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chapter 4 almost rewritten

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16
tex/chapters/experiments.tex
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@@ -68,19 +68,19 @@
% error development over time while walking along a path
\begin{figure}
\input{gfx/eval/error_timed_nexus}
\caption{Development of the error while walking along
%path 1 (upper) and
path 4 (lower) using the Motorola Nexus 6.
Path 4 shows increasing errors for our methods when leaving the shortest path (3) and when facing multimodalities between two
staircases just before the destination (9).}
\caption{Development of the error while walking along Path 4 using the Motorola Nexus 6.
When leaving the suggested route (3), the error of shortest path \refeq{eq:transShortestPath}
and multipath \refeq{eq:transMultiPath} increases.
The same issues arise when facing multimodalities between two staircases just before the destination (9).}
\label{fig:errorTimedNexus}
\end{figure}
% detailed analysis of path 4
\begin{figure}
\input{gfx/eval/path_nexus_detail}
\caption{Detailed path analysis depicting the individual segments of path 4. Their corresponding error contribution can
be seen in fig. \ref{fig:errorTimedNexus}. Even though the shortest path suggested by the system is ignored multiple
times ($3'$ and $3''$) our approach is still able to improve the overall localisation error.}
\caption{Detailed path analysis depicting the individual segments of path 4 using \refeq{eq:transMultiPath}.
Their corresponding error can be seen in fig. \ref{fig:errorTimedNexus}. Even though the shortest path
suggested by the system is ignored multiple times ($3'$ and $3''$) our approach is still able to improve
the overall localisation error.}
\label{fig:nexusPathDetails}
\end{figure}
%

302
tex/chapters/grid.tex
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@@ -7,11 +7,11 @@
To sample only transitions that are actually feasible
within the environment, we utilize a \SI{20}{\centimeter}-gridded graph
$G = (V,E)$, $v_{x,y,z} \in V$, $e_{v_{x,y,z}}^{v_{x',y',z'}} \in E$
$G = (V,E)$ with vertices $v \in V$ and undirected edges $e \in E$
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}
@@ -24,40 +24,45 @@
\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 \SI{20}{\centimeter} bounding-box around each vertex.
$[ \vec{\spoint}_1 \vec{\spoint}_2 ]$ 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 \SI{20}{\centimeter} bounding-box around each
node's centre $\fPos{v} = (x,y,z)^T$.
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.
Walking the grid is now possible by moving along adjacent nodes into a given walking-direction
until a desired distance $\gDist$ is reached \cite{Ebner-15}.
In order to use meaningful headings $\gHead$ and distances $\gDist$
(matching the pedestrian's real heading and walking speed) for each transition,
we use the current sensor-readings $\mObsVec_{t}$ for hinted instead of randomized adjustments:
%
New states $\mStateVec_{t}$ may now be sampled by starting at the vertex for $\fPos{\mStateVec_{t-1}}$ and
walking along adjacent nodes into a given walking-direction $\gHead$ until a distance $\gDist$ is
reached \cite{Ebner-15}.
Both, heading and distance, are supplied by the current sensor readings $\mObsVec_{t}$
and thus reflect the pedestrian's real heading and walking speed including uncertainty.
Working with relative sensor readings, the state's heading is updated during each transition:
%
\begin{align}
\mState_{t}^{\mStateHeading} = \gHead &= \mState_{t-1}^{\mStateHeading} + \mObs_t^{\mObsHeading} + \mathcal{N}(0, \sigma_{\gHead}^2) \\
\gDist &= \mObs_t^{\mObsSteps} \cdot \SI{0.7}{\meter} + \mathcal{N}(0, \sigma_{\gDist}^2)
\gHead &= \mState_{t}^{\mStateHeading} = \mState_{t-1}^{\mStateHeading} + \mObs_t^{\mObsHeading} + \mathcal{N}(0, \sigma_{\gHead}^2) \\
\gDist &= \mObs_t^{\mObsSteps} \cdot \SI{0.7}{\meter} + \mathcal{N}(0, \sigma_{\gDist}^2)
.
