Biomedical Engineering Reference
In-Depth Information
some psychologically motivated examples), or in case of partly missing informa-
tion (like in problems of inpainting). Then simple segmentation techniques fail
and image analysis becomes a difficult task. In all these situations, the subjective
surface method can help significantly. It is an evolutionary method based on nu-
merical solution of time-dependent highly nonlinear partial differential equations
(PDEs), solving a Riemannian mean curvature flow of a graph problem. The seg-
mentation result is obtained as a “steady state” of this evolution. In case of a large
dataset (3D images or image sequences), where the amount of processed informa-
tion is huge, a discretization of the partial differential equation leads to systems of
equations with a huge amount of unknowns. Then parallel implementation of the
method is necessary, first, due to a large memory requirement, and second, due
to a necessity for fast computing times. For both purposes, an implementation
on the massively parallel processor (MPP) architecture using the message passing
interface (MPI) standard is a favourable solution.
2.
MATHEMATICAL MODELS IN IMAGE SEGMENTATION
Image segmentation based on the subjective surface method is related to geo-
desic (or conformal) mean curvature flow of level sets (level curves in case of 2D
images and level surfaces in case of 3D images). Let us outline first the curve and
surface evolution models and their level set formulations, preceding the subjective
surface method.
A simple approach to image segmentation (similar to various discrete region-
growing algorithms) is to place a small seed, e.g., a small circle in the 2D case, or
a small ball in the 3D case, inside the object and then evolving this
segmentation
curve
or
segmentation surface
to find automatically the object boundary (cf. [1]).
Complex mathematical models as well as Lagrangean numerical schemes have
been suggested and studied for evolving curves and surfaces over the last two
decades (see, e.g., [2, 3, 4, 5, 6, 7, 8, 9, 10]). For moving curves and surfaces the
robust level set models and methods were introduced (see, e.g., [11, 12, 13, 14,
15, 16, 17, 18]). A basic idea in the level set methods is that the moving curve or
surface corresponds to the evolution of a particular level curve or level surface of
the so-called level set function
u
that solves some form of the following general
level set equation:
u
t
=
F
|∇
u
|
, where
F
represents the normal component of the
velocity of this motion.
The first segmentation level set model with the speed of the segmentation
curve (surface) modulated by
F
I
0
|
), where
G
σ
is a smoothing
kernel and
g
is a smooth edge detector function, e.g.,
g
(
s
)=1
/
(1 +
Ks
2
), and
was given in [19] and [20]. Due to the shape of the Perona-Malik function
g
,
the moving curve is strongly slowed down in a neighbourhood of an edge, and a
“steady state” of the segmentation curve is taken as the boundary of segmented
object. However, if an edge is crossed during evolution (e.g., in a noisy image),
=
g
0
≡
g
(
|∇
G
σ
∗
