Better subsection 4.2

tex/chapters/4_ftmloc.tex

@@ -12,6 +12,7 @@ After measuring several distances to different anchor points one can calculate h
 %TODO Alles mit 2D, weil halt
 
 \subsection{Multilateration}
+\label{sec:multilateration}
 %Ganz kurz erläutern was Multilateration eigentlich ist. in 2D min 3 aps und in 3D min. 4D. Aber grundsätzlich gilt: viel hilft viel.
 %Es ist uns klar, dass Trilat nichts taugt aber ist halt der einfachste Schritt
 %Typische Nachteile: Wenn Schnittpunkt nicht analytisch exakt bestimmt werden können
@@ -83,18 +84,37 @@ However, GDOP still gives a good first impression of the theoretical suitability
 %Vorteil: Nicht ideale Schnittpunkte sind kein Problem, weil die Dichte sowas abbilden kann
 %FTM Vorteil: Fehlen von Messungen kann probabilistisch erfasst werden indem Streuung der NV größer wird / Oder es enstehen einfach mehrere Hypthesen über die Position
 Probabilistic frameworks allow to natively incorporate faulty or inaccurate measurements into the position model.
-While multilateration produces a single point as position estimate the probabilistic framework allows to describe the estimated position as probability density function.
-This approach has the advantage to model inaccurate measurments as variances and quantify them.
+While multilateration produces a single point as position estimate a probabilistic framework allows to describe the estimated position in terms of probability density functions.
+This approach has the advantage to model inaccurate measurements as variances and quantify them.
 
 Assuming that FTM measurements follow a Gaussian distribution, the measured distance $d_i$ between the sender and AP $i$ is described by $\mathcal{N}(d^*_i, \sigma^2_i)$ where $d^*_i=\|\mPosVec_i - \mPosVec \|$ is the actual distance between the AP position $\mPosVec_i$ and a given position $\mPosVec$. $\sigma^2_i$ is the variance of the measurement.
-Visualized on a two dimensional plane this produces a ring shaped density function where the cross section of the ring is the well known Gaussian curve.
-
-Multiple distance measurements can be combined by a joint density function, if the individual distance measurements to the APs are statistical independent.
-The probability to observe a distance measurement $d$ at position $\mPosVec$ is given as
+Hence, the probability to observe a distance $d_i$ measured to the AP $i$ at position $\mPosVec$ is given by the density function:
 \begin{equation}
-  p(d | \mPosVec ) = \prod_i \frac{1}{\sqrt{2\pi\sigma^2_i}} \exp \left( - \frac{(d - d^*_i)^2}{2\sigma^2_i} \right)
+\label{eq:distDensity}
+  p(d_i | \mPosVec ) = \frac{1}{\sqrt{2\pi\sigma^2_i}} \exp \left( - \frac{(d_i - d^*_i)^2}{2\sigma^2_i} \right)
 \end{equation}
 
+Visualized on a two dimensional plane this produces a ring shaped density function where the cross section of the ring is the well known Gaussian curve.
+Multiple distance measurements can be combined by a joint density function, if the individual distance measurements to the individual APs are statistical independent.
+The areas where the individual density functions overlap have higher likelihood to observe the given distance.
+Thus, the geometrical considerations of \autoref{sec:multilateration} still apply.
+
+% + Dynamic:
+In order to estimate the position of a moving pedestrian at each discrete timestep $\Delta t$ the most likely position given the observed distances is computed.
+Note that multiple distance measurements can be available if the measurement rate is higher than the update rate $\Delta t$.
+It is also possible that no measurements are available because the sight to a particular AP is blocked.
+Hence, the distance measurements for a given AP $i$ are given with the vector $\vec{d}_i$ of length equal to the number of successful measurements.
+
+The probability to observe the given distances $\vec{d}_{1:i} =(\vec{d}_1, \dots, \vec{d}_i)$ to each AP at the position $\mPosVec$ is given by the joint probability density function
+\begin{equation}
+  p(\vec{d}_{1:i} | \mPosVec ) = \prod_i p(\vec{d}_i | \mPosVec ) = \prod_i\prod_{m=1}^{M_i} p(d_{i,m} | \mPosVec )
+\end{equation}
+where $M_i$ is the number of successful measurements to AP $i$.
+For each AP its measurements form a joint distribution by itself.
+This allows to integrate several measurements per update time step.
+An alternative approach could be to compute the mean measurement over all $M_i$ distances in $\vec{d}_i$ and use the density in \eqref{eq:distDensity} directly.
+
+
 \subsection{Particle Filter}
 %Warum auch noch PF? Weil... die meisten lokalisierungs systeme diesen als nicht-linearen schätzer benutzen. er ist vielseitig und kann einfach mit anderen sensoren kombiniert werden. ist das gängigste sensor fusionsverfahren. die dichte wird in samples repräsentiert und ist damit nur eine approximatino der wahrsch. dichte.. dadurch ganz ander repräsentation als probabilistic teil. außerdem ist es ein FILTER, hat also vergangenheit mit drin.
 %Ganz schnell nochmal den PF einführen mit standard formel. ein größerer absatz.