Resistance-Geodesic Distance and Its Use in Image Segmentation

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Measuring the distance is an important task in many clustering and image-segmentation algorithms. The value of the distance decides whether two image points belong to a single or, respectively, to two different image segments. The Euclidean distance is used quite often. In more complicated cases, measuring the distances along the surface that is defined by the image function may be more appropriate. The geodesic distance, i.e. the shortest path in the corresponding graph, has become popular in this context. The problem is that it is determined on the basis of only one path that can be viewed as infinitely thin and that can arise accidentally as a result of imperfections in the image. Considering the k shortest paths can be regarded as an effort towards the measurement of the distance that is more reliable. The drawback remains that measuring the distance along several paths is burdened with the same problems as the original geodesic distance. Therefore, it does not guarantee significantly better results. In addition to this, the approach is computationally demanding. This paper introduces the resistance-geodesic distance with the goal to reduce the possibility of using a false accidental path for measurement. The approach can be briefly characterised in such a way that the path of a certain chosen width is sought for, which is in contrast to the geodesic distance. Firstly, the effective conductance is computed for each pair of the neighbouring nodes to determine the local width of the path that could possibly run through the arc connecting them. The width computed in this way is then used for determining the costs of arcs; the arcs whose use would lead to a small width of the final path are penalised. The usual methods for computing the shortest path in a graph are then used to compute the final distances. The corresponding theory and the experimental results are presented in this paper.