A specific fine-grained identification model for plasma-treated rice growth using multiscale shortcut convolutional neural network

Wenzhuo Chen; Yuan Wang; Xiaojiang Tang; Pengfei Yan; Xin Liu; Lianfeng Lin; Guannan Shi; Eric Robert; Feng Huang; Wenzhuo Chen; Yuan Wang; Xiaojiang Tang; Pengfei Yan; Xin Liu; Lianfeng Lin; Guannan Shi; Eric Robert; Feng Huang

doi:10.3934/mbe.2023448

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 6: 10223-10243. doi: 10.3934/mbe.2023448

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Research article

A specific fine-grained identification model for plasma-treated rice growth using multiscale shortcut convolutional neural network

1.
College of Information and Electrical Engineering, China Agricultural University, Beijing, 10083, China
2.
College of Science, China Agricultural University, Beijing 100083, China
3.
GREMI, UMR 7344, CNRS/Université d'Orléans, 45067 Orléans Cedex France
4.
LE STUDIUM Loire Valley Institute for Advanced Studies, Centre-Val de Loire region, France
^#The same contribution

Received: 31 December 2022 Revised: 12 March 2023 Accepted: 19 March 2023 Published: 30 March 2023

As an agricultural innovation, low-temperature plasma technology is an environmentally friendly green technology that increases crop quality and productivity. However, there is a lack of research on the identification of plasma-treated rice growth. Although traditional convolutional neural networks (CNN) can automatically share convolution kernels and extract features, the outputs are only suitable for entry-level categorization. Indeed, shortcuts from the bottom layers to fully connected layers can be established feasibly in order to utilize spatial and local information from the bottom layers, which contain small distinctions necessary for fine-grain identification. In this work, 5000 original images which contain the basic growth information of rice (including plasma treated rice and the control rice) at the tillering stage were collected. An efficient multiscale shortcut CNN (MSCNN) model utilizing key information and cross-layer features was proposed. The results show that MSCNN outperforms the mainstream models in terms of accuracy, recall, precision and F1 score with 92.64%, 90.87%, 92.88% and 92.69%, respectively. Finally, the ablation experiment, comparing the average precision of MSCNN with and without shortcuts, revealed that the MSCNN with three shortcuts achieved the best performance with the highest precision.

Keywords:

Citation: Wenzhuo Chen, Yuan Wang, Xiaojiang Tang, Pengfei Yan, Xin Liu, Lianfeng Lin, Guannan Shi, Eric Robert, Feng Huang. A specific fine-grained identification model for plasma-treated rice growth using multiscale shortcut convolutional neural network[J]. Mathematical Biosciences and Engineering, 2023, 20(6): 10223-10243. doi: 10.3934/mbe.2023448

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Abstract

1. Introduction

Since the 1960s, the rapid development of high-speed rail has made it a very important means of transportation. However, the vibration will be caused because of the contact between the wheels of the train and the train tracks during the operation of the high-speed train. Therefore, the analytical vibration model can be mathematically summarized as a quadratic palindromic eigenvalue problem (QPEP) (see ^[1,2])

$\begin{equation*} (\lambda^2 A_1+\lambda A_0+A_1^\top)x = 0, \end{equation*}$

with $A_i \in {\mathbb{R}}^{n \times n}, \ i = 0, 1$ and $A_0^\top = A_0.$ The eigenvalues $\lambda$ , the corresponding eigenvectors $x$ are relevant to the vibration frequencies and the shapes of the vibration, respectively. Many scholars have put forward many effective methods to solve QPEP ^[3,4,5,6,7]. In addition, under mild assumptions, the quadratic palindromic eigenvalue problem can be converted to the following linear palindromic eigenvalue problem (see ^[8])

$\begin{equation} Ax = \lambda A^\top x, \end{equation}$

(1)

with $A \in {\mathbb{R}}^{n \times n}$ is a given matrix, $\lambda\in {\mathbb{C}}$ and nonzero vectors $x\in {\mathbb{C}}^{n}$ are the wanted eigenvalues and eigenvectors of the vibration model. We can obtain $\frac{1}{\lambda} x^\top A^\top = x^\top A$ by transposing the equation (1). Thus, $\lambda$ and $\frac{1}{\lambda}$ always come in pairs. Many methods have been proposed to solve the palindromic eigenvalue problem such as URV-decomposition based structured method ^[9], QR-like algorithm ^[10], structure-preserving methods ^[11], and palindromic doubling algorithm ^[12].

On the other hand, the modal data obtained by the mathematical model are often evidently different from the relevant experimental ones because of the complexity of the structure and inevitable factors of the actual model. Therefore, the coefficient matrices need to be modified so that the updated model satisfies the dynamic equation and closely matches the experimental data. Al-Ammari ^[13] considered the inverse quadratic palindromic eigenvalue problem. Batzke and Mehl ^[14] studied the inverse eigenvalue problem for T-palindromic matrix polynomials excluding the case that both $+1$ and $-1$ are eigenvalues. Zhao et al. ^[15] updated $\ast$ -palindromic quadratic systems with no spill-over. However, the linear inverse palindromic eigenvalue problem has not been extensively considered in recent years.

In this work, we just consider the linear inverse palindromic eigenvalue problem (IPEP). It can be stated as the following problem:

Problem IPEP. Given a pair of matrices $(\Lambda, X)$ in the form

$\begin{equation*} \Lambda = \text{diag} \{\lambda_{1}, \cdots, \lambda_{p}\} \in {\mathbb{C}}^{p \times p}, \end{equation*}$

and

$\begin{equation*} X = [x_{1}, \cdots, x_{p}] \in {\mathbb{C}}^{n \times p}, \end{equation*}$

where diagonal elements of $\Lambda$ are all distinct, $X$ is of full column rank $p$ , and both $\Lambda$ and $X$ are closed under complex conjugation in the sense that $\lambda_{2i} = \bar{\lambda}_{2i-1} \in {\mathbb{C}}$ , $x_{2i} = \bar{x}_{2i-1} \in {\mathbb{C}}^{n}$ for $i = 1, \cdots, m$ , and $\lambda_{j} \in {\mathbb{R}}$ , $x_{j} \in {\mathbb{R}}^{n}$ for $j = 2m+1, \cdots, p$ , find a real-valued matrix $A$ that satisfy the equation

$\begin{equation} AX = A^\top X\Lambda. \end{equation}$

(2)

Namely, each pair $(\lambda_{t}, x_{t})$ , $t = 1, \cdots, p$ , is an eigenpair of the matrix pencil

$P(\lambda) = A x- \lambda A^\top x.$

It is known that the mathematical model is a "good" representation of the system, we hope to find a model that is closest to the original model. Therefore, we consider the following best approximation problem:

Problem BAP. Given $\tilde{A} \in {\mathbb{R}}^{n \times n}$ , find $\hat{A} \in {\mathcal{S}}_{A}$ such that

$\begin{equation} \|\hat{A}-\tilde{A}\| = \min\limits_{{A} \in {\mathcal{S}_{A}}} \|A-\tilde{A}\|, \end{equation}$

(3)

where $\| \cdot \|$ is the Frobenius norm, and ${\mathcal{S}}_{A}$ is the solution set of Problem IPEP.

In this paper, we will put forward a new direct method to solve Problem IPEP and Problem BAP. By partitioning the matrix $\Lambda$ and using the QR-decomposition, the expression of the general solution of Problem IPEP is derived. Also, we show that the best approximation solution $\hat{A}$ of Problem BAP is unique and derive an explicit formula for it.

