}x(t)+e^{-3t}x(t)]) \end{equation}$

(6.1)

with boundary condition

$\begin{equation} {\sum\limits^{m}_{k = 1}\bigg[\frac{1}{k}-\frac{1}{k+1}\bigg]} x(\tau_k ) = x_{0},\; a_{k} > 0\; \; \tau_{k}\in[0,1]. \end{equation}$

(6.2)

Let

$\begin{eqnarray*} f_{1}(t,x(t),I^{\gamma}f_{2}(t,D^{\alpha}x(t))) = t^{4}e^{-t}+\frac{x(t)}{\sqrt{t+2}}+\frac{1}{3}I^{\gamma}(\cos(5t+1)+\frac{1}{9}[t^{5}\sin D^{\frac{1}{3}}x(t)+e^{-3t}x(t)]), \end{eqnarray*}$

then

$\begin{eqnarray*} |f_{1}(t,x(t),I^{\gamma}f_{2}(t,D^{\alpha}x(t))) = |t^{4}e^{-t}+\frac{x(t)}{\sqrt{2t+4}}+\\ \frac{1}{4}(\frac{4}{3}I^{\gamma}(\cos(5t+1)+\frac{1}{9}[t^{5}\sin D^{\frac{1}{3}}x(t)+e^{-3t}x(t)]))| \end{eqnarray*}$

and also

$\begin{eqnarray*} &&|I^{\gamma}f_{2}(t,D^{\alpha}x(t))|\leq \frac{1}{4}(\frac{4}{3}I^{\gamma}|\cos(5t+1)+\frac{1}{3}[t^{5}\sin D^{\frac{1}{3}}x(t)+e^{-3t}x(t)]|. \end{eqnarray*}$

It is clear that the assumptions (I) and (II) of Theorem 2.1 are satisfied with $a_{1}(t) = t^{4}e^{-t}\in L^{1}[0, 1]$ , $a_{2}(t) = \frac{4}{3}I^{\gamma}|(\cos(5t+1)|\in L^{1}[0, 1]$ , and let $\alpha = \frac{1}{3}$ , $\gamma = \frac{2}{3}$ , then $2K_{1}\gamma+K_{1}K_{2}\alpha = \frac{1}{2} < \alpha\gamma\Gamma(2-\alpha) = \frac{1}{2}\Gamma(2-\alpha)$ . Therefore, by applying Theorem 2.1, the nonlocal problems (6.1) and (6.2) has a continuous solution.

7. Conclusions

In this paper, we have studied a boundary value problem of fractional order differential inclusion with nonlocal, integral and infinite points boundary conditions. We have prove some existence results for that a single nonlocal boundary value problem, in of proving some existence results for a boundary value problem of fractional order differential inclusion with nonlocal, integral and infinite points boundary conditions. Next, we have proved the existence of maximal and minimal solutions. Then we have established the sufficient conditions for the uniqueness of solutions and continuous dependence of solution on some initial data and on the coefficients $a_k$ are studied. Finally, we have proved the existence of a nonlocal boundary value problem with Riemann-Stieltjes integral condition and with infinite-point boundary condition. An example is given to illustrate our results.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	R. Rombach, A. Blattmann, D. Lorenz, P. Esser, B. Ommer, High-resolution image synthesis with latent diffusion models, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 10684–10695.
[2]	Y. Cao, S. Li, Y. Liu, Z. Yan, Y. Dai, P. S. Yu, et al., A comprehensive survey of AI-generated content (aigc): A history of generative AI from GAN to ChatGPT, preprint, arXiv: 2303.04226.
[3]	R. Bommasani, D. A. Hudson, E. Adeli, R. Altman, S. Arora, S. V. Arx, et al., On the opportunities and risks of foundation models, preprint, arXiv: 2108.07258.
[4]	L. Zhang, A. Rao, M. Agrawala, Adding conditional control to text-to-image diffusion models, in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2023), 3836–3847.
[5]	X. Wang, L. Xie, C. Dong, Y. Shan, Real-ESRGAN: Training real-world blind super-resolution with pure synthetic data, in Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (ICCVW), IEEE, (2021), 1905–1914.
[6]	H. Jonathan, J. Ajay, A. Pieter, Denoising diffusion probabilistic models, in Advances in Neural Information Processing Systems, Curran Associates, Inc., 33 (2020), 6840–6851.
[7]	I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, et al., Generative adversarial nets, in Advances in Neural Information Processing Systems, Curran Associates, Inc., 27 (2014), 1–9.
[8]	Y. LeCun, S. Chopra, R. Hadsell, M. Ranzato, F. Huang, A tutorial on energy-based learning, Predict. Struct. Data, 1 (2006), 1–59.
[9]	J. Zhou, Z. Wu, Z. Jiang, K. Huang, K. Guo, S. Zhao, Background selection schema on deep learning-based classification of dermatological disease, Comput. Biol. Med., 149 (2022), 105966. https://doi.org/10.1016/j.compbiomed.2022.105966 doi: 10.1016/j.compbiomed.2022.105966
[10]	Q. Su, F. Wang, D. Chen, G. Chen, C. Li, L. Wei, Deep convolutional neural networks with ensemble learning and transfer learning for automated detection of gastrointestinal diseases, Comput. Biol. Med., 150 (2022), 106054. https://doi.org/10.1016/j.compbiomed.2022.106054 doi: 10.1016/j.compbiomed.2022.106054
[11]	G. Liu, Q. Ding, H. Luo, M. Sha, X. Li, M. Ju, Cx22: A new publicly available dataset for deep learning-based segmentation of cervical cytology images, Comput. Biol. Med., 150 (2022), 106194. https://doi.org/10.1016/j.compbiomed.2022.106194 doi: 10.1016/j.compbiomed.2022.106194
[12]	L. Xu, R. Magar, A. B. Farimani, Forecasting COVID-19 new cases using deep learning methods, Comput. Biol. Med., 144 (2022), 105342. https://doi.org/10.1016/j.compbiomed.2022.105342 doi: 10.1016/j.compbiomed.2022.105342
[13]	D. P. Kingma, M. Welling, Auto-encoding variational bayes, preprint, arXiv: 1312.6114.
[14]	A. Radford, J. Wu, R. Child, D. Luan, D. Amodei, I. Sutskever, Language models are unsupervised multitask learners, OpenAI Blog, 1 (2019), 9.
[15]	H. Huang, P. S. Yu, C. Wang, An introduction to image synthesis with generative adversarial nets, preprint, arXiv: 1803.04469.
[16]	M. Mirza, S. Osindero, Conditional generative adversarial nets, preprint, arXiv: 1411.1784.
