From 9d3efd10c7d991586dae60088d78bf4b7f8fe5ed Mon Sep 17 00:00:00 2001 From: kazu Date: Thu, 5 May 2016 10:24:00 +0200 Subject: [PATCH] changed chapter order (filtering) --- tex/chapters/filtering.tex | 160 ++++++++++++++++++------------------- 1 file changed, 79 insertions(+), 81 deletions(-) diff --git a/tex/chapters/filtering.tex b/tex/chapters/filtering.tex index 5065777..dbb73b8 100644 --- a/tex/chapters/filtering.tex +++ b/tex/chapters/filtering.tex @@ -1,84 +1,82 @@ -%\section{Filtering} -% -% \label{sec:filtering} -% -% \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.} -% -\section{Evaluation} +\section{Filtering} - \commentByFrank{brauchen wir hier noch was (kurze einleitung) oder passt das so?} + \commentByFrank{eval und transition tauschen von der reihenfolge?} - \subsection{Barometer} - \label{sec:sensBaro} - % - The probability of currently residing on a floor is evaluated using the smartphone's barometer. - Environmental influences are circumvented by using relative pressure values instead of absolute ones. - To reduce the impact of noisy sensors, we calculate the average $\overline{\mObsPressure}$ of several - sensor readings, carried out while the pedestrian chooses his destination. This average serves as relative base - for all future measurements. Likewise, we estimate the sensor's uncertainty $\sigma_\text{baro}$ for later use - within the evaluation step. - - In order to evaluate the relative pressure readings, we need a prediction to compare them with. Therefore, each - transition from $\mStateVec_{t-1}$ to $\mStateVec_t$ estimates the state's relative pressure prediction - $\mStatePressure$ by tracking every height-change ($z$-axis): - % - \begin{equation} - \mState_{t}^{\mStatePressure} = \mState_{t-1}^{\mStatePressure} + \Delta z \cdot b - ,\enskip - \Delta z = \mState_{t-1}^{z} - \mState_{t}^z - ,\enskip - b \in \R - \enspace . - \label{eq:baroTransition} - \end{equation} - % - In \refeq{eq:baroTransition}, $b$ denotes the common pressure change in $\frac{\text{hPa}}{\text{m}}$. - The evaluation step for time $t$ compares every predicted relative pressure $\mState_t^{\mStatePressure}$ with the observed - one $\mObs_t^{\mObsPressure}$ using a normal distribution with the previously estimated $\sigma_\text{baro}$: - % - \begin{equation} - p(\mObsVec_t \mid \mStateVec_t)_\text{baro} = \mathcal{N}(\mObs_t^{\mObsPressure} \mid \mState_t^{\mStatePressure}, \sigma_\text{baro}^2) \enspace. - \label{eq:baroEval} - \end{equation} - % - % - % - \subsection{Wi-Fi \& iBeacons} - % - The smartphone's \docWIFI{} and \docIBeacon{} component provides an absolute location estimation by - measuring the signal-strengths of nearby transmitters. The positions of detected \docAP{}s (\docAPshort{}) and \docIBeacon{}s - are known beforehand. Using the wall-attenuation-factor signal strength prediction model \cite{Ebner-15}, we are able to - compare each measurement with a corresponding estimation. To infer this estimation, the prediction model - uses the 3D distance $d$ and the number of floors $\Delta f$ between transmitter and the state-in-question $\mStateVec$: - % - \begin{equation} - P_r(d, \Delta f) = \mTXP - 10 \mPLE \log_{10}{\frac{\mMdlDist}{\mMdlDist_0}} + \Delta{f} \mWAF \enspace , - \label{eq:waf} - \end{equation} - % - In \refeq{eq:waf}, there are three more parameters per \docAPshort{}. The signal-strength $\mTXP$ measurable at a distance - $\mMdlDist_0$ (usually \SI{1}{\meter}), a path-loss exponent $\mPLE$ describing the transmitter's environment and the attenuation - per floor $\mWAF$. - To reduce the system's setup time, we use the same three values for all \docAP{}s at the cost of accuracy. - All parameters are chosen empirically. Further details on how to determine this parameters exactly, - can be found in \cite{PathLossPredictionModelsForIndoor}. + \subsection{Evaluation} + + \commentByFrank{brauchen wir hier noch was (kurze einleitung) oder passt das so?} + + \subsubsection{Barometer} + \label{sec:sensBaro} + % + The probability of currently residing on a floor is evaluated using the smartphone's barometer. + Environmental influences are circumvented by using relative pressure values instead of absolute ones. + To reduce the impact of noisy sensors, we calculate the average $\overline{\mObsPressure}$ of several + sensor readings, carried out while the pedestrian chooses his destination. This average serves as relative base + for all future measurements. Likewise, we estimate the sensor's uncertainty $\sigma_\text{baro}$ for later use + within the evaluation step. - The same holds for the \docIBeacon{} component, except $\mTXP$, - which is broadcasted by each beacon. However, as \docIBeacon{}s cover only a small area, $\mPLE$ is usually much smaller compared - to the one needed for \docWIFI{}. + In order to evaluate the relative pressure readings, we need a prediction to compare them with. Therefore, each + transition from $\mStateVec_{t-1}$ to $\mStateVec_t$ estimates the state's relative pressure prediction + $\mStatePressure$ by tracking every height-change ($z$-axis): + % + \begin{equation} + \mState_{t}^{\mStatePressure} = \mState_{t-1}^{\mStatePressure} + \Delta z \cdot b + ,\enskip + \Delta z = \mState_{t-1}^{z} - \mState_{t}^z + ,\enskip + b \in \R + \enspace . + \label{eq:baroTransition} + \end{equation} + % + In \refeq{eq:baroTransition}, $b$ denotes the common pressure change in $\frac{\text{hPa}}{\text{m}}$. + The evaluation step for time $t$ compares every predicted relative pressure $\mState_t^{\mStatePressure}$ with the observed + one $\mObs_t^{\mObsPressure}$ using a normal distribution with the previously estimated $\sigma_\text{baro}$: + % + \begin{equation} + p(\mObsVec_t \mid \mStateVec_t)_\text{baro} = \mathcal{N}(\mObs_t^{\mObsPressure} \mid \mState_t^{\mStatePressure}, \sigma_\text{baro}^2) \enspace. + \label{eq:baroEval} + \end{equation} + % + % + % + \subsubsection{Wi-Fi \& iBeacons} + % + The smartphone's \docWIFI{} and \docIBeacon{} component provides an absolute location estimation by + measuring the signal-strengths of nearby transmitters. The positions of detected \docAP{}s (\docAPshort{}) and \docIBeacon{}s + are known beforehand. Using the wall-attenuation-factor signal strength prediction model \cite{Ebner-15}, we are able to + compare each measurement with a corresponding estimation. To infer this estimation, the prediction model + uses the 3D distance $d$ and the number of floors $\Delta f$ between transmitter and the state-in-question $\mStateVec$: + % + \begin{equation} + P_r(d, \Delta f) = \mTXP - 10 \mPLE \log_{10}{\frac{\mMdlDist}{\mMdlDist_0}} + \Delta{f} \mWAF \enspace , + \label{eq:waf} + \end{equation} + % + In \refeq{eq:waf}, there are three more parameters per \docAPshort{}. The signal-strength $\mTXP$ measurable at a distance + $\mMdlDist_0$ (usually \SI{1}{\meter}), a path-loss exponent $\mPLE$ describing the transmitter's environment and the attenuation + per floor $\mWAF$. + To reduce the system's setup time, we use the same three values for all \docAP{}s at the cost of accuracy. + All parameters are chosen empirically. Further details on how to determine this parameters exactly, + can be found in \cite{PathLossPredictionModelsForIndoor}. + + The same holds for the \docIBeacon{} component, except $\mTXP$, + which is broadcasted by each beacon. However, as \docIBeacon{}s cover only a small area, $\mPLE$ is usually much smaller compared + to the one needed for \docWIFI{}. + + As transmitters are assumed to be statistically independent, the overall probability to measure their predictions at a given location is: + % + \begin{equation} + \mProb(\mObsVec_t \mid \mStateVec_t)_\text{wifi} = + \prod\limits_{i=1}^{n} \mathcal{N}(\mRssi_\text{wifi}^{i} \mid P_{r}(\mMdlDist_{i}, \Delta{f_{i}}), \sigma_{\text{wifi}}^2) \enspace . + \label{eq:wifiTotal} + \end{equation} + + - As transmitters are assumed to be statistically independent, the overall probability to measure their predictions at a given location is: - % - \begin{equation} - \mProb(\mObsVec_t \mid \mStateVec_t)_\text{wifi} = - \prod\limits_{i=1}^{n} \mathcal{N}(\mRssi_\text{wifi}^{i} \mid P_{r}(\mMdlDist_{i}, \Delta{f_{i}}), \sigma_{\text{wifi}}^2) \enspace . - \label{eq:wifiTotal} - \end{equation} - - - -\section{Transition} -\label{sec:transition} + \subsection{Transition} + \label{sec:transition} The transition-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 \cite{Ebner-16}. @@ -94,14 +92,14 @@ \commentByFrank{ist das verstaendlich oder schon zu kurz?} - \subsection{Pedestrian's Destination} + \subsubsection{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 his chosen destination with a ratio of $0.9:0.1$ over those, departing from the destination \cite{Ebner-16}. The underlying shortest-path uses Dijkstra's algorithm with special weight (distance) metric, considering special architectural facts: - \subsection{Architectural Facts} + \subsubsection{Architectural Facts} Normally, the shortest-path calculated for a narrow grid would stick unnaturally close to obstacles like walls. To ensure realistic (human like) path estimations, we include architectural knowledge within Dijkstra's edge-weight function \cite{Ebner-16}: Each vertex's distance from the nearest wall is used to artificially increase the edge-weight and thus prevent the shortest-path @@ -109,7 +107,7 @@ and favoured by decreasing their edge-weight. - \subsection{Step- \& Turn-Detection} + \subsubsection{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 $\gDist$ by assuming a fixed step-size with some deviation: @@ -138,7 +136,7 @@ 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} + \subsubsection{Activity-Detection} Additionally we perform a simple activity detection for the pedestrian, able to distinguish between several actions $\mObsActivity \in \{ \text{unknown}, \text{standing}, \text{walking}, \text{stairs\_up}, \text{stairs\_down} \}$.