toni first_draft

tex/chapters/grid.tex

@@ -11,7 +11,7 @@
 	derived from the buildings floorplan as described in section \ref{sec:relatedWork}.
 	However, we add improved $z$-transitions by also modelling realistic
 	stairwells using nodes and edges, depicted in fig. \ref{fig:gridStairs}.
-	
+	%
 	\begin{figure}
 		\centering
 		\input{gfx/grid/grid}
@@ -22,12 +22,12 @@
 			}
 		\label{fig:gridStairs}
 	\end{figure}
-	
-	
+	%	
 	Stairs are defined using three points $\vec{\spoint}_1, \vec{\spoint}_2, \vec{\spoint}_3 \in \R^3$ whereby the segment
 	$[ \vec{\spoint}_1 \vec{\spoint}_2 ]$ describes the starting-edge, and $[ \vec{\spoint}_2 \vec{\spoint}_3 ]$ the stair's direction
 	(see fig. \ref{fig:gridStairs}). The grid-vertices corresponding to the starting-edge are determined using an intersection of
 	the segment $[ \vec{\spoint}_1 \vec{\spoint}_2 ]$ with the bounding-box for each vertex.
+	\commentByToni{Welche Bounding-box? hab ich was ueberlesen?}
 	
 	To reduce the system's memory footprint, we search for the largest connected region within the graph and
 	remove all nodes and edges that are not connected to this region.
@@ -46,11 +46,12 @@
 	%
 	During a walk, each edge has an assigned probability $p(e)$ which depends on a chosen implementation.
 	This probability describes aspects such as the likelihood for walking into the edge's direction $\angle e$
-	given the current heading heading $\gHead$. Furthermore, we will incorporate additional prior knowledge to
+	given the current heading $\gHead$. Furthermore, we will incorporate additional prior knowledge to
 	favour some vertices/edges. For each single step on the graph, we calculate $p(e)$ for all available edges,
 	and, hereafter, randomly draw the to-be-walked edge depending on those probabilities. The random walk ends,
 	as soon as the distance $d$ is reached. The latter depends on the number of detected steps
 	$\mObsSteps$ and assumes an average step-size of \SI{0.7}{\meter}.
+	\commentByToni{steps steps... Allgemein finde ich den Absatz etwas komisch. erst wird p(e) eingefuehrt, dann kommt auf einmal prior knowledge und dann wie wir p(e) berechnen? reihenfolge passt da nicht so und macht es undeutlch. Hier soll doch eigentlich nur erklaert werden wie wir ueber den graphen laufen. egal ob prior oder nicht. laufen muessen wir ja}
 
 	For comparison purpose we define a simple weighting method that assigns a probability to each edge
 	just based on the deviation from the currently estimated heading $\gHead$:
@@ -83,6 +84,7 @@
 		larger open spaces, somewhere far from the walls. In order to calculate paths that resemble such a walking
 		behaviour, an importance factor is derived for each vertex within the graph. Those will be used to 
 		adjust weight between two vertices, needed by the shortest-path algorithm. 
+		\commentByToni{The edge's weight?? vielleicht?}
 		
 		To downvote vertices near walls, we need to get the distance of each vertex from its nearest wall.
 		We therefore build an inverted version $G' = (V', E')$ of the graph $G$, just containing walls and other obstacles.
@@ -113,7 +115,7 @@
 	\label{sec:doorDetection}
 				
 		Doors are usually anchored between two 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.
+		around the door, its surrounding walls describe a flat ellipse with the same centre as the door itself.
 		It is thus possible to detect doors within the floorplan using a PCA.
 		
 		To decide whether a vertex $v_{x,y,z}$ within the (non-inverted) grid $G$ belongs to a door, we use $k$-NN to fetch its
@@ -121,18 +123,19 @@
 		If the distance $\| \vec{c} - v_{x,y,z} \|$ between the centroid and the vertex-in-question is above a certain threshold,
 		the node does not belong to a door.
 		\todo{Distanzformel centroid/vertex. ideen? $\|v1 - v2\|$ oder $d(v1, v2)$?}
+		\commentByToni{Beides ist gebraeuchlich und $d$ ist kuerzer. von daher wuerde ich $d$ empfehlen.} 
 			
 		Assuming the distance is fine, we compare the two eigenvalues $\{\lambda_1, \lambda_2 \mid \lambda_1 > \lambda_2\}$,
 		determined by the PCA. If their ratio $\frac{\lambda_1}{\lambda_2}$ is above a certain threshold (flat ellipse)
 		the node-in-question belongs to a door or some kind of narrow passage.
-		
+		%
 		\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}
-		
+		%
 		Fig. \ref{fig:doorPCA} depicts all three cases where
 		(left) the node is part of a door,
 		(middle) the distance between node and k-NN centroid is above the threshold and
@@ -246,7 +249,7 @@
 			This new reference node $\pathRef$ serves as a comparison base:
 			\commentByToni{Allgemein mal zur Schreibweise der Vertices. Irgendwie finde ich dieses $v_{x,y,z}$ nicht so gut. Ich denke jeder sieht das wir 3D haben und deswegen könntem man doch schlicht $v$, $v'$, $\hat{v}$ ... nutzen, oder was denkst du?}
 			\commentByFrank{war der vorschlag von frank d. letztes mal, weil man an vertices nicht einfach attribute (x,y,z) anhaengen kann wie wir es bei $\mObsVec$, $\mStateVec$ haben.}
-			
+			%
 			\begin{equation}
 				\begin{split}
 					p(v_{x',y',z'} \mid v_{x,y,z})
@@ -259,7 +262,7 @@
 				\end{split}
 				\label{eq:transShortestPath}
 			\end{equation}
-			
+			%
 			Eq. \eqref{eq:transShortestPath} combines the simple transition \refeq{eq:transSimple} with 
 			a second probability, downvoting all nodes that are farther away from the reference $\pathRef$
 			than the previous step. Put another way: grid-steps increasing the distance to the reference