645 lines
19 KiB
TeX
645 lines
19 KiB
TeX
\documentclass[10pt]{beamer}
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\usetheme[everytitleformat=regular]{m}
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% Costumizing the m-theme here
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% End Costumizing
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\newcommand{\R}{\mathbb{R}}
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\newcommand{\N}{\mathbb{N}}
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\newcommand{\NDist}{\mathcal{N}}
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\newcommand{\UDist}{\mathcal{U}}
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\newcommand{\qTurn}{\theta}
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\newcommand{\qBaro}{\hat\rho_{\text{rel}}}
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\newcommand{\oWifi}{s_{\text{wifi}}}
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\newcommand{\oBeacons}{s_{\text{beacons}}}
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\newcommand{\oStep}{n_\text{step}}
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\newcommand{\oTurn}{\Delta\theta}
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\newcommand{\oBaro}{\rho_{\text{rel}}}
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\newcommand{\ispace}{\vspace{2mm}}
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\newcommand{\vecB}[2]{\begin{pmatrix} #1\\ #2 \end{pmatrix}}
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\newcommand{\matD}[4]{\begin{bmatrix} #1 & #2 \\ #3 & #4 \end{bmatrix}}
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\title{On Monte Carlo Smoothing in Multi Sensor Indoor Localisation}
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\date{\today}
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\newcommand{\insertmail}{toni.fetzer@fhws.de}
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\author{T. Fetzer$^\star$, F. Ebner$^\star$, F. Deinzer$^\star$, L. K\"oping$^\dagger$, M. Grzegorzek$^\dagger$}
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\date{\today}
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\institute{ $^\star$ University of Applied Sciences W\"urzburg - Schweinfurt \\
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$^\dagger$ University of Siegen - Pattern Recognition Group}
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%\input{misc/keywords}
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%\input{misc/functions}
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\begin{document}
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\maketitle
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\begin{frame}
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\frametitle{Table of Contents}
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\setbeamertemplate{section in toc}[sections numbered]
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\tableofcontents[hideallsubsections]
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\end{frame}
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\section{General Idea \& Motivation}
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\begin{frame}[fragile]
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\frametitle{Indoor Localisation System}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{gfx/info_graphic}
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\end{figure}%
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Multimodal Distribution}
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\begin{figure}
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\centering
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\def\svgwidth{0.9\columnwidth}
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\input{gfx/multimodalpath.eps_tex}
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\end{figure}%
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Research Objectives}
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\textbf{Goal:} Provide and discuss Monte-Carlo smoothing methods in the context of indoor localisation.
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\newline
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\newline
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\textbf{General Assumptions:}
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\begin{itemize}
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\item A time-sequential, non-linear and non-Gaussian state-space
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\newline $\rightarrow$ Monte-Carlo methods for approximation
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\item A given indoor localisation system based on a statistical sensor fusion
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\item No multi-target tracking (more than one person)
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\end{itemize}
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\end{frame}
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%Section: Transition Model
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\section{Forward Propagation}
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\begin{frame}
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\frametitle{Recursive Density Estimation}
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\begin{itemize}
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\item<1-> Current State\\
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$\vec{q} = (x,y,z, \qTurn, \qBaro)^T, \enskip{}
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\overbrace{x,y,z \in \R}^{\text{position}}, \enskip
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\overbrace{\qTurn \in \R}^{\text{heading}},\enskip{}
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\overbrace{\qBaro \in \R}^{\text{rel. pressure}}
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$ \\
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$\vec{q}_0 = $ uniformly distributed, $\qBaro = 0$
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\ispace
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\item<2-> Observation\\
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$\vec{o} = (\vec{\oWifi}, \vec{\oBeacons}, \oStep, \oTurn, \oBaro, \Omega)$
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\ispace
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\item<3-> \small$
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\underbrace{ p(\vec{q}_t\mid \vec{o}_{1:t})}_{\text{estimation}}
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\propto %
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\underbrace{ p(\vec{o}_t \mid \vec{q}_t) }_{\text{evaluation}}%
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\int
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\underbrace{ p(\vec{q}_t \mid \vec{q}_{t-1}, \vec{o}_{t-1}) }_{\text{transition}}%
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\underbrace{ p(\vec{q}_{t-1} \mid \vec{o}_{1:t-1})}_{\text{recursion}}%
