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\documentclass{mcmthesis}
%\setlength{\parskip}{0.1em} 
 %\documentclass[CTeX = true]{mcmthesis}  % 当使用 CTeX 套装时请注释上一行使用该行的设置
\mcmsetup{tstyle=\color{black}\bfseries,%修改题号，队号的颜色和加粗显示，黑色可以修改为 black
        tcn = H034, problem = B, %修改队号，参赛题号
        sheet = true, titleinsheet = true, keywordsinsheet = true,%修改sheet显示信息
        titlepage = false, abstract = true}

  %四款字体可以选择
  \usepackage{comment}
  \usepackage{times}
  %\usepackage{newtxtext,newtxmath} %CTeX 无此字体，可用 txfonts 替代，请使用新版 TeXLive.
  %\usepackage{palatino}
  %\usepackage{txfonts}
  \usepackage{hyperref} % 引入超链接包
 \usepackage{algorithm}%自己加的写伪代码用
\usepackage{algorithmic}%同上
%\usepackage{algpseudocode}%同上
\usepackage{graphicx}
\usepackage{subcaption}%画图用
\usepackage{indentfirst}  %首行缩进，注释掉，首行就不再缩进。
\usepackage{lipsum}
\usepackage{enumitem}
\title{\textbf{Mastering the Magnetic Realm: Unveiling Earth's Secrets through Multi-Sensor Synergy}}
\author{\small \href{https://www.latexstudio.net/}
  {\includegraphics[width=7cm]{mcmthesis-logo}}}
\date{\today}
\begin{document}
In order to better describe the model, we assign the following symbols meanings:
\begin{table}[H]  % 强制表格固定在当前位置
  \caption{Notations Table}  % 插入图表标题
  \begin{center}
    \centering % 表格居中
    \begin{tabularx}{\linewidth}{lX}  % 使用 \linewidth 限制表格宽度
        \toprule[1.5pt]
        \textbf{Notations}  & \textbf{Explanation} \\ 
        \midrule[1pt]
        $\mathcal{L}$ & Loss function: difference between predicted and actual values. \\
        $N$ & Total number of time points or samples. \\
        $Y_{1}(t), (i=1)$ & Observed value at time $t$ for Sensor 1. \\
        $\hat{Y}_{1}(t), (i=1)$ & Predicted value at time $t$ for Sensor 1. \\
        $A_{i}(t), (i=1, 2, 3, \dots)$ & Additional signal of Sensor $i$ at time $t$. \\
        $Y_{i}(t), (i=1, 2, 3, \dots)$ & Actual measurement from Sensor $i$. \\
        $\hat{Y}_{i}(t), (i=1, 2, 3, \dots)$ & Predicted measurement for Sensor $i$. \\
        $N_{\text{time}}$ & Total number of time intervals. \\
        $t_{0,j}$ & Start time of the $j$-th interval. \\
        $t_{0,j}+24$ & End time of the $j$-th interval. \\
        $\Delta M_{x,i}(t), \Delta M_{y,i}(t), \Delta M_{z,i}(t)$ & Change in geomagnetic field for Sensor $i$. \\
        $B_{1}, B_{2}, \dots, B_{6}$ & Symbols for different concepts. \\
        $G_1, G_2, G_3$ & Components of the objective function. \\
        $B_3, B_4, B_5, B_6$ & Objective functions in Models II and III. \\
        $\text{MSE}$ & Mean Squared Error: evaluates filtering performance. \\
        $\text{SNR}$ & Signal-to-Noise Ratio: measures signal quality. \\
        $\hat{x}(t)$ & Reconstructed signal: denoised version of $f(t)$. \\
        $x(t)$ & Raw signal with potential noise. \\
        $T$ & Noise threshold in wavelet transformation. \\
        $x_1(t)$ & Geomagnetic signal recorded by Sensor 1. \\
        $\hat{x}_i(t)$ & Signals from other sensors. \\
        $y(t)$ & Magnetic signal from the target object. \\
        \bottomrule[1.5pt]
    \end{tabularx}
    \label{tab:notations} % 设置标签以便引用
  \end{center}
\end{table}


\vspace{-10pt}  % 减少与后续内容的垂直间距
 % 清空浮动队列，确保表格后紧跟正文

\section{Model I: Comprehensive Filtering Algorithm Based on ResNet-Autoencoder Convolutional Network and W-SG}

    \begin{algorithm}[H]
\caption{Signal Preprocessing and Autoencoder Processing}
\textbf{Input:}\\
\setlength{\parindent}{1em} 
\indent Raw signal $X \in \mathbb{R}^{N \times M}$ \\
\indent Savitzky-Golay filter parameters $(\text{window\_length}, \text{polyorder})$ \\
\indent Wavelet decomposition parameters $(\text{wavelet}, \text{level})$ \\
\textbf{Output:}\\
\indent Filtered signal $X_{\text{filtered}} \in \mathbb{R}^{N \times M}$ \\
\indent Latent representation $Z \in \mathbb{R}^{N \times K}$ \\
\indent Reconstructed signal $X_{\text{reconstructed}} \in \mathbb{R}^{N \times M}$ \\
\begin{algorithmic}[1]
\STATE \textbf{Step 1: Encoder Module.}
$$
\text{Encoder}(X) = 
\begin{cases} 
    \text{Conv1D}(X, 64, \text{kernel\_size}=3, \text{padding}=1) \rightarrow \text{ReLU} \\
    \text{Conv1D}(64, 32, \text{kernel\_size}=3, \text{padding}=1) \rightarrow \text{ReLU} \\
    \text{Conv1D}(32, 16, \text{kernel\_size}=3, \text{padding}=1)
\end{cases}
$$
\STATE \textbf{Step 2: Decoder Module.}
$$
\text{Decoder}(Z) = 
\begin{cases} 
    \text{Conv1D}(Z, 32, \text{kernel\_size}=3, \text{padding}=1) \rightarrow \text{ReLU} \\
    \text{Conv1D}(32, 64, \text{kernel\_size}=3, \text{padding}=1) \rightarrow \text{ReLU} \\
    \text{Conv1D}(64, M, \text{kernel\_size}=3, \text{padding}=1)
\end{cases}
$$

\STATE \textbf{Step 3: Smooth Signal Using Savitzky-Golay Filter.}
$$
X_{\text{smooth}}[i] = \text{Savgol}(X[i], \text{window\_length}, \text{polyorder}), \; \forall i \in \{1, \dots, N\}
$$

\STATE \textbf{Step 4: Perform Wavelet Decomposition and Thresholding.}
$$
\text{coeffs} = \text{wavedec}(X_{\text{smooth}}, \text{wavelet}, \text{level})
$$
$$
\text{threshold} = \sqrt{2 \ln(N)} \cdot \frac{\text{median}(|\text{coeffs}[-1]|)}{0.6745}
$$
$$
\text{denoised\_coeffs}[k] = \text{threshold}(\text{coeffs}[k], \text{threshold}, \text{mode}=\text{soft}), \; \forall k
$$

\STATE \textbf{Step 5: Reconstruct Signal After Filtering.}
$$
X_{\text{filtered}} = \text{waverec}(\text{denoised\_coeffs}, \text{wavelet})
$$

\STATE \textbf{Step 6: Encode and Reconstruct.}
$$
Z = \text{Encoder}(X_{\text{filtered}})
$$
$$
X_{\text{reconstructed}} = \text{Decoder}(Z) + X_{\text{filtered}}
$$

\STATE \textbf{Output:} Return $X_{\text{filtered}}$, $Z$, and $X_{\text{reconstructed}}$.
\end{algorithmic}
\end{algorithm}

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