Thesis (PDF) - Signal & Image Processing Lab

36 CHAPTER 2. THEORETICAL BACKGROUND
The trivial supremum ⊔ of two alternating sequences V1 and V2 is given by:
�
⎧
⎪⎨ V1, V2 ⊑ V1,
V1 ⊔ V2 = V2,
⎪⎩
∃,
V1 ⊑ V2,
otherwise.
(2.25)
Table 2.2 exemplifies calculations of the infimum and the supremum of pairs of
alternating sequences.
Relation Infimum Supremum
0 {9, −2, 8, −4} ⊑ {9, −2, 8, −4, 7} {9, −2, 8, −4} {9, −2, 8, −4, 7}
1 {−1, 7, −9, 2} �⊑ {−5, 4, −7, 1, −3} {−1} � ∃
2 {7, −2, 5} ⊑ {7, −2, 6, −1, 10} {7, −2, 5} {7, −2, 6, −1, 10}
3 {7, −2, 7} �⊑ {7, −2, 6, −1, 10} {7, −2, 6} � ∃
4 {1, −3, 5} �⊑ {−4, 6, −2, 8, −3} {0} � ∃
Table 2.2: Infimum and supremum of pairs of alternating sequences.
The following operator can now be defined:
ˆεB(f) = V −1 {⊓y∈B sV{fy}} , (2.26)
where fy denotes the translation of f by y: fy(x) = f(x − y). It can be shown that
the above operator is an erosion in the complete inf-semilattices of BTV-transform
images. The adjoint dilation is given by
ˆδB(f) = V −1 {⊔y∈BV{fy}} , (2.27)
Like for any erosion in a complete inf-semilattice, the opening operator ˆγB asso-
ciated to ˆεB is well defined and is given by ˆγB = ˆ δB ˆεB.