chapter 4 almost rewritten
changed gfx
This commit is contained in:
@@ -68,19 +68,19 @@
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% error development over time while walking along a path
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% error development over time while walking along a path
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\begin{figure}
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\begin{figure}
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\input{gfx/eval/error_timed_nexus}
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\input{gfx/eval/error_timed_nexus}
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\caption{Development of the error while walking along
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\caption{Development of the error while walking along Path 4 using the Motorola Nexus 6.
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%path 1 (upper) and
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When leaving the suggested route (3), the error of shortest path \refeq{eq:transShortestPath}
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path 4 (lower) using the Motorola Nexus 6.
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and multipath \refeq{eq:transMultiPath} increases.
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Path 4 shows increasing errors for our methods when leaving the shortest path (3) and when facing multimodalities between two
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The same issues arise when facing multimodalities between two staircases just before the destination (9).}
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staircases just before the destination (9).}
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\label{fig:errorTimedNexus}
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\label{fig:errorTimedNexus}
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\end{figure}
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\end{figure}
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% detailed analysis of path 4
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% detailed analysis of path 4
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\begin{figure}
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\begin{figure}
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\input{gfx/eval/path_nexus_detail}
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\input{gfx/eval/path_nexus_detail}
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\caption{Detailed path analysis depicting the individual segments of path 4. Their corresponding error contribution can
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\caption{Detailed path analysis depicting the individual segments of path 4 using \refeq{eq:transMultiPath}.
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be seen in fig. \ref{fig:errorTimedNexus}. Even though the shortest path suggested by the system is ignored multiple
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Their corresponding error can be seen in fig. \ref{fig:errorTimedNexus}. Even though the shortest path
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times ($3'$ and $3''$) our approach is still able to improve the overall localisation error.}
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suggested by the system is ignored multiple times ($3'$ and $3''$) our approach is still able to improve
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the overall localisation error.}
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\label{fig:nexusPathDetails}
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\label{fig:nexusPathDetails}
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\end{figure}
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\end{figure}
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%
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%
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@@ -7,11 +7,11 @@
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To sample only transitions that are actually feasible
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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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within the environment, we utilize a \SI{20}{\centimeter}-gridded graph
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$G = (V,E)$, $v_{x,y,z} \in V$, $e_{v_{x,y,z}}^{v_{x',y',z'}} \in E$
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$G = (V,E)$ with vertices $v \in V$ and undirected edges $e \in E$
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derived from the buildings floorplan as described in section \ref{sec:relatedWork}.
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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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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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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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\begin{figure}
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\centering
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\centering
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\input{gfx/grid/grid}
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\input{gfx/grid/grid}
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@@ -24,40 +24,45 @@
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\end{figure}
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\end{figure}
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%
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%
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Stairs are defined using three points $\vec{\spoint}_1, \vec{\spoint}_2, \vec{\spoint}_3 \in \R^3$ whereby the segment
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Stairs are defined using three points $\vec{\spoint}_1, \vec{\spoint}_2, \vec{\spoint}_3 \in \R^3$ whereby the segment
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$[ \vec{\spoint}_1 \vec{\spoint}_2 ]$ describes the starting-edge, and $[ \vec{\spoint}_2 \vec{\spoint}_3 ]$ the stair's direction
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$[ \vec{\spoint}_1 \vec{\spoint}_2 ]$ describes the starting-edge, and $[ \vec{\spoint}_2 \vec{\spoint}_3 ]$ the stair's
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(see fig. \ref{fig:gridStairs}). The grid-vertices corresponding to the starting-edge are determined using an intersection of
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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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the segment $[ \vec{\spoint}_1 \vec{\spoint}_2 ]$ with the \SI{20}{\centimeter} bounding-box around each vertex.
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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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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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remove all nodes and edges that are not connected to this region.
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Walking the grid is now possible by moving along adjacent nodes into a given walking-direction
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New states $\mStateVec_{t}$ may now be sampled by starting at the vertex for $\fPos{\mStateVec_{t-1}}$ and
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until a desired distance $\gDist$ is reached \cite{Ebner-15}.
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walking along adjacent nodes into a given walking-direction $\gHead$ until a distance $\gDist$ is
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In order to use meaningful headings $\gHead$ and distances $\gDist$
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reached \cite{Ebner-15}.
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(matching the pedestrian's real heading and walking speed) for each transition,
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Both, heading and distance, are supplied by the current sensor readings $\mObsVec_{t}$
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we use the current sensor-readings $\mObsVec_{t}$ for hinted instead of randomized adjustments:
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and thus reflect the pedestrian's real heading and walking speed including uncertainty.
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%
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Working with relative sensor readings, the state's heading is updated during each transition:
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%
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\begin{align}
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\begin{align}
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\mState_{t}^{\mStateHeading} = \gHead &= \mState_{t-1}^{\mStateHeading} + \mObs_t^{\mObsHeading} + \mathcal{N}(0, \sigma_{\gHead}^2) \\
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\gHead &= \mState_{t}^{\mStateHeading} = \mState_{t-1}^{\mStateHeading} + \mObs_t^{\mObsHeading} + \mathcal{N}(0, \sigma_{\gHead}^2) \\
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\gDist &= \mObs_t^{\mObsSteps} \cdot \SI{0.7}{\meter} + \mathcal{N}(0, \sigma_{\gDist}^2)
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\gDist &= \mObs_t^{\mObsSteps} \cdot \SI{0.7}{\meter} + \mathcal{N}(0, \sigma_{\gDist}^2)
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.