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d\vec{q}_{t-1}%
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$
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Observation}
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\begin{itemize}
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\item<1-> a location's probability based on the current sensor readings
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\begin{equation*}
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\begin{split}
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p(\vec{o}_t \mid \vec{q}_t) =&\\
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&p(\vec{o}_t \mid \vec{q}_t)_{\text{wifi}} \\
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&p(\vec{o}_t \mid \vec{q}_t)_{\text{beacons}} \\
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&p(\vec{o}_t \mid \vec{q}_t)_{\text{baro}} \\
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\end{split}
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\end{equation*}
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\ispace
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\item<1-> assuming statistical independence
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\item<1-> \textit{step- and turn detection are used within the transition}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Observation - Wi-Fi/iBeacons}
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\begin{itemize}
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%\only<1>{
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\item<1->
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$p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}=$
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$p(\vec{\oWifi} \mid \vec{q}_t) = \prod_{\oWifi} \NDist(s_i \mid \overbrace{P_r(d_i, \Delta f_i)}^\text{model prediction}, \sigma_{\text{wifi}}^2)$,\\
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\ispace
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\only<1>{%
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\small{\textit{probability to measure all currently received signal-strengths $\vec{\oWifi}$ at a location $\vec{q}_t$, by comparing them with corresponding estimations from a prediction model}}%
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%\vspace{2.9cm}%
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}
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%}
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\item<2->
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3D signal strength prediction\\\ispace
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$
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P_r(d,\Delta f) =
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\underbrace{P_0}_{\text{reference}}\enskip
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\underbrace{- 10 \gamma \cdot \log_{10}(\tfrac{d}{d_0})}_{\text{attenuation per meter}}\enskip
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\underbrace{+ \Delta f \lambda}_\text{floor attenuation}
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\underbrace{+ X}_{\text{ noise }}
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$
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%\\\ispace$
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% X \sim \NDist(0,\sigma^2_{\text{wifi}}),\enskip
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% \underbrace{\Delta f \in \N}_{\text{number of floors}},\enskip
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% \underbrace{\lambda \approx -8}_{\text{attenuation per floor}}
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%$
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%\ispace
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\newline
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\raisebox{5.0cm}{
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%\only<2>{ \vspace{4.0cm} }
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\only<3>{ \includegraphics[width = 0.35\textwidth]{gfx/wifi1.png} }%
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\only<4->{ \includegraphics[width = 0.35\textwidth]{gfx/wifi2.png} }%
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\only<5>{ \includegraphics[width = 0.35\textwidth]{gfx/wifi3.png} }%
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\only<6->{ \includegraphics[width = 0.35\textwidth]{gfx/wifi4.png} }%
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}
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%\vspace{6mm}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Observation - Barometer}
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\begin{itemize}
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\item<1->
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$p(\vec{o}_t \mid \vec{q}_t)_{\text{baro}} = $
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$\NDist(o_t^{\oBaro} \mid q_t^{\qBaro}, \sigma_{\text{baro}}^2)$\\
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\ispace
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\small{\textit{probability to measure the pressure $o_t^{\oBaro}$ (relative to the start) at a location $\vec{q}_t$}, by comparing it with the corresponding prediction}
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\item<2->
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each transition performs a relative pressure prediction:\\
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\ispace
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$q_t^{\qBaro} = q_{t-1}^{\qBaro} + \Delta z \cdot b$, \enskip
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$\underbrace{\Delta z = q_{t-1}^z - q_{t}^z}_{\text{height change}}$, \enskip
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$\underbrace{b \in \R}_{\text{pressure change / meter}}$\\
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%
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\vspace{5mm}
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\begin{figure}
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\centering
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\includegraphics[width = 0.4\textwidth]{gfx/baroChange}
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\end{figure}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Transition - Floorplan}
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\only<1>{%
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1) start with the building's floorplan\\%
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\includegraphics[width = 1.0\textwidth]{gfx/step1}%
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}%
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\only<2>{%
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2) divide into cells and remove those intersecting with walls\\%
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\includegraphics[width = 1.0\textwidth]{gfx/step2}%
