Image Processing Reference
In-Depth Information
We emphasize that it is the coordinate transformation that determines what
I
20
and
I
11
represent and detect. Central to the generalized structure tensor is the har-
monic function pair
ξ
(
x, y
) and
η
(
x, y
), which creates new coordinate curves to rep-
resent the points of the 2D plane. An image
f
(
x, y
) can always be expressed by such
a coordinate pair
ξ
(
x, y
) and
η
(
x, y
) as long as the transformation from (
x, y
) to
(
ξ, η
) is one-to-one and onto. The deformation by itself does not create new gray
tones, i.e., no new function values of
f
are created, but rather it is the isogray curves
of
f
that are deformed. The harmonic coordinate transformations deform the ap-
pearance of the target patterns to make the detection process mathematically more
tractable. In the principle suggested by theorem 11.1, these transformations are not
applied to an image because they are implicitly encoded in the utilized complex fil-
ters. The deformations occur only in the idea, when designing the detection scheme
and computing the filters.
11.4 Discrete Approximation of GST
For computation of the generalized structure tensor parameters, the ILST of the im-
age
f
,
(
f
)(
x, y
)=(
D
x
f
+
iD
y
f
)
2
(11.53)
is needed. This is, however, the same complex image that is used to compute the
ordinary structure tensor, discussed in Eq. (10.79) and in theorem 10.4. Accordingly,
provided that the image is densely sampled to a sufficient degree, the ILST image
can be obtained through:
(
f
)(
x, y
)=(
D
x
f
(
x, y
)+
iD
y
f
(
x, y
))
2
|
(
x,y
)=(
x
j
,y
j
)
=[(
f
x
j
+
if
y
j
)]
2
(11.54)
where (
x
j
,y
j
) is a point on the grid on which the original discrete image
f
is defined.
Below we discuss how to estimate the GST elements on the discrete Cartesian grid,
(
x
j
,y
j
).
Case 1: Estimation of
I
20
with known analytic expression of
ξ
(
x, y
)
Assuming that an analytic expression of
ξ
(
x, y
) is known explicitly, it follows from
this that
(
ξ
)(
x, y
) is known. An approximation of
I
20
can be obtained by substitut-
ing the reconstructed (from its samples)
(
f
),
(
f
)(
x, y
)=
j
ψ
(
x
−
x
j
,y
−
y
j
)
(
f
)(
x
j
,y
j
)
(11.55)
into Eq. (11.45)
I
20
=
j
(
f
)(
x
j
,y
j
)
y
j
)[
(
ξ
)(
x, y
)
]
∗
dxdy
ψ
(
x
−
x
j
,y
−
|
(
ξ
)(
x, y
)
|
=
j
(
f
)(
x
j
,y
j
)(
w
20
)
∗
(11.56)
j