[17]	L. A. Gatys, A. S. Ecker, M. Bethge, Image style transfer using convolutional neural networks, in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2016), 2414–2423. https://doi.org/10.1109/CVPR.2016.265
[18]	S. Agarwal, N. Snavely, I. Simon, S. M. Seitz, R. Szeliski, Building rome in a day, in 2009 IEEE 12th International Conference on Computer Vision, IEEE, (2009), 72–79. https://doi.org/10.1109/ICCV.2009.5459148
[19]	L. Yang, T. Yendo, M. P. Tehrani, T. Fujii, M. Tanimoto, Probabilistic reliability based view synthesis for FTV, in 2010 IEEE International Conference on Image Processing, IEEE, (2010), 1785–1788. https://doi.org/10.1109/ICIP.2010.5650222
[20]	Y. Zheng, G. Zeng, H. Li, Q. Cai, J. Du, Colorful 3D reconstruction at high resolution using multi-view representation, J. Visual Commun. Image Represent., 85 (2022), 103486. https://doi.org/10.1016/j.jvcir.2022.103486 doi: 10.1016/j.jvcir.2022.103486
[21]	J. Deng, W. Dong, R. Socher, L. Li, L. Kai, F. Li, ImageNet: A large-scale hierarchical image database, in 2009 IEEE Conference on Computer Vision and Pattern Recognition, IEEE, (2009), 248–255. https://doi.org/10.1109/CVPR.2009.5206848
[22]	S. Christoph, B. Romain, V. Richard, G. Cade, W. Ross, C. Mehdi, et al., Laion-5b: An open large-scale dataset for training next generation image-text models, in Advances in Neural Information Processing Systems, Curran Associates, Inc., 35 (2022), 25278–25294.
[23]	S. M. Mohammad, S. Kiritchenko, Wikiart emotions: An annotated dataset of emotions evoked by art, in Proceedings of the 11th Edition of the Language Resources and Evaluation Conference (LREC-2018), (2018), 1–14.
[24]	M. Ben, P. P. Srinivasan, M. Tancik, J. T. Barron, R. Ramamoorthi, R. Ng, NeRF: Representing scenes as Neural Radiance Fields for view synthesis, in European Conference on Computer Vision, Springer, (2020), 405–421. https://doi.org/10.1007/978-3-030-58452-8_24
[25]	S. Huang, Q. Li, J. Liao, L. Liu, L. Li, An overview of controllable image synthesis: Current challenges and future trends, SSRN, 2022.
[26]	A. Tsirikoglou, G. Eilertsen, J. Unger, A survey of image synthesis methods for visual machine learning, Comput. Graphics Forum, 39 (2020), 426–451. https://doi.org/10.1111/cgf.14047 doi: 10.1111/cgf.14047
[27]	H. Ren, G. Stella, B. S. Sami, Controllable GAN synthesis using non-rigid structure-from-motion, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2023), 678–687.
[28]	J. Zhang, A. Siarohin, Y. Liu, H. Tang, N. Sebe, W. Wang, Training and tuning generative neural radiance fields for attribute-conditional 3D-aware face generation, preprint, arXiv: 2208.12550.
[29]	J. Ko, K. Cho, D. Choi, K. Ryoo, S. Kim, 3D GAN inversion with pose optimization, in Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), IEEE, (2023), 2967–2976.
[30]	S. Yang, W. Wang, B. Peng, J. Dong, Designing a 3D-aware StyleNeRF encoder for face editing, in ICASSP 2023–2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, (2023), 1–5. https://doi.org/10.1109/ICASSP49357.2023.10094932
[31]	J. Collins, S. Goel, K. Deng, A. Luthra, L. Xu, E. Gundogdu, et al., ABO: Dataset and benchmarks for real-world 3D object understanding, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2022), 21126–21136.
[32]	B. Yang, Y. Zhang, Y. Xu, Y. Li, H. Zhou, H. Bao, et al., Learning object-compositional Neural Radiance Field for editable scene rendering, in 2021 IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2021), 13759–13768. https://doi.org/10.1109/ICCV48922.2021.01352
[33]	M. Niemeyer, A. Geiger, GIRAFFE: Representing scenes as compositional generative neural feature fields, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2021), 11453–11464.
[34]	J. Zhu, C. Yang, Y. Shen, Z. Shi, B. Dai, D. Zhao, et al., LinkGAN: Linking GAN latents to pixels for controllable image synthesis, in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2023), 7656–7666.
[35]	R. Gross, I. Matthews, J. Cohn, T. Kanade, S. Baker, Multi-PIE, in 2008 8th IEEE International Conference on Automatic Face and Gesture Recognition, IEEE, (2008), 1–8. https://doi.org/10.1109/AFGR.2008.4813399
[36]	M. Boss, R. Braun, V. Jampani, J. T. Barron, C. Liu, H. P. A. Lensch, NeRD: Neural reflectance decomposition from image collections, in 2021 IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2021), 12664–12674. https://doi.org/10.1109/ICCV48922.2021.01245
[37]	X. Yan, Z. Yuan, Y. Du, Y. Liao, Y. Guo, Z. Li, et al., CLEVR3D: Compositional language and elementary visual reasoning for question answering in 3D real-world scenes, preprint, arXiv: 2112.11691.
[38]	A. Dai, A. X. Chang, M. Savva, M. Halber, T. Funkhouser, M. Nießner, ScanNet: Richly-annotated 3D reconstructions of indoor scenes, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2017), 5828–5839.
[39]	T. Zhou, T. Richard, F. John, F. Graham, S. Noah, Stereo magnification: Learning view synthesis using multiplane images, ACM Trans. Graphics, 37 (2018), 1–12. https://doi.org/10.1145/3197517.3201323 doi: 10.1145/3197517.3201323
[40]	A. X. Chang, T. A. Funkhouser, L. J. Guibas, P. Hanrahan, Q. Huang, Z. Li, et al., ShapeNet: An information-rich 3D model repository, preprint, arXiv: 1512.03012.
[41]	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, IEEE, (2012), 3354–3361. https://doi.org/10.1109/CVPR.2012.6248074
[42]	H. Caesar, V. Bankiti, A. H. Lang, S. Vora, V. E. Liong, Q. Xu, et al., NuScenes: A multimodal dataset for autonomous driving, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2020), 11621–11631.
[43]	S. K. Ramakrishnan, A. Gokaslan, E. Wijmans, O. Maksymets, A. Clegg, J. M. Turner, et al., Habitat-Matterport 3D dataset (HM3D): 1000 large-scale 3D environments for embodied AI, in Thirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 2), (2021), 1–12.