\end{align}
%
During a walk, each edge has an assigned probability $p(e)$ which depends on
the its direction $\angle e$ and the current heading $\gHead$.
We will use $p(e)$ to incorporate prior path knowledge to
favour some vertices/edges. For each single movement on the graph,
we calculate $p(e)$ for all adjacent 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. $d$ depends on the number of detected steps
$\mObsVec_t^{\mObsSteps}$ and assumes an average step-size of \SI{0.7}{\meter}.
During the random walk, each edge has its own probability $p(e)$
which e.g. depends on the edge's direction $\angle e$ and the
pedestrian's current heading $\gHead$.
Furthermore, $p(e)$ is used to incorporate prior path knowledge to
favour edges leading towards the pedestrian's desired target.
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 $\gHead$ omitting additional prior knowledge:
\commentByFrank{das erste $=$ ist komisch. ideen?}
\commentByToni{Find ich jetzt nicht tragisch. Eher notwendig fuers Verstaendnis.}
For each single movement on the graph, we calculate $p(e)$ for all edges
connected to a vertex $v$, and, hereafter, randomly draw the to-be-walked edge
depending on those probabilities. This step is repeated until the sum
of the length of all used edges exceeds $d$. The latter depends on the number of
detected steps $\mObs_t^{\mObsSteps}$ and assumes an average
step-size of \SI{0.7}{\meter}.
To quantify the improvement prior knowledge is able to provide,
we define a simple reference for $p(e)$ that assigns a probability to each edge
just based on its deviation from the currently estimated heading $\gHead$:
%
\begin{equation}
p(e) = p(e \mid \gHead) = \mathcal{N} (\angle e \mid \gHead, \sigma_\text{dev}^2).
\label{eq:transSimple}
@@ -79,34 +84,24 @@
\label{sec:wallAvoidance}
As discussed in section \ref{sec:relatedWork}, simply applying a shortest-path algorithm such as Dijkstra or
A* using the previously created graph would obviously lead to non-realistic paths sticking to the walls and
walking many diagonals. Pedestrian's however, walk either somewhere near (but not close to) a wall or, for
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 the distance-weight between two vertices, needed by the shortest-path algorithm.
A* using the previously created graph would obviously lead to non-realistic paths sticking to walls and
walking many diagonals. Pedestrian's however, will probably keep a small gap between themselves and
nearby walls. To calculate paths that resemble this behaviour, an importance-factor is derived for
each vertex. Those will be used to modify the euclidean distance $\fDistance{u}{v}$ between two vertices
$u,v$, examined by the shortest-path algorithm.
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.
A nearest-neighbour search \cite{Cover1967}
%$\mNN(v_{x,y,z}, G')$
will then provide the nearest wall-vertex
$v'_{x,y,z} \in V'$ from the inverted graph. To get a smooth gradient, the wall avoidance
is calculated using a normal distribution with the distance from the nearest wall
and a deviation of \SI{0.5}{\meter}:
%
%\begin{equation}
% d_{v, v'} = \| v_{x,y,z}, v'_{x,y,z}, \enskip 0.0 < d_{v, v'} < 2.2 \\
%\end{equation}
%\begin{equation}
%\begin{array}{ll}
% \text{wa}_{x,y,z} = & - 0.30 \cdot \mathcal{N}(d_{v, v'} \mid 0.0, 0.5^2) \\
% & + 0.15 \cdot \mathcal{N}(d_{v, v'} \mid 0.9, 0.5^2) \\
% & + 0.15 \cdot \mathcal{N}(d_{v, v'} \mid 2.2, 0.5^2)
%\end{array}
%\label{eq:wallAvoidance}
%\end{equation}
To downvote vertices near walls, we need to determine the distance of each vertex from its nearest wall.
We therefore derive an inverted version $G' = (V', E')$ of the graph $G$, just describing walls and other
obstacles. A nearest-neighbour search \cite{Cover1967} within $V'$ provides the vertex $v'$
nearest to $v$:
\begin{equation}
\text{wa}_{x,y,z} = \mathcal{N}( \| v_{x,y,z} - v'_{x,y,z} \| \mid 0.0, 0.5^2) \\
v' = \fNN{v}{V'} \enskip .