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}%
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\only<3>{%
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3) add edges to all (available) adjacent cells\\%
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\includegraphics[width = 1.0\textwidth]{gfx/step3}%
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}%
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\only<4>{%
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4) add stairs and remove unreachable cells\\%
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\includegraphics[width = 1.0\textwidth]{gfx/step4}%
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}%
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\end{frame}
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\newcommand{\leHeading}{\theta_{\text{walk}}}
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%\newcommand{\leDistance}{d_{\text{walk}}}
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\newcommand{\leDistance}{d}
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\begin{frame}
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\frametitle{Transition - Random Walk}
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\begin{minipage}{0.49\textwidth}
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$p(\vec{q}_t \mid \vec{q}_{t-1}, \vec{o}_{t-1})$:
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\begin{enumerate}
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\item get node $\vec{q}_{t-1}$ belongs to
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\item draw distance $\leDistance$ to walk%\\ \textit{depends on the number of detected steps}
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\item repeat until $\leDistance$ is reached
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\begin{enumerate}
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\item draw edge $e_{i,j}$ according to its probability $p(e_{i,j})$
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\item walk along the edge
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\item $\leDistance = \leDistance - \|e_{i,j}\|$
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\end{enumerate}
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\end{enumerate}
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\end{minipage}
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\begin{minipage}{0.49\textwidth}
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\begin{figure}
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\includegraphics[width = 1.0\textwidth]{gfx/walk}
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\end{figure}
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\end{minipage}
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\end{frame}
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\begin{frame}
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\frametitle{Transition - Random Walk}
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\begin{itemize}
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\item<1-> distance to walk\\
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\ispace
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$%
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\leDistance =
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\underbrace{{o}_{t-1}^{\oStep}}_\text{steps detected} \cdot
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\underbrace{s_\text{step}}_\text{step size} +
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\underbrace{\mathcal{N}(0, \sigma^2_{\leDistance})}_\text{uncertainty}
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$\newline\newline
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\item<2-> pedestrian's heading\\
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\ispace
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$p(e_{i,j})_\text{turn} = p(e_{i,j} \mid \leHeading) = \NDist(\angle e_{i,j} \mid \leHeading, \sigma^2_{\text{dev}} )$\\
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\ispace
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$%
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\underbrace{\leHeading = {q}_{t}^{\qTurn}}_\text{current heading} =
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\underbrace{{q}_{t-1}^{\qTurn}}_\text{previous heading} +
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\underbrace{{o}_{t-1}^{\oTurn}}_\text{sensor readings} +
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\underbrace{\mathcal{N}(0, \sigma^2_{\leHeading})}_\text{uncertainty}
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$\\
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Transition - Activity Recognition}
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\begin{itemize}
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\item additionally activity detection with\\
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\ispace
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$\Omega \in \{ \tt{unknown}, \tt{standing}, \tt{walking}, \tt{stairs\_up}, \tt{stairs\_down} \}$\newline
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\item edges $e_{i,j}$ matching the currently detected
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activity are favoured using $p(e_{i,j})_\text{act} = 0.8$ and $0.2$ otherwise
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\end{itemize}
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\end{frame}
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%Section: Backward Propagation
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\section{Backward Propagation}
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\begin{frame}[fragile]
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\frametitle{Basics of Particle Smoothing}
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\begin{figure}
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\centering
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\def\svgwidth{0.9\columnwidth}
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\input{gfx/basicssmoothing.eps_tex}
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\end{figure}%
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Forward-Backward Smoothing}
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\begin{algorithm}[H]
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\caption{Forward-Backward Smoother}
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\label{alg:forward-backwardSmoother}
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\begin{algorithmic}[1] % The number tells where the line numbering should start
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\Statex{\textbf{Input:} Prior $\mu(\vec{X}^i_1)$}
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\Statex{~}
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\For{$t = 1$ \textbf{to} $T$} \Comment{Filtering}
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\State{Perform particle filtering to obtain the weighted trajectories $ \{ W^i_t, \vec{X}^i_t\}^N_{i=1}$}
