\section{Introduction}

Since the advent of smartphones, location aware apps and services are ubiquitous and have become a natural part of our lives. Whether driving a car, jogging or shopping in the streets, GNSS-based applications are making orientation easier, point the way and even track our fitness achievements. But as soon as we drive into an underground car park or visit a shopping mall, they perform poorly. That is because satellite signals are to weak to pass through obstacles like buildings' walls. Moreover, their accuracy is not sufficient for individual parking spaces or rooms. Therefore, many different solutions for localizing a moving object within buildings have been developed in recent years \cite{}. Especially the hard problem of pedestrian localization and navigation has lately attracted a lot of interest. 

Most modern indoor localisation systems primarily use smartphones for determining the position of a pedestrian. Especially the phone's inertial measurement unit (IMU) as well as external information like Wi-Fi or Bluetooth are used for collecting the necessary data. Additionally, environmental knowledge is often incorporated by using floor maps. This combination of highly different sensor types is also known as sensor fusion. Here probabilistic methods like particle filters or Kalman filters or often used to approximate a probability distribution describing the uncertainties of the system. This procedure can be separated into two probabilistic models: The transition model represents the dynamics of the system and predicts the next accessible locations, while the evaluation model estimates a probability that the position also corresponds to the current sensor measurement.
 %Therefore, the most accurate position is represented by a peak of the probability distribution. 
In our previous work we were able to present such a localisation system based on all the above mentioned sensors including the phone's barometer \cite{Ebner-15}. 

In pedestrian navigation, the human movement underlies the characteristics of walking speed and walking direction. Additionally, environmental restrictions need to be considered as well, for example, walking through walls is in most cases impossible. Therefore, incorporating environmental knowledge is a necessary and gainful step. Like other systems, we are using a graph-based approach for this. The main advantage of such an approach is that the graph only samples valid locations. The unique feature of our approach is the way in how we model the human movement. This is done by using random walks on graphs, which are based upon the heading of the pedestrian. However, this suffers from several drawbacks, we want to address within this work. 

The transition model presented in \cite{Ebner-15} uses discrete floors. Changing the floor on a discrete basis is like jumping down the staircase. This does not resemble real world floor changes and it could be shown that a correct estimation strongly depends on the quality of $z$-transitions. To address this problem we extended the graph by identically shaped stairs, allowing a step-wise transition in the $z$-direction.


However, we also discovered that correct estimation
strongly depends on the quality of z-transitions. If one is
missed or incorrectly detected, the estimation only slowly re-
covers. The presented discrete transition does not resemble real
world floor changes and thus not always works as expected.

Our current transition model uses discrete floors. Chang-
ing the floor on a discrete basis is like jumping down the
staircase. Such a transition would e.g. require the barometer
to detect an immediate pressure change. In reality, the pressure
slowly increases while walking down the stairs. This traps the
density on the previous floor until the pedestrian has reached
the end of the staircase and the sensor values actually match
a floor-change. However, until then, the density might have
already passed the stairwell and thus has no chance of changing
the floor

senkrechte stockwerke, wehcseln schwer blabal.. Therefore, we extend the graph by additional non-discrete nodes which resemble the shape of the stairs.

blumenverteilung, kurven laufen fällt schwer... bessers ziehen.

considering a navigation scenario... we present a novel method for pedestrian navigation by using pathfinding methods. to achieve a preferably realistic path, areas near a wall are less likely to be choosen for the path then a door or a small hallway. ... probability map/graph ... dijkstra doof... deswegen abstände zur wand mit einbeziehen.



\begin{itemize}
	\item Hinführen zum Thema 1/4 + Abstract (haben so wenig platz nur 8 seiten...)
	\item "In unseren vergangenen Arbeiten konnten wir zeigen ... aber ..."
	\item Probleme ganz konkret aufzählen und gleich die lösungen zu den problemen
	  \subitem diskrete stockwerke, senkrecht 
	  \subitem kurven laufen bereitet große probleme
	  \subitem lokalisationsergebnisse sind zwar gut, aber vor allem an treppen instabil und räume werden schwer erkannt.
	\item Was machen wir jetzt besser / anders. Hinführen zum Thema
	    \subitem Kontinuierliche Stockwerke
	    \subitem Bessers ziehen in der Transition 
	    \subitem zusätzliche Gewichtung der Knoten anstatt nur annahme des geradeaus laufens mit geringer chance auf turns. 
	    \subitem Annahme der Navigation
	\item Aufbau der Arbeit (falls platz, haben nur 8 Seiten)
\end{itemize}

Since the estabishlement of mobile 




\cite{Nurminen14-MMF}