2. The solution of Problem IPEP

We first rearrange the matrix $\Lambda$ as

$\begin{equation} \begin{array}{ccc} \Lambda = \left[ \begin{array}{ccc} 1 & 0 & \ \ \ 0 \\ 0 & \Lambda_1 & \ \ \ 0 \\ 0 & 0 & \ \ \ \Lambda_2 \\ \end{array} \right]&\begin{array}{c} t \\ 2s \\ 2(k+2l) \\ \end{array} \\ \begin{array}{ccc} \ \ \ \ \ \ \ \ \ \ \ t & \ 2s & 2(k+2l) \\ \end{array}& \end{array}, \end{equation}$

(4)

where $t+2s+2(k+2l) = p$ , $t = 0 \ or \ 1,$

$\begin{eqnarray*} && \Lambda_1 = \text{diag} \{\lambda_1, \lambda_2, \cdots, \lambda_{2s-1}, \lambda_{2s}\}, \lambda_i \in \mathbb{R}, \ \lambda_{2i-1}^{-1} = \lambda_{2i}, \ 1\leq i\leq s, \\ && \Lambda_2 = \text{diag} \{\delta_1, \cdots, \delta_k, \delta_{k+1}, \delta_{k+2}, \cdots, \delta_{k+2l-1}, \delta_{k+2l}\}, \ \delta_j \in \mathbb{C}^{2 \times 2}, \end{eqnarray*}$

with

$\begin{eqnarray*} &&\delta_j = \left[ \begin{array}{cc} \alpha_j+\beta_{j} i & 0 \\ 0 & \alpha_j-\beta_{j} i \\ \end{array} \right], \ i = \sqrt{-1}, \ 1\leq j\leq k+2l, \\ &&\delta_j^{-1} = \bar{\delta}_j, \ 1 \leq j \leq k, \\ &&\delta_{k+2j-1}^{-1} = \delta_{k+2j}, \ 1 \leq j \leq l, \end{eqnarray*}$

and the adjustment of the column vectors of $X$ corresponds to those of $\Lambda$ .

Define $T_p$ as

$\begin{equation} T_p = \text{diag}\left\{ I_{t+2s}, \frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1 & -i \\ 1 & i \\ \end{array}\right], \cdots, \frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1 & -i \\ 1 & i \\ \end{array}\right]\right\} \in \mathbb{C}^{p \times p}, \end{equation}$

(5)

where $i = \sqrt{-1}$ . It is easy to verify that $T_p^\mathrm{H} T_p = I_p$ . Using this matrix of (5), we obtain

$\begin{equation} \tilde{\Lambda} = T_p^\mathrm{H} \Lambda T_p = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \Lambda_1 & 0 \\ 0 & 0 & \tilde{\Lambda}_2 \\ \end{array} \right], \end{equation}$

(6)

$\begin{equation} \tilde{X} = XT_p = [x_t, \cdots, x_{t+2s}, \sqrt{2}y_{t+2s+1}, \sqrt{2}z_{t+2s+1}, \cdots, \sqrt{2}y_{p-1}, \sqrt{2}z_{p-1}], \end{equation}$

(7)

where

$\begin{equation*} \tilde{\Lambda}_2 = \text{diag} \left\{ \left[\begin{array}{cc} \alpha_1 & \beta_1 \\ -\beta_1 & \alpha_1 \\ \end{array}\right], \cdots, \left[\begin{array}{cc} \alpha_{k+2l} & \beta_{k+2l} \\ -\beta_{k+2l} & \alpha_{k+2l} \\ \end{array}\right] \right\}\triangleq \text{diag} \{\tilde{\delta}_1, \cdots, \tilde{\delta}_{k+2l}\}, \end{equation*}$

and $\tilde{\Lambda}_2 \in {\bf \mathbb{R}}^{2(k+2l) \times 2(k+2l)}$ , $\tilde{X} \in {\bf \mathbb{R}}^{n \times p}$ . $y_{t+2s+j}$ and $z_{t+2s+j}$ are, respectively, the real part and imaginary part of the complex vector $x_{t+2s+j}$ for $j = 1, 3, \cdots, 2(k+2l)-1$ . Using (6) and (7), the matrix equation (2) is equivalent to

$\begin{equation} A\tilde{X} = A^\top \tilde{X}\tilde{\Lambda}. \end{equation}$

(8)

Since $\text{rank}(X) = \text{rank}(\tilde{X}) = p$ . Now, let the QR-decomposition of $\tilde{X}$ be

$\begin{equation} \tilde{X} = Q\left[ \begin{array}{c} R \\ 0 \\ \end{array} \right], \end{equation}$

(9)

where $Q = [Q_1, Q_2]\in \mathbb{R}^{n \times n}$ is an orthogonal matrix and $R \in \mathbb{R}^{p \times p}$ is nonsingular. Let

$\begin{equation} \begin{array}{cc} Q^\top AQ = \left[ \begin{array}{cc} A_{11}& A_{12} \\ A_{21}& A_{22} \\ \end{array}\right]&\begin{array}{c} p \\ n-p \\ \end{array} \\ \begin{array}{cc} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p & \ n-p \\ \end{array}& \end{array}. \end{equation}$

(10)

Using (9) and (10), then the equation of (8) is equivalent to

$\begin{eqnarray} && A_{11}R = A_{11}^\top R\tilde{\Lambda}, \end{eqnarray}$

(11)

$\begin{eqnarray} && A_{21}R = A_{12}^\top R\tilde{\Lambda}. \end{eqnarray}$

(12)

Write

$\begin{equation} \begin{array}{ccc} R^\top A_{11}R\triangleq F = \left[ \begin{array}{ccc} f_{11} & F_{12} & \ \ \ F_{13} \\ F_{21} & F_{22} & \ \ \ F_{23} \\ F_{31} & F_{32} & \ \ \ F_{33} \\ \end{array} \right]&\begin{array}{c} t \\ 2s \\ 2(k+2l) \\ \end{array} \\ \begin{array}{ccc} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t & \ \ \ 2s & \ 2(k+2l) \\ \end{array}& \end{array}, \end{equation}$

(13)

then the equation of (11) is equivalent to

$\begin{eqnarray} && F_{12} = F_{21}^\top \Lambda_1, \ F_{21} = F_{12}^\top, \end{eqnarray}$

(14)

$\begin{eqnarray} && F_{13} = F_{31}^\top \tilde{\Lambda}_2, \ F_{31} = F_{13}^\top, \end{eqnarray}$

(15)

$\begin{eqnarray} && F_{23} = F_{32}^\top \tilde{\Lambda}_2, \ F_{32} = F_{23}^\top \Lambda_1, \end{eqnarray}$

(16)

$\begin{eqnarray} && F_{22} = F_{22}^\top \Lambda_1, \end{eqnarray}$

(17)

$\begin{eqnarray} && F_{33} = F_{33}^\top \tilde{\Lambda}_2. \end{eqnarray}$

(18)

Because the elements of $\Lambda_1, \tilde{\Lambda}_2$ are distinct, we can obtain the following relations by Eqs (14)-(18)

$\begin{eqnarray} &&F_{12} = 0, \ F_{21} = 0, \ F_{13} = 0, \ F_{31} = 0, \ F_{23} = 0, \ F_{32} = 0, \end{eqnarray}$

(19)

$\begin{eqnarray} &&F_{22} = \text{diag} \left\{ \left[\begin{array}{cc} 0 & h_{1} \\ \lambda_1 h_{1} & 0 \\ \end{array}\right], \cdots, \left[\begin{array}{cc} 0 & h_{s} \\ \lambda_{2s-1} h_{s} & 0 \\ \end{array}\right] \right\}, \end{eqnarray}$

(20)

$\begin{eqnarray} &&F_{33} = \text{diag}\left\{G_{1}, \cdots, G_{k}, \left[\begin{array}{cc} 0 & G_{k+1} \\ G_{k+1}^\top\tilde{\delta}_{k+1} & 0 \\ \end{array}\right], \cdots, \left[\begin{array}{cc} 0 & G_{k+l} \\ G_{k+l}^\top\tilde{\delta}_{k+2l-1} & 0 \\ \end{array}\right]\right\}, \end{eqnarray}$

(21)

where

$\begin{eqnarray*} && G_i = a_iB_i, \ G_{k+j} = a_{k+2j-1}D_1+a_{k+2j}D_2, \ G_{k+j}^\top = G_{k+j}, \\ && B_i = \left[\begin{array}{cc} 1 & \frac{1-\alpha_{i}}{\beta_{i}} \\ -\frac{1-\alpha_{i}}{\beta_{i}} & 1 \\ \end{array}\right], \ D_1 = \left[\begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array}\right], \ D_2 = \left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array}\right], \end{eqnarray*}$

and $1\leq i\leq k, 1\leq j\leq l$ . $h_{1}, \cdots, h_{s}, a_{1}, \cdots, a_{k+2l}$ are arbitrary real numbers. It follows from Eq (12) that

$\begin{equation} A_{21} = A_{12}^\top E, \end{equation}$

(22)

where $E = R\tilde{\Lambda}R^{-1}$ .

