\documentclass[preview]{standalone}
\usepackage{amsmath}
\usepackage[flushleft]{threeparttable}
\usepackage{makecell,booktabs}
\begin{document}
\title{Multi-sensor Fusion Rules}
\newsavebox\independent
\begin{lrbox}{\independent}
$\begin{aligned}
P_f &= (\sum_{i=1}^{n} P_i^{-1})^{-1} \\
\hat{x}_f &= P_f (\sum_{i=1}^{n} P_i^{-1} \hat{x}_i)
\end{aligned} $
\end{lrbox}
%-------------------------------
\newsavebox\correlated
\begin{lrbox}{\correlated}
$ \begin{aligned}
P_f &= (e^T \Sigma^{-1} e)^{-1} \\
\hat{x}_f &= P_f (e^T \Sigma^{-1} \hat{x})
\end{aligned} $
\end{lrbox}
%-------------------------------
\newsavebox\uc
\begin{lrbox}{\uc}
$ \begin{aligned}
P_f &= (\sum_{i=1}^{n} \omega_i P_i^{-1})^{-1} \\
\hat{x}_f &= P_f (\sum_{i=1}^{n} \omega_i P_i^{-1} \hat{x}_i)
\end{aligned} $
\end{lrbox}
%-------------------------------
\begin{table}
\caption{Multi-sensor Fusion Rules}
\centering
\begin{threeparttable}
\begin{tabular}{cc@{\qquad}c}
Types of Estimation Errors & Fusion Rules & Comments \\ \midrule\midrule
\makecell{No Correlations \\(Independent)} & \usebox{\independent} & Optimal \\
\cmidrule(l r){1-3}
\makecell{Known Correlations \\ (Correlated)} & \usebox{\correlated} \tnote{*} & Optimal \\ \cmidrule(l r){1-3}
Unknown Correlations & \usebox{\uc} \tnote{**} & \makecell{Suboptimal }\\ \midrule\midrule
\end{tabular}
\begin{tablenotes}
\item[*] $e=[I, \cdots, I]^T$, $\Sigma=(P_{ij}), \; i,j=1,\cdots, n$, and $\hat{x}=[\hat{x}_1^T, \cdots, \hat{x}_n^T]^T$.
\item[**] Covariance intersection rule, where $\omega_i\in [0\,,1],\; \sum_{i=1}^n \omega_i=1$, and $\omega_i=\arg\min_{\omega_i\in[0\,,1]} \text{tr}\{P_{f}\}$.
\end{tablenotes}
\end{threeparttable}
\end{table}
\end{document}
