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tex v2 - without experiments

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toni
2017-05-10 23:33:01 +02:00
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4
code/filter/Logic.h
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@@ -153,6 +153,10 @@ struct PFTransSimple : public K::ParticleFilterTransition<MyState, MyControl>{
for (int i = 0; i < Settings::numParticles; ++i) {
K::Particle<MyState>& p = particles[i];
// update the baromter
float deltaZ_cm = p.state.positionOld.inMeter().z - p.state.position.inMeter().z;
p.state.relativePressure += deltaZ_cm * 0.105f;
double diffHeight = p.state.position.inMeter().z + height_m.draw();
double newHeight_cm = p.state.position.z_cm;
if(diffHeight > 9.1){

2
tex/bare_conf.tex
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@@ -201,6 +201,8 @@
\input{chapters/system}
\input{chapters/kld}
\input{chapters/method}
\input{chapters/experiments}

4
tex/chapters/abstract.tex
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@@ -6,9 +6,9 @@ The sample impoverishment problem can therefore be described as a too small part
Restrictive transition models, as they are used in indoor localisation, also enhance this effect significantly.
However, an accurate position estimation requires a certain degree of focus and thus behaves contrary to the need of diversity.
The proposed method is able to deal with the trade-off between the need of diversity and focus by deploying an interacting multiple model particle filter (IMMPF) for jump Markov non-linear systems.
Therefore we propose a new method that is able to deal with the trade-off between the need of diversity and focus by deploying an interacting multiple model particle filter (IMMPF) for jump Markov non-linear systems.
We combine two similar particle filters using a non-trivial Markov switching process, depending upon the Kullback-Leibler divergence and a Wi-Fi quality factor. The main benefit of this
approach is the easy adaptation to other localization approaches based on particle filters.
approach is an easy adaptation to other localisation approaches based on particle filters.
%One with a very restrictive transition scheme, providing very accurate results. The other with more flexible and simple dynamics, resulting in a higher sample diversity.

27
tex/chapters/conclusion.tex
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@@ -1,20 +1,13 @@
\section{Conclusion}
In this work we presented an approach for mixing two different localisation schemes using an IMMPF and a non-trivial Markov switching process, which is easy to adapt to many existing systems.
By mixing two particle sets based upon the Kullback-Leibler divergence and a Wi-Fi quality factor, we were able to satisfy the need of diversity and focus to recover from sample impoverishment in context of indoor localisation.
It was shown, that the here presented approach is able to improve the robustness, while keeping the error low.
However, in some rare situations given bad Wi-Fi readings we were not able to increase the results as usual.
This requires further investigations regarding the Wi-Fi quality factor.
In this work we presented .. which is easy to adapt to many existing localisation systems.
combining different filter shemes using an IMMPF
enables us to combine beliebie transition models.
Looking at the results / experiments
we were able to satisfy the need of diversity and focues to reduce the recover from sample impoverishment in context of indoor localization.
The benefits of our approach demonstrated
This further
future work completely different localisation approaches, not only transitions.
more then 2 filters
a more advanced wi-fi quality factor
incorporating a smoothing filter as mode, so we are able to draw new particles from a smoothed particle set. ..
Finally, the possibility of combining different localisation models enables many new approaches and techniques.
By incorporating completely different modes, not only transitions, the robustness and accuracy can be further increased.
This would additionally allow an on-the-fly switching between sensor models, e.g. different signal strength methods.
Such a modular solution could be able to fit any environment and thus form a highly flexible and adjustable localisation system.
However, adjusting the Markov switching process to any number of modes is no easy task and therefore requires intensive future work.

132
tex/chapters/experiments.tex
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@@ -5,51 +5,41 @@
\begin{figure}
\centering
\input{gfx/eval/paths.tex}
\caption{Lorem Ipsum is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry's standard dummy }
\caption{The three paths that were part of the experiments. Starting positions are marked with black circles. The red squares illustrate the \docWIFI{} quality in this sector. The intensity of red indicates a low coverage and thus a bad quality for localisation.}
\label{fig:paths}
\end{figure}
%
%Gebäude
All upcoming experiments were carried out on four floors (0 to 3) of a \SI{77}{m} x \SI{55}{m} sized faculty building.
It includes several staircases and elevators and has a ceiling height of about \SI{3}{m}.
Nevertheless, the grid was generated for the complete campus and thus outdoor areas like the courtyard are also walkable.
As Wi-Fi is attenuated by obstacles and walls, it does not provide a consistent quality over the complete building.
In fig. \ref{fig:paths} we illustrate the quality obtained by the wall attenuation factor model presented earlier.
Here, the intensity of red indicates a low coverage and thus a bad quality for localisation.
To obtain this information we interpolated the Wi-Fi quality factor given by all test walks using $l_{\text{max}} = \SI{-75}{\dBm}$ and $l_{\text{min}} = \SI{-90}{\dBm}$.
As mentioned before, we omit any time-consuming calibration processes and use the same values for all access-points. That would be $P_{0_{\text{wifi}}} = \SI{-46}{\dBm}, \mPLE_{\text{wifi}} = \SI{2.7}{}, \mWAF_{\text{wifi}} = \SI{8}{\dB}$.
The position of the access-points (about five per floor) is known beforehand.
Due to legal terms, we are not allowed to depict their positions and therefore omit this information within the figures.
The gridded graph was generated for the complete campus and thus outdoor areas like the courtyard are also walkable.
As \docWIFI{} is attenuated by obstacles and walls, it does not provide a consistent quality over the complete building.
To get an idea about the \docWIFI{} quality, we interpolated the \docWIFI{} quality factor $q(\mObsVec_t^{\mRssiVec_\text{wifi}})$, with $l_{\text{max}} = \SI{-75}{\dBm}$ and $l_{\text{min}} = \SI{-90}{\dBm}$, between some measurements using a nearest-neighbour procedure.
In fig. \ref{fig:paths} the resulting colourized floorplan is illustrated.
Here, the intensity of red indicates a low signal strength and thus a bad quality for localising a pedestrian within this area using \docWIFI{}.
% gewählte parameter (auch mal die optimieren wifi parameter testen)
%Pfade
We arranged three distinct walks (see also fig. \ref{fig:paths}).
The measurements for the walks were recorded using a Motorola Nexus 6 at 2.4 GHz band only.
The computation was done offline as described in algorithm \ref{fig:paths}.
For each walk we deployed $50$ MC runs using 5000 Particles for each mode.
Instead of an initial position and heading, all walks start with a uniform distribution (random position and heading) as prior.
For the filtering we used $\sigma_\text{wifi} = 8.0$ as uncertainties, both growing with each measurement's age.
While the pressure change was assumed to be \SI{0.105}{$\frac{\text{\hpa}}{\text{\meter}}$}, all other barometer-parameters are determined automatically.
The step size $\mStepSize$ for the transition was configured to be \SI{70}{\centimeter} with an allowed derivation of \SI{10}{\percent}.
The heading deviation was set to \SI{25}{\degree}.
The pedestrian's position (state) was estimated using the weighted arithmetic mean of
the particle set.
For each walk we deployed $50$ runs using 5000 particles for each mode.
Instead of an initial position and heading, all walks start with a uniform distribution (random position and heading) as prior $q_1$.
In the graph-based transition of the dominant filter, the to-be-walked distance is given by the number of steps using a step size of \SI{70}{\centimeter} with an allowed deviation of \SI{10}{\percent}.
The deviation for the walking direction was set to \SI{25}{\degree}.
Since the simple transition randomly scatters particles within a specific range, we choose a covariance matrix that allows a variance of \SI{200}{\centimeter} in $x$- and $y$-direction for the multivariate normal distribution.
Here, floors were changed by deploying a discrete distribution for every floor level, providing a chance of \SI{27}{\percent} for changing one floor and \SI{5}{\percent} for two floors in a particular $z$-direction.
% wie für die kld gezogen? begründen warum wir nun keine parzenschätzung machen (weil ähnliche ergebnisse)
To calculate \eqref{equ:KLD} and thus the Kullback-Leibler divergence, we need to sample densities from both modes likewise.
The grid is suitable for this purpose.
However, sampling at any vertex $\mVertexA$ of the grid, given just a set of random variables (particles), is not the easiest task.
We need to estimate the posterior distribution given by the respective particle sets.
A common way is to deploy a kernel density estimation using a Gaussian distribution as kernel.