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.
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\end{align}
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\end{align}
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%
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%
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During a walk, each edge has an assigned probability $p(e)$ which depends on
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During the random walk, each edge has its own probability $p(e)$
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the its direction $\angle e$ and the current heading $\gHead$.
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which e.g. depends on the edge's direction $\angle e$ and the
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We will use $p(e)$ to incorporate prior path knowledge to
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pedestrian's current heading $\gHead$.
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favour some vertices/edges. For each single movement on the graph,
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Furthermore, $p(e)$ is used to incorporate prior path knowledge to
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we calculate $p(e)$ for all adjacent edges, and, hereafter, randomly draw the
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favour edges leading towards the pedestrian's desired target.
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to-be-walked edge depending on those probabilities. The random walk ends,
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as soon as the distance $d$ is reached. $d$ depends on the number of detected steps
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$\mObsVec_t^{\mObsSteps}$ and assumes an average step-size of \SI{0.7}{\meter}.
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For comparison purpose we define a simple weighting method that assigns a probability to each edge
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For each single movement on the graph, we calculate $p(e)$ for all edges
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just based on the deviation from the currently estimated heading $\gHead$ omitting additional prior knowledge:
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connected to a vertex $v$, and, hereafter, randomly draw the to-be-walked edge
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depending on those probabilities. This step is repeated until the sum
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\commentByFrank{das erste $=$ ist komisch. ideen?}
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of the length of all used edges exceeds $d$. The latter depends on the number of
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\commentByToni{Find ich jetzt nicht tragisch. Eher notwendig fuers Verstaendnis.}
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detected steps $\mObs_t^{\mObsSteps}$ and assumes an average
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step-size of \SI{0.7}{\meter}.
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To quantify the improvement prior knowledge is able to provide,
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we define a simple reference for $p(e)$ that assigns a probability to each edge
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just based on its deviation from the currently estimated heading $\gHead$:
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%
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\begin{equation}
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\begin{equation}
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p(e) = p(e \mid \gHead) = \mathcal{N} (\angle e \mid \gHead, \sigma_\text{dev}^2).
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p(e) = p(e \mid \gHead) = \mathcal{N} (\angle e \mid \gHead, \sigma_\text{dev}^2).
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\label{eq:transSimple}
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\label{eq:transSimple}
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@@ -79,34 +84,24 @@
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\label{sec:wallAvoidance}
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\label{sec:wallAvoidance}
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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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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 the walls and
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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. Pedestrian's however, walk either somewhere near (but not close to) a wall or, for
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walking many diagonals. Pedestrian's however, will probably keep a small gap between themselves and
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larger open spaces, somewhere far from the walls. In order to calculate paths that resemble such a walking
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nearby walls. To calculate paths that resemble this behaviour, an importance-factor is derived for
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behaviour, an importance-factor is derived for each vertex within the graph. Those will be used to
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each vertex. Those will be used to modify the euclidean distance $\fDistance{u}{v}$ between two vertices
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adjust the distance-weight between two vertices, needed by the shortest-path algorithm.
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$u,v$, examined by the shortest-path algorithm.
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To downvote vertices near walls, we need to get the distance of each vertex from its nearest wall.
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To downvote vertices near walls, we need to determine the distance of each vertex from its nearest wall.
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We therefore build an inverted version $G' = (V', E')$ of the graph $G$, just containing walls and other obstacles.
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We therefore derive an inverted version $G' = (V', E')$ of the graph $G$, just describing walls and other
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A nearest-neighbour search \cite{Cover1967}
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obstacles. A nearest-neighbour search \cite{Cover1967} within $V'$ provides the vertex $v'$
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%$\mNN(v_{x,y,z}, G')$
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nearest to $v$:
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will then provide the nearest wall-vertex
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$v'_{x,y,z} \in V'$ from the inverted graph. To get a smooth gradient, the wall avoidance
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is calculated using a normal distribution with the distance from the nearest wall
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and a deviation of \SI{0.5}{\meter}:
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%
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%\begin{equation}
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% d_{v, v'} = \| v_{x,y,z}, v'_{x,y,z}, \enskip 0.0 < d_{v, v'} < 2.2 \\
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%\end{equation}
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%\begin{equation}
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%\begin{array}{ll}
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% \text{wa}_{x,y,z} = & - 0.30 \cdot \mathcal{N}(d_{v, v'} \mid 0.0, 0.5^2) \\
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% & + 0.15 \cdot \mathcal{N}(d_{v, v'} \mid 0.9, 0.5^2) \\
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% & + 0.15 \cdot \mathcal{N}(d_{v, v'} \mid 2.2, 0.5^2)
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%\end{array}
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%\label{eq:wallAvoidance}
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%\end{equation}
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\begin{equation}
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\begin{equation}
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\text{wa}_{x,y,z} = \mathcal{N}( \| v_{x,y,z} - v'_{x,y,z} \| \mid 0.0, 0.5^2) \\
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v' = \fNN{v}{V'} \enskip .
