the image intensities in the region x : ϕ xðÞ> 0fgand
x : ϕ xðÞ< 0
fg
, respectively. Such optimal constants can
be far away from the original image data, if the intensities
outside or inside the contour C ¼ x : ϕ xðÞ¼0fgare not
homogeneous. As a result, the C V model generally fails to
segment images with intensity inhomogeneity.
(PS) model
Intensity inhomogeneity can be addressed by more
sophisticated models than PC models. Vese and Chan
[21] and Tsai et al. [22] independently propose two simi-
lar region-based models for general images. These mod-
els, widely known as (PS) models , aims at expressing the
intensities inside and outside the contour as (PS) func-
tions instead of constants. The following energy func-
tional was defined:
E
PS
ðu
þ
; u
; ϕÞ¼
Z
Ω
u
þ
Ijj
2
HðϕÞdx þ μ
Z
Ω
ru
þ
jj
2
HðϕÞdx
þ
Z
Ω
u
I
jj
2
1 H ϕðÞðÞdx þ μ
Z
Ω
ru
jj
2
1 H ϕðÞðÞdx
þν
Z
Ω
rH ϕðÞ
jj
dx
ð4Þ
where u
+
and u
-
are smooth functions approximating the
image I inside and outside the contour, respectively,
which are obtained by solving the following two damped
Poisson equations:
u
þ
I ¼ μΔu
þ
in x : ϕ xðÞ> 0
fg
;
@u
þ
@n
¼ 0 on x : ϕ xðÞ¼0
fg
ð5Þ
u
I ¼ μΔu
in x : ϕ x
ðÞ
< 0fg;
@u
@n
¼ 0 on x : ϕ xðÞ¼0
fg
ð6Þ
The PS model is also implemented by an alternative
procedure: for each iteration and the corresponding level
set function ϕ
n
, we firs t obtain u
þ
ϕ
n
ðÞand u
ϕ
n
ðÞby
solving Equations (5) and (6), then obtain ϕ
nþ1
by min-
imizing the functional E
PS
u
þ
ϕ
n
ðÞ; u
ϕ
n
ðÞ; ϕðÞwith re-
spect to ϕ . Repeat the process until the zero-level set of
ϕ
nþ1
is exactly on the object boundary.
The solution of the PS model lead to a PS approxima-
tion of the original image I(x):
uxðÞ¼u
þ
H ϕ xðÞðÞþu
1 H ϕ xðÞðÞðÞð7Þ
where u
+
and u
-
are obtained by solving Equations (5) and
(6). In the PS model, two coupled equations must be solved
to obtain u
+
and u
-
before each iteration, and the computa-
tional cost is very expensive. Moreover, in the implementa-
tion of PS model, u
+
and u
-
must be extended to the whole
image domain, which is difficult to implement and also
increases the computational cost. In summary, the high
complexity limits the application of PS model in practice.
RSF model
In order to improve the performance of the global CV
[12] and PS models on [21,22] images with inhomogen-
eity, Li et al. [14,26] recently proposed a novel region-
based activ e contour model in a variational level set for-
mulation. They introduced a kernel function and defined
the following energy functional:
E
RSF
ðf
1
; f
2
; ϕÞ¼λ
1
ZZ
K
σ
x yðÞIyðÞf
1
xðÞ
jj
2
H ϕ yðÞðÞdy
dx
þλ
2
ZZ
K
σ
x yðÞIyðÞf
1
xðÞ
jj
2
1 H ϕ yðÞðÞðÞdy
dx
þν
Z
Ω
rH ϕ x
ðÞðÞ
jjdx þ μ
Z
Ω
1
2
rϕ x
ðÞ
jj1
ðÞ
2
dx
ð8Þ
where K
σ
is a Gaussian kernel with standard deviation σ,
and f
1
(x)andf
2
(x) are two smooth functions that ap-
proximate the local image intensities inside and outside
the contour, respectively. They are computed as:
f
1
xðÞ¼
Z
Ω
K
σ
x yðÞIyðÞH ϕ yðÞðÞdy
Z
Ω
K
σ
x yðÞH ϕ yðÞðÞdy
; f
2
xðÞ
¼
Z
Ω
K
σ
x yðÞIyðÞ1 H ϕ yðÞðÞðÞdy
Z
Ω
K
σ
x yðÞ1 H ϕ yðÞðÞðÞdy
ð9Þ
The RSF model is implemented via an alternative pro-
cedure: for each iteration and the corresponding level
set function ϕ
n
, we first compute the fitting values
f
1
ϕ
n
ðÞand f
2
ϕ
n
ðÞ, then obtain ϕ
nþ1
by minimizing
E
RSF
f
1
ϕ
n
ðÞ; f
2
ϕ
n
ðÞ; ϕðÞwith respect to ϕ. This process is
repeated until the zero-level set of ϕ
nþ1
is exactly on the
object boundary.
Like the PS model, the solution of the RSF model also
lead to a piecewise smooth approximatio n of the original
image I (x):
u xðÞ¼f
1
H
E
ϕ xðÞðÞþf
2
1 H
E
ϕ xðÞðÞðÞð10Þ
However, the smooth functions f
1
and f
2
are computed
directly from (9) but no longer obtained by solving
equations.
The RSF model improves the global PC and PS models
on intensity inhomogeneity and is more computationally
efficient than the PS models. Nevertheless, four convolu-
tions to be computed at each iteration still demand a high
computational cost. Besides, the RSF model is to some ex-
tent sensitive to contour initialization (initial locations,
sizes and shapes); it obtained different segmentation with
Wang and He EURASIP Journal on Image and Video Processing 2012, 2012:16 Page 3 of 13
http://jivp.eurasipjournals.com/content/2012/1/16