Weakly supervised training for eye fundus lesion segmentation in patients with diabetic retinopathy

Yu Li; Meilong Zhu; Guangmin Sun; Jiayang Chen; Xiaorong Zhu; Jinkui Yang; Yu Li; Meilong Zhu; Guangmin Sun; Jiayang Chen; Xiaorong Zhu; Jinkui Yang

doi:10.3934/mbe.2022248

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 5: 5293-5311. doi: 10.3934/mbe.2022248

Previous Article Next Article

Research article Special Issues

Weakly supervised training for eye fundus lesion segmentation in patients with diabetic retinopathy

1.
Faculty of Information Technology, Beijing University of Technology, Beijing 100124, China
2.
School of Computer Science and Engineering, The Chinese University of Hong Kong, Hong Kong, China
3.
Beijing Tongren Hospital, Beijing 100730, China
4.
Beijing Institute of Diabetes Research, Beijing 100730, China

Academic Editor: Simon James Fong

Received: 18 December 2021 Revised: 05 March 2022 Accepted: 16 March 2022 Published: 24 March 2022

Objective

Diabetic retinopathy is the leading cause of vision loss in working-age adults. Early screening and diagnosis can help to facilitate subsequent treatment and prevent vision loss. Deep learning has been applied in various fields of medical identification. However, current deep learning-based lesion segmentation techniques rely on a large amount of pixel-level labeled ground truth data, which limits their performance and application. In this work, we present a weakly supervised deep learning framework for eye fundus lesion segmentation in patients with diabetic retinopathy.

Methods

First, an efficient segmentation algorithm based on grayscale and morphological features is proposed for rapid coarse segmentation of lesions. Then, a deep learning model named Residual-Attention Unet (RAUNet) is proposed for eye fundus lesion segmentation. Finally, a data sample of fundus images with labeled lesions and unlabeled images with coarse segmentation results is jointly used to train RAUNet to broaden the diversity of lesion samples and increase the robustness of the segmentation model.

Results

A dataset containing 582 fundus images with labels verified by doctors, including hemorrhage (HE), microaneurysm (MA), hard exudate (EX) and soft exudate (SE), and 903 images without labels was used to evaluate the model. In ablation test, the proposed RAUNet achieved the highest intersection over union (IOU) on the labeled dataset, and the proposed attention and residual modules both improved the IOU of the UNet benchmark. Using both the images labeled by doctors and the proposed coarse segmentation method, the weakly supervised framework based on RAUNet architecture significantly improved the mean segmentation accuracy by over 7% on the lesions.

Significance

This study demonstrates that combining unlabeled medical images with coarse segmentation results can effectively improve the robustness of the lesion segmentation model and proposes a practical framework for improving the performance of medical image segmentation given limited labeled data samples.

Keywords:

Citation: Yu Li, Meilong Zhu, Guangmin Sun, Jiayang Chen, Xiaorong Zhu, Jinkui Yang. Weakly supervised training for eye fundus lesion segmentation in patients with diabetic retinopathy[J]. Mathematical Biosciences and Engineering, 2022, 19(5): 5293-5311. doi: 10.3934/mbe.2022248

Related Papers:

[1]	Muhammad Asif Zahoor Raja, Adeeba Haider, Kottakkaran Sooppy Nisar, Muhammad Shoaib . Intelligent computing knacks for infected media and time delay impacts on dynamical behaviors and control measures of rumor-spreading model. AIMS Biophysics, 2024, 11(1): 1-17. doi: 10.3934/biophy.2024001
[2]	Julia C. Arciero, Paola Causin, Francesca Malgaroli . Mathematical methods for modeling the microcirculation. AIMS Biophysics, 2017, 4(3): 362-399. doi: 10.3934/biophy.2017.3.362
[3]	Mario Lefebvre . A Wiener process with jumps to model the logarithm of new epidemic cases. AIMS Biophysics, 2022, 9(3): 271-281. doi: 10.3934/biophy.2022023
[4]	Ramesh Rijal, Hari Prasad Lamichhane, Kiran Pudasainee . Molecular structure, homo-lumo analysis and vibrational spectroscopy of the cancer healing pro-drug temozolomide based on dft calculations. AIMS Biophysics, 2022, 9(3): 208-220. doi: 10.3934/biophy.2022018
[5]	Dafina Xhako, Niko Hyka, Elda Spahiu, Suela Hoxhaj . Medical image analysis using deep learning algorithms (DLA). AIMS Biophysics, 2025, 12(2): 121-143. doi: 10.3934/biophy.2025008
[6]	T. M. Medvedeva, A. K. Lüttjohann, M. V. Sysoeva, G. van Luijtelaar, I. V. Sysoev . Estimating complexity of spike-wave discharges with largest Lyapunov exponent in computational models and experimental data. AIMS Biophysics, 2020, 7(2): 65-75. doi: 10.3934/biophy.2020006
[7]	Kieran Greer . New ideas for brain modelling 6. AIMS Biophysics, 2020, 7(4): 308-322. doi: 10.3934/biophy.2020022
[8]	Dina Falah Noori Al-Sabak, Leila Sadeghi, Gholamreza Dehghan . Predicting the correlation between neurological abnormalities and thyroid dysfunction using artificial neural networks. AIMS Biophysics, 2024, 11(4): 403-426. doi: 10.3934/biophy.2024022
[9]	Soumyajit Podder, Abhishek Mallick, Sudipta Das, Kartik Sau, Arijit Roy . Accurate diagnosis of liver diseases through the application of deep convolutional neural network on biopsy images. AIMS Biophysics, 2023, 10(4): 453-481. doi: 10.3934/biophy.2023026
[10]	Arjun Acharya, Madan Khanal, Rajesh Maharjan, Kalpana Gyawali, Bhoj Raj Luitel, Rameshwar Adhikari, Deependra Das Mulmi, Tika Ram Lamichhane, Hari Prasad Lamichhane . Quantum chemical calculations on calcium oxalate and dolichin A and their binding efficacy to lactoferrin: An in silico study using DFT, molecular docking, and molecular dynamics simulations. AIMS Biophysics, 2024, 11(2): 142-165. doi: 10.3934/biophy.2024010

Abstract

Objective

Methods

Results

Significance

1. Introduction

Numerous studies on recurrent neural networks (RNNs), including those on bidirectional associative memory neural networks, cellular neural networks, and Hopfield neural networks, among others, have been proposed. In the meantime, neural networks (NNs) have been employed as a tool to address issues that have arisen in associative memory, signal processing, image processing, static image treatment, pattern recognition, and optimization. Additionally, it seems that time delays are crucial to networked control systems, chemical reactions, communication systems, etc. The authors of [1], [2] investigated time-delayed artificial neural network electronic implementations. Several studies have looked into different sorts of delays inside NNs in this context; see the papers [38]–[40] and references therein. The passivity theory is widely applied in many engineering problems. Indeed, it is intimately related to the circuit analysis which is a useful and significant tool to analyze the stability of nonlinear systems, signal processing and chaos control. Thus, it has been employed in various fields of science and engineering [7], [8], [44]–[48]. In [9], the authors proposed neural adaptive output feedback control based on passivity with adaptive set-point regulation of nonlinear uncertain non-negative and compartmental systems. In the last decade, great attention has been paid to the passivity analysis of RNNs with delay-independent [10], [49] and delay-dependent [11]–[14], [36]. In [13], the author studied the passivity analysis of Markovian jump neural network with leakage time varying delay, discrete and distributed time varying delays. We can see the discussion of the extended dissipativity of discrete-time neural networks NNs with time-varying delay in [55]. However, comparatively less interest has been shown towards the passivity analysis of mode dependent delay on neutral type neural networks (NNNs) with Markovian jumping (Mj). On the other hand, a Markov jump system is a special classes of a hybrid system. Indeed, they have great ability to model the dynamical systems and their application can be found in manufacturing systems, economic systems, network control systems, modeling production system, communication systems and so on. In the recent years, several results are reported on the stability analysis for neural networks with Markovian jump parameters, see the references [16]–[19], [41]–[43]. In [20], the authors studied global exponential estimates of delayed stochastic NNs with Markovian switching by constructing with positive definite matrices in stochastic Lyapunov functional which are dependent on the system mode and a triple-integral term. The ℋ_∞ synchronization issue for singularly perturbed coupled neural networks (SPCNNs) affected by both nonlinear constraints and gain uncertainties was explored in [57] using a novel double-layer switching regulation containing Markov chain and persistent dwell-time switching regulation (PDTSR). Convolutional Neural Networks (CNNs) are efficient tools for pattern recognition applications. More on this topic can be seen in [58], [60]. An exponential synchronization problem for the multi-weighted complex dynamical network (MCDN) with hybrid delays on a time scale is investigated in [61]. We can see the establishment of fixed-point and coincidence-point consequences in generalized metric spaces in [59]. The nonfragile ℋ_∞ synchronization issue for a class of discrete-time Takagi–Sugeno (T–S) fuzzy Markov jump systems was investigated in [56]. In [21], the authors have studied stochastic NNNs with mixed time-delays under adaptive synchronization. Additionally, the problem of state estimation of RNNs with Mj parameters and mixed delays based on mode-dependent approach was investigated in [22].

