Biomedical Engineering Reference
In-Depth Information
co-volume level set method based on semi-implicit, i.e. linear, time discretiza-
tion was given and studied in [25]. In [25], the method was applied to image
smoothing nonlinear diffusion level set equation; here we apply the method to
image segmentation and completion of missing boundaries.
Let us note that Eq. (11.8) can be rewritten into an advection-diffusion
form as
∇
u
|∇
u
|
u
t
=
g
0
+∇
g
0
|∇
u
|∇ ·
·∇
u
.
(11.9)
Various finite difference schemes [7-9, 30, 31, 48-50] are usually based on this
form using upwinding in advection term and explicit time stepping. Our co-
volume technique relies on discretization of the basic form (11.8), or more pre-
cisely on its regularization (11.2), and we use its integral (weak, variational)
formulation. In such a way, the discretization scheme naturally respects a varia-
tional structure of the problem, it gives clear discrete form of local mass balance,
and it naturally fulfills discrete minimum-maximum principle (L
∞
-stability).
The semi-implicit discretization in time yields such stability property (i.e. no
spurious oscillations appear in our solution) for any length of discrete time
step. This is a main advantage in comparison with explicit time stepping, where
the stability is often achieved only under severe time step restriction. Since in
nonlinear diffusion problems (such as the level set equation), the coefficients
depend on the solution itself and thus they must be recomputed in every dis-
crete time update, an overall CPU time for explicit scheme can be tremendous.
On the other hand, the implicit time stepping as in [56], although uncondition-
ally stable, leads to solution of nonlinear systems in every discrete time up-
date. For the level-set-like problems, there is no guarantee for convergence of
a fast Newton solver, and fixed-point-like iterations are very slow [56]. From
this point of view, the semi-implicit method seems to be optimal regarding sta-
bility and efficiency. In every time update we solve linear system of equations
which can be done efficiently using, e.g., suitable preconditioned iterative linear
solvers.
In Section 11.2 we discuss various curve evolution and level set models
leading to segmentation Eqs. (11.8) and (11.2). In Section 11.3 we introduce our
semi-implicit co-volume level set method for solving these equations and discuss
some of its theoretical properties and implementation aspects. In Section 11.4
we discuss numerical experiments.