The density of a specific point $\hat\mStateVec_{t} = \fPos{\mVertexA}$ is then given by
%
\begin{equation}
p(\hat\mStateVec_{t} \mid m_t, \mObsVec_{1:t}) = \sum_{i=1}^{N_{m_t}} \mathcal{N}(d^i_{\text{KL}} \mid 0, \sigma_{\text{KL}})
\enspace ,
\end{equation}
%
while $d^i_{\text{KL}}$ is the euclidean distance between the considered point's $\hat\mStateVec_{t}$ and all particles $\fPos{\vec{X}_t^{i,m_t}}$ of the mode. The variance $\sigma_{\text{KL}}$ is set to \SI{1}{m}.
It is well known, that the computation of the kernel density estimation is rather slow, thus we also used a much simpler estimation by assuming a multivariate Gaussian distribution for both modes.
Here, the mean is given by weighted arithmetic mean of the particles and the variance is defined by the sample covariance matrix.
Calculating a meaningful $D_{\text{KL}}$, both estimation methods performed almost identical and therefore we used the multivariate Gaussian distribution for both modes with $\lambda = 0.03$ for the upcoming experimental discussion.
We omit any time-consuming calibration processes and therefore use the same parameters for all \docWIFI{} access-points, similar to \cite{Ebner-15}.
The position of the access-points (about five per floor) is known beforehand.
Due to legal terms, we are not allowed to depict their positions and therefore omit this information within the figures.
To model the uncertainty of the \docWIFI{}, a white Gaussian noise with $\sigma_\text{wifi} = \SI{8}{\dBm}$, growing with each measurement's age, was chosen.
The pressure change was assumed to be \SI{0.105}{$\frac{\text{\hpa}}{\text{\meter}}$}, all other barometer-parameters are determined automatically as described in \cite{Fetzer2016OMC}.
As expected before, the kernel density estimation in \eqref{eq:KDE} and the much simpler multivariate normal distribution estimation are both producing similar results in context of the divergence measure given in \eqref{equ:KLD}.
For this reason and because of the faster processing time, especially in context of calculating on a smartphone, we estimated the posteriors of the modes by approximating a multivariate normal distribution (cf. section \ref{sec:divergence}).
For calculating \eqref{equ:KLD} we set $\lambda = 0.03$ based on best practices.
Finally, the pedestrian's most likely position (state) was then estimated using the weighted arithmetic mean of the particle set.
% ground truth
The ground truth is measured by recording a timestamp at marked spots on the walking route. When passing a marker, the pedestrian clicked a button on the smartphone application.
@@ -65,24 +55,24 @@ The approximation error is then calculated by comparing the interpolated ground
\centering
\input{gfx/eval/path3.tex}
\input{gfx/eval/path3-kld.tex}
\caption{Lorem Ipsum is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry's standard dummy }
\caption{Exemplary results on path 3 for the common particle filter using the graph-based (red) or simple transition model (blue) and our IMMPF approach (green). The Kullback-Leibler divergence $D_{\text{KL}}$ between the standalone filters (purple) proves itself as a good indicator, if one filter gets stuck or loses track.}
\label{fig:path3}
\end{figure}
%
At first, we discuss the results of path 3, starting at the left-hand side of the building.
Exemplary estimation results, using the modes standalone and combined within the IMMPF, can be seen in fig. \ref{fig:path3}.
As mentioned above, every run of a walk starts with a uniform distribution as prior.
Due to a low Wi-Fi coverage at the starting point, the pedestrian's position is falsely estimated into a room instead of the corridor.
All three filters are able to overcome this false detection.
However, the common particle filter (red) gets then indissoluble captured within a room, because of its restrictive behaviour and the aftereffects of the initial Wi-Fi readings.
It provides an \SI{75}{\percent}-quantil of $\tilde{x}_{0.75} = \SI{3884}{\centimeter}$ and got captured in \SI{100}{\percent} of all runs.
As expected and discussed earlier, the simple transition (blue) is less prone to bad observations and provides not so accurate, but very robust results of $\tilde{x}_{0.75}= \SI{809}{\centimeter}$ and a standard deviation over all results of $\bar{\sigma} = \SI{529}{\centimeter}$.
It provides an \SI{75}{\percent}-quantil of $\tilde{x}_{75} = \SI{3884}{\centimeter}$ and got captured in \SI{100}{\percent} of all runs.
As expected and discussed earlier, the simple transition (blue) is less prone to bad observations and provides not so accurate, but very robust results of $\tilde{x}_{75}= \SI{809}{\centimeter}$ and a standard deviation over all results of $\bar{\sigma} = \SI{529}{\centimeter}$.
Looking at $D_{\text{KL}}$ over time confirms our assumption made in section \ref{sec:immpf}.
It is clearly visible, that the Kullback-Leibler divergence between both modes (purple) is a very good indicator to observe, if the dominant filter gets stuck or loses track.
The graph-based filter (red) gets stuck and is not able to recover, starting on from this point, the Kullback-Leibler divergence $D_{\text{KL}}$ (purple) further increases due to the growing distance between both filters (blue and red).
It is clearly visible, that this divergence between both filters is a very good indicator to observe, if a filter gets stuck or loses track.
Following, the IMMPF (green) results in a very natural and straight path estimation and a low $D_{\text{KL}}$ between modes and no sticking.
The benefits of mixing both filtering schemes within the scenario of path 3 are thus obvious.
The IMMPF filters with an error of $\tilde{x}_{0.75} = \SI{667}{\centimeter}$ and $\bar{\sigma} = \SI{558}{\centimeter}$.
The IMMPF filters with an error of $\tilde{x}_{75} = \SI{667}{\centimeter}$ and $\bar{\sigma} = \SI{558}{\centimeter}$.
% zeigen das schlechtes wi-fi (zu hohe diversity) behoben wird.
% bild: lauf auf der rechten seite des gebäudes zeige mit und ohne wifi faktor (schlechtes wifi einzeichnen)
@@ -90,7 +80,7 @@ The IMMPF filters with an error of $\tilde{x}_{0.75} = \SI{667}{\centimeter}$ an
\centering
\input{gfx/eval/path2.tex}
\input{gfx/eval/path2-wifi-quality.tex}
\caption{Lorem Ipsum is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry's standard dummy }
\caption{Comparison of the estimation results on path 2 with (green) and without (red) the Wi-Fi quality factor in the Markov transition matrix. The low Wi-Fi quality and thus high errors between the \SI{80}{th} and \SI{130}{th} second are caused by the high attenuation and low signal coverage inside the zig-zag stairwell on the building's backside.}
\label{fig:path2}
\end{figure}
%
@@ -99,11 +89,11 @@ Here, the overall Wi-Fi quality is rather low, especially in the zig-zag stairwe
Path 2 starts in the second floor, walking town the centred stairs into the first floor, then making a right turn and walking the stairs down to zero floor, from there we walk back to second floor using the zig-zag stairwell and after finally crossing a room we are back at the start.
This is a very challenging scenario, at first the estimation got stuck on the first floor in a room's corner and after that the Wi-Fi is highly attenuated.
Looking at fig. \ref{fig:path2}, one can observe the impact of the Wi-Fi quality factor within the Markov transition matrix.
Without it, the position estimation (red) is drifting in the courtyard, missing the stairwell and producing high errors between \SI{80}{th} and \SI{130}{th} second.
Without it, the position estimation (red) is drifting in the courtyard, missing the stairwell and producing high errors between the \SI{80}{th} and \SI{130}{th} second.
As described before, the bad Wi-Fi readings are causing $D_{\text{KL}}$ to grow.
It follows that the accurate dominant filter draws new particles from the uncertain support and therefore worsen the position estimation.
In this scenario it is cold comfort that the system is able to recover thanks to its high diversity during situations with uncertain measurements.
Only by adding the Wi-Fi quality factor the system is able to improve the approximated path (green) and the overall estimation results from $\tilde{x}_{0.75} = \SI{1278}{\centimeter}$ with $\bar{\sigma} = \SI{948}{\centimeter}$ to $\tilde{x}_{0.75} = \SI{811}{\centimeter}$ with $\bar{\sigma} = \SI{340}{\centimeter}$.
Only by adding the Wi-Fi quality factor the system is able to improve the approximated path (green) and the overall estimation results from $\tilde{x}_{75} = \SI{1278}{\centimeter}$ with $\bar{\sigma} = \SI{948}{\centimeter}$ to $\tilde{x}_{75} = \SI{953}{\centimeter}$ with $\bar{\sigma} = \SI{543}{\centimeter}$.