Theorem 1. Suppose that $\Lambda = \mathit{\text{diag}} \{\lambda_{1}, \cdots, \lambda_{p}\} \in {\mathbb{C}}^{p \times p}$ , $X = [x_{1}, \cdots, x_{p}] \in {\mathbb{C}}^{n \times p}$ , where diagonal elements of $\Lambda$ are all distinct, $X$ is of full column rank $p$ , and both $\Lambda$ and $X$ are closed under complex conjugation in the sense that $\lambda_{2i} = \bar{\lambda}_{2i-1} \in {\mathbb{C}}$ , $x_{2i} = \bar{x}_{2i-1} \in {\mathbb{C}}^{n}$ for $i = 1, \cdots, m$ , and $\lambda_{j} \in {\mathbb{R}}$ , $x_{j} \in {\mathbb{R}}^{n}$ for $j = 2m+1, \cdots, p$ . Rearrange the matrix $\Lambda$ as $(4)$ , and adjust the column vectors of $X$ with corresponding to those of $\Lambda$ . Let $\Lambda, X$ transform into $\tilde{\Lambda}, \tilde{X}$ by $(6)-(7)$ and QR-decomposition of the matrix $\tilde{X}$ be given by $(9)$ . Then the general solution of $(2)$ can be expressed as

$\begin{equation} {\mathcal{S}}_{A} = \left\{A \left| A = Q\left[ \begin{array}{cc} R^{-\top}\left[ \begin{array}{ccc} f_{11} & 0 & 0 \\ 0 & F_{22} & 0 \\ 0 & 0 & F_{33} \\ \end{array} \right]R^{-1} & A_{12} \\ A_{12}^\top E & A_{22} \\ \end{array} \right]Q^\top \right. \right\}, \end{equation}$

(23)

where $E = R\tilde{\Lambda}R^{-1},$ $f_{11}$ is arbitrary real number, $A_{12} \in {\mathbb{R}}^{p \times (n-p)}, A_{22}\in {\mathbb{R}}^{(n-p) \times (n-p)}$ are arbitrary real-valued matrices and $F_{22}, F_{33}$ are given by $(20)-(21)$ .

3. The solution of Problem BAP

In order to solve Problem BAP, we need the following lemma.

Lemma 1. ^[16] Let $A, B$ be two real matrices, and $X$ be an unknown variable matrix. Then

$\begin{eqnarray*} && \frac{\partial tr(BX)}{\partial X} = B^\top, \ \frac{\partial tr(X^\top B^\top)}{\partial X} = B^\top, \ \frac{\partial tr(AXBX)}{\partial X} = (BXA+AXB)^\top, \\ && \frac{\partial tr(AX^\top BX^\top)}{\partial X} = BX^\top A+AX^\top B, \ \frac{\partial tr(AXBX^\top)}{\partial X} = AXB+A^\top XB^\top. \end{eqnarray*}$

By Theorem $1$ , we can obtain the explicit representation of the solution set ${\mathcal{S}}_{A}$ . It is easy to verify that $\mathcal{S}_A$ is a closed convex subset of ${\bf \mathbb{R}}^{n \times n}\times {\bf \mathbb{R}}^{n \times n}.$ By the best approximation theorem (see Ref. ^[17]), we know that there exists a unique solution of Problem BAP. In the following we will seek the unique solution $\hat{A}$ in $\mathcal{S}_A.$ For the given matrix $\tilde{A} \in {\bf \mathbb{R}}^{n \times n},$ write

$\begin{equation} \begin{array}{cc} Q^\top \tilde{A}Q = \left[ \begin{array}{cc} \tilde{A}_{11}& \tilde{A}_{12} \\ \tilde{A}_{21}& \tilde{A}_{22} \\ \end{array}\right]&\begin{array}{c} p \\ n-p \\ \end{array} \\ \begin{array}{cc} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p & \ n-p \\ \end{array}& \end{array}, \end{equation}$

(24)

then

$\begin{eqnarray*} \|A-\tilde{A}\|^2 & = & \left\|\left[ \begin{array}{cc} R^{- \top}\left[ \begin{array}{ccc} f_{11} & 0 & 0 \\ 0 & F_{22} & 0 \\ 0 & 0 & F_{33} \\ \end{array} \right]R^{-1}-\tilde{A}_{11} & A_{12}-\tilde{A}_{12} \\ A_{12}^\top E-\tilde{A}_{21} & A_{22}-\tilde{A}_{22} \\ \end{array} \right]\right\|^2\\ & = &\left\| R^{- \top}\left[ \begin{array}{ccc} f_{11} & 0 & 0 \\ 0 & F_{22} & 0 \\ 0 & 0 & F_{33} \\ \end{array} \right]R^{-1}-\tilde{A}_{11}\right\|^2\\ &+& \| A_{12}-\tilde{A}_{12}\|^2+\|A_{12}^\top E-\tilde{A}_{21} \|^2+\|A_{22}-\tilde{A}_{22}\|^2. \end{eqnarray*}$

Therefore, $\|A-\tilde{A}\| = \min$ if and only if

$\begin{eqnarray} && \left\|R^{-\top}\left[ \begin{array}{ccc} f_{11} & 0 & 0 \\ 0 & F_{22} & 0 \\ 0 & 0 & F_{33} \\ \end{array} \right]R^{-1}-\tilde{A}_{11}\right\|^2 = \min, \end{eqnarray}$

(25)

$\begin{eqnarray} && \| A_{12}-\tilde{A}_{12}\|^2+\|A_{12}^\top E-\tilde{A}_{21} \|^2 = \min, \end{eqnarray}$

(26)

$\begin{eqnarray} &&A_{22} = \tilde{A}_{22}. \end{eqnarray}$

(27)

Let

$\begin{equation} \begin{array}{c} R^{-1} = \left[ \begin{array}{c} R_1 \\ R_2 \\ R_3 \\ \end{array} \right] \end{array}, \end{equation}$

(28)

then the relation of (25) is equivalent to

$\begin{equation} \|R_1^\top f_{11}R_1+ R_2^\top F_{22}R_2 + R_3^\top F_{33} R_3- \tilde{A}_{11} \|^2 = \min. \end{equation}$

(29)

Write

$\begin{equation} R_1 = \left[r_{1, t}\right], \ R_2 = \left[ \begin{array}{c} r_{2, 1} \\ \vdots \\ r_{2, 2s}\\ \end{array} \right], \ R_3 = \left[ \begin{array}{c} r_{3, 1} \\ \vdots \\ r_{3, k+2l} \\ \end{array} \right], \end{equation}$

(30)

where $r_{1, t} \in {\mathbb{R}}^{t \times p}, r_{2, i} \in {\mathbb{R}}^{1 \times p}, r_{3, j} \in {\mathbb{R}}^{2 \times p}, \ i = 1, \cdots, 2s, \ j = 1, \cdots, k+2l$ .

