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propose a novel slice-by-slice correction scheme to the resulting images obtained
in the first stage of the segmentation. We assume that the coronary arteries can be
modelled as a tree structure, originating from the ostium (located in the descenting
aorta at the top of the heart) and that their transaxial cross-sectional segments
taken from the top to the bottom can only split but not merge over the frame
sequence (see Fig.
5.6
).
It can be observed from the zoomed image at the right of Fig.
5.5
b that the non-
arterial vessel appears darker than the coronary, and therefore, it can be distin-
guished from the coronary by intensity difference. However, two-phase level set
methods (i.e. using one level set function for segmentation) can only separate two
homogenous regions. Images with more than one object regions cannot be cor-
rectly segmented using such models. In this research, we employ a multiple
regions competition-based level sets method to correct the resulting image slice-
by-slice. In the proposed method, each homogenous region (object) is represented
by a level set function, and thus, other non-arterial structures can be identified and
removed from the segmentation. As described by Brox and Weickert [
25
], for a
fixed number of objects to be segmented, the evolution equations of multiple
regions level sets for image segmentation can be defined as:
0
@
1
A
o
E
ð
/
i
Þ
ot
¼ H
0
ð
/
i
Þ
e
i
max
H
0
ð
/
j
Þ
[ 0
j
6
¼i
ð
e
j
;
e
i
1
Þ
ð
5
:
9
Þ
r
/
k
jr
/
k
j
e
k
¼ log p
k
þ
2
div
Since the term log(p
1
) is always negative, the evolution of a contour based on
this force alone will eventually lead to the level set becoming negative every-
where, and thus eventually shrinking to a single point. The maximum operator
ensures that the contour expands outwards with a constant speed, when there is no
competition around the zero level set of the current embedding function. If there
are multiple regions within a narrow band of current zero levels of the embedding
function, the contour will evolve according to the maximal force in the interface.
However, the scheme would quickly expand the current contour if only one curve
is present, thus, moving the contour away from its initial location and capturing
undesired boundaries. By making use of the fact that the segmentation obtained in
the first stage is almost accurate, we allow the curve to evolve according to
multiple-phase energy when there is no competition nearby, while, when multiple
contours are presented in an interface, the points in the interface will move
together subject to the strongest force across all the regions. The steps of the slice-
by-slice correction algorithm are as follows:
1. Determine the first axial image which contains coronary arteries using the
segmented images obtained in the first stage.
2. Assign each connected object to a level set function in the starting slice.
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