However, this is far from perfect and in some cases ($\sim \SI{9}{\percent}$) the quality factor was not able to prevent the estimation to drift in the courtyard.
This solely happened when particles were sampled directly onto the courtyard while changing from first to zero floor.
Those particles then received a high weight due to the attenuated measurements, causing a weight degeneracy.
@@ -115,43 +105,49 @@ Adapting the bounds $l_{\text{max}}$ and $l_{\text{min}}$ of the quality factor
\centering
\input{gfx/eval/path1.tex}
\input{gfx/eval/path1-time.tex}
\caption{Lorem Ipsum is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry's standard dummy }
\caption{Estimation results and error development while walking alongside path 1. In \SI{20}{\percent} of cases, the the graph-based particle filter failed to detect the first floor change. Therefore, a good (blue) and a bad (red) result are shown. The here presented approach (green) never lost track.}
\label{fig:path1}
\end{figure}
%
An exemplary result for path 1 is illustrated in fig. \ref{fig:path1}.
The path starts on the first floor and finishes on the third after walking two straight stairs.
Using the grid-based particle filter for localisation, we were able to provide an very accurate path (blue) in \SI{80}{\percent} of the MC runs providing $\tilde{x}_{0.75} = \SI{526}{\centimeter}$ with $\bar{\sigma} = \SI{316}{\centimeter}$.
Using the graph-based particle filter for localisation, we were able to obtain a very accurate path (blue) in \SI{80}{\percent} of the runs providing $\tilde{x}_{75} = \SI{526}{\centimeter}$ with $\bar{\sigma} = \SI{316}{\centimeter}$.
Due to a lack of particles near the stairs, the other \SI{20}{\percent} failed to detect the first floor change (red).
Using our approach (green), we were able detect all floor changes and thus never lost track.
It performs with $\tilde{x}_{0.75} = \SI{544}{\centimeter}$ and $\bar{\sigma} = \SI{281}{\centimeter}$.
It performs with $\tilde{x}_{75} = \SI{544}{\centimeter}$ and $\bar{\sigma} = \SI{281}{\centimeter}$.
Those very similar estimation results confirm the efficiency of the mixing and how it is able to keep the accuracy while providing a higher robustness against failures.
\todo{mehr die ergebnisse von bild 5 diskutieren. an manchen stellen verlieren wir genauigkeit, an anderen wird es besser.}
% gegenüberstellung aller pfade und werte in tabelle
\definecolor{header}{rgb}{.8, .8, .8}
\begin{table}
\caption{Median error for all conducted walks.}
\label{tbl:err}
\centering
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}
\hline
& \multicolumn{3}{c}{Path 1} & \multicolumn{3}{|c|}{Path 2} & \multicolumn{3}{|c|}{Path 3}\\
\hline
& $\bar{x}$ & $\bar{\sigma}$ & $\tilde{x}_{0.75}$ & $\bar{x}$ & $\bar{\sigma}$ & $\tilde{x}_{0.75}$ & $\bar{x}$ & $\bar{\sigma}$ & $\tilde{x}_{0.75}$ \\
\hline
PF_{\text{grid}}& $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ \\
\hline
PF_{\text{simple}}& $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ \\
\hline
IMMPF & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ & $x$ \\
\hline
\end{tabular}
\end{table}
% \begin{table}
% \caption{Resulting Errors for all conducted walks in meter.}
% \label{tbl:err}
% \centering
% \scalebox{0.93}{
% \begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}
% \hline
% & \multicolumn{3}{c|}{Path 1} & \multicolumn{3}{c|}{Path 2} & \multicolumn{3}{c|}{Path 3}\\
% \hline
% & $\bar{x}$ & $\bar{\sigma}$ & $\tilde{x}_{75}$ & $\bar{x}$ & $\bar{\sigma}$ & $\tilde{x}_{75}$ & $\bar{x}$ & $\bar{\sigma}$ & $\tilde{x}_{75}$ \\
% \hline
% $\text{PF}_{\text{graph}}$ & $4.0$ & $3.2$ & $5.3$ & $8.2$ & $4.0$ & $10.7$ & $30.3$ & $12.8$ & $38.8$ \\
% \hline
% $\text{PF}_{\text{simple}}$ & $4.9$ & $2.8$ & $6.2$ & $7.3$ & $2.9$ & $9.4$ & $6.8$ & $5.4$ & $8.1$ \\
% \hline
% IMMPF & $4.2$ & $2.8$ & $5.4$ & $7.7$ & $5.4$ & $9.5$ & $6.3$ & $5.6$ & $6.7$ \\
% \hline
% \end{tabular}
% }
% \end{table}
An overview of all localisation results can be seen in table \ref{tbl:err}.
Again, it should be noted that the localisation system used for the experiments is very basic and can be seen as a slimmed version of our previous works \cite{}.
%An overview of all localisation results can be seen in table \ref{tbl:err}.
The here presented walks were selected because they fail in some way using a restrictive transition model and thus are well suited to represent the benefits and drawbacks of the IMMPF approach.
%So the results of table \ref{tbl:err} should not be seen as best case localization results, but more as proofing robustness while providing room for further improvements.
In this context, it should be noted that the localisation system used for the experiments is very basic and can be seen as a slimmed version of our previous works \cite{Fetzer2016OMC, Ebner-16}.
Optimizing the Wi-Fi parameters and adding additional methods will improve the localisation results significantly.
Especially, the graph-based transition model allows many optimizations and performance boosts.
More importantly, the here presented approach was able to recover in all situations and thus never got completely stuck within a demarcated area.
All results were similar or more accurate then the ones provided by the standalone filters, even when the localisation did not suffer from any problems.

45
tex/chapters/introduction.tex
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@@ -1,30 +1,30 @@
\section{Introduction}
Localising pedestrians inside buildings can be considered as a time-sequential, non-linear and non-Gaussian state estimation problem.
Such problems are often solved by using Bayesian filter, which update the state estimation recursively with every new incoming measurement.
Such problems are often solved by using Bayesian filters, which update the state estimation recursively with every new incoming measurement.
A powerful method to obtain numerical results for this approach are particle filters.
%einführen von partikel filter ganz allgemein
Especially in indoor localisation, particle filter can lately be considered as the standard method for solving complex non-linear problems \cite{Doucet11:ATO}.
Especially in indoor localisation, particle filters can lately be considered as the standard method for solving complex non-linear problems \cite{Doucet11:ATO}.
By using a set of weighted random samples (particles), they approximate a probability distribution describing the pedestrian's possible whereabouts and therefore the uncertainty of the system.
In its most basic form, the particle filter operates three main steps:
At first, new particles are drawn according to some importance distribution, those particles are then weighted by an incremental importance weight distribution and finally a resampling step is deployed to prevent that only a small number of particles have a signifcant weight and all the other will have negligible small weights instead \cite{orhan2012particle}.
In its most basic form, the particle filter is based on three main steps:
At first, new particles are drawn according to some importance distribution, those particles are then weighted by an incremental importance weight distribution and finally a resampling step is deployed to prevent that only a small number of particles have a signifcant weight and all the other will have negligible small weights instead \cite{chen2003bayesian}.
%transition und evaluation einführen
In practice the importance distribution is often represented by the state transition, modelling the dynamics of the system.
In practice, the importance distribution is often represented by the state transition, modelling the dynamics of the system.
A new weight is then obtained by the state evaluation given different sensor measurements.
Most localisation approaches differ mainly in how the transition and evaluation steps are implemented and the available sensors are incorporated \cite{Nurminen13-PSI, Ebner2016OPN, Hilsenbeck2014}.
However, as \cite{Li2014} already mentioned, particle filter (and nearly all of its modifications) continue to suffer from two notorious problems: sample degeneracy and impoverishment.
Most localisation approaches differ mainly in how the transition and evaluation steps are implemented and the available sensors are incorporated \cite{Nurminen13-PSI, Ebner-16, Hilsenbeck2014}.
However, as \cite{Li2014} already mentioned, particle filters (and nearly all of its modifications) continue to suffer from two notorious problems: sample degeneracy and impoverishment.
As one can imagine, after a few iterations with continuously reweighting particles, the weight will concentrate on a few particles only.
This is why the resampling step was presented in the first place.
Here, a new set of equally weighted particles is drawn by multiplying high weighted particles while abandoning low weighted ones.
However, this leads to an decreasing diversity of particles after a resampling step, also known as sample impoverishment.