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\end{equation}
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To get a smooth gradient around walls, the avoidance-factor
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is derived using a normal distribution with a deviation of \SI{0.5}{\meter}:
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%
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\begin{equation}
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\fWA{v} = \mathcal{N}( \fDistance{v}{\fNN{v}{V'}} \mid 0.0, 0.5^2) \\
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\label{eq:wallAvoidance}
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\label{eq:wallAvoidance}
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\end{equation}
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\end{equation}
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%
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%
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@@ -114,9 +109,9 @@
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%While this approach provides good results for most areas, doors are downvoted by
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%While this approach provides good results for most areas, doors are downvoted by
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%\refeq{eq:wallAvoidance}, as they have only vertices that are close to walls.
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%\refeq{eq:wallAvoidance}, as they have only vertices that are close to walls.
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%Door detection and upvoting thus is the next conducted step.
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%Door detection and upvoting thus is the next conducted step.
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While effectively rendering wall-regions less likely, \refeq{eq:wallAvoidance}
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While rendering wall-regions less likely, \refeq{eq:wallAvoidance}
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will obviously have the same effect on all doors located within the building.
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will obviously have the same effect on doors as they are just a small gap between
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Therefore, a door-detection is necessary, to upvote them again.
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consecutive walls. Therefore, a door-detection is necessary, to upvote them again.
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@@ -124,90 +119,103 @@
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\label{sec:doorDetection}
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\label{sec:doorDetection}
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To automatically detect doors within the floorplan, we utilize the fact that doors are usually
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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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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 describe a flat ellipse with the door as its
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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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centre.
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%It is thus possible to detect doors within the floorplan using a PCA.
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%It is thus possible to detect doors within the floorplan using a PCA.
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To decide whether a vertex $v_{x,y,z}$ within the (non-inverted) grid $G$ belongs to a door,
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To decide whether a vertex $v$ belongs to a door, we use \mbox{$k$-NN} to fetch its $k$ nearest neighbours
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we use $k$-NN to fetch its $k$ nearest neighbours $\hat{V}$ within the inverted grid $G'$,
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$\{v'_1, v'_2, \dots, v'_k\} \in V'$ within the inverted grid $G'$, denoting nearby walls.
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describing the walls nearby. Hereafter we determine the centroid $\vec{c} \in \R^3$
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%Hereafter we determine the centroid $\vec{c} \in \R^3$
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and 2D covariance $(x,y)$ for those vertices.
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%and 2D covariance $(x,y)$ for those vertices.
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Hereafter we determine their centroid $\vec{c}$ and covariance $\mat{\Sigma}$:
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\begin{equation}
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\vec{c} = \frac{1}{k}\sum_{i=1}^{k} \fPos{v_i}
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,\smallskip
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\mat{\Sigma} = \frac{1}{k} \sum_{i=1}^{k} \varphi \cdot \varphi^T
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,\smallskip
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\varphi = \fPos{v_i}-\vec{c} \enskip.
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\end{equation}
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%
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%
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Using the PCA, we examine the two eigenvalues $\{\lambda_1, \lambda_2 \mid \lambda_1 > \lambda_2\}$,
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For $\mat{\Sigma}$, the two largest eigenvalues $\{\lambda_1, \lambda_2 \mid \lambda_1 > \lambda_2\}$
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for the covariance matrix. If their ratio $\frac{\lambda_1}{\lambda_2}$ is above a certain
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are calculated. If their ratio $^{\lambda_1}/_{\lambda_2}$ is above a certain
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threshold, the neighbourhood describes a flat ellipse and thus either a straight wall or door.
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threshold, the neighbourhood describes a flat ellipse and thus either a door or a straight wall
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%
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%
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To prevent a vertex $v_{x,y,z}$ adjacent to such straight walls from also being detected,
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To filter the latter, we enforce the euclidean distance \mbox{$\| \fPos{v} - \vec{c} \|$} between
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we ensure the distance $\| \vec{c} - v_{x,y,z} \|$ between the centroid and the vertex is
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the centroid and the vertex to be very small. Hereafter, only vertices located directly within a
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below a certain threshold. Hereafter, only vertices located within the door itself remain.
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door itself remain.
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%
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%
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\begin{figure}
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\begin{figure}
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\includegraphics[width=\columnwidth]{door_pca}
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\includegraphics[width=\columnwidth]{door_pca}
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\caption{Detect doors within the floorplan using $k$-NN and PCA.
