Thesis (PDF) - Signal & Image Processing Lab

2.2. KNOWN TREE REPRESENTATIONS 17
merged to create region 5. Then, region 5 is merged with region 3 to create region 6.
Finally, region 6 is merged with region 4 and this creates region 7, corresponding to
the region of support of the whole image. In this example, the merging sequence is:
(1; 2)|(5; 3)|(6; 4). This merging sequence progressively defines the Binary Partition
Tree as shown in Fig. 2.2.
2.2.3 Tree of Shapes
The tree of shapes was proposed by P. Monasse in [20]. The tree of shapes represents
an image as a hierarchy of shapes, were dark and light shapes are treated in the
same way. It is build according to the inclusion order. Therefore each father vertex
area includes also all sons area. An article by Monasse and Guichard [12] includes a
compact definition of the tree of shapes. The following paragraphs are based on this
paper.
This tree is based on upper and lower level sets: Let E be an arbitrary set, and
f : E ↦→ IR. The upper level sets (or just level sets) of f are the collection of sets
{θt(f)}, t ∈ IR, given by:
The sets {θt(f) c } are called the lower level sets.
to:
θt(f) △ = {x ∈ E|f(x) ≥ t}. (2.1)
Given the collection of levels sets of f, the latter can be reconstructed according
f(x) = sup{t ∈ IR|x ∈ θt(f)}. (2.2)
Note that the level sets are nested; the family of upper (resp. lower) level sets is
decreasing (resp. increasing):
∀λ � t, θλ(f) ⊃ θt(f), θλ(f) c ⊂ θt(f) c
(2.3)
The relation (2.3) states that the level sets are nested. When going from the whole
level sets to their connected components, these relations are of course still true. Now,
a connected component can contain several connected components. These inclusions
can be represented into a tree, as shown in Fig. 2.3(a). As we can see, the connected
components of upper and lower level sets trees differ. In addition, we see that we