A grey-level image may be seen as a topographic relief, where the grey level of a pixel is interpreted as its altitude in the relief. A drop of water falling on a topographic relief flows along a path to finally reach a local minimum. Intuitively, the watershed of a relief correspond to the limits of the adjacent catchment basins of the drops of water.
In image processing, different watershed lines may be computed. In graphs, some may be defined on the nodes, on the edges, or hybrid lines on both nodes and edges. Watersheds may also be defined in the continuous domain [1]. There are also many different algorithms to compute watersheds.
For a segmentation purpose, the length of the gradient is interpreted as elevation information.
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The idea has been introduced in 1979 by S. Beucher and C. Lantuéjoul in [2]. It consists in placing a water source in each regional minimum, to flood the relief from sources, and build barriers when different sources are meeting. The resulting set of barriers constitutes a watershed by flooding.
Intuitively, a drop of water falling on a topographic relief flows towards the "nearest" minimum. The "nearest" miniumum is that minimum which lies at the end of the path of steepest descent. In terms of topography, this occurs if the point lies in the catchment basin of that minimum. The previous definition does not verify this condition.
S. Beucher and F. Meyer introduced in [3] an algorithmic inter-pixel definition of the watershed, given the following procedure:
1. Label each minimum with a distinct label. Initialize a set S with the labeled nodes.
2. Extract from S a node x of minimal altitude F, that is to say F(x) = min{F(y)|y ∈ S}. Attribute the label of x to each non-labeled node y adjacent to x, and insert y in S.
3. Repeat Step 2. Until S is empty.
Previous notions focus on catchment basins, but not to the produced separating line. The topological watershed was introduced by M. Couprie and G. Bertrand in 1997[4], and beneficiate of the following fundamental property. A function W is a watershed of a function F if and only if W ≤ F and W preserves the contrast between the regional minima of F; where the contrast between two regional minima M1 and M2 is defined as the minimal altitude to which one must climb in order to go from M1 to M2[5].
Different approaches may be employed to use the watershed principle for image segmentation.
One of the most common watershed algorithms was introduced by F. Meyer in the early 90's.
The algorithm works on a gray scale image. During the successive flooding of the grey value relief, watersheds with adjacent catchment basins are constructed. This flooding process is performed on the gradient image, i.e. the basins should emerge along the edges. Normally this will lead to an over-segmentation of the image, especially for noisy image material, e.g. medical CT data. Either the image must be pre-processed or the regions must be merged on the basis of a similarity criterion afterwards.
The non-labeled pixels are the watershed lines.
Watersheds as optimal spanning forest have been introduced by Jean Cousty et al.[6] They establish the consistency of these watersheds: they can be equivalently defined by their “catchment basins” (through a steepest descent property) or by the “dividing lines” separating these catchment basins (through the drop of water principle). Then they prove, through an equivalence theorem, their optimality in terms of minimum spanning forests. Afterward, they introduce a linear-time algorithm to compute them. It is worthwhile to note that similar properties are not verified in other frameworks and the proposed algorithm is the most efficient existing algorithm, both in theory and practice.
In 2007, C. Allène et al.[7] established links relating Graph Cuts to optimal spanning forests. More precisely, they show that when the power of the weights of the graph is above a certain number, the cut minimizing the graph cuts energy is a cut by maximum spanning forest.
The image foresting transform (IFT) of Falcao et al.[8] is a procedure computing shortest path forests. It has been proved by J. Cousty et al.[9] that when the markers of the IFT corresponds to extrema of the weight function, the cut induced by the forest is a watershed cut.
The random walker algorithm is a segmentation algorithm solving the combinatorial Dirichlet problem, adapted to image segmentation by L. Grady in 2006 [10]. In 2009, C. Couprie et al. proved that when the power of the weights of the graph converge toward infinity, the cut minimizing the random walker energy is a cut by maximum spanning forest [11].
A hierarchic watershed transformation converts the result into a graph display (i.e. the neighbor relationships of the segmented regions are determined) and applies further watershed transformations recursively.