However, some of the researchers discussed the robust passive filtering for NNNs with delays in [25]. In [26], the authors investigated the global asymptotic stability of NNNs with delays by utilizing the Lyapunov-Krasovskii functional (LKF) and the linear matrix inequality approach. While employing the method of Lyapunov–Krasovskii functional, we necessarily need these three steps for the derivation of a global asymptotic stability criterion: constructing a Lyapunov–Krasovskii functional, estimating the derivative of the Lyapunov–Krasovskii functional, and formulating a global asymptotic stability criterion. You will get an overview of recent developments in each of the above steps if you refer to [54].The author in [52] studied passivity and exponential passivity for NNNs with various delays. In [51], the authors investigated the robust passivity analysis of mixed delayed NNs with distributed time-varying delays. The exponential passivity of discrete-time switched NNs with transmission delay was studied in [53]. Recently, the authors in [5] studied passivity analysis for NNNs with Mj parameters and time delay in the leakage term. New delay-dependent passivity conditions are derived in terms of LMIs with a proper construction of LKF, and it can be checked easily via standard numerical packages. However, triple and quadruple integrals have not been taken into account to derive the passivity conditions and, moreover, the mode-dependent time delays have not been included in [5]. Recently, a novel Lyapunov functional with some terms involving triple or quadruple integrals are taken into account to study the state estimation problem with mode-dependent approach in [22]. Motivated by the above discussion, the main purpose of this paper is to study the global passivity of Mj for NNNs with leakage and mode-dependent delay terms. By construction of a new LKF involving mode-dependent Lyapunov matrices, some sufficient conditions are derived in terms of LMIs. For the sake of illustration, a numerical example is given to demonstrate the usefulness and effectiveness of the presented results. Unlike previous results, we will introduce an improved Lyapunov–Krasovskii functional with triple and quadruple integrals for deriving the reported stability results in this paper. Based on this discussion, our technique not only provides different approach but also gives less conservative conditions than those studied in [5], [22]. The rest of this paper is organized as follows. The problem and some preliminaries are introduced in Section 2. In Section 3, the main results are stated and proved. Some sufficient conditions for global passivity results are developed here. In Section 4, an illustrative example is provided to demonstrate the effectiveness of the proposed criteria. We conclude the results of this paper in Section 5.

Notations: Throughout this paper the following notations are used: $ℝ^{n}$ - n-dimensional Euclidean space; $ℝ^{n \times n}$ - the set of all $n \times n$ real matrices; $d i a g (\dots)$ - a block diagonal matrix; I -the identity matrix with compatible dimensions; $C^{T}$ - the transpose of C; X and Y are symmetric matrices, where $X \geq Y$ (similarly $X > Y$ )- $X - Y$ is a positive semi-definite (similarly positive definite); $(Ω, ℱ, 𝒫)$ - a complete probability space with a natural filtration ${ℱ_{t}}_{t \geq 0}$ ; $E [\cdot]$ - expectation operator with respect to the given probability measure $𝒫$ ; $𝒞 ([- d,0]; ℝ^{n})$ - the family of continuously differentiable function; $‖ φ ‖ = m a x {m a x_{- τ \leq θ \leq 0} | φ (θ) |, m a x_{- d \leq θ \leq 0} | φ^{'} (θ) |}$ ; $𝒞_{ℱ_{0}}^{2} ([- d,0]; ℝ^{n})$ - the family of bounded $ℱ_{0}$ -measurable; $𝒞 ([- d,0]; ℝ^{n})$ -valued stochastic variables $ξ = {ξ (θ) : - d \leq θ \leq 0}$ such that $\int_{- d}^{0} E | ξ (θ {) |}^{2} d s < \infty$ ; *-the symmetric block in one symmetric matrix.

2. Problem formulation and preliminaries

Let ${\nabla (t), t \in ℤ^{+}}$ be right-continuous on $(Ω, ℱ, 𝒫)$ . Here $(Ω, ℱ, 𝒫)$ is a complete probability space with Markov chain, $\nabla (t)$ takes the values from a finite state space $𝒮 = {1,2, \dots, N}$ with generator $Γ = (π_{i j})_{N \times N}$ given by

$P {\nabla (t + Δ t) = j | \nabla (t) = i} = {\begin{array}{l} π_{i j} Δ t + o (Δ t), i \neq j, \\ 1 + π_{i i} Δ t + o (Δ t), i = j, \end{array}$

where $Δ t > 0$ and ${l i m}_{Δ t \to 0} \frac{o (Δ t)}{Δ t} = 0$ , $π_{i j} \geq 0$ $(i \neq j)$ is the transition rate from i to j, and $π_{i i} = - \sum_{j = 1, j \neq i}^{N} π_{i j}$ .

We consider Mode-dependent Markov jump $N N N s$ with mixed time-delays:

${\begin{array}{l} \dot{x} (t) & = & - 𝒜 (\nabla (t)) x (t - σ (\nabla (t))) + ℬ (\nabla (t)) ℋ (x (t)) + 𝒞 (\nabla (t)) ℋ (x (t - τ (t, \nabla (t)))) \\ + 𝒟 (\nabla (t)) \dot{x} (t - h (t, \nabla (t))) + ℰ (\nabla (t)) \int_{t - d (t, \nabla (t))}^{t} ℋ (x (s)) d s + u (t) \\ y (t) & = & ℋ (x (t)) \end{array}$ (2.1)

where $x (t) = [x_{1} (t), x_{2} (t), \dots, x_{n} (t {)]}^{T} \in ℝ^{n}$ is the state vector linked with n neurons. The diagonal matrix $𝒜 (\nabla (t)) = d i a g (A_{1} (\nabla (t)), A_{2} (\nabla (t)), \dots, A_{n} (\nabla (t)))$ has positive entries $A_{i} (\nabla (t)) > 0$ $(i = 1,2, \dots, n)$ . $ℬ (\nabla (t))$ , $𝒞 (\nabla (t))$ , $𝒟 (\nabla (t))$ , $ℰ (\nabla (t))$ are known appropriate dimensional constant matrices. Here the neuron activation function is $ℋ (x (t)) = [ℋ_{1} (x_{1} (t)), ℋ_{2} (x_{2} (t)), \dots, ℋ_{n} (x_{n} (t {))]}^{T}$ . $u (t)$ denotes a constant input. $τ (t, \nabla (t))$ , $h (t, \nabla (t))$ , $d (t, \nabla (t))$ are mode dependent discrete, neutral and distributed delays, respectively and $σ (\nabla (t))$ is the mode dependent leakage delay.

Throughout this paper, we assume the following.

Assumption 1. For any $j = 1,2, \dots, n$ , $ℋ_{j} (0) = 0$ and there exist constants ${\hat{l}}_{j}^{-}$ and ${\hat{l}}_{j}^{+}$ such that

${\hat{l}}_{j}^{-} \leq \frac{ℋ_{j} (γ_{1}) - ℋ_{j} (γ_{2})}{γ_{1} - γ_{2}} \leq {\hat{l}}_{j}^{+},$ (2.2)

where $γ_{1}, γ_{2} \in ℝ$ , and $γ_{1} \neq γ_{2}$ .

For the sake of convenience, we denote $𝒜 (\nabla (t) = i) = 𝒜_{i}, ℬ (\nabla (t) = i) = ℬ_{i}, 𝒞 (\nabla (t) = i) = 𝒞_{i}$ , $𝒟 (\nabla (t) = i) = 𝒟_{i}$ , $ℰ (\nabla (t) = i) = ℰ_{i}$ , respectively.

System (2.1) can be rewritten as

${\begin{array}{l} \dot{x} (t) & = & - 𝒜_{i} x (t - σ_{i}) + ℬ_{i} ℋ (x (t)) + 𝒞_{i} ℋ (x (t - τ_{i} (t))) + 𝒟_{i} \dot{x} (t - h_{i} (t)) \\ + ℰ_{i} \int_{t - d_{i} (t)}^{t} ℋ (x (s)) d s + u (t) \\ y (t) & = & ℋ (x (t)) \end{array}$ (2.3)

and the parameters associated with time delays are assumed to satisfy following:

$0 \leq τ_{i} (t) \leq τ_{i}, {\dot{τ}}_{i} (t) \leq τ_{µ_{i}}, 0 \leq h_{i} (t) \leq h_{i}, {\dot{h}}_{i} (t) \leq h_{µ_{i}}, 0 \leq d_{i} (t) \leq d_{i}, {\dot{d}}_{i} (t) \leq d_{µ_{i}}, σ_{i} > 0$ (2.4)

where $τ_{i}, h_{i}, d_{i}, τ_{µ_{i}}, h_{µ_{i}}$ and $d_{µ_{i}}$ are some real constants and $τ = \underset{i \in S}{m a x} {τ_{i}}$ , $h = \underset{i \in S}{m a x} {h_{i}}$ , $d = \underset{i \in S}{m a x} {d_{i}}$ , $σ = \underset{i \in S}{m a x} {σ_{i}}$ .

Now we can see a few necessary lemmas and a definition.

$(b - a) [\int_{a}^{b} y {(s)}^{T} M y (s) d s] \geq {[\int_{a}^{b} y (s) d s]}^{T} M [\int_{a}^{b} y (s) d s] .$

holds.

Proof. By Schur complement,

$[\begin{matrix} y {(s)}^{T} M y (s) & y {(s)}^{T} \\ y (s) & M^{- 1} \end{matrix}] \geq 0, s \in [a, b] .$

On integration from a to b yields, $[\begin{matrix} \int_{a}^{b} y {(s)}^{T} M y (s) d s & \int_{a}^{b} y {(s)}^{T} d s \\ \int_{a}^{b} y (s) d s & (b - a) M^{- 1} \end{matrix}]$ $\geq 0, s \in [a, b] .$ Now using the Schur complement on this inequality, we obtain our desired result. □

$\pm 2 x^{T} y \leq x^{T} ℳ x + y^{T} ℳ^{- 1} y .$

Lemma 3. [31] (Schur complement) Given $Ω_{1}, Ω_{2}$ and $Ω_{3}$ are constant matrices with appropriate dimensions, where $Ω_{1}, Ω_{2} > 0$ are symmetric matrices, then

$Ω_{1} + Ω_{3}^{T} Ω_{2}^{- 1} Ω_{3} < 0 \Leftrightarrow [\begin{matrix} Ω_{1} & Ω_{3}^{T} \\ * & - Ω_{2} \end{matrix}] < 0, o r [\begin{matrix} - Ω_{2} & Ω_{3} \\ * & Ω_{1} \end{matrix}] < 0.$

Definition 1. [32] If there exists a scalar $ν \geq 0$ such that $\forall t_{p} \geq 0$ and for all solutions of (2.1), the following inequality holds under zero initial conditions,

$2 \int_{0}^{t_{p}} E {y {(s)}^{T} u (s)} d s \geq - γ \int_{0}^{t_{p}} E {u {(s)}^{T} u (s)} d s,$ (2.5)

then the system (2.3) is said to be passive.