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\EndFor
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\For{ $i = 1$ \textbf{to} $N$} \Comment{Initialization}
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\State{Set $W^i_{T \mid T} = W^i_T$}
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\EndFor
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\For{$t = T-1$ \textbf{to} $1$} \Comment{Smoothing}
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\For{$i = 1$ \textbf{to} $N$}
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\State{Compute the weights
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\fcolorbox{mOrange}{mBGColor}{
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$
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W^i_{t \mid T} = W^i_t \left[ \sum^N_{j=1} W^j_{t+1 \mid T} \frac{p(\vec{X}^j_{t+1} \mid \vec{X}^i_t)}{\sum^N_{k=1} W^k_t ~ p(\vec{X}^j_{t+1} \mid \vec{X}^k_t)} \right]
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$}
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}
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\EndFor
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\EndFor
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\end{algorithmic}
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\end{algorithm}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Backward Simulation}
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\begin{algorithm}[H]
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\caption{Backward Simulation Smoothing}
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\label{alg:backwardSimulation}
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\begin{algorithmic}[1] % The number tells where the line numbering should start
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\Statex{\textbf{Input:} Prior $\mu(\vec{X}^i_1)$}
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\Statex{~}
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\For{$t = 1$ \textbf{to} $T$} \Comment{Filtering}
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\State{Perform particle filtering to obtain the weighted trajectories $ \{ W^i_t, \vec{X}^i_t\}^N_{i=1}$}
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\EndFor
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\For{ $k = 1$ \textbf{to} $N_{\text{sample}}$}
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\State{Choose $\tilde{\vec{q}}^k_T = \vec{X}^i_T$ with probability $W^i_T$} \Comment{Initialize realization}
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\For{$t = T-1$ \textbf{to} $1$} \Comment{Smoothing}
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\For{$j = 1$ \textbf{to} $N$}
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\State{Compute the weights \fcolorbox{mOrange}{mBGColor}{$W^j_{t \mid t+1} = W^j_t ~ p(\tilde{\vec{q}}_{t+1} \mid \vec{X}^j_{t})$}}
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\EndFor
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\State{\fcolorbox{mOrange}{mBGColor}{Choose $\tilde{\vec{q}}^k_t = \vec{X}^j_t$ with probability $W^j_{t\mid t+1}$}}
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\EndFor
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\State{$\tilde{\vec{q}}^k_{1:T} = (\tilde{\vec{q}}^k_1, \tilde{\vec{q}}^k_2, ..., \tilde{\vec{q}}^k_T)$ is one approximate realization from $p(\vec{q}_{1:T} \mid \vec{o}_{1:T})$}
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\EndFor
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\end{algorithmic}
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\end{algorithm}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Backward Transition}
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\begin{columns}[T,onlytextwidth]
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\column{0.55\textwidth}
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Very simple model for calculating the state transition $p(q_{t+1} \mid q_{t})$:
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\begin{itemize}
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\setlength{\itemindent}{-0.5cm}
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\item Linear distance: $p(\vec{q}_{t+1} \mid \vec{q}_t)_{\text{step}} = $\\
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\ispace
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$%
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\mathcal{N}(\Delta d_t \mid \mu_{\text{step}}, \sigma_{\text{step}}^2)
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$\newline
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\item Heading change: $p(\vec{q}_{t+1} \mid \vec{q}_t, \vec{o}_t)_{\text{turn}} = $\\
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\ispace
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$%
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\mathcal{N}(\Delta\alpha_t \mid \oTurn, \sigma^2_{\text{turn}})
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$\newline
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\item Height change: $p(\vec{q}_{t+1} \mid \vec{q}_t, \vec{o}_t)_{\text{baro}} = $\\
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|
\ispace
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|
$
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|
\mathcal{N}(\Delta z \mid \mu_z, \sigma^2_{z})
|
|
$
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|
|
|
\end{itemize}
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|
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|
%\column{0.05\textwidth}
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|
|
|
\column{0.60\textwidth}
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|
|
|
\centering
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|
\input{gfx/backwardTransition.eps_tex}{}
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|
|
|
\end{columns}
|
|
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\frametitle{Backward Transition}
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|
\begin{itemize}
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|
\item<1-> the probability density of the smoothing transition is then given by
|
|
\begin{equation*}
|
|
\arraycolsep=1.2pt
|
|
\begin{array}{ll}
|
|
p(\vec{q}_{t+1} \mid \vec{q}_t, \vec{o}_t) =
|
|
&p(\vec{q}_{t+1} \mid \vec{q}_t)_{\text{step}}\\
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|
&p(\vec{q}_{t+1} \mid \vec{q}_t, \vec{o}_t)_{\text{turn}}\\
|
|
&p(\vec{q}_{t+1} \mid \vec{q}_t, \vec{o}_t)_{\text{baro}}
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|
\end{array}
|
|
\enspace .
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|
\end{equation*}
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|
\ispace
|
|
\item<1-> assuming again statistical independence
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|
\item<1-> important: to do all this, we need to save all particles and their weight at each time step while filtering.