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\caption{Detect doors within the floorplan using $k$-NN to get the centroid $\vec{c}$ and covariance $\mat{\Sigma}$
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While the white nodes are walkable, the black ones represent walls. The grey node is the one in question.}
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of the wall-vertices (black) near $v$. While the left version is fine, the $v$ in the middle is too far
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from $\vec{c}$ and the right one has an invalid eigenvalue-ratio.}
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\label{fig:doorPCA}
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\label{fig:doorPCA}
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\end{figure}
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\end{figure}
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%
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%
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Fig. \ref{fig:doorPCA} depicts all three cases where
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Fig. \ref{fig:doorPCA} depicts all three cases where
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(left) the vertex is part of a door,
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(left) the vertex is part of a door,
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(middle) the distance between node and centroid is above the threshold and
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(middle) the distance between node and centroid is above the threshold and
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(right) the ration between $\lambda_1$ and $\lambda_2$ is below the threshold.
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(right) the ratio between $\lambda_1$ and $\lambda_2$ is below the threshold.
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For smooth importance-gradients around doors, we again use a distribution based on
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For smooth gradients around doors, we again use a distribution based on
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the distance of a vertex $v_{x,y,z}$ from its nearest door and a deviation
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the distance of a vertex $v$ from its nearest door and a deviation
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of \SI{1.0}{\meter} to determine its importance-factor:
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of \SI{1.0}{\meter}:
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%
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%
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%\commentByFrank{distanzrechnung: formel ok?}
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%\commentByFrank{distanzrechnung: formel ok?}
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\begin{equation}
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\begin{equation}
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\text{dd}_{x,y,z} = \mathcal{N}( \| \vec{c} - v_{x,y,z} \| \mid 0.0, 1.0^2 )
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\fDD{v} = \mathcal{N}( \| \fPos{v} - \vec{c} \| \mid 0.0, 1.0^2 )
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\label{eq:doorDetection}
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\label{eq:doorDetection}
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\end{equation}
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\end{equation}
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%
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The final importance for each node is now calculated using \refeq{eq:wallAvoidance}
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The final importance combines \refeq{eq:wallAvoidance}
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and \refeq{eq:doorDetection}:
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and \refeq{eq:doorDetection}:
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%
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%
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\begin{equation}
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\begin{equation}
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\text{imp}_{x,y,z} = 1.0 - \text{wa}_{x,y,z} + \text{dd}_{x,y,z} \enspace .
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\fImp{v} = 1.0 - \fWA{v} + \fDD{v} \enspace .
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\label{eq:finalImp}
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\end{equation}
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\end{equation}
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%
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%
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While most vertices receive a factor of $1.0$, wall-regions get lower and
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While most nodes receive a neutral factor of $1.0$, wall-regions get lower and
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door-regions higher values, depicted in fig. \ref{fig:importance}.
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door-regions higher values, depicted in fig. \ref{fig:importance}.
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%
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%
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\begin{figure}
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\begin{figure}
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\includegraphics[width=\columnwidth]{floorplan_importance}
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\includegraphics[width=\columnwidth]{floorplan_importance}
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\caption{The calculated importance-factor for each vertex. While the black wall-elements denote
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\caption{The calculated importance-factor \refeq{eq:finalImp} for each vertex.
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a small importance, the yellow door-regions receive much higher values.}
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While the dark wall-elements denote a small importance, the yellow areas around doors and narrow
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passages depict a high importance.}
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\label{fig:importance}
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\label{fig:importance}
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\end{figure}
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\end{figure}
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\subsection{Path Estimation}
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\subsection{Path Estimation}
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\label{sec:pathEstimation}
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\label{sec:pathEstimation}
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\commentByFrank{ueberleitung}
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For routing the pedestrian towards his desired target, a modified version
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To estimate the shortest path to the pedestrian's desired target, we use 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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of Dijkstra's algorithm. 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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we swap start/end and do not terminate the calculation until 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.
|
||||||
Thus, every node in the grid knows the 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{equation}
|
||||||
\begin{split}
|
\begin{split}
|
||||||
\text{weight}(v_{x,y,z}, v_{x',y',z'}) =
|
\delta(v_1, v_2) =
|
||||||
\frac
|
\frac
|
||||||
{ \| v_{x,y,z} - v_{x',y',z'} \| }
|
{ \| v_1 - v_2 \| }
|
||||||
{ \text{imp}_{x',y',z'} }
|
{ \fImp{v_2} }
|
||||||
.
|
\enskip.
|
||||||
\end{split}
|
\end{split}
|
||||||
|
\label{eq:edgeWeight}
|
||||||
\end{equation}
|
\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
|
%whereby $\text{stretch}(\cdots)$ is a scaling function (linear or non-linear) used to adjust
|
||||||
%the impact of the previously calculated importance-factors.
|
%the impact of the previously calculated importance-factors.
|
||||||
%
|
%
|
||||||
Fig. \ref{fig:multiHeatMap} depicts the difference between the shortest path calculated without (dashed) and
|
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}
|
%\begin{figure}
|
||||||
% \includegraphics[angle=-90, width=\columnwidth, trim=20 19 17 9, clip]{floorplan_paths}
|
% \includegraphics[angle=-90, width=\columnwidth, trim=20 19 17 9, clip]{floorplan_paths}
|
||||||
@@ -219,101 +227,89 @@
|
|||||||
|
|
||||||
\subsection{Guidance}
|
\subsection{Guidance}
|
||||||
|
|
||||||
Based on the previous calculations, we propose two approaches to utilize the prior
|
Based on the previous considerations, we propose two approaches to utilize prior
|
||||||
knowledge within the transition model.
