stretched gfx (less height)
removed some words for a better text-flow
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@@ -1,10 +1,10 @@
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\section{Transition Model}
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\label{sec:trans}
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%
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\newcommand{\spoint}{l}
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\newcommand{\gHead}{\theta_\text{walk}}
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\newcommand{\gDist}{d_\text{walk}}
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%
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To sample only transitions that are actually feasible
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within the environment, we utilize a \SI{20}{\centimeter}-gridded graph
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$G = (V,E)$ with vertices $v_i \in V$ and undirected edges $e_{i,j} \in E$
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@@ -12,7 +12,7 @@
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derived from the buildings floorplan as described in section \ref{sec:relatedWork}.
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However, we add improved $z$-transitions by also modelling realistic
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stairwells using nodes and edges, depicted in fig. \ref{fig:gridStairs}.
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%
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\begin{figure}
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\centering
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\input{gfx/grid/grid}
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@@ -29,7 +29,6 @@
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direction (see fig. \ref{fig:gridStairs}). The grid-vertices corresponding to the starting-edge are determined using an
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intersection of the segment $[ \vec{\spoint}_1 \vec{\spoint}_2 ]$ with the \SI{20}{\centimeter} bounding-box around each
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node's centre $\fPos{v} = (x,y,z)^T$.
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To reduce the system's memory footprint, we search for the largest connected region within the graph and
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remove all nodes and edges that are not connected to this region.
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@@ -72,14 +71,14 @@
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p(\mEdgeAB) = p(\mEdgeAB \mid \gHead) = \mathcal{N} (\angle \mEdgeAB \mid \gHead, \sigma_\text{dev}^2).
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\label{eq:transSimple}
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\end{equation}
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%
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%
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%
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%
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%
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\section{Navigational Knowledge}
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\label{sec:nav}
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%
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Considering navigation, a pedestrian wants to reach a well-known destination which represents additional
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prior knowledge. Most probably, the user will stick to the path presented by
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a navigation system. However, some deviations like chatting to someone or taking another route
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@@ -88,7 +87,7 @@
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\subsection{Wall Avoidance}
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\label{sec:wallAvoidance}
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%
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%As discussed in section \ref{sec:relatedWork}, simply applying a shortest-path algorithm such as Dijkstra or
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%A* using the previously created graph would obviously lead to non-realistic paths sticking to walls and
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%walking many diagonals. Pedestrians however, will probably keep a small gap between themselves and
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@@ -96,7 +95,7 @@
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%To calculate paths that resemble this behaviour, an importance-factor is derived for
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%each vertex. Those will be used to modify the weight $\fDistance{v}{v'}$ between two vertices
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%$v,v'$, examined by the shortest-path algorithm.
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%
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Shortest-path algorithms such as Dijkstra use a scalar weight $\fDistance{v_1}{v_2}$ between two vertices
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to determine the path with the lowest overall weight.
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As discussed in section \ref{sec:relatedWork}, simply using the distance
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@@ -130,12 +129,12 @@
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While rendering wall-regions less likely, \refeq{eq:wallAvoidance}
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will obviously have the same effect on doors as they are just a small gap between
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consecutive walls. Therefore, a door-detection is necessary, to upvote them again.
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%
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%
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%
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\subsection{Door Detection}
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\label{sec:doorDetection}
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%
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To automatically detect doors within the floorplan, we utilize the fact that doors are usually
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anchored between two straight walls and have a normed width. Examining the region directly
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around it, the door and its surrounding walls thus describe a flat ellipse with the door as its centre.
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@@ -205,11 +204,11 @@
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passages depict a high importance.}
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\label{fig:importance}
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\end{figure}
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%
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\subsection{Path Estimation}
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\label{sec:pathEstimation}
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For routing the pedestrian towards his desired target, a modified version
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%
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To route the pedestrian towards his desired target, a modified version
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of Dijkstra's algorithm is used. Instead of calculating the shortest path from the start to the end,
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the direction is inverted and the calculated terminates as soon as every single node was evaluated.
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Hereafter, every node in the grid knows the distance and shortest path to the pedestrian's target.
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@@ -237,32 +236,31 @@
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%
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Fig. \ref{fig:multiHeatMap} depicts the difference between the shortest path calculated without (dashed) and
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with importance-factors (solid), where the latter is clearly more realistic.
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%
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%\begin{figure}
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% \includegraphics[angle=-90, width=\columnwidth, trim=20 19 17 9, clip]{floorplan_paths}
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% \caption{Comparision of shortest-path calculation without (dotted) and with (solid) importance-factors
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% use for edge-weight-adjustment.}
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% \label{fig:shortestPath}
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%\end{figure}
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%
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%
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\subsection{Guidance}
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%
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Based on the previous considerations, we propose two approaches to utilize prior
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knowledge within the transition.
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\subsubsection{Shortest Path}
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%
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\newcommand{\pathCentroid}{{\vec{\overline{c}}_{t-1}}}
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\newcommand{\pathDev}{\sigma_{t-1}}
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\newcommand{\pathRef}{v_\text{ref}}
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%
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Before every transition, the centre-position $\pathCentroid = \fPos{\mStateVec_{t-1}^*}$ of the current sample-set, where
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\begin{equation}
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\mStateVec_{t-1}^* = \underset{\mStateVec_{t-1}}{\argmax} \enspace p(\mStateVec_{t-1} | \mObsVec_{t-1})
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\end{equation}
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represents the most proper state of the posterior distribution at time $t-1$, is calculated.
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\begin{equation}
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\mStateVec_{t-1}^* = \underset{\mStateVec_{t-1}}{\argmax} \enspace p(\mStateVec_{t-1} | \mObsVec_{t-1})
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\end{equation}
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represents the most proper state of the posterior distribution at time $t-1$, is calculated.
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%
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%
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%%\commentByFrank{avg-state vom sample-set. frank d. meinte ja hier muessen wir aufpassen. bin noch unschluessig wie.}
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@@ -280,13 +278,13 @@ represents the most proper state of the posterior distribution at time $t-1$, is
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We thus calculate the standard deviation of the distance of all sample-positions
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$\fPos{\mStateVec_{t-1}}$ from aforementioned centre $\pathCentroid$.
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%\commentByFrank{so klarer? platz fuer groese Eq. fehlt und Notation zum ansprechen jedes einzelnen Particles vermeide ich lieber...}
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%
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%\begin{equation}
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% d_\text{cen} = \| pos(q_{t-1}) - \pathCentroid \|
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% \sigma_\text{cen} = stdDev(distance)
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%\end{equation}
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After advancing the starting-vertex by three times this deviation
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After advancing the starting-vertex by three times this deviation
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we get the new point $\pathRef$ that is: part of the shortest path, outside of the sample-set
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and closer (but not too close) to the desired destination.
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%
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@@ -313,10 +311,10 @@ represents the most proper state of the posterior distribution at time $t-1$, is
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\end{equation}
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%\commentByFrank{$\mUsePath$ als variable}
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%
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%
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%
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\subsubsection{Multipath}
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%
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The shortest-path algorithm mentioned in \ref{sec:pathEstimation} already calculated the distance
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$\fLength{\mVertexA}{\mVertexDest}$ % = \sum_{i=s}^{e-1} \| v_{i} - v_{i+1} \| $
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for the path from $\mVertexA$ to the pedestrian's destination $\mVertexDest$.
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