Biomedical Engineering Reference
In-Depth Information
introducing nonedge constraints [86, 87], it is necessary and equally important
to improve the edge estimation process itself.
As pointed out by the pioneering work of Mallat
et al.
[16], first- or second-
derivative-based wavelet functions can be used for multiscale edge detection.
Most multiscale edge detectors smooth the input signal at various scales and
detect sharp variation locations (edges) from their first or second derivatives.
Edge locations are related to the extrema of the first derivative of the signal
and the zero crossings of the second derivative of the signal. In [16], it was
also pointed out that first-derivative wavelet functions are more appropriate for
edge detection since the magnitude of wavelet modulus represents the relative
“strength” of the edges, and therefore enable to differentiate meaningful edges
from small fluctuations caused by noise.
Using the first derivative of a smooth function
θ
(
x
,
y
) as the mother wavelet
of a multiscale expansion results in a representation where the two components
of wavelet coefficients at a certain scale
s
are related to the gradient vector of
the input image
f
(
x
,
y
) smoothed by a dilated version of
θ
(
x
,
y
) at scale
s
:
W
s
f
(
x
,
y
)
W
s
f
(
x
,
y
)
=
s
∇
(
f
∗
θ
s
)(
x
,
y
)
.
(6.45)
The direction of the gradient vector at a point (
x
,
y
) indicates the direction in
the image plane along which the directional derivative of
f
(
x
,
y
) has the largest
absolute value. Edge points (local maxima) can be detected as points (
x
0
,
y
0
)
such that the modulus of the gradient vector is maximum in the direction toward
which the gradient vector points in the image plane. Such computation is closely
related to a Canny edge detector [88]. Extension to higher dimension is quite
straightforward.
Figure 6.24 provides an example of a multiscale edge detection method based
on a first derivative wavelet function.
To further improve the robustness of such a multiscale edge detector, Mallat
and Zhong [16] also investigated the relations between singularity (Lipschitz
regularity) and the propagation of multiscale edges across wavelet scales. In
[89], the dyadic expansion was extended to an
M
-band expansion to increase
directional selectivity. Also, continuous scale representation was used for better
adaptation to object sizes [90]. Continuity constraints were applied to fully re-
cover a reliable boundary delineation from 2D and 3D cardiac ultrasound in [91]
