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added missing legend to gfx

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fixed some typos and refactored some sentences
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FrankE
2016-02-15 17:11:03 +01:00
parent ac542ba634
commit 54ab3d8dbe
9 changed files with 124 additions and 75 deletions
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37
tex/chapters/grid.tex
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@@ -2,8 +2,8 @@
\label{sec:trans}
\newcommand{\spoint}{l}
\newcommand{\gHead}{\theta}
\newcommand{\gDist}{d}
\newcommand{\gHead}{\theta_\text{walk}}
\newcommand{\gDist}{d_\text{walk}}
To sample only transitions that are actually feasible
within the environment, we utilize a \SI{20}{\centimeter}-gridded graph
@@ -26,8 +26,8 @@
Stairs are defined using three points $\vec{\spoint}_1, \vec{\spoint}_2, \vec{\spoint}_3 \in \R^3$ whereby the segment
$[ \vec{\spoint}_1 \vec{\spoint}_2 ]$ describes the starting-edge, and $[ \vec{\spoint}_2 \vec{\spoint}_3 ]$ the stair's direction
(see fig. \ref{fig:gridStairs}). The grid-vertices corresponding to the starting-edge are determined using an intersection of
the segment $[ \vec{\spoint}_1 \vec{\spoint}_2 ]$ with the bounding-box for each vertex.
\commentByToni{Welche Bounding-box? hab ich was ueberlesen?}
the segment $[ \vec{\spoint}_1 \vec{\spoint}_2 ]$ with the \SI{20}{\centimeter} bounding-box around each vertex.
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.
@@ -44,17 +44,17 @@
.
\end{align}
%
During a walk, each edge has an assigned probability $p(e)$ which depends on a chosen implementation.
This probability describes aspects such as the likelihood for walking into the edge's direction $\angle e$
given the current heading $\gHead$. Furthermore, we will incorporate additional prior knowledge to
favour some vertices/edges. For each single step on the graph, we calculate $p(e)$ for all available edges,
and, hereafter, randomly draw the to-be-walked edge depending on those probabilities. The random walk ends,
as soon as the distance $d$ is reached. The latter depends on the number of detected steps
$\mObsSteps$ and assumes an average step-size of \SI{0.7}{\meter}.
\commentByToni{steps steps... Allgemein finde ich den Absatz etwas komisch. erst wird p(e) eingefuehrt, dann kommt auf einmal prior knowledge und dann wie wir p(e) berechnen? reihenfolge passt da nicht so und macht es undeutlch. Hier soll doch eigentlich nur erklaert werden wie wir ueber den graphen laufen. egal ob prior oder nicht. laufen muessen wir ja}
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}.
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 $\gHead$:
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.}
@@ -82,9 +82,8 @@
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 weight between two vertices, needed by the shortest-path algorithm.
\commentByToni{The edge's weight?? vielleicht?}
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.
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.
@@ -144,7 +143,7 @@
Like before, we apply a distribution based on the distance from the nearest door to determine
an importance-factor for each node:
%
\commentByFrank{distanzrechnung: formel}
\commentByFrank{distanzrechnung: formel ok?}
\begin{equation}
\text{dd}_{x,y,z} = 0.8 \cdot \mathcal{N}( \| \vec{c} - v_{x,y,z} \| \mid 0.0, 1.0 )
\end{equation}
@@ -271,7 +270,7 @@
\subsubsection{Multipath}
The Dijkstra calculation 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.
We thus apply the same assumption as above and downvote grid-steps not decreasing
the distance to the destination: