Image Processing Reference
In-Depth Information
U
2
)=1
,
cos(
ϕ
)=
U
,a
U
/
(
U
∗
a
U
)=
a
U
,
U
/
(
a
(3.59)
to the effect that
V
is an ideal match if cos(
ϕ
) is 1.
Since all occurrences of
U
in the manuscript must be tested, the norm has to be
calculated for all (same size) subimages in the picture of the manuscript. The result
ofEq.(3.58)isillustratedinFig.3.7byshowingtheautomaticallyfoundpositionsof
three different characters, “alaph”, “mim”, and “tau”. The more “redish” an image,
the closer cos(
ϕ
) is to 1 at that location (middle). As example, two positions of
“Alaph” character are indicated.
With the same threshold of 0.925 applied to the three directional cosine images
(only one is shown in Fig. 3.7), all locations of the three individual characters could
be found and marked with respective colors. An “alaph” template painted the black
pixelvaluesintheoriginalimageasred,ifthedirectionalcosinecorrespondingtothe
template center was above the threshold. A similar procedure was applied to other
two templates to obtain the result at the bottom.
A
2
A
3
A
2
A
1
0.844 0.778 0.826
Exercise 3.7.
We compared the four images in Fig. 3.4 by computing the directional
cosines Eq. (3.57) between the image
A
1
and the images
A
2
-
A
4
. We obtained the
directional cosines, listed above.Would you prefer the directional cosines (i.e., the
Schwartz inequality ratios) to the triangle inequality ratios to measure similarities
between images?
HINT: Use average relative similarties in your judgement.
In summary, if we have a Hilbert space then we have a vector space on which we
have a scalar product. The existence of a scalar product always allows us to derive
a useful norm, the
L
2
norm. In turn, this norm allows to measure distances between
points in the Hilbert space. Furthermore, we can also measure the angles between
vectors by a scalar product.
