Biomedical Engineering Reference
In-Depth Information
The first step of our approach is to generate the distance map inside the brain
images as shown in Figure 10a. The second step is to use this distance map to gen-
erate iso-surfaces as shown in Figure 10b-c. Note that the number of iso-surfaces,
which is not necessarily the same for both images, depends on the accuracy and
speed required by the user. The third step consists is finding the correspondences
between the iso-surfaces. The final step is evolution of the iso-surfaces. Here,
our goal is to deform the iso-surfaces in the first dataset (target image I t ( · ))to
match the iso-surfaces in the second dataset (source image I s ( · )). Before stating
the evolution equation, let us define the following:
φ I n iso ( ·
) are the iso-surfaces on the target image I t ( · ), with n iso =
1 ,...,N iso being the index of the iso-surfaces and ν the iteration step,
φ I m iso ( · ) are the iso-surfaces on the source image I s ( · ), where m iso =
1 ,...,M iso is the index of the iso-surfaces,
S ( h, γ h ) denotes the Euclidean distance between an iso-surface voxel h on
I t ( · ), and its corresponding iso-surface voxel γ h on I s ( · ); γ h is searched
for within a local window centered at the position of h in I s ( · ); note also
that γ h may be the same for different h ,
S I n iso ,n iso 1 ( h )is theEuclidian distance between φ I n iso ( h, ν )and φ I n iso 1 ( h, ν )
at each iteration ν ,
V ( · ) is the propagation speed function.
One major step in the propagation model is selection of the propagation speed
function V . This selection must satisfy the following conditions:
V ( h )=0 , if S ( h, γ h )=0 ,
(7)
V ( h ) min( S ( h, γ h ) ,S I n iso ,n iso 1 ( h ) ,S I n iso ,n iso +1 ( h )) , if S ( h, γ h
) =0. (8)
The latter condition, known as the smoothness constraint, prevents the current
point from cross-passing the closest neighbor surfaces, as shown in Figure 10d.
Note that the function
V ( h ) = exp( β ( h ) ·
S ( h, γ h )) 1
(9)
satisfies the above conditions, where β ( h ) is the propagation term such that, at
each iso-surface point h
I t ,
ln[min( S ( h, γ h ) ,S I n iso ,n iso 1 ( h ) ,S I n iso ,n iso +1 ( h ))+1]
S ( h, γ h )
β ( h )
.
(10)
 
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