\end{equation}
To get a smooth gradient around walls, the avoidance-factor
is derived using a normal distribution with a deviation of \SI{0.5}{\meter}:
%
\begin{equation}
\fWA{v} = \mathcal{N}( \fDistance{v}{\fNN{v}{V'}} \mid 0.0, 0.5^2) \\
\label{eq:wallAvoidance}
\end{equation}
%
@@ -114,9 +109,9 @@
%While this approach provides good results for most areas, doors are downvoted by
%\refeq{eq:wallAvoidance}, as they have only vertices that are close to walls.
%Door detection and upvoting thus is the next conducted step.
While effectively rendering wall-regions less likely, \refeq{eq:wallAvoidance}
will obviously have the same effect on all doors located within the building.
Therefore, a door-detection is necessary, to upvote them again.
While rendering wall-regions less likely, \refeq{eq:wallAvoidance}
will obviously have the same effect on doors as they are just a small gap between
consecutive walls. Therefore, a door-detection is necessary, to upvote them again.
@@ -124,90 +119,103 @@
\label{sec:doorDetection}
To automatically detect doors within the floorplan, we utilize the fact that doors are usually
anchored between two (straight) walls and have a normed width. Examining the region directly
around it, the door and its surrounding walls describe a flat ellipse with the door as its
centre.
anchored between two straight walls and have a normed width. Examining the region directly
around it, the door and its surrounding walls thus describe a flat ellipse with the door as its centre.
%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 $k$ nearest neighbours $\hat{V}$ within the inverted grid $G'$,
describing the walls nearby. Hereafter we determine the centroid $\vec{c} \in \R^3$
and 2D covariance $(x,y)$ for those vertices.
To decide whether a vertex $v$ belongs to a door, we use \mbox{$k$-NN} to fetch its $k$ nearest neighbours
$\{v'_1, v'_2, \dots, v'_k\} \in V'$ within the inverted grid $G'$, denoting nearby walls.
%Hereafter we determine the centroid $\vec{c} \in \R^3$
%and 2D covariance $(x,y)$ for those vertices.
Hereafter we determine their centroid $\vec{c}$ and covariance $\mat{\Sigma}$:
\begin{equation}
\vec{c} = \frac{1}{k}\sum_{i=1}^{k} \fPos{v_i}
,\smallskip
\mat{\Sigma} = \frac{1}{k} \sum_{i=1}^{k} \varphi \cdot \varphi^T
,\smallskip
\varphi = \fPos{v_i}-\vec{c} \enskip.
\end{equation}
%
Using the PCA, we examine the two eigenvalues $\{\lambda_1, \lambda_2 \mid \lambda_1 > \lambda_2\}$,
for the covariance matrix. If their ratio $\frac{\lambda_1}{\lambda_2}$ is above a certain
threshold, the neighbourhood describes a flat ellipse and thus either a straight wall or door.
For $\mat{\Sigma}$, the two largest eigenvalues $\{\lambda_1, \lambda_2 \mid \lambda_1 > \lambda_2\}$
are calculated. If their ratio $^{\lambda_1}/_{\lambda_2}$ is above a certain
threshold, the neighbourhood describes a flat ellipse and thus either a door or a straight wall
%
To prevent a vertex $v_{x,y,z}$ adjacent to such straight walls from also being detected,
we ensure the distance $\| \vec{c} - v_{x,y,z} \|$ between the centroid and the vertex is
below a certain threshold. Hereafter, only vertices located within the door itself remain.
To filter the latter, we enforce the euclidean distance \mbox{$\| \fPos{v} - \vec{c} \|$} between
the centroid and the vertex to be very small. Hereafter, only vertices located directly within a
door itself remain.
