Digital Signal Processing Reference
In-Depth Information
where the translated image
f
b
(
t
)
=
f
(
t
−
b
)
=
f
(
t
1
−
b
1
,
t
2
−
b
2
) and the struc-
turing element
has a shape usually chosen according to some a priori knowl-
edge about the geometry of relevant and irrelevant image structures.
The dilation is commonly known as “fill,” “expand,” or “grow.” It can be used
to fill “holes” of size equal to or smaller than the structuring element. Used with
binary images, where each pixel is either 1 or 0, dilation is similar to convolution.
At each pixel of the image, the origin of the structuring element is overlaid. If
the image pixel is nonzero, each pixel of the structuring element is added to the
result using the “or” logical operator.
B
Erosion
consists of replacing each pixel of an image by the infimum
3
of its neigh-
bors within the structuring element
B
:
E
B
(
f
)
=
f
−
b
,
b
∈B
where the translated image
f
−
b
(
t
)
b
2
).
Erosion is the dual of dilation. It does to the background what dilation does
to the foreground. This operator is commonly known as “shrink” or “reduce.” It
can be used to remove islands smaller than the structuring element. At each pixel
of the image, the origin of the structuring element is overlaid. If each nonzero
element of the structuring element is contained in the image, the output pixel is
set to 1; otherwise, it is set to zero.
=
f
(
t
+
b
)
=
f
(
t
1
+
b
1
,
t
2
+
B
Opening
is defined as an erosion with the structuring element
followedbya
dilation with the reflected
4
structuring element
˜
B
:
O
B
(
f
)
=D
(
E
B
(
f
))
.
˜
B
Closing
consists of a dilation with the structuring element
B
followedbyanero-
˜
sion with the reflected structuring element
B
:
C
B
(
f
)
= E
(
D
B
(
f
))
.
˜
B
In a more general way,
opening
and
closing
refer to morphological filters that re-
spect some specific properties (Serra 1982; Breen et al. 2000; Soille 2003). Such mor-
phological filters were used for removing “cirruslike” emissions from far-infrared
extragalactic Infrared Astronomical Satellite (IRAS) fields (Appleton et al. 1993)
and for astronomical image compression (Huang and Bijaoui 1991).
4.3.2 Lifting Scheme and Mathematical Morphology
A nonredundant multiscale morphological transform can easily be built via the lift-
ing scheme by introducing morphological operators. For example, the Haar mor-
phological transform, also called the
G transform
,is
c
j
[
l
]
(
c
j
[
l
])
w
j
+
1
[
l
]
=
−P
(4.8)
c
j
[
l
]
c
j
+
1
[
l
]
=
+U
(
w
j
+
1
[
l
])
,
3
For discrete grayscale levels, the infimum is superseded by the pointwise minimum.
4
Reflection is essential here; see Serra (1982) and Soille (2003). For a symmetric structuring element,
the reflected version is the element itself.
