Thesis (PDF) - Signal & Image Processing Lab

26 CHAPTER 2. THEORETICAL BACKGROUND
of the sequence s.
For instance, {−1, +1, −1} � {−1, +1, −1, +1, +1}, and {} � {1, 1}.
Proposition 7. The prefix relation � is a partial ordering, and (S, �) is a complete
inf-semilattice, where the null sequence {} is the least element, and the infimum s ▽ r
is given by the greatest common prefix, i.e.:
where
(s ▽ r)n = sn, n = 1, . . . , L(s, r), (2.11)
L(s, r) = sup {n ∈ IN, n ≤ min [ℓ(s), ℓ(r)] | sn = rn} . (2.12)
The proof is given in [17].
In order to facilitate notation in the sequel, trailing zeros can be added to all
binary sequences. I.e., let Z be the operator that maps each sequence into an infi-
nite counterpart by appending an infinite number of zeros to it. The operator Z is
invertible, and the inverse Z −1 consists of removing all zero elements of an appended
sequence. Considering the set Z(S) of all 0-appended sequences, then it also forms a
complete inf-semilattice with respect to the order given by the concatenation �◦Z −1 .
In the remainder of the section, sequences are referred in this complete inf-semilattice
only, and use � to denote the partial order of 0-appended sequences as well. More-
over, the length of a 0-appended sequence will be by convention the length of its finite
counterpart, i.e., ℓ(s) actually meaning ℓ (Z −1 s).
2.4.3 Images of Shape Sequences
Images of Shape Sequences are defined in [17] by Keshet.
Definition 8. Let the image of shape sequences associated with an image f be the
mapping sf : E ↦→ Z(S), such that sf(x) is the sequence of differences s(τ) for all
shapes that contain τ(x), ordered from the largest to the smallest, followed by zeros.
For instance, referring to the image in Fig. 2.4(a), and its modified tree in Fig. 2.5(b),
for all x in region G, set sf(x) = {+1, +1, −1, −1, 0, . . .}, for all x in D and not in F,
sf(x) = {+1, −1, 0, . . .}, and for all x in F, sf(x) = {+1, −1, +1, +1, 0, . . .}.