\section{Related Work}
\label{sec:relatedWork}

We consider indoor localization to be a time-sequential, non-linear and non-Gaussian state estimation problem. 
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.

In context of indoor localisation, particle filter approximate a probability distribution describing the pedestrian's possible whereabouts by using a set of weighted random samples (particles).
Here, new particles are drawn according to some importance distribution, often represented by the state transition, which models the dynamics of the system.
Those particles are then weighted by the state evaluation given different sensor measurements.
A resampling step is deployed to prevent that only a small number of particles have a signifcant weight \cite{chen2003bayesian}.
Most localisation approaches differ mainly in how the transition and evaluation steps are implemented and the available sensors are incorporated \cite{Fetzer-16, Ebner-16, Hilsenbeck2014}.
Additionally, within this paper we present a method, which is designed to run solely on a smartphone.

In its most basic form, the state transition is given by..

\todo{nochmal mit frank klären was wir jetzt GENAU machen.}

These disadvantages can be avoided by using spatial models
like indoor graphs. Besonders geometric spatial models sind beliebt

\todo{kurz auf voronoi eingehen mit neueren papern und dann auf grid basierte eingehen. schreiben das wir in previous work auch solche benutzt haben, aber das problem ist halt der gigantische speicheraufwand. deshalb haben wir uns für triangle based entscheiden, die erstellung ist einfacher, die verfahren sind aus der spieletheorie bekannt und erfolgreich im einatz. natürlich ist das ganze ein wenig rechenaufwendiger, da nun bla und blub gemacht werden muss, jedoch ist das laufen realisischer und nicht auf 45 grad winkel begrenzt. es wird also eine höhere genaugikeit erwartet, bei stark reduzierten speicher und zugrifssbedarf auf das netz.}

The state evaluation process depends highly on the used sensors. 
However, most smartphone-based localization systems are using wireless technologies like Wi-Fi and Bluetooth. 
Here one can mainly differ between fingerprintg and signal-strength based solutions.
For example \cite{} used a fingerprinting based method to...  however, high setup time. 
\cite{} uses a robot for this, however in old buildings not every area is easily accesable for robots due to absätze und kleine treppen. außerdem sehr teuer der ganze mist. 
wir haben ansonsten immer signalstrength basierte systeme genommen, welche aber eine simple line of sight annahme machen, außerdem haben wir nur eine materialkonstante angenommen, was für gebäude mit unterschiedlichen baumaterialen nicht klappen kann da das signal durch bla und blub abgelenkt wird. deshalb wird in dieser arbeit ein kompromiss zwischen beiden verwendet anhand eines optimierungsverfahren. ein vorteil der dabei entsteht, die position der ap's kann uns egal sein. da diese geschätzt werden. 
\todo{gibt es dazu related work?} 


Besides well chosen probabilistic models, the system's performance is also highly affected by handling problems which are .. based on the nature of particle filters. One very affecting problem is the before mentioned sample impoverishment. In blabal \cite{} this problems was tackled by and. In \cite{} we deployed a ... . However, deploying a IMMPF is in most cased not a necassary step, thus we present i much simple, but also very heuristic model within this paper. 

Finally, as the name recursive state estimation states, it requires to find the most probable state within the state space, to provide the “best estimate” of the underlying problem.
In the discrete manner of a sample representation this is often done by providing a single value, also known as sample statistic, to serve as a “best guess”. 
This value is then calculated by means of simple parametric point estimators, e.g. the weighted-average over all samples, the sample with the highest weight or by assuming other parametric statistics like normal distributions 
However in complex situtations like a multimodal representatio of the posterior, such methods fail to provide an accurate statement about the most probable state.
A well known solution is KDE. 
For example \cite{} used a ... in .... However it is obvious that this method has a massive computation time and is thus not practicle for smartphone-based solutions.

Within this paper we use a rapid bla und blub, what was recently presented in \cite{}. 

\todo{umschreiben mit entsprechenden cites und auf particles }

\todo{mal die letzten beiden IPIN Jahre durchstöbern und deren system raussuchen. \\
dabei vor allem mit dem fokus, nicht sehr flexibel, braucht fertige ap positionen etc draufschauen \\ 
danach ein wenig schaun, ob es andere gibt die einzelne verfahren, wie wir sie haben ähnlich machen \\
nicht verbergen das wir hier viel aus unseren eigenen paper zehren, also ruhig citen.}

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