Image Processing Reference
In-Depth Information
image operations
. Two types of compositions -
parallel composition
and
serial
composition
- are important for our objectives.
Definition 2.8 (Parallel composition).
A parallel composition (by a point-
wise operation “
∗
”) of arbitrary two operators
O
1
and
O
2
that share a com-
mon domain
P
0
is defined as follows:
(
O
1
∗
O
2
)(
F
)
≡
O
1
(
F
)
∗
O
2
(
F
)
,
∀
F
∈P
0
.
(2.19)
O
2
in general, and a more ap-
propriate symbol may be used instead of the asterisk to designate a particular
type of composition (e.g., (
O
1
+
O
2
)(
The parallel composition is denoted by
O
1
∗
F
)
≡
O
1
(
F
)+
O
2
(
F
)).
The parallel composition
O
1
∗
O
2
represents the processing that generates
an image
H
by applying the pointwise operation
∗
to the result of the operator
O
1
appliedtoanimage
F
and that of the operator
O
2
applied to an image
G
(Fig. 2.11 (a)).
Definition 2.9 (Serial composition).
Serial composition
of two arbitrary
operators
O
1
and
O
2
, denoted by
O
1
·
O
2
,isdefinedasfollows:
(
O
1
·
O
2
)(
F
)
≡
O
1
(
O
2
(
F
))
,
(2.20)
where the domain of
O
1
is assumed to be coincident with the range of
O
2
.
The domain of
O
1
·
O
2
is equal to the domain of
O
2
; the serial composition
O
1
·
O
2
means the processing that is equivalent to applying the operator
O
1
to the output of the operator
O
2
(Fig. 2.11 (b)).
Remark 2.5.
More specifically, a serial composition
O
1
·
O
2
can be defined
well if the range of the operator
O
2
is contained in the domain of the operator
O
1
. The requirement that the domain of
O
1
should be coincident with the
range of
O
2
is added only for the sake of simplicity in theoretical analysis.
Two basic operators, the
inverse
and the
power
of an operator, are defined
using these compositions as follows.
Definition 2.10 (Inverse, the power of an operator).
The result of
n
times of serial compositions of the same operator
O
is called
n
-th
power
of
the operator
O
and is denoted by
O
n
.Thatis,
O
n
≡
O
n−
1
,n
O
·
≥
2
,
(2.21)
whereweassumethat
O
n−
1
always exists and its range is included in the
domain of
O
for all
n
2
.
For a given operator
O
, if there exists an operator
O
such that
≥
O
O
=
I
(= identity operator)
,
·
O
=
O
·
(2.22)
then
O
is called the
inverse operator
of
O
and is denoted by
O
−
1
.
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