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So far, the candidates are all two-dimensional. Lesions actually extend through the
spine in three dimensions. Therefore the next step is to merge the two-dimensional
candidates that belong to the same lesion into a single three-dimensional
blob
. Two
candidates A and B are merged if and only if
1. They lie in adjacent slices z and z +1
2. For all 2-D candidates C on slice(z + 1) and all D on slice(z), it is true that
ð
pr A B
ðÞ=
aA
ðÞþ
pr B A
ðÞ=
aB
ðÞ
Þ=
2
[
ð
pr A C
ðÞ=
aA
ðÞþ
pr C A
ðÞ=
aC
ðÞ
Þ=
2
and pr A B
ð
ðÞ=
aA
ðÞþ
pr B A
ðÞ=
aB
ðÞ
Þ=
2
[
ð
pr D B
ðÞ=
aD
ðÞþ
pr B D
ðÞ=
aB
ðÞ
Þ=
2
;
where slice(z) is the CT slice at height z, pr X (Y) is the fraction of candidate Y that
overlaps with X when projected into the slice of X, and a(X) is the area of
candidate X. In other words, the average projectional overlap of A and B is
greater than the average projectional overlap of A with any other candidate in the
same slice as B, and also greater than the average projectional overlap of B with
any candidate in the same slice as A.
After the lesions are detected in 3D, a level set algorithm is applied to obtain the
3D segmentation so that characteristic features can be derived. Level sets are
evolving interfaces (contours or surfaces) that can expand, contract, and even split or
merge. Level set methods are part of the family of segmentation algorithms that rely
on the propagation of an approximate initial boundary under the in
uence of images
forces [ 45 ]. The underlying idea behind the level set method is to embed the moving
interfaces as the zero level set of a higher dimensional function
fl
x
;
t
Þ
,de
ned as
x ; t Þ ¼ d
ð 15 Þ
where
d is the signed distance to the interface from point x. That is, x is outside
the interface when
x
;
t
Þ [
0, inside the interface when
x
;
t
Þ \
0, and on the
interface when
x
;
t
Þ ¼
0. The evolution of
x
;
t
Þ
can be represented by a partial
differential equation:
o /
o
x 0 t
t þr/
ðÞ ¼
0
ð 16 Þ
x 0 t
¼ r/ is the normal
direction, and then the above equation becomes the level set equation:
o /
o
De
ne the scalar speed
field F as F
¼
n
ðÞ
, where n
t þ
F
jr/j ¼
0
ð 17 Þ
usually, the speed function F can be written as an explicit level set scheme:
F
¼
F prop þ
F curv þ
F adv
ð 18 Þ
where F prop is the propagation expansion speed, F curv is the speed on the curvature
ʺ
, and F adv is the advection speed. Combining Eqs. 17 and 18 , the
final equation for
level set segmentation can be written as:
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