[44]	D. Scharstein, R. Szeliski, A taxonomy and evaluation of dense two-frame stereo correspondence algorithms, Int. J. Comput. Vision, 47 (2002), 7–42. https://doi.org/10.1023/A:1014573219977 doi: 10.1023/A:1014573219977
[45]	D. Scharstein, R. Szeliski, High-accuracy stereo depth maps using structured light, in 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, IEEE, (2003), 1. https://doi.org/10.1109/CVPR.2003.1211354
[46]	D. Scharstein, C. Pal, Learning conditional random fields for stereo, in 2007 IEEE Conference on Computer Vision and Pattern Recognition, IEEE, (2007), 1–8. https://doi.org/10.1109/CVPR.2007.383191
[47]	H. Hirschmuller, D. Scharstein, Evaluation of cost functions for stereo matching, in 2007 IEEE Conference on Computer Vision and Pattern Recognition, IEEE, (2007), 1–8. https://doi.org/10.1109/CVPR.2007.383248
[48]	D. Scharstein, H. Hirschmüller, Y. Kitajima, G. Krathwohl, N. Nesic, X. Wang, et al., High-resolution stereo datasets with subpixel-accurate ground truth, in 36th German Conference on Pattern Recognition, Springer, (2014), 31–42. https://doi.org/10.1007/978-3-319-11752-2_3
[49]	N. Silberman, D. Hoiem, K. Pushmeet, R. Fergus, Indoor segmentation and support inference from rgbd images, in Computer Vision–ECCV 2012: 12th European Conference on Computer Vision, Springer, (2012), 746–760. https://doi.org/https://doi.org/10.1007/978-3-642-33715-4_54
[50]	K. Guo, P. Lincoln, P. Davidson, J. Busch, X. Yu, M. Whalen, et al., The Relightables: Volumetric performance capture of humans with realistic relighting, ACM Trans. Graphics, 38 (2019), 1–19. https://doi.org/10.1145/3355089.3356571 doi: 10.1145/3355089.3356571
[51]	A. Horé, D. Ziou, Image quality metrics: PSNR vs. SSIM, in 2010 20th International Conference on Pattern Recognition, IEEE, (2010), 2366–2369. https://doi.org/10.1109/ICPR.2010.579
[52]	Z. Wang, A. C. Bovik, H. R. Sheikh, E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600–612. https://doi.org/10.1109/TIP.2003.819861 doi: 10.1109/TIP.2003.819861
[53]	R. Zhang, P. Isola, A. A. Efros, E. Shechtman, O. Wang, The unreasonable effectiveness of deep features as a perceptual metric, in Proceedings of the IEEE conference on computer vision and pattern recognition, IEEE, (2018), 586–595.
[54]	T. Salimans, I. Goodfellow, W. Zaremba, V. Cheung, A. Radford, X. Chen, Improved techniques for training GANs, in Advances in Neural Information Processing Systems, Curran Associates, Inc., 29 (2016), 1–9.
[55]	M. Heusel, H. Ramsauer, T. Unterthiner, B. Nessler, S. Hochreiter, GANs trained by a two time-scale update rule converge to a local nash equilibrium, in Advances in Neural Information Processing Systems, Curran Associates Inc., 30 (2017), 1–12.
[56]	M. Bińkowski, D. J. Sutherland, M. Arbel, A. Gretton, Demystifying MMD GANs, in International Conference on Learning Representations, 2018.
[57]	Z. Shi, S. Peng, Y. Xu, Y. Liao, Y. Shen, Deep generative models on 3D representations: A survey, preprint, arXiv: 2210.15663.
[58]	R. Huang, S. Zhang, T. Li, R. He, Beyond face rotation: Global and local perception GAN for photorealistic and identity preserving frontal view synthesis, in Proceedings of the IEEE International Conference on Computer Vision (ICCV), IEEE, (2017), 2439–2448.
[59]	B. Zhao, X. Wu, Z. Cheng, H. Liu, Z. Jie, J. Feng, Multi-view image generation from a single-view, in Proceedings of the 26th ACM International Conference on Multimedia, ACM, (2018), 383–391. https://doi.org/10.1145/3240508.3240536
[60]	K. Regmi, A. Borji, Cross-view image synthesis using conditional GANs, in 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2018), 3501–3510. https://doi.org/10.1109/CVPR.2018.00369
[61]	K. Regmi, A. Borji, Cross-view image synthesis using geometry-guided conditional GANs, Comput. Vision Image Understanding, 187 (2019), 102788. https://doi.org/10.1016/j.cviu.2019.07.008 doi: 10.1016/j.cviu.2019.07.008
[62]	F. Mokhayeri, K. Kamali, E. Granger, Cross-domain face synthesis using a controllable GAN, in 2020 IEEE Winter Conference on Applications of Computer Vision (WACV), IEEE, (2020), 241–249. https://doi.org/10.1109/WACV45572.2020.9093275
[63]	X. Zhu, Z. Yin, J. Shi, H. Li, D. Lin, Generative adversarial frontal view to bird view synthesis, in 2018 International Conference on 3D Vision (3DV), IEEE, (2018), 454–463. https://doi.org/10.1109/3DV.2018.00059
[64]	H. Ding, S. Wu, H. Tang, F. Wu, G. Gao, X. Jing, Cross-view image synthesis with deformable convolution and attention mechanism, in Pattern Recognition and Computer Vision, Springer, (2020), 386–397. https://doi.org/10.1007/978-3-030-60633-6_32
[65]	B. Ren, H. Tang, N. Sebe, Cascaded cross MLP-Mixer GANs for cross-view image translation, in British Machine Vision Conference, (2021), 1–14.
[66]	J. Zhu, T. Park, P. Isola, A. A. Efros, Unpaired image-to-image translation using cycle-consistent adversarial networks, in 2017 IEEE International Conference on Computer Vision (ICCV), IEEE, (2017), 2242–2251. https://doi.org/10.1109/ICCV.2017.244
[67]	M. Yin, L. Sun, Q. Li, Novel view synthesis on unpaired data by conditional deformable variational auto-encoder, in Computer Vision–ECCV 2020, Springer, (2020), 87–103. https://doi.org/10.1007/978-3-030-58604-1_6
[68]	X. Shen, J. Plested, Y. Yao, T. Gedeon, Pairwise-GAN: Pose-based view synthesis through pair-wise training, in Neural Information Processing, Springer, (2020), 507–515. https://doi.org/10.1007/978-3-030-63820-7_58
[69]	E. R. Chan, M. Monteiro, P. Kellnhofer, J. Wu, G. Wetzstein, pi-GAN: Periodic implicit generative adversarial networks for 3D-aware image synthesis, in 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2021), 5795–5805. https://doi.org/10.1109/CVPR46437.2021.00574
[70]	S. Cai, A. Obukhov, D. Dai, L. V. Gool, Pix2NeRF: Unsupervised conditional $\pi$ -GAN for single image to Neural Radiance Fields translation, in 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 3971–3980. https://doi.org/10.1109/CVPR52688.2022.00395
[71]	T. Leimkhler, G. Drettakis, FreeStyleGAN, ACM Trans. Graphics, 40 (2021), 1–15. https://doi.org/10.1145/3478513.3480538 doi: 10.1145/3478513.3480538
[72]	S. C. Medin, B. Egger, A. Cherian, Y. Wang, J. B. Tenenbaum, X. Liu, et al., MOST-GAN: 3D morphable StyleGAN for disentangled face image manipulation, in Proceedings of the AAAI Conference on Artificial Intelligence, AAAI Press, 36 (2022), 1962–1971. https://doi.org/10.1609/aaai.v36i2.20091
[73]	R. Or-El, X. Luo, M. Shan, E. Shechtman, J. J. Park, I. Kemelmacher-Shlizerman, StyleSDF: High-resolution 3D-consistent image and geometry generation, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 13503–13513.
