Convex Hulls and Metric Gabriel Graphs - Cellular Automata in Image Processing and Geometry - page 182

Fig. 10.4 Convergence to many convex hulls

the wanted result. This second stage is simply a modified version of the majority
rule described earlier, which is enough because the result of the first stage is a con-
nected region. However, the transition from the first to the second stage requires a
mechanism that ensure a coherent global transition to the second stage, as it is not
desirable to be in the shrinking and growing stages at the same time. There are of
course some little technical difficulties, and the reader is redirected to the original
articles for more details. An example of initial configuration is shown in Fig. 10.5,
along with the result of the first and second stages on it.

10.4 The Metrical Point of View

The solution described in the previous section is a great improvement on the major-
ity rule since it handles an arbitrary set of seeds. However, one might ask whether it
is possible to get rid of the initial region and simply build the region from the seeds.
In the case of the Euclidean convex hull, we noted in Sect. 10.2, the definition of
the convex hull asks to add all required segments to the initial set of seeds in order

a Initial configuration bAfterErosion c After Expansion

Fig. 10.5 Stages from wrapped seeds to their convex hull: The initial wrapping (a) is shrunk
into (b) and is then grown to convex hull (c)