|
knowledge within the transition.
|
||||||
|
|
||||||
\subsubsection{Shortest Path}
|
\subsubsection{Shortest Path}
|
||||||
|
|
||||||
Before every transition, the centroid $\vec{c}$ of the current sample-set $\Upsilon_{t-1}$,
|
\newcommand{\pathCentroid}{{\vec{\overline{c}}_{t-1}}}
|
||||||
representing the posterior distribution at time $t-1$, is calculated:
|
\newcommand{\pathDev}{\sigma_{t-1}}
|
||||||
%
|
\newcommand{\pathRef}{\hat{v}}
|
||||||
\begin{equation}
|
|
||||||
\vec{c} = \frac
|
Before every transition, the centroid $\pathCentroid$ of the current
|
||||||
%{ \sum_{\mStateVec_{t-1}} (\mState_{t-1}^x, \mState_{t-1}^y, \mState_{t-1}^z)^T }
|
sample-set, representing the posterior distribution at time $t-1$, is calculated.
|
||||||
{ \sum_{i=1}^N \Upsilon_{t-1}^{x,y,z} }
|
|
||||||
{N}
|
|
||||||
\end{equation}
|
|
||||||
%
|
%
|
||||||
oder
|
|
||||||
%
|
%
|
||||||
\begin{equation}
|
%\commentByFrank{avg-state vom sample-set. frank d. meinte ja hier muessen wir aufpassen. bin noch unschluessig wie.}
|
||||||
\vec{c} = \frac
|
%\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}
|
||||||
{ \sum_{i=1}^N \{(\mState_{t-1}^x, \mState_{t-1}^y, \mState_{t-1}^z)^T\}^i }
|
|
||||||
{N}
|
|
||||||
\end{equation}
|
|
||||||
%
|
%
|
||||||
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}
|
As new states $\mStateVec_{t}$ should approach the pedestrian's destination
|
||||||
\vec{c} = \frac
|
we use a reference $\pathRef$ all states try to reach. This references must
|
||||||
%{ \sum_{\mStateVec_{t-1}} (\mState_{t-1}^x, \mState_{t-1}^y, \mState_{t-1}^z)^T }
|
both, part of the shortest path and located somewhere outside of the sample-set.
|
||||||
{ \sum_{i=1}^N \Upsilon_{t-1}^{i} }
|
%
|
||||||
{N}
|
We thus calculate the standard deviation of the distance of all samples from the centre
|
||||||
\end{equation}
|
$\pathCentroid$. 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
|
||||||
\newcommand{\pathRef}{v_{\hat{x},\hat{y},\hat{z}}}
|
and closer (but not too close) to the desired destination.
|
||||||
\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}
|
Hereafter, the simple transition \refeq{eq:transSimple} is combined with a second probability,
|
||||||
|
downvoting all grid-steps that depart from $\pathRef$.
|
||||||
This center is used as starting-point for the shortest path. As it is not necessarily part of
|
Finally, \refeq{eq:transShortestPath} provides a metric tending towards the reference while
|
||||||
the grid, its nearest-grid-neighbor is determined and used instead.
|
still allowing the pedestrian to leave the shortest path:
|
||||||
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.}
|
|
||||||
%
|
%
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\begin{split}
|
\begin{split}
|
||||||
p(e) &=
|
p(e) &=
|
||||||
p(v_{x',y',z'} \mid v_{x,y,z})\\
|
p(v' \mid v) =
|
||||||
&= \mathcal{N} (\angle [ v_{x,y,z} v_{x',y',z'} ] \mid \gHead, \sigma_\text{dev}^2) \cdot \alpha \\
|
\mathcal{N} (\angle e \mid \gHead, \sigma_\text{dev}^2) \cdot \alpha \\
|
||||||
\alpha &=
|
\alpha &=
|
||||||
\begin{cases}
|
\begin{cases}
|
||||||
0.9 & \| v_{x',y',z'} - \pathRef \| < \| v_{x,y,z} - \pathRef \| \\
|
0.9 & \| v' - \pathRef \| < \| v - \pathRef \| \\
|
||||||
0.1 & \text{else}
|
0.1 & \text{else}
|
||||||
\end{cases}
|
\end{cases}
|
||||||
\end{split}
|
\end{split}
|
||||||
|
.
|
||||||
\label{eq:transShortestPath}
|
\label{eq:transShortestPath}
|
||||||
\end{equation}
|
\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}
|
\subsubsection{Multipath}
|
||||||
|
|
||||||
The shortest-path algorithm mentioned in \ref{sec:pathEstimation} already calculated the
|
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.
|
$\fLength{v}{\dot{v}}$ % = \sum_{i=s}^{e-1} \| v_{i} - v_{i+1} \| $
|
||||||
We thus apply the same assumption as above and downvote grid-steps not decreasing
|
for the path from $v$ to the pedestrian's destination $\dot{v}$.
