introduction done

competition/tex/chapters/introduction.tex

@@ -2,34 +2,59 @@
 
 The navigation system is based on our previous works, primarily on the approach presented in \cite{ebner-15}. 
 For this, we have been awarded the best overall paper award at IPIN 2015 in Banff, Canada. 
-Since then, we extended our approach by prior navigation knowledge using realistic human walking paths \cite{} and smoothing methods \cite{}. 
+Since then, we extended our approach by prior navigation knowledge using realistic human walking paths \cite{ebner-16} and smoothing methods \cite{fetzer-16}. 
 Additionally, a self-developed map editor allows for creating advanced 3D maps and realistically shaped stairs.
 Compared to many other systems, we avoid any time-consuming fingerprinting and calibration processes and are able to start with a uniform distribution over the whole building.
-All calculations are computed in real time on a commercial smartphone, in most of our examples this is the Motorola Nexus 6.
+All calculations are computed in real time on a commercial smartphone, in most of our examples this is the Motorola Nexus 6 or the Samsung Galaxy S5. 
 The system is implemented in C++ using the Qt framework and OpenCL. 
 
 An overview of all involved components and the sensor fusion procedure can be seen in fig. \ref{}. 
 Here, the smartphone provides all necessary measurements and no additional device is needed.
+The readings of all those sensors are fused using recursive density estimation, directly on the phone:
+%		
+\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}
+%	
+where $\mObsVec_{1:t} = \mObsVec_{1}, \mObsVec_{1}, ..., \mObsVec_{t}$ is a series of observations up to time $t$. 
+	The hidden state $\mStateVec$ is given by
+	\begin{equation}
+		\mStateVec = (x, y, z, \mStateHeading, \mStatePressure),\enskip
+		x, y, z, \mStateHeading, \mStatePressure \in \R \enspace,
+	\end{equation}
+	%
+	where $x, y, z$ represent the position in 3D space, $\mStateHeading$ the user's heading and $\mStatePressure$ the relative atmospheric pressure prediction in hectopascal (hPa). 
+	The recursive part of the density estimation contains all information up to time $t-1$.
+	Furthermore, the state transition $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ models the pedestrian's movement and is carried out using random walks on a graph, which is built offline, and uses the building's floorplan \cite{ebner-16}. 
+
+	Containing all relevant sensor measurements to evaluate the current state, the observation vector is defined as follows:
+	%
+	\begin{equation}
+		\mObsVec = (\mRssiVec_\text{wifi}, \mRssiVec_\text{ib}, \mObsHeading, \mObsSteps, \mObsPressure) \enspace,
+	\end{equation}
+	%
+	where $\mRssiVec_\text{wifi}$ and $\mRssiVec_\text{ib}$ contain the measurements of all nearby \docAP{}s (\docAPshort{})
+	and \docIBeacon{}s, respectively. Both serve as absolute positioning component. $\mObsHeading$ and $\mObsSteps$ describe the relative angular change and the number of steps detected for the pedestrian.	
+	If the smartphone provides a barometer, $\mObsPressure$ is used as an additional, relative verification for the current $z$-component of the pedestrian's location.
+%
+The recursive density estimation of eq. \eqref{eq:recursiveDensity} is implemented using a particle-filter with the state transition as proposal density.
+This ensures valid position estimations even if a sensor is defect or is not provided by the smartphone itself.
 
 
-This ensures sensor is defect or is not provided by the smartphone itself
 
-\begin{itemize}
-	\item Hinfuehren zum System
-	\item aus welchen arbeiten fuegt sich das system zusammen?
-	\item grober ueberblick ueber die einzelnen komponenten und sensoren
-	\item modulare uebersicht ueber das gesamte system.  (denis bild + smoothing und prior)
-	\item particle filter mit formeluebersicht und was fusioniert wird
+\section{Prior Arrangements}
+System setup is very easily and no fingerprinting is required.
 	\begin{figure}[h!]
 		\centering%
 		\includegraphics[trim=99 0 0 0, clip, width=8.2cm]{editor1.png}%
 	\end{figure}
-\end{itemize}
-
-\cite{ebner-15}
-
-\section{Prior Arrangements}
-System setup is very easily and no fingerprinting is required.
 
 \begin{itemize}
 \item Map building: Grobe Beschreibung, Funktionen und Moeglichkeiten des Map Builders. bildchen