Image Processing Reference
In-Depth Information
to estimate both the background and the car motion using the affine model which in
turn has helped to find the motion boundaries [61]. The boundary estimation aided
by motion parameters is far more accurate than only using static image frames (e.g.,
see the antenna of the car which has also a cable, loosely attached to it), although not
perfect (e.g., the dark region behind the car is not well segmented).
12.9 Motion Estimation by Differentials in Two Frames
We begin with writing down the
velocity of a particle
,
v
=(
v
x
,v
y
)
T
,movinginthe
(
x, y
)-plane, as defined in mechanics:
v
=
d
s
(
t
)
dt
=(
dx
(
t
)
dt
,
dy
(
t
)
dt
)
T
=(
v
x
,v
y
)
T
(12.93)
where
s
=(
x, y
)
T
is the coordinate of the particle and
t
is the time coordinate. Let
the function
f
(
x, y, t
) represent the spatio-temporal image of such moving parti-
cles,x where
f
is the gray intensity. A change of the gray intensity can be expected
when changing the coordinates
s
=(
x, y
)
T
and/or
t
. From calculus we can conclude
that a small intensity change in
f
can be achieved by small changes of all three vari-
ables of
f
. To be exact, the change
df
is controllable by the independent changes
dx, dy, dt
via:
df
=
dt
∂f
∂t
+
dx
∂f
∂x
+
dy
∂f
(12.94)
∂y
or equivalently,
df
dt
=
∂f
∂t
+
dx
dt
∂x
+
dy
∂f
∂f
∂y
(12.95)
dt
Here we assumed that
f
is differentiable, i.e., all of its partial derivatives exist con-
tinuously. It is of great interest to find a path
s
(
t
) such that
f
does not change at all,
i.e.,
df
dt
=0
(12.96)
This path will yield the isocurves of
f
(
x, y, t
), which in turn offers an opportunity to
track points. Under the condition that the path is unique, this is the same as having
tracked a point if the BCC holds [110, 122], see Definition (12.2). This conclusion
is reasonable because the image of a moving 3D point is a moving 2D point, which
normally changes its gray value only insignificantly, at least during a sufficiently
short observation time:
∂f
∂t
+
v
x
∂f
∂x
+
v
y
∂f
BCC:
=0
(12.97)
∂y
Conversely, if an image
f
(
x, y, t
) satisfies this equation we can conclude that the
motion satisfies the BCC [110, 157]. This is why the equation is frequently quoted as
the
BCC equation
in image analysis studies.
