Image Processing Reference
In-Depth Information
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FIGURE 1.19: (a) Connected components of foreground points (differently
colored) and background points (white) of the 2-D point set in an (8,4) grid,
and (b) corresponding adjacency tree. (See color insert.)
the root of the tree. This component surrounds every other component
of the picture.
2. One of the vertices of an edge would be a foreground component and
the other would be a background component.
3. If a node X lies in a path from Y to the root of the tree, X surrounds
Y .
1.5 The Euler Characteristics
In continuous 2-D space, a polyhedral set is defined as a subset of a plane
that is the union of all the points, closed line segments, and closed triangles.
In 3-D continuous space, in addition to these primitives, we also include all
the closed tetrahedra in a 3-D polyhedral set. A topologically invariant feature
of a polyhedral set is its Euler characteristics, which is a number and defined
as follows.
Definition 1.1. Let the Euler number of a polyhedral set S be denoted as
χ(S), then:
1. χ(Φ) = 0, where Φ is the null set.
