\subsection{Transition}
	
	To enhance the quality of the proposal distribution, the transition step is 
	based on random walks along a \SI{20}{\centimeter}-gridded graph $G = (V,E)$
	with vertices $v_i \in V$ and undirected edges $e_{i,j} \in E$
	derived from the buildings floorplan (figure \ref{fig:graphOverview}).
	This ensures that only valid movements can be sampled from the previous state.%
	
	\begin{figure}
		%\noindent\hspace{1mm}
		\input{gfx/graphOverview}
		\caption{The floorplan-based graph that is used for the transition step.}
		\label{fig:graphOverview}
	\end{figure}
	
	The graph is built once and offline using the floorplan created by our editor.
	Besides realistic stairwells, additional semantic information (e.g. doors)
	can be included. Hereafter, the built graph is transmitted to the smartphone
	and is used during the online phase.
	
	If the pedestrian's destination is know beforehand, this information can
	be included as prior knowledge for the random walk: A shortest-path
	calculation imposes constraints by favouring
	moves (edges) that approach the desired destination (pedestrian sticking to the shortest path)
	over movements that depart from the destination.
	
	To ensure that the calculated shortest path is realistic (resembles human walking paths)
	each node within the graph contains a weight, denoting the likelihood for being visited
	by the pedestrian. Using this approach, nodes near walls receive a lower likelihood.
	During the path-calculation this importance is hereafter used to artificially increase the
	weight $\delta(\mEdgeAB)$ between two nodes. This ensures that the resulting path is 
	farther away from obstacles and looks much more realistic, as can be seen in figure \ref{fig:graphPaths}.
	
	\begin{figure}
		%\noindent\hspace{1mm}
		\input{gfx/graphPaths}
		\caption{%
			Shortest path calculation based on the underlying graph.
			Just using the distance between two nodes as weight $\delta(\mEdgeAB)$ results in very unrealistic walking paths (blue).
			Artificially increasing this weight for edges near walls, creates much better path estimations (green).%
		}
		\label{fig:graphPaths}
	\end{figure}