Here, a new set of equally weighted particles is drawn by duplicating highly weighted particles while abandoning low weighted ones.
However, this leads to a decreasing diversity of particles after a resampling step, also known as sample impoverishment.
This high concentration of particles follows a bad approximation of the underlying probability distribution and therefore worse estimation results.
The effect of impoverishment is not solely caused by resampling only.
The effect of impoverishment is not solely caused by resampling.
Restrictive transition models, as they are used in indoor localisation applications, also enhance this effect significantly.
An example is illustrated in fig. \ref{fig:multimodalPath}, where a graph-based structure prohibits walking through walls and considers the human movement speed.
An example is illustrated in fig. \ref{fig:multimodalPath}, where the state dynamics prohibit walking through walls and consider the human movement speed.
%
\begin{figure}[t]
\centering
@@ -35,23 +35,24 @@ An example is illustrated in fig. \ref{fig:multimodalPath}, where a graph-based
\label{fig:multimodalPath}
\end{figure}
%
Due to uncertain measurements the posterior distribution of the particle filter is captured within a room.
Due to uncertain measurements, the posterior distribution of the particle filter is captured within a room.
Between time $t-1$ and $t$, the resampling step abandons all particles on the corridor and drawing new particles outside the room is not possible due to the restricted transition.
At this point, standard filtering methods are not able to recover.
A simple solution would be drawing a handful new particles randomly in the building.
However, it is obvious that this leads to a higher uncertainty and possible a highly multimodal posterior distribution.
Additionally, very uncertain absolute measurements, like attenuated Wi-Fi signals, can cause unpredictable jumps to such a newly drawn position, which would otherwise be not possible.
Especially, methods using relative measurements like pedestrian dead reckoning approaches are losing their importance.
However, it is obvious that this leads to a higher uncertainty and possibly a multimodal posterior distribution.
Additionally, very uncertain absolute measurements, like attenuated Wi-Fi signals, can cause unpredictable jumps to such a newly drawn position, which would otherwise not be possible.
Especially, methods using relative measurements like pedestrian dead reckoning are losing their importance.
As mentioned before, sample degeneracy and impoverishment are a pair of contradictions that can be described as a trade-off between the need of diversity and the need of focus \cite{Li2014}.
We tackle this problem in context of indoor localisation by deploying an interacting multiple model particle filter (IMMPF) for jump Markov non-linear systems \cite{Driessen2005}.
This enables a merging between posterior probability distributions approximated by particle filters, refereed as modes within this context.
We combine two similar particle filters using a non-trivial Markov switching process, depending upon the Kullback-Leibler divergence between the modes and a Wi-Fi quality factor.
One with a very restrictive transition scheme, providing very accurate results.
The other with more flexible and simple dynamics, resulting in a higher sample diversity.
Both are then successfully combined, to satisfy the need of diversity and the need of focus.
The main benefit of this approach is that it can be easily adapted to other existing localization approaches based on particle filters.
We tackle this problem in the context of indoor localisation by deploying an interacting multiple model particle filter (IMMPF) for jump Markov non-linear systems \cite{Driessen2005}.
This enables merging of posterior probability distributions, approximated by particle filters.
Within this context a particle filter is also refereed to as a mode of the IMMPF.
We combine two modes using a non-trivial Markov switching process, depending upon the Kullback-Leibler divergence between them and a Wi-Fi quality factor.
One mode with a very restrictive transition scheme, providing very accurate localisation results.
The other with more flexible, robust and simple transition, resulting in a higher sample diversity.
The modes are then successfully combined, to satisfy \textit{both}, the need of diversity and the need of focus.
This approaches main benefit is, that it can be easily adapted to other existing methods based on particle filters.

78
tex/chapters/method.tex
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@@ -1,22 +1,17 @@
\section{IMMPF and Mixing}
\label{sec:immpf}
In the previous section we have introduced a standard particle filter, an evaluation step and two different transition models.
Using this, we are able to implement two different localisation schemes.
One providing a high diversity with a robust, but uncertain position estimation.
The other keeps the localisation error low by using a very realistic propagation model, while being prone to sample impoverishment \cite{Ebner-15}.
In the following, we will combine those filters using the Interacting Multiple Model Particle Filter (IMMPF) and a non-trivial Markov switching process.
%Einführen des IMMPF
Consider a jump Markov non-linear system that is represented by different particle filters as state space description, where the characteristics change in time according to a Markov chain.
The posterior distribution is then described by
%
\begin{equation}
p(\mStateVec_{t}, m_t \mid \mObsVec_{1:t}) = P(m_k \mid \mObsVec_{1:t}) p(\mStateVec_{t} \mid m_t, \mObsVec_{1:t})
p(\mStateVec_{t}, m_t \mid \mObsVec_{1:t}) = P(m_t \mid \mObsVec_{1:t}) p(\mStateVec_{t} \mid m_t, \mObsVec_{1:t})
\label{equ:immpfPosterior}
\end{equation}
%
where $m_t\in M\subset \mathbb{N}$ is the modal state of the system \cite{Driessen2005}.
where $m_t\in M\subset \mathbb{N}$ is the modal state of the system and thus describes the current mode (particle filter) \cite{Driessen2005}.
The notation $P(\cdot)$ provides a discrete probability distribution.
Given \eqref{equ:immpfPosterior} and \eqref{equ:bayesInt}, the mode conditioned filtering stage can be written as
%
\begin{equation}
@@ -31,7 +26,7 @@ Given \eqref{equ:immpfPosterior} and \eqref{equ:bayesInt}, the mode conditioned
\label{equ:immpfFiltering}
\end{equation}
%
and the posterior mode probabilities are calculated by
and the posterior mode probabilities, providing how likely it is that a considered mode represents the system's state, are calculated by
%
\begin{equation}
p(m_t \mid \mObsVec_{1:t}) \propto p(\mStateVec_{t} \mid m_t, \mObsVec_{1:t-1}) P(m_t \mid \mObsVec_{1:t-1})
@@ -40,7 +35,7 @@ and the posterior mode probabilities are calculated by
\end{equation}
%
It should be noted that \eqref{equ:immpfFiltering} and \eqref{equ:immpModeProb} are not normalized and thus such a step is required.
To provide a solution for $P(m_t \mid \mObsVec_{1:t-1})$, the recursion for $m_t$ in \eqref{equ:immpfPosterior} is now derived by the mixing stage \cite{Driessen2005}.
To provide a solution for the probability distribution $P(m_t \mid \mObsVec_{1:t-1})$, the recursion for $m_t$ in \eqref{equ:immpfPosterior} is now derived by the so called mixing stage \cite{Driessen2005}.
Here, we compute
%
\begin{equation}
@@ -68,14 +63,26 @@ and
\end{equation}
%
where \eqref{equ:immpModeMixing} is a weighted sum of distributions and the weights are provided through \eqref{equ:immpModeMixing2}.
The transition probability $P(m_{t+1} = k \mid m_t = l)$ is given by the Markov transition matrix $[\Pi_t]_{kl}$.
Sampling from \eqref{equ:immpModeMixing} is done by first drawing a modal state $m_t$ from $P(m_t \mid m_{t+1}, \mObsVec_{1:t})$ and then drawing a state $\mStateVec_{t}$ from $p(\mStateVec_{t} \mid m_t, \mObsVec_{1:t})$ in dependence to that $m_t$.
To find a solution for $P(m_t \mid \mObsVec_{1:t})$, an estimate of the posterior probability $p(m_t \mid \mObsVec_{1:t})$ in \eqref{equ:immpModeProb} can be calculated according to
%
\begin{equation}
P(m_t \mid \mObsVec_{1:t}) = \frac{\omega_t^{m_t} P(m_t \mid \mObsVec_{1:t-1})}{\sum_{m=1}^M \omega_t^{m_t} P(m_t \mid \mObsVec_{1:t-1})}
\enspace .
\label{equ:immpMode2}
\end{equation}
%
Here, $\omega_t^{m_t}$ is the unnormalized weight given by the state evaluation of the respective mode $m_t$.
The initial mode probabilities $P(m_1 \mid \mObsVec_{1})$ have to be defined beforehand.
The transition probability $P(m_{t+1} = k \mid m_t = l)$ in \eqref{equ:immpModeMixing3} is given by the Markov transition matrix $[\Pi_t]_{kl}$.
The matrix $\Pi_t$ is a real square matrix, with each row summing to 1.
It provides the probability of moving from $m_t$ to $m_{t+1}$ in one time step.
