Thesis (PDF) - Signal & Image Processing Lab

18 CHAPTER 2. THEORETICAL BACKGROUND
end up with a non natural description of the inclusion. Naturally, in the example of
Fig. 2.3(a), one would have expected to have the two small squares included into the
gray rectangle, and included into the white background. But the inclusion for these
trees is mostly driven by the gray level rather than by the geometrical inclusion.
Finally, we see that we need both trees if we want to have the two small squares
represented, since each of them appears in one description, and not in the other.
Now, instead of upper and lower sets, lets us introduce a term of shape. A shape
is a connected component of upper or lower set without any holes in it. If a hole
exists, it is filled. The shape corresponds to the connected component and its filled
”holes”.
The sorting of shapes can then be made thanks to their geometrical inclusions.
We can then create a tree structure as follows: each node corresponds to a shape;
descendants are the shapes included into it, and the parent is the smallest shape that
contains it (see Fig. 2.3(b) ). Each shape can specify either a gray level difference
between it and its father or an absolute gray level.
This will give us one single inclusion tree describing the image, in which a white
object on black background is represented in the same manner as a black object on
a white background.
2.3 Theoretical Background on semilattices
Mathematical Morphology is a nonlinear image processing theory, which was based
on complete lattices. In [21] Keshet has extended its scope to complete semilattices,
which are more general. This section provides a brief overview of Mathematical
Morphology on complete semilattices.
2.3.1 Complete Semilattices and Lattices
A partially ordered set A is a set associated with a binary operator ≤, satisfying the
following properties for any x, y, z ∈ A: reflexivity (x ≤ x), anti-symmetry (x ≤
y, y ≤ x ⇒ x = y), and transitivity (x ≤ y, y ≤ z ⇒ x ≤ z). In a partially ordered
set A, the least majorant ∨X (also called supremum) of a subset X ⊆ A is defined
as an element a0 ∈ A, such that: