Image Processing Reference
In-Depth Information
∂x
=
∂η
∂ξ
∂ξ
∂y
=
∂η
∂x
∂y
,
−
(11.2)
then
η
is said to be the
conjugate harmonic function
of
ξ
. Equivalently, the pair
(
ξ, η
) is said to be a
harmonic function pair
(HFP). Conversely, if two functions
with continuous second-order partial derivatives satisfy Eq. (11.2), then both are
harmonic, i.e., fulfill the Laplace equation. This is seen by applying
∂
∂x
∂
and
∂y
,
respectively, to the Cauchy-Riemann equations, which yield:
∂
2
ξ
∂x
2
∂
2
η
∂x∂y
,
∂
2
ξ
∂y
2
∂
2
η
∂y∂x
=
=
−
(11.3)
∂
2
η
∂x∂y
=
∂
2
η
wherewehave
∂y∂x
because of the continuity assumption on the second-
order derivatives. Accordingly, the conjugate harmonic function of a known har-
monic function,
ξ
, is found by solving the Laplace equation using the Cauchy-
Riemann equations as boundary conditions. These equations stipulate that the gradi-
ents of a harmonic function pair are orthogonal to each other at every point, i.e., they
are
locally orthogonal
.
An analytic function is generally a complex function and is characterized by the
fact that it has complex derivatives of all orders. It can be shown that the imaginary
part of any analytic function is the conjugate harmonic function of the real part. With-
out loss of generality, we can assume both
ξ
and
η
to be single-valued by imposing
proper restrictions. Then by definition Eq. (11.2), an HFP curve pair,
ξ
0
=
ξ
(
x, y
)
(11.4)
η
0
=
η
(
x, y
)
(11.5)
has orthogonal gradients at the same point. For nontrivial
ξ
(
x, y
) and
η
(
x, y
), Eqs.
(11.4) and (11.5) define a
coordinate transformation
(CT) which is invertible almost
everywhere.
Let an image be represented by the real function
f
1
(
x, y
). Another representation
of the image
f
2
can be obtained by means of a CT using the HFP,
f
1
(
x
(
ξ, η
)
,y
(
ξ, η
)) =
f
2
(
ξ, η
)
(11.6)
As before, the term image will refer to a subimage.
Definition 11.1.
The image
f
(
ξ, η
)
is said to be linearly symmetric in the coordinates
(
ξ, η
)
if there exists a one-dimensional function
g
such that
f
(
ξ, η
)=
g
(
aξ
+
bη
)
(11.7)
for some real constants
a
and
b
. Here
ξ
(
x, y
)
,η
(
x, y
)
is a HF
P, and th
e symmetry
direction vector,
(
a, b
)
, has its length normalized to unity, i.e.,
√
a
2
+
b
2
=1
.
The notion “linearly symmetric in (
ξ, η
)” is, in analogy with Chap. 10, motivated
by the fact that the isocurves of such images are
parallel lines
in the
ξη
coordinates.
Likewise, such images have a high concentration of their spectral power along a line
through the origin.
Starting with the trivial unity transformation we give examples of CTs for pattern
families that can be modeled and detected by the toolbox to be presented.
