Thesis (PDF) - Signal & Image Processing Lab

2.4. SHAPE TREE SEMILATTICE BACKGROUND 27
Let the operator, that maps f to sf, to be denoted by U.
The original function f can be reconstructed from s by:
f(x) =
∞�
[sf(x)]n. (2.13)
n=1
Therefore, by defining, U ∗ {s} △ = � ∞
n=1 [s(x)]n for an arbitrary image of sequences
s : E ↦→ Z(S), we have U −1 {r} = U ∗ {r} for any image of sequences r in the range of
U.
The datum of the image of sequences sf associated to f is identical to that of the
tree of shapes of f. Indeed, all pixels in a shape have a common sequence prefix, and
all connected components of pixels sharing a common prefix is a shape.
2.4.4 Inf-Semilattice of Images of Shape Sequences
In [17], Keshet defines a type of complete inf-semilattice called Semilattice of Images
of Shape Sequences and a set of morphological operations in it.
Definition 9. Let the binary relation ⊑ be given by:
f ⊑ g ⇐⇒ sf(x) � sg(x), ∀x ∈ E. (2.14)
Select a fixed point in E, and denote it by o.
Theorem 1. Let ˜ FIN(E) be the set of all discrete natural functions, for which f(o) =
0. Then according to [17], ( ˜ FIN(E), ⊑) is a complete inf-semilattice, with infimum
given by
[f ⊓ g](x) =
∞�
[sf(x) ▽ sg(x)] n . (2.15)
n=1
Notice that, even though the binary case provides us with a lattice, in the gray-
scale case the supremum is not well defined, which leaves us with an inf-semilattice
only. The reason for that stems from the inf-semilattice structure of the set of se-
quences. Indeed, if f and g are given for instance by the images in Fig. 2.6(a) and (b),
respectively, then the infimum f ▽ g is given by the image in Fig. 2.6(c), because the