\section{Related Work}
\label{sec:relatedWork}
% 3/4 Seite ca.

%kurze einleitung zum smoothing
Sequential MC filters, like the aforementioned particle filter, use all observations $\mObsVec_{1:t}$ until the current time $t$ for computing an estimation of the state $\mStateVec_t$. 
In a Bayesian setting, this can be formalized as the computation of the posterior distribution $p(\mStateVec_t \mid \mObsVec_{1:t})$ using a sample set of $N$ independent random variables, $\vec{X}^i_{t} \sim p(\mStateVec_t \mid \mObsVec_{1:t})$ for $i = 1,...,N$ for approximation. 
Due to importance sampling, a weight $W^i_t$ is assigned to each sample $\vec{X}^i_{t}$.
In the context of particle filtering $\{W^i_{1:t}, \vec{X}^i_{1:t} \}_{i=1}^N$ is a weighted set of samples, also called particles.
Therefore a particle is a representation of one possible system state $\mStateVec$.
By considering a situation given all observations $\vec{o}_{1:T}$ until a time step $T$, where $t \ll T$, standard filtering methods are not able to make use of this additional data for computing $p(\mStateVec_t \mid \mObsVec_{1:T})$.
This problem can be solved with a smoothing algorithm.

Within this work we utilise two types of smoothing: fixed-lag and fixed-interval 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. 
This makes the fixed-lag smoother able to run online. 
On the other hand, fixed-interval smoothing requires all observations until time $T$ and therefore only runs offline, after the filtering procedure is finished \cite{chen2003bayesian}. 


%historie des smoothings und entwicklung der methoden.
The origin of MC smoothing can be traced back to Genshiro Kitagawa. 
In his work \cite{kitagawa1996monte} he presented the simplest form of smoothing as an extension to the particle filter. 
This algorithm is often called the filter-smoother since it runs online and a smoothing is provided while filtering.
%\commentByFrank{das mit dem weighted paths irritiert mich etwas. war das original work auch fuer etwas, wo pfade im spiel waren? weils halt gar so gut passt. ned dass da begrifflichkeiten durcheinander kommen. beim lesen fehlt mir das beim 1. anlauf was damit gemeint ist}
This approach uses the particle filter steps to update weighted paths $\{(W^i_t, \vec{X}_{1:t}^i)\}^N_{i=1}$, producing an accurate approximation of the filtering posterior $p(\vec{q}_{t} \mid  \vec{o}_{1:t})$ with a computational complexity of only $\mathcal{O}(N)$. 
However, it gives a poor representation of previous states due to a monotonic decrease of distinct particles caused by resampling of each weighted path \cite{Doucet11:ATO}.
Based on this, more advanced methods like the forward-backward smoother \cite{doucet2000} and backward simulation \cite{Godsill04:MCS} were developed.
Both methods are running backwards in time to reweight a set of particles recursively by using future observations. 
Algorithmic details will be shown in section \ref{sec:smoothing}.

%wo werden diese eingesetzt, paar beispiele. offline, online 
%\commentByFrank{wenn du meinst, 'bei indoor wirds NICHT verwendet' dann ist 'as' das falsche. wuerde auch 'got' statt 'gets' verwenden}
In recent years, smoothing got attention mainly in other areas than indoor localisation. 
The early work of \cite{isard1998smoothing} demonstrates the possibilities of smoothing for visual tracking.
They used a combination of the CONDENSATION particle filter with a forward-backward smoother. 
Based on this pioneering approach, many different solutions for visual and multi-target tracking have been developed \cite{Perez2004}.
For example, in \cite{Platzer:2008} a particle smoother is used to reduce multimodalities in a blood flow simulation for human vessels. The authors of \cite{Hu2014} use a smoother to overcoming the problem of particle impoverishment while predicting the Remaining Useful Life (RUL) of equipment (e.g. a Lithium-ion battery).

%smoothing im bezug auf indoor
Nevertheless, there are some promising approaches for indoor localisation systems as well. 
For example \cite{Nurminen2014} deployed a fixed-interval forward-backward smoother to improve the position estimation for non-real-time applications. 
They combined \docWIFI{}, step and turn detection, a simple line-of-sight model for floor plan restrictions and the barometric change within a particle filter. 
The state transition samples a new state based on the heading change, altitude change and a fixed step length. 
The experiments of \cite{Nurminen2014} clearly emphasize the benefits of smoothing techniques. The estimation error could be decreased significantly. 
However, a fixed-lag smoother was discussed only in theory.

In the work of \cite{Paul2009} both fixed-interval and fixed-lag smoothing were presented. 
They implemented \docWIFI{}, binary infrared motion sensors, binary foot-switches and a potential field for floor plan restrictions. 
Those sensors were incorporated using a sigma-point Kalman filter in combination with a forward-backward smoother.
It was also proven by \cite{Paul2009}, that the fixed-lag smoother is slightly less accurate than the fixed-interval smoother, as one would expect from the theoretical foundation.
Unfortunately, even a sigma-point Kalman filters is after all just a linearisation and therefore not as flexible and suited for the complex problem of indoor localisation as a non-linear estimator like a particle filter. 
%\commentByToni{Kann das jemand nochmal verifizieren? Das mit dem Kalman Filter. Danke.}
%\commentByLukas{Ich wuerde den Satz ganz weglassen. Ansonsten musst du angeben, wo die eigentlichen Probleme liegen, also z.B. in welchen konkreten Situation das Kalman Filter nicht mehr funktioniert usw. So ist es jetzt erstmal nur eine Behauptung ohne jeglichen Hintergrund.}
%\commentByToni{Ich bin mir nicht sicher ob das eine Behauptung ohne jeglichen Hintergrund ist. Meiner Meinung nach ist das ziemlich weitreichend bekannt. Finde den Satz persoenlich ganz gut, weil er uns deutlich von dieser Arbeit abgrenzt und das ist wichtig.}
%\commentByFrank{hab zwar ka was das ist, aber vermutlich ist es auch normal-dist also unimodal? dann verweisen wir doch einfach auf fig1 mit dem zusatz: 'sowas geht garned erst'}.
%\commentByToni{Sigma Point Kalman Filter können mit Multimodalitäten umgehen... auch wenn sie linearisieren. frag nicht wie das gehen soll.}
Additionally, the \docWIFI{} RSSI model requires known calibration points and is deployed using a remarkable number of access points for very small spaces.  
In our opinion this is not practical and does not suite real-world conditions.
Since humans with a specific destination in mind do not tend to change their directions randomly, we would further recommend adding a PDR-based transition to draw samples in a more directed manner instead of scattering them randomly in every direction.

%\commentByFrank{algorithmS?}
%\commentByFrank{'is able to use', oder 'will use'? gehts um die eval (will use), oder generell um die theorie und moeglichkeiten (is able to)}
%\commentByFrank{man koennte die reihenfolge vlt umstellen, erst die ganzen filtering sachen beschreiben, map, activity, ... und on top of that two smoothing algorithms both implemented as fixed-interval and fixed-lag?}
The here presented approach will use two different smoothing algorithm, both implemented as fixed-interval and fixed-lag versions. 
Further, our localisation system presented in \cite{Ebner-16} enables us to walk stairs and thus going into the third dimension.
Therefore, a regularly tessellated graph is utilised to avoid walls, detecting doors and recognizing stairs. 
Within this work, this is additionally supported by a simple classification that detects the activities unknown, standing, walking and walking stairs. 
%Finally, we incorporate prior navigation knowledge by using syntactically calculated realistic human walking paths \cite{Ebner-16}. 
%This method makes use of the given destination and thereby provides a more targeted movement. 