Let

$\begin{equation} \left\{ \begin{array}{rcl} && J_t = r_{1, t}^\top r_{1, t}, \\ && J_{t+i} = \lambda_{2i-1}r_{2, 2i}^\top r_{2, 2i-1}+r_{2, 2i-1}^\top r_{2, 2i} \ (1 \leq i \leq s), \\ && J_{r+i} = r_{3, i}^\top B_i r_{3, i} \ (1 \leq i \leq k), \\ && J_{r+k+2i-1} = r_{3, k+2i}^\top D_1 \tilde{\delta}_{k+2i-1}r_{3, k+2i-1}+r_{3, k+2i-1}^\top D_1 r_{3, k+2i} \ (1 \leq i \leq l), \\ && J_{r+k+2i} = r_{3, k+2i}^\top D_2 \tilde{\delta}_{k+2i-1}r_{3, k+2i-1}+r_{3, k+2i-1}^\top D_2 r_{3, k+2i} \ (1 \leq i \leq l), \end{array} \right. \end{equation}$

(31)

with $r = t+s, q = t+s+k+2l.$ Then the relation of (29) is equivalent to

$\begin{eqnarray*} && g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l}) = \\ &&\|f_{11}J_t+h_1J_{t+1}+\cdots+h_sJ_{r}+a_1J_{r+1}+\cdots+a_{k+2l}J_{q}-\tilde{A}_{11}\|^2 = \text{min}, \end{eqnarray*}$

that is,

$\begin{eqnarray*} && g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l}) \\ && = \text{tr} [(f_{11}J_t+h_1J_{t+1}+\cdots+h_sJ_{r}+a_1J_{r+1}+\cdots+a_{k+2l}J_{q}-\tilde{A}_{11})^\top \\ && (f_{11}J_t+h_1J_{t+1}+\cdots+h_sJ_{r}+a_1J_{r+1}+\cdots+a_{k+2l}J_{q}-\tilde{A}_{11})]\\ && = f_{11}^2c_{t, t}+2f_{11}h_1c_{t, t+1}+\cdots+2f_{11}h_sc_{t, r}+2f_{11}a_1c_{t, r+1}+\cdots+2f_{11}a_{k+2l}c_{t, q}-2f_{11}e_t\\ && +h_1^2c_{t+1, t+1}+\cdots+2h_1h_sc_{t+1, r}+2h_1a_1c_{t+1, r+1}+\cdots+2h_1a_{k+2l}c_{t+1, q}-2h_1e_{t+1} \\ && +\cdots \\ && +h_s^2c_{r, r}+2h_sa_1c_{r, r+1}+\cdots+2h_sa_{k+2l}c_{r, q}-2h_se_{r}\\ && +a_1^2c_{r+1, r+1}+\cdots+2a_1a_{k+2l}c_{r+1, q}-2a_1e_{r+1}\\ && +\cdots \\ && +a_{k+2l}^2c_{q, q}-2a_{k+2l}e_{q}+\text{tr} (\tilde{A}_{11}^\top \tilde{A}_{11}), \end{eqnarray*}$

where $c_{i, j} = \text{tr} (J_i^\top J_j), e_i = \text{tr} (J_i^\top\tilde{A}_{11}) (i, j = t, \cdots, t+s+k+2l)$ and $c_{i, j} = c_{j, i}$ .

Consequently,

$\begin{eqnarray*} \frac{\partial g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l})}{\partial f_{11}}&& = 2f_{11}c_{t, t}+2h_1c_{t, t+1}+\cdots+2h_sc_{t, r}+2a_1c_{t, r+1}\\ &&+\cdots+2a_{k+2l}c_{t, q}-2e_t, \\ \frac{\partial g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l})}{\partial h_{1}}&& = 2f_{11}c_{t+1, t}+2h_1c_{t+1, t+1}+\cdots+2h_sc_{t+1, r}+2a_1c_{t+1, r+1}\\ &&+\cdots+2a_{k+2l}c_{t+1, q}-2e_{t+1}, \\ &&\cdots\\ \frac{\partial g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l})}{\partial h_{s}}&& = 2f_{11}c_{r, t}+2h_1c_{r, t+1}+\cdots+2h_sc_{r, r}+2a_1c_{r, r+1}\\ &&+\cdots+2a_{k+2l}c_{r, q}-2e_{r}, \\ \frac{\partial g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l})}{\partial a_{1}}&& = 2f_{11}c_{r+1, t}+2h_1c_{r+1, t+1}+\cdots+2h_sc_{r+1, r}+2a_1c_{r+1, r+1}\\ &&+\cdots+2a_{k+2l}c_{r+1, q}-2e_{r+1}, \\ &&\cdots\\ \frac{\partial g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l})}{\partial a_{k+2l}}&& = 2f_{11}c_{q, t}+2h_1c_{q, t+1}+\cdots+2h_sc_{q, r}+2a_1c_{q, r+1}\\ &&+\cdots+2a_{k+2l}c_{q, q}-2e_{q}. \end{eqnarray*}$

Clearly, $g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l}) = \text{min}$ if and only if

$\frac{\partial g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l})}{\partial f_{11}} = 0, \cdots, \frac{\partial g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l})}{\partial a_{k+2l}} = 0.$

Therefore,

$\begin{equation} \begin{split} &f_{11}c_{t, t}+h_1c_{t, t+1}+\cdots+h_sc_{t, r}+a_1c_{t, r+1}+\cdots+a_{k+2l}c_{t, q} = e_t, \\ &f_{11}c_{t+1, t}+h_1c_{t+1, t+1}+\cdots+h_sc_{t+1, r}+a_1c_{t+1, r+1}+\cdots+a_{k+2l}c_{t+1, q} = e_{t+1}, \\ &\cdots\\ &f_{11}c_{r, t}+h_1c_{r, t+1}+\cdots+h_sc_{r, r}+a_1c_{r, r+1}+\cdots+a_{k+2l}c_{r, q} = e_{r}, \\ &f_{11}c_{r+1, t}+h_1c_{r+1, t+1}+\cdots+h_sc_{r+1, r}+a_1c_{r+1, r+1}+\cdots+a_{k+2l}c_{r+1, q} = e_{r+1}, \\ &\cdots\\ &f_{11}c_{q, t}+h_1c_{q, t+1}+\cdots+h_sc_{q, r}+a_1c_{q, r+1}+\cdots+a_{k+2l}c_{q, q} = e_{q}. \end{split} \end{equation}$

(32)

If let

$C = \left[ \begin{array}{ccccccc} c_{t, t}&c_{t, t+1}&\cdots&c_{t, r}&c_{t, r+1}&\cdots&c_{t, q}\\ c_{t+1, t}&c_{t+1, t+1}&\cdots&c_{t+1, r}&c_{t+1, r+1}&\cdots&c_{t+1, q}\\ \vdots&\vdots& &\vdots&\vdots& &\vdots\\ c_{r, t}&c_{r, t+1}&\cdots&c_{r, r}&c_{r, r+1}&\cdots&c_{r, q}\\ c_{r+1, t}&c_{r+1, t+1}&\cdots&c_{r+1, r}&c_{r+1, r+1}&\cdots&c_{r+1, q}\\ \vdots&\vdots& &\vdots&\vdots& &\vdots\\ c_{q, t}&c_{q, t+1}&\cdots&c_{q, r}&c_{q, r+1}&\cdots&c_{q, q}\\ \end{array} \right], \ h = \left[ \begin{array}{c} f_{11}\\ h_1\\ \vdots\\ h_s\\ a_1\\ \vdots\\ a_{k+2l}\\ \end{array} \right], \ e = \left[ \begin{array}{c} e_t\\ e_{t+1}\\ \vdots\\ e_{r}\\ e_{r+1}\\ \vdots\\ e_{q}\\ \end{array} \right],$

where $C$ is symmetric matrix. Then the equation (32) is equivalent to

$\begin{equation} C h = e, \end{equation}$

(33)

and the solution of the equation (33) is

$\begin{equation} h = C^{-1} e. \end{equation}$

(34)

Substituting (34) into (20)-(21), we can obtain $f_{11}, F_{22}$ and $F_{33}$ explicitly. Similarly, the equation of (26) is equivalent to

$\begin{eqnarray*} &&g(A_{12}) = \text{tr} (A_{12}^\top A_{12})+\text{tr} (\tilde{A}_{12}^\top \tilde{A}_{12})-2\text{tr} (A_{12}^\top \tilde{A}_{12})\\ &&+\text{tr} (E^\top A_{12}A_{12}^\top E)+\text{tr} (\tilde{A}_{21}^\top \tilde{A}_{21})-2\text{tr} (E^\top A_{12}\tilde{A}_{21}). \end{eqnarray*}$

