Biomedical Engineering Reference
In-Depth Information
The idea to use Riemannian mean curvature flow of graphs to compute
the so-called subjective contours [29] originates in [48-50]. The subjective sur-
faces method, developed there, has been successfully used to complete missing
boundaries of objects in digital 2D and 3D data sets and thus it is a powerful
method for segmentation of highly noisy, e.g. medical, images. In this chapter
we follow the same idea.
Initially, a “point-of-view” surface, given by an observer (user) chosen fix-
ation point inside the image, is taken as
u
0
(see e.g. Fig. 11.11 (top right)).
Then this initial state of the segmentation function is evolved by Eq. (11.2), un-
til the so-called subjective surface arises (see e.g. Fig. 11.11 ( bottom) right or
Fig. 11.14 (top row)). For small
ε
, the subjective surface closes gaps in image
object boundaries and is stabilized, i.e. almost does not change by further evolu-
tion, so it is easy to stop the segmentation process. The idea to follow evolution
of the graph of segmentation function [48-50] and not to follow evolution of a
particular level set of
u
is new in comparison with other level set methods used
in image segmentation (cf. [6-9, 30, 31, 36]). In standard level set approach, the
redistancing [42, 55] is used to keep unit slope along the level set of interest
(e.g. along segmentation curve). In such an approach the evolution of
u
itself
is forgotten at every redistancing step. Such solution prevents steepening of
u
and one cannot obtain the subjective surfaces. In our computational method we
do not impose any specific requirements (e.g., redistancing) to solution of the
level set equation, the numerically computed segmentation function can natu-
rally evolve to a “piecewise constant steady state” result of the segmentation
process.
For numerical solution of the nonlinear diffusion equation (11.2), governing
Riemannian mean curvature flow of graphs, we use semi-implicit complemen-
tary volume (called also co-volume or finite volume-element) method. Since
(11.2) is regularization of (11.8), for the curvature driven level set flow (11.8) or
for some other form of the level set equation (11.1), the method can be used as
well (cf. [21, 25]).
For time discretization of nonlinear diffusion equations, there are basically
three possibilities: implicit, semi-implicit, or explicit schemes. For spatial dis-
cretization usually finite difference, finite volume, or finite element method is
used. The co-volume technique is a combination of finite element and finite vol-
ume methods. Implicit, i.e. nonlinear, time discretization, and co-volume tech-
niques for solution of the level set equation were introduced in [56]. The efficient
