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% \title{Information Theoretic Approach to Multi-sensor Perception Problems}
% \author[1]{Creed Jones}
% \author[1]{Paul Plassmann}
% \affil[1]{The Bradley Department of Electrical and Computer Engineering}
\author{}
\date{\vspace{-10ex}}
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\begin{document}
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% \maketitle
\begin{center}\large{\textbf{Title: Very Cool Title}}\end{center}
% \begin{center}\large{\textbf{Information Theoretic Approach to Solve Gait Analysis Problem with Multi-Sensor Perception}}\end{center}
This document briefly presents a solution to indoor occupant localization problem with an unconventional (from perception viewpoint), unintrusive (from privacy perspective), and unconstrained (from mode of deployment aspect) approach~\cite{alajlouni2019new}.
The proposed solution does not require occupants to carry special devices, markers, or beacons; hence, it is a passive approach while the sensible area (field-of-view) cannot be occluded.
In this work, the structural vibration signal captured by floor-mounted accelerometers due to an occupant's footfall patterns is used to determine the whereabouts of an occupant in a room under measurement uncertainty.
\noindent\textbf{\underline{Contributions}}: The original contributions of this study is as given below:
\vspace{-1ex}
\begin{itemize}
\item A measurement model that models different sensing uncertainties, \vspace{-2ex}
\item A framework that works out uncertainty bounds of vibro-localization techniques, \vspace{-2ex}
\item Employment of multi-sensing principles to achieve minimal localization uncertainty, \vspace{-2ex}
\item Multiple validation studies based on simulation and experimental data.
\end{itemize}
\vspace{-4ex}\subsubsection*{Method}\vspace{-1ex}
The main contribution of this study is creating a step localization framework $\vect{h}_i(\cdot)$ that can localize the heel-strike events at $\vect{x}_i$ with the measurement of an imperfect sensor, i.e., it yields noisy and likely bias-drifted measurements $\vect{z}_i$ that is located at $\vect{s}_i$.
% The proposed framework decomposes step localization problem into two smaller problems: estimation of the distance $d_i = g_d\left(\vect{z}_i; \vect{\beta}_d\right)$, and directionality $\theta_i = g_\theta\left(\vect{z}_i; \vect{\beta}_\theta\right)$ of the impact location to the sensor.
With the given representation, the location estimate $\vect{x}_i$ is given by when the occupant's true heel-strike location is $\vect{x}_t$:
\begin{equation*}
\vect{x}_i = \vect{x}_t + \vect{\chi}_i = \vect{h}_i(\vect{z}_i; \vect{\beta}_d, \vect{\beta}_{\theta})
\end{equation*}
where the vector of imperfect time-domain vibro-measurements of a single-axis accelerometer is given by $\vect{z}_i = \left(z_i[1], \ldots, z_i[n]\right)^\top \in \mathbb{R}^n$ between time steps $k = \{1, \ldots, n\}$ for all sensors $i = \{1, \ldots, m\}$.
% are modeled as the combination of the true vibro-measurement that the sensor is supposed to register, $z_t[k]$, and random effects of sensor imperfections, $\zeta[k]$.
% \begin{equation*}
% \vect{z} = \{z[k]:k = \{1,\ldots n\}\}, \qquad \text{where } z[k] = z_t[k] + \zeta[k], \text{ and }\zeta[k] \sim \N{\delta}{\sigma_{\zeta}}.
