Image Processing Reference
In-Depth Information
symmetry. Conversely, if
S
has an eigenvalue that vanishes, there is no unique veloc-
ity that can be estimated from the image measurements. Having an overdetermined
system of equations and yet not being able to solve for the velocity may appear coun-
terintuitive, but it is explained by the fact that the 2D pattern
f
(
x
k
,y
l
,t
0
) is linearly
symmetric when an eigenvalue of
S
vanishes. Such images have parallel lines as
isocurves, that is, if translated along these lines no difference in gray values will be
noticed. Accordingly, this situation is the same as the one in the example of Fig. 12.2
(left), where a translating line has been depicted. Because lines and edges are com-
mon in real images, singular structure tensors are also common in image sequences.
In consequence, an appropriate nonsingularity test before solving Eq. (12.101) must
be applied.
The solution of Eq. (12.101) necesitates the estimation of the structure tensor for
a 2D image neighborhood, which we know how to do from Sect. 10.11. However,
the vector
d
is also needed. It is customary to estimate it via a temporal difference
∂f
(
x
k
,y
k
,t
0
)
∂t
=
f
(
x
k
,y
k
,t
1
)
−
f
(
x
k
,y
k
,t
0
)
(12.104)
between two successive image frames [157]. Accordingly, the optical flow technique
discussed above is possible to compute from just two frames.
The technique described in this section and the tensor approach discussed in
Sects. 12.5 and 12.6 are related since they are both gradient-based. The main dif-
ference is that the tensor approach solves the regression problem in the TLS sense,
whereas here it is solved in the MS sense, meaning that the regression error in the
time direction is assumed to be noise-free, see Sect. 10.10. Because of this, the es-
timation is not independent of the coordinate system, in contrast to the tensor aver-
aging approach. A second difference lies in that the tensor approach uses 3D gradi-
ents in which both types of translations are jointly represented, whereas Lucas and
Kanade's approach uses 2D gradients and frame differences to estimate the transla-
tion of points, which decreases its noise tolerance. However, it should be pointed out
that the 2D approach is computationally less demanding and therefore faster.
12.10 Motion Estimation by Spatial Correlation
We assume that the discrete image frame
f
(
x
k
,y
k
,t
0
), which is typically a local
image patch, satisfies the BCC constraint while it undergoes a translation with the
displacement vector
v
=(
v
x
,v
y
)
T
. In other words, the 2D pattern
f
(
x
k
,y
k
,t
0
) and
f
(
x
k
,y
k
,t
1
) are the same or change insignificantly except for a translational CT in
the spatial coordinates,
s
k
=(
x
k
,y
k
)
T
:
s
(
t
1
)=
s
(
t
0
)+
v
(
t
1
−
t
0
)=
s
(
t
0
)
−
v
(12.105)
where we wrote the position of a point in a 2D frame as a function of the time,
s
(
t
),
and assumed that the temporal sampling period is normalized, (
t
1
−t
0
)=1. Because
the two image frames satisfy the BCC, one can write
