Image Processing Reference
In-Depth Information
G
(
k
x
,k
y
,k
t
)=
G
(
k
x
,k
y
)
(12.65)
The inclination of the plane, Eq. (12.64), is steered by the velocity,
v
, in that it
controls the normal of the plane,
w
w
=
v
1
(12.66)
This normal vector, and thereby the velocity, can be estimated from the image data
by the least significant eigenvector of the structure tensor,
k
3
. By construction, the
third element of
w
in Eq. (12.66) must be 1 for its first two elements to equal to
v
.
Accordingly, we obtain
v
from
k
3
as follows.
v
=(
k
x
k
t
,
k
y
k
t
)
T
k
3
=(
k
x
,k
y
,k
t
)
⇒
(12.67)
Some studies refer to Eq. (12.62) as “a tilting of
G
”. The product
Gδ
is thus a cut
of the cylinder
G
consisting of the 2D FT of the still pattern by a thin oblique plane,
δ
, representing the motion plane, Eq. (12.64). Evidently, such a cut is not a true tilting
of
G
(
k
) in the FT domain because the coordinates of
G
(
k
) would have undergone
an orthogonal CT, which Eq. (12.62) does not depict. Instead, the transformation is
an inverse projection, i.e., the FT of the still image is a projection of the motion
plane to the
k
x
,k
y
plane. Even if the still image is a band-limited function, without
a limitation on the speed, a translation can tilt the motion plane to reach arbitrary
high values for
k
t
. Accordingly, a translation is a band-enlarging operation for band-
limited functions.
The velocity in Eq. (12.67) requires that
k
t
0 the velocity risks
increasing beyond every bound, causing numerical instability in computations. The
question for what patterns
k
t
becomes small, and thereby the unambiguous motion
estimation becomes unstable, is discussed in the next paragraph.
=0. When
k
t
→
The Velocity of Two Nonparallel Lines and Sensitivity Analysis
We assume now that we have an image,
g
(
x
0
,y
0
)
(12.68)
defined on
E
2
, but that this image contains two sets of isocurves with distinct direc-
tions,
7
e.g., two sets of nonparallel lines in the image
g
(
x
0
,y
0
). Accordingly, the 2D
vectors
a
1
,
a
2
represent the respective normal vectors of the assumed line sets,
a
1
=(cos(
θ
1
)
,
sin(
θ
1
))
T
,
a
2
= (cos(
θ
2
)
,
sin(
θ
2
))
T
,
where
θ
1
=
θ
2
.
(12.69)
7
Such an image is not linearly symmetric in 2D since it contains more than one direction.
For this reason
g
is not a function originally defined on 1D originally but instead on 2D.
Sums of two linearly symmetric functions with different directions are examples of such
images.