3. Main results

Now, we denote

${\hat{L}}_{1} = d i a g {{\hat{l}}_{1}^{-} {\hat{l}}_{1}^{+}, {\hat{l}}_{2}^{-} {\hat{l}}_{2}^{+}, \dots, {\hat{l}}_{m}^{-} {\hat{l}}_{m}^{+}}, a n d {\hat{L}}_{2} = d i a g {\frac{{\hat{l}}_{1}^{-} + {\hat{l}}_{1}^{+}}{2}, \frac{{\hat{l}}_{2}^{-} + {\hat{l}}_{2}^{+}}{2}, \dots, \frac{{\hat{l}}_{m}^{-} + {\hat{l}}_{m}^{+}}{2}} .$

$[\begin{matrix} X_{i} & N_{i} \\ X_{i} \end{matrix}] \geq 0,$ (3.1)

$\sum_{j = 1}^{N} π_{i j} V_{j} - V \leq 0,$ (3.2)

$\sum_{j = 1, j \neq i}^{N} π_{i j} 𝒢_{j} - 𝒢 \leq 0,$ (3.3)

$Φ = [\begin{matrix} Ω & Γ^{T} \\ - \frac{1}{π_{i j}} K_{j} \end{matrix}] < 0,$ (3.4)

where $𝒢_{j}$ in (3.3) respectively represents $Q_{j}$ , $W_{j}$ , $R_{j}$ , $S_{j}$ , $U_{j}$ , $X_{j}$ , $Y_{j}$ , $T_{j}$ , $Z_{j}$ , $L_{j}$ and correspondingly 𝒢 represents $Q$ , $W$ , $R$ , $S$ , $U$ , $X$ , $Y$ , $T$ , $Z$ , (e.q.,when 𝒢_j is $Q_{j}$ , $𝒢$ is $Q$ ) and

$Ω = (ϑ_{i, j})_{15 \times 15},$

$ϑ_{1,1} = - P_{i} 𝒜_{i} - 𝒜_{i}^{T} P_{i} + π_{i i} P_{i} + \sum_{j \neq i} π_{i j} P_{j} + \sum_{j \neq i} π_{i j} K_{j} + Q_{1 i} + τ Q_{1} + W_{i} + τ W + R_{i} + σ R$

$+ τ_{i} U_{i} + \frac{τ^{2}}{2} U - \frac{1}{τ_{i}} X_{i} - \frac{1}{h_{i}} Y_{i} - 2 T_{i} - 2 Z_{i} - 2 L_{i} - {\hat{L}}_{1} H_{1 i} - {\hat{L}}_{1} H_{3 i} - {\hat{L}}_{1} H_{4 i}$

$- {\hat{L}}_{1} H_{5 i} - {\hat{L}}_{1} H_{6 i} - 2 σ^{2} M, ϑ_{1,2} = - \frac{1}{τ_{i}} N_{i}^{T} + \frac{1}{τ_{i}} X_{i}, ϑ_{1,3} = \frac{1}{τ_{i}} N_{i}^{T} + {\hat{L}}_{2} H_{3 i},$

$ϑ_{1,4} = \frac{2}{τ_{i}} T_{i}, ϑ_{1,5} = P_{i} ℬ_{i} + Q_{2 i} + Q_{2} + {\hat{L}}_{2} H_{1 i} + J_{1} ℬ_{i}, ϑ_{1,6} = P_{i} 𝒞_{i} + J_{1} 𝒞_{i},$

$ϑ_{1,7} = - J_{1} 𝒜_{i} - 𝒜_{i}^{T} J_{1}^{T} + {\hat{L}}_{2} H_{5 i}, ϑ_{1,8} = 𝒜_{i}^{T} P_{i} 𝒜_{i} - π_{i i} P_{i} 𝒜_{i} - \frac{2}{σ_{i}} L_{i}, ϑ_{1,9} = 2 σ M,$

$ϑ_{1,10} = P_{i}, ϑ_{1,11} = P_{i} 𝒟_{i} + J_{1} 𝒟_{i}, ϑ_{1,12} = \frac{1}{h_{i}} Y_{i} + {\hat{L}}_{2} H_{4 i}, ϑ_{1,13} = \frac{2}{h_{i}} Z_{i},$

$ϑ_{1,14} = J_{1} ℰ_{i}, ϑ_{1,15} = - J_{1} + {\hat{L}}_{2} H_{6 i}, ϑ_{2,2} = - (1 - τ_{µ_{i}}) Q_{1 i} - {\hat{L}}_{1} H_{2 i} - \frac{2}{τ_{i}} X_{i} + \frac{1}{τ_{i}} N_{i}^{T},$

$ϑ_{2,3} = \frac{1}{τ_{i}} X_{i} - \frac{1}{τ_{i}} N_{i}^{T}, ϑ_{2,6} = - (1 - τ_{µ_{i}}) Q_{1 i} + {\hat{L}}_{2} H_{2 i}, ϑ_{3,3} = - W_{i} - \frac{1}{τ_{i}} X_{i} - H_{3 i},$

$ϑ_{4,4} = - \frac{1}{τ_{i}} U_{i} - \frac{2}{τ_{i}^{2}} T_{i}, ϑ_{5,5} = Q_{3 i} + Q_{3} + d_{i} V_{i} + \frac{d^{2}}{2} V - H_{1 i}, ϑ_{5,8} = - ℬ_{i}^{T} P_{i} 𝒜_{i},$

$ϑ_{5,15} = ℬ_{i}^{T} J_{2}^{T}, ϑ_{6,6} = - (1 - τ_{µ_{i}}) Q_{3 i} - H_{2 i}, ϑ_{6,8} = - 𝒞_{i}^{T} P_{i} 𝒜_{i}, ϑ_{6,15} = 𝒞_{i}^{T} J_{2}^{T},$

$ϑ_{7,7} = - R_{i} - H_{5 i}, ϑ_{7,15} = - 𝒜_{i}^{T} J_{2}^{T}, ϑ_{8,8} = π_{i i} 𝒜_{i}^{T} P_{i} 𝒜_{i} - \frac{2}{σ_{i}^{2}} L_{i}, ϑ_{8,10} = - 𝒜_{i}^{T} P_{i},$

$ϑ_{8,11} = - 𝒜_{i}^{T} P_{i} 𝒟_{i}, ϑ_{8,14} = - 𝒜_{i}^{T} P_{i} ℰ_{i}, ϑ_{9,9} = \sum_{j \neq i} π_{i j} 𝒜_{j}^{T} P_{j} 𝒜_{j} - 2 M, ϑ_{10,10} = - γ I,$

$ϑ_{10,15} = J_{2}^{T}, ϑ_{11,11} = - S_{i} (1 - h_{µ_{i}}), ϑ_{11,15} = 𝒟_{i}^{T} J_{2}^{T}, ϑ_{12,12} = - \frac{1}{h_{i}} Y_{i} - H_{4 i},$

$ϑ_{13,13} = - \frac{2}{h_{i}^{2}} Z_{i}, ϑ_{14,14} = - \frac{(1 - d_{µ_{i}})}{d_{i}} V_{i}, ϑ_{14,15} = ℰ_{i}^{T} J_{2}^{T}, ϑ_{15,15} = S_{i} + h S + τ_{i} X_{i} + \frac{τ^{2}}{2} X$

$+ h_{i} Y_{i} + \frac{h^{2}}{2} Y + \frac{τ_{i}^{2}}{2} T_{i} + \frac{τ^{3}}{6} T + \frac{h_{i}^{2}}{2} Z_{i} + \frac{h^{3}}{6} Z + \frac{σ_{i}^{2}}{2} L_{i} + \frac{σ^{3}}{6} L + \frac{σ^{4}}{2} Z - (J_{2} + J_{2}^{T}) - H_{6 i},$

$Γ^{T} = [{\underset{}{\underset{︸}{0.....0}}}_{8 e l e m e n t s} \sum_{j \neq i} π_{i j} {(𝒜_{j}^{T} P_{j})}^{T} {\underset{}{\underset{︸}{0.....0}}}_{15 e l e m e n t s}]^{T},$

and the other coefficients are zero.

Proof. Here, we consider LKF candidate:

$V (x_{t}, i, t) = \sum_{κ = 1}^{13} V_{κ} (x_{t}, i, t),$ (3.5)

where

$V_{1} (x_{t}, i, t) = {[x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]}^{T} P_{i} [x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s],$

$V_{2} (x_{t}, i, t) = \int_{t - τ_{i} (t)}^{t} ζ^{T} (s) Q_{i} ζ (s) d s + \int_{- τ}^{0} \int_{t + θ}^{t} ζ^{T} (s) Q ζ (s) d s d θ,$

$V_{3} (x_{t}, i, t) = \int_{t - τ_{i}}^{t} x^{T} (s) W_{i} x (s) d s + \int_{- τ}^{0} \int_{t + θ}^{t} x^{T} (s) W x (s) d s d θ,$

$V_{4} (x_{t}, i, t) = \int_{t - σ_{i}}^{t} x^{T} (s) R_{i} x (s) d s + \int_{- σ}^{0} \int_{t + θ}^{t} x^{T} (s) R x (s) d s d θ,$

$V_{5} (x_{t}, i, t) = \int_{t - h_{i}}^{t} x^{T} (s) S_{i} x (s) d s + \int_{- h}^{0} \int_{t + θ}^{t} x^{T} (s) S x (s) d s d θ,$

$V_{6} (x_{t}, i, t) = \int_{- d_{i} (t)}^{0} \int_{t + θ}^{t} ℋ^{T} (x (s)) V_{i} ℋ (x (s)) d s d θ + \int_{- d}^{0} \int_{θ}^{0} \int_{t + β}^{t} ℋ^{T} (x (s)) V ℋ (x (s)) d s d β d θ,$

$V_{7} (x_{t}, i, t) = \int_{- τ_{i}}^{0} \int_{t + θ}^{t} x^{T} (s) U_{i} x (s) d s d θ + \int_{- τ}^{0} \int_{θ}^{0} \int_{t + β}^{t} x^{T} (s) U x (s) d s d β d θ,$

$V_{8} (x_{t}, i, t) = \int_{- τ_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) X_{i} \dot{x} (s) d s d θ + \int_{- τ}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) X \dot{x} (s) d s d β d θ,$

