Digital Signal Processing Reference
In-Depth Information
image will be used to compare the various alternatives of morphological operators
and to illustrate the interest of such operators.
The dilation and erosion of an image
f
(
x
)
∈
F(
E
,
PDS
(
n
))
by structuring ele-
∨
and infimum
∧
are respectively given by
ment
B
according to supremum
A
δ
B
,
(
f
)(
x
)
=
∨
:
A
∨
=
[
f
(
z
)
]
,
z
∈
B
x
.
(1.24)
z
A
∈
B
x
ε
B
,
(
f
)(
x
)
=
∧
:
A
∧
=
[
f
(
z
)
]
,
z
.
(1.25)
z
Considering for instance
B
equal to a square of 3
×
3 pixels, four examples of the
dilation
δ
B
,
(
of the image test are given in Fig.
1.7
d, e.
It is compared in particular the supremum and the infimum defined by the lexico-
graphic total ordering
f
)
and the erosion
δ
B
,
(
f
)
1
lex
(priority given to the energy then to the eccentricity), the
lexicographic total ordering
≤
2
lex
(inverted priorities), the spectral sup/inf on geomet-
ric mean basis, and the asymptotic sup/inf using the counter-harmonic mean (with
P
≤
10). Effects of dilation/erosion operators are easily interpreted according to
the underlying supremum/infimum. Dilation and erosion are the basic operators to
define useful morphological filters. The two basic morphological filters, as prod-
=±
ucts of adjunction
δ
B
,
, ε
B
,
, are the opening
γ
B
,
(
)
=
δ
B
,
ε
B
,
(
)
and the
f
f
)
=
ε
B
,
δ
B
,
(
)
. Opening and closing and idempotent operators
closing
ϕ
B
,
(
f
f
(stable under the iteration), i.e.,
γ
B
,
γ
B
,
(
)
=
γ
B
,
(
f
f
)
, the opening (closing) is
an anti-extensive (extensive) operator; i.e.,
f
.We
will not insist again on the fact that these algebraic properties are only valid in case
where the pair of dilation/erosion forms an adjunction. The geometrical meaning of
the opening (closing) is to remove image structures of high (low) value, according to
the ordering, which are thinner than the structuring element
B
, the image structures
larger than
B
preserve their values. Figure
1.7
f, g give respectively the results of
closings and openings of the test image.
The obtained results produce quite different PDS
≥
γ
B
,
(
f
)
and
f
≤
ϕ
B
,
(
f
)
(
)
matrix-valued images, but at
first sight we cannot say which one is better or which one is more robust since that
will depend on the application. In order to complete this comparison, we propose to
compute other morphological filters based on dilation/erosion. For instance, using
the Riemannian distance for PDS
n
matrices, as defined in Eq. (
1.8
), we can easily
evaluates the norm of the morphological gradient as the image
(
n
)
d
Rie
δ
B
,
(
)
.
B
,
(
f
)(
x
)
=
f
)(
x
), ε
B
,
(
f
)(
x
Figure
1.8
d depicts the results for the four strategies of dilation/erosion, which can
be interpreted with the help of the images (b) and (c) of spectral energy and spectral
eccentricity. Obviously, the lexicographic approaches are more selective in terms
of the contours detected in comparison with the spectral or the counter-harmonic
