\newcommand{\mPosAP}{\varrho}							% char for access point position vector
\newcommand{\mPos}{\rho}								% char for positions


\newcommand{\mPosVec}{\vec{\mPos}}						% position vector
\newcommand{\mRssi}{s}							% client signal strength measurements
\newcommand{\mRssiVec}{\vec{s}}							% client signal strength measurements
\newcommand{\mRssiVecWiFi}{\vec{s}}							% client signal strength measurements

\newcommand{\mPLE}{\ensuremath{\gamma}}					% path-loss exponent
\newcommand{\mTXP}{\ensuremath{P_0}}					% tx-power
\newcommand{\mWAF}{\ensuremath{\beta}}					% wall attenuation factor
\newcommand{\mGaussNoise}{\ensuremath{\mathcal{X}}}

\newcommand{\mMdlDist}{\ensuremath{d}}					% distance used within propagation models

\newcommand{\mState}{q}									% state variable
\newcommand{\mStateVec}{\vec{q}}						% state vector variable
\newcommand{\mObs}{o}									% observation variable
\newcommand{\mObsVec}{\vec{o}}							% observation vector variable
\newcommand{\mObsWifi}{\vec{o}_{\text{wifi}}}			% wifi observation

\newcommand{\mPressure}{\rho}
\newcommand{\mObsPressure}{\mPressure_\text{rel}}				% symbol for observation pressure
\newcommand{\mStatePressure}{\hat{\mPressure}_\text{rel}}		% symbol for state pressure

\newcommand{\mHeading}{\theta}
\newcommand{\mObsHeading}{\Delta\mHeading}						% symbol used for the observation heading
\newcommand{\mStateHeading}{\mHeading}							% symbol used for the state heading

\newcommand{\mSteps}{n_\text{steps}}
\newcommand{\mObsSteps}{\mSteps}

\newcommand{\mActivity}{\Omega}
\newcommand{\mObsActivity}{\mActivity}

\newcommand{\R}{\mathbb{R}}
%\newcommand{\N}{\mathbb{N}}


\section{Indoor Lokalisierung}

	\subsection{Überblick}
	\begin{frame}[fragile]
	  \frametitle{Indoor Lokalisierung}

	  \begin{figure}
		\centering
		\includegraphics[width=\linewidth]{gfx/info_graphic}
	  \end{figure}%   
	  
	\end{frame}

	\begin{frame}[fragile]
		\frametitle{Indoor Lokalisierung}
	
		\small
		
		\begin{minipage}{0.49\textwidth}
		Unbekannter Zustand
			\begin{equation*}
				\mStateVec = (\underbrace{x, y, z}_{\text{Position}}, \underbrace{\mStateHeading}_{\text{Richtung}}),\enskip
				x, y, z, \mStateHeading \in \R
			\end{equation*}
		\vspace{1mm}
		\end{minipage}
		%
		\begin{minipage}{0.49\textwidth}
			Smartphone Sensordaten
			\begin{equation*}
				\mObsVec = (\mRssiVec_\text{wifi}, \mRssiVec_\text{beacon}, \mObsSteps, \mObsHeading, \mStateHeading, \mObsActivity, ) 
			\end{equation*}
		\vspace{1mm}
		\end{minipage}
		
		Sensorfusion über rekursive Dichteschätzung
		\begin{equation*}
			\arraycolsep=1.2pt
			%\begin{array}{ll}
				p(\mStateVec_{t} \mid \mObsVec_{1:t}) \propto
					\underbrace{p(\mObsVec_{t} \mid \mStateVec_{t})}_{\text{evaluation}}
					\int \underbrace{p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})}_{\text{transition}}
					\underbrace{p(\mStateVec_{t-1} \mid \mObsVec_{1:t-1})d\vec{q}_{t-1}}_{\text{recursion}}
			%\end{array}
			%\label{equ:bayesInt}
		\end{equation*}
		%\vspace{1mm}
		%
		\begin{equation*}
			\begin{aligned}
				p(\vec{o}_t \mid \vec{q}_t) &= 
				\,p(\mRssiVec_\text{wifi} \mid \vec{q}_t)%_\text{wifi}
				\,p(\mRssiVec_\text{beacon} \mid \vec{q}_t)%_\text{beacon}
				\,p(\mStateHeading \mid \vec{q}_t)%_\text{kompass}
				\,p(\mObsActivity \mid \vec{q}_t)%_\text{activity}
				\footnote{\tiny Annahme: Sensoren sind statistisch unabhängig}
				\\
				p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1}) &= \text{Bewegungsmodell mit Zusatzwissen}
			\end{aligned}
		\end{equation*}
	
	\end{frame}

	\subsection{Bewegungsmodell}
	\begin{frame}[fragile]
		\frametitle{Bewegungsmodell}%   $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$  }
		%\emph{Wo könnte ich in 10 Sekunden sein?}
		
		%
		\centering\includegraphics<1>[height=6cm]{gfx/plan1.pdf}%
		\centering\includegraphics<2>[height=6cm]{gfx/plan2.pdf}%
		\centering\includegraphics<3>[height=6cm]{gfx/importance.pdf}%
		\centering\includegraphics<4>[height=6cm]{gfx/dijkstra.pdf}%
		\centering\includegraphics<5>[height=6cm]{gfx/map1}%
		
		
	\end{frame}

	\subsection{Sensorik}
	\begin{frame}[fragile]
		\frametitle{Sensorik - Wi-Fi}
		
		\begin{equation*}
			p(\vec{o}_t \mid \vec{q}_t)_\text{wifi} = 
			p(\mRssiVecWiFi \mid \mPosVec) =
			\prod_{\mRssi_{i} \in \mRssiVec{}} p(\mRssi_{i} \mid \mPosVec)
			\footnote{\tiny Annahme: Messungen von Access-Points sind statistisch unabhängig}
			,\enskip
			%\mPos = (x,y,z)^T
			\mPosVec \in \R^3
			%\label{eq:wifiObs}
		\end{equation*}
		%
		\begin{equation*}
			p(\mRssi_i \mid \mPosVec) = 
			\mathcal{N}(\mRssi_i \mid \mu_{i,\mPosVec}, \sigma_{i,\mPosVec}^2)
			%\label{eq:wifiProb}
		\end{equation*}
		%
		\begin{equation*}
			\mRssi = {\mTXP{}} - 10 \mPLE{} + \log_{10} \frac{d}{d_0} + n \beta + \mGaussNoise{}
			%\label{eq:logDistModel}
		\end{equation*}

		\includegraphics[width=4.5cm]{gfx/fingerprints.pdf}
		\includegraphics[width=4.5cm]{gfx/opt.pdf}
		\includegraphics[width=4.5cm]{gfx/wifiMultimodality.pdf}
		
	\end{frame}