Applying Lemma 1, we obtain

$\frac{\partial g(A_{12})}{\partial A_{12}} = 2A_{12}-2\tilde{A}_{12}+2EE^\top A_{12}-2E\tilde{A}_{21}^\top,$

setting $\frac{\partial g(A_{12})}{\partial A_{12}} = 0$ , we obtain

$\begin{equation} A_{12} = (I_p+EE^\top)^{-1}(\tilde{A}_{12}+E\tilde{A}_{21}^\top), \end{equation}$

(35)

Theorem 2. Given $\tilde{A} \in {\mathbb{R}}^{n \times n}$ , then the Problem BAP has a unique solution and the unique solution of Problem BAP is

$\begin{equation} \hat{A} = Q\left[ \begin{array}{cc} R^{-\top}\left[ \begin{array}{ccc} f_{11} & 0 & 0 \\ 0 & F_{22} & 0 \\ 0 & 0 & F_{33} \\ \end{array} \right]R^{-1} & A_{12} \\ A_{12}^\top E & \tilde{A}_{22} \\ \end{array} \right]Q^\top, \end{equation}$

(36)

where $E = R\tilde{\Lambda} R^{-1}$ , $F_{22}, F_{33}, A_{12}, \tilde{A}_{22}$ are given by $(20), (21), (35), (24)$ and $f_{11}, h_{1}, \cdots, h_s, a_1, \cdots, a_{k+2l}$ are given by $(34)$ .

4. A numerical example

Based on Theorems $1$ and $2$ , we can describe an algorithm for solving Problem BAP as follows.

Algorithm 1.

1) Input matrices $\Lambda$ , $X$ and $\tilde{A}$ ;

2) Rearrange $\Lambda$ as (4), and adjust the column vectors of $X$ with corresponding to those of $\Lambda$ ;

3) Form the unitary transformation matrix $T_p$ by (5);

4) Compute real-valued matrices $\tilde{\Lambda}, \tilde{X}$ by (6) and (7);

5) Compute the QR-decomposition of $\tilde{X}$ by (9);

6) $F_{12} = 0, F_{21} = 0, F_{13} = 0, F_{31} = 0, F_{23} = 0, F_{32} = 0$ by (19) and $E = R\tilde{\Lambda} R^{-1}$ ;

7) Compute $\tilde{A}_{ij} = Q_i^\top \tilde{A}Q_j, i, j = 1, 2$ ;

8) Compute $R^{-1}$ by (28) to form $R_1, R_2, R_3$ ;

9) Divide matrices $R_1, R_2, R_3$ by (30) to form $r_{1, t}, r_{2, i}, r_{3, j},$ $i = 1, \cdots, 2s, j = 1, \cdots, k+2l$ ;

10) Compute $J_i,$ $i = t, \cdots, t+s+k+2l,$ by (31);

11) Compute $c_{i, j} = \mbox{tr} (J_i^\top J_j), e_i = \mbox{tr} (J_i^\top\tilde{A}_{11}),$ $i, j = t, \cdots, t+s+k+2l$ ;

12) Compute $f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l}$ by (34);

13) Compute $F_{22}, F_{33}$ by (20), (21) and $A_{22} = \tilde{A}_{22}$ ;

14) Compute $A_{12}$ by (35) and $A_{21}$ by (22);

15) Compute the matrix $\hat{A}$ by (36).

Example 1. Consider a $11$ -DOF system, where

$\tilde{A} = \left[ {\begin{array}{rrrrrrrrrrr} 96.1898 & 18.1847 & 51.3250 & 49.0864 & 13.1973 & 64.9115 & 62.5619 & 81.7628 & 58.7045 & 31.1102 & 26.2212 \\ 0.4634 & 26.3803 & 40.1808 & 48.9253 & 94.2051 & 73.1722 & 78.0227 & 79.4831 & 20.7742 & 92.3380 & 60.2843 \\ 77.4910 & 14.5539 & 7.5967 & 33.7719 & 95.6135 & 64.7746 & 8.1126 & 64.4318 & 30.1246 & 43.0207 & 71.1216 \\ 81.7303 & 13.6069 & 23.9916 & 90.0054 & 57.5209 & 45.0924 & 92.9386 & 37.8609 & 47.0923 & 18.4816 & 22.1747 \\ 86.8695 & 86.9292 & 12.3319 & 36.9247 & 5.9780 & 54.7009 & 77.5713 & 81.1580 & 23.0488 & 90.4881 & 11.7418 \\ 8.4436 & 57.9705 & 18.3908 & 11.1203 & 23.4780 & 29.6321 & 48.6792 & 53.2826 & 84.4309 & 97.9748 & 29.6676 \\ 39.9783 & 54.9860 & 23.9953 & 78.0252 & 35.3159 & 74.4693 & 43.5859 & 35.0727 & 19.4764 & 43.8870 & 31.8778 \\ 25.9870 & 14.4955 & 41.7267 & 38.9739 & 82.1194 & 18.8955 & 44.6784 & 93.9002 & 22.5922 & 11.1119 & 42.4167 \\ 80.0068 & 85.3031 & 4.9654 & 24.1691 & 1.5403 & 68.6775 & 30.6349 & 87.5943 & 17.0708 & 25.8065 & 50.7858 \\ 43.1414 & 62.2055 & 90.2716 & 40.3912 & 4.3024 & 18.3511 & 50.8509 & 55.0156 & 22.7664 & 40.8720 & 8.5516 \\ 91.0648 & 35.0952 & 94.4787 & 9.6455 & 16.8990 & 36.8485 & 51.0772 & 62.2475 & 43.5699 & 59.4896 & 26.2482 \\ \end{array}} \right],$

the measured eigenvalue and eigenvector matrices $\Lambda$ and $X$ are given by

$\begin{eqnarray*} &&\Lambda = \mbox{diag} \{1.0000, \ -1.8969, \ -0.5272, \ -0.1131+0.9936i, -0.1131-0.9936i, \\ &&1.9228+2.7256i, \ 1.9228-2.7256i, \ 0.1728-0.2450i, \ 0.1728+0.2450i\}, \end{eqnarray*}$

and

$\begin{eqnarray*} &&X = \left[ {\begin{array}{rrrrr} -0.0132& -1.0000& 0.1753& 0.0840 + 0.4722i& 0.0840 - 0.4722i \\ -0.0955& 0.3937& 0.1196& -0.3302 - 0.1892i& -0.3302 + 0.1892i \\ -0.1992& 0.5220& -0.0401& 0.3930 - 0.2908i& 0.3930 + 0.2908i \\ 0.0740& 0.0287& 0.6295& -0.3587 - 0.3507i& -0.3587 + 0.3507i \\ 0.4425& -0.3609& -0.5745& 0.4544 - 0.3119i& 0.4544 + 0.3119i \\ 0.4544& -0.3192& -0.2461& -0.3002 - 0.1267i& -0.3002 + 0.1267i \\ 0.2597& 0.3363& 0.9046& -0.2398 - 0.0134i& -0.2398 + 0.0134i \\ 0.1140& 0.0966& 0.0871& 0.1508 + 0.0275i& 0.1508 - 0.0275i \\ -0.0914& -0.0356& -0.2387& -0.1890 - 0.0492i& -0.1890 + 0.0492i \\ 0.2431& 0.5428& -1.0000& 0.6652 + 0.3348i& 0.6652 - 0.3348i \\ 1.0000& -0.2458& 0.2430& -0.2434 + 0.6061i& -0.2434 - 0.6061i \\ \end{array}} \right. \\ &&\left. {\begin{array}{rrrr} 0.6669 + 0.2418i& 0.6669 - 0.2418i& 0.2556 - 0.1080i& 0.2556 + 0.1080i \\ -0.1172 - 0.0674i& -0.1172 + 0.0674i& -0.5506 - 0.1209i& -0.5506 + 0.1209i \\ 0.5597 - 0.2765i& 0.5597 + 0.2765i& -0.3308 + 0.1936i& -0.3308 - 0.1936i \\ -0.7217 - 0.0566i& -0.7217 + 0.0566i& -0.7306 - 0.2136i& -0.7306 + 0.2136i \\ 0.0909 + 0.0713i& 0.0909 - 0.0713i& 0.5577 + 0.1291i& 0.5577 - 0.1291i \\ 0.1867 + 0.0254i& 0.1867 - 0.0254i& 0.2866 + 0.1427i& 0.2866 - 0.1427i \\ -0.5311 - 0.1165i& -0.5311 + 0.1165i& -0.3873 - 0.1096i& -0.3873 + 0.1096i \\ 0.2624 + 0.0114i& 0.2624 - 0.0114i& -0.6438 + 0.2188i& -0.6438 - 0.2188i \\ -0.0619 - 0.1504i& -0.0619 + 0.1504i& 0.2787 - 0.2166i& 0.2787 + 0.2166i \\ 0.3294 - 0.1718i& 0.3294 + 0.1718i& 0.9333 + 0.0667i& 0.9333 - 0.0667i \\ -0.4812 + 0.5188i& -0.4812 - 0.5188i& 0.6483 - 0.1950i& 0.6483 + 0.1950i \\ \end{array}} \right]. \end{eqnarray*}$

