From f8d5449dbcf6c0e34dcb77a5933bcacda9249e9a Mon Sep 17 00:00:00 2001 From: Toni Date: Mon, 11 Jul 2016 17:51:32 +0200 Subject: [PATCH] added smoothing and performance --- competition/tex/chapters/components.tex | 2 +- competition/tex/chapters/introduction.tex | 9 +------ competition/tex/chapters/performance.tex | 32 +++++++++++++++++++++++ competition/tex/chapters/smoothing.tex | 23 ++++++++++++++++ 4 files changed, 57 insertions(+), 9 deletions(-) create mode 100644 competition/tex/chapters/performance.tex create mode 100644 competition/tex/chapters/smoothing.tex diff --git a/competition/tex/chapters/components.tex b/competition/tex/chapters/components.tex index 811e3c2..1037c8a 100644 --- a/competition/tex/chapters/components.tex +++ b/competition/tex/chapters/components.tex @@ -27,4 +27,4 @@ By assuming statistical independence of all sensors, the probability density of \input{chapters/wifi.tex} \input{chapters/stepturn.tex} \input{chapters/graph.tex} - + \input{chapters/smoothing.tex} diff --git a/competition/tex/chapters/introduction.tex b/competition/tex/chapters/introduction.tex index 676a3a0..8082ad8 100644 --- a/competition/tex/chapters/introduction.tex +++ b/competition/tex/chapters/introduction.tex @@ -68,11 +68,4 @@ System setup is very easily and no fingerprinting is required. \input{chapters/components.tex} - - -\begin{itemize} - \item Fixed-lag smoother -\end{itemize} - -\section{Performance Overview} -Wie toll sind wir? kurzer ueberblick der ergebnisse in einer tabelle und paar worte dazu. eventl graphic. +\input{chapters/performance.tex} diff --git a/competition/tex/chapters/performance.tex b/competition/tex/chapters/performance.tex new file mode 100644 index 0000000..6d15ca1 --- /dev/null +++ b/competition/tex/chapters/performance.tex @@ -0,0 +1,32 @@ +\section{Performance Overview} +% all paths we evaluated + \begin{figure} + \input{gfx/paths} + \caption{The four paths that were part of the evaluation. + Starting positions are marked with black circles. + For a better visualisation they were slightly shifted to avoid overlapping.} + %\commentByFrank{font war korrekt, aber die groesse war zu gross im vgl. zu den anderen} + \label{fig:paths} + \end{figure} +% +To give a brief overview of the system's performance we look back at the evaluation provided in \cite{ebner-16}. +Here, 4 distinct walks were conducted within the faculty building (cf. fig. \ref{fig:paths}). +No smoothing was carried out. +We used \SI{7500}{particles} as realization and calculated the weighted arithmetic mean of the particles as state estimation. +The ground truth was measured by recording a timestamp at marked spots on the walking route, similar as described in the competition guidelines. +Starting uniformly distributed, the median error for all conducted walks are listed in table \ref{tbl:errNexus} for the Motorola Nexus 6 and the Samsung Galaxy S5. +Additionally performing a smoothing step, would further improve the results and reduces temporal errors, as shown in \cite{fetzer-16}. +% + \begin{table}[h] + \caption{Median error for all conducted walks.} + \label{tbl:errNexus} + \centering + \begin{tabular}{|l|c|c|c|c|} + \hline + \textbf{Device:} & Path1 & Path2 & Path3 & Path4 \\\hline + Motorola Nexus 6 & \SI{2.62}{\meter} & \SI{2.14}{\meter} & \SI{2.46}{\meter} & \SI{2.75}{\meter} \\\hline + Samsung Galaxy S5 & \SI{ 6.35}{\meter} & \SI{4.21}{\meter} & \SI{5.03}{\meter} & \SI{6.79}{\meter} \\\hline + \end{tabular} + \end{table} + + diff --git a/competition/tex/chapters/smoothing.tex b/competition/tex/chapters/smoothing.tex new file mode 100644 index 0000000..7e68298 --- /dev/null +++ b/competition/tex/chapters/smoothing.tex @@ -0,0 +1,23 @@ +\subsection{Fixed-lag smoothing} + +Within \cite{fetzer-16} we added an additional smoothing step to the localisation procedure. +In contrast to normal filtering, smoothing methods are able to incorporate future measurements instead of just using current and past data. +Therefore, they are able to compute probability distributions in the form of $p(\mStateVec_t \mid \mObsVec_{1:T})$. +Especially interesting for real-time applications is the so-called fixed-lag smoothing. +In fixed-lag smoothing, one tries to estimate the current state, given measurements up to a time $t + \tau$, where $\tau$ is a predefined lag. +By running backwards in time, they are able to remove multimodalities and improve the overall localisation result. +We can distinguish between two different smoothing algorithms: Forward-backward smoothing \cite{doucet2000} and backward simulation \cite{Godsill04:MCS}. +Both perform very similar and are reweighting possible states based on a smoothing transition model. + +The smoothing transition model calculates the probability of being in a state $\vec{q}_{t+1}$ in regard to previous states and the pedestrian's walking behaviour. +Therefore, we compare the distance, angle and height between $\vec{q}_{t+1}$ and $\vec{q}_{t}$ in regard to the measurements gettered at time $t$. +The resulting likelihood is then used for reweighting. + +%By writing +%\begin{equation} +%p(\vec{q}_{t+1} \mid \vec{q}_t, \mObsVec_t)_{\text{step}} = \mathcal{N}(\Delta d_t \mid \mu_{\text{step}}, \sigma_{\text{step}}^2) +%\label{eq:smoothingTransDistance} +%\end{equation} +%we receive a statement about how likely it is to cover a distance $\Delta d_t$ between two states $\vec{q}_{t+1}$ and $\vec{q}_{t}$. +%In the easiest case, $\Delta d_t$ is the euclidean distance between two states. +%The average step length $\mu_{\text{step}}$ is based on the pedestrian's walking speed and $\sigma_{\text{step}}^2$ denotes the step length's variance.