$V_{9} (x_{t}, i, t) = \int_{- h_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Y_{i} \dot{x} (s) d s d θ + \int_{- h}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) Y \dot{x} (s) d s d β d θ,$

$V_{10} (x_{t}, i, t) = \int_{- τ_{i}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) T_{i} \dot{x} (s) d s d β d θ + \int_{- τ}^{0} \int_{θ}^{0} \int_{β}^{0} \int_{t + α}^{t} {\dot{x}}^{T} (s) T \dot{x} (s) d s d α d β d θ,$

$V_{11} (x_{t}, i, t) = \int_{- h_{i}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) Z_{i} \dot{x} (s) d s d β d θ + \int_{- h}^{0} \int_{θ}^{0} \int_{β}^{0} \int_{t + α}^{t} {\dot{x}}^{T} (s) Z \dot{x} (s) d s d α d β d θ,$

$V_{12} (x_{t}, i, t) = \int_{- σ_{i}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) L_{i} \dot{x} (s) d s d β d θ + \int_{- σ}^{0} \int_{θ}^{0} \int_{β}^{0} \int_{t + α}^{t} {\dot{x}}^{T} (s) L \dot{x} (s) d s d α d β d θ,$

$V_{13} (x_{t}, i, t) = σ^{2} \int_{- σ}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) M \dot{x} (s) d s d β d θ,$

where $ζ^{T} (t) = [x^{T} (t), ℋ^{T} (x (t {))]}^{T} .$

From (2.1), we get

$ℒ V (x_{t}, i, t) = \sum_{κ = 1}^{13} ℒ V_{κ} (x_{t}, i, t),$ (3.6)

where

$ℒ V_{1} (x_{t}, i, t) = 2 {[x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]}^{T} P_{i} \frac{d}{d t} [x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]$

$+ \sum_{j = 1}^{N} π_{i j} {[x (t) - 𝒜_{j} \int_{t - σ_{j}}^{t} x (s) d s]}^{T} P_{j} [x (t) - 𝒜_{j} \int_{t - σ_{j}}^{t} x (s) d s],$

$\leq 2 {[x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]}^{T} P_{i} [- 𝒜_{i} x (t) + ℬ_{i} ℋ (x (t)) + 𝒞_{i} ℋ (x (t - τ_{i} (t)))$

$+ 𝒟_{i} \dot{x} (t - h_{i} (t)) + ℰ_{i} \int_{t - d_{i} (t)}^{t} ℋ (x (s)) d s + u (t)]$

$+ π_{i i} {[x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]}^{T} P_{i} [x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]$

$+ \sum_{j \neq i} π_{i j} [x^{T} (t) P_{j} x (t) + \int_{t - σ_{j}}^{t} x^{T} (s) d s 𝒜_{j} P_{j} K_{j}^{- 1} P_{j} 𝒜_{j} \int_{t - σ_{j}}^{t} x (s) d s$

$+ \int_{t - σ}^{t} x^{T} (s) d s 𝒜_{j}^{T} P_{j} 𝒜_{j} \int_{t - σ}^{t} x (s) d s],$

$ℒ V_{2} (x_{t}, i, t) \leq ζ^{T} (t) Q_{i} ζ (t) - ζ^{T} (t - τ_{i} (t)) Q_{i} ζ (t - τ_{i} (t)) (1 - τ_{µ_{i}}) + \sum_{j = 1}^{N} π_{i j} \int_{t - τ_{j} (t)}^{t} ζ^{T} (s) Q_{j} ζ (s) d s$

$+ τ ζ^{T} (t) Q ζ (t) - \int_{t - τ}^{t} ζ^{T} (s) Q ζ (s) d s,$

$ℒ V_{3} (x_{t}, i, t) = x^{T} (t) W_{i} x (t) - x^{T} (t - τ_{i}) W_{i} x (t - τ_{i}) + \sum_{j = 1}^{N} π_{i j} \int_{t - τ_{j}}^{t} x^{T} (s) W_{j} x (s) d s$

$+ τ x^{T} (t) W x (t) - \int_{t - τ}^{t} x^{T} (s) W x (s) d s,$

$ℒ V_{4} (x_{t}, i, t) = x^{T} (t) R_{i} x (t) - x^{T} (t - σ_{i}) R_{i} x (t - σ_{i}) + \sum_{j = 1}^{N} π_{i j} \int_{t - σ_{j}}^{t} x^{T} (s) R_{j} x (s) d s$

$+ σ x^{T} (t) R x (t) - \int_{t - σ}^{t} x^{T} (s) R x (s) d s,$

$ℒ V_{5} (x_{t}, i, t) \leq {\dot{x}}^{T} (t) S_{i} \dot{x} (t) - {\dot{x}}^{T} (t - h_{i} (t)) S_{i} \dot{x} (t - h_{i} (t)) (1 - h_{µ_{i}}) + \sum_{j = 1}^{N} π_{i j} \int_{t - h_{j} (t)}^{t} {\dot{x}}^{T} (s) S_{j} \dot{x} (s) d s$

$+ h {\dot{x}}^{T} (t) S \dot{x} (t) - \int_{t - h}^{t} {\dot{x}}^{T} (s) S \dot{x} (s) d s,$

$ℒ V_{6} (x_{t}, i, t) = d_{i} (t) ℋ^{T} (x (t)) V_{i} ℋ (x (t)) - (1 - d_{µ_{i}}) \int_{t - d_{i} (t)}^{t} ℋ^{T} (x (s)) V_{i} ℋ (x (s)) d s$

$+ \sum_{j = 1}^{N} π_{i j} \int_{- d_{j} (t)}^{0} \int_{t + θ}^{t} ℋ^{T} (x (s)) V_{j} ℋ (x (s)) d s d θ$

$+ \frac{d^{2}}{2} ℋ^{T} (x (t)) V ℋ (x (t)) - \int_{- d}^{0} \int_{t + θ}^{t} ℋ^{T} (x (s)) V ℋ (x (s)) d s d θ,$

$ℒ V_{7} (x_{t}, i, t) = τ_{i} x^{T} (t) U_{i} x (t) - \int_{t - τ_{i}}^{t} x^{T} (s) U_{i} x (s) d s + \sum_{j = 1}^{N} π_{i j} \int_{- τ_{j}}^{0} \int_{t + θ}^{t} x^{T} (s) U_{j} x (s) d s d θ$

$+ \frac{τ^{2}}{2} x^{T} (t) U x (t) - \int_{- τ}^{0} \int_{t + θ}^{t} x^{T} (s) U x (s) d s d θ,$

$ℒ V_{8} (x_{t}, i, t) = τ_{i} {\dot{x}}^{T} (t) X_{i} \dot{x} (t) - \int_{t - τ_{i}}^{t} {\dot{x}}^{T} (s) X_{i} \dot{x} (s) d s + \sum_{j = 1}^{N} π_{i j} \int_{- τ_{j}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) X_{j} \dot{x} (s) d s d θ$

$+ \frac{τ^{2}}{2} {\dot{x}}^{T} (t) X \dot{x} (t) - \int_{- τ}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) X \dot{x} (s) d s d θ,$

$ℒ V_{9} (x_{t}, i, t) = h_{i} {\dot{x}}^{T} (t) Y_{i} \dot{x} (t) - \int_{t - h_{i}}^{t} {\dot{x}}^{T} (s) Y_{i} \dot{x} (s) d s + \sum_{j = 1}^{N} π_{i j} \int_{- h_{j}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Y_{j} \dot{x} (s) d s d θ$

$+ \frac{h^{2}}{2} {\dot{x}}^{T} (t) Y \dot{x} (t) - \int_{- h}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Y \dot{x} (s) d s d θ,$

$ℒ V_{10} (x_{t}, i, t) = \frac{τ_{i}^{2}}{2} {\dot{x}}^{T} (t) T_{i} \dot{x} (t) - \int_{- τ_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) T_{i} \dot{x} (s) d s d θ + \frac{τ^{3}}{6} {\dot{x}}^{T} (t) T \dot{x} (t)$

$+ \sum_{j = 1}^{N} π_{i j} \int_{- τ_{j}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) T_{j} \dot{x} (s) d s d β d θ - \int_{- τ}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) T \dot{x} (s) d s d β d θ,$

$ℒ V_{11} (x_{t}, i, t) = \frac{h_{i}^{2}}{2} {\dot{x}}^{T} (t) Z_{i} \dot{x} (t) - \int_{- h_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Z_{i} \dot{x} (s) d s d θ + \frac{h^{3}}{6} {\dot{x}}^{T} (t) Z \dot{x} (t)$

$+ \sum_{j = 1}^{N} π_{i j} \int_{- h_{j}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) Z_{j} \dot{x} (s) d s d β d θ - \int_{- h}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) Z \dot{x} (s) d s d β d θ,$

$ℒ V_{12} (x_{t}, i, t) = \frac{σ_{i}^{2}}{2} {\dot{x}}^{T} (t) L_{i} \dot{x} (t) - \int_{- σ_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) L_{i} \dot{x} (s) d s d θ + \frac{σ^{3}}{6} {\dot{x}}^{T} (t) L \dot{x} (t)$

$+ \sum_{j = 1}^{N} π_{i j} \int_{- σ_{j}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) L_{j} \dot{x} (s) d s d β d θ - \int_{- σ}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) L \dot{x} (s) d s d β d θ,$

$ℒ V_{13} (x_{t}, i, t) = \frac{σ^{4}}{2} {\dot{x}}^{T} (t) M \dot{x} (t) - σ^{2} \int_{- σ}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) M \dot{x} (s) d s d θ .$

Using the upper bounds of discrete, neutral, distributed time-varying delays, leakage delays, Lemma 2 and with $π_{i i} < 0,$ the following relationship is obtained

