Image Processing Reference
In-Depth Information
Fig. 3.2. Norms under scaling of 2D vector-valued images are illustrated by a digital color
image. On the
left
the original image
A
, having three color components at each image point,
with the norm
A
= 874, is shown. On the
right
, 0
.
5
A
with the norm
0
.
5
A
= 437 is
displayed
Example 3.2. The left image in Fig. 3.2 has three real valued color components at
each pixel, i.e., RGB color values. Each color component varies between 0 and 1,
representing the lowest and highest color component value, respectively. The image
is thus a vector-valued image that we can call
A
. The darker right image in Fig. 3.2
is0
.
5
A
,i.e.,itisobtainedbymultiplyingallcolorcomponentsbythescalar0.5.The
normof
A
iscomputed tobe874, byusing theexpression inEq.(3.20),whereas the
norm of0
.
5
A
using the same expression is computed to be 437. As expected from a
true norm, scaling all vector pixels by the scalar 0.5 results in a scaling of the image
norm with the same scalar.
Exercise3.5.
If the image
A
shown in Fig. 3.2 had been white, i.e., all three color
components were equal to 1.0, then one obtains
= 1536
, where the norm is in
the sense of Eq. (3.20). How many pixels are there in
A
? Can you find how many
rows and columns there are in
A
?
HINT: Use a ruler.
A
The images in Fig. 3.3 illustrate the triangle inequality of the norm. They rep-
resent, clockwise from top left,
A
1
,
A
2
,
A
3
, and
A
1
+
A
2
+
A
3
, respectively.
At any image point the pixel is a vector consisting of three color components.
The norms of these images are
A
1
= 517,
A
2
= 527,
A
3
= 468, and
= 874, respectively. As expected from a norm, we obtain
A
1
+
A
2
+
A
3
≤
A
1
+
A
2
+
A
3
A
1
+
A
2
+
A
3
.
To illustrate both the triangle inequality and the scaling property of the norms,
we study the quotient
Q
Q
(
A
1
,
A
2
)=
A
1
+
A
2
A
1
A
2
≤
1
(3.21)
+
which equals 1 if both
A
1
and
A
2
are vectors that share the same direction. In other
words, if
