Image Processing Reference
In-Depth Information
exp(
D
ζ
)=exp(
cos
θD
ξ
) exp(
sin
θD
η
)
=exp(
sin
θD
η
) exp(
cos
θD
ξ
)
(11.16)
Paraphrasing,
D
ξ
and
D
η
together act as an operator basis pair for
any
translation
along a “line” (which is a linear combination of
ξ, η
) [31], which in turn makes
D
ζ
a classical
directional gradient,
but in the curvilinear coordinates (
ξ, η
). The
corresponding one-parameter Lie group of transformation is given by
T
(
ξ, η,
1
)=
ξ
=
ξ
+
cos
θ,
(11.17)
η
=
η
+
sin
θ,
with
θ
being a “directional” constant, characterizing a unique family of curves, which
are also the invariants of the operator
D
ζ
, Eq. (11.15).
−
ξ
sin
θ
+
η
cos
θ
= constant
(11.18)
The converse is also true, i.e., Eq. (11.18) uniquely represents
D
ζ
. In Fig. 11.3 we
show two curve families generated by using Eq. (11.18) for various
θ
s. T
he bott
om
row shows log(
x
+
iy
)=
ξ
(
x, y
)+
iη
(
x, y
), while the top row shows
√
x
+
iy
=
ξ
(
x, y
)+
iη
(
x, y
). Notice that when changing
θ
we generate the members of the
same pattern family, whereas when changing the
ξ, η
pair we generate completely
new families of patterns.
11.3 The Generalized Structure Tensor (GST)
We wish to know how well an arbitrary image can be described by means of analytic
curves. In our approach
ξ
and
η
are known a priori and are conjugate harmonic pairs,
whereas neither
θ
nor the gray values are known. We will attempt to fit a family of
curves as defined by Eq. (11.18) to an arbitrary image by finding its “closest member”
to the image. We do this by minimizing the following error or energy:
e
(
θ
)=
2
dξdη
=
2
dξdη
|
D
ζ
f
(
ξ, η
)
|
|
(cos
θD
ξ
+sin
θD
η
)
f
(
ξ, η
)
|
(11.19)
with respect to
θ
. Since we do not know if the image is really a member of the space
linearly spanned by the
ξ
,
η
pair, the next best thing is to find a member of this family
closest to our image and see if the error is small enough. Notice that
e
(
θ
) is a norm
(in the sense of
L
2
), and from this it follows that when
e
(
θ
) is zero for some
θ
, the
image
f
fulfills
D
ζ
f
(
x, y
)=0, almost for all
x, y
. By using Eq. (11.12), Eq. (11.19)
can be seen as the total error in a small translation. Furthermore, the error is the total
least square error as in the ordinary linear symmetry error function discussed in Sect.
10.10 except that the coordinates are now the curve pair
ξ, η
.
Example 11.5. To fix the ideas, we return to Example 11.2 and note that the opera-
tors
∂
∂ξ
∂
∂η
and
have the curves