$\begin{array}{l} \sum_{j \neq i} (- 2 π_{i j} x^{T} (t) P_{j} 𝒜_{j} \int_{t - σ_{j}}^{t} x (s) d s) \\ \leq \sum_{j \neq i} π_{i j} (x^{T} (t) K_{j} x (t) + \int_{t - σ_{j}}^{t} x^{T} (s) d s 𝒜_{j}^{T} P_{j} K_{j}^{- 1} P_{j} 𝒜_{j} \int_{t - σ_{j}}^{t} x (s) d s) \\ \leq \sum_{j \neq i} π_{i j} (x^{T} (t) K_{j} x (t) + \int_{t - σ}^{t} x^{T} (s) d s 𝒜_{j}^{T} P_{j} K_{j}^{- 1} P_{j} 𝒜_{j} \int_{t - σ}^{t} x (s) d s), \end{array}$ (3.7)

$\begin{array}{l} \sum_{j \neq i} π_{i j} (\int_{t - σ_{j}}^{t} x^{T} (s) d s) 𝒜_{j}^{T} P_{j} 𝒜_{j} (\int_{t - σ_{j}}^{t} x (s) d s) \\ \leq \sum_{j \neq i} π_{i j} (\int_{t - σ}^{t} x^{T} (s) d s) 𝒜_{j}^{T} P_{j} 𝒜_{j} (\int_{t - σ}^{t} x (s) d s) . \end{array}$ (3.8)

Similarly,

$\begin{array}{l} \sum_{j = 1}^{N} π_{i j} \int_{t - τ_{j} (t)}^{t} ζ^{T} (s) Q_{j} ζ (s) d s \leq \sum_{j = 1, j \neq i}^{N} π_{i j} \int_{t - τ_{j} (t)}^{t} ζ^{T} (s) Q_{j} ζ (s) d s \\ \leq \sum_{j = 1, j \neq i}^{N} π_{i j} \int_{t - τ}^{t} ζ^{T} (s) Q_{j} ζ (s) d s \\ \leq \int_{t - τ}^{t} ζ^{T} (s) Q ζ (s) d s, \end{array}$ (3.9)

$\sum_{j = 1}^{N} π_{i j} \int_{t - τ_{j}}^{t} x^{T} (s) W_{j} x (s) d s \leq \int_{t - τ}^{t} x^{T} (s) W x (s) d s,$ (3.10)

$\sum_{j = 1}^{N} π_{i j} \int_{t - σ_{j}}^{t} x^{T} (s) R_{j} x (s) d s \leq \int_{t - σ}^{t} x^{T} (s) R x (s) d s,$ (3.11)

$\sum_{j = 1}^{N} π_{i j} \int_{t - h_{j} (t)}^{t} {\dot{x}}^{T} (s) S_{j} \dot{x} (s) d s \leq \int_{t - h}^{t} {\dot{x}}^{T} (s) S \dot{x} (s) d s,$ (3.12)

$\sum_{j = 1}^{N} π_{i j} \int_{- d_{j} (t)}^{0} \int_{t + θ}^{t} ℋ^{T} (x (s)) V_{j} ℋ (x (s)) d s d θ \leq - \int_{- d}^{0} \int_{t + θ}^{t} ℋ^{T} (x (s)) V ℋ (x (s)) d s d θ,$ (3.13)

$\sum_{j = 1}^{N} π_{i j} \int_{- τ_{j}}^{0} \int_{t + θ}^{t} x^{T} (s) U_{j} x (s) d s d θ \leq \int_{- τ}^{0} \int_{t + θ}^{t} x^{T} (s) U x (s) d s d θ,$ (3.14)

$\sum_{j = 1}^{N} π_{i j} \int_{- τ_{j}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) X_{j} \dot{x} (s) d s d θ \leq \int_{- τ}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) X \dot{x} (s) d s d θ,$ (3.15)

$\sum_{j = 1}^{N} π_{i j} \int_{- h_{j}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Y_{j} \dot{x} (s) d s d θ \leq \int_{- h}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Y \dot{x} (s) d s d θ,$ (3.16)

$\sum_{j = 1}^{N} π_{i j} \int_{- τ_{j}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) T_{j} \dot{x} (s) d s d β d θ \leq \int_{- τ}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) T \dot{x} (s) d s d β d θ,$ (3.17)

$\sum_{j = 1}^{N} π_{i j} \int_{- h_{j}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) Z_{j} \dot{x} (s) d s d β d θ \leq \int_{- h}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) Z \dot{x} (s) d s d β d θ,$ (3.18)

$\sum_{j = 1}^{N} π_{i j} \int_{- σ_{j}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) L_{j} \dot{x} (s) d s d β d θ \leq \int_{- σ}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) L \dot{x} (s) d s d β d θ .$ (3.19)

By using Lemma 1, we get

$- (1 - d_{µ_{i}}) \int_{t - d_{i} (t)}^{t} ℋ^{T} (x (s)) V_{i} ℋ (x (s)) d s \leq - \frac{(1 - d_{µ_{i}})}{d_{i}} \int_{t - d_{i} (t)}^{t} ℋ^{T} (x (s)) d s V_{i} \int_{t - d_{i} (t)}^{t} ℋ (x (s)) d s,$

$- \int_{t - τ_{i}}^{t} x^{T} (s) U_{i} x (s) d s \leq - \frac{1}{τ_{i}} \int_{t - τ_{i}}^{t} x^{T} (s) d s U_{i} \int_{t - τ_{i}}^{t} x (s) d s,$

$- \int_{t - h_{i}}^{t} {\dot{x}}^{T} (s) Y_{i} \dot{x} (s) d s \leq - \frac{1}{h_{i}} \int_{t - h_{i}}^{t} {\dot{x}}^{T} (s) d s Y_{i} \int_{t - h_{i}}^{t} \dot{x} (s) d s,$

$- \int_{- τ_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) T_{i} \dot{x} (s) d s d θ \leq - \frac{2}{τ_{i}^{2}} \int_{- τ_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) d s d θ T_{i} \int_{- τ_{i}}^{0} \int_{t + θ}^{t} \dot{x} (s) d s d θ,$

$- \int_{- h_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Z_{i} \dot{x} (s) d s d θ \leq - \frac{2}{h_{i}^{2}} \int_{- h_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) d s d θ Z_{i} \int_{- h_{i}}^{0} \int_{t + θ}^{t} \dot{x} (s) d s d θ,$

$- \int_{- σ_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Z_{i} \dot{x} (s) d s d θ \leq - \frac{2}{σ_{i}^{2}} \int_{- σ_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) d s d θ Z_{i} \int_{- σ_{i}}^{0} \int_{t + θ}^{t} \dot{x} (s) d s d θ,$

$\int_{- σ}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) M \dot{x} (s) d s d θ \leq - \frac{2}{σ^{2}} \int_{- σ}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) d s d θ Z_{i} \int_{- σ}^{0} \int_{t + θ}^{t} \dot{x} (s) d s d θ .$

Note that from (3.1) and using the reciprocally convex technique in [34], we obtain

$- \int_{t - τ_{i}}^{t} {\dot{x}}^{T} (s) X_{i} \dot{x} (s) d s \leq - \int_{t - τ_{i}}^{t - τ_{i} (t)} {\dot{x}}^{T} (s) X_{i} \dot{x} (s) d s - \int_{t - τ_{i} (t)}^{t} {\dot{x}}^{T} (s) X_{i} \dot{x} (s) d s$

$\leq - \frac{1}{τ_{i}} ϖ^{T} (t) [\begin{matrix} R_{i} & X_{i} \\ R_{i} \end{matrix}] ϖ (t),$

where $ϖ (t) = [x^{T} (t - τ_{i} (t)) - x^{T} (t - τ_{i}), x^{T} (t) - x^{T} (t - τ_{i} (t))]$ . For positive diagonal matrices $H_{1 i}, H_{2 i}, H_{3 i}, H_{4 i}, H_{5 i}, H_{6 i}$ , by Assumption 1, we get

${[\begin{matrix} x (t) \\ ℋ (x (t)) \end{matrix}]}^{T} [\begin{matrix} {\hat{L}}_{1} H_{1 i} & - {\hat{L}}_{2} H_{1 i} \\ * & H_{1 i} \end{matrix}] [\begin{matrix} x (t) \\ ℋ (x (t)) \end{matrix}] \leq 0,$ (3.20)

${[\begin{matrix} x (t - τ_{i} (t)) \\ ℋ (x (t - τ_{i} (t))) \end{matrix}]}^{T} [\begin{matrix} {\hat{L}}_{1} H_{2 i} & - {\hat{L}}_{2} H_{2 i} \\ * & H_{2 i} \end{matrix}] [\begin{matrix} x (t - τ_{i} (t)) \\ ℋ (x (t - τ_{i} (t))) \end{matrix}] \leq 0,$ (3.21)

${[\begin{matrix} x (t) \\ x (t - τ_{i}) \end{matrix}]}^{T} [\begin{matrix} {\hat{L}}_{1} H_{3 i} & - {\hat{L}}_{2} H_{3 i} \\ * & H_{3 i} \end{matrix}] [\begin{matrix} x (t) \\ x (t - τ_{i}) \end{matrix}] \leq 0,$ (3.22)

${[\begin{matrix} x (t) \\ x (t - h_{i}) \end{matrix}]}^{T} [\begin{matrix} {\hat{L}}_{1} H_{4 i} & - {\hat{L}}_{2} H_{4 i} \\ * & H_{4 i} \end{matrix}] [\begin{matrix} x (t) \\ x (t - h_{i}) \end{matrix}] \leq 0,$ (3.23)

${[\begin{matrix} x (t) \\ x (t - σ_{i}) \end{matrix}]}^{T} [\begin{matrix} {\hat{L}}_{1} H_{5 i} & - {\hat{L}}_{2} H_{5 i} \\ * & H_{5 i} \end{matrix}] [\begin{matrix} x (t) \\ x (t - σ_{i}) \end{matrix}] \leq 0,$ (3.24)

${[\begin{matrix} x (t) \\ \dot{x} (t) \end{matrix}]}^{T} [\begin{matrix} {\hat{L}}_{1} H_{6 i} & - {\hat{L}}_{2} H_{6 i} \\ * & H_{6 i} \end{matrix}] [\begin{matrix} x (t) \\ \dot{x} (t) \end{matrix}] \leq 0.$ (3.25)