Sampling from \eqref{equ:immpModeMixing} is done by first drawing a modal state $m_t$ from $P(m_t \mid m_{t+1}, \mObsVec_{1:t})$ and then drawing a state $\mStateVec_{t}$ from $p(\mStateVec_{t} \mid m_t, \mObsVec_{1:t})$ in regard to that $m_t$.
In context of particle filtering, this means that \eqref{equ:immpModeMixing} enables us to pick particles from all available modes in regard to the discrete distribution $P(m_t \mid m_{t+1}, \mObsVec_{1:t})$.
Further, the number of particles in each mode can be selected independently of the actual mode probabilities.
Algorithm \ref{alg:immpf} shows the complete IMMPF procedure in detail.
As prior knowledge, $M$ initial probabilities $P(m_1 \mid \mObsVec_{1})$ and initial distributions $p(\mStateVec_{1} \mid m_1, \mObsVec_{1})$ providing a particle set $\{W^i_{1}, \vec{X}^i_{1} \}_{i=1}^N$ are available.
The mixing step requires that the independent running filtering process are all finished.
The mixing step requires that the independently running filtering processes are all finished.
\begin{algorithm}[t]
\caption{IMMPF Algorithm}
@@ -106,31 +113,32 @@ The mixing step requires that the independent running filtering process are all
%grundidee warum die matrix so gewählt wird.
With the above, we are finally able to combine the two filters described in section \ref{sec:rse}.
The basic idea of our approach is to utilize the restrictive filter as the dominant one, providing the state estimation for the localisation.
Due to its robustness and good diversity the other, more permissive one, is then used as support for possible sample impoverishment.
If we recognize that the dominant filter gets stuck or loses track, particles from the supporting filter will be picked with a higher probability while mixing the new particle set for the dominant filter.
With the above, we are finally able to combine the two filters described in section \ref{sec:rse} and realize the considerations made in section \ref{sec:divergence}.
Within the IMMPF we utilize the restrictive graph-based filter as the \textit{dominant} one, providing the state estimation for the localisation.
Due to its robustness and good diversity the simple, more permissive filter, is then used as \textit{support} for possible sample impoverishment.
%kld
This is achieved by measuring the Kullback-Leibler divergence $D_{\text{KL}}(P \|Q)$ between both filtering posterior distributions $p(\mStateVec_{t} \mid m_t, \mObsVec_{1:t})$.
The Kullback-Leibler divergence is a non-symmetric non-negative difference between two probability distributions $Q$ and $P$.
It is also stated as the amount of information lost when $Q$ is used to approximate $P$ \cite{Sun2013}.
We set $D_{\text{KL}} = D_{\text{KL}}(P \|Q)$, while $P$ is the dominant and $Q$ the supporting filter.
Since the supporting filter is more robust and ignores all environmental restrictions, we are able to make a statement whether state estimation is stuck due to sample impoverishment or not by looking at the positive exponential distribution
As a reminder, both filters (modes) are running in parallel for the entire estimation life cycle.
If we recognize that the dominant filter diverges from the supporting filter and thus got stuck or lost track, particles from the supporting filter will be picked with a higher probability while mixing the new particle set for the dominant filter.
As seen before, the Markov transition matrix $\Pi_t$ provides the probability $P(m_{t+1} \mid m_t)$ for transitioning between modes.
%In our approach those modes are the dominant graph-based filter and the supporting simple filter.
The dominant filter's probability to draw particles from its own posterior is given by the positive exponential distribution
%
\begin{equation}
f(D_{\text{KL}}, \lambda) = e^{-\lambda D_{\text{KL}}}
\enspace .
\label{equ:KLD}
\end{equation}
%
If $D_{\text{KL}}$ increases to a certain point, \eqref{equ:KLD} provides a probability that allows for mixing the particle sets.
Therefore, drawing particles from the support is given by $1 - f(D_{\text{KL}}, \lambda)$.
If the Kullback-Leibler divergence $D_{\text{KL}}$ increases to a certain point, \eqref{equ:KLD} provides a probability that allows for mixing the particle sets.
$\lambda$ depends highly on the respective filter models and is therefore chosen heuristically.
In most cases $\lambda$ tends to be somewhere between $0.01$ and $0.10$.
However, it is obvious that \eqref{equ:KLD} only works reliable if the measurement noises are within reasonable limits, because the support filter depends solely on them.
Especially Wi-Fi serves as the main source for estimation and thus attenuated or bad Wi-Fi readings are causing $D_{\text{KL}}$ to grow, even if the dominant filter provides a good position estimation.
However, \eqref{equ:KLD} only works reliable if the measurement noise is within reasonable limits, because the support filter using the simple transition depends solely on them.
Especially \docWIFI{} serves as the main source for estimation and thus attenuated or bad \docWIFI{} readings are causing bad estimation results for the supporting filter.
This further leads to a growing $D_{\text{KL}}$, even if the dominant filter provides a good position estimation.
In such scenarios a lower diversity and higher focus of the particle set, as given by the dominant filter, is required.
We achieves this by introducing a Wi-Fi quality factor, allowing the support filter to pick particles from the dominant filter and prevent the later from doing it vice versa.
We achieves this by introducing a \docWIFI{} quality factor, allowing the support filter to pick particles from the dominant filter and prevent the later from doing it vice versa.
The quality factor is defined by
%
\begin{equation}
@@ -150,13 +158,13 @@ The quality factor is defined by
\label{eq:wifiQuality}
\end{equation}
%
where $\bar\mRssi_\text{wifi}$ is the average of all signal-strength measurements received from the observation $\mObsVec_t$. An upper and lower bound is given by $l_\text{max}$ and $l_\text{min}$.
where $\bar\mRssi_\text{wifi}$ is the average of all signal strength measurements received from the observation $\mObsVec_t^{\mRssiVec_\text{wifi}}$. An upper and lower bound is given by $l_\text{max}$ and $l_\text{min}$.
%matrix
To incorporate all this within the IMMPF, we utilize a non-trivial Markov switching process.
This is done by updating the Markov transition matrix $\Pi_t$ at every time step $t$.
As reminder, $\Pi_t$ highly influences the mixing process in \eqref{equ:immpModeMixing2}.
Considering the above presented measures, $\Pi_t$ is two-dimensional and given by
As reminder, $\Pi_t$ highly influences the mixing process in \eqref{equ:immpModeMixing2} and is responsible for providing the probability of moving from $m_t$ to $m_{t+1}$ in one time step.
Considering the measures \eqref{equ:KLD} and \eqref{eq:wifiQuality} presented above, the $2$x$2$ matrix $\Pi_t$ is given by
%
\begin{equation}
\Pi_t =
@@ -169,6 +177,12 @@ Considering the above presented measures, $\Pi_t$ is two-dimensional and given b
\end{equation}
%
This matrix is the centrepiece of our approach.
It is responsible for controlling and satisfying the need of diversity and the need of focus for the whole localization approach.
It is responsible for controlling and satisfying the need of diversity and the need of focus for the whole localisation approach.
How $\Pi_t$ works is straightforward.
If the dominant graph-based filter suffers from sample impoverishment (get stuck or lose track), the probability in $[\Pi_t]_{12} = (1 - f(D_{\text{KL}}))$ increases with diverging support filter.
Consequently, the number of particles, the dominant filter draws from the supporting filter, also increases by $[\Pi_t]_{12} \cdot 100\%$.
Similar behaviour applies to the \docWIFI{} quality factor $q(\mObsVec_t^{\mRssiVec_\text{wifi}})$.
If the quality is low, the supporting filter regains focus by obtaining particles from the dominant's posterior.
Therefore, recovering from sample impoverishment or degeneracy depends to a large extent on $\Pi_t$.

34
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@@ -4,43 +4,47 @@
%klassisch resampling
A common way to handle degeneracy and impoverishment is to apply suitable resampling methods.
The four most popular and well established approaches found in literature are multinomial, stratified, systematic and residual resampling.
The four most popular and well established approaches found in literature are multinomial-, stratified-, systematic- and residual resampling.
They are also referred to as traditional methods, since a single distribution is used for resampling and the number of times a particle is re-drawn is always proportional to is weight \cite{Li2015b}.
%advanced resampling
A more advanced method, with an adaptive particle size instead of a fixed one, is KLD-resampling.
It determines the number of particles to resample so that the Kullback-Leibler divergence between the distribution before resampling and after resampling does not exceed a pre-specified error bound \cite{Sun2013}.
