shorten the method

torgersen-award-manuscript/Ambarkutuk-Extended-Abstract.log

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torgersen-award-manuscript/Ambarkutuk-Extended-Abstract.tex

@@ -13,7 +13,7 @@
 \pagenumbering{gobble}
 \author{\vspace{-10ex}}
 \date{\vspace{-10ex}}
-\renewcommand{\baselinestretch}{0.93}
+\renewcommand{\baselinestretch}{.98}
 
 % \linenumbers
 \input{etc/definitions}
@@ -26,17 +26,17 @@
 
 % \begin{center}\large{\textbf{Information Theoretic Approach to Solve Gait Analysis Problem with Multi-Sensor Perception}}\end{center}
 \vspace{-2ex}\subsubsection*{Introduction}\vspace{-1ex}
-Existing methods for solving the indoor occupant location problem with passive perception schemes (determining, for example, the location of footsteps of a person without an active emitter walking across a room) suffer from high uncertainty.
+Existing methods for solving the indoor occupant location problem with passive perception schemes (determining, for example, the location of thefootsteps of a person without an active emitter walking across a room) suffer from high uncertainty.
 In my work, I present a novel probabilistic localization framework that tackles both model imperfections and measurement uncertainty to improve the precision of computed localization estimates.
 
-The key idea is the relaxation of the normality assumption for each individual sensor's estimate, and the employment of maximum likelihood cost function approach to combine these individual sensor distributions.
+The key ideas are (1) the relaxation of the normality assumption for each individual sensor's estimate, and (2) the employment of maximum likelihood cost function approach to combine these individual sensor distributions.
 While the proposed technique requires the solution of a non-linear equation, it provides significantly precise location estimates.
 Specifically, this work studies the structural vibration signal captured by floor-mounted accelerometers\footnote{In this document, the words ``sensor'' and ``accelerometer'' are used interchangably.} due to an occupant's footfall patterns to determine the whereabouts of an occupant in a room.
 
-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; however, the measurements in-hand are uncertain and governing phenomenon, i.e., wave propagation, is signicantly complex.
-The proposed solution is unconventional (with relaxed assumptions), unintrusive (from privacy perspective), and unconstrained (from mode of deployment aspect) approach.
+% 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; however, the measurements in-hand are uncertain and governing phenomenon, i.e., wave propagation, is signicantly complex.
+% The proposed solution is unconventional (with relaxed assumptions), unintrusive (from privacy perspective), and unconstrained (from mode of deployment aspect) approach.
 
-\noindent\textbf{\underline{Contributions}}: The original contributions of this study is as given below:
+\noindent\textbf{\underline{Contributions}}: The original contributions of this study are given below:
 \vspace{-1ex}
 \begin{itemize}
     % \item A measurement model that models different sensing uncertainties, \vspace{-2ex}
@@ -46,50 +46,60 @@ The proposed solution is unconventional (with relaxed assumptions), unintrusive
 \end{itemize}
 
 \vspace{-4ex}\subsubsection*{Method}\vspace{-1ex}
-An overview of the proposed technique is shown in \Cref{fig:overview}.
-Briefly, the proposed technique uses $m$ number of acccelerometers' vibro-measurements separately then combines the results with a Sensor Fusion algorithm to obtain the consensus among the sensors' belief about the occupant location.
 \input{fig_system}
+An overview of the proposed framework is shown in \Cref{fig:overview}.
+When an occupant takes a step, the footfall pattern of the occupant applies some forcing, i.e., the ground reaction force, on the floor which generates a structural vibrations wave in it.
+This wave is then sensed by $m$ number of accelerometers yielding to a vibro-measurement vector $\vect{z}_i = (z_i[1], \ldots z_i[n])^\top \in \mathbb{R}^n$ $\forall i \in \{1,\ldots, m\}$ for each sensor indexed with $i$ obtained at discrete time steps at $\forall k \in \{1,\ldots, n\}$.
+% The measurement of $i^{th}$ sensor for discrete time steps $k=\{1,\ldots, n\}$ constitute the elements of the vibro-measurement vector $\vect{z}_i$, i.e., $\vect{z}_i = \left(z_i[1], \ldots, z_i[n]\right)^\top \in \mathbb{R}^n$.
+It is well-known in the literature that the vibro-measurements are often disturbed with random measurement errors and drifted due to sensor bias.
+In order to tackle such disturbances, we employ a probablistic approach to see how likely it is to obtain measurement vector $\vect{z}_i$ when its \gls{pdf} is given as $\f{\vect{Z}_i}$. 
 
