Biomedical Engineering Reference
In-Depth Information
The outer probability combines the region information of the bone and
background tissues. If we assume this distributions to be independent (i.e.,
C
1
∩
C
2
=∅
), then
P
out
=
P
(
I
(
x
)
=
i
|
C
1
∪
C
2
)
=
P
(
I
(
x
)
=
i
|
C
1
)
+
P
(
I
(
x
)
=
i
|
C
2
)
(5.18)
5.2.4.3
Considerations of Implementation
kNN Algorithm.
For the construction of the classifier, six images were selected,
and a total of
N
=
1830 points were manually picked and labelled. Image features
were computed with a Gaussian aperture of
σ
=
0
.
8 mm. In our approach, we
used
k
=
10 as size of the kNN neighborhood.
For the-extraction of the nearest neighbors, a brute-force computation is
highly time-consuming, specially if the size of the training set or the dimen-
sion of the feature vector are large. For this reason, we used the approxi-
mate nearest neighbors (ANN) algorithm [30], that computes efficiently the k-
nearest neighbors from a given vector. The ANN library implements a number of
different data structures, based on kd-trees and box-decomposition trees, and
employs different search strategies for speeding-up the search process.
Numerical Schemes for GAR.
The level set equation
φ
t
=
F
|∇
φ
|
is an hyper-
bolic equation that belongs to the family of Hamilton-Jacobi equations. Only in
one dimension, the level set equation is an hyperbolic conservation law. Never-
theless, the numerical methodology developed to solve conservation laws can
be generalized to obtain numerical schemes that solve Hamilton-Jacobi equa-
tions and, in particular, the level set Equation [31]. The Courant-Friedrichs-Levy
(CFL) condition is used to compute the optimal time step for the GAR level
set equation [32]. The term
ζ
(
k
out
−
k
in
)
|∇
φ
|
represents a motion in the normal
direction. An upwind numerical scheme is used to select the correct solution.
The term
g
κ
|∇
φ
|
represents a motion involving mean curvature. This term acts
as a viscous force so a scheme with central differences is sufficient to select
the correct solution. The term
∇
g
∇
φ
represents a motion in an externally gen-
erated velocity field. This term is also called advection term. Upwind numerical
differences are also used to select the correct solution in this case.
Initialization and Reinitialization with the Fast Marching Method.
The
Fast Marching algorithm is an efficient method to compute the solution of the
Eikonal equation
F
·|∇
T
|=
1 with initial condition
T
0
=
0. If
F
=
1, the solution