A more advanced method, with an adaptive number of particles instead of a fixed one, is KLD-resampling.
It determines the number of particles to resample so that the Kullback-Leibler divergence between the distribution before resampling and after resampling does not exceed a pre-specified value \cite{Sun2013}.
The key idea is to choose a small number of samples if the distribution is focused on a small part of the state space and a large number of particles if the distribution is much more spread out and requires a higher diversity of samples.
The problem of sample degeneracy and impoverishment is therefore encountered by adapting the number of particles depend upon the systems current uncertainty.
In \cite{Li2015b} an overview of different resampling approaches are given.
%allgemien auf andere methoden überleiten
As seen, resampling methods are able to reduce impoverishment to a certain degree by themselves.
However, in practice sample impoverishment is also a problem of environmental restrictions and system dynamics.
As seen in the example of KLD-resampling, some resampling methods are able to reduce impoverishment to a certain degree by themselves.
However, in practice, sample impoverishment is also a problem of environmental restrictions and system dynamics.
Here, classical resampling schemes fail, since they are not able to propagate new particles into the state space.
More promising and intelligent solutions are given by techniques of Particle Distribution Optimization (PDO).
These variations of techniques are acting in different ways to optimize the spatial distribution of particles and are particularly effective in alleviating sample degeneracy and impoverishment \cite{Li2014}.
For example in \cite{Xiaoqin2008} a Particle Swarm Optimization is used as importance distribution for visual tracking.
Particles are iteratively updated according to their own experience and the experience of the swarm (or neighboring particles).
This allows for a multi-layer importance sampling and incorporation of the current measurement into the importance distribution, dealing with the sample impoverishment.
This allows for a multi-layer importance sampling and incorporation of the current measurements into the importance distribution, dealing with the sample impoverishment.
Other PDO methods are presented in \cite{Li2014}.
%hinführen zu IMM
In context of this work, our aim is to present a general solution that can be easily adapted to common localisation systems.
In context of this work, our aim is to present a general solution that can easily be adapted to common localisation systems.
A promising approach for an easy to deploy PDO are Interacting Multiple Models (IMM) \cite{Bar-Shalom1988}.
IMM are able to mix appropriate dynamical systems based on a Bayesian probability metric and Gaussian noise.
Therefore, a set of modes like Kalman Filters are running in parallel.
IMMs are able to mix appropriate systems based on a Bayesian probability metric and Gaussian noise.
Therefore, a set of modes, like Kalman filters, are running in parallel.
The mixing between modes is done by using a Markov Chain process, providing a probability for every mode and a transition matrix for switching between them.
The most proper mode is then chosen for the current state estimation, what allows
the right choice to the right time.
For example \cite{Zhang2013} uses IMM to switch between a line-of-sight and a non-line-of-sight filtering procedure for indoor localisation.
Thereby, they are able to provide a robust and stable position estimation in both environments.
the right choice for every instant in time.
For example, \cite{Zhang2013} deploys an IMM for a time difference of arrival (TDOA) based localisation using an ultrasonic system.
Here, line-of-sight (LOS) and non-line-of-sight (NLOS) measurement noises are considered through switching between two extended Kalman filters, one for each condition.
Thereby, they are able to provide a robust and stable position estimation with high accuracy in both LOS and NLOS noise scenarios.
An extension to particle filters and therefore to non-linear and non-Gaussian system was presented by \cite{Boers2003}.
An extension to particle filters, and therefore to non-linear and non-Gaussian system, was presented by \cite{Boers2003}.
The so called Interacting Multiple Model Particle Filter (IMMPF) was then further developed by \cite{Driessen2005}, adding a direct sampling approach.
This allows a merging between different particle filters by providing a possibility for each filter to additional sample particles from all available particle sets and not just from its own.
It is obvious that the possibility to draw from other particle sets is based on the mode probability and the transition matrix provided by the Markov Chain process and therefore does not violate the Markov property.
This allows a merging between different particle filters by providing a possibility for each filter to sample additional particles from all available particle sets and not just from its own.
It is obvious that the possibility to draw from other particle sets is based on the mode's probability and the transition matrix provided by the Markov Chain process and therefore does not violate the Markov property.
Now, the key idea of this work is to satisfy the trade-off between diversity and focus by using appropriate modes within the IMMPF.
Warum? Weil die meinsten loca systeme auf particle filtern basieren und deswegen bietet es sich an. es erlaubt bereits vorhandene methoden die auf die jeweils einzeln auf die probleme eingehen zu kombinieren und so ein hybrid zu schaffen.
%Therefore, two different dynamical models are utilized and a novel approach for a non-trivial Markov switching Process based on Kullback-Leibler divergence and a Wi-Fi quality factor are presented.

242
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@@ -3,7 +3,7 @@
In this section, we present two common localisation schemes based on particle filtering using two different transition models for propagating new states and an identical evaluation step for udpating the weights.
As mentioned before, we consider indoor localisation as a time-sequential, non-linear and non-Gaussian state estimation problem.
%As mentioned before, we consider indoor localisation as a time-sequential, non-linear and non-Gaussian state estimation problem.
Recursive filters, like the aforementioned particle filter, use all observations $\mObsVec_{t}$ until the current time $t$ for computing an estimation of the hidden state $\mStateVec_{t}$.
In a Bayesian setting, this can be formalized as the computation of the posterior distribution:
%
@@ -19,11 +19,8 @@ In a Bayesian setting, this can be formalized as the computation of the posterio
\end{equation}
%
Here, the previous observation $\mObsVec_{t-1}$ is included into the state transition \cite{Ebner-15}.
For approximating $p(\mStateVec_{t} \mid \mObsVec_{1:t})$ with a particle filter, a sample set of $N$ independet random variables, $\vec{X}^i_{t} \sim p(\mStateVec_t \mid \mObsVec_{1:t})$ for $i = 1,...,N$, is used.
Due to importance sampling, a weight $W^i_t$ is assigned to each sample $\vec{X}^i_{t}$.
A particle set of the filter is then given by $\{W^i_{1:t}, \vec{X}^i_{1:t} \}_{i=1}^N$.
The transition is used as proposal distribution, also known as CONDENSATION algorithm \cite{Isard98:CCD}.
This algorithm also performs a traditional resampling step.
%This algorithm also performs a traditional resampling step.
For indoor localisation we define the hidden state $\mStateVec$ as follows:
\begin{equation}
@@ -32,16 +29,18 @@ For indoor localisation we define the hidden state $\mStateVec$ as follows:
\end{equation}
%
where $x, y, z$ represent the position in 3D space, $\mStateHeading$ the user's heading and $\mStatePressure$ the relative atmospheric pressure prediction in hectopascal (hPa).
A particle is therefore a weighted representation of one possible system state $\mStateVec$. All relevant sensor measurements are incorporated into the observation $\mObsVec$ given by
All relevant sensor measurements are incorporated into the observation $\mObsVec$ given by
%
\begin{equation}
\mObsVec = (\mObsHeading, \mObsSteps, \mRssiVec_\text{wifi}, \mObsPressure) \enspace,
\end{equation}
%
Here, $\mObsHeading$ and $\mObsSteps$ describe the relative angular change and the number of steps detected for the pedestrian.
Further, $\mRssiVec_\text{wifi}$ and $\mRssiVec_\text{ib}$ contain the measurements of all nearby \docAP{}s (\docAPshort{}).
Further, $\mRssiVec_\text{wifi}$ contains the measurements of all nearby \docAP{}s (\docAPshort{}).
Finally, $\mObsPressure$ is the relative barometric pressure with respect to a fixed reference.
%\subsection{Evaluation}
We assume a statistical independence of all sensors. The probability density of the state evaluation is then given by the following components:
%
\begin{equation}
@@ -54,127 +53,156 @@ We assume a statistical independence of all sensors. The probability density of
\label{eq:evalBayes}
\end{equation}
%
The current pressure value is evaluated using $p(\vec{o}_t \mid \vec{q}_t)_\text{baro}$ and absolute position information is given by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for \docWIFI{}.
%The current pressure value is evaluated using $p(\vec{o}_t \mid \vec{q}_t)_\text{baro}$ and absolute position information is given by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for \docWIFI{}.
\subsection{Evaluation}
%Barometer
First, the smartphone’s barometer is used to infer the likeliness of the current $z$-location.
Due to noisy sensors, we calculate the average $\overline{\mObsPressure}$ of several sensor readings and the sensor's uncertainty $\sigma_\text{baro}$.