[74]	X. Zheng, Y. Liu, P. Wang, X. Tong, SDF-StyleGAN: Implicit SDF-based StyleGAN for 3D shape generation, Comput. Graphics Forum, 41 (2022), 52–63. https://doi.org/10.1111/cgf.14602 doi: 10.1111/cgf.14602
[75]	Y. Deng, J. Yang, J. Xiang, X. Tong, GRAM: Generative radiance manifolds for 3D-aware image generation, in 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2022), 10663–10673. https://doi.org/10.1109/CVPR52688.2022.01041
[76]	J. Xiang, J. Yang, Y. Deng, X. Tong, GRAM-HD: 3D-consistent image generation at high resolution with generative radiance manifolds, in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2023), 2195–2205.
[77]	E. R. Chan, C. Z. Lin, M. A. Chan, K. Nagano, B. Pan, S. D. Mello, et al., Efficient geometry-aware 3D generative adversarial networks, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 16123–16133.
[78]	X. Zhao, F. Ma, D. Güera, Z. Ren, A. G. Schwing, A. Colburn, Generative multiplane images: Making a 2D GAN 3D-aware, in Computer Vision–ECCV 2022, Springer, (2022), 18–35. https://doi.org/10.1007/978-3-031-20065-6_2
[79]	H. A. Alhaija, A. Dirik, A. Knrig, S. Fidler, M. Shugrina, XDGAN: Multi-modal 3D shape generation in 2D space, in British Machine Vision Conference, (2022), 1–14.
[80]	K. Zhang, G. Riegler, N. Snavely, V. Koltun, NeRF++: Analyzing and improving Neural Radiance Fields, preprint, arXiv: 2010.07492.
[81]	D. Rebain, W. Jiang, S. Yazdani, K. Li, K. M. Yi, A. Tagliasacchi, DeRF: Decomposed radiance fields, in 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2021), 14148–14156. https://doi.org/10.1109/CVPR46437.2021.01393
[82]	K. Park, U. Sinha, J. T. Barron, S. Bouaziz, D. B. Goldman, S. M. Seitz, et al., Nerfies: Deformable Neural Radiance Fields, in 2021 IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2021), 5845–5854. https://doi.org/10.1109/ICCV48922.2021.00581
[83]	J. Li, Z. Feng, Q. She, H. Ding, C. Wang, G. H. Lee, MINE: Towards continuous depth MPI with NeRF for novel view synthesis, in 2021 IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2021), 12558–12568. https://doi.org/10.1109/ICCV48922.2021.01235
[84]	K. Park, U. Sinha, P. Hedman, J. T. Barron, S. Bouaziz, D. B. Goldman, et al., HyperNeRF: A higher-dimensional representation for topologically varying Neural Radiance Fields, ACM Trans. Graphics, 40 (2021), 1–12. https://doi.org/10.1145/3478513.3480487 doi: 10.1145/3478513.3480487
[85]	T. Chen, P. Wang, Z. Fan, Z. Wang, Aug-NeRF: Training stronger Neural Radiance Fields with triple-level physically-grounded augmentations, in 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 15170–15181. https://doi.org/10.1109/CVPR52688.2022.01476
[86]	T. Kaneko, AR-NeRF: Unsupervised learning of depth and defocus effects from natural images with aperture rendering Neural Radiance Fields, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 18387–18397.
[87]	X. Li, C. Hong, Y. Wang, Z. Cao, K. Xian, G. Lin, SymmNeRF: Learning to explore symmetry prior for single-view view synthesis, in Proceedings of the Asian Conference on Computer Vision (ACCV), (2022), 1726–1742.
[88]	K. Zhou, W. Li, Y. Wang, T. Hu, N. Jiang, X. Han, et al., NeRFLix: High-quality neural view synthesis by learning a degradation-driven inter-viewpoint mixer, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 12363–12374.
[89]	Z. Wang, S. Wu, W. Xie, M. Chen, V. A. Prisacariu, NeRF–: Neural Radiance Fields without known camera parameters, preprint, arXiv: 2102.07064.
[90]	B. Mildenhall, P. P. Srinivasan, R. Ortiz-Cayon, N. K. Kalantari, R. Ramamoorthi, R. Ng, et al., Local light field fusion: Practical view synthesis with prescriptive sampling guidelines, ACM Trans. Graphics, 38 (2019), 1–14. https://doi.org/10.1145/3306346.3322980 doi: 10.1145/3306346.3322980
[91]	Q. Meng, A. Chen, H. Luo, M. Wu, H. Su, L. Xu, et al., GNeRF: GAN-based Neural Radiance Field without posed camera, in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2021), 6351–6361.
[92]	R. Jensen, A. Dahl, G. Vogiatzis, E. Tola, H. Aanæs, Large scale multi-view stereopsis evaluation, in Proceedings of the 2014 IEEE Conference on Computer Vision and Pattern Recognition, IEEE, (2014), 406–413. https://doi.org/10.1109/CVPR.2014.59
[93]	Y. Jeong, S. Ahn, C. Choy, A. Anandkumar, M. Cho, J. Park, Self-calibrating Neural Radiance Fields, in 2021 IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2021), 5826–5834. https://doi.org/10.1109/ICCV48922.2021.00579
[94]	A. Knapitsch, J. Park, Q. Zhou, V. Koltun, Tanks and temples: Benchmarking large-scale scene reconstruction, ACM Trans. Graphics, 36 (2017), 1–13. https://doi.org/10.1145/3072959.3073599 doi: 10.1145/3072959.3073599
[95]	W. Bian, Z. Wang, K. Li, J. Bian, V. A. Prisacariu, NoPe-NeRF: Optimising Neural Radiance Field with no pose prior, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 4160–4169.
[96]	P. Truong, M. Rakotosaona, F. Manhardt, F. Tombari, SPARF: Neural Radiance Fields from sparse and noisy poses, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 4190–4200.
[97]	J. Straub, T. Whelan, L. Ma, Y. Chen, E. Wijmans, S. Green, et al., The replica dataset: A digital replica of indoor spaces, preprint, arXiv: 1906.05797.
[98]	J. Y. Zhang, G. Yang, S. Tulsiani, D. Ramanan, NeRS: Neural reflectance surfaces for sparse-view 3D reconstruction in the wild, in Conference on Neural Information Processing Systems, Curran Associates, Inc., 34 (2021), 29835–29847.
[99]	S. Seo, D. Han, Y. Chang, N. Kwak, MixNeRF: Modeling a ray with mixture density for novel view synthesis from sparse inputs, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 20659–20668.