Hence, for any matrices J₁, J₂ of appropriate dimensions, we get

$\begin{array}{l} 0 = 2 [x^{T} (t) J_{1} + {\dot{x}}^{T} (t) J_{2}] [- 𝒜_{i} x (t - σ_{i}) + ℬ_{i} ℋ (x (t)) + 𝒞_{i} ℋ (x (t - τ_{i} (t))) \\ + 𝒟_{i} \dot{x} (t - h_{i} (t)) + ℰ_{i} \int_{t - d_{i} (t)}^{t} ℋ (x (s)) d s + u (t) - \dot{x} (t)] . \end{array}$ (3.26)

Using (3.6) and adding (3.20)-(3.26), we have

$ℒ V (x_{t}, i, t) - 2 y^{T} (t) u (t) - γ u^{T} (t) u (t) \leq ℒ {η^{T} (t) Φ η (t)},$ (3.27)

where

$η^{T} (t) = [x^{T} (t) x^{T} (t - τ_{i} (t)) x^{T} (t - τ_{i}) \int_{t - τ_{i}}^{t} x^{T} (s) d s ℋ^{T} (x (t)) ℋ^{T} (x (t - τ_{i} (t)))$

$x^{T} (t - σ_{i}) \int_{t - σ_{i}}^{t} x^{T} (s) d s \int_{t - σ}^{t} x^{T} (s) d s u^{T} (t) {\dot{x}}^{T} (t - h_{i} (t)) x^{T} (t - h_{i})$

$\int_{t - h_{i}}^{t} x^{T} (s) d s \int_{t - d_{i} (t)}^{t} ℋ^{T} (x (s)) d s {\dot{x}}^{T} (t)] .$

Hence from equation (3.4) we have,

$ℒ V (x_{t}, i, t) - 2 y {(t)}^{T} u (t) - γ u {(t)}^{T} u (t) \leq 0.$

Now, to show the passivity of the delayed NNs in (2.3), we take

$J (t_{p}) = 𝔼 {\int_{0}^{t_{p}} [- γ u {(t)}^{T} u (t) - 2 y {(t)}^{T} u (t)] d t}$ (3.28)

where $t_{p} \geq 0$ .

From Dynkin's formula, we get

$𝔼 [\int_{0}^{t_{p}} ℒ V (x_{t}, i, t) d t] = 𝔼 [V (x_{t_{p}}, i, t_{p})] - 𝔼 [V (x_{0}, \nabla (0),0)] .$

Therefore,

$\begin{array}{l} J (t_{p}) = 𝔼 {\int_{0}^{t_{p}} [- γ u {(t)}^{T} u (t) - 2 y {(t)}^{T} u (t) + ℒ V (x_{t}, i, t)] d t} - 𝔼 [\int_{0}^{t_{p}} ℒ V (x_{t}, i, t) d t] \\ = 𝔼 {\int_{0}^{t_{p}} [- γ u {(t)}^{T} u (t) - 2 y {(t)}^{T} u (t) + ℒ V (x_{t}, i, t)] d t} \\ - 𝔼 [V (x_{t}, i, t)] + 𝔼 [V (x_{0}, \nabla (0),0)] . \end{array}$ (3.29)

By applying lemma 3 to (3.4), we have

$Φ < 0.$ (3.30)

Thus, if (3.30) holds, then $𝔼 [V (x_{t_{p}}, i, t_{p})] \geq 0$ and $V (x_{0}, \nabla (0),0) = 0$ holds with zero initial conditions. From (3.30), it follows that $J (t_{p}) \leq 0$ for any $t_{p} \geq 0$ , which implies (2.5) is satisfied and hence the delayed NNs (2.3) is locally passive.

Now, we prove the global passivity of the system.

By taking expectation of (3.27) and then integration from 0 to t we get,

$\int_{0}^{t} 𝔼 [ℒ V (x_{s}, r (s), s)] d s - 2 \int_{0}^{t} 𝔼 [y^{T} (s) u (s)] d s - γ \int_{0}^{t} 𝔼 [u^{T} (s) u (s)] d s \leq \int_{0}^{t} 𝔼 [η^{T} (s) Φ η (s)] d s .$

Then by Dynkin's formula,

$𝔼 [ℒ V (x_{t}, i, t)] - 𝔼 [ℒ V (x_{0}, \nabla (0),0)] - 2 \int_{0}^{t} 𝔼 [y^{T} (s) u (s)] d s - γ \int_{0}^{t} 𝔼 [u^{T} (s) u (s)] d s$

$\leq \int_{0}^{t} 𝔼 [η^{T} (s) Φ η (s)] d s .$

Hence,

$\begin{matrix} 𝔼 [ℒ V (x_{t}, i, t)] - \int_{0}^{t} 𝔼 [η^{T} (s) Φ η (s)] d s \leq 𝔼 [ℒ V (x_{0}, \nabla (0),0)] + 2 \int_{0}^{t} 𝔼 [y^{T} (s) u (s)] d s \\ + γ \int_{0}^{t} 𝔼 [u^{T} (s) u (s)] d s \\ < \infty, t \geq 0. \end{matrix}$ (3.31)

By Jenson's inequality and (3.6), we get

$\begin{array}{l} E {∥ 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s ∥}^{2} = E {[𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]}^{T} [𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s] \\ \leq λ_{m a x} (𝒜_{i}^{2}) E {[\int_{t - σ_{i}}^{t} x (s) d s]}^{T} [\int_{t - σ_{i}}^{t} x (s) d s] \\ \leq \frac{λ_{m a x} (𝒜_{i}^{2})}{λ_{m i n} (R_{i})} {[\int_{t - σ_{i}}^{t} E x (s) d s]}^{T} R_{i} [\int_{t - σ_{i}}^{t} E x (s) d s] \\ \leq σ_{i} \frac{λ_{m a x} (𝒜_{i}^{2})}{λ_{m i n} (R_{i})} {\int_{t - σ_{i}}^{t} E x^{T} (s) R_{i} x (s) d s} \\ \leq σ_{i} \frac{λ_{m a x} (𝒜_{i}^{2})}{λ_{m i n} (R_{i})} E V_{4} (x_{t}, i, t) \\ \leq σ \frac{λ_{m a x} (𝒜_{i}^{2})}{λ_{m i n} (R_{i})} E V (x_{t}, i, t) \\ \leq σ \frac{λ_{m a x} (𝒜_{i}^{2})}{λ_{m i n} (R_{i})} E V (x_{0}, 𝒜 (0),0), t \geq 0. \end{array}$ (3.32)

Similarly, it follows from the definition of $V_{1} (x_{t}, i, t)$ that

$E {∥ x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s ∥}^{2} = E {[𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]}^{T} [𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]$

$\leq \frac{E V_{1} (x_{t}, i, t)}{λ_{m i n} (P_{i})}$

$\leq \frac{E V (x_{t}, i, t)}{λ_{m i n} (P_{i})}$

$\leq \frac{E V (x_{0}, \nabla (0),0)}{λ_{m i n} (P_{i})}, t \geq 0.$

Hence, it can be obtained that

$\begin{array}{l} E {∥ x (t) ∥}^{2} = E {∥ x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s + 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s ∥}^{2} \\ \leq 2 E {∥ A_{i} \int_{t - σ_{i}}^{t} x (s) d s ∥}^{2} + 2 E {∥ x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s ∥}^{2} \\ \leq 2 σ \frac{λ_{m a x} (𝒜_{i}^{2})}{λ_{m i n} (R_{i})} E V (x_{0}, \nabla (0),0) + 2 \frac{E V (x_{0}, \nabla (0),0)}{λ_{m i n} (P_{i})} < \infty, t \geq 0, \end{array}$ (3.33)

$\begin{array}{l} E V (x_{0}, \nabla (0),0) \\ \leq {2 λ_{\underset{i \in S}{m a x}} (P_{i}) (1 + σ_{i}^{2} \underset{i \in S}{m a x} 𝒜_{i}) + τ \underset{i \in S}{m a x} {λ_{m a x} (Q_{i})} + τ^{2} λ_{m a x} (Q) + τ \underset{i \in S}{m a x} {λ_{m a x} (W_{i})} \\ + τ^{2} λ_{m a x} (W) + σ \underset{i \in S}{m a x} {λ_{m a x} (R_{i})} + σ^{2} λ_{m a x} (R) + h \underset{i \in S}{m a x} {λ_{m a x} (S_{i})} + h^{2} λ_{m a x} (S) \\ + d^{2} \underset{i \in S}{m a x} {λ_{m a x} (V_{i})} + d^{3} λ_{m a x} (V) + τ^{2} \underset{i \in S}{m a x} {λ_{m a x} (U_{i})} + τ^{3} λ_{m a x} (U) + τ^{2} \underset{i \in S}{m a x} {λ_{m a x} (X_{i})} \\ + τ^{3} λ_{m a x} (X) + h^{2} \underset{i \in S}{m a x} {λ_{m a x} (Y_{i})} + h^{3} λ_{m a x} (Y) + τ^{3} \underset{i \in S}{m a x} {λ_{m a x} (T_{i})} + τ^{4} λ_{m a x} (T) \\ + h^{3} \underset{i \in S}{m a x} {λ_{m a x} (Z_{i})} + h^{4} λ_{m a x} (Z) + σ^{3} \underset{i \in S}{m a x} {λ_{m a x} (L_{i})} + σ^{4} λ_{m a x} (L) + σ^{5} λ_{m a x} (M)} < \infty . \end{array}$ (3.34)

From (3.33) and (3.34), we get that the solution of the system (2.3) is locally passive. Then the solutions $x (t) = x (t,0, φ)$ of system (2.3) is bounded on $[0, \infty)$ . The solution $x (t)$ on $[0, \infty)$ is uniformly continuous because $\dot{x} (t)$ is bounded on $[0, \infty)$ . Further, from the equation (3.31), the following holds:

$λ_{m i n} (Φ) \int_{0}^{t} E [x^{T} (s) x (s)] d s \leq 𝔼 [ℒ V (x_{t}, i, t)] - \int_{0}^{t} 𝔼 [η^{T} (s) Φ η (s)] d s$

$\leq 𝔼 [ℒ V (x_{0}, \nabla (0),0)] + 2 \int_{0}^{t} 𝔼 [y^{T} (s) u (s)] d s$

$+ γ \int_{0}^{t} 𝔼 [u^{T} (s) u (s)] d s$

$< \infty, t \geq 0.$

From Barbalats lemma [30], $𝔼 [{‖ x (t) ‖}^{2}] \to 0$ as $t \to \infty$ holds. Hence the proof is complete.

