Image Processing Reference
In-Depth Information
Fig. 3.3. Norms under addition of 2D vector-valued images are illustrated by adding three
digital color images. The
bottom right
is the addition of the other images. The norms, counter
clockwise from
top left
, are
A
1
= 517,
A
2
= 527,
A
3
= 468,
A
1
+
A
2
+
A
3
=
874, respectively
A
2
=
α
A
1
(3.22)
where
α
isapositivescalar,thenthequotient
Q
willequal1,whichisaconsequence
of the triangle inequality and the scaling property of the studied vector space. When
A
1
and
A
2
are those displayed in Fig. 3.2, the quotient
Q
equals to 1, because by
computationweobtain
= 437.Inview
ofthenormrepresentedbyEq.(3.20)andthetriangleinequality,relation(3.19),this
is expected.
However,becauseofthecontinuityof
Q
w.r.t.
A
2
,evenwhen
A
2
approximately
equals to
α
A
1
, we may expect that
Q
will be close to 1. To see whether this is
true, we study the corresponding quotients for the four images in Fig. 3.4, which are
Q
(
A
1
,
A
2
)=0
.
960,
Q
(
A
1
,
A
3
)=0
.
943,
Q
(
A
1
,
A
4
)=0
.
955. These suggest that
image
A
1
was likely obtained by multiplying
A
2
with a positive constant, although,
infact,thisisnottrue.Theimages
A
1
and
A
2
areinsteaddifferentviewsofthesame
scene, and are not obtainable from each other by scaling the color components. As
suggested already by theexample, i.e.,that allthreequotients areclosetoeach other
andnotfarawayfrom1,thequotient
Q
asameanstoestablishgeneralsimilaritiesof
imagesmaynotbereliable,althoughitcanbeusedtotestcertaintypesofsimilarities
A
1
+
A
2
= 1311,
A
1
= 874,and
A
2
