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\State \Call{Send\_Pos}{$x_t, y_t$}  \EndProcedure  \end{algorithmic}  \end{algorithm}\section{Filtering and Smoothing}  \label{sec:filtering}  Computer vision-based estimations are usually noisy or ambiguous, and  so is the proposed system: beginning with the estimations of the  homographies, the ground truth is already based on possibly faulty  labelings. Texton histograms obtained during flight will not perfectly  match the ones in the training data set: blur, lighting settings,  viewing angles and other variables change the shape of the texton  histograms.  To filter out outliers and smooth the estimations, a popular filter  choice is the Kalman filter. However, the Kalman filter is not able to  represent multimodal probability distributions. This makes it rather  unsuitable for the presented approach: if the $k$NN model outputs two  predictions, one would need to use average of these predictions and  feed this value to the Kalman Filter. This approach can lead to biased  predictions, especially, if the the model outputs belong to distant  locations--- to similar texton distributions at these positions.  Instead, the more powerful general Bayesian filter could simultaneously keep  account of both possible locations and resolve the ambiguity as soon  as one location can be favored. In this case, the predictions of the  $k$ neighbors are directly fed into the particle filter without  averaging them first. The filter is able to smooth the estimations,  handle uncertainty, and simultaneously keep track of several competing  position estimations. Since the calculation of a full Bayesian filter is computationally intractable, a particle filter which is based on sampling was used as approximation.   In general, particle filters estimate the posterior probability of the  state given observations. Specifically, this means that finding the  position of the UAV can be described as $p(X_t \mid Z_t)$, where $X_t$  is the state vector at time $t$ ($x,y$-position, heading, speed,  acceleration) and $Z_t$ are the measurements ($z_1, ..., z_t$, where  each $z_i$ represents the $x,y$ output of the proposed algorithm) up  to time $t$. The state vector is \emph{hidden}, since the variables  cannot be measured directly, therefore the situation can be described  using a hidden Markov model. Instead, noisy or ambiguous data can be  obtained through the proposed algorithm.  The weighted particles are a discrete approximation of the posterior  probability function ($pdf$) of the state vector.  Particle filters have several advantages. First, one can represent  uncertainty by the variance of the state variables of the  particles. Second, the particle filter allows for \emph{sensor fusion}, and  can integrate IMU data or optical flow into the position estimation.  A major disadvantage is the rather high computational complexity. This  can be circumvented by reducing the amount of particles (trading off  speed and accuracy)---allowing for adapting the computational payload  to the used processor.  The used particle filter is initialized using 100 particles at random  $x, y$-positions. To incorporate the distances, the sensor model $p(z  \midx) = \frac{p(x \mid z)p(z)}{p(x)}$ is used, where $\textbf{x} =  ((x_1, y_1), (x_2, y_2), \ldots, (x_k, y_k))^T$. A two-dimensional  Gaussian model was used for each point. The parameters of the Gaussian  have been determined by comparing of the positions based on the motion  tracking system with the predictions of the proposed system. This  results in values for the variances in $x$ and $y$, the correlation  $\rho$ between $x$ and $y$. The mean values $\mu$ were set to zero  (no-systematic bias). Figure~\ref{fig:measurementmodel} shows the  results of one such evaluation.  \begin{figure}[h!]  \begin{center}  \includegraphics[width=0.7\columnwidth]{figures/measurement_model/default_figure}  \caption{{\label{fig:measurementmodel} Measurement model showing the delta x and delta y  positions%  }}  \end{center}  \end{figure}  This allows to make use of the information in all $k$ neighbors and  keep track of a multimodal distribution. While keeping track of a  multimodal distribution allows for incorporating several possible  states of the system, the problem arises of which mode is the best  one. Using a weighted average of the modes would again introduce the  problem, that the weighted average falls into a low density  region. Therefore, the maximum a posteriori estimate, as described in  \cite{driessen2008map} is used. This approach uses the following  formula to obtain the MAP estimate:  Therefore, the final position estimate is equal to the position of one  of the particles.  \begin{align}  s_k^{MAP} &= \argmax_{s_k}{p(s_k \mid Z_k)}\\  &= \argmax_{s_k}{p(z_k \mid s_k) p(s_k \mid Z_{k-1})}   \end{align}  Therefore, the MAP estimator in our case is  \begin{align}  s_k^{MAP} = \argmax_{s_k} \lambda^M(s_k)  \end{align}  with  \begin{align}  \lambda^M(s_k) = p(z_k \mid s_k) \sum_{j=1}^Mp(s_k \mid s_{k-1}^j)w^j_{k-1}  \end{align}  This function is now only evaluated at a finite, chosen number of  states, the particles, using  \begin{align}  \hat{s}_k^{MAP} = \argmax_{s_k \in \{s_k^i \mid i=1,\ldots,N\}} \lambda^M(s_k)  \end{align}  In this formula, $p(z_k \mid s_k)$ is the likelihood of the particle  $s_k$ given the current measurement $z_k$. In our setting, this  probability is equal to the weight of the particle $z_k$, therefore  $p(z_k \mid s_k) = w^i_k$.  The estimation of \emph{uncertainty} is a core part of the proposed  approach, being important for safety and accuracy. Therefore,  uncertainty was modeled using the spread of the particles.  In every time step, the particles of the filter get updated based on  the optical flow estimates. These estimates are noisy, as illustrated  in Figure~\ref{fig:edgeflow}. Additionally, optical flow estimates  aggregate noise over time, since each estimate is dependent on the  previous one, leading to drift (\emph{relative position  estimates}). In contrast, the machine learning-based method makes  independent predictions. While this allow for avoiding accumulating  errors, the predictions do not dependent on each other, and might be  `jumping' between two points. To combine the advantages of both  methods, and leverage out the disadvantages, the particle filter is  used.  An idea was to include the similarity to the neighbors as confidence  value, thus reducing the measurement noise, if a high similarity  between current histogram and a training histogram is  achieved. However, we found no correlation between these  variables. Figure~\ref{fig:cor_sim_measurement} displays the  dependence structure.