Image Processing Reference
In-Depth Information
is the projection of the continuous kernel w 11 ( x, y ) onto the space of band-limited
signals. Using the kernel w 1 j in Eq. (11.65) yields a biased estimate of I 11 as shown
below. The discrete kernel w 1 j is the perfect sampling of w 11 ( x, y ). The magnitudes
of the two kernels fulfill the inequality
|w 2 j |≤w 11 j which is a weaker relation-
ship than the equality relationship in the continuous case, Eq. (11.64). Consequently,
when
w 20
j
< w 11
j
|
|
even for one kernel coefficient j , we get
( f )( x j ,y j ) w 2 j
w 20
j
|
I 20 |
=
|
|≤
|
( f )( x j ,y j )
||
|
j
j
<
j
= I 11
w 11
j
|
( f )( x j ,y j )
|
(11.67)
will not attain the value I 11 even if the image is linearly
symmetric in harmonic coordinates. By using the triangle inequality, it can be shown
that
This implies that
|I 20 |
will attain the upper bound only when the discrete ILST of the image and
the kernel coefficients are collinear. This behavior is similar to the continuous case,
as can be seen by applying the triangle inequality to Eq. (11.45) and comparing the
result to (11.47). In order to avoid the bias introduced by the discretization process,
we will use the discrete kernel,
|
I 20 |
w 11
j
w 20
j
=
|
|
(11.68)
instead of Eq. (11.66) to compute
I 11 =
j
w 20
j
|
( f )( x j ,y j )
||
|
(11.69)
where w 20 is assumed to be available through either of the processes described in
Eqs. (11.57), (11.59), and (11.61).
11.5 The Generalized Hough Transform (GHT)
When an analytic expression for a target curve or a collection of curves is not avail-
able, provided that there is a discrete version of the target, ξ , it is still possible to
detect it in discrete images. However, in this section we consider the alternative ap-
proach, the generalized Hough transform (GHT). In the subsequent section, we will
establish that GST is a GHT, with the additional capability to recognize antitargets
during the target recognition. We will dwell only on the case when an analytic ex-
pression for the target curve is not available. The discussion when the target curve is
analytically available is analogous and is omitted.
The chief tool to achieve machine recognition of general curves in images is the
GHT [12, 111] which has been extensively studied [54, 118, 126, 177]. GHT is pop-
ular for its robustness, because even when the occurrence of a target curve is only
partial in an image, for some reason, including occlusion, the method can find the
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