Image Processing Reference
In-Depth Information
a Example of a metric Gabriel center
b Example of a cell which is not a metric Gabriel center
Fig. 10.17 Relation between distance fields, metric Gabriel balls, and union of balls
from the seeds, not the actual seeds position. Taking the minimally-valued cells in
this neighborhood of radius 1, we can see that they can be partitioned in two sets of
distance 2, which means that this partition forms a diameter for the neighborhood.
In fact, this neighborhood is a metric Gabriel ball for a dilation of the union balls of
of radius 3 centered at each of the three seeds, as shown in the third configuration.
In Fig. 10.17b, we can see the converse. The first configuration shows a cell,
indicated with a little dot, that is not a metric Gabriel center. Indeed, the two seeds
do not form a diameter for this ball. When considering only the neighborhood of this
cell, in the second configuration, there is no way to partition the three minimally-
valued cells of the neighborhood in two sets of distance 2.
To summarize, the distance field values in the neighborhood of a cell allows it to
detect whether it is a metric Gabriel center, and using this detection for the detection
of middle required in Sect. 10.4.2.1 and Sect. 10.4.2.2, we know that sufficiently
many pairs of seeds will be connected to obtained only one connected component
and construct the complete convex hull.
10.6
The Complete Cellular Automaton
We are now in position to write the complete cellular automaton constructing the
convex hull for arbitrary set of seeds, and for any cellular spaces 3 .Itissimplya
3
For the sake of brevity, some parts are not sufficiently general to be used without modifica-
tion in all cellular spaces, but we believe that the main “understanding” is in this restriction
version. We hope that any reader who needs the extra missing piece of generality will be
able to guess it.
 
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