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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}
\vspace{-5ex}\subsubsection*{Introduction}\vspace{-1ex}
% \gls{ga} is the study of human locomotion which is widely acknowledged in indicating degenerative diseases affecting motor functions, namely, Alzeihmer's, Parkinson's, Multiple Sclerosis.
% Being the current de-facto standard technique, visual perception techniques~\cite{moeslund2001survey, Munoz2018, Bei2018}, including \gls{mocap} systems, have seen great attention due to its ability to provide accurate \gls{ga} measurements~\cite{Li2019}.
% However, their cost (due to highly sensitive narrow-band infrared cameras~\cite{Lee2009}), cumbersome nature (requiring patient-specific calibrations), and limited deployment ability (needing to place special reflective markers on the patients body~\cite{Ceseracciu2014}) render them impractical for unrestricted and unconstrained use of such devices in everyday life.
This document briefly presents a solution to \gls{ga} problem with an unconventional (from perception viewpoint), unintrusive (from privacy perspective), and unconstrained (from place of deployment aspect) approach~\cite{alajlouni2019new}.
In this work, the structural vibration signal captured by floor-mounted accelerometers due to an occupant's footfall patterns in a room is used to determine where the ``heel-strike'' events occur thereby estimating the spatio- and temporal-parameters of human gait.
\vspace{-2ex}\subsubsection*{List of Contributions of This Study}\vspace{-1ex}
\begin{enumerate}
\item An end-to-end framework that allows the derivation of energy-based vibro-localization uncertainty given the sensor imperfections. \vspace{-2ex}
\item A localization technique which exploits ranging and bearing information separately.\vspace{-2ex}
\item Multiple validation studies based on simulation and experimental data.
\end{enumerate}
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\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=\textwidth]{signal.eps}
\caption{$k$ vs. $z_i[k]$ and $z_{t,i}[k]$}
\label{fig:three sin x}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=\textwidth]{fd.eps}
\caption{$d_i = g_d(\vect{z}_i; \vect{\beta}_d)$}
% \label{}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=\textwidth]{ftheta.eps}
\caption{$\theta_i = g_\theta(\vect{z}_i; \vect{\beta}_\theta)$}
% \label{}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=\textwidth]{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 $\vect{s}_i$.
}
\label{fig:xy_mle}
\end{figure}
\vspace{-2ex}\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 $\vect{x}_t$:
\vect{x}_i = \vect{x}_t + \vect{\chi}_i = \vect{h}_i(\vect{z}_i; \vect{\beta}_d, \vect{\beta}_{\theta}) = \vect{s}_i + g_d(\vect{z}_i; \vect{\beta}_d) \begin{bmatrix}
\cos{g_\theta(\vect{z}_i; \vect{\beta}_{\theta})} \\ \sin{g_\theta(\vect{z}_i; \vect{\beta}_{\theta})}
\end{bmatrix}
\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*}
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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