Biomedical Engineering Reference
In-Depth Information
the major blood vessels, while edges in
Fig. 6.4-7
D do not
enclose blood vessels completely.
The edge-based techniques are computationally fast
and do not require a priori information about image
content. The common problem of edge-based segmen-
tation is that often the edges do not enclose the object
completely. To form closed boundaries surrounding re-
gions, a postprocessing step of linking or grouping edges
that correspond to a single boundary is required. The
simplest approach to edge linking involves examining
pixels in a small neighborhood of the edge pixel (3
3,
5
5, etc.) and linking pixels with similar edge magni-
tude and/or edge direction. In general, edge linking is
computationally expensive and not very reliable. One
solution is to make the edge linking semiautomatic and
allow a user to draw the edge when the automatic tracing
becomes ambiguous. For example, Wang
et al.
developed
a hybrid algorithm (for MR cardiac cineangiography) in
which a human operator interacts with the edge tracing
operation by using anatomic knowledge to correct errors
[121]
. A technique of
graph searching
for border de-
tection has been used in many medical applications
[6,
14, 64, 81, 105, 106, 112]
. In this technique each image
pixel corresponds to a graph node and each path in
a graph corresponds to a possible edge in an image. Each
node has a cost associated with it, which is usually cal-
culated using the local edge magnitude, edge direction,
and a priori knowledge about the boundary shape or lo-
cation. The cost of a path through the graph is the sum of
costs of all nodes that are included in the path. By finding
the optimal low-cost path in the graph, the optimal
border can be defined. The graph searching technique is
very powerful, but it strongly depends on an application-
specific cost function. A review of graph searching algo-
rithms and cost function selection can be found in
Ref.
[107]
.
Since the peaks in the first-order derivative corre-
spond to zeros in the second-order derivative, the
Laplacian
operator (which approximates second-order
derivative) can also be used to detect edges
[16, 36, 96]
.
The Laplace operator
V
2
of a function
f
(
x, y
)isdefinedas
Figure 6.4-8 Results of Laplacian and Laplacian of Gaussian
(LoG) applied to the original image shown in Fig. 7A. (A) 33
Laplacian image, (B) result of a 77 Gaussian smoothing
followed by a 77 Laplacian, (C) zero-crossings of the Laplacian
image A, (D) zero-crossings of the LoG image B.
crossings).
Figure 6.4-8
A shows a result of a 3
3
Laplacian applied to the image in
Fig. 6.4-7
A. The zero
crossings of the Laplacian are shown in
Fig. 6.4-8
C.
All edge detection methods that are based on a gradi-
ent or Laplacian are very sensitive to noise. In some ap-
plications, noise effects can be reduced by smoothing the
image before applying an edge operation. Marr and
Hildreth
[72]
proposed smoothing the image with
a Gaussian filter before application of the Laplacian (this
operation is called Laplacian of Gaussian, LoG).
Figure 6.4-8
B shows the result of a 7
7 Gaussian
followed by a 7
7 Laplacian applied to the original
image in
Fig. 6.4-7
A. The zero crossings of the LoG op-
erator are shown in
Fig. 6.4-8
D. The advantage of LoG
operator compared to a Laplacian is that the edges of the
blood vessels are smoother and better outlined. How-
ever, in both
Figs 6.4-8
C and D, the nonsignificant edges
are detected in regions of almost constant gray level. To
solve this problem, the information about the edges
obtained using first and second derivatives can be com-
bined
[107]
. This approach was used by Goshtasby and
Turner
[38]
to extract the ventricular chambers in flow-
enhanced MR cardiac images. They used a combination
of zero crossings of the LoG operator and local maximum
of the gradient magnitude image, followed by the curve-
fitting algorithm.
The Marr-Hildreth operator was used by Bomans
et al.
[12]
to segment the MR images of the head. In
a study of coronary arteriograms, Sun
et al.
[110]
used
a directional low-pass filter to average image intensities in
the direction parallel to the vessel border. Other edge-
finding algorithms can be found in Refs.
[24, 30, 36, 96]
.
V
2
fðx; yÞ¼
v
2
fðx; yÞ
vx
2
þ
v
2
fðx; yÞ
vy
2
:
(6.4.7)
The Laplacian is approximated in digital images by an
N
by
N
convolution mask
[96, 107]
. Here are three ex-
amples of 3
3 Laplacian masks that represent different
approximations of the Laplacian operator:
0
10
14
1
0
10
1
1
1
18
1
1
1
1
1
21
24
2
1
21
The image edges can be found by locating pixels where
the Laplacian makes a transition through zero (zero