This average serves as relative base for all future measurements and is carried out while the pedestrian chooses his destination \cite{Fetzer2016OMC}.
The evaluation step for time $t$ is given by
%
\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}
%
The smartphone's barometer is used to infer the likeliness of the current $z$-location in $p(\vec{o}_t \mid \vec{q}_t)_\text{baro}$ and thus enables to walk stairs or to drive elevators.
Here, every predicted relative pressure $\mState_t^{\mStatePressure}$ is compared with the observed one $\mObs_t^{\mObsPressure}$ using a normal distribution.
The state's relative pressure prediction $\mStatePressure$ is estimated within each transition from $\mStateVec_{t-1}$ to $\mStateVec_t$ by tracking every height-change ($z$-axis):
The state's relative pressure prediction $\mStatePressure$ is estimated within each transition from $\mStateVec_{t-1}$ to $\mStateVec_t$ by tracking the pressure between every height-change on the $z$-axis.
Absolute position information is given by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for \docWIFI{}.
We are using the wall attenuation factor model based on Friis transmission equation to predict an \docAP{}'s (\docAPshort{}) signal strength at an arbitrary position $\fPos{\mStateVec_t} = (x, y,z)^T$.
This predicted signal strength is then matched against the current observation $\mObs_t^{\mRssiVec_\text{wifi}}$ received from this particular \docAPshort{}, providing a likelihood of the pedestrian being at $\fPos{\mStateVec_t}$.
The positions of detected \docAPshort{}'s are known beforehand.
The main advantage of this approach is that no time-consuming initial calibration phase and updates in case of infrastructural changes are needed.
%Barometer
%Due to noisy sensors, we calculate the average $\overline{\mObsPressure}$ of several sensor readings and the sensor's uncertainty $\sigma_\text{baro}$.
%This average serves as relative base for all future measurements and is carried out while the pedestrian chooses his destination \cite{Fetzer2016OMC}.
%The evaluation step for time $t$ is given by
%
% \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}
%
%Here, every predicted relative pressure $\mState_t^{\mStatePressure}$ is compared with the observed one $\mObs_t^{\mObsPressure}$ using a normal distribution.
%The state's relative pressure prediction $\mStatePressure$ is estimated within each transition from $\mStateVec_{t-1}$ to $\mStateVec_t$ 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}
% \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}
%
Here, $b$ denotes the common pressure change in $\frac{\text{hPa}}{\text{m}}$.
%Here, $b$ denotes the common pressure change in $\frac{\text{hPa}}{\text{m}}$.
%WI-FI
Signal strength models are a well-established and popular method for estimating a pedestrian's position in indoor environments.
We are using the wall attenuation factor model based on Friis transmission equation to predict an \docAP{}’s (\docAPshort{}) signal strength at an arbitrary position $\mStateVec_t$ \cite{Ebner-15}.
Here, the positions of detected \docAP{}s (\docAPshort{}) are known beforehand.
The main advantage of this approach is that no time-consuming initial calibration phase and updates in case of infrastructural changes are needed.
Using the 3D distance $d$ and the number of floors $\Delta f$ between the transmitter and the state-in-question, it can be described by
%Signal strength models are a well-established and popular method for estimating a pedestrian's position in indoor environments.
%We are using the wall attenuation factor model based on Friis transmission equation to predict an \docAP{}’s (\docAPshort{}) signal strength at an arbitrary position $\mStateVec_t$ \cite{Ebner-15}.
%Here, the positions of detected \docAP{}s (\docAPshort{}) are known beforehand.
%The main advantage of this approach is that no time-consuming initial calibration phase and updates in case of infrastructural changes are needed.
%Using the 3D distance $d$ and the number of floors $\Delta f$ between the transmitter and the state-in-question, it can be described by
%
\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}
% \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}
%
where $\mTXP$ contains the AP’s signal strength at $\mMdlDist_0$ and $\mPLE$ models the signal’s depletion with growing distance.
The attenuation per floor is described by $\mWAF$.
To reduce the system's setup time, we use the same values for all \docAP{}s at the cost of accuracy.
%where $\mTXP$ contains the AP’s signal strength at $\mMdlDist_0$ and $\mPLE$ models the signal’s depletion with growing distance.
%The attenuation per floor is described by $\mWAF$.
%To reduce the system's setup time, we use the same values for all \docAP{}s at the cost of accuracy.
By assuming statistical independence, the overall probability can be determined using
%By assuming statistical independence, the overall probability can be determined using
%
\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}
The uncertainty of the measurements is given by $\Delta{f_{i}}$. More details on this approach and possible extensions can be found in \cite{Ebner-15} and \cite{Ebner-17}.
% \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}
%The uncertainty of the measurements is given by $\Delta{f_{i}}$. More details on this approach and possible extensions can be found in \cite{Ebner-15} and \cite{Ebner-17}.
\subsection{Transition}
%\subsection{Graph-based Transition}
%Keine Überschriften. Transition über liste trennen
As mentioned before, we are searching for a solution to satisfy both, sample diversity and focus.
In the following, two very different transition models, each providing one of this abilities, are presented.
Therefore, we utilize two very different transition models, each providing one of these abilities.
\begin{itemize}
\item The \textit{graph-based transition} samples only new states that are allowed by a gridded graph, which is generated from the building's floorplan. Thus walking through walls, ceilings or obstacles is prohibited. Additionally, the human movement is considered by randomly walking along adjacent edges into a given direction until a to-be-walked distance is reached. To provide a direction and distance, turns and steps are detected using the smartphone's IMU. All this promises a very focused propagation for new states and draws only valid movements.
%
\item \textit{The simple transition} is a continuous model, that draws new states depending on a random direction and distance provided by a multivariate normal distribution. That means, no environmental knowledge and no statement about the pedestrian's (real) movement are considered.
\end{itemize}
For further explanations and details of the localisation filter please refer to \cite{Ebner-15}.
The first transition model is based upon random walks on a graph $G=(V,E)$ with vertices $\mVertexA \in V$ and undirected edges $\mEdgeAB \in E$, which is generated from the buildings floorplan \cite{Ebner-16}.
Starting at the vertex of the position $\fPos{\mStateVec_{t-1}} = (x, y,z)^T$ a new particle is sampled by walking along adjacent nodes into a given walking-direction $\gHead$ until a distance $\gDist$ is reached \cite{Ebner-15}.
During the random walk, each edge has its own probability $p(\mEdgeAB)$ which depends on the edge’s direction $\angle \mEdgeAB$ and the pedestrian’s current heading $\gHead$.
The to-be-walked edge is thus drawn according to their resemblance:
%
\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 $\sigma^2$.
Given the above, we are now able to implement two different localisation schemes, one for each transition model presented.
The graph-based transition keeps the localisation error low by using a very realistic propagation model, while being prone to sample impoverishment.
On the other hand, the simple transition provides a high diversity with a robust, but uncertain position estimation.
Both are evaluating a state $\mStateVec_{t}$ using \eqref{eq:evalBayes}.
To provide $\gHead$ and $\gDist$, 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:
%
\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 over 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 current state's heading:
%
\begin{equation}
\gHead = \mState_{t}^{\mStateHeading} = \mState_{t-1}^{\mStateHeading} + \mObs_{t-1}^{\mObsHeading} + \mathcal{N}(0, \sigma_{\gHead}^2) \\
\end{equation}
%
All this promises a very focused propagation for new particles and draws only valid movements, as ambient conditions (walls, doors, stairs, etc.) are considered.
Additionally, the graph-based approach offers plenty of scope for further extension as can be seen in \cite{Ebner2016OPN}.
%
%In the following, two very different transition models, each providing one of this abilities, are presented.
The second transition model is very simple and thus often used in scenarios with little or rough information provided by sensors.
Especially in cases without any regular statements about the pedestrian's movement.
This continuous model just moves into a random direction, ignoring the graph and thus any floorplan knowledge:
%The first transition model is based upon random walks on a gridded graph $G=(V,E)$ with vertices $\mVertexA \in V$ and undirected edges $\mEdgeAB \in E$, which is generated from the buildings floorplan \cite{Ebner-16}.
%Starting at the vertex of the position $\fPos{\mStateVec_{t-1}} = (x, y,z)^T$ a new particle is sampled by walking along adjacent nodes into a given walking-direction $\gHead$ until a distance $\gDist$ is reached \cite{Ebner-15}.
%During the random walk, each edge has its own probability $p(\mEdgeAB)$ which depends on the edge’s direction $\angle \mEdgeAB$ and the pedestrian’s current heading $\gHead$.