[100]	A. Cao, R. D. Charette, SceneRF: Self-supervised monocular 3D scene reconstruction with radiance fields, in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2023), 9387–9398.
[101]	J. Behley, M. Garbade, A. Milioto, J. Quenzel, S. Behnke, C. Stachniss, et al., SemanticKITTI: A dataset for semantic scene understanding of lidar sequences, in 2019 IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2019), 9296–9306. https://doi.org/10.1109/ICCV.2019.00939
[102]	J. Chen, W. Yi, L. Ma, X. Jia, H. Lu, GM-NeRF: Learning generalizable model-based Neural Radiance Fields from multi-view images, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 20648–20658.
[103]	T. Yu, Z. Zheng, K. Guo, P. Liu, Q. Dai, Y. Liu, Function4D: Real-time human volumetric capture from very sparse consumer rgbd sensors, in 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2021), 5742–5752. https://doi.org/10.1109/CVPR46437.2021.00569
[104]	B. Bhatnagar, G. Tiwari, C. Theobalt, G. Pons-Moll, Multi-Garment net: Learning to dress 3D people from images, in 2019 IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2019), 5419–5429. https://doi.org/10.1109/ICCV.2019.00552
[105]	W. Cheng, S. Xu, J. Piao, C. Qian, W. Wu, K. Lin, et al., Generalizable neural performer: Learning robust radiance fields for human novel view synthesis, preprint, arXiv: 2204.11798.
[106]	S. Peng, Y. Zhang, Y. Xu, Q. Wang, Q. Shuai, H. Bao, et al., Neural Body: Implicit neural representations with structured latent codes for novel view synthesis of dynamic humans, in 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2021), 9050–9059. https://doi.org/10.1109/CVPR46437.2021.00894
[107]	B. Mildenhall, P. Hedman, R. Martin-Brualla, P. P. Srinivasan, J. T. Barron, NeRF in the dark: High dynamic range view synthesis from noisy raw images, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 16190–16199.
[108]	L. Ma, X. Li, J. Liao, Q. Zhang, X. Wang, J. Wang, et al., Deblur-NeRF: Neural Radiance Fields from blurry images, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 12861–12870.
[109]	X. Huang, Q. Zhang, Y. Feng, H. Li, X. Wang, Q. Wang, Hdr-NeRF: High dynamic range Neural Radiance Fields, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2022), 18398–18408.
[110]	P. Naama, T. Tali, K. Simon, NAN: Noise-aware NeRFs for burst-denoising, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 12672–12681.
[111]	J. T. Barron, B. Mildenhall, D. Verbin, P. P. Srinivasan, P. Hedman, Mip-NeRF 360: Unbounded anti-aliased Neural Radiance Fields, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 5470–5479.
[112]	Y. Xiangli, L. Xu, X. Pan, N. Zhao, A. Rao, C. Theobalt, et al., BungeeNeRF: Progressive Neural Radiance Field for extreme multi-scale scene rendering, in Computer Vision–ECCV 2022, Springer, (2022), 106–122. https://doi.org/10.1007/978-3-031-19824-3_7
[113]	Google, Google earth studio, 2018. Available from: https://www.google.com/earth/studio/.
[114]	M. Tancik, V. Casser, X. Yan, S. Pradhan, B. P. Mildenhall, P. Srinivasan, et al., Block-NeRF: Scalable large scene neural view synthesis, in 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 8238–8248. https://doi.org/10.1109/CVPR52688.2022.00807
[115]	L. Xu, Y. Xiangli, S. Peng, X. Pan, N. Zhao, C. Theobalt, et al., Grid-guided Neural Radiance Fields for large urban scenes, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 8296–8306.
[116]	T. Haithem, R. Deva, S. Mahadev, Mega-NERF: Scalable construction of large-scale NeRFs for virtual fly-throughs, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 12922–12931.
[117]	C. Choi, S. M. Kim, Y. M. Kim, Balanced spherical grid for egocentric view synthesis, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 16590–16599.
[118]	A. Yu, R. Li, M. Tancik, H. Li, R. Ng, A. Kanazawa, PlenOctrees for real-time rendering of Neural Radiance Fields, in 2021 IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2021), 5732–5741. https://doi.org/10.1109/ICCV48922.2021.00570
[119]	C. Sun, M. Sun, H. Chen, Direct voxel grid optimization: Super-fast convergence for radiance fields reconstruction, in 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 5449–5459. https://doi.org/10.1109/CVPR52688.2022.00538
[120]	L. Liu, J. Gu, K. Z. Lin, T. Chua, C. Theobalt, Neural sparse voxel fields, in Advances in Neural Information Processing Systems, Curran Associates, Inc., 33 (2020), 15651–15663.
[121]	Y. Yao, Z. Luo, S. Li, J. Zhang, Y. Ren, L. Zhou, et al., BlendedMVS: A large-scale dataset for generalized multi-view stereo networks, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2020), 1790–1799.
[122]	V. Sitzmann, J. Thies, F. Heide, M. Nießner, G. Wetzstein, M. Zollhöfer, DeepVoxels: Learning persistent 3D feature embeddings, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2019), 2437–2446.
[123]	H. Wang, J. Ren, Z. Huang, K. Olszewski, M. Chai, Y. Fu, et al., R2L: Distilling Neural Radiance Field to neural light field for efficient novel view synthesis, in Computer Vision–ECCV 2022, Springer, (2022), 612–629. https://doi.org/10.1007/978-3-031-19821-2_35
[124]	T. Neff, P. Stadlbauer, M. Parger, A. Kurz, J. H. Mueller, C. R. A. Chaitanya, et al., DONeRF: Towards real-time rendering of compact Neural Radiance Fields using depth oracle networks, Comput. Graphics Forum, 40 (2021), 45–59. https://doi.org/10.1111/cgf.14340 doi: 10.1111/cgf.14340
[125]	K. Wadhwani, T. Kojima, SqueezeNeRF: Further factorized FastNeRF for memory-efficient inference, in 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), IEEE, (2022), 2716–2724. https://doi.org/10.1109/CVPRW56347.2022.00307
[126]	Z. Chen, T. Funkhouser, P. Hedman, A. Tagliasacchi, MobileNeRF: Exploiting the polygon rasterization pipeline for efficient neural field rendering on mobile architectures, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2023), 16569–16578.
[127]	Y. Chen, X. Chen, X. Wang, Q. Zhang, Y. Guo, Y. Shan, et al., Local-to-global registration for bundle-adjusting Neural Radiance Fields, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2023), 8264–8273.
[128]	C. Sbrolli, P. Cudrano, M. Frosi, M. Matteucci, IC3D: Image-conditioned 3D diffusion for shape generation, preprint, arXiv: 2211.10865.
[129]	J. Gu, Q. Gao, S. Zhai, B. Chen, L. Liu, J. Susskind, Learning controllable 3D diffusion models from single-view images, preprint, arXiv: 2304.06700.
