Image Processing Reference
In-Depth Information
Exercise 3.6.
Prove that the Schwartz inequality holds for arrays with one or two
indices.
The theorem says
|
u
v
≤
|
u
,
v
1
(3.55)
which in turn yields that
u
,
v
−
1
≤
≤
1
(3.56)
u
v
When the dimension of the vector space is large (
N>
3) the concept of angle is
difficult to imagine. The following definition helps to bridge this difficulty. The idea
is to use the amplification factor that is needed to turn the Schwartz inequality into
an equality to pinpoint the angle between two vectors. This is possible because the
amplification factor is always in the interval [
−
1
,
1], as shown above.
Definition 3.5.
The angle
ϕ
between the two vectors
u
and
v
is determined via
cos
ϕ
=
u
,
v
(3.57)
u
v
Angle computation requires a scalar product. If we have a Hilbert space, however, it
comes with a method that produces the scalar product.
The Schwartz inequality, Eqs. (3.54-3.57), with a suitable scalar product e.g.,
(3.44)-(3.45) or (3.52)-(3.53) are extensively used in machine vision. In particular
Eq. (3.57) is known as the
normalized correlation
, where
U
is a template or an image
of an object that is searched for, and
V
is a test image which is usually a subpart of an
image that is matched with the template. The direction cosine between the template
and the test vector is measured by the expression in Eq. (3.57). When the two patterns
are vectors that share the same direction, then the system decides for a match. This
angular closeness is often tested by checking whether the absolute value of Eq. (3.57)
is above a threshold, which is empirically decided e.g., by inspecting the image data
manually or by using statistical tools such as a classifier or neural network.
Example 3.5.
ConsiderthephotographofthetextshowninFig.3.7.Thedigitalpho-
tographs of pages of topics or manuscripts such as this can be conveniently stored
in computers. Yet, to let computers process the stored data, e.g., to search for a par-
ticular word in such images, demands that the letters and words be located and rec-
ognized automatically by computers themselves. The goal is to automatically locate
and identify all characters of the relevant alphabet in digital pictures of text. This is
knownastheoptical character recognition (OCR)probleminimageanalysis.Weuse
theSchwartzinequalitytoconstructasimpleOCRsystemforthedisplayedscriptas
summarized next.
Let
U
be the character to be searched for, and
V
be the test image that is any
subimage of Fig. 3.7 having the same size as
U
. The angle
ϕ
between
U
and
V
is
determined via the equation