%
\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.}
\caption{Detect doors within the floorplan using $k$-NN to get the centroid $\vec{c}$ and covariance $\mat{\Sigma}$
of the wall-vertices (black) near $v$. While the left version is fine, the $v$ in the middle is too far
from $\vec{c}$ and the right one has an invalid eigenvalue-ratio.}
\label{fig:doorPCA}
\end{figure}
%
Fig. \ref{fig:doorPCA} depicts all three cases where
(left) the vertex is part of a door,
(middle) the distance between node and centroid is above the threshold and
(right) the ration between $\lambda_1$ and $\lambda_2$ is below the threshold.
(right) the ratio between $\lambda_1$ and $\lambda_2$ is below the threshold.
For smooth importance-gradients around doors, we again use a distribution based on
the distance of a vertex $v_{x,y,z}$ from its nearest door and a deviation
of \SI{1.0}{\meter} to determine its importance-factor:
For smooth gradients around doors, we again use a distribution based on
the distance of a vertex $v$ from its nearest door and a deviation
of \SI{1.0}{\meter}:
%
%\commentByFrank{distanzrechnung: formel ok?}
\begin{equation}
\text{dd}_{x,y,z} = \mathcal{N}( \| \vec{c} - v_{x,y,z} \| \mid 0.0, 1.0^2 )
\fDD{v} = \mathcal{N}( \| \fPos{v} - \vec{c} \| \mid 0.0, 1.0^2 )
\label{eq:doorDetection}
\end{equation}
The final importance for each node is now calculated using \refeq{eq:wallAvoidance}
%
The final importance combines \refeq{eq:wallAvoidance}
and \refeq{eq:doorDetection}:
%
\begin{equation}
\text{imp}_{x,y,z} = 1.0 - \text{wa}_{x,y,z} + \text{dd}_{x,y,z} \enspace .
\fImp{v} = 1.0 - \fWA{v} + \fDD{v} \enspace .
\label{eq:finalImp}
\end{equation}
%
While most vertices receive a factor of $1.0$, wall-regions get lower and
While most nodes receive a neutral factor of $1.0$, wall-regions get lower and
door-regions higher values, depicted in fig. \ref{fig:importance}.
%
\begin{figure}
\includegraphics[width=\columnwidth]{floorplan_importance}
\caption{The calculated importance-factor for each vertex. While the black wall-elements denote
a small importance, the yellow door-regions receive much higher values.}
\caption{The calculated importance-factor \refeq{eq:finalImp} for each vertex.
While the dark wall-elements denote a small importance, the yellow areas around doors and narrow
passages depict a high importance.}
\label{fig:importance}
\end{figure}
\subsection{Path Estimation}
\label{sec:pathEstimation}
\commentByFrank{ueberleitung}
To estimate the shortest path to the pedestrian's desired target, we use a modified version
of Dijkstra's algorithm. Instead of calculating the shortest path from the start to the end,
we swap start/end and do not terminate the calculation until every single node was evaluated.
Thus, every node in the grid knows the shortest path to the pedestrian's target.
For routing the pedestrian towards his desired target, a modified version
of Dijkstra's algorithm is used. Instead of calculating the shortest path from the start to the end,
the direction is inverted and the calculated terminates as soon as every single node was evaluated.
Hereafter, every node in the grid knows the distance and shortest path to the pedestrian's target.
As weighting-function we use
To get realistic path suggestions, we use the importance-factors to adjust the edge-weight
$\delta(v_1, v_2)$ for the Dijkstra:
%
\begin{equation}
\begin{split}
\text{weight}(v_{x,y,z}, v_{x',y',z'}) =
\delta(v_1, v_2) =
\frac
{ \| v_{x,y,z} - v_{x',y',z'} \| }
{ \text{imp}_{x',y',z'} }
.
{ \| v_1 - v_2 \| }
{ \fImp{v_2} }
\enskip.
\end{split}
\label{eq:edgeWeight}
\end{equation}
%
Eq. \eqref{eq:edgeWeight} artificially increases the euclidean distance between $v_1, v_2$ when
$v_2$ approaches a wall and decreases it when encountering a door.
%
%whereby $\text{stretch}(\cdots)$ is a scaling function (linear or non-linear) used to adjust
%the impact of the previously calculated importance-factors.
