Graphics Reference
In-Depth Information
of contours can be described by moments of area. Thus, the change in moments
of area can be used in terms of the ane transformation parameters.
With the origin at the centroid of the contour of interest so that the first
moments are zero, the above equation with
f
=
x
and
f
=
y
shows that the
centroid of the deformed shape specifies the mean translation [
u
0
,v
0
].
f
=1
shows that the divergence of the image velocity field can be estimated as the
derivative of scaled area,
dI
0
dt
=
I
0
(
u
x
+
v
y
)
(5.59)
and
da
(
t
)
dt
=
a
(
t
)
div
v
.
(5.60)
To get additional constraints, one can increase the order of moments. So, if we
get six linearly independent equations, we can solve for the ane transforma-
tion parameters and combine the coecients to recover the differential invari-
ants. The error between the transformed and observed image contours helps
to check the validity of the ane transformation. Note that certain contours
in practice may lead to equations that are not independent or ill-conditioned.
Under such circumstances, the normal components of image velocity are not
sucient to recover the true image velocity field globally. Waxman and Whon
[172] termed this problem as the “aperture problem in the large.” This was
investigated in the article [20]. However, it is always possible to recover the
divergence from a closed contour.
Tracking Closed Contours
B-spline snakes are used to localize and track closed image contours. We can
write the B-spline curve in the way,
x
(
s
)=
i
=1
f
i
(
s
)
V
i
,
(5.61)
where
f
i
s are the spline basis functions and
V
i
s are the control points of the
curve and
s
is a parameter, not necessarily arc length. The snakes are initial-
ized as points in the center of the image and are forced to expand radially
outwards until they are near the edge and the image forces stabilize the snake
close to a high contrast closed contour. Subsequent image motion is automat-
ically tracked by the snake. B-spline snakes have local control and continuity.
The enclosed area is a function of control points and also applies to other
moments.
From Green's theorem in the plane, the area enclosed by a curve with
parametrization
x
(
s
)and
y
(
s
) is given by
a
=
s
N
s
o
x
(
s
)
y
(
s
)
ds.
(5.62)
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