toni/IPIN2016
2
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

frank d annotations

Browse Source
This commit is contained in:
toni
2016-05-08 20:22:02 +02:00
parent 61685b4723
commit 769d78d7f6
5 changed files with 13 additions and 15 deletions
Download Patch File Download Diff File Expand all files Collapse all files

14
tex/chapters/smoothing.tex
View File

@@ -2,7 +2,7 @@
\label{sec:smoothing}
The main purpose of this work is to provide MC smoothing methods in context of indoor localisation.
As mentioned before, those algorithms are able to compute probability distributions in the form of $p(\mStateVec_t \mid \mObsVec_{1:T})$ and are therefore able to make use of future observations between $t$ and $T$, where $t << T$.
As mentioned before, those algorithms are able to compute probability distributions in the form of $p(\mStateVec_t \mid \mObsVec_{1:T})$ and are therefore able to make use of future observations between $t$ and $T$, where $t \ll T$.
%Especially fixed-lag smoothing is very promising in context of pedestrian localisation.
In the following we discuss the algorithmic details of the forward-backward smoother and the backward simulation.
Further, a novel approach for incorporating them into the localisation system is shown.
@@ -38,7 +38,6 @@ The weights are obtained through the backward recursion in line 9.
\begin{algorithmic}[1] % The number tells where the line numbering should start
\For{$t = 1$ \textbf{to} $T$} \Comment{Filtering}
\State{Obtain the weighted trajectories $ \{ W^i_t, \vec{X}^i_t\}^N_{i=1}$}
\todo{Filtering hier genauer beschreiben?}
\EndFor
\For{ $i = 1$ \textbf{to} $N$} \Comment{Initialization}
\State{Set $W^i_{T \mid T} = W^i_T$}
@@ -64,7 +63,7 @@ By reweighting the filter particles, the FBS improves the simple filter-smoother
\subsection{Backward Simulation}
For smoothing applications with a high number of particles, it is often not necessary to use all particles for smoothing.
This decision can for example be made due to a high sample impoverishment and/or highly accurate sensors.
This decision can, for example, be made due to a high sample impoverishment and/or highly accurate sensors.
By choosing a good sub-set for representing the posterior distribution, it is theoretically possible to further improve the estimation.
Therefore, \cite{Godsill04:MCS} presented the backward simulation (BS). Where a number of independent sample realisations
@@ -100,15 +99,14 @@ Here, $\tilde{\vec{q}}_t$ is a random sample drawn approximately from $p(\vec{q}
For example $\tilde{\vec{q}}_t$ could be chosen by selecting particles within a cumulative frequency.
Therefore $\tilde{\vec{q}}_{1:T} = (\tilde{\vec{q}}_{1}, \tilde{\vec{q}}_{2}, ...,\tilde{\vec{q}}_{T})$ is one particular sample
realisation from $p(\vec{q}_{1:T} \mid \vec{o}_{1:T})$.
Further independent realisations are obtained by repeating the algorithm until the desired number $N_{\text{sample}}$ is reached.
Further independent realisations are obtained by repeating the algorithm until the desired number of realisations $N_{\text{sample}}$ is reached.
The computational complexity for one particular realisation is $\mathcal{O}(N)$.
However, the computations are then repeated for each realisation drawn \cite{Godsill04:MCS}.
\subsection{Transition for Smoothing}
As seen above, both algorithms are reweighting particles based on a state transition model.
Unlike the transition presented in section \ref{sec:transition}, it is not possible to just draw a set of new samples.
Here, $p(\vec{q}_{t+1} \mid \vec{q}_{t})$ needs to provide the probability of the \textit{known} future state $\vec{q}_{t+1}$ under the condition of the current state $\vec{q}_{t}$.
In case of indoor localisation using particle filtering, it is necessary to not only provide the probability of moving to a particle's position under the condition of its ancestor, but also of all other particles at time $t$.
Here, $p(\vec{q}_{t+1} \mid \vec{q}_{t})$ needs to provide the probability of the \textit{known} future state $\vec{q}_{t+1}$ under the condition of its ancestor $\vec{q}_{t}$.
The smoothing transition model therefore calculates the probability of being in a state $\vec{q}_{t+1}$ in regard to previous states and the pedestrian's walking behaviour.
This means that a state $\vec{q}_t$ is more likely if it is a proper ancestor (realistic previous position) of a future state $\vec{q}_{t+1}$.
In the following a simple and inexpensive approach for receiving this information will be described.
@@ -119,7 +117,7 @@ p(\vec{q}_{t+1} \mid \vec{q}_t, \mObsVec_t)_{\text{step}} = \mathcal{N}(\Delta d
\label{eq:smoothingTransDistance}
\end{equation}
we receive a statement about how likely it is to cover a distance $\Delta d_t$ between two states $\vec{q}_{t+1}$ and $\vec{q}_{t}$.
In the easiest case, $\Delta d_t$ is the linear distance between two states.
In the easiest case, $\Delta d_t$ is the euclidean distance between two states.
Of course, based on the graph structure, one could calculate the shortest path between both and sum up the respective edge lengths.
However, this requires tremendous calculation time for negligible improvements.
Therefore this is not further discussed within this work.
@@ -156,7 +154,7 @@ Looking at \refeq{eq:smoothingTransDistance} to \refeq{eq:smoothingTransPressure
\begin{equation}
\arraycolsep=1.2pt
\begin{array}{ll}
p(\vec{q}_{t+1} \mid \vec{q}_t) =
p(\vec{q}_{t+1} \mid \vec{q}_t, \mObsVec_t) =
&p(\vec{q}_{t+1} \mid \vec{q}_t, \mObsVec_t)_{\text{step}}\\
&p(\vec{q}_{t+1} \mid \vec{q}_t, \mObsVec_t)_{\text{turn}}\\
&p(\vec{q}_{t+1} \mid \vec{q}_t, \mObsVec_t)_{\text{baro}}