\section{Component Description}
	
	Our indoor localisation solely uses the sensors provided by almost each commodity smartphone.
	The readings of all those sensors are fused using recursive density estimation, directly on the phone:
	
	\commentByFrank{state beschreiben: x, y, z, heading. oder machst du das schon weiter oben? dann kann vermutlicha uch die formel hier weg}
	
	\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}} \enspace,
		\end{array}
		\label{eq:recursiveDensity}
	\end{equation}
	
	\docWIFI{} and (if available) \docIBeacon{}s serve as absolute positioning component. If the smartphone provides
	a barometer, its measurements are used as an additional, relative verification for the current $z$-component
	of the pedestrian's location.
	
	The transition in \refeq{eq:recursiveDensity} is carried out using random walks on a graph, which is built offline, and uses
	the building's floorplan. During the localisation process, the smartphone's IMU (accelerometer, gyroscope) is used to constrain the random walk
	in both, distance and heading.
	
	The recursive density estimation is implemented using a particle-filter.
	
	\input{chapters/barometer.tex}
	\input{chapters/wifi.tex}
	\input{chapters/stepturn.tex}
	\input{chapters/graph.tex}
	