Remark 1. Recently, studies on passivity analysis for neural networks of neutral type with Markovian jumping parameters and time delay in the leakage term were conducted in [5]. By constructing proper Lyapunov–Krasovskii functional, new delay-dependent passivity conditions are derived in terms of LMIs and it can easily be checked using standard numerical packages. Moreover, it is well known that the passivity behaviour of neural networks is very sensitive to the time delay in the leakage term. Triple and quadruple integrals have not been taken into account to derive the passivity conditions in [5]. Mode-dependent time delays were not included in [5]. Very recently, a mode-dependent approach is proposed by constructing a novel Lyapunov functional, where some terms involving triple or quadruple integrals are taken into account to study the state estimation problem in [22]. Motivated by this reason, we have introduced improved Lyapunov–Krasovskii functional with triple and quadruple integrals for deriving the reported stability results in this paper. Based on this discussion, our results will give less conservative results than those studied in [5], [22].

4. Numerical example

In this section, a numerical example is provided to demonstrate the validity of the proposed theorems.

Example 1. Consider a 2-D Mode-dependent Markov jump NNNs with mixed time-delays (2.3) with the following parameters

$𝒜_{1} = [\begin{matrix} 8.4 & 0 \\ 0 & 9 \end{matrix}], 𝒜_{2} = [\begin{matrix} 7.8 & 0 \\ 0 & 8.5 \end{matrix}], ℬ_{1} = [\begin{matrix} - 0.21 & - 0.19 \\ - 0.24 & 0.1 \end{matrix}], ℬ_{2} = [\begin{matrix} 0.9 & - 0.9 \\ 0.5 & - 0.8 \end{matrix}], 𝒞_{1} = [\begin{matrix} - 0.09 & - 0.2 \\ 0.2 & 0.1 \end{matrix}], 𝒞_{2} = [\begin{matrix} 0.1 & 0.1 \\ 0.2 & 0.3 \end{matrix}], 𝒟_{1} = [\begin{matrix} - 0.2 & 0 \\ 0.2 & - 0.09 \end{matrix}], 𝒟_{2} = [\begin{matrix} 0.1 & 0 \\ 0.5 & - 0.1 \end{matrix}], ℰ_{1} = [\begin{matrix} - 0.5 & 0 \\ 0 & - 0.5 \end{matrix}], ℰ_{2} = [\begin{matrix} 0.1 & - 0.02 \\ - 0.2 & 0.07 \end{matrix}], {\hat{L}}_{1} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}], {\hat{L}}_{2} = [\begin{matrix} 0.25 & 0 \\ 0 & 0.25 \end{matrix}] .$

Take $ℋ_{1} (s) = ℋ_{2} (s) = t a n h (s)$ , $τ_{1} (t) = τ_{2} (t) = h_{1} (t) = h_{2} (t) = d_{1} (t) = d_{2} (t) = 0.1 c o s t + 0.4$ , $σ_{1} = σ_{2} = 0.1$ , $τ_{µ_{1}} = τ_{µ_{2}} = h_{µ_{1}} = h_{µ_{2}} = d_{µ_{1}} = d_{µ_{2}} = 0.1$ . $Γ = [\begin{matrix} - 7 & 7 \\ 6 & - 6 \end{matrix}] .$

By using the MATLAB LMI toolbox, we can obtain the following feasible solution for the LMIs (3.1)–(3.4):

$P_{1} = [\begin{matrix} 0.0157 & 0.0008 \\ 0.0008 & 0.0143 \end{matrix}], P_{2} = [\begin{matrix} 0.0068 & 0.0004 \\ 0.0004 & 0.0060 \end{matrix}], Q_{11} = [\begin{matrix} 0.2401 & 0.0007 \\ 0.0007 & 0.2592 \end{matrix}], Q_{12} = [\begin{matrix} 0.3770 & - 0.0192 \\ - 0.0192 & 0.3992 \end{matrix}], Q_{21} = [\begin{matrix} - 0.4101 & 0.0043 \\ 0.0201 & - 0.4457 \end{matrix}], Q_{22} = [\begin{matrix} - 0.6428 & 0.0773 \\ 0.0231 & - 0.6959 \end{matrix}], Q_{31} = [\begin{matrix} 1.3786 & 0.0133 \\ 0.0133 & 1.4063 \end{matrix}], Q_{32} = [\begin{matrix} 1.9101 & - 0.0284 \\ - 0.0284 & 1.9903 \end{matrix}], W_{1} = [\begin{matrix} 0.1039 & 0.0088 \\ 0.0088 & 0.1018 \end{matrix}], W_{2} = [\begin{matrix} 0.1517 & 0.0121 \\ 0.0121 & 0.1487 \end{matrix}], R_{1} = [\begin{matrix} 0.1758 & 0.0160 \\ 0.0160 & 0.1721 \end{matrix}], R_{2} = [\begin{matrix} 0.5675 & 0.0333 \\ 0.0333 & 0.5563 \end{matrix}], S_{1} = [\begin{matrix} 1.0817 & - 0.0075 \\ - 0.0075 & 0.7204 \end{matrix}], S_{2} = [\begin{matrix} 1.2231 & 0.0093 \\ 0.0093 & 0.9256 \end{matrix}], V_{1} = [\begin{matrix} 10.7113 & 0.5692 \\ 0.5692 & 9.9973 \end{matrix}], V_{2} = [\begin{matrix} 10.8483 & 0.5953 \\ 0.5953 & 10.2619 \end{matrix}], U_{1} = [\begin{matrix} 0.5120 & 0.0419 \\ 0.0419 & 0.5056 \end{matrix}], U_{2} = [\begin{matrix} 0.6238 & 0.0492 \\ 0.0492 & 0.6133 \end{matrix}], X_{1} = [\begin{matrix} 21.0470 & - 0.1153 \\ - 0.1153 & 21.2070 \end{matrix}], X_{2} = [\begin{matrix} 18.5642 & - 0.3139 \\ - 0.3139 & 18.6620 \end{matrix}], Y_{1} = [\begin{matrix} 27.5946 & - 0.0019 \\ - 0.0019 & 26.6557 \end{matrix}], Y_{2} = [\begin{matrix} 28.4115 & 0.1096 \\ 0.1096 & 27.4131 \end{matrix}], T_{1} = [\begin{matrix} 3.0928 & 0.0943 \\ 0.0943 & 3.0505 \end{matrix}], T_{2} = [\begin{matrix} 3.8182 & 0.0431 \\ 0.0431 & 3.7941 \end{matrix}], Z_{1} = [\begin{matrix} 3.0867 & 0.1080 \\ 0.1080 & 3.0591 \end{matrix}], Z_{2} = [\begin{matrix} 3.8647 & 0.0504 \\ 0.0504 & 3.8446 \end{matrix}], L_{1} = [\begin{matrix} 0.0068 & 0.0001 \\ 0.0001 & 0.0068 \end{matrix}], L_{2} = [\begin{matrix} 0.0059 & 0.0004 \\ 0.0004 & 0.0062 \end{matrix}], K_{1} = [\begin{matrix} 1.3461 & - 0.1410 \\ - 0.1410 & 1.3280 \end{matrix}], K_{2} = [\begin{matrix} 0.0919 & 0.0135 \\ 0.0135 & 0.0907 \end{matrix}], Q_{1} = [\begin{matrix} 4.0913 & - 0.1850 \\ - 0.1850 & 4.4495 \end{matrix}], Q_{2} = [\begin{matrix} - 6.6332 & 0.7686 \\ 0.3444 & - 7.3948 \end{matrix}], γ = 0.2272.$

Continuing in this way, the remaining feasible matrices are obtained. This shows that the given system (2.3) is globally passive in the mean square.

5. Conclusion

In this paper, passivity analysis of Markovian jumping NNNs with time delays in the leakage term is considered. Delay-mode-dependent passivity conditions are derived by taking the inherent characteristic of such kinds of NNs into account. An improved LKF, with the triple integral terms and quadruple integrals, is constructed and the results are derived in terms of linear matrix inequalities. The information of the mode-dependent of all delays have been taken into account in the constructed LKF and derived novel stability criterion. Theoretical results are validated through a numerical example.

