Image Processing Reference
In-Depth Information
extreme points are very useful for shape description and matching, since they represent
significant information with reduced data.
Curvature is normally defined by considering a parametric form of a planar curve. The
parametric contour
v
(
t
) =
x
(
t
)
U
x
+
y
(
t
)
U
y
describes the points in a continuous curve as the
end points of the position vector. Here, the values of
t
define an arbitrary parameterisation,
the unit vectors are again
U
x
= [1, 0] and
U
y
= [0, 1]. Changes in the position vector are
given by the tangent vector function of the curve
v
(
t
). That is,
˙
v
() =
˙
˙
t xt U yt
x y
.
This vectorial expression has a simple intuitive meaning. If we think of the trace of the
curve as the motion of a point and
t
is related to time, then the tangent vector defines the
instantaneous motion. At any moment, the point moves with a speed given by
| ( )| =
()
+
()
v
t
˙
˙
2
˙
2
(
t
) = tan
-1
(()/())
˙
˙
yt xt
. The curvature at a point
v
(
t
) describes the changes in the direction ϕ (
t
) with respect to changes in arc length. That
is,
x
( ) +
t
y
( )
t
in the direction
ϕ
dt
ds
ϕ
()
κ
() =
t
(4.30)
where
s
is arc length, along the edge itself. Here
ϕ
is the angle of the tangent to the curve.
That is,
is the gradient direction defined in Equation 4.13. That is, if
we apply an edge detector operator to an image, then we have for each pixel a gradient
direction value that represents the normal direction to each point in a curve. The tangent to
a curve is given by an orthogonal vector. Curvature is given with respect to arc length
because a curve parameterised by arc length maintains a constant speed of motion. Thus,
curvature represents changes in direction for constant displacements along the curve. By
considering the chain rule, we have
ϕ
=
±
90
, where
dt
dt
ϕ
()
dt
ds
κ
() =
t
(4.31)
The differential
ds
/
dt
defines the change in arc length with respect to the parameter
t
. If we
again consider the curve as the motion of a point, then this differential defines the instantaneous
change in distance with respect to time. That is, the instantaneous speed. Thus,
v
t
˙
˙
2
˙
2
ds
/
dt
=
| ( )| =
x
( ) +
t
y
( )
t
(4.32)
and
dt
/
ds
= 1/
xt yt
˙
2
() +
˙
2
()
(4.33)
(
t
) = tan
-1
(()/())
yt xt
, then the curvature at a point
v
(
t
) in Equation
˙
˙
By considering that
ϕ
4.31 is given by
˙ ˙
˙ ˙
xt yt yt xt
xt yt
() () -
() ()
κ
() =
t
(4.34)
[
˙
2
( ) +
˙
2
( )]
3/2
This relationship is called the
curvature function
and it is the standard measure of curvature
for
planar
curves (Apostol, 1966). An important feature of curvature is that it relates the
derivative of a tangential vector to a normal vector. This can be explained by the simplified
Serret-Frenet equations (Goetz, 1970) as follows. We can express the tangential vector in
polar form as
˙
˙
v
( ) = |
t
v
( )|(cos(
t
ϕ
( )) + sin(
t
j
ϕ
( )))
t
(4.35)
