Graphics Reference
In-Depth Information
5.7.5 Extraction of Differential Invariants
There are a number of ways to extract differential invariants from the image.
Differential invariants of the image velocity field characterize the changes in
apparent shape due to relative motion between the viewer and scene. It is
possible to recover the normal image velocity component from local measure-
ments at a curve [163, 76]. It is shown that this information is sucient to
estimate differential invariants within closed curves. The moments of area of
a contour are defined in terms of an area integral with boundaries defined by
the contour in the image plane;
I
f
=
fdxdy,
(5.56)
a
(
t
)
where
a
(
t
) is the area of a contour of interest at time
t
and
f
is a scalar
function of image position (
x, y
). For example,
f
= 1 gives the zero order
moment of area (labeled as
I
o
). This is the area of the contour. Similarly,
when
f
=
x
or
f
=
y
, we get first order moments about the
x
or
y
axis in
the image plane. Moments of area can be measured through their temporal
derivatives in the following way:
fdxdy
dt
(
I
f
)=
dt
d
a
(
t
)
(5.57)
=
[
f
v
.
n
p
]
ds.
c
(
t
)
v
.
n
p
is the normal component of the image velocity
v
atapointonthe
contour. We, therefore, note that the temporal derivatives of moments of area
are simply the effect of integration of the normal image velocities at a contour
weighted by a scalar
f
(
x, y
). By Green's theorem, an integral over the contour
c
(
t
) can be expressed as an integral over the area enclosed by the contour
a
(
t
).
Therefore,
dt
(
I
f
)=
a
(
t
)
d
div
(
f
v
)]
dxdy
=
a
(
t
)
[
fdiv
v
+
v
.gradf
]
dxdy
=
a
(
t
)
[
fdiv
v
+
f
x
u
+
f
y
v
]
dxdy
(5.58)
=
u
0
a
(
t
)
f
x
dxdy
+
u
x
a
(
t
)
[
xf
x
+
f
]
dxdy
+
u
y
yf
x
dxdy
+
v
0
a
(
t
)
f
y
dxdy
+
v
x
[
xf
y
+
f
]
dxdy
+
v
y
a
(
t
)
(
t
)
[
yf
y
+
f
]
dxdy,
a
(
t
)
where we get the last line using equation (5.36). We, therefore, see that the
image velocity field deforms the shape of contours in the image and the shape
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