Biomedical Engineering Reference
In-Depth Information
2.
ACTIVE CONTOUR MODEL: THEORY
This section will elaborate the theory of the active contour model. For easy
reference, the definitions and notations used here are defined in the next subsection.
2.1. Definitions and Notations
2
will denote the Euclidean
plane. An
image
is considered to be embedded on a rectangular subspace
R
We use
is to denote a real number line, and
⊂
2
.
Over
R
, intensity values are acquired at every point with integral coordinates com-
monly referred to as
pixels
. A point in
R
will be represented as a two-dimensional
position vector
u
=(
x, y
), where
x, y
denote the x- and y-coordinate values of
u
.
Let
f
:
R
→
[0
,
1
,
2
,
3
, ...,
MaxIntensity] denote the
intensity function
for a given
image.
A
parametric curve
or
spline
is represented as a function
τ
:[0
,
1]
→
2
.A
curve is closed if the initial and terminal points are identical, i.e.,
τ
(0) =
τ
(1).A
point on the curve will be denoted by
∈
[0
,
1] denotes
the arc-length parameter, and
x
(
s
) and
y
(
s
) refer to its location in the xy-plane. Al-
though in a continuous space any real value in [0
,
1] may be assigned to the parame-
ter
s
, in the digital world only discrete values can be used. The snake is an ordered
sequence of discrete points on a curve at a regular interval
δ<
1
.
0, where
δ
isafi-
nitely small positive number. The points
...,
τ
(
s
)=(
x
(
s
)
,y
(
s
)), where
s
τ
(
−
2
δ
)
,
τ
(
−
δ
)
,
τ
(0)
,
τ
(
δ
)
,
τ
(2
δ
)
, ...
on a snake
at an interval of
δ
will be referred to as
control points
.
Let a contiguous set of pixels belonging to the structure of interest, sharing
some similar attributes, be called the
foreground
and be denoted as
O
τ
⊂
R
, where
R
is the image space. Any pixel
c
O
is called an object pixel. On the other
hand, any pixel
c
∈
R
−
O
, i.e., belonging to the image space
R
but not belonging
to the object space
O
,isa
background
pixel. The task of image segmentation is to
identify the foreground
O
from the image space
R
. This requires representation
of the foreground region into a compact geometric form.
∈
2.2. Basic Snake Theory
Snakes are planar deformable contours that are useful in several image analysis
tasks. In many images, the boundaries are not well delineated due to degradation
by regional blurring, noise, and other artifacts. Despite these difficulties, human
vision and perception interpolate between missing boundary segments. An active
contour model is intended at inculcating this property of the human vision system.
So the snake framework is formulated such that it approximates the locations and
shapes of object boundaries in images based on the assumption that boundaries
are piecewise continuous or smooth.
The mathematical basis for active contour models owes its foundation to the
principle of unification of physics and optimization theory [12]. The laws of