[130]	T. Anciukevičius, Z. Xu, M. Fisher, P. Henderson, H. Bilen, N. J. Mitra, et al., RenderDiffusion: Image diffusion for 3D reconstruction, inpainting and generation, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 12608–12618.
[131]	J. Xiang, J. Yang, B. Huang, X. Tong, 3D-aware image generation using 2D diffusion models, in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2023), 2383–2393.
[132]	R. Liu, R. Wu, B. V. Hoorick, P. Tokmakov, S. Zakharov, C. Vondrick, Zero-1-to-3: Zero-shot one image to 3D object, preprint, arXiv: 2303.11328.
[133]	E. R. Chan, K. Nagano, M. A. Chan, A. W. Bergman, J. J. Park, A. Levy, et al., Generative novel view synthesis with 3D-aware diffusion models, in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2023), 4217–4229.
[134]	A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, et al., An image is worth 16 x 16 words: Transformers for image recognition at scale, in International Conference on Learning Representations, 2021.
[135]	A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, et al., Attention is all you need, in Proceedings of the 31st International Conference on Neural Information Processing Systems, Curran Associates Inc., (2017), 6000–6010.
[136]	P. Nguyen-Ha, L. Huynh, E. Rahtu, J. Heikkila, Sequential view synthesis with transformer, in Proceedings of the Asian Conference on Computer Vision (ACCV), 2020.
[137]	J. Yang, Y. Li, L. Yang, Shape transformer nets: Generating viewpoint-invariant 3D shapes from a single image, J. Visual Commun. Image Represent., 81 (2021), 103345. https://doi.org/10.1016/j.jvcir.2021.103345 doi: 10.1016/j.jvcir.2021.103345
[138]	J. Kulhánek, E. Derner, T. Sattler, R. Babuška, ViewFormer: NeRF-free neural rendering from few images using transformers, in Computer Vision–ECCV 2022, Springer, (2022), 198–216. https://doi.org/10.1007/978-3-031-19784-0_12
[139]	P. Zhou, L. Xie, B. Ni, Q. Tian, CIPS-3D: A 3D-aware generator of GANs based on conditionally-independent pixel synthesis, preprint, arXiv: 2110.09788.
[140]	X. Xu, X. Pan, D. Lin, B. Dai, Generative occupancy fields for 3D surface-aware image synthesis, in Advances in Neural Information Processing Systems, Curran Associates, Inc., 34 (2021), 20683–20695.
[141]	Y. Lan, X. Meng, S. Yang, C. C. Loy, B. Dai, Self-supervised geometry-aware encoder for style-based 3D GAN inversion, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 20940–20949.
[142]	S. Li, J. van de Weijer, Y. Wang, F. S. Khan, M. Liu, J. Yang, 3D-aware multi-class image-to-image translation with NeRFs, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 12652–12662.
[143]	M. Shahbazi, E. Ntavelis, A. Tonioni, E. Collins, D. P. Paudel, M. Danelljan, et al., NeRF-GAN distillation for efficient 3D-aware generation with convolutions, in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV) Workshops, IEEE, (2023), 2888–2898.
[144]	A. Kania, A. Kasymov, M. Ziba, P. Spurek, HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN, preprint, arXiv: 2301.11631.
[145]	A. R. Bhattarai, M. Nießner, A. Sevastopolsky, TriPlaneNet: An encoder for EG3D inversion, preprint, arXiv: 2303.13497.
[146]	N. Müller, Y. Siddiqui, L. Porzi, S. R. Bulo, P. Kontschieder, M. Nießner, Diffrf: Rendering-guided 3D radiance field diffusion, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2023), 4328–4338.
[147]	D. Xu, Y. Jiang, P. Wang, Z. Fan, Y. Wang, Z. Wang, NeuralLift-360: Lifting an in-the-wild 2D photo to a 3D object with 360deg views, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 4479–4489.
[148]	H. Chen, J. Gu, A. Chen, W. Tian, Z. Tu, L. Liu, et al., Single-stage diffusion NeRF: A unified approach to 3D generation and reconstruction, in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2023), 2416–2425.
[149]	J. Gu, A. Trevithick, K. Lin, J. Susskind, C. Theobalt, L. Liu, et al., NerfDiff: Single-image view synthesis with NeRF-guided distillation from 3D-aware diffusion, in International Conference on Machine Learning, PMLR, (2023), 11808–11826.
[150]	D. Wang, X. Cui, S. Salcudean, Z. J. Wang, Generalizable Neural Radiance Fields for novel view synthesis with transformer, preprint, arXiv: 2206.05375.
[151]	K. Lin, L. Yen-Chen, W. Lai, T. Lin, Y. Shih, R. Ramamoorthi, Vision transformer for NeRF-based view synthesis from a single input image, in 2023 IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), IEEE, (2023), 806–815. https://doi.org/10.1109/WACV56688.2023.00087
[152]	J. Liu, Q. Nie, Y. Liu, C. Wang, NeRF-Loc: Visual localization with conditional Neural Radiance Field, preprint, arXiv: 2304.07979.
[153]	Y. Liao, K. Schwarz, L. Mescheder, A. Geiger, Towards unsupervised learning of generative models for 3D controllable image synthesis, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2020), 5871–5880.
[154]	T. Nguyen-Phuoc, C. Richardt, L. Mai, Y. Yang, N. Mitra, BlockGAN: Learning 3D object-aware scene representations from unlabelled images, in Advances in Neural Information Processing Systems, Curran Associates, Inc., 33 (2020), 6767–6778.
[155]	X. Pan, B. Dai, Z. Liu, C. C. Loy, P. Luo, Do 2D GANs know 3D shape? Unsupervised 3D shape reconstruction from 2D image GANs, in International Conference on Learning Representations, 2021.
[156]	A. Tewari, M. B. R, X. Pan, O. Fried, M. Agrawala, C. Theobalt, Disentangled3D: Learning a 3D generative model with disentangled geometry and appearance from monocular images, in 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 1506–1515. https://doi.org/10.1109/CVPR52688.2022.00157
[157]	S. Kobayashi, E. Matsumoto, V. Sitzmann, Decomposing NeRF for editing via feature field distillation, in Advances in Neural Information Processing Systems, Curran Associates, Inc., 35 (2022), 23311–23330.
[158]	X. Zhang, K. Abhijit, F. Thomas, G. Leonidas, S. Hao, G. Kyle, Nerflets: Local radiance fields for efficient structure-aware 3D scene representation from 2D supervision, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 8274–8284.
[159]	C. Zheng, W. Lin, F. Xu, EditableNeRF: Editing topologically varying Neural Radiance Fields by key points, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 8317–8327.
[160]	J. Zhang, L. Yang, MonodepthPlus: Self-supervised monocular depth estimation using soft-attention and learnable outlier-masking, J. Electron. Imaging, 30 (2021), 023017. https://doi.org/10.1117/1.JEI.30.2.023017 doi: 10.1117/1.JEI.30.2.023017
[161]	R. Liang, J. Zhang, H. Li, C. Yang, Y. Guan, N. Vijaykumar, SPIDR: SDF-based neural point fields for illumination and deformation, preprint, arXiv: 2210.08398.