-The main contribution of this study is creating a step localization framework $\vect{h}_i(\cdot)$ that provide a \gls{pdf} representing where the heel-strike location events $\vect{x}_i$ occured with the measurement of an imperfect sensor, i.e., the measurements $\vect{z}_i$ might be perturbed with random measurement errors and likely drift due to sensor bias.
-% 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 $\vect{x}_i$ with its corresponding localization error $\vect{\chi}$ are 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})
+    \boxed{
+        \vect{x}_i = \vect{x}_{true} + \vect{\chi}_i = \vect{h}_i(\vect{z}_i; \vect{\beta})
+    }
 \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 $i = \{1, \ldots, m\}$.
-It should be noted that the calibration vector $\vect{\beta}$ represents some parameters describing the characteristics of the wave propagation phenomenon occuring in the floor and is assumed to be known during the employment of the technique.
-% 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}
-The proposed localization technique yields a \gls{pdf} of location estimate $f_{\vect{X}_i}\left(\vect{x}_i \right)$ by using the localization framework $\vect{h}_i\left(\cdot\right)$ and the \gls{pdf} of the vibro-measurements $\f{\vect{Z}_i}$, which are straightforward to obtain and are unique to each individual sensor.
-The \gls{pdf} $f_{\vect{X}_i}\left(\vect{x} \right)$ inherently assigns a probability to any arbitrary location vector $\vect{x}$ in the localization space, i.e., sensors' belief about the occupant location.
-% The proposed method is able to derive the theoretical \gls{pdf} of location estimates 
-% $(\vect{x}_1, \ldots, \vect{x}_m)$ 
-% Specifically, each sensor's likelihood function represents where the sensor ``thinks'' the occupant's foot landed.
-By combining the \gls{pdf} of each sensor, the joint-\gls{pdf} is obtained where the peak (mode) of the joint-\gls{pdf} is finally determined as the location estimation.
-In short, the generation of this joint-\gls{pdf} is called ``Sensor Fusion'' in the literature where the importance of each sensor's \gls{pdf} is scaled according the information that the \gls{pdf} carries.
-
-% 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.
+Equation above forms the backbone of the localization framework $\vect{h}_i(\cdot)$ where the calibration vector $\vect{\beta}$ represents some parameters describing the characteristics of the wave propagation phenomenon occuring in the floor and is assumed to be known during the employment of the technique.
+With the given representation, the location estimate $\vect{x}_i$ with its corresponding localization error $\vect{\chi}_i$ are given by when the occupant's true heel-strike location is $\vect{x}_{true}$.
+In short, the localization framework embraces the measurement error in vector $\vect{z}_i$, imperfections in $\vect{beta}$ and $\vect{h}_i(\cdot)$ by considering \glspl{pdf} $\f{\vect{Z}_i}$.
+Therefore, it yields another set of \glspl{pdf}  denoting each sensor's belief about occupant location $\f{\vect{X}_i}$.
+In essence, \glspl{pdf} $f_{\vect{X}_i}\left(\vect{x} \right)$ inherently assigns a probability to any arbitrary location vector $\vect{x}$ in the localization space forming sensors' belief about the occupant location.
+Sequentially, these \glspl{pdf} are combined within a Sensor Fusion algorithm to obtain the consensus among the sensors' belief $\f{\vect{X_1}, \ldots, \vect{X}_m}$.
+
+
+% 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}
+% The proposed localization technique yields a \gls{pdf} of location estimate $f_{\vect{X}_i}\left(\vect{x}_i \right)$ by using the localization framework $\vect{h}_i\left(\cdot\right)$ and the \gls{pdf} of the vibro-measurements $\f{\vect{Z}_i}$, which are straightforward to obtain and are unique to each individual sensor.
+
+% % The proposed method is able to derive the theoretical \gls{pdf} of location estimates 
+% % $(\vect{x}_1, \ldots, \vect{x}_m)$ 
+% % Specifically, each sensor's likelihood function represents where the sensor ``thinks'' the occupant's foot landed.
+% By combining the \gls{pdf} of each sensor, the joint-\gls{pdf} is obtained where the peak (mode) of the joint-\gls{pdf} is finally determined as the location estimation.
+% In short, the generation of this joint-\gls{pdf} is called ``Sensor Fusion'' in the literature where the importance of each sensor's \gls{pdf} is scaled according the information that the \gls{pdf} carries.
+
+% % 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]{0.49\textwidth}
+    \begin{subfigure}[t]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{k_50.png}
         \caption{\gls{pdf} of each sensor's belief about occupant location}
         \label{fig:pdf}
     \end{subfigure}
     \hfill
-    \begin{subfigure}[b]{0.49\textwidth}
+    \begin{subfigure}[t]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{k_50.png}
         \caption{Joint-\gls{pdf} showing a consensus among the sensors}
         \label{fig:joint_pdf}
     \end{subfigure}
-
+    \hfill
     % \caption{\textbf{(a)}: This figure overlays the \gls{pdf} of each sensor about the occupant location together. The red crosses denote the sensor location, and the black dot shows the actual impact location. As can be seen in the figure, as the distance between the impact and the sensor increases, the spread of the \gls{pdf} increases, meaning the sensor is increasingly becoming uncertain about the occupant location.
     % \textbf{(b)}}
     \caption{Results of the localization framework $\vect{h}_i(\cdot)$ and Sensor Fusion}