The steepest watershed: from graphs to images

The watershed transform is a powerful and popular tool for segmenting objects whose contours appear as crest lines on a gradient image : it associates to a topographic surface a partition into catchment basins, defined as attraction zones of a drop of water falling on the relief and following a line of steepest descent. To each regional minimum corresponds a catchment basin. Points from where several distinct minima may be reached are problematic as it is not clear to which catchment basin they should be assigned. Such points belong to watershed zones, which may be thick. Watershed zones are empty if for each point, there exists a unique steepest path towards a unique minimum. Unfortunately, the classical watershed algorithm accept too many steep trajectories, as they use too small neighborhoods for estimating their steepness. In order to nevertheless produce a unique partition they do arbitrary choices, out of control of the user. Finally, their shortsidedness results in unprecise localisation of the contours. We propose an algorithm without myopia, which considers the total length of a trajectory for estimating its steepness ; more precisely, a lexicographic order relation of infinite depth is defined for comparing non ascending paths and chosing the steepest. For the sake of generality, we consider topographic surfaces defined on node weighted graphs. This allows to easily adapt the algorithms to images defined on any type of grids in any number of dimensions. The graphs are pruned in order to eliminate all downwards trajectories which are not the steepest. An iterative algorithm with simple neighborhood operations performs the pruning and constructs the catchment basins. The algorithm is then adapted to gray tone images. The neighborhood relations of each pixel are determined by the grid structure and are fixed ; the directions of the lowest neighbors of each pixel are encoded as a binary number. Like that, the graph may be recorded as an image. A pair of adaptative erosions and dilations prune the graph and extend the catchment basins. As a result one obtains a precise detection of the catchment basin and a graph of the steepest trajectories. A last iterative ar X iv :1 20 4. 21 34 v1 [ cs .C V ] 1 0 A pr 2 01 2 algorithm allows to follow these downwards trajectories in order to detect particular structures such as rivers or thalweg lines of the topographic sur-