## 2 Method and Approach
### Constrained Path Estimation with Homotopic Shrinking  
The first task is the automatic extraction of the individual markers from the RGB map. Color coding is used to extract the individual tasks from the map. It is important that the individual segments have a unique uniform RGB coding and do not allow any color gradation to the neighboring region. The RGB colors can now be segmented into a passable region, a non-passable region, and tasks. In Figure 2 the passable region is shown in green, purple, and blue as disconnected regions and the non-travelable region is white. Similar coding is also found in the broader map in Section 3.1. The passable and non-passable regions can now be stored in a digital image. A digital image \(\wp\) can be stored via a quadruple with  
\[\wp=(V,m,n,B) \tag{1}\]  
where the two-dimensional elements of \(V=\mathbb{Z}^{2}\) are called pixels of the image [44]. \(B\) stands for the pixels in the background and \(F=V-B\) for the pixels of the image and therefore of the foreground. The neighborhood can be defined by the configuration \((m,n)=(4,8)\) or \((m,n)=(8,4)\). The neighborhood of a pixel \(p_{i}\) with a pixel \(p_{j}\) can be classified by a common edge \(n=4\) or a point contact \(n=8\)[44]. If two pixels share a common edge, they are 4-adjacent to each other. All pixels that are n-adjacent to a point \(p_{i}\) are stored in the set \(N_{n}(p_{i})\). A pixel chain \(J=\{p_{i},\ldots,p_{i}\}\) with \(J\in F\) consists of 8-adjacent points, where the consecutive pixels \(p_{k}\) and \(p_{k+1}\) are 8-adjacent to each other [44]. In Figure 2, the green pixel chain represents both a 4-adjacent and an 8-adjacent chain, while the purple pixel chain represents only an 8-adjacent chain. Since the blue region does not allow a chain due to the massive faces, this region must be thinned to a skeleton. To obtain a skeleton, the boundary of \(F\) is eroded step by step until only a one-pixel thin line is left. During erosion, the state of a so-called simple point must be checked for each pixel, which can be eroded in terms of topology preservation [45]. A pixel at the boundary of the foreground \(p\in F\) is simple and thus erodible if the topological properties do not change after the erosion of this point. The topological properties of the image are preserved if the number of objects, cavities, and holes remain the same after erosion [13; 45]. All points that do not meet this condition must not be eroded. This erosion can now be applied iteratively until there are no more simple points. If only the condition of the simple point is checked during the erosion, the erosion method is a so-called homotopy-preserving shrinking [45]. If now additional so-called endpoints \(W_{e}\) are declared as also not erodible, then it is a homotopy-preserving thinning method. Figure 2: Path generation with shrinking and thinning: image with different regions, erosion due to anchor point thinning, erosion due to anchor point shrinking. In this work, a so-called homotopy-preserving anchor point shrinkage is chosen for path-finding derived from thinning method of [13], due to its desirable properties of the centered path and topology preservation. Therefore, a simple point condition is checked by considering the preservation of Euler characteristics and the connectivity via an adjacency tree. While the thinning method of [13] uses other shape preservation criteria such as endpoints for the curve skeletons or surface skeletons, this work is restricted exclusively to homotopic shrinkage with so-called anchor points. Anchor points are points \(a\in A\) with \(A\subset F\) that must not be removed during erosion. Once the process has converged, the skeleton is obtained from the remaining pixels with \(S\subset F\). Figure 2 shows a comparison between anchor point thinning and anchor point shrinking of a simple example image. While thinning almost preserves the green and purple pixel chains, shrinking reduces them to a single pixel.