\section{Floorplan/Path/whatever (RENAME)}

	\subsection{wall avoidance}
	
		\begin{figure}
			\includegraphics[angle=-90, width=\columnwidth, trim=15 15 15 15]{floorplan_importance}
		\end{figure}
	
	
	\subsection{door detection}
	
		invert grid,
		find k-NN within inversion
		calculate centroid, covariance and eigenvalues
		when centroid OK and eigenvalues OK: is door/passage
	
		\commentByFrank {
		doors serve 3 purposes:
			- reverting the downvote due to "near walls"
			- hint for a pathless-movement-model how to leave rooms
			- also detecting narrow passages (center the path)
		}
		
		Doors are usually anchored between two (thin) walls and have a normed width. Examining only a limited region
		around the door, its surrounding walls describe a flat ellipse with the same center as the door itself. It is thus
		possible to detect doors within the floorplan using the PCA \todo{cite}. To describe the surroundings of a door,
		we create and inverted version of our grid whereby the contained nodes denote walls and other obstacles.
		To decide whether a node within the (non-inverted) grid belongs to a door, we use $k$-NN \todo{cite} to fetch its
		$k$ nearest neighbors within the inverted grid. If the distance \todo{equation} between the centroid of those neighbors
		and the node-in-question is above certain threshold, the node does not belong to a door. Assuming the distance was fine,
		we compare the two eigenvalues $\{e_1, e_2 \mid e_1 > e_2\}$ , determined by the PCA. If their ratio $\frac{e_1}{e_2}$ is above
		a certain threshold (flat ellipse)
		the node-in-question belongs to a door. Fig. \ref{fig:doorPCA} depicts all three cases where the node is part of a door (left),
		the distance between node and k-NN centroid is above the threshold (middle) and the ration between $e_1$ and $e_2$ is below the
		threshold (right).

		\begin{figure}
			\includegraphics[width=\columnwidth]{door_pca}
			\caption{Detect doors within the floorplan using k-NN and PCA. While the white nodes are walkable, the black ones represent walls. The grey node is the one in question.}
			\label{fig:doorPCA}
		\end{figure}
		
		
	\subsection{pathfinding}
		
		\begin{figure}
			\includegraphics[angle=-90, width=\columnwidth, trim=15 15 15 15]{floorplan_paths}
		\end{figure}
		
		\begin{figure}
			\includegraphics[angle=-90, width=\columnwidth, trim=15 15 15 15]{floorplan_dijkstra_heatmap}
		\end{figure}
		
