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added recent filtering.tex

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kazu
2016-04-23 21:24:56 +02:00
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98
tex/chapters/filtering.tex
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@@ -4,10 +4,9 @@
\commentByToni{Bin mir nicht sicher ob wir diese Section überhaupt brauchen. Könnte man bestimmt auch einfach unter Section 3 packen. Aber dann können wir ungestört voneinander schreiben.}
\subsection{Evaluation}
\section{Barometer}
\section{Evaluation}
\subsection{Barometer}
\label{sec:sensBaro}
%
The probability of currently residing on a given floor is evaluated using the smartphone's barometer.
@@ -75,23 +74,88 @@
%
\subsection{Transition}
The transition step depends on random walks on a graph, generated from the buildings floorplan
\todo{cite}. This setup allows only valid movements, as ambient conditions (walls, doors, etc.) are considered.
\section{Transition}
The distribution $p(\mStateVec_{t} \mid \mStateVec_{t-1})$ is sampled via random walks on a graph
$G=(V,E)$, which is generated from the buildings floorplan \todo{FUSION2016}.
$p(\mStateVec_{t} \mid \mStateVec_{t-1})$ is determined by walking along adjacent edges $\mEdgeAB$ connecting
two vertices $\mVertexA, \mVertexB \in V$ until a certain distance $\gDist$ is reached.
Thus, the position of any $\mStateVec$ is represented by the position $\fPos{\mVertexA}$ of the corresponding vertex.
This approach draws only valid movements, as ambient conditions (walls, doors, stairs, etc.) are considered.
Furthermore, we assume the pedestrian's desired destination to be known beforehand. This prior knowledge is evaluated
during the random walk, to favour movements approaching the chosen destination.
To ensure the transition step provides a viable posterior distribution, we include some sensors directly into the transition step.
Adding them to the evaluation instead, would lead to sample impoverishment when using Monte Carlo methods.
\subsection{Step- \& Turn-Detection}
%
Steps and turns are detected using the smartphone's IMU and are implemented as described in \cite{Ebner-15}.
While sampling, to-be-walked edges are not chosen uniformly, but depending on a probability $p(\mEdgeAB)$.
The latter depends on several constraints and recent sensor-readings from the smartphone. Using sensors
directly within the transition step provides a more robust posterior distribution. Adding them to the evaluation
instead, would lead to sample impoverishment due to the used Monte Carlo methods.
\subsection{Pedestrian's Destination}
We assume the pedestrian's desired destination to be known beforehand. This prior knowledge is incorporated
during the random walk using $p(\mEdgeAB)_\text{path}$, which is a simple heuristic, favouring movements (edges)
approaching the chosen destination with a ratio of $0.9:0.1$ over those, departing from the destination
\cite{FUSION2016}. The underlying shortest-path is based on a special distance metric, considering special
architectural facts:
\subsection{Architectural Facts}
To ensure realistic path estimations, we include additional architectural knowledge.
Each vertex's distance from the nearest wall is used to prevent the shortest-path from clinging to walls.
Likewise, his distance from the nearest door (favour ).
\subsection{Step- \& Turn-Detection}
Steps and turns are detected using the smartphone's IMU, implemented as described in \cite{Ebner-15}.
The number of steps detected since the last transition is used to estimate the to-be-walked distance $\gDist$
assuming a fixed step-size with some deviation:
%
\begin{equation}
\gDist = \mObs_{t-1}^{\mObsSteps} \cdot \mStepSize + \mathcal{N}(0, \sigma_{\gDist}^2)
\enspace .
\end{equation}
%
Turn-Detection supplies the magnitude of the detected heading change by integrating the gyroscope's change
since the last transition. Together with some deviation and the state's previous heading, the magnitude is
used to estimate the state's current heading:
%
\begin{equation}
\gHead = \mState_{t}^{\mStateHeading} = \mState_{t-1}^{\mStateHeading} + \mObs_{t-1}^{\mObsHeading} + \mathcal{N}(0, \sigma_{\gHead}^2) \\
\end{equation}
%
During the random walk, edges should satisfy the heading:
%
\begin{equation}
p(\mEdgeAB)_\text{head} = p(\mEdgeAB \mid \gHead) = \mathcal{N} (\angle \mEdgeAB \mid \gHead, \sigma_\text{head}^2)
\enspace .
\label{eq:transHeading}
\end{equation}
%
While the distribution \refeq{eq:transHeading} does not integrate to $1.0$ due to circularity of angular
data, in our case, the normal distribution can be assumed as sufficient for small enough $\sigma^2$.
\subsection{Activity-Detection}
\todo{write}
Additionally we perform a simple activity detection for the pedestrian, able to distinguish between
standing, walking, walking stairs upwards and downwards. Likewise, this knowledge
is evaluated when walking the grid: Edges $\mEdgeAB$ matching the currently detected
activity are favoured using $p(\mEdgeAB)_\text{act} = 0.8$ and $0.2$ otherwise.
\begin{equation}
p(\mEdgeAB)_\text{act} =
\begin{cases}
0.8 & \text{stairs\_up}, \fPos{\mVertexB}_z > \fPos{\mVertexA}_z \\
0.2 & \text{stairs\_up}, \fPos{\mVertexB}_z \le \fPos{\mVertexA}_z \\
\cdots
\end{cases}
\end{equation}
\commentByFrank{das switch ist wahrscheinlich unnoetig und der text reicht}
\commentByFrank{hier passen die sachen vom lukas. kurze beschreibung der beiden geschaetzten verteilungen etc}

3
tex/misc/functions.tex
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@@ -57,6 +57,9 @@
\newcommand{\mVertexB}{v_j}
\newcommand{\mEdgeAB}{e_{i,j}}
\newcommand{\mVertexDest}{v_\text{dest}}
\newcommand{\gDist}{d_\text{step}}
\newcommand{\gHead}{\theta_\text{walk}}
\newcommand{\mUsePath}{\kappa}