Thesis (PDF) - Signal & Image Processing Lab

28 CHAPTER 2. THEORETICAL BACKGROUND
central regions in those images are characterized by the sequences {+1, −1, 0, . . .} and
{+1, +1, 0, . . .}, respectively, whose infimum is given by {+1, 0, . . .}. The supremum
f △ g, on the other hand, does not exist in this case.
Figure 2.6: (a)-(b) Two grayscale images, and (c) their infimum according to ⊑. The
supremum does not exist in this case.
Here is another way of understanding why the binary case lends itself to a lattice
framework: Binary images are represented by alternating binary sequences starting
with +1. These are fully characterized by their length alone, and are completely
ordered.
Even though working with an inf-semilattice means that dilations can not be
defined in all cases, the inf-semilattice theory described in [4, 5] assures that a great
deal can be accomplished nevertheless. Not only can erosions be defined, but also
openings, internal gradients, white top-hats, skeletons, openings by reconstruction,
etc.
Let us define the translation-invariant erosion:
ˆεS(f) = U −1 � ▽ y∈ ˇ SU{fy} � , (2.16)
where fy denotes the translation of f by y: fy(x) = f(x − y). Fig. 2.7 shows an
example of using the above “shape erosion” on a gray-scale image, compared to using
the standard gray-scale erosion.
The adjoint dilation is given by
ˆδS(f) = ▽{g | f ⊑ ˆεS(g)}, (2.17)