Digital Signal Processing Reference
In-Depth Information
Theory of morphological operators has been formulated in the general framework
of complete lattices [
23
]: a complete lattice
(L,
≤
)
is a partially ordered set
L
with
, a supremum written
, and an infimum written
, such that every
order relation
≤
subset of
L
has a supremum (smallest upper bound) and an infimum (greatest lower
bound). Let
L
be a complete lattice. A dilation
δ
:
L
→
L
is a mapping commuting
with suprema, i.e.,
δ
i
X
i
=
i
δ (
X
i
)
.Anerosion
ε
:
L
→
L
commutes with
infima, i.e.,
δ
i
X
i
=
i
δ (
X
i
)
. Then the pair
(ε, δ)
is called an adjunction on
L
if for very
X
. Mathematical morphology is
also characterized by its domain of invariance in the complete lattice
,
Y
∈
L
, it holds:
δ(
X
)
≤
Y
⇔
X
≤
ε(
Y
)
L)
of the space
of image values. Morphological operators
ψ(
·
)
:
L
→
L
commutate with a group
of transformations
G
(
·
)
:
L
→
L
of image values, i.e., for any
f
(
x
)
∈
F(
E
, L)
we
have
ψ(
G
(
f
))(
x
)
=
G
(ψ(
f
))(
x
)
or
f
(
x
)
−→
ψ(
f
)(
x
)
G
(
f
)(
x
)
−→
ψ(
G
(
f
))(
x
)
◦
ψ
is equivalent to the invariance of the
Obviously the commutativity of the product
G
ordering
≤
under the transformation
G
(
·
)
. The group of invariant transformations
G
, e.g., in gray level
images, morphological operators commute with anamorphosis (i.e.,
G
(
·
)
depends on the physical properties of each particular
L
(
·
)
is a strictly
increasing mapping).
Dilation and erosion can be also computed using an eikonal PDE [
2
]:
∂
u
t
=±∇
u
,
(1.3)
with initial conditions
u
leads to the dilation and the
sign—to an erosion using an isotropic structuring element. Some advantages of the
continuous formulation are, on the one hand, the fact that required elements (partial
derivatives and Euclidean norm) do not required an ordering and, on the other hand,
as other standard methods for numerical solutions of PDEs, the continuous approach
allows for sub-pixel accuracy of morphological operators.
In addition, dilation and erosion can be also studied in the framework of convex
analysis, as the supremum/infimum convolution in the
(
x
,
y
,
0
)
=
f
(
x
,
y
)
. The sign
+
algebras,
with the corresponding connection with the Le
ge
ndre transf
or
m [
26
]. More precisely,
the two basic morphological mappings
(
max
,
+
)
/
(
min
,
+
)
F(
E
,
R
)
→
F(
E
,
R
)
are given respectively
by
δ
b
(
f
)(
x
)
=
(
f
⊕
b
)(
x
)
=
sup
h
E
(
f
(
x
−
h
)
+
b
(
h
)) ,
(1.4)
∈
and
ε
b
(
f
)(
x
)
=
(
f
b
)(
x
)
=
h
∈
E
(
inf
f
(
x
+
h
)
−
b
(
h
)) .
(1.5)
where
the
canonical
family
of
structuring
functions
are
the
paraboloids
2
)
=−
x
b
a
(
x
.
2
a
