As previously seen, e.g. in section 2.3, it is crucial to identify connected (or continuous) contours. Up to now, we had to threshold the topological gradient with a not too small value, in order to identify connected contours, but this leads to thick identified edges, and also to consider more noisy points as potential edges.
We noticed that the edges correspond to valley lines of the topological gradient. It is of course possible to identify them by adapting the threshold coefficient, but we propose here to use the minimal path and fast marching techniques for identifying the valley lines of the topological gradient [48,49,54,56,118,91,107].
In the following, we consider any of the previous image processing problems. We only assume that the topological gradient 
has been defined and computed everywhere. The goal is to identify the valley lines corresponding to the most negative parts of the topological gradient.
This section summarizes the study presented in [2], in which several numerical experiments are shown in the case of segmentation and inpainting.