diff --git a/tex/chapters/conclusion.tex b/tex/chapters/conclusion.tex index 8622cec..8143464 100644 --- a/tex/chapters/conclusion.tex +++ b/tex/chapters/conclusion.tex @@ -1,2 +1,6 @@ \section{Conclusion} +\section{Future Work} + \commentByFrank{balance zwischen den einzelnen wahrscheinlichkeiten ist oft ein schmaler grad. wieviel turn erlauben, wieviel auf den pfad zwingen. das verbesern} + \commentByFrank{position der APs wissen ist viel arbeit. vereinfachen durch test-walks auf vorgegebenen pfaden -> numerisch optimieren wo APs sind} + diff --git a/tex/chapters/experiments.tex b/tex/chapters/experiments.tex index 73b3f02..a5480b9 100644 --- a/tex/chapters/experiments.tex +++ b/tex/chapters/experiments.tex @@ -9,6 +9,9 @@ \item Vergleich zum Schluss das neue System mit dem Alten um eine schöne Conclusion der Verbesserungen einzuleiten. \end{itemize} +\commentByFrank{we start with a uniform distribution $\mStateVec_0$} + +\commentByFrank{hinweis auf die verschiedenen geraete (smartphones) und unterschiede, wlan/baro} \commentByFrank{sensorausfall simulieren, z.b. in der mitte, oder auf einer treppe} @@ -23,4 +26,4 @@ mit pfad laeuft es falsch, weil die andere treppe kuerzer zum ziel ist und das w \commentByFrank{zu grosser einfluss vom pfad ist also kein allheilmittel.. kann, wie beim treppenhaus, auch nach hinten los gehen} -\commentByFrank{path1: bad start due to nearby AP and bad parameters (path-loss too high)} +\commentByFrank{path1: bad start due to nearby AP and bad parameters (path-loss too high): high starting errors: median better} diff --git a/tex/chapters/floorplan.tex b/tex/chapters/floorplan.tex index 4fdbfeb..47fc2e1 100644 --- a/tex/chapters/floorplan.tex +++ b/tex/chapters/floorplan.tex @@ -1,53 +1,4 @@ -\section{Floorplan/Path/whatever (RENAME)} +\section{Flooplan - REMOVE} - \subsection{wall avoidance} - \begin{figure} - \includegraphics[angle=-90, width=\columnwidth, trim=15 15 15 15]{floorplan_importance} - \end{figure} - - - \subsection{door detection} - - invert grid, - find k-NN within inversion - calculate centroid, covariance and eigenvalues - when centroid OK and eigenvalues OK: is door/passage - - \commentByFrank { - doors serve 3 purposes: - - reverting the downvote due to "near walls" - - hint for a pathless-movement-model how to leave rooms - - also detecting narrow passages (center the path) - } - - Doors are usually anchored between two (thin) walls and have a normed width. Examining only a limited region - around the door, its surrounding walls describe a flat ellipse with the same center as the door itself. It is thus - possible to detect doors within the floorplan using the PCA \todo{cite}. To describe the surroundings of a door, - we create and inverted version of our grid whereby the contained nodes denote walls and other obstacles. - To decide whether a node within the (non-inverted) grid belongs to a door, we use $k$-NN \todo{cite} to fetch its - $k$ nearest neighbors within the inverted grid. If the distance \todo{equation} between the centroid of those neighbors - and the node-in-question is above certain threshold, the node does not belong to a door. Assuming the distance was fine, - we compare the two eigenvalues $\{e_1, e_2 \mid e_1 > e_2\}$ , determined by the PCA. If their ratio $\frac{e_1}{e_2}$ is above - a certain threshold (flat ellipse) - the node-in-question belongs to a door. Fig. \ref{fig:doorPCA} depicts all three cases where the node is part of a door (left), - the distance between node and k-NN centroid is above the threshold (middle) and the ration between $e_1$ and $e_2$ is below the - threshold (right). - - \begin{figure} - \includegraphics[width=\columnwidth]{door_pca} - \caption{Detect doors within the floorplan using k-NN and PCA. While the white nodes are walkable, the black ones represent walls. The grey node is the one in question.} - \label{fig:doorPCA} - \end{figure} - - - \subsection{pathfinding} - - \begin{figure} - \includegraphics[angle=-90, width=\columnwidth, trim=15 15 15 15]{floorplan_paths} - \end{figure} - - \begin{figure} - \includegraphics[angle=-90, width=\columnwidth, trim=15 15 15 15]{floorplan_dijkstra_heatmap} - \end{figure} diff --git a/tex/chapters/grid.tex b/tex/chapters/grid.tex index 4cf9361..88ef73a 100644 --- a/tex/chapters/grid.tex +++ b/tex/chapters/grid.tex @@ -115,8 +115,8 @@ % \vec{c} = \frac{ \sum_{v_{x,y,z} \in N'} v_{x,y,z} }{k} %\end{equation} - Assuming the distance was fine, we compare the two eigenvalues $\{e_1, e_2 \mid e_1 > e_2\}$ , determined by the PCA. - If their ratio $\frac{e_1}{e_2}$ is above a certain threshold (flat ellipse) + Assuming the distance was fine, we compare the two eigenvalues $\{\lambda_1, \lambda_2 \mid \lambda_1 > \lambda_2\}$ , determined by the