% \end{equation*}
% \vspace{-5ex}
\begin{tikzpicture}[
roundnode/.style={circle, draw=green!60, fill=green!5, very thick, minimum size=7mm},
squarednode/.style={rectangle, draw=red!60, fill=red!5, very thick, minimum size=5mm},
pdfnode/.style={circle, draw=blue!60, fill=blue!5, very thick},
]
%Nodes
\node[roundnode](impact) {Heel-strike};
\node[roundnode](sensor1)[right=of impact] {Sensor $i$};
\node[squarednode](h1)[right=of sensor1]{$\vect{h}_i(\cdot)$};
\node[pdfnode](fx1)[right=of h1]{PDF $\f{\vect{X}_i}$};
% %Lines
\draw[->] (sensor1.east) -- (h1.west);
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% \end{subfigure}
% \begin{subfigure}[b]{0.3\textwidth}
% \centering
% \includegraphics[height=1.5in]{signal.eps}
% \caption{$k$ vs. $z_i[k]$ and $z_{t,i}[k]$}
% \label{}
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% \begin{subfigure}[b]{0.3\textwidth}
% \centering
% \includegraphics[height=0.1\textheight]{layout.eps}
% \caption{$\vect{x}_i = \vect{h}_i\left(\vect{z}_i; \vect{\beta}_d, \vect{\beta}_\theta \right)$}
% % \label{}
% \end{subfigure}
% \caption{
% \textbf{(a)} Time-domain vibro-measurements of a noisy accelerometer
% \textbf{(b)} $g_d(\vect{z}_i; \vect{\beta}_d)$
% \textbf{(c)} $g_{\theta}(\vect{z}_i; \vect{\beta}_{\theta})$
% \textbf{(d)} $\vect{x}_i = \vect{h}_i\left(\vect{z}_i; \vect{\beta}_d, \vect{\beta}_\theta \right)$ in coordinate system of sensor located at $\vect{s}_i$.
% }
\end{figure}
Due to stochastic nature of the problem, we model the location estimate $\vect{x}$ as a variate of a random variable $\vect{X}$.
Therefore, we seek to derive the \gls{pdf} $\f{\vect{X}}$ defines the statistical properties of the location estimates.
Specifically, each sensor's likelihood function represents where the sensor ``thinks'' the occupant's foot landed.
By combining the likelihood function of each sensor, the joint likelihood function is obtained where the peak (mode) of the joint likelihood function is finally determined as the location estimation.
% Due to stochastic nature of the problem, we employ a probabilistic approach such that the localization framework $\vect{h}(\cdot)$ yields to a likelihood function defined over the localization space.
% Specifically, each sensor's likelihood function represents where the sensor ``thinks'' the occupant's foot landed.
% By combining the likelihood function of each sensor, the joint likelihood function is obtained where the peak (mode) of the joint likelihood function is finally determined as the location estimation.
\begin{figure}[!t]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{k_50.png}
\caption{}
% \label{fig:three sin x}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{k_50.png}
\caption{}
% \label{fig:three sin x}
\end{subfigure}
\caption{\textbf{(a)}\textbf{(b)}}
\vspace{-2ex}\subsubsection*{Results}\vspace{-1ex}
\Cref{fig:xy_mle2} demonstrates the error statistics of the estimated heel-strike locations as a function of the location the heel-strike locations.
The left and center plot shows these error by only using the structural vibration and visual signal only.
As can been seen from the left and center plots, the error characteristics of these sensing modalities are significantly different.
When the result of these sensors are fused, the right plot is obtained where the SF algorithm greatly benefits from structural vibration and visual signals.
\lipsum[1-2]
% \begin{figure}[!h]
% \centering
% \begin{subfigure}[b]{0.3\textwidth}
% \centering
% \includegraphics[width=\textwidth]{example-image-a}
% \caption{}
% % \label{fig:three sin x}
% \end{subfigure}
% \hfill
% \begin{subfigure}[b]{0.3\textwidth}
% \centering
% \includegraphics[width=\textwidth]{example-image-b}
% \caption{}
% % \label{fig:five over x}
% \end{subfigure}
% \caption{
% \textbf{(a)}
% \textbf{(b)}
% }
% \label{fig:xy_mle3}
% \end{figure}
% \Cref{fig:xy_mle3} demonstrates the error statistics of the estimated heel-strike locations as a function of the location the heel-strike locations.
% The left and center plot shows these error by only using the structural vibration and visual signal only.
% As can been seen from the left and center plots, the error characteristics of these sensing modalities are significantly different.
% When the result of these sensors are fused, the right plot is obtained where the SF algorithm greatly benefits from structural vibration and visual signals.
\vspace{-2ex}\subsubsection*{Conclusions and Future Work}\vspace{-1ex}
\lipsum[1]
% In this document, two different research studies conducted in VTSIL that involve the SF discipline were presented.
% Overall, SF techniques are robust to many erroneous factors while capturing the sensor uncertainties to adaptively tune its internal parameters.
% We believe that SF has the potential to provide crucial information to the solutions to complex problems such as structural health monitoring of complex structures, vibration control, non-field-of-view perception, etc.
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% \section*{Appendix}
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