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\section{Map evaluation}  \label{sec:mapeval}  \subsection{Synthetic Data Generation}  \label{sec:syntheticdatageneration}  In the scope of this thesis, an application to simulate different  camera positions during flight was created. It generates synthetic  image patches based on perspective transformations of a map  image. Examples of generated images are displayed in  Figure~\ref{fig:montage}. The application allows for comparing and  predicting the performance of different maps. The software is written  in \CC{} and OpenCV~3.0.0.  The software is able to generate a specified amount of image patches  using random values for rotational angles, translational shifts, as  well as blur, contrast and brightness intensity. The values are  sampled from different probability distributions, see  Table~\ref{tab:distributions} for a summary. An additional graphical  user interface (GUI) displays the result of applied transformations  and saves the generated images.  To simulate camera movements in 3D space, a 2D to 3D projection of the  image is performed first. Then, by building separate rotation matrices  $R_x$, $R_y$, and $R_z$ around the axes $x$, $y$, and $z$, the  rotations can be performed separately. Next the rotation matrix $R$ is  created by multiplying the separate matrices, i.e.,  $R = R_x \times R_y \times R_z$. The 3D translation matrix is  multiplied by the transposed rotation matrix. This step is crucial to  rotate the \emph{camera model} and not the image itself. Finally,  after performing all steps, a projection from 3D space to 2D is  applied, to obtain the transformed image.  \begin{table}[h!]  \centering  \begin{tabular}{llllll}  \toprule  \multicolumn{6}{c}{Distribution} \\  \multicolumn{3}{c}{Uniform ($\mathcal{U}$)} & \multicolumn{3}{c}{Normal ($\mathcal{N}$)} \\  \cmidrule(r){1-3}\cmidrule(r){4-6}  Parameter & Min & Max & Parameter & M & STD \\  \cmidrule(r){1-3}\cmidrule(r){4-6}  Yaw & $0$ & $360$ & Roll & $90$ & $3$ \\  Translation X & $100$ & $500$ & Pitch & $90$ & $4$ \\  Translation Y & $100$ & $500$ & Brightness & $2$ & $0.1$ \\  Height & $100$ & $700$ & & & \\  Blur & $1$ & $10$ & & & \\  Contrast & $2$ & $3$ & & & \\  \bottomrule  \end{tabular}  \caption[Distributions for the different  parameters of the synthetic data augmentation tool.]{The table shows the used distributions for the different  parameters of the synthetic data augmentation tool.}  \label{tab:distributions}  \end{table}  \begin{figure}[h!]  \begin{center}  \includegraphics[width=0.7\columnwidth]{figures/montage/default_figure}  \caption{{\label{fig:montage} 16 example images generated by the synthetic data generation  tool. The black parts are the parts beyond the image borders of  the underlying image (i.e., the part where pixel maps to) or the  ones that could not be identified during the mosaic making  process.%  }}  \end{center}  \end{figure}  \subsection{Evaluation Scheme}  \label{sec:evaluationscheme}  The performance of a method might largely depend  on the environment it is used in. The evaluation of a map is  difficult, since the obtained histograms during a real flight depend  on many factors: motion blur, distance to the map and rotations  proportional to the map.  Therefore, we propose an initial evaluation scheme for given  maps. This scheme assigns a global fitness value to a given map,  proportional to the expected accuracy if it is used in the physical  world. Additionally, it allows to inspect the given map and detect the  regions that are responsible for the overall fitness value.  In the first step of the map evaluation procedure, $N$ different  patches of a given map are generated using the tool \emph{draug}  (Section~\ref{sec:draug}). We propose the following loss function  ($L$) for evaluating a given map ($\mathcal{M}$):  \begin{align}  L(\mathcal{M}) &= \sum_{i = 1}^{N} \sum_{j = 1}^{N} \ell(d_a(h_i, h_j), d_e(h_i, h_j))  \end{align}  \begin{align}  \ell(x, y) &= x - y\\  d_a(h_i, h_j) &= \text{cosine\_similarity}(h_i, h_j)\\  d_e(h_i, h_j) &= f_X(pos_i) = f_X(x_i, y_i)\\  \end{align}  \begin{align}  \mu = pos_j = (x_j, y_j)\\  \Sigma =  \begin{bmatrix}  \rho & 0\\  0 & \rho\\  \end{bmatrix}  \end{align}  The idea behind the function $L$ is that histograms in closeby areas  should be similar and the similarity should decrease the further away  two positions are. This is modeled as a 2-dimensional Gaussian with 0  covariance (Figure~\ref{fig:model}). The variance is depended on the  desired accuracy ($\rho$): the lower the variance, the more punctuated  a certain location is but also the higher the risk that a totally  wrong measurement occurs. The following visualization are based on  color histograms (and not texton histograms) for easier visual  analysis.  \begin{figure}[h!]  \begin{center}  \includegraphics[width=0.7\columnwidth]{figures/model-crop/default_figure}  \caption{{\label{fig:model} Ideal  histogram similarity for a given position. Histograms taken at  positions close to $\textbf{x} = (400, 300)$ should be similar to  this histogram. The further away the position of a certain  histogram, the lower the ideal similarity should be.%  }}  \end{center}  \end{figure}  \begin{figure}[h!]  \begin{center}  \includegraphics[width=0.7\columnwidth]{figures/mosaic_enlarged/default_figure}  \caption{{\label{fig:map} This figure shows a map with a repeating pattern: two  yellow rectangles.%  }}  \end{center}  \end{figure}  \begin{figure}[h!]  \begin{center}  \includegraphics[width=0.7\columnwidth]{figures/repeating-crop/default_figure}  \caption{{\label{fig:repeating} Repeating map. Heatmap colors shows losses per region  (smoothed using a Gaussian filter). Mean loss: 67%  }}  \end{center}  \end{figure}  In future research, the ``bad regions'' could be optimized using an  optimization approach such as an evolutionary algorithm. It also  allows to show the similarities for a fixed position, and the loss for  a fixed positions based on the expected similarity and the actual  similarity.