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+\section{Goal Detection}
+\label{p sec goal detect}
+
+The goal detection presented itself as a more difficult task. The color of the
+goal is white, and there are generally many white areas in the image from the
+robot camera, which have area larger than that of the image of the goal, for
+example the white field lines and the big white wall in the room with the
+field. To deal with the multitude of the possible goal candidates, we
+propose the following heuristic algorithm.
+
+\begin{figure}[ht]
+  \includegraphics[width=\textwidth]{\fig goal-detection}
+  \caption{Goal detection}
+  \label{p figure goal-detection}
+\end{figure}
+
+First, all contours around white areas are extracted by using a procedure
+similar to that described in the section \ref{p sec ball detection}. Unlike in
+the ball detection, the resulting binary mask undergoes some slight erosions
+and dilations, since in the goal shape detection the noise is undesired. Next,
+the \textit{candidate preselection} takes place. During this stage only $N$
+contours with the largest areas are considered further (in our experiments it
+was empirically determined that $N=5$ provides good results). Furthermore, all
+convex contours are rejected, since the goal is a highly non-convex shape.
+After that, a check is performed, how many points are necessary to approximate
+the remaining contours. The motivation behind this is the following: It is
+clearly visible that the goal shape can be perfectly approximated by a line
+with 8 straight segments. On an image from the camera, the approximation is
+almost perfect when using only 6 line segments, and in some degenerate cases
+when the input image is noisy, it might be necessary to use 9 line segments to
+approximate the shape of the goal. Any contour that requires a different number
+of line segments to be approximated is probably not the goal. The preselection
+stage ends here, and the remaining candidates are passed to the scoring
+function.
+
+The scoring function calculates, how different the properties of the
+candidates are from the properties, that an idealized goal contour is expected
+to have. The evaluation is happening based on two properties. The first
+property is based on the observation, that the area of the goal contour is much
+smaller than the area of its \textit{enclosing convex hull} \cite{convex-hull}.
+The second observation is that all points of the goal contour must lie close to
+the enclosing convex hull. The mathematical formulation of a corresponding
+scoring function can then look like the following:
+
+\begin{equation*}
+  S(c)=\frac{A(c)}{A(Hull(c))}+\displaystyle\sum_{x_i \in c}\min_{h \in Hull(c)
+  }(||x_i-h||)
+\end{equation*}
+
+The contour, that minimizes the scoring function, while keeping its value under
+a certain threshold is considered the goal. If no contour scores below the
+threshold, the algorithm assumes that no goal was found. An important note
+is that the algorithm is designed in such a way, that the preselection and
+scoring are modular, which means that the current simple scoring function can
+later be replaced by a function with a better heuristic, or even by some
+function that employs machine learning models.
+
+Our tests have shown, that when the white color is calibrated correctly, the
+algorithm can detect the goal almost without mistakes, when the goal is present
+in the image. Most irrelevant candidates are normally discarded in the
+preselection stage, and the scoring function improves the robustness further.
+Figure \ref{p figure goal-detection} demonstrates the algorithm in action. On
+the right is the binary mask with all found contours. On the left are the goal,
+and one contour that passed preselection but was rejected during scoring.
+
+One downside of this algorithm is that in some cases the field lines
+might appear to have the same properties, that the goal contour is expected to
+have, therefore the field lines can be mistaken for the goal. We will describe,
+how we dealt with this problem, in the section \ref{p sec field detect}.
+
+\section{Field Detection}
+\label{p sec field detect}
+
+The algorithm for the field detection is very similar to the ball detection
+algorithm, but some concepts introduced in the section \ref{p sec goal detect}
+are also used here. This algorithm extracts the biggest green area in the
+image, finds its enclosing convex hull, and assumes everything inside the hull
+to be the field. In here, when we extract the field, we apply strong Gaussian
+blurring and erosions-dilations combination to the binary mask, so that the
+objects on the field are properly consumed.
+
+\begin{figure}[ht]
+  \includegraphics[width=\textwidth]{\fig field-detection}
+  \caption{Field detection}
+  \label{p figure field-detection}
+\end{figure}
+
+Such rather simple field detection has two important applications. The first
+one is that the robot should be aware, where the field is, so that it doesn't
+try to walk away from the field. Due to time constraints, we didn't implement
+this part of the behavior. The second application of field detection is the
+improvement of the quality of goal and ball recognition. As was mentioned in
+the section on ball detection, the current algorithm might get confused, if
+there are any red objects in the robot's field of view. However, there
+shouldn't be any red objects on the field, except the ball itself. So, if
+everything that's not on the field is ignored, when trying to detect the ball,
+the probability of identifying a wrong object decreases. On the other hand, the
+problem with the goal detection algorithm was that it could be distracted by
+the field lines. So, if everything on the field is ignored for goal
+recognition, then the accuracy can be improved.
+
+\begin{figure}[ht]
+  \includegraphics[width=\textwidth]{\fig combined-detection}
+  \caption{Using field detection to improve ball and goal detection}
+  \label{p figure combined-detection}
+\end{figure}