Image Processing Reference
In-Depth Information
foreground or it does not. The background of the image is not considered. Conse-
quently, the erosion may only model minterms that do not have negation. This limi-
tation is related to the lattice structure representation of the increasing function.
Once an input is found into which the structuring element will fit (i.e., its output is
1), it may be safely assumed that all inputs above it in the lattice also have an output
of 1. However, for the nonincreasing function no such order may be assumed.
In order to produce a morphological representation of a nonincreasing function
such as the one above, it is necessary to use the hit-or-miss transform . 7 In this oper-
ation, the kernel of structuring elements is split into two parts: foreground and
background. They are linked in pairs—one from the foreground and one from the
background. The output is true only if the foreground structuring element fits the
foreground of the image while at the same time the background structuring element
fits the background. While this would be an AND function in Boolean algebra, in
set theory terminology it is an intersection
.
The only problem in the example just described is caused by the operator below,
which must be decomposed into a foreground and background element as shown.
Background
Foreground
SE, g i
SE, b i
The cells of the structuring element without negation are placed in the foreground
set. The cells with negation are inverted and placed in the background set. These are
applied to the background of the image.
The structuring elements without negation, i.e., those corresponding to increas-
ing functions, simply have an empty background set. Therefore, in morphological
and set notation the operation is written as
[
]
(
)
(
)
I
Θ
b I
Θ
g
,
(5.9)
i
i
i
where b i and g i are the corresponding structuring element pairs in the foreground
and background sets respectively,
I is the in-
verted image such that the structuring element g i is applied to the background pix-
els.
Θ
is the morphological erosion and
Increasing filters tend to work well in removing certain types of signal-inde-
pendent additive noise. Bear in mind that the more black pixels there are in the in-
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