Image Processing Reference
In-Depth Information
with
D
ξ
1
=
D
x
D
ξ
2
=
D
y
D
ξ
3
=
xD
x
+
yD
y
D
ξ
4
=
yD
x
+
xD
y
D
ξ
5
=
xD
x
−
−
(12.86)
yD
y
D
ξ
6
=
yD
x
+
xD
y
D
ξ
7
=
D
t
under which a spatio-temporal image generated by the affine coordinate transforma-
tion is invariant:
D
ζ
f
(
x, y, t
)=
j
k
j
D
ξ
j
f
(
x, y, t
)=0
(12.87)
From this equation we will attempt to solve the unknown parameter vector:
k
7
)
T
,
k
=(
k
1
,
···
with
k
=1
.
(12.88)
The solution is evidently not possible if we only know the 7D measurement vector
D
ξ
f
defined as
D
ξ
f
=(
D
ξ
1
f, D
ξ
2
f,···D
ξ
7
f
)
T
(12.89)
at a single point
r
=(
x, y, t
)
t
. Equation (12.87) is effectively an equation of a hy-
perplane in
E
7
.
k
T
D
ξ
f
(
r
)=0
(12.90)
The equation is satisfied in many points ideally because
f
is in that case a spatio-
temporal image that is truly generated from a static image by use of an
affine coordi-
nate transformation
. In that case
F
will be concentrated to a 6D plane, see theorem
12.1 and lemma 12.7. Accordingly, estimating the
affine motion parameters
is a lin-
ear symmetry problem in
E
7
. It can be solved in the TLS error sense by minimizing
the error
e
(
k
)=
2
dxdydt
=
k
T
[
k
T
D
ξ
f
(D
ξ
f
)(
D
T
f
)
dxdydt
]
k
=
k
T
Sk
(12.91)
ξ
under the constraint
k
=1. Here, the matrix
S
is the structure tensor defined as
S
=
(D
ξ
f
)(
D
T
f
)
dxdydt
(12.92)
ξ
The TLS estimate of the parameter vector
k
is given by the least significant eigen-
vector of
S
. The necessary differential operators and the integrals are possible to
implement by use of Eqs. (12.86), and (12.89) via separable convolutions with ker-
nels derived from Gaussians [59]. Estimating
S
by sampling the structure tensor in
N
D is discussed in Sect. 12.7. Figure 12.5 illustrates a motion image sequence. The
car is moving to the left whereas a video camera tracks the car and keeps it approx-
imately at the center of the camera view. As a result there are two motion regions,
the car's and the background's. The eigenvectors of Eq. (12.92) are computed twice,