[162]	Y. Zhang, X. Huang, B. Ni, T. Li, W. Zhang, Frequency-modulated point cloud rendering with easy editing, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 119–129.
[163]	J. Chen, J. Lyu, Y. Wang, NeuralEditor: Editing Neural Radiance Fields via manipulating point clouds, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2023), 12439–12448.
[164]	J. Zhu, Z. Zhang, C. Zhang, J. Wu, A. Torralba, J. B. Tenenbaum, et al., Visual object networks: Image generation with disentangled 3D representations, in Advances in Neural Information Processing Systems, Curran Associates, Inc., 31 (2018).
[165]	A. Mirzaei, T. Aumentado-Armstrong, M. A. Brubaker, J. Kelly, A. Levinshtein, K. G. Derpanis, et al., Reference-guided controllable inpainting of Neural Radiance Fields, in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2023), 17815–17825.
[166]	Y. Yin, Z. Fu, F. Yang, G. Lin, OR-NeRF: Object removing from 3D scenes guided by multiview segmentation with Neural Radiance Fields, preprint, arXiv: 2305.10503.
[167]	H. G. Kim, M. Park, S. Lee, S. Kim, Y. M. Ro, Visual comfort aware-reinforcement learning for depth adjustment of stereoscopic 3D images, in Proceedings of the AAAI Conference on Artificial Intelligence, AAAI Press, 35 (2021), 1762–1770. https://doi.org/10.1609/aaai.v35i2.16270
[168]	R. Jheng, T. Wu, J. Yeh, W. H. Hsu, Free-form 3D scene inpainting with dual-stream GAN, in British Machine Vision Conference, 2022.
[169]	Q. Wang, Y. Wang, M. Birsak, P. Wonka, BlobGAN-3D: A spatially-disentangled 3D-aware generative model for indoor scenes, preprint, arXiv: 2303.14706.
[170]	J. Gu, L. Liu, P. Wang, C. Theobalt, StyleNeRF: A style-based 3D aware generator for high-resolution image synthesis, in Tenth International Conference on Learning Representations, (2022), 1–25.
[171]	C. Wang, M. Chai, M. He, D. Chen, J. Liao, CLIP-NeRF: Text-and-image driven manipulation of Neural Radiance Fields, in 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 3825–3834. https://doi.org/10.1109/CVPR52688.2022.00381
[172]	K. Kania, K. M. Yi, M. Kowalski, T. Trzciński, A. Tagliasacchi, CoNeRF: Controllable Neural Radiance Fields, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, IEEE, (2022), 18623–18632.
[173]	V. Lazova, V. Guzov, K. Olszewski, S. Tulyakov, G. Pons-Moll, Control-NeRF: Editable feature volumes for scene rendering and manipulation, in 2023 IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), IEEE, (2023), 4329–4339. https://doi.org/10.1109/WACV56688.2023.00432
[174]	Y. Yuan, Y. Sun, Y. La, Y. Ma, R. Jia, L. Gao, NeRF-Editing: Geometry editing of Neural Radiance Fields, in 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 18332–18343. https://doi.org/10.1109/CVPR52688.2022.01781
[175]	C. Sun, Y. Liu, J. Han, S. Gould, NeRFEditor: Differentiable style decomposition for full 3D scene editing, preprint, arXiv: 2212.03848.
[176]	Z. Wang, Y. Deng, J. Yang, J. Yu, X. Tong, Generative deformable radiance fields for disentangled image synthesis of topology-varying objects, Comput. Graphics Forum, 41 (2022), 431–442. https://doi.org/10.1111/cgf.14689 doi: 10.1111/cgf.14689
[177]	K. Tertikas, D. Paschalidou, B. Pan, J. J. Park, M. A. Uy, I. Emiris, et al., PartNeRF: Generating part-aware editable 3D shapes without 3D supervision, preprint, arXiv: 2303.09554.
[178]	C. Bao, Y. Zhang, B. Yang, T. Fan, Z. Yang, H. Bao, et al., SINE: Semantic-driven image-based NeRF editing with prior-guided editing field, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2023), 20919–20929.
[179]	D. Cohen-Bar, E. Richardson, G. Metzer, R. Giryes, D. Cohen-Or, Set-the-Scene: Global-local training for generating controllable NeRF scenes, in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV) Workshops, IEEE, (2023), 2920–2929.
[180]	A. Mirzaei, T. Aumentado-Armstrong, K. G. Derpanis, J. Kelly, M. A. Brubaker, I. Gilitschenski, et al., SPIn-NeRF: Multiview segmentation and perceptual inpainting with Neural Radiance Fields, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2023), 20669–20679.
[181]	O. Avrahami, D. Lischinski, O. Fried, Blended diffusion for text-driven editing of natural images, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 18208–18218.
[182]	A. Nichol, P. Dhariwal, A. Ramesh, P. Shyam, P. Mishkin, B. McGrew, et al., GLIDE: Towards photorealistic image generation and editing with text-guided diffusion models, in Proceedings of the 39th International Conference on Machine Learning, PMLR, (2022), 16784–16804.
[183]	G. Couairon, J. Verbeek, H. Schwenk, M. Cord, DiffEdit: Diffusion-based semantic image editing with mask guidance, in the Eleventh International Conference on Learning Representations, 2023.
[184]	E. Sella, G. Fiebelman, P. Hedman, H. Averbuch-Elor, Vox-E: Text-guided voxel editing of 3D objects, in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2023), 430–440.
[185]	A. Haque, M. Tancik, A. A. Efros, A. Holynski, A. Kanazawa, Instruct-NeRF2NeRF: Editing 3D scenes with instructions, in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2023), 19740–19750.
[186]	Y. Lin, H. Bai, S. Li, H. Lu, X. Lin, H. Xiong, et al., CompoNeRF: Text-guided multi-object compositional NeRF with editable 3D scene layout, preprint, arXiv: 2303.13843.
[187]	R. Martin-Brualla, N. Radwan, M. S. M. Sajjadi, J. T. Barron, A. Dosovitskiy, D. Duckworth, NeRF in the wild: Neural Radiance Fields for unconstrained photo collections, in 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2021), 7206–7215. https://doi.org/10.1109/CVPR46437.2021.00713
[188]	M. Boss, A. Engelhardt, A. Kar, Y. Li, D. Sun, J. T. Barron, et al., SAMURAI: Shape and material from unconstrained real-world arbitrary image collections, in Advances in Neural Information Processing Systems, Curran Associates, Inc., 35 (2022), 26389–26403.
[189]	C. Choi, J. Kim, Y. M. Kim, IBL-NeRF: Image-based lighting formulation of Neural Radiance Fields, preprint, arXiv: 2210.08202.
[190]	Z. Yan, C. Li, G. H. Lee, NeRF-DS: Neural Radiance Fields for dynamic specular objects, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 8285–8295.