|
||||||
the distance to the destination:
|
We thus apply the same assumption as \refeq{eq:transShortestPath} and downvote edges
|
||||||
|
not decreasing the distance to the destination:
|
||||||
|
%
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\begin{split}
|
\begin{split}
|
||||||
p(e) &=
|
p(e) &=
|
||||||
p(v_{x',y',z'} \mid v_{x,y,z})\\
|
p(v' \mid v) =
|
||||||
& = \mathcal{N} (\angle [ v_{x,y,z} v_{x',y',z'} ] \mid \gHead, \sigma_\text{dev}^2) \cdot \alpha \\
|
\mathcal{N} (\angle e \mid \gHead, \sigma_\text{dev}^2) \cdot \alpha \\
|
||||||
\alpha &=
|
\alpha &=
|
||||||
\begin{cases}
|
\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}
|
0.1 & \text{else}
|
||||||
\end{cases}
|
\end{cases}
|
||||||
\end{split}
|
\end{split}
|
||||||
\label{eq:transMultiPath}
|
\label{eq:transMultiPath}
|
||||||
\end{equation}
|
\end{equation}
|
||||||
|
%
|
||||||
Fig. \ref{fig:multiHeatMap} shows the heat-map of visited vertices after several \SI{125}{\meter}
|
Fig. \ref{fig:multiHeatMap} shows a heat-map of how often vertices were visited after several
|
||||||
walks simulating slight, random heading changes.
|
\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}
|
\begin{figure}
|
||||||
\includegraphics[width=\columnwidth]{floorplan_dijkstra_heatmap}
|
\includegraphics[width=\columnwidth, trim=4 8 4 4]{floorplan_dijkstra_heatmap}
|
||||||
\caption{Heat-Map of visited vertices after several walks using \refeq{eq:transMultiPath}.
|
\caption{Heat-Map of visited vertices after several walks using \refeq{eq:transMultiPath} with colours
|
||||||
Additionally shows the shortest path calculation without (dashed) and with (solid) importance-factors
|
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.}
|
used for edge-weight-adjustment.}
|
||||||
\label{fig:multiHeatMap}
|
\label{fig:multiHeatMap}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|||||||
@@ -1,10 +1,10 @@
|
|||||||
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|
%%CreationDate: Wed Feb 17 20:50:33 2016
|
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%%EndComments
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||||||
save
|
save
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Before Width: | Height: | Size: 29 KiB After Width: | Height: | Size: 35 KiB |
@@ -1,7 +1,7 @@
|
|||||||
%!PS-Adobe-2.0 EPSF-2.0
|
%!PS-Adobe-2.0 EPSF-2.0
|
||||||
%%Title: error_timed_nexus.tex
|
%%Title: error_timed_nexus.tex
|
||||||
%%Creator: gnuplot 5.0 patchlevel 1
|
%%Creator: gnuplot 5.0 patchlevel 1
|
||||||
%%CreationDate: Tue Feb 16 19:06:27 2016
|
%%CreationDate: Wed Feb 17 14:47:42 2016
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%%EndComments
|
%%EndComments
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@@ -441,7 +441,7 @@ SDict begin [
|
|||||||
/Author (kazu)
|
/Author (kazu)
|
||||||
% /Producer (gnuplot)
|
% /Producer (gnuplot)
|
||||||
% /Keywords ()
|
% /Keywords ()
|
||||||
/CreationDate (Tue Feb 16 19:06:27 2016)
|
/CreationDate (Wed Feb 17 14:47:42 2016)
|
||||||
/DOCINFO pdfmark
|
/DOCINFO pdfmark
|
||||||
end
|
end
|
||||||
} ifelse
|
} ifelse
|
||||||
|
|||||||
@@ -83,9 +83,9 @@ set object 10 rectangle from 142,0 to 150,15 fs solid noborder fc rgb "#dddddd"
|
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set label 10 "\\footnotesize{10}" at 146,10 center rotate by 0 front
|
set label 10 "\\footnotesize{10}" at 146,10 center rotate by 0 front
|
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|
|
||||||
set object 900 rectangle at graph 0.5, 0.94 size graph 0.65,0.12 fs solid fc rgb "#ffffff" lc rgb "#000000"