%
Fig. \ref{fig:multiHeatMap} depicts the difference between the shortest path calculated without (dashed) and
with importance-factors (solid), where the latter version is clearly more realistic.
with importance-factors (solid), where the latter is clearly more realistic.
%\begin{figure}
% \includegraphics[angle=-90, width=\columnwidth, trim=20 19 17 9, clip]{floorplan_paths}
@@ -219,101 +227,89 @@
\subsection{Guidance}
Based on the previous calculations, we propose two approaches to utilize the prior
knowledge within the transition model.
Based on the previous considerations, we propose two approaches to utilize prior
knowledge within the transition.
\subsubsection{Shortest Path}
Before every transition, the centroid $\vec{c}$ of the current sample-set $\Upsilon_{t-1}$,
representing the posterior distribution at time $t-1$, is calculated:
%
\begin{equation}
\vec{c} = \frac
%{ \sum_{\mStateVec_{t-1}} (\mState_{t-1}^x, \mState_{t-1}^y, \mState_{t-1}^z)^T }
{ \sum_{i=1}^N \Upsilon_{t-1}^{x,y,z} }
{N}
\end{equation}
\newcommand{\pathCentroid}{{\vec{\overline{c}}_{t-1}}}
\newcommand{\pathDev}{\sigma_{t-1}}
\newcommand{\pathRef}{\hat{v}}
Before every transition, the centroid $\pathCentroid$ of the current
sample-set, representing the posterior distribution at time $t-1$, is calculated.
%
oder
%
\begin{equation}
\vec{c} = \frac
{ \sum_{i=1}^N \{(\mState_{t-1}^x, \mState_{t-1}^y, \mState_{t-1}^z)^T\}^i }
{N}
\end{equation}
%\commentByFrank{avg-state vom sample-set. frank d. meinte ja hier muessen wir aufpassen. bin noch unschluessig wie.}
%\commentByToni{Das ist gar nicht so einfach... wir haben nie ein Sample Set eingefuehrt. Nicht mal einen Sample. Wir haben immer nur diesen State... Man könnte natuerlich einfach sagen das $\Upsilon_t$ an set of random samples representing the posterior distribution ist oder einfach nur ein set von partikeln. habs mal eingefuegt wie ich denke}
%
oder: the centroid $\vec{c}$ of the current sample-set's 3D positions $\Upsilon_{t-1}^i = \{(\mState_{t-1}^x, \mState_{t-1}^y, \mState_{t-1}^z)^T\}_{i=0}^N$ is calculated:
This centre serves as the starting point for the shortest-path calculation.
As it is not necessarily part of the grid, its nearest-grid-neighbour is determined and used instead.
This vertex is located somewhere within the sample-set and already knows the way to the pedestrian's destination.
%
\begin{equation}
\vec{c} = \frac
%{ \sum_{\mStateVec_{t-1}} (\mState_{t-1}^x, \mState_{t-1}^y, \mState_{t-1}^z)^T }
{ \sum_{i=1}^N \Upsilon_{t-1}^{i} }
{N}
\end{equation}
\newcommand{\pathRef}{v_{\hat{x},\hat{y},\hat{z}}}
\commentByFrank{avg-state vom sample-set. frank d. meinte ja hier muessen wir aufpassen. bin noch unschluessig wie.}
\commentByToni{Das ist gar nicht so einfach... wir haben nie ein Sample Set eingefuehrt. Nicht mal einen Sample. Wir haben immer nur diesen State... Man könnte natuerlich einfach sagen das $\Upsilon_t$ an set of random samples representing the posterior distribution ist oder einfach nur ein set von partikeln. habs mal eingefuegt wie ich denke}
This center is used as starting-point for the shortest path. As it is not necessarily part of
the grid, its nearest-grid-neighbor is determined and used instead.
The resulting vertex already knows its way to the pedestrian's destination, but is located somewhere
within the sample-set. We thus calculate the standard deviation for the distance
of all samples from the centre. After advancing the starting-vertex by three times this deviation
we get a new point that is: part of the shortest path, outside of the sample-set and closer to the
desired destination.