Using Algorithm 1, we obtain the unique solution of Problem BAP as follows:

$\begin{eqnarray*} \hat{A} = \left[ {\begin{array}{rrrrrrrrrrr} 34.2563 & 41.7824 & 33.3573 & 33.6298 & 23.8064 & 42.0770 & 50.0641 & 37.5705 & 31.0908 & 48.6169 & 19.0972\\ 18.8561 & 35.2252 & 35.9592 & 44.3502 & 31.9918 & 55.2920 & 55.3052 & 54.3793 & 31.3909 & 60.8345 & 16.9540\\ 29.6359 & 7.6805 & 19.1249 & 17.7183 & 16.7082 & 40.0636 & 18.2916 & 49.9437 & 37.6913 & 15.6027 & 4.9603\\ 58.8782 & 51.4906 & 47.8974 & 35.6985 & 45.6889 & 56.0434 & 53.0908 & 56.5402 & 55.5120 & 38.3447 & 35.8894\\ 33.4087 & 46.9635 & 9.7767 & 41.4215 & 51.4466 & 52.1058 & 65.6724 & 60.1293 & 5.8061 & 62.0139 & 16.5231\\ 31.6580 & 51.2359 & 24.7978 & 65.5567 & 61.7840 & 62.5494 & 58.9363 & 74.7099 & 52.2105 & 55.8532 & 44.3925\\ 19.2961 & 51.2333 & 22.4280 & 56.9340 & 42.6348 & 45.8453 & 56.3729 & 61.5555 & 31.6836 & 67.9525 & 40.2012\\ 41.2796 & 71.3821 & 34.4140 & 33.2817 & 77.4393 & 60.8944 & 32.1411 & 108.5056 & 49.6078 & 19.8351 & 85.7434\\ 64.0890 & 57.6524 & 19.1280 & 25.0394 & 39.0524 & 66.7740 & 20.9023 & 48.8512 & 14.4695 & 18.9284 & 24.8348\\ 37.2550 & 32.3254 & 38.3534 & 59.7358 & 33.5902 & 54.0265 & 50.7770 & 70.2011 & 65.4159 & 58.0720 & 40.0652\\ 28.1301 & 14.7638 & 8.9507 & 20.0963 & 25.5907 & 59.6940 & 30.8558 & 66.8781 & 30.4807 & 23.6107 & 12.9984\\ \end{array}} \right], \end{eqnarray*}$

and

$\|\hat{A}X-\hat{A}^\top X\Lambda\| = 8.2431\times 10^{-13}.$

Therefore, the new model $\hat{A}X = \hat{A}^\top X\Lambda$ reproduces the prescribed eigenvalues (the diagonal elements of the matrix $\Lambda$ ) and eigenvectors (the column vectors of the matrix $X$ ).

Example 2. (Example 4.1 of ^[12]) Given $\alpha = \cos(\theta)$ , $\beta = \sin(\theta)$ with $\theta = 0.62$ and $\lambda_1 = 0.2, \lambda_2 = 0.3, \lambda_3 = 0.4$ . Let

$\begin{equation*} J_0 = \left[ {\begin{array}{rr} 0_2 & \Gamma\\ I_2 & I_2\\ \end{array}} \right], \ J_s = \left[ {\begin{array}{cc} 0_3 & \mbox{diag} \{\lambda_1, \lambda_2, \lambda_3 \}\\ I_3 & 0_3\\ \end{array}} \right], \end{equation*}$

where $\Gamma = \left[{\begin{array}{cc} \alpha & -\beta\\ \beta & \alpha\\ \end{array}} \right].$ We construct

$\begin{equation*} \tilde{A} = \left[ {\begin{array}{rr} J_0 & 0\\ 0 & J_s\\ \end{array}} \right], \end{equation*}$

the measured eigenvalue and eigenvector matrices $\Lambda$ and $X$ are given by

$\begin{eqnarray*} \Lambda = \mbox{diag} \{5, 0.2, 0.8139+0.5810i, 0.8139-0.5810i\}, \end{eqnarray*}$

and

$\begin{eqnarray*} X = \left[ {\begin{array}{rrrr} -0.4155& 0.6875& -0.2157 - 0.4824i& -0.2157 + 0.4824i\\ -0.4224& -0.3148& -0.3752 + 0.1610i& -0.3752 - 0.1610i\\ -0.0703& -0.6302& -0.5950 - 0.4050i& -0.5950 + 0.4050i\\ -1.0000& -0.4667& 0.2293 - 0.1045i& 0.2293 + 0.1045i\\ 0.2650& 0.3051& -0.2253 + 0.7115i& -0.2253 - 0.7115i\\ 0.9030& -0.2327& 0.4862 - 0.3311i& 0.4862 + 0.3311i\\ -0.6742& 0.3132& 0.5521 - 0.0430i& 0.5521 + 0.0430i\\ 0.6358& 0.1172& -0.0623 - 0.0341i& -0.0623 + 0.0341i\\ -0.4119& -0.2768& 0.1575 + 0.4333i& 0.1575 - 0.4333i\\ -0.2062& 1.0000& -0.1779 - 0.0784i& -0.1779 + 0.0784i\\ \end{array}} \right]. \end{eqnarray*}$

Using Algorithm 1, we obtain the unique solution of Problem BAP as follows:

$\begin{equation*} \hat{A} = \left[ {\begin{array}{rrrrrrrrrr} -0.1169& -0.2366& 0.6172& -0.7195& -0.0836& 0.2884& 0.0092& -0.0490& -0.0202& 0.0171\\ -0.0114& -0.0957& 0.1462& 0.6194& 0.3738& -0.1637& 0.1291& -0.0071& 0.0972& 0.1247\\ 0.7607& -0.0497& 0.5803& -0.0346& 0.0979& 0.2959& 0.0937& -0.1060& 0.1323& -0.0339\\ -0.0109& 0.6740& -0.3013& 0.7340& 0.1942& -0.0872& 0.0054& 0.0051& 0.0297& 0.0814\\ 0.1783& 0.2283& 0.2643& 0.0387& 0.0986& -0.3125& -0.0292& 0.2926& -0.0717& -0.0546\\ 0.0953& 0.1027& 0.0360& 0.2668& -0.2418& 0.1206& 0.1406& -0.0551& 0.3071& 0.2097\\ -0.0106& -0.2319& 0.1946& -0.0298& -0.1935& 0.0158& -0.0886& 0.0216& -0.0560& 0.2484\\ 0.1044& 0.1285& 0.1902& 0.2277& 0.6961& 0.1657& 0.0728& -0.0262& -0.0831& -0.0001\\ 0.0906& 0.0021& 0.0764& -0.1264& 0.2144& 0.6703& -0.0850& 0.0764& -0.0104& -0.0149\\ -0.1245& 0.0813& 0.1952& -0.0784& 0.0760& -0.0875& 0.7978& -0.0093& 0.0206& -0.1182\\ \end{array}} \right], \end{equation*}$