|
set object 900 rectangle at graph 0.5, 0.94 size graph 0.65,0.12 fs solid fc rgb "#ffffff" lc rgb "#000000"
|
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set label 901 "\\textcolor[rgb]{0,0,0}{\\footnotesize{simple \\eqref{eq:transSimple}}}" at graph 0.28, 0.94 center front
|
set label 901 "\\textcolor[rgb]{0,0,0}{\\footnotesize{simple}}" at graph 0.28, 0.94 center front
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||||||
set label 902 "\\textcolor[rgb]{0,0,0.8}{\\footnotesize{multi \\eqref{eq:transMultiPath}}}" at graph 0.49, 0.94 center front
|
set label 902 "\\textcolor[rgb]{0,0,0.8}{\\footnotesize{multi}}" at graph 0.49, 0.94 center front
|
||||||
set label 903 "\\textcolor[rgb]{0.8,0,0}{\\footnotesize{shortest \\eqref{eq:transShortestPath}}}" at graph 0.71, 0.94 center front
|
set label 903 "\\textcolor[rgb]{0.8,0,0}{\\footnotesize{shortest}}" at graph 0.71, 0.94 center front
|
||||||
|
|
||||||
unset key
|
unset key
|
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set yrange[0:13]
|
set yrange[0:13]
|
||||||
|
|||||||
@@ -110,9 +110,9 @@
|
|||||||
\put(4057,1559){\makebox(0,0){\strut{}\footnotesize{8}}}%
|
\put(4057,1559){\makebox(0,0){\strut{}\footnotesize{8}}}%
|
||||||
\put(4421,1559){\makebox(0,0){\strut{}\footnotesize{9}}}%
|
\put(4421,1559){\makebox(0,0){\strut{}\footnotesize{9}}}%
|
||||||
\put(4702,1559){\makebox(0,0){\strut{}\footnotesize{10}}}%
|
\put(4702,1559){\makebox(0,0){\strut{}\footnotesize{10}}}%
|
||||||
\put(1793,1846){\makebox(0,0){\strut{}\textcolor[rgb]{0,0,0}{\footnotesize{simple \eqref{eq:transSimple}}}}}%
|
\put(1793,1846){\makebox(0,0){\strut{}\textcolor[rgb]{0,0,0}{\footnotesize{simple}}}}%
|
||||||
\put(2682,1846){\makebox(0,0){\strut{}\textcolor[rgb]{0,0,0.8}{\footnotesize{multi \eqref{eq:transMultiPath}}}}}%
|
\put(2682,1846){\makebox(0,0){\strut{}\textcolor[rgb]{0,0,0.8}{\footnotesize{multi}}}}%
|
||||||
\put(3614,1846){\makebox(0,0){\strut{}\textcolor[rgb]{0.8,0,0}{\footnotesize{shortest \eqref{eq:transShortestPath}}}}}%
|
\put(3614,1846){\makebox(0,0){\strut{}\textcolor[rgb]{0.8,0,0}{\footnotesize{shortest}}}}%
|
||||||
}%
|
}%
|
||||||
\gplbacktext
|
\gplbacktext
|
||||||
\put(0,0){\includegraphics{error_timed_nexus}}%
|
\put(0,0){\includegraphics{error_timed_nexus}}%
|
||||||
|
|||||||
@@ -70,7 +70,7 @@ set multiplot layout 1,1 scale 2.4,2.4 offset 0,0.1
|
|||||||
splot \
|
splot \
|
||||||
"data/floors.dat" with lines lc rgb "#cccccc" notitle,\
|
"data/floors.dat" with lines lc rgb "#cccccc" notitle,\
|
||||||
"data/est_bergwerk_path4_nexus_multi.dat" using 1:2:3:(f4(column(0))) with lines lw 8 palette notitle,\
|
"data/est_bergwerk_path4_nexus_multi.dat" using 1:2:3:(f4(column(0))) with lines lw 8 palette notitle,\
|
||||||
"data/est_bergwerk_path4_nexus_multi.dat" with lines lw 2 lc rgb "#000099" title "\\footnotesize{multi \\eqref{eq:transMultiPath}}",\
|
"data/est_bergwerk_path4_nexus_multi.dat" with lines lw 2 lc rgb "#000099" title "\\footnotesize{multi}",\
|
||||||
"data/path4.dat" with lines title "\\footnotesize{ground truth}" dashtype 3 lw 2 lc rgb "#000000"
|
"data/path4.dat" with lines title "\\footnotesize{ground truth}" dashtype 3 lw 2 lc rgb "#000000"
|
||||||
|
|
||||||
unset multiplot
|
unset multiplot
|
||||||
|
|||||||
@@ -1,7 +1,7 @@
|
|||||||
%!PS-Adobe-2.0 EPSF-2.0
|
%!PS-Adobe-2.0 EPSF-2.0
|
||||||
%%Title: path_nexus_detail.tex
|
%%Title: path_nexus_detail.tex
|
||||||
%%Creator: gnuplot 5.0 patchlevel 1
|
%%Creator: gnuplot 5.0 patchlevel 1
|
||||||
%%CreationDate: Tue Feb 16 19:12:37 2016
|