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.}
As new states $\mStateVec_{t}$ should approach the pedestrian's destination
we use a reference $\pathRef$ all states try to reach. This references must
both, part of the shortest path and located somewhere outside of the sample-set.
%
We thus calculate the standard deviation of the distance of all samples from the centre
$\pathCentroid$. After advancing the starting-vertex by three times this deviation
we get a new point $\pathRef$ that is: part of the shortest path, outside of the sample-set
and closer (but not too close) to the desired destination.
%
Hereafter, the simple transition \refeq{eq:transSimple} is combined with a second probability,
downvoting all grid-steps that depart from $\pathRef$.
Finally, \refeq{eq:transShortestPath} provides a metric tending towards the reference while
still allowing the pedestrian to leave the shortest path:
%
\begin{equation}
\begin{split}
p(e) &=
p(v_{x',y',z'} \mid v_{x,y,z})\\
&= \mathcal{N} (\angle [ v_{x,y,z} v_{x',y',z'} ] \mid \gHead, \sigma_\text{dev}^2) \cdot \alpha \\
p(v' \mid v) =
\mathcal{N} (\angle e \mid \gHead, \sigma_\text{dev}^2) \cdot \alpha \\
\alpha &=
\begin{cases}
0.9 & \| v_{x',y',z'} - \pathRef \| < \| v_{x,y,z} - \pathRef \| \\
0.9 & \| v' - \pathRef \| < \| v - \pathRef \| \\
0.1 & \text{else}
\end{cases}
\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
are unlikely but not impossible.
\subsubsection{Multipath}
The shortest-path algorithm mentioned in \ref{sec:pathEstimation} already calculated the
cumulative distance $\text{cdist}_{x,y,z}$ to the pedestrian's target for each vertex.
We thus apply the same assumption as above and downvote grid-steps not decreasing
the distance to the destination:
$\fLength{v}{\dot{v}}$ % = \sum_{i=s}^{e-1} \| v_{i} - v_{i+1} \| $
for the path from $v$ to the pedestrian's destination $\dot{v}$.
We thus apply the same assumption as \refeq{eq:transShortestPath} and downvote edges
not decreasing the distance to the destination:
%
\begin{equation}
\begin{split}
p(e) &=
p(v_{x',y',z'} \mid v_{x,y,z})\\
& = \mathcal{N} (\angle [ v_{x,y,z} v_{x',y',z'} ] \mid \gHead, \sigma_\text{dev}^2) \cdot \alpha \\
p(v' \mid v) =
\mathcal{N} (\angle e \mid \gHead, \sigma_\text{dev}^2) \cdot \alpha \\
\alpha &=
\begin{cases}
0.9 & \text{cdist}_{x',y',z'} < \text{cdist}_{x,y,z} \\
0.9 & \fLength{v'}{\dot{v}} < \fLength{v}{\dot{v}} \\
0.1 & \text{else}
\end{cases}
\end{split}
\label{eq:transMultiPath}
\end{equation}
Fig. \ref{fig:multiHeatMap} shows the heat-map of visited vertices after several \SI{125}{\meter}
walks simulating slight, random heading changes.
%
Fig. \ref{fig:multiHeatMap} shows a heat-map of how often vertices were visited after several
\SI{125}{\meter} walks. The colours from cold to hot indicate that both possible paths
are covered and slight deviations from the shortest version are possible.
%
\begin{figure}
\includegraphics[width=\columnwidth]{floorplan_dijkstra_heatmap}
\caption{Heat-Map of visited vertices after several walks using \refeq{eq:transMultiPath}.
Additionally shows the shortest path calculation without (dashed) and with (solid) importance-factors
\includegraphics[width=\columnwidth, trim=4 8 4 4]{floorplan_dijkstra_heatmap}
\caption{Heat-Map of visited vertices after several walks using \refeq{eq:transMultiPath} with colours
from cold to hot. Both possible paths are covered and slight deviations are possible.
Additionally shows the shortest-path calculation without (dashed) and with (solid) importance-factors
used for edge-weight-adjustment.}
\label{fig:multiHeatMap}
\end{figure}