Biomedical Engineering Reference
In-Depth Information
The function map topological enhancement can be described as follows. Given the
max-tree representation
of the function map
T
and a parameter
θ
that represents
a value that we consider to be high enough to determine a set of seed connected
components, a safety set is defined
S
T
C
h
. For instance, we can
assume that regions with values over 90% are reliable enough to represent safety
areas that are used to initialize the method. The method enhances topologically
all the connected components that contain any of the safety areas by weighting the
value of the connected component by a function
f
(
h
), where
h
is the threshold
parameter for the connected component. The function
f
is monotonically increas-
ing and is desired to be S-shaped. Let's define the set of connected components
that have a node in
S
T
:
=
{
|
h>θ
}
C
i
,
C
i
C
h
|
C
h
⊃
=
{
∀
∈
S
T
}
S
N
.
Then, the topologically enhanced function map is defined as:
L
(
h
)=(
f
(
h
)
·
S
N
(
h
))
∪
S
N
(
h
)
,
where (
f
(
h
)
·
S
N
(
h
)) is the product of the value of each of the connected com-
ponents of
S
N
at level
h
with the value of a function at the threshold level. The
result of the product is an “enhanced” set o
f co
mponents in which the value of
each of the components has been increased.
S
N
is the complementary set of
S
N
.
The resulting enhanced map
L
is the union of the enhanced set and the comple-
mentary set. The effect of this enhancing process is to increase topologically the
value of the neighboring connected components of the high likelihood value areas
while preserving the rest of the map intact. A final step is performed so that the
resulting values after enhancing never surpass the value of the connected com-
ponent at level
θ
. Therefore, each connected component the value of which is
over
θ
is constrained to the value at level
θ
. Remember that Figure 10c shows the
non-constrained enhanced map, and Figure 10d depicts the final result. As can be
observed in Figure 10d, the topologically enhanced area has steeper slopes and
therefore defines better the contours of the region. On the other hand, the low-
value region is kept at the same level. The shape of the function
f
(
h
) determines
the degree of enhancement. Usually this function parameters are related to the
inflexion point and the slope. In our experiments we have used an exponential
function.
Figure 11 shows the result of the enhancing process when applied to the
original likelihood map. Figure 11a shows the original image, Figure 11b depicts
the likelihood map, and Figure 11c shows the enhanced map. Note that the process
enhances the regions near the areas of maximum likelihood while preserving the
rest of the regions untouched.
