Image Processing Reference
In-Depth Information
direction of - ϕ
(
x
,
y
)) giving κ
(
x
,
y
). In this case we consider that the curve is given by
-ϕ
x
(
t
) =
x
+
t
cos(- ϕ
(
x
,
y
)) and
y
(
t
) =
y
+
t
sin(-ϕ
(
x
,
y
)). Thus,
(,
xy
)
-
Mx
x
My
x
My
y
Mx
y
(4.50)
Two
further
measures can be obtained by considering a forward
and a backward
differential
along the
normal
to the edge. These differentials cannot be related to the actual definition
of curvature, but can be explained intuitively. If we consider that curves are more than one
pixel wide, then differentiation along the edge will measure the difference between the
gradient angle between interior and exterior borders of a wide curve. In theory, the tangent
angle should be the same. However, in discrete images there is a change due to the
measures in a window. If the curve is a straight line, then the interior and exterior borders
are the same. Thus, gradient direction normal to the edge does not change locally. As we
bend a straight line, we increase the difference between the curves defining the interior and
exterior borders. Thus, we expect the measure of gradient direction to change. That is, if
we differentiate along the normal direction, we maximise detection of gross curvature. The
value
1
2
2
=
My
-
MxMy
-
Mx
+
MxMy
3
2
2
2
(
Mx
+
My
)
κ
⊥ϕ
(
x
,
y
) is obtained when
x
(
t
) =
x
+
t
sin(
(
x
,
y
)) and
y
(
t
) =
y
+
t
cos(
(
x
,
y
)). In this
case,
(,
xy
)
My
x
My
x
My
y
Mx
y
1
2
2
=
Mx
-
MxMy
-
MxMy
+
My
3
2
2
2
(
Mx
+
My
)
(4.51)
In a
backward
formulation along a
normal
direction to the edge, we obtain:
κ
(,
xy
)
-
My
x
Mx
x
My
y
Mx
y
1
2
2
=
-
Mx
+
MxMy
-
MxMy
+
My
3
2
(
Mx
2
+
My
2
)
(4.52)
This was originally used by Kass (1988) as a means to detect
line terminations
, as part of
a feature extraction scheme called snakes (active contours) which are covered in Chapter
6. Code
4.18
shows an implementation of the four measures of curvature. The function
Gradient
is used to obtain the gradient of the image and to obtain its derivatives. The
output image is obtained by applying the function according to the selection of parameter
op
.
Let us see how the four functions for estimating curvature from image intensity perform
for the image given in Figure
4.32
. In general, points where the curvature is large are
highlighted by each function. Different measures of curvature, Figure
4.34
, highlight differing
points on the feature boundary. All measures appear to offer better performance than that
derived by reformulating hysteresis thresholding, Figure
4.32
,
and by fitting cubic polynomials,
Figure
4.33
. Though there is little discernible performance advantage between the direction
of differentiation. As the results in Figure
4.34
suggest, detecting curvature directly from