%%CreationDate: Wed Feb 17 14:45:48 2016
|
||||||
%%DocumentFonts:
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%%DocumentFonts:
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%%BoundingBox: 50 50 302 226
|
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%%EndComments
|
%%EndComments
|
||||||
@@ -441,7 +441,7 @@ SDict begin [
|
|||||||
/Author (kazu)
|
/Author (kazu)
|
||||||
% /Producer (gnuplot)
|
% /Producer (gnuplot)
|
||||||
% /Keywords ()
|
% /Keywords ()
|
||||||
/CreationDate (Tue Feb 16 19:12:37 2016)
|
/CreationDate (Wed Feb 17 14:45:48 2016)
|
||||||
/DOCINFO pdfmark
|
/DOCINFO pdfmark
|
||||||
end
|
end
|
||||||
} ifelse
|
} ifelse
|
||||||
@@ -553,14 +553,14 @@ stroke
|
|||||||
LTb
|
LTb
|
||||||
LCb setrgbcolor
|
LCb setrgbcolor
|
||||||
LTb
|
LTb
|
||||||
103 89 N
|
1687 89 N
|
||||||
0 440 V
|
0 440 V
|
||||||
1409 0 V
|
-175 0 V
|
||||||
0 -440 V
|
0 -440 V
|
||||||
103 89 L
|
175 0 V
|
||||||
Z stroke
|
Z stroke
|
||||||
103 529 M
|
1687 529 M
|
||||||
1409 0 V
|
-175 0 V
|
||||||
stroke
|
stroke
|
||||||
0.80 0.80 0.80 C 2555 1288 M
|
0.80 0.80 0.80 C 2555 1288 M
|
||||||
-50 -15 V
|
-50 -15 V
|
||||||
@@ -4668,19 +4668,19 @@ LC2 setrgbcolor
|
|||||||
221 0 V
|
221 0 V
|
||||||
stroke
|
stroke
|
||||||
LCw setrgbcolor
|
LCw setrgbcolor
|
||||||
1.000 103 89 1409 440 BoxColFill
|
1.000 1687 89 -175 440 BoxColFill
|
||||||
1.000 UL
|
1.000 UL
|
||||||
LTb
|
LTb
|
||||||
LCb setrgbcolor
|
LCb setrgbcolor
|
||||||
LTb
|
LTb
|
||||||
103 89 N
|
1687 89 N
|
||||||
0 440 V
|
0 440 V
|
||||||
1409 0 V
|
-175 0 V
|
||||||
0 -440 V
|
0 -440 V
|
||||||
103 89 L
|
175 0 V
|
||||||
Z stroke
|
Z stroke
|
||||||
103 529 M
|
1687 529 M
|
||||||
1409 0 V
|
-175 0 V
|
||||||
stroke
|
stroke
|
||||||
0.80 0.80 0.80 C 1.000 UL
|
0.80 0.80 0.80 C 1.000 UL
|
||||||
LTb
|
LTb
|
||||||
|
|||||||
@@ -84,11 +84,11 @@
|
|||||||
}%
|
}%
|
||||||
\gplgaddtomacro\gplfronttext{%
|
\gplgaddtomacro\gplfronttext{%
|
||||||
\csname LTb\endcsname%
|
\csname LTb\endcsname%
|
||||||
\put(1027,419){\makebox(0,0)[r]{\strut{}\footnotesize{multi \eqref{eq:transMultiPath}}}}%
|
\put(1027,419){\makebox(0,0)[r]{\strut{}\footnotesize{multi}}}%
|
||||||
\csname LTb\endcsname%
|
\csname LTb\endcsname%
|
||||||
\put(1027,199){\makebox(0,0)[r]{\strut{}\footnotesize{ground truth}}}%
|
\put(1027,199){\makebox(0,0)[r]{\strut{}\footnotesize{ground truth}}}%
|
||||||
\csname LTb\endcsname%
|
\csname LTb\endcsname%
|
||||||
\put(1027,419){\makebox(0,0)[r]{\strut{}\footnotesize{multi \eqref{eq:transMultiPath}}}}%
|
\put(1027,419){\makebox(0,0)[r]{\strut{}\footnotesize{multi}}}%
|
||||||
\csname LTb\endcsname%
|
\csname LTb\endcsname%
|
||||||
\put(1027,199){\makebox(0,0)[r]{\strut{}\footnotesize{ground truth}}}%
|
\put(1027,199){\makebox(0,0)[r]{\strut{}\footnotesize{ground truth}}}%
|
||||||
\csname LTb\endcsname%
|
\csname LTb\endcsname%
|
||||||
|
|||||||
@@ -41,6 +41,13 @@
|
|||||||
|
|
||||||
\newcommand{\mNN}{\text{nn}}
|
\newcommand{\mNN}{\text{nn}}
|
||||||
\newcommand{\mKNN}{\text{knn}}
|
\newcommand{\mKNN}{\text{knn}}
|
||||||
|
\newcommand{\fPos}[1]{\textbf{pos}(#1)}
|
||||||
|
\newcommand{\fDistance}[2]{\delta(#1, #2)}
|
||||||
|
\newcommand{\fWA}[1]{\text{wa}(#1)}
|
||||||
|
\newcommand{\fDD}[1]{\text{dd}(#1)}
|
||||||
|
\newcommand{\fImp}[1]{\text{imp}(#1)}
|
||||||
|
\newcommand{\fNN}[2]{\text{nn}(#1, #2)}
|
||||||
|
\newcommand{\fLength}[2]{\text{length}(#1, #2)}
|
||||||
|
|
||||||
%\newcommand{\docIBeacon}{iBeacon}
|
%\newcommand{\docIBeacon}{iBeacon}
|
||||||
|
|
||||||
|
|||||||
Reference in New Issue
Block a user