[191]	D. Guo, L. Zhu, S. Ling, T. Li, G. Zhang, Q. Yang, et al., Face illumination normalization based on Generative Adversarial Network, Nat. Comput., 22 (2022), 105–117. https://doi.org/10.1007/s11047-022-09892-4 doi: 10.1007/s11047-022-09892-4
[192]	Z. Cui, L. Gu, X. Sun, Y. Qiao, T. Harada, Aleth-NeRF: Low-light condition view synthesis with concealing fields, preprint, arXiv: 2303.05807.
[193]	A. R. Nandhini, V. P. D. Raj, Low-light image enhancement based on generative adversarial network, Front. Genet., 12 (2021), 799777. https://doi.org/10.3389/fgene.2021.799777 doi: 10.3389/fgene.2021.799777
[194]	W. Kim, R. Lee, M. Park, S. Lee, Low-light image enhancement based on maximal diffusion values, IEEE Access, 7 (2019), 129150–129163. https://doi.org/10.1109/ACCESS.2019.2940452 doi: 10.1109/ACCESS.2019.2940452
[195]	P. Ponglertnapakorn, N. Tritrong, S. Suwajanakorn, DiFaReli: Diffusion face relighting, preprint, arXiv: 2304.09479.
[196]	M. Guo, A. Fathi, J. Wu, T. Funkhouser, Object-centric neural scene rendering, preprint, arXiv: 2012.08503.
[197]	Y. Wang, W. Zhou, Z. Lu, H. Li, UDoc-GAN: Unpaired document illumination correction with background light prior, in Proceedings of the 30th ACM International Conference on Multimedia, ACM, (2022), 5074–5082. https://doi.org/10.1145/3503161.3547916
[198]	J. Ling, Z. Wang, F. Xu, ShadowNeuS: Neural SDF reconstruction by shadow ray supervision, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2023), 175–185.
[199]	V. Rudnev, M. Elgharib, W. Smith, L. Liu, V. Golyanik, C. Theobalt, NeRF for outdoor scene relighting, in Computer Vision–ECCV 2022, Springer, (2022), 615–631. https://doi.org/10.1007/978-3-031-19787-1_35
[200]	C. Higuera, B. Boots, M. Mukadam, Learning to read braille: Bridging the tactile reality gap with diffusion models, preprint, arXiv: 2304.01182.
[201]	T. Guo, D. Kang, L. Bao, Y. He, S. Zhang, NeRFReN: Neural Radiance Fields with reflections, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2022), 18409–18418.
[202]	C. LeGendre, W. Ma, G. Fyffe, J. Flynn, L. Charbonnel, J. Busch, et al., DeepLight: Learning illumination for unconstrained mobile mixed reality, in 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2019), 5911–5921. https://doi.org/10.1109/CVPR.2019.00607
[203]	W. Ye, S. Chen, C. Bao, H. Bao, M. Pollefeys, Z. Cui, et al., IntrinsicNeRF: Learning intrinsic Neural Radiance Fields for editable novel view synthesis, in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2023), 339–351.
[204]	M. Boss, V. Jampani, R. Braun, C. Liu, J. T. Barron, H. P. A. Lensch, Neural-PIL: Neural pre-integrated lighting for reflectance decomposition, in Advances in Neural Information Processing Systems, Curran Associates, Inc., 34 (2021), 10691–10704.
[205]	S. Saito, T. Simon, J. Saragih, H. Joo, PIFuHD: Multi-level pixel-aligned implicit function for high-resolution 3D human digitization, in 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2020), 81–90. https://doi.org/10.1109/CVPR42600.2020.00016
[206]	H. Tang, S. Bai, L. Zhang, P. H. Torr, N. Sebe, XingGAN for person image generation, in Computer Vision–ECCV 2020, Springer, (2020), 717–734. https://doi.org/10.1007/978-3-030-58595-2_43
[207]	Y. Ren, X. Yu, J. Chen, T. H. Li, G. Li, Deep image spatial transformation for person image generation, in 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2020), 7687–7696. https://doi.org/10.1109/CVPR42600.2020.00771
[208]	Y. Liu, Z. Qin, T. Wan, Z. Luo, Auto-painter: Cartoon image generation from sketch by using conditional wasserstein generative adversarial networks, Neurocomputing, 311 (2018), 78–87. https://doi.org/10.1016/j.neucom.2018.05.045 doi: 10.1016/j.neucom.2018.05.045
[209]	H. Li, AI synthesis for the metaverse: From avatars to 3D scenes, Stanford University, Stanford Talks, 2022. Available from: https://talks.stanford.edu/hao-li-pinscreen-on-ai-synthesis-for-the-metaverse-from-avatars-to-3d-scenes/.
[210]	S. Murray, A. Tallon, Mapping gothic france, Columbia University, Media Center for Art History, 2023. Available from: https://mcid.mcah.columbia.edu/art-atlas/mapping-gothic.
[211]	Y. Xiang, C. Lv, Q. Liu, X. Yang, B. Liu, M. Ju, A creative industry image generation dataset based on captions, preprint, arXiv: 2211.09035.
[212]	C. Tatsch, J. A. Bredu, D. Covell, I. B. Tulu, Y. Gu, Rhino: An autonomous robot for mapping underground mine environments, in 2023 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM), IEEE, (2023), 1166–1173. https://doi.org/10.1109/AIM46323.2023.10196202
[213]	Y. Tian, L. Li, A. Fumagalli, Y. Tadesse, B. Prabhakaran, Haptic-enabled mixed reality system for mixed-initiative remote robot control, preprint, arXiv: 2102.03521.
[214]	G. Pu, Y. Men, Y. Mao, Y. Jiang, W. Ma, Z. Lian, Controllable image synthesis with attribute-decomposed GAN, IEEE Trans. Pattern Anal. Mach. Intell., 45 (2023), 1514–1532. https://doi.org/10.1109/TPAMI.2022.3161985 doi: 10.1109/TPAMI.2022.3161985
[215]	X. Wu, Y. Zhang, Q. Li, Y. Qi, J. Wang, Y. Guo, Face aging with pixel-level alignment GAN, Appl. Intell., 52 (2022), 14665–14678. https://doi.org/10.1007/s10489-022-03541-0 doi: 10.1007/s10489-022-03541-0
[216]	D. Sero, A. Zaidi, J. Li, J. D. White, T. B. G. Zarzar, M. L. Marazita, et al., Facial recognition from dna using face-to-dna classifiers, Nat. Commun., 10 (2019), 1. https://doi.org/10.1038/s41467-018-07882-8 doi: 10.1038/s41467-018-07882-8
[217]	M. Nicolae, M. Sinn, M. Tran, B. Buesser, A. Rawat, M. Wistuba, et al., Adversarial robustness toolbox v1.0.0, 2018. Available from: https://github.com/Trusted-AI/adversarial-robustness-toolbox.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)