67 lines
5.8 KiB
TeX
67 lines
5.8 KiB
TeX
\section{Introduction}
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Localising pedestrians inside buildings can be considered as a time-sequential, non-linear and non-Gaussian state estimation problem.
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Such problems are often solved by using Bayesian filter, which update the state estimation recursively with every new incoming measurement.
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A powerful method to obtain numerical results for this approach are particle filters.
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%einführen von partikel filter ganz allgemein
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Especially in indoor localisation, particle filter can lately be considered as the standard method for solving complex non-linear problems \cite{Doucet11:ATO}.
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By using a set of weighted random samples, they approximate a probability distribution describing the pedestrian's possible whereabouts and therefore the uncertainty of the system.
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In its most basic form, the particle filter operates three main steps:
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At first, new samples are drawn according to some importance distribution, those samples are then weighted by an incremental importance weight distribution and finally a resampling step is deployed to prevent that only a small number of samples have a signifcant weight and all the other will have negligible small weights instead \cite{orhan2012particle}.
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%transition und evaluation einführen
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In practice the importance distribution is often represented by the state transition, modelling the dynamics of the system.
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A new weight is then obtained by the state evaluation given different sensor measurements.
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Most localisation approaches differ mainly in how the transition and evaluation steps are implemented and the available sensors are incorporated \cite{Nurminen13-PSI, Ebner-15, Hilsenbeck2014}.
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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.
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As one can imagine, after a few iterations with continuously reweighting samples, the weight will concentrate on a few samples only.
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This is why the resampling step was presented in the first place.
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Here, a new set of equally weighted samples is drawn by multiplying high weighted samples while abandoning low weighted ones.
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However, this leads to an decreasing diversity of samples after a resampling step, also known as sample impoverishment.
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This high concentration of samples follows a bad approximation of the underlying probability distribution and therefore worse estimation results.
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The effect of impoverishment is not solely caused by resampling only.
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Restrictive transition models, as they are used in indoor localisation applications, also enhance this effect significantly.
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An example is illustrated in fig. \ref{fig:multimodalPath}, where a graph-based structure prohibits walking through walls and considers the human movement speed.
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%
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\begin{figure}[t]
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\centering
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\def\svgwidth{0.9\columnwidth}
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\input{gfx/multimodalPath.eps_tex}
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\caption[An example of the occurrence of sample impoverishment.]{
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An example of the occurrence of sample impoverishment enhanced by a restrictive transition model that prevents sampling through walls. At time $t-1$ the approximated position (green line) drifts apart from the ground truth (black line) due to uncertain measurements. The posterior distribution is then captured within the room and not able to recover by itself. }
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\label{fig:multimodalPath}
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\end{figure}
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%
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Due to uncertain measurements the posterior distribution of the particle filter is captured within a room.
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Between time $t-1$ and $t$, the resampling step abandons all samples on the corridor and drawing new samples outside the room is not possible due to the restricted transition.
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At this point, standard filtering methods are not able to recover.
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A simple solution would be drawing a handful new samples randomly in the building.
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However, it is obvious that this leads to a higher uncertainty and possible a highly multimodal posterior distribution.
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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.
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Especially, methods using relative measurements like pedestrian dead reckoning approaches are losing their importance.
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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}.
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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}.
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This enables a merging between posterior probability distributions approximated by particle filters, refereed as modes within this context.
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combining two particles filters, one with a very restrictive transition scheme sehr genaue ergebnisse, and one with a more flexible but schlechtere ergebnisse scheme werden gewinnrbingen gemixed within the IMMPF,
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Therefore a non-trivial Markov switching process, depending upon the Kullback-Leibler divergence between the modes and a
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The main benefit of this approach is that it be easily adapted to other existing localization approaches based on particle filters.
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%paar notizen:
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%jennsen shannon divergence ist zwar symmetrisch und die wurzel davon zählt als metrik, ist aber in unserem kontext eher inpraktikabel da der upperbound bei ln(2) liegt.. das sorgt für ... deshalb nehmen wir die einfach kld! diese ist nach oben hin offen und somit erlaubt diese eine bessere aussage nicht nur darüber wie unterschiedlich die beiden verteilungen sind, sondern auch wie weit sie sich im wahrscheinlichkeitsraum voneinander entfernt befinden. "direct divergence measure"
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%aus gründen der simplicity haben wir alle modelle so einfach wie möglich gehalten um nur den vorzug der neuen methode zu erhalten. für weitere informationen, optimale parameter ... siehe paper, paper paper von uns.
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