45 lines
3.3 KiB
TeX
45 lines
3.3 KiB
TeX
\section{Related Work}
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\label{sec:relatedWork}
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% 3/4 Seite ca.
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%kurze einleitung zum smoothing
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Filtering algorithm, like aforementioned particle filter, use all observations $\mObsVec_{1:t}$ until the current time $t$ for computing an estimation of the state $\mStateVec_t$.
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In a Bayesian setting, this can be formalized as the computation of the posterior distribution $p(\mStateVec_t \mid \mObsVec_{1:t})$.
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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 $p(\mStateVec_t \mid \mObsVec_{1:T})$.
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This problem can be solved with a smoothing algorithm.
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Within this work we utilise two types of smoothing: fixed-lag and fixed-interval smoothing.
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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.
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This makes the fixed-lag smoother able to run online.
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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}.
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%historie des smoothings und entwicklung der methoden.
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The origin of MC smoothing can be traced back to Genshiro Kitagawa.
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In his work \cite{kitagawa1996monte} he presented the simplest form of smoothing as an extension to the particle filter.
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This algorithm is often called the filter-smoother since it runs online and a smoothing is provided while filtering.
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This approach uses the particle filter steps to update weighted paths $\{(\vec{q}_{1:t}^i , w^i_t)\}^N_{i=1}$, producing an accurate approximation of the filtering posterior $p(\vec{q}_{t} \mid \vec{o}_{1:t})$ with a computational complexity of only $\mathcal{O}(N)$.
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However, it gives a poor representation of previous states due a monotonic decrease of distinct particles caused by resampling of each weighted path \cite{Doucet11:ATO}.
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Based on this, more advanced methods like the forward-backward smoother \cite{doucet2000} and backward simulation \cite{Godsill04:MCS} were developed.
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Both methods are running backwards in time to reweight a set of particles recursively by using future observations.
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Algorithmic details will be shown in section \ref{sec:smoothing}.
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%wo werden diese eingesetzt, paar beispiele. offline, online
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In recent years, smoothing gets attention mainly in other areas as indoor localisation.
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The early work of \cite{isard1998smoothing} demonstrates the possibilities of smoothing for visual tracking.
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They used a combination of the CONDENSATION particle filter with a forward-backward smoother.
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Based on this pioneering approach, many different solutions for visual and multi-target tracking have been developed \cite{Perez2004}.
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For example, in \cite{Platzer:2008} a particle smoother is used to reduce multimodalities in a blood flow simulation for human vessels. Or \cite{}
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Nevertheless, their are some promising approach for indoor localisation systems as well. For example ...
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%smoothing im bezug auf indoor
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Smoothing solutions in indoor localisation werden bisher nicht wirklich behandelt. das liegt hauptsächlich daran das es sehr langsam ist \cite{}. es gibt ansätze von ... und ... diese benutzen blah und blah. wir machen das genauso/besser.
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