%The to-be-walked edge is thus drawn according to their resemblance:
%
\begin{equation}
\begin{split}
\mProb(\mStateVec_{t} \mid \mStateVec_{t-1}) &=
\mathcal{N}\left(
\fPos{\mStateVec_{t}}
\mid{}
\fPos{\mStateVec_{t-1}},
\mat{\Sigma}_{\text{move}}
\right),\\
\mat{\Sigma}_{\text{move}} &=
\begin{pmatrix}
\sigma_{\text{move}} & 0 & 0\\
0 & \sigma_{\text{move}} & 0\\
0 & 0 & \sigma_{\text{floor}}\\
\end{pmatrix}
\end{split}
\label{eq:simpleTrans}
\end{equation}
% \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}
%
The only restriction made, is that newly drawn particles need to be somewhere in between the graphs boundaries and therefore have a valid vertex for $\fPos{\mStateVec_{t}}$.
If the particle does not satisfy this condition, the position of nearest available vertex is chosen instead.
This ensures that a particle resides always on a valid vertex $\mVertexA$, what will be of importance for the upcoming IMMPF.
%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 $\sigma^2$.
%To provide $\gHead$ and $\gDist$, 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:
%
% \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 over 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 current state's heading:
%
% \begin{equation}
% \gHead = \mState_{t}^{\mStateHeading} = \mState_{t-1}^{\mStateHeading} + \mObs_{t-1}^{\mObsHeading} + \mathcal{N}(0, \sigma_{\gHead}^2) \\
% \end{equation}
%
%All this promises a very focused propagation for new particles and draws only valid movements, as ambient conditions (walls, doors, stairs, etc.) are considered.
%Additionally, the graph-based approach offers plenty of scope for further extension as can be seen in \cite{Ebner-16}.
%\subsection{Simple Transition}
%The second transition model is very simple and thus often used in scenarios with little or rough information provided by sensors.
%Especially in cases without any regular statements about the pedestrian's movement.
%This continuous model just moves into a random direction, ignoring the graph and thus any floorplan %knowledge:
% %
% \begin{equation}
% \begin{split}
% \mProb(\mStateVec_{t} \mid \mStateVec_{t-1}) &=
% \mathcal{N}\left(
% \fPos{\mStateVec_{t}}
% \mid{}
% \fPos{\mStateVec_{t-1}},
% \mat{\Sigma}_{\text{move}}
% \right),\\
% \mat{\Sigma}_{\text{move}} &=
% \begin{pmatrix}
% \sigma_{\text{move}} & 0 & 0\\
% 0 & \sigma_{\text{move}} & 0\\
% 0 & 0 & \sigma_{\text{floor}}\\
% \end{pmatrix}
% \end{split}
% \label{eq:simpleTrans}
% \end{equation}
%
%The only restriction made, is that newly drawn particles need to be somewhere in between the graphs boundaries and therefore have a valid vertex for $\fPos{\mStateVec_{t}}$.
%If the particle does not satisfy this condition, the position of nearest available vertex is chosen instead.
%This ensures that a particle resides always on a valid vertex $\mVertexA$, what will be of importance for the upcoming IMMPF.
%For further explanations of the filtering process for indoor localisation please refer to \cite{Ebner-15}.

37
tex/egbib.bib
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@@ -1721,8 +1721,10 @@ doi={10.1109/PLANS.2008.4570051},}
booktitle={Indoor Positioning and Indoor Navigation (IPIN), Int. Conf. on},
title={{Multi Sensor 3D Indoor Localisation}},
year={2015},
publisher = {IEEE},
address = {Banff, Canada},
IGNOREmonth={October},
pages={1-10},
pages={},
}
@inproceedings{Ebner-16,
@@ -2746,7 +2748,7 @@ keywords = {Artificial intelligence,Impoverishment,Machine learning,Markov Chain
mendeley-groups = {IPIN 2017},
number = {8},
pages = {3944--3954},
title = {{Fight sample degeneracy and impoverishment in particle filters: A review of intelligent approaches}},
title = {{Fight Sample Degeneracy and Impoverishment in Particle Filters: A Review of Intelligent Approaches}},
volume = {41},
year = {2014}
}
@@ -2761,7 +2763,7 @@ journal = {Radar, Sonar and Navigation, IEE Proceedings -},
mendeley-groups = {IPIN 2017},
number = {5},
pages = {323--326},
title = {{Efficient particle filter for jump Markov nonlinear systems}},
title = {{Efficient Particle Filter for Jump Markov Nonlinear Systems}},
volume = {152},
year = {2005}
}
@@ -2772,10 +2774,9 @@ author = {Li, Tiancheng and Boli{\'{c}}, Miodrag and Djuri{\'{c}}, Petar M.},
doi = {10.1109/MSP.2014.2330626},
issn = {10535888},
journal = {IEEE Signal Processing Magazine},
month = {may},
number = {3},
pages = {70--86},
title = {{Resampling Methods for Particle Filtering: Classification, implementation, and strategies}},
title = {{Resampling Methods for Particle Filtering: Classification, Implementation, and Strategies}},
volume = {32},
year = {2015}
}
@@ -2789,10 +2790,9 @@ isbn = {1350-911X},
issn = {0013-5194},
journal = {Electronics Letters},
mendeley-groups = {IPIN 2017},
month = {jun},
number = {12},
pages = {740--742},
title = {{Adapting sample size in particle filters through KLD-resampling}},
title = {{Adapting Sample Size in Particle Filters through KLD-resampling}},
volume = {49},
year = {2013}
}
@@ -2807,10 +2807,9 @@ isbn = {1063-6919},
issn = {1063-6919},
keywords = {Bayes methods,Bayesian inference view,Bayesian methods,Inference algorithms,Monte Carlo methods,Motion measurement,Nonhomogeneous media,Particle filters,Particle swarm optimization,Particle tracking,Video sequences,image sampling,image sequences,importance sampling,motion estimation,multilayer importance sampling,particle filtering (numerical methods),particle swarm optimisation,sequential particle swarm optimization,temporal continuity information,unscented particle filter,video sequence,video signal processing,visual tracking},
mendeley-groups = {IPIN 2017},
month = {jun},
pages = {1--8},
publisher = {IEEE},
title = {{Sequential particle swarm optimization for visual tracking}},
title = {{Sequential Particle Swarm Optimization for Visual Tracking}},
year = {2008}
}
@@ -2824,10 +2823,9 @@ isbn = {9781467315906},
issn = {00189456},
keywords = {Indoor localization,Kalman filter,M-estimator,interacting multiple model (IMM),time difference of arrival (TDOA),ultrasound},
mendeley-groups = {IPIN 2017},
month = {aug},
number = {8},
pages = {2205--2214},
title = {{TDOA-Based localization using interacting multiple model estimator and ultrasonic transmitter/receiver}},
title = {{TDOA-Based Localization using Interacting Multiple Model Estimator and Ultrasonic Transmitter/Receiver}},
volume = {62},
year = {2013}
}
@@ -2843,7 +2841,7 @@ journal = {IEEE Transactions on Automatic Control},
mendeley-groups = {IPIN 2017},
number = {8},
pages = {780--783},
title = {{Interacting multiple model algorithm for systems with Markovian switching coefficients.}},
title = {{Interacting Multiple Model Algorithm for Systems with Markovian Switching Coefficients.}},
volume = {33},
year = {1988}
}
@@ -2858,7 +2856,7 @@ journal = {October},
mendeley-groups = {IPIN 2017},
number = {5},
pages = {344--349},
title = {{Interacting multiple model particle filter}},
title = {{Interacting Multiple Model Particle Filter}},
volume = {150},
year = {2003}
}
@@ -2869,23 +2867,10 @@ year = {2003}
booktitle = {Indoor Positioning and Indoor Navigation (IPIN), Int. Conf. on},
editor = {},
year = {2016},
month = {October},
publisher = {IEEE},
pages = {},
address = {Madrid, Spain},
issn = {}
}
@inproceedings{Ebner2016OPN,
author = {F. Ebner and T. Fetzer and M. Grzegorzek and F. Deinzer},
title = {{On Prior Navigation Knowledge in Multi Sensor Indoor Localisation}},
booktitle = {International Conference on Information Fusion (FUSION 2016)},
editor = {},
year = {2016},
month = {July},
publisher = {IEEE},
address = {Heidelberg, Germany},
pages = {},
isbn = {}
}

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