Artificial vision is a relatively new field, and sometimes imports ideas from older sciences, basically, reasoning by analogy. This is the case of entities as ridges and valleys, which has been imported from geomorphology, mainly from the sub-field known as hydrology. The words 'ridge' and 'valley' do not refer here to any kind of special line but to a region whose shape remembers that with the same name in a relief. In the field of hydrology one of the most studied concepts is the drainage pattern of a landscape, directly related to its ridges and valleys. In the last century and beginning of the present there was a big discussion about how to mathematically define the lines sketching the drainage pattern of a landscape. Artificial vision has imported ideas from geomorphology: we can see a 2-dimensional image as the sampling of a landscape, and then extract the topographic structure of the image to single out interesting features in it. In the topographic structure is included the drainage pattern, therefore, the discussion of how characterize it has also been imported from hydrology to artificial vision. Depending on what mathematical definition is used, the characterized feature is not really the actual drainage pattern of the underlying landscape. However, in my opinion, this is a void discussion: most of the times, when a definition fails to detect the lines forming the drainage pattern, this really doesn't matter since the feature we are looking for in the image is not its drainage pattern. The problem most of the times arises just as a question of names and sometimes for using a bad analogy between image features and hydrologic landscape concepts. Therefore, at least from a pragmatic viewpoint, the only thing that we should check is whether our definition of lines sketching the ridges/valleys is suitable to extract from the image the desired features.
In section 2 I overview the source of the analogy: the hydrology of a landscape and the importance of drainage patterns in this field. In section 3, I give some motivations for extracting lines that are or resemble the drainage patterns. I use the word 'resemble' in the sense of being lines that somehow sketch the ridges/valleys of a landscape, without necessarily coincide with its drainage pattern. Finally, in section 4 we will review different definitions that attempt to characterize mathematically the drainage patterns, such that they are used to extract interesting features of images, despite the fact that they don't really characterize the true drainage pattern of the underlying landscape. In fact, these characterizations give rise to entities such that, points and even curve segments of the landscape are, simultaneously, instances of them.
Along this overview, I will constrain myself to images 
, mostly for d=2 (earth landscape) and being 
and 
incommensurable spaces, but I want also to notice that ridge/valley-like structures (to define in section 2) can be also defined for and being commensurable axes (e.g. range images) [27, 69] and, in general, for hypersurfaces 
[38, 91, 31, 88, 89, 32, 62, 90, 10].