Image Processing Reference
In-Depth Information
addition of a set of elements. For a set of image pixels (
x
,
y
) of image
A
,
f
(
x
,
y
)
and structuring element
B
, dilation is defined as
DABAB
(,)
=⊕
=+ +
{(
xpyp xy FppB
,
) :( ,) ,(
∈
, ) }
∈
(9.1)
x
y
xx
∪
=
(
Ab
+
)
bB
∈
Erosion removes the structures of certain shapes and sizes, given by a struc-
turing element, and it shrinks the objects. It is an operation that combines
two sets using vector subtraction of a set of elements, or said in another way,
erosion of set
A
by set
B
is the set intersection of all negative translates of set
A
by elements of set
B
or a set of all positions where set
B
fits inside set
A
.
Erosion of image
A
by structuring element
B
is defined as
{
}
AB
Θ
=
forevery
bB
∈
,
exists an
aA
∈
such that
xab
= −
{
}
AB xbA
Θ
=+∈
forever
y
bB
∈
(9.2)
∪
(
Ab
)
−
bB
∈
Again, dilation and erosion can be combined to get an opening or closing
operation.
Opening is erosion followed by dilation. Opening can be used to remove
small objects, protrusions from objects and connections between objects:
AB AB B
=−⊕
(9.3)
(
)
Closing is dilation followed by erosion. It removes all holes and gaps in the
image objects:
AB ABB
i
=⊕
)
Θ
(9.4)
(
9.2.1 Greyscale Mathematical Morphology
Greyscale morphological operations are an extension of binary morphological
operations to greyscale images. The structuring element may be flat where
the intensity variation is not continuous or non-flat where the intensity of the
structuring element is continuous. If
a
(
x
,
y
) and
b
(
x
,
y
) are the greyscale image
and flat structuring element, greyscale dilation is defined as the maximum
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