PCA. + If their ratio $\frac{\lambda_1}{\lambda_2}$ is above a certain threshold (flat ellipse) the node-in-question belongs to a door or some kind of narrow passage. \begin{figure} @@ -141,6 +141,7 @@ \subsection{path estimation} + \label{sec:pathEstimation} Based on aforementioned assumptions, the final importance for each node is @@ -168,7 +169,7 @@ \begin{split} \text{weight}(v_{x,y,z}, v_{x',y',z'}) = \frac - { \text{dist}(v_{x,y,z}, v_{x',y',z'}) } + { \| v_{x,y,z} - v_{x',y',z'} \| } { \text{stretch}(\text{imp}_{x',y',z'}) } , \end{split} @@ -197,34 +198,76 @@ Based on the previous calculations, we propose two approaches to incorporate the prior knowledge into the transiton model. - During every transition, the first algorithm calculates the centroid $\vec{c}$ of the current sample-set: - \begin{equation} - \vec{c} = \frac - { \sum_{\mStateVec_{t-1}} (\mState_{t-1}^x, \mState_{t-1}^y, \mState_{t-1}^z)^T } - {N} - \end{equation} - 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 used instead. - The resulting node already knows its way to the pedestrian's destination, but is located somewhere - within the deviation of the sample set. After slightly advancing it by a fixed value of about \SI{5}{\meter} - we get a new point outside of the sample-set and closer to the desired destination. - This new reference node serves as a comparison base + \subsubsection{Shortest Path} - \begin{equation} - p(e) = - \end{equation} + Before every transition, the centroid $\vec{c}$ of the current sample-set is calculated: + \todo{summe laesst sich so nicht schreiben. ideen?} + \begin{equation} + \vec{c} = \frac + { \sum_{\mStateVec_{t-1}} (\mState_{t-1}^x, \mState_{t-1}^y, \mState_{t-1}^z)^T } + {N} + \end{equation} + + \newcommand{\pathRef}{v_{\hat{x},\hat{y},\hat{z}}} + 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 used instead. + The resulting node already knows its way to the pedestrian's destination, but is located somewhere + within the deviation of the sample set. After slightly advancing it by a fixed value of about \SI{5}{\meter} + we get a new point outside of the sample-set and closer to the desired destination. + This new reference node $\pathRef$ serves as a comparison base: + + \todo{bessere ideen fuer die schreibweise?} + \begin{equation} + \begin{split} + p(v_{x',y',z'} \mid v_{x,y,z}) + = N(\angle [ v_{x,y,z} v_{x',y',z'} ] \mid \gHead, \sigma_\text{dev}^2) \cdot \alpha \\ + \alpha = + \begin{cases} + 1.0 & \| v_{x',y',z'} - \pathRef \| < \| v_{x,y,z} - \pathRef \| \\ + 0.2 & \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. - - \begin{figure} - \includegraphics[angle=-90, width=\columnwidth, trim=20 19 17 9, clip]{floorplan_dijkstra_heatmap} - \end{figure} + + \subsubsection{Multipath} + + The Dijkstra calculation mentioned in \ref{sec:pathEstimation} already calculated the + cumulative distance $\text{cdst}_{x,y,z}$ to the pedestrian's target for each vertex. + We thus apply the same assumption as above and downvote steps not decreasing + the distance to the destination: + + \begin{equation} + \begin{split} + p(v_{x',y',z'} \mid v_{x,y,z}) + = N(\angle [ v_{x,y,z} v_{x',y',z'} ] \mid \gHead, \sigma_\text{dev}^2) \cdot \alpha \\ + \alpha = + \begin{cases} + 1.0 & \text{cdst}_{x',y',z'} < \text{cdst}_{x,y,z} \\ + 0.2 & \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. + + \begin{figure} + \includegraphics[angle=-90, width=\columnwidth, trim=20 19 17 9, clip]{floorplan_dijkstra_heatmap} + \caption{Heat-Map of visited vertices after several walks using \refeq{eq:transMultiPath}} + \label{fig:multiHeatMap} + \end{figure} \commentByFrank{angular-change probability as polar-plot} \commentByFrank{describe the multi-path version} - - \commentByFrank{describe the single-path version} - \commentByFrank{exp-dist for distance to the path. more distance = less-likely} - \commentByFrank{lambda-factor controls the allowed deviation from the shortest-path} + diff --git a/tex/gfx/grid/grid.eps b/tex/gfx/grid/grid.eps new file mode 100644 index 0000000..e35a24e --- /dev/null +++ b/tex/gfx/grid/grid.eps @@ -0,0 +1,3122 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: cairo 1.14.4 (http://cairographics.org) +%%CreationDate: Mon Feb 8 10:53:53 2016 +%%Pages: 1 +%%DocumentData: Clean7Bit +%%LanguageLevel: 2 +%%BoundingBox: 0 0 360 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pressure + \newcommand{\mHeading}{\theta} \newcommand{\mObsHeading}{\Delta\mHeading} % symbol used for the observation heading \newcommand{\mStateHeading}{\mHeading} % symbol used for the state heading +\newcommand{\mSteps}{s} +\newcommand{\mObsSteps}{\mSteps} + +\newcommand{\mNN}{\text{nn}} +\newcommand{\mKNN}{\text{knn}} + %\newcommand{\docIBeacon}{iBeacon} % for equation references