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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}
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\date{\vspace{-10ex}}

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% \maketitle
\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}

\begin{figure}[!b]
    \centering
    \includegraphics[width=\linewidth]{holistic_error_combined_675inch.png}
    \caption{\textbf{Left:} Error statistics of heel-strike localization by using the structural vibration signal.
    \textbf{Center:} Error statistics of heel-strike localization by using the visual signal.
    \textbf{Right:} Error statistics of heel-strike localization with SF.}
    \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 decomposes step localization problem into two smaller problems: estimation of the distance and bearing (directionality) of the impact location to the sensor.
Assume the sensor $i$ that is located at $\vect{s}_i$ in the localization space, the sensor is able to locate each step in its local coordinates by using functions $g_d\left(\cdot\right)$ and $g_\theta\left(\cdot \right)$.
With the given representation, the statistical properties of the location estimates $\vect{x}_i$ of an unreliable sensor located at $\vect{s}_i$ 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; g_d, g_\theta, \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 time-domain vibro-measurements of a single-axis accelerometer $\vect{z}_i = \left(z_i[1], \ldots, z_i[n]\right)^\top \in \mathbb{R}^n$ is obtained by a sensor with index $i^{th}$ which is placed in the room between time steps $k = \{1, \ldots, n\}$ for $i = \{1, \ldots, m\}$. 
% 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.

\vspace{-2ex}\subsubsection*{Results}\vspace{-1ex}
\begin{figure}[!t]
    \centering
    \includegraphics[width=\linewidth]{holistic_error_combined_675inch.png}
    \caption{\textbf{Left:} Error statistics of heel-strike localization by using the structural vibration signal.
    \textbf{Center:} Error statistics of heel-strike localization by using the visual signal.
    \textbf{Right:} Error statistics of heel-strike localization with SF.}
    \label{fig:xy_mle2}
\end{figure}

\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. 

\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{holistic_error_combined_675inch.png}
    \caption{\textbf{Left:} Error statistics of heel-strike localization by using the structural vibration signal.
    \textbf{Center:} Error statistics of heel-strike localization by using the visual signal.
    \textbf{Right:} Error statistics of heel-strike localization with SF.}
    \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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