Image Processing Reference
In-Depth Information
When the line sets translate with a common velocity vector
v
so that a point at
s
0
moves to
s
:
v
=(
v
x
,v
y
)
T
s
=(
x, y
)
T
=
s
0
+
t
v
=(
x
0
+
v
x
t, y
0
+
v
y
t
)
T
⇒
(12.70)
We obtain by this CT an image
f
that is continuous in
x, y, t
f
(
x, y, t
)=
g
(
x
−
v
x
t, y
−
v
y
t
)
(12.71)
in analogy with the discussion in the previous section. Because (
s
s
0
)=
t
v
,
is what we are interested in, the orthogonal projections of the (common) velocity
v
on the normal directions of the lines are the individual speeds that would have
been perceived by the observer, if there was only one direction in the image. In that
case, the components of the velocity in the directions of
a
1
, and
a
2
and the lines
themselves would define two sets of oblique planes containing the lines. The normal
vectors of these planes generated by the motion of each line are given by
−
k
1
=(cos(
θ
1
)
,
sin(
θ
1
)
,
k
2
=(cos(
θ
2
)
,
sin(
θ
2
)
,
−
a
T
1
v
)
T
−
a
T
2
v
)
T
and
according to the discussion preceeding Eq. (12.53). The cross-product of these 3D
normals will then be orthogonal to both, yielding the vector pointing in the direction
of the intersection (lines) of the nonparallel plane sets.
⎛
⎞
⎛
⎞
cos(
θ
1
)
sin(
θ
1
)
cos(
θ
2
)
sin(
θ
2
)
k
3
=
k
1
×
k
2
=
⎝
⎠
×
⎝
⎠
−
v
x
cos(
θ
1
)
−
v
y
sin(
θ
1
)
−
v
x
cos(
θ
2
)
−
v
y
sin(
θ
2
)
⎛
⎞
sin(
θ
1
)(
−
v
x
cos(
θ
2
)
−
v
y
sin(
θ
2
))
−
sin(
θ
2
)(
−
v
x
cos(
θ
1
)
−
v
y
sin(
θ
1
))
⎝
⎠
=
−
cos(
θ
1
)(
−
v
x
cos(
θ
2
)
−
v
y
sin(
θ
2
)) + cos(
θ
2
)(
−
v
x
cos(
θ
1
)
−
v
y
sin(
θ
1
))
cos(
θ
1
) sin(
θ
2
)
−
sin(
θ
1
)cos(
θ
2
)
⎛
⎞
−
v
x
cos(
θ
2
) sin(
θ
1
)+
v
x
sin(
θ
2
)cos(
θ
1
)
v
y
sin(
θ
2
)cos(
θ
1
)
⎝
⎠
=
v
y
sin(
θ
1
)cos(
θ
2
)
sin(
θ
2
−
−
θ
1
)
⎛
⎞
θ
1
)=
v
1
sin(
θ
2
−
v
x
v
y
1
⎝
⎠
sin(
θ
2
−
=
θ
1
)
The vector
k
3
can be estimated by the least significant eigenvector of the structure
tensor of the function
f
(
x, y, t
), i.e.,
k
3
, up to a scale.
⎛
⎞
v
1
sin(
θ
2
−
k
x
k
y
k
t
k
3
=
k
3
⎝
⎠
⇔
θ
1
)=
(12.72)
This reconfirms the velocity estimate computations suggested in the previous para-
graph, Eq. (12.67), as
v
=(
k
x
k
t
,
k
y
k
t
)
T
(12.73)
