Image Processing Reference
In-Depth Information
where
w
20
j
is given by:
=
w
20
j
ψ
(
x − x
j
,y− y
j
) exp(
i
2tan
−
1
(
ξ
x
,ξ
y
))
dxdy
(11.57)
Here, we observe that the discrete kernel of
I
20
, which is
w
2
j
, is obtained by pro-
jecting the continuous kernel, Eq. (11.45),
w
20
(
x, y
)=
(
ξ
)(
x, y
)
|
=exp(
i
2tan
−
1
(
ξ
x
,ξ
y
))
(11.58)
(
ξ
)(
x, y
)
|
onto the space of band-limited signals. We note that the continuous kernel has mod-
ulus 1 except at
(
ξ
) is undefined. At these points,
w
20
can
safely be assumed to be 0, as the values of the integrand on a set of points with zero
measure do not affect
I
20
. Technically, Eq. (11.57) is a lowpass filtering followed by
discretization, which is also known as perfect sampling. Thus, Eq. (11.56) is essen-
tially a matching of the direction of the basis tangent vector field with the tangent
vector field of the image. This observation will prove to be useful in Sect. (11.4).
Naturally, the closed form of the integral in Eq. (11.57) is not possible to obtain for
most
ξ
s. However,
w
2
j
can be computed numerically and off line, e.g. [19], for pat-
tern recognition purposes. In one important case, when
ξ
=
x
, though, Eq. (11.57)
can be derived analytically and reduces to
w
2
j
=1which gives the ordinary struc-
ture tensor kernel for the image. Assuming that we will need
I
20
for a local image,
w
2
j
will however, be a window function, e.g., a Gaussian. There are other nontrivial
ξ
(
x, y
)s yielding analytically tractable kernels, those with polynomial derivatives.
We will discuss this class further in Sect. 11.7.
Is it really worth computing
w
2
j
exactly through Eq. (11.57) to obtain a useful
approximation of
I
20
? The answer to this question depends on the application at
hand. The computation of
w
2
j
through Eq. (11.57) and then substituting it in Eq.
(11.56) yields robust approximations of
I
20
since the weight zero is automatically
given to the appropriate points of the kernel at the same time as all kernel coefficients
vary smoothly. If the digitized image
f
j
, represents a small neighborhood, it might be
worth computing
w
2
j
in the aforementioned “orthodox” fashion (i.e., by projection
on the band-limited functions), as this yields less biased estimates. However, this
may not be worth doing if the number of singularity points is negligible compared to
the total number of the image points, since the bias of the singularity points will be
negligible. In this case one can use the approximation:
w
j
=
exp[
i
2tan
−
1
(
ξ
x
,ξ
y
)]
∇ξ
(
x, y
)=0, where
|
x
=
x
j
,y
=
y
j
,
if
∇
ξ
(
x
j
,y
j
)
=0;
(11.59)
0
,
if
∇ξ
(
x
j
,y
j
)=0.
Case 2: Estimation of
I
20
with unknown analytic expression of
ξ
Often a digital image of a prototype pattern
ξ
j
is all that is known, and one would
like to know whether
ξ
j
occurs in a discrete image
f
j
or not. One cannot assume
that isocurves of the prototype are sampled isocurves of harmonic functions, i.e., the