|
|
\end{itemize}
|
|
\end{frame}
|
|
|
|
%Section Evaluation
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|
\section{Evaluation}
|
|
|
|
\begin{frame}[fragile]
|
|
\frametitle{Test Environment}
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\scalebox{1.2}{
|
|
\input{gfx/paths}}
|
|
\end{figure}
|
|
%
|
|
|
|
\end{frame}
|
|
|
|
|
|
\begin{frame}[fragile]
|
|
\frametitle{Fixed-interval Smoothing - Path 2}
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\scalebox{1.2}{
|
|
\input{gfx/path2_interval_compare}}
|
|
%\caption{a) Exemplary results for path 2 where BS (blue) and filtering (green) using 2500 particles and 500 sample realisations. b) A situation where smoothing provides a worse error in regard to the ground truth, but obviously a more realistic path.}
|
|
%\label{fig:int_path2}
|
|
\end{figure}
|
|
%
|
|
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]
|
|
\frametitle{Fixed-interval Smoothing - UAH (This building)}
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=0.8\textwidth]{gfx/uah_live.png}
|
|
\end{figure}
|
|
%
|
|
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]
|
|
\frametitle{Fixed-lag Smoothing - Path 4}
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\scalebox{1.2}{
|
|
\input{gfx/path4_lag_comp}}
|
|
%\caption{Estimation results on path 4 for the filter and both smoothers using fixed-lag smoothing with $\tau = 5$. For a better visualisation, the segments are divided using an outline of alternating grey levels. The corresponding segment-error can be seen in fig. \ref{fig:lag_error_path4}.}
|
|
%\label{fig:lag_comp_path4}
|
|
\end{figure}
|
|
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]
|
|
\frametitle{Fixed-lag Smoothing - Path 4}
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\scalebox{1.2}{
|
|
\input{gfx/error_timed}}
|
|
%\caption{Estimation results on path 4 for the filter and both smoothers using fixed-lag smoothing with $\tau = 5$. For a better visualisation, the segments are divided using an outline of alternating grey levels. The corresponding segment-error can be seen in fig. \ref{fig:lag_error_path4}.}
|
|
%\label{fig:lag_comp_path4}
|
|
\end{figure}
|
|
|
|
\end{frame}
|
|
|
|
\begin{frame}[fragile]
|
|
\frametitle{Overall Results}
|
|
|
|
Overall estimation results for all paths on 10 MC runs:
|
|
|
|
\centering
|
|
\begin{tabular}{ |l |c |c |}
|
|
\hline
|
|
& Galaxy S5 & Nexus 6 \\ \hline
|
|
Filtering & \SI{7.92}{m} & \SI{5.50}{m} \\\hline
|
|
Forwards-Backward & \SI{6.48}{m} & \SI{4.47}{m} \\\hline
|
|
Backward Simulation & \SI{6.68}{m} & \SI{4.80}{m} \\
|
|
\hline
|
|
\end{tabular}
|
|
|
|
\begin{itemize}
|
|
\item even with 50 particles backward simulation is still able to provide good results.
|
|
\item bigger lags improve the results with increasing temporal error
|
|
\end{itemize}
|
|
|
|
\end{frame}
|
|
|
|
|
|
%Section Conclusion
|
|
\section{Conclusion}
|
|
|
|
\begin{frame}[fragile]
|
|
\frametitle{Conclusion}
|
|
\textcolor{mLightBrown}{What you have seen:}
|
|
\begin{itemize}
|
|
\item Successful use of \textbf{different smoothing algorithm} in context of indoor localisation
|
|
\item A \textbf{very simple} backward transition model
|
|
\item \textbf{Practical} and \textbf{theoretical comparison} between the single approaches
|
|
\end{itemize}
|
|
\begin{columns}[T,onlytextwidth]
|
|
\column{0.5\textwidth}
|
|
|
|
\textcolor{pro}{What's good:}
|
|
\begin{itemize}
|
|
\item Realistic paths and good error compensation
|
|
\item \textbf{Low computational cost} and \textbf{low complexity} by using backward simulation
|
|
\end{itemize}
|
|
\column{0.05\textwidth}
|
|
\column{0.5\textwidth}
|
|
|
|
\textcolor{con}{What's future work?}
|
|
\begin{itemize}
|
|
\item Dynamic-lag smoothing
|
|
\item Prediction of future observations to reduce the estimation lag
|
|
\end{itemize}
|
|
\end{columns}
|
|
|
|
\end{frame}
|
|
|
|
|
|
\begin{frame}[fragile]{}
|
|
\setbeamercolor{frametitle}{use=mDarkTeal,fg=mDarkTeal, bg=mWhite}
|
|
\vfill
|
|
\begin{center}
|
|
\vspace{1em}
|
|
\usebeamerfont{section title}
|
|
Thank you. Any questions?
|
|
\end{center}
|
|
\vfill
|
|
|
|
Toni Fetzer \\
|
|
\insertmail \\
|
|
University of Applied Sciences W\"urzburg - Schweinfurt
|
|
\end{frame}
|
|
|
|
|
|
%\begin{frame}[allowframebreaks]
|
|
|
|
% \frametitle{References}
|
|
|
|
% \bibliography{demo}
|
|
% \bibliographystyle{abbrv}
|
|
|
|
%\end{frame}
|
|
|
|
\end{document}
|