Digital Signal Processing Reference
In-Depth Information
6) Else keep the point. The number of passes across direction should be chosen according
to the level of accuracy desired.
Second-order approaches to edge detection
Some edge-detection operators are instead based upon second-order derivatives of the
intensity. This essentially captures the rate of change in the intensity gradient. Thus, in
the ideal continuous case, detection of zero-crossings in the second derivative captures
local maxima in the gradient.
The early Marr-Hildreth operator is based on the detection of zero-crossings of the
Laplacian operator applied to a Gaussian-smoothed image. It can be shown, however,
that this operator will also return false edges corresponding to local minima of the
gradient magnitude. Moreover, this operator will give poor localization at curved edges.
Hence, this operator is today mainly of historical interest.
Differential edge detection
A more refined second-order edge detection approach which automatically detects edges
with sub-pixel accuracy, uses the following
differential approach
of detecting zero-
crossings of the second-order directional derivative in the gradient direction:
Following the differential geometric way of expressing the requirement of non-maximum
suppression proposed by Lindeberg, let us introduce at every image point a local
coordinate system (
u
,
v
), with the
v
-direction parallel to the gradient direction. Assuming
that the image has been presmoothed by Gaussian smoothing and a scale-space
representation
L
(
x
,
y
;
t
) at scale
t
has been computed, we can require that the gradient
magnitude of the scale-space representation, which is equal to the first-order directional
derivative in the
v
-direction
L
v
, should have its first order directional derivative in the
v
-
direction equal to zero
while the second-order directional derivative in the
v
-direction of
L
v
should be negative,
i.e.,
Written out as an explicit expression in terms of local partial derivatives
L
x
,
L
y
...
L
yyy
, this
edge definition can be expressed as the zero-crossing curves of the differential invariant
that satisfy a sign-condition on the following differential invariant
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