Image Processing Reference
In-Depth Information
n
O
i
=
O
1
+
O
2
+
...
+
O
n
=(
...
((
O
1
+
O
2
)+
...
)+
O
n
,
(2.24)
i
=1
n
O
i
=
O
n
·
O
n−
1
·
...
·
O
2
·
O
1
=(
...
(
O
3
·
(
O
2
·
O
1
))
...
)
.
(2.25)
i
=1
These algebraic operations among image operators have the following prop-
erty.
Property 2.1.
For arbitrary image operators
O
1
,
O
2
,and
O
3
, the following
relations hold:
(
O
1
·
O
2
)
·
O
3
=
O
1
·
(
O
2
·
O
3
)
(associative law)
,
(2.26)
(
O
1
∗
O
2
)
·
O
3
=(
O
1
·
O
3
)
∗
(
O
2
·
O
3
) (right distributive law)
,
(2.27)
where we assume that domains and ranges of relating operators satisfy ap-
propriate conditions such that compositions “
·
”and“
∗
” in these equations
are meaningful (Fig. 2.11 (d), (e)).
Remark 2.6.
The left distributive law
O
3
·
(
O
1
∗
O
3
)=(
O
3
·
O
1
)
∗
(
O
3
·
O
1
)
(2.28)
does not hold true in general, and requires proof of correctness for individual
cases. In serial composition,
O
1
·
O
2
is not always equal to
O
2
·
O
1
.
2.3.5 Basic operators
Let us introduce here a few basic image operators that are relatively simple
but important in the subsequent parts of the text.
(a) Monotonic operator
Definition 2.11 (Monotonicity).
An operator
O
with the domain
P
is said
to be
monotonic
if the following relation holds:
O
(
F
)
≥
O
(
G
)
,
∀
F
∈P
,
∀
G
∈P
such that
F
≥
G
.
(2.29)
For example, an operator
O
1
which maps a set of binary images onto a set
of binary images may replace some of 1-voxels by 0-voxels, but never changes
0-voxels. Then the operator
O
1
is monotonic if that figures (a set of all 1-
pixels) in a binary image
F
cover figures in
G
implies that figures in the
output of the operator
O
1
for
F
cover figures in the output of
O
1
for
G
for
arbitrary binary images
. The local minimum filter and the local
maximum filter are examples of the monotonic operator. Thinning discussed
in Chapter 4 and 5 is not monotonic, although it never changes 0-voxels. In
fact, a circle is reduced to an isolated single pixel by thinning, but a long
stick-like figure that is smaller than the circle may be converted to a long thin
line segment by thinning.
F
and
G
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