References

[1]	R. K. Meleppat, K. E. Ronning, S. J. Karlen, K. K. Kothandath, M. E. Burns, E. N. Pugh Jr, et al., In situ morphologic and spectral characterization of retinal pigment epithelium organelles in mice using multicolor confocal fluorescence imaging, Invest. Ophthalmol. Visual Sci., 16 (2020). https://doi.org/10.1167/iovs.61.13.1 doi: 10.1167/iovs.61.13.1
[2]	R. K. Meleppat, K. E. Ronning, S. J. Karlen, M. E. Burns, E. N. Pugh Jr, R. J. Zawadzki, In vivo multimodal retinal imaging of disease-related pigmentary changes in retinal pigment epithelium, Sci. Rep., 11 (2021), 16252. https://doi.org/10.1038/s41598-021-95320-z doi: 10.1038/s41598-021-95320-z
[3]	S. Fu, Analysis of 56 cases of type 2 diabetes mellitus with ocular lesions as the first manifestation, Clin. Focus, 22 (2007), 256-257. https://doi.org/10.3969/j.issn.1004-583X.2007.04.013 doi: 10.3969/j.issn.1004-583X.2007.04.013
[4]	V. Gulshan, L. Peng, M. Coram, M. C. Stumpe, D. Wu, A. Narayanaswamy, et al., Development and validation of a deep learning algorithm for detection of diabetic retinopathy in retinal fundus photographs, JAMA, 316 (2016), 2402-2410. https://doi.org/10.1001/jama.2016.17216 doi: 10.1001/jama.2016.17216
[5]	S. Sengupta, A. Singh, H. A. Leopold, T. Gulati, V. Lakshminarayanan, Ophthalmic diagnosis using deep learning with fundus images - A critical review, Artif. Intell. Med., 102 (2020), 101758. https://doi.org/10.1016/j.artmed.2019.101758 doi: 10.1016/j.artmed.2019.101758
[6]	J. Son, J. Y. Shin, H. D. Kim, K. Jung, K. Park, S. J. Park, Development and validation of deep learning models for screening multiple abnormal findings in retinal fundus images, Ophthalmology, 127 (2020), 85-94. https://doi.org/10.1016/j.ophtha.2019.05.029 doi: 10.1016/j.ophtha.2019.05.029
[7]	L. M. Devi, K. Wahengbam, A. D. Singh, Dehazing buried tissues in retinal fundus images using a multiple radiance pre-processing with deep learning based multiple feature-fusion, Opt. Laser Technol., 138 (2021), 106908. https://doi.org/10.1016/j.optlastec.2020.106908 doi: 10.1016/j.optlastec.2020.106908
[8]	A. V. Varadarajan, P. Bavishi, P. Ruamviboonsuk, P. Chotcomwongse, S. Venugopalan, A. Narayanaswamy, et al., Predicting optical coherence tomography-derived diabetic macular edema grades from fundus photographs using deep learning, Nat. Commun., 11 (2020), 130. https://doi.org/10.1038/s41467-019-13922-8 doi: 10.1038/s41467-019-13922-8
[9]	H. N. Veena, A. Muruganandham, T. S. Kumaran, A review on the optic disc and optic cup segmentation and classification approaches over retinal fundus images for detection of glaucoma, SN Appl. Sci., 2 (2020), 1476. https://doi.org/10.1007/s42452-020-03221-z doi: 10.1007/s42452-020-03221-z
[10]	Q. Wu, A. Cheddad, Segmentation-based deep learning fundus image analysis, in 2019 Ninth International Conference on Image Processing Theory, Tools and Applications (IPTA), (2019), 1-5. https://doi.org/10.1109/IPTA.2019.8936078
[11]	S. Guo, T. Li, H. Kang, N. Li, Y. Zhang, K. Wang, L-Seg: An end-to-end unified framework for multi-lesion segmentation of fundus images, Neurocomputing, 349 (2019), 52-63. https://doi.org/10.1016/j.neucom.2019.04.019 doi: 10.1016/j.neucom.2019.04.019
[12]	C. Playout, R. Duval, F. Cheriet, A novel weakly supervised multitask architecture for retinal lesions segmentation on fundus images, IEEE Trans. Med. Imaging, 38 (2019), 2434-2444. https://doi.org/10.1109/tmi.2019.2906319 doi: 10.1109/tmi.2019.2906319
[13]	R. Wang, B. Chen, D. Meng, L. Wang, Weakly supervised lesion detection from fundus images, IEEE Trans. Med. Imaging, 38 (2019), 1501-1512. https://doi.org/10.1109/TMI.2018.2885376 doi: 10.1109/TMI.2018.2885376
[14]	K. Shankar, A. R. W. Saitet, D. Gupta, S. K. Lakshmanaprabu, A. Khanna, H. M. Pandey, Automated detection and classification of fundus diabetic retinopathy images using synergic deep learning model, Pattern Recognit. Lett., 133 (2020), 210-216. https://doi.org/10.1016/j.patrec.2020.02.026 doi: 10.1016/j.patrec.2020.02.026
[15]	M. Hire, S. Shinde, Ant colony optimization based exudates segmentation in retinal fundus images and classification, in 2018 Fourth International Conference on Computing Communication Control and Automation (ICCUBEA), (2018), 1-6. https://doi.org/10.1109/ICCUBEA.2018.8697727
[16]	E. Imani, H. Pourreza, A novel method for retinal exudate segmentation using signal separation algorithm, Comput. Methods Programs Biomed., 133 (2016), 195-205. https://doi.org/10.1016/j.cmpb.2016.05.016 doi: 10.1016/j.cmpb.2016.05.016
[17]	V. Sathananthavathi, G. Indumathi, R. Rajalakshmi, Abnormalities detection in retinal fundus images, in 2017 International Conference on Inventive Communication and Computational Technologies (ICICCT), (2017), 89-93. https://doi.org/10.1109/ICICCT.2017.7975165
[18]	O. Oktay, J. Schlemper, L. L. Folgoc1, M. Lee, M. Heinrich, K. Misawa, et al., Attention U-Net: Learning where to look for the pancreas, preprint, arXiv: 1804.03999v3.
[19]	N. Ilyasova, A. Shirokanev, N. Demin, R. Paringer, Graph-based segmentation for diabetic macular edema selection in OCT images, in 2019 Fifth International Conference on Frontiers of Signal Processing (ICFSP), (2019), 77-81. https://doi.org/10.1109/ICFSP48124.2019.8938047
[20]	M. Tavakoli, S. Jazani, M. Nazar, Automated detection of microaneurysms in color fundus images using deep learning with different preprocessing approaches, in Medical Imaging 2020: Imaging Informatics for Healthcare, Research, and Applications, 11318 (2020), 113180E. https://doi.org/10.1117/12.2548526
[21]	Z. Yu, C. Feng, M. Y. Liu, S. Ramalingam, CASENet: deep category-aware semantic edge detection, in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2017), 1761-1770. https://doi.org/10.1109/CVPR.2017.191
[22]	J. Mo, L. Zhang, Y. Feng, Exudate-based diabetic macular edema recognition in retinal images using cascaded deep residual networks, Neurocomputing, 290 (2018), 161-171. https://doi.org/10.1016/j.neucom.2018.02.035 doi: 10.1016/j.neucom.2018.02.035
[23]	N. Kasabov, N. M. Scott, E. Tu, S. Marks, N. Sengupta, E. Capecci, et al., Evolving spatio-temporal data machines based on the NeuCube neuromorphic framework: Design methodology and selected applications, Neural Networks, 78 (2016), 1-14. https://doi.org/10.1016/j.neunet.2015.09.011 doi: 10.1016/j.neunet.2015.09.011
[24]	L. K. Abood, Contrast enhancement of infrared images using Adaptive Histogram Equalization (AHE) with Contrast Limited Adaptive Histogram Equalization (CLAHE), Iraqi J. Phys., 16 (2018). https://doi.org/10.30723/ijp.v16i37.84 doi: 10.30723/ijp.v16i37.84
[25]	O. Ramos-Soto, E. Rodríguez-Esparza, S. E. Balderas-Mata, D. Oliva, A. E. Hassanien, R. K. Meleppat, et al., An efficient retinal blood vessel segmentation in eye fundus images by using optimized top-hat and homomorphic filtering, Comput. Methods Programs Biomed., 201 (2021), 105949. https://doi.org/10.1016/j.cmpb.2021.105949 doi: 10.1016/j.cmpb.2021.105949
[26]	X. Fan, J. Gong, Y. Yan, Red lesion detection in fundus images based on convolution neural network, in 2019 Chinese Control And Decision Conference (CCDC), (2019), 5661-5666. https://doi.org/10.1109/CCDC.2019.8833280
[27]	P. Maragos, 3.3 - Morphological filtering for image enhancement and feature detection, in Handbook of Image and Video Processing (Second Edition), (2005), 135-156. https://doi.org/10.1016/B978-012119792-6/50072-3
[28]	L. Cheng, J. Xiong, L. He, Non-gaussian statistical timing analysis using second-order polynomial fitting, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 28 (2009), 130-140. https://doi.org/10.1109/TCAD.2008.2009143 doi: 10.1109/TCAD.2008.2009143
[29]	A. Valizadeh, Z. J. Wang, Minimum mean square error detector for multimessage spread spectrum embedding, in 2009 Sixteenth IEEE International Conference on Image Processing (ICIP), (2009), 121-124. https://doi.org/10.1109/ICIP.2009.5414115
[30]	O. Ronneberger, U-Net convolutional networks for biomedical image segmentation, in Bildverarbeitung für die Medizin 2017, (2017), 3. https://doi.org/10.1007/978-3-662-54345-0_3
[31]	L. Han, Y. Chen, J. Li, B. Zhong, Y. Lei, M. Sun, Liver segmentation with 2.5D perpendicular UNets, Comput. Electr. Eng., 91 (2021), 107118. https://doi.org/10.1016/j.compeleceng.2021.107118 doi: 10.1016/j.compeleceng.2021.107118
[32]	Y. Zhang, H. Lai, W. Yang, Cascade UNet and CH-UNet for thyroid nodule segmentation and benign and malignant classification, in MICCAI 2020: Segmentation, Classification, and Registration of Multi-modality Medical Imaging Data, (2021), 129-134. https://doi.org/10.1007/978-3-030-71827-5_17
[33]	Z. H. Zhou, A brief introduction to weakly supervised learning, Natl. Sci. Rev., 5 (2018), 44-53. https://doi.org/10.1093/nsr/nwx106 doi: 10.1093/nsr/nwx106
[34]	T. Li, Y. Gao, K. Wang, S. Guo, H. Liu, H. Kang, Diagnostic assessment of deep learning algorithms for diabetic retinopathy screening, Inf. Sci., 501 (2019), 511-522. https://doi.org/10.1016/j.ins.2019.06.011 doi: 10.1016/j.ins.2019.06.011
[35]	E. Decencière, X. Zhang, G. Cazuguel, B. Lay, B. Cochener, C. Trone, et al., Feedback on a publicly distributed image database: the messidor database, Image Anal. Stereol., 33 (2014), 231-234. https://doi.org/10.5566/ias.1155 doi: 10.5566/ias.1155
[36]	P. Porwal, S. Pachade, M. Kokare, G. Deshmukh, F. Mériaudeau, IDRiD: Diabetic retinopathy - segmentation and grading challenge, Med. Image Anal., 59 (2020), 101561.
[37]	L. C. Chen, Y. Zhu, G. Papandreou, F. Schroff, H. Adam, Encoder-decoder with atrous separable convolution for semantic image segmentation, in Computer Vision - ECCV 2018, 833-851. https://doi.org/10.1007/978-3-030-01234-2_49

Reader Comments

Your name:*

Email:*
© 2022 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)