and

$\|\hat{A}X-\hat{A}^\top X\Lambda\| = 1.7538\times 10^{-8}.$

5. Concluding remarks

In this paper, we have developed a direct method to solve the linear inverse palindromic eigenvalue problem by partitioning the matrix $\Lambda$ and using the QR-decomposition. The explicit best approximation solution is given. The numerical examples show that the proposed method is straightforward and easy to implement.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	A. M. Khaneghah, L. M. Martins, A. M. Von Hertwig, R. Bertoldo, A. S. Sant'Ana, Deoxynivalenol and its masked forms: Characteristics, incidence, control and fate during wheat and wheat-based products processing - A review, Trends Food Sci. Technol., 71 (2018), 13−24. https://doi.org/10.1016/j.tifs.2017.10.012 doi: 10.1016/j.tifs.2017.10.012
[2]	N.S. Poluxeni, M. Sotirios, K. Chrysanthi, S. Panagiotis, H. Luc, Chemical pesticides and human health: the urgent need for a new concept in agriculture, Front. Public Health, 4 (2016) 148−148. https://doi.org/10.3389/fpubh.2016.00148 doi: 10.3389/fpubh.2016.00148
[3]	X. Lei, R. Qiu, Evaluation of food security in China based entropy TOPSIS model and the diagnosis of its obstacle factors, J. China Agric. Univ., 27 (2022), 1−14. https://doi.org/10.11841/j.issn.1007-4333.2022.12.01 doi: 10.11841/j.issn.1007-4333.2022.12.01
[4]	Y. T. Hui, D. C. Wang, Y. You, C. Y. Shao, C. S. Zhong, H. D. Wang, Effect of low temperature plasma treatment on biological characteristics and yield components of wheat seeds (Triticum aestivum L.), Plasma Chem. Plasma Process., 40 (2020), 1555−1570. https://doi.org/10.1007/s11090-020-10104-z doi: 10.1007/s11090-020-10104-z
[5]	H. Liu, Y. H. Zhang, H. Yin, W. X. Wang, X. M. Zhao, Y. G. Du, Alginate oligosaccharides enhanced triticum aestivum L. tolerance to drought stress, Plant Physiol. Biochem., 62 (2013), 33−40. https://doi.org/10.1016/j.plaphy.2012.10.012 doi: 10.1016/j.plaphy.2012.10.012
[6]	B. Šerá, P. r Špatenka, M. l Šerý, N. Vrchotová, I. a Hrušková, Influence of plasma treatment on wheat and oat germination and early growth, IEEE Trans. Plasma Sci., 38 (2010), 2963−2968. https://doi.org/10.1109/TPS.2010.2060728 doi: 10.1109/TPS.2010.2060728
[7]	R. Thirumdas, A. Kothakota, U. Annapure, K. Siliveru, R. Blundell, R. Gatt, et al., Plasma activated water (PAW) Chemistry, physico-chemical properties, applications in food and agriculture, Trends Food Sci. Technol., 77 (2018), 21−31. https://doi.org/10.1016/j.tifs.2018.05.007 doi: 10.1016/j.tifs.2018.05.007
[8]	L. Tonks, Oscillations in ionized gases, Plasma and Oscillations, Elsevier, 1961,122−139. https://doi.org/10.1016/B978-1-4831-9913-9.50014-5
[9]	B. Zhao, J. S. Feng, X. Wu, S. C. Yan, A survey on deep learning-based fine-grained object classification and semantic segmentation, Int. J. Autom. Comput., 14 (2017), 119−135. https://doi.org/10.1007/s11633-017-1053-3 doi: 10.1007/s11633-017-1053-3
[10]	A. Srivastava, E. Han, V. Kumar, V. Singh, Parallel formulations of decision-tree classification algorithms, High Performance Data Mining, Springer, Boston, 1999,237−261. https://doi.org/10.1007/0-306-47011-X_2
[11]	G. D. Guo, H. Wang, D. Bell, Y. X. Bi, KNN model-based approach in classification, in OTM Confederated International Conferences CoopIS, DOA, and ODBASE, (2003), 986−996. https://doi.org/10.1007/978-3-540-39964-3_62
[12]	A. Tharwat, A. E. Hassanien, B. E. Elnaghi, A BA-based algorithm for parameter optimization of Support Vector Machine, Pattern Recognit. Lett., 93 (2017), 13−22. https://doi.org/10.1016/j.patrec.2016.10.007 doi: 10.1016/j.patrec.2016.10.007
[13]	N. Coskun, T. Yildirim, The effects of training algorithms in MLP network on image classification, in Proceedings of the International Joint Conference on Neural Networks, (2003), 1223−1226.
[14]	J. Deng, J. Krause, F. F. Li, Fine-grained crowdsourcing for fine-grained recognition, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2013), 580−587. https://doi.org/10.1109/CVPR.2013.81
[15]	E. Gavves, B. Fernando, C. G. Snoek, A. W. Smeulders, T. Tuytelaars, Fine-grained categorization by alignments, in Proceedings of the IEEE International Conference on Computer Vision, (2013), 1713−1720. https://doi.org/10.1109/ICCV.2013.215
[16]	K. M. He, X. Y. Zhang, S. Q. Ren, J. Sun, Deep residual learning for image recognition, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2016), 770−778. https://doi.org/10.1109/CVPR.2016.90
[17]	G. Huang, Z. Liu, L. Van Der Maaten, K. Q. Weinberger, Densely connected convolutional networks, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2017), 4700−4708. https://doi.org/10.1109/cvpr.2017.243
[18]	S. Ioffe, C. Szegedy, Batch normalization: Accelerating deep network training by reducing internal covariate shift, in Proceedings of the 32nd International Conference on Machine Learning, (2015), 448−456.
[19]	A. Krizhevsky, I. Sutskever, G. E. Hinton, ImageNet classification with deep convolutional neural networks, Commun. ACM, 60 (2017), 84-90. https://doi.org/10.1145/3065386 doi: 10.1145/3065386
[20]	S. Jin, H. X. Yao, X. S. Sun, S. C. Zhou, L. Zhang, X. S. Hua, Deep saliency hashing for fine-grained retrieval, IEEE Trans. Image Process., 29 (2020), 5336−5351. https://doi.org/10.1109/TIP.2020.2971105 doi: 10.1109/TIP.2020.2971105
[21]	Y. Jing, W. Wang, L. Wang, T. N. Tan, Learning aligned image-text representations using graph attentive relational net-work, IEEE Trans. Image Process., 30 (2021), 1840−1852. https://doi.org/10.1109/TIP.2020.3048627 doi: 10.1109/TIP.2020.3048627
[22]	L. L. Zhang, J. Liu, M. N. Luo, X. J. Chang, Q. H. Zheng, Deep semisupervised zero-shot learning with maximum mean discrepancy, Neural Comput., 30 (2018), 1426−1447. https://doi.org/10.1162/neco_a_01071 doi: 10.1162/neco_a_01071
[23]	K. Liu, D. Liu, L. Li, N. Yan, H. Q. Li, Semantics-to-signal scalable image compression with learned revertible representations, Int. J. Comput. Vis., 129 (2021), 2605−2621. https://doi.org/10.1007/s11263-021-01491-7 doi: 10.1007/s11263-021-01491-7
[24]	L. Qi, X. Q. Lu, X. L. Li, Exploiting spatial relation for fine-grained image classification, Pattern Recognit., 91 (2019), 47−55. https://doi.org/10.1016/j.patcog.2019.02.007 doi: 10.1016/j.patcog.2019.02.007
[25]	L. Wang, K. He, X. Feng, X. T. Ma, Multilayer feature fusion with parallel convolutional block for fine-grained image classification, Appl. Intell., 52 (2022), 2872−2883. https://doi.org/10.1007/s10489-021-02573-2 doi: 10.1007/s10489-021-02573-2
[26]	M. Srinivas, Y. Y. Lin, H. Y. M. Liao, Deep dictionary learning for fine-grained image classification, in 2017 IEEE International Conference on Image Processing, (2017), 835−839. https://doi.org/10.1109/ICIP.2017.8296398
