Biomedical Engineering Reference
In-Depth Information
c
t
¼
G
ðc;
I
Þþ
H
ðc;
U
Þ
(16)
!
Z
2
2
G
¼ dcð
x
Þ
O
y
Bð
x
;
y
Þdcð
y
Þðð
I
ð
y
Þ
u
l
Þ
ð
I
ð
y
Þ
v
l
Þ
Þ
dy
þ lk
(17)
U
2
H
¼
K
U
ðcð
x
Þ
U
ð
x
ÞÞ
ð
x
Þ
(18)
The
Eð
Þ
in (
6
) is the spatially varying width of the region of convergence
from the ground truth segmentation. Its dependence on spatial location means
that in certain places
c
is tolerant of large perturbations while in others care must
be taken (i.e., increased user input is required) to drive
c
toward
c
∗
and assure
convergence. Selecting a local active contour energy functional allows the user
to concentrate input on a particularly sensitive region of
c
without worrying about
global effects.
x
2.3 User Interaction
In this section, we describe in detail how the user participates in the feedback loop
shown in Fig.
4
. First, the user provides a rough contour initialization and allows
the contour to evolve for a set time
t
. In real time, the result is computed and
displayed. Then, as shown in the bottom loop in Fig.
4
, the user manually touches
up the contour only in regions where it did not appear to move toward the object
boundaries; this modified contour
c
(
x
,
t
k
+
) serves as the initialization at the next
time step
t
+
D
t
, and the region of user input is recorded. The user compares
visually his knowledge of the desired segmentation
c
D
ð
x
Þ
to the current segmenta-
tion
cð
. If there are errors, two events can explain them: some regions of
c
0
were outside of the interval in (
6
), or the time
x
;
t
Þ
D
t
was too short for the level
set to converge.
After each time interval of
t
, the automatic algorithm returns a segmentation
for visualization enabling the user to optionally generate another pulse function
g
k
in (
8
) or simply continue the segmentation. Further inputs by the user occur only in
places where corrections are desired, and through the accumulated function of
inputs
U
D
ð
x
;
t
Þ
, the algorithm learns regions where significant input has been
provided over time (i.e., the energy in (
11
) is not discriminative in these regions
of the image and user input should dominate here). A detailed convergence analysis
of this approach is presented in [
6
]. Currently, segmentation is done slice by
slice for the image volume although the same approach can be extended to 3D.
Consistency between slices in maintained by the user. Once slice
k
is completed, a
“copy and paste” operation is performed to transfer this 2D label map onto slice
k
+ 1; it serves as the initialization for the segmentation of slice
k
+1.
