Biomedical Engineering Reference
In-Depth Information
the hypersurface is in the vicinity of the object boundary. The hypersurface evo-
lution can be obtained by calculating the gradient flow of the proposed energy
functional. In the chapter, both two-phase and multiphase methods have been
investigated. In particular, for brain MR images segmentation, we performed seg-
mentation using a three-phase level set model. We obtained a united form of the
evolution equation from which we can derive geodesic active regions model when
we replace J-divergence with KL-divergence. Setting the region variance to a con-
stant gives the same evolution equation as in [9]. Besides, only mean and variance
need to be calculated for the Gaussian distribution assumption. It is straightfor-
ward in terms of implementation and computing efficiency. Experimental results
have been validated for the proposed algorithm.
Future work will focus on using nonparametric statistical methods instead of
the Gaussian distribution hypothesis, although the latter can fulfill most of the
MRI segmentation requirements. To reduce computation time, a new energy term
|−
1)
2
d
from [32] will be added to our proposed energy, which will force
the level set close to a signed distance function and therefore completely eliminate
the need for re-initialization during evolution.
Ω
(
|∇
φ
x
5.
ACKNOWLEDGMENTS
The authors wish to thank Dr. T. Bailloeul for his many valuable sugges-
tions and comments about the chapter and pleasant discussions about the level set
method.
APPENDIX A
1.
DERIVATION OF LEVEL SET EVOLUTION EQUATION
We now deduce the gradient flow minimizing the functional of Eq. (11). The
energy functional consists of region- and edge-based energy. To obtain the gradient
flow, one can directly compute the first variation by the shape derivative introduced
in [33]. Another approach is to calculate the first variation with the level set
representation [9]. The proposed energy functional is as follows:
s
E
(
φ
1
,φ
2
,...,φ
n
)=
λ
i
·
D
i
·
G
i
(
φ
1
,φ
2
,...,φ
n
)
d
x
Ω
i
=1
n
+
ν
j
·
g
(
|∇
I
(Γ
j
)
|
)
|∇
H
(
φ
j
)
|
d
x
,
Ω
j
=1
where
D
i
is the J-divergence between the probability of region
R
i
and local region
probability,
n
is the number of the level set function, and
s
is the number of regions.