[27]	L. Liao, R. M. Hu, J. Xiao, Q. Wang, J. Xiao, J. Chen, Exploiting effects of parts in fine-grained categorization of vehicles, in 2015 IEEE international conference on image processing, (2015), 745−749. https://doi.org/10.1109/ICIP.2015.7350898
[28]	K. Wang, M. Z. Liu, YOLOv3-MTis A YOLOv3 using multi-target tracking for vehicle visual detection, Appl. Intell., 52 (2022), 2070−2091. https://doi.org/10.1007/s10489-021-02491-3 doi: 10.1007/s10489-021-02491-3
[29]	S. M. Pan, W. Q. Feng, Y. W. Chong, Attribute-guided global and part-level identity network for person re-identification, Int. J. Pattern Recognit. Artif. Intell., 36 (2022), 2250011. https://doi.org/S0218001422500112 doi: 10.1142/S0218001422500112
[30]	C. Wang, J. Y. Sun, S. W. Ma, Y. Q. Lu, W. Liu, Multi-stream network for human-object interaction detection, Int. J. Pattern Recognit. Artif. Intell., 35 (2021), 2150025. https://doi.org/10.1142/S0218001421500257 doi: 10.1142/S0218001421500257
[31]	Z. Q. Lin, S. M. Mu, F. Huang, K. A. Mateen, M. J. Wang, W. L. Gao, et al., A unified matrix-based convolutional neural network for fine-grained image classification of wheat leaf diseases, IEEE Access, 7 (2019), 11570−11590. https://doi.org/10.1109/ACCESS.2019.2891739 doi: 10.1109/ACCESS.2019.2891739
[32]	Z. Q. Lin, S. M. Mu, A. J. Shi, C. Pang, X. X. Sun, A novel method of maize leaf disease image identification based on a multichannel convolutional neural network, Trans. ASABE, 61 (2018), 1461−1474. https://doi.org/10.13031/trans.12440 doi: 10.13031/trans.12440
[33]	H. Lu, Z. G. Cao, Y. Xiao, Z. W. Fang, Y. J. Zhu, Fine-grained maize cultivar identification using filter-specific convolutional activations, in 2016 IEEE International Conference on Image Processing, (2016), 3718−3722. https://doi.org/10.1109/ICIP.2016.7533054
[34]	X. P. Zhang, H. K. Xiong, W. G. Zhou, Q. Tian, Fused one-vs-all features with semantic alignments for fine-grained visual categorization, IEEE Trans. Image Process., 25 (2015), 878−892. https://doi.org/10.1109/TIP.2015.2509425 doi: 10.1109/TIP.2015.2509425
[35]	X. S. Wei, C. W. Xie, J. X. Wu, C. H. Shen, Mask-CNN is Localizing parts and selecting descriptors for fine-grained bird species categorization, Pattern Recognit., 76 (2018), 704−714. https://doi.org/10.1016/j.patcog.2017.10.002 doi: 10.1016/j.patcog.2017.10.002
[36]	L. Qi, X. Q. Lu, X. L. Li, Exploiting spatial relation for fine-grained image classification, Pattern Recognit., 91 (2019), 47−55. https://doi.org/10.1016/j.patcog.2019.02.007 doi: 10.1016/j.patcog.2019.02.007
[37]	Y. Zhang, X. S. Wei, J. X. Wu, J. F. Cai, J. B. Lu, V. A. Nguyen, et al., Weakly supervised fine-grained categorization with part-based image representation, IEEE Trans. Image Process., 25 (2016), 1713−1725. https://doi.org/10.1109/TIP.2016.2531289 doi: 10.1109/TIP.2016.2531289
[38]	S. L. Huang, Z. Xu, D. C. Tao, Y. Zhang, Part-Stacked CNN for fine-grained visual categorization, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2016), 1173−1182. https://doi.org/10.1109/CVPR.2016.132
[39]	S. H. Lee, C. S. Chan, S. J. Mayo, P. Remagnino, How deep learning extracts and learns leaf features for plant classification, Pattern Recognit., 71 (2017), 1−13. https://doi.org/10.1016/j.patcog.2017.05.015 doi: 10.1016/j.patcog.2017.05.015
[40]	M. Rohrbach, A. Rohrbach, M. Regneri, S. Amin, M. Andriluka, M. Pinkal, et al., Recognizing fine-grained and composite activities using hand-centric features and script data, Int. J. Comput. Vision., 119 (2016), 346−373. https://doi.org/10.1007/s11263-015-0851-8 doi: 10.1007/s11263-015-0851-8
[41]	S. Cai, W. Zuo, Z. Lei, Higher-order integration of hierarchical convolutional activations for fine-grained visual categorization, in Proceedings of the IEEE International Conference on Computer Vision, (2017), 511−520.
[42]	Q. Hu, H. Wang, T. Li, C. Shen, Deep CNNs with spatially weighted pooling for fine-grained car recognition, IEEE. Intell Transp., 91 (2019), 47−55. https://doi.org/10.1016/j.patcog.2019.02.007 doi: 10.1016/j.patcog.2019.02.007
[43]	P. J. Burt, E. H. Adelson, Readings in computer vision, Elsevier, Piscataway, 1987,671−679. https://doi.org/10.1016/B978-0-08-051581-6.50065-9
[44]	C. Farabet, C. Couprie, L. Najman, Y. LeCun, Learning hierarchical features for scene labeling, IEEE Trans. Pattern Anal. Mach. Intell., 35 (2012), 1915−1929. https://doi.org/10.1109/TPAMI.2012.231 doi: 10.1109/TPAMI.2012.231
[45]	B. Hariharan, P. Arbeláez, R. Girshick, J. Malik, Hyper columns for object segmentation and fine-grained localization, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2015), 447−456. https://doi.org/10.1109/CVPR.2015.7298642
[46]	J. Weber, J. Malik, Robust computation of optical flow in a multi-scale differential framework, Int. J. Comput. Vis., 14 (1995), 67−81. https://doi.org/10.1007/BF01421489 doi: 10.1007/BF01421489
[47]	H. L. Zheng, J. L. Fu, T. Mei, J.B. Luo, Learning multi-attention convolutional neural network for fine-grained image recognition, in Proceedings of the IEEE international conference on computer vision, (2017), 5209−5217. https://doi.org/10.1109/ICCV.2017.557
[48]	T. Y. Lin, A. RoyChowdhury, S. Maji, Bilinear CNN models for fine-grained visual recognition, in Proceedings of the IEEE International Conference on Computer Vision, (2015), 1449−1457. https://doi.org/10.1109/ICCV.2015.170
[49]	A. Fawzi, H. Samulowitz, D. Turaga, P. Frossard, Adaptive data augmentation for image classification, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2013), 580−587. https://doi.org/10.1109/ICIP.2016.7533048
[50]	R. Dellana, K. Roy, Data augmentation in CNN-based periocular authentication, in 2016 6th International Conference on Information Communication and Management, (2016), 141−145. https://doi.org/10.1109/INFOCOMAN.2016.7784231
[51]	J. Johnson, A. Karpathy, F. F. Li, Densecap is Fully convolutional localization networks for dense captioning, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2016), 4565−4574. https://doi.org/10.1109/CVPR.2016.494
[52]	N. S. Keskar, D. Mudigere, J. Nocedal, M. Smelyanskiy, P. T. P. Tang, On large-batch training for deep learning is generalization gap and sharp minima, (2016). https://doi.org/10.48550/arXiv.1609.04836
[53]	H. Li, Z. Xu, G. Taylor, T. Goldstein, Visualizing the loss landscape of neural nets, in 32nd Conference on Neural Information Processing Systems, 31 (2018).
[54]	P. Goyal, P. Dollár, R. Girshick, P. Noordhuis, L. Wesolowski, A. Kyrola, et al., Accurate, large minibatch SGD is Training ImageNet in 1 hour, (2017). https://doi.org/10.48550/arXiv.1706.02677
[55]	K. Simonyan, A. Zisserman, Very deep convolutional networks for large-scale image recognition, in Proceedings of International Conference on Learning Representations, (2015). https://doi.org/10.48550/arXiv.1409.1556

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