Image Processing Reference
In-Depth Information
2.
good localisation with minimal distance between detected and true edge position;
3.
single response to eliminate multiple responses to a single edge.
The first requirement aims to
reduce
the response to noise. This can be effected by
optimal smoothing; Canny was the first to demonstrate that Gaussian filtering is optimal
for edge detection (within his criteria). The second criterion aims for accuracy: edges are
to be detected, in the right place. This can be achieved by a process of
non-maximum
suppression
(which is equivalent to peak detection). Non-maximum suppression retains
only those points at the top of a ridge of edge data, whilst suppressing all others. This
results in thinning: the output of non-maximum suppression is thin lines of edge points, in
the right place. The third constraint concerns location of a single edge point in response to
a change in brightness. This is because more than one edge can be denoted to be present,
consistent with the output obtained by earlier edge operators.
Recalling that the Gaussian operator
g
(
x
,
y
) is given by:
2
2
-(
xy
+
)
2
gxy
(,
) =
e
(4.14)
2
By differentiation, for unit vectors
U
x
= [1, 0] and
U
y
= [0, 1] along the co-ordinate axes,
we obtain:
gxy
x
(,
)
gxy
y
(,
)
gxy
(,
) =
U
+
U
x
y
(4.15)
2
2
2
2
-(
xy
+
)
-(
xy
+
)
y
x
2
2
= -
e
U
-
e
U
2
2
x
y
2
2
Equation 4.15 gives a way to calculate the coefficients of a template that combines first-
order differentiation with Gaussian smoothing. This is a smoothed image, and so the edge
will be a ridge of data. In order to mark an edge at the correct point (and to reduce multiple
response), we can convolve an image with an operator which gives the first derivative in a
direction normal to the edge. The maximum of this function should be the peak of the edge
data, where the gradient in the original image is sharpest, and hence the location of the
edge. Accordingly, we seek an operator,
G
n
, which is a first derivative of a Gaussian
function
g
in the direction of the normal,
n
⊥
:
y
G
n
=
(4.16)
n
where
n
⊥
can be estimated from the first-order difference of the Gaussian function
g
convolved with the image
P
, and scaled appropriately as:
( )
|( )|
P
P
g
g
n
=
(4.17)
The location of the true edge point is then at the maximum point of
G
n
convolved with the
image. This maximum is when the differential (along
n
⊥
) is zero:
(
G
P
)
(4.18)
n
n
