rnal of Electrical Systems and Signals, Vol, 2, No, J
Accelerated Local Binary Fitting Scheme for Medical Images Segmentation Moh am mad Bagher Kham echian and Mahd i Saadatmand-Tarzj an
Ab stract. Geom etric and geodesic active contours are typical approaches for medical image segm entation , Specially, local binary fitt ing (LBF) effectively takes advantage of the local intensi ty average in the energy functional to ove rcome segmentation difficulties caused by intensity in homogeneity and ruptured edges. Despite promising results, the convergence rate of LBF is too slow . In this pape r, we proposed a new effici ent implementation of LBF based on the additive operator splitting scheme. In more detail, the multi-dimensional deformation equation of LBF is decomposed into some one -dimensional equations which can be efficiently solved by Tomas' algorithm. Experimental resu lts demonstrated that the proposed algorithm performs better than LBF in terms of both CPU time and solution quality. Ke ywords: Geometric active contours, leve l set, local binary fitting, additive operator splitting, image segmentation. 1. Introduction
Deformable mo dels are frequently-used and we ll-known approaches in medical image segmentation. They were pr imarily introduced by Kass et al. [ I] based on evolving/deforming a parametric act ive contour (Snake) in [he image domain to minimize the interna l and external energy functionals. The former makes the curve smoot h while the later mov es it toward the interested features in the g~ doma in. In orde r to handle topological changes and n "Teasnlg numerical stability, Seth ian et al. [2,3] proposed ceometric active contours based on the level-set funct ion. Primari ly, the edge-based geom etric active contours, stablished by Casllese et al. [4,5] , Kichenassamy et al. [6], and Siddiqi et al. [7] effectively emplo y the object boundary information for segmentation . However, the tive contour without edges (ACWE) is the first region based deform able mod el. It takes advantage of average gray-levels of the internal and external regions of the curve for evolving the act ive contour based on the Mumfor d-Shah energy funct ional [8]. Afterwards, they extended the ir work for segmentation of the multi-phase homogeneous and inhomogeneous images [9]. Despite significant advantages, both regio n-based and edge-based deformable models su ffer considerabl e difficu lties in segm entation of medical images due to boundary rup tur ing, gray- level inh omogeneity, and imag ing artifacts , . Ianuscript received May 24, 2013; revis ed No vembe r 13, 2013; accepted November' 27, 2013 . Th e authors are with the Department of Electrica l Engineering, Ferdowsi University of Mas hhad, Iran. Th e corresponding aut hor' s ema il is:
[email protected].
Recently, patch-based deformable models have been proposed as a soluti on [11- 15]. Gener ally, they effectively employ local edge and region information by taking advantage of appropriate kem el functi ons. For example, Lankton et al. [ 13] proposed a natural framework that allo ws any region-based segmentation energy to be re formulated in a local way . Then, they effectively used the framework with th ree wel l-known region-based schem es for segmentation of differen t medical images. Li et al. [14] introduced a patch-ba sed method based on the k-mea ns clustering fr amework for multi-phase segmentation and bias correction of inho mogeneous medical images. In another work , they presented the local binary fitting (LB F) which empl oys local intens ity average to cope intensity inhomogeneity and ruptured edges. Although LBF provides remarkable results in medical image s segm entatio n, it is affl icted by low convergence rate and sensitivity to parameters, mainly because of using inefficient re initia lizat ion process (in each step, during evolution) and small time-step size (necess ary to guarantee the evolution stability).
1.1. A OS-Based Imp lementations To speed-up the evolution of active con tours, Weickert et al. [ 16] propo sed the addit ive operator splittin g (AOS) method. AOS is an effic ient numerical framework to accelerate curvature-based methods. They demonstrated that AOS can ex pedite typical edge-based active contours with the coefficient of 9. The basic idea of AOS is decomposition of a mult i-dimensional problem into some one-dimensional sub-problems which can be solved sign ificantl y more efficient. The final multi-dimensional solution is appro ximated as the average of all given one dimensional solutions. A number of active contours took advantage of AO S to speed up their implementation . For example, Paragioset ai. [ 17] comb ined the geo desic active contour [4] with the gradient vector flow [18] and utilized AOS for efficient implementation. Gol denb erg et al. [19] took advantage of both AOS and narrow band technique (NRT) [20] for efficient tracking in co lor vid eos. TIley used the Fast marching method [21] for accelerating the re-in itialization of the distance function. Han et at. [24] also employed AOS w ith the geodesic active contour . In another work, Zheng et at. [22] improved implementation of the Mum ford-Shah energy functional [10] by AOS and NR T methods. Also, Leo et al. [23] use d Zheng's approach for segme ntation of red cells in microsco pic images of the urinary sediment. As anoth er example , Jeon et al. [25] could successfully speed up AC WE by using AOS for segmentation of mult i-phase images .
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/.2. ALBF In this paper, we proposed an extended version of ADS for efficient implementation of LBF. It has been successfully used for segmentation of a number of different medical images. In comparison with LBF, the proposed accelerated version of LBF (referred to as AL BF) converged at-least 5 times faster with equivalent solution quality. Furtherm ore, experimental results demonstrated that compared to ACW E and a frequently-used version of the edge-based geometric active contour, ALB F could superiorly segment the blood pool boundary of the left ventricle in II cardiac magnetic resonance (MR) images in terms of both the area-similarity and shape-similarity measures . The desired boundary of each benchmark image was manually delineated by a cardiologist expert.
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In Eq. ( I), the functions J;(x ) and f 2(x ) locally approximate lI(X) in the inside and outside regions of the active contour, respectively. They can be optimally obtained by K (j( x) .
I
( x) =
I
[At eJ (tp( X))II( X)]
- --=-- - -- ---=-
i = 1,2
(5)
K (j( x) . M £ ,i (tp(x»
Finally, according to the Euler-Lagrange equation, LBF evolution equation is given by [1 I]
(6)
/.3. Pap er Outline
The remainder of the paper is organized as follows. Section 2 brietly presents the LBF scheme. The ADS framework is stated in Section 3. Section 4 is devoted to state the proposed method in detail. Experimental results are given in Section 5 and finally, conclusions are drawn in Section 6. 2. Local Binary Fitting Scheme
Let's assume the image II : n ---1 R with the domain n c If and gray-level U(X)E [0.1] for all x =(X,y )E n. The energy functional of LBF is given by [ I I] J (tp,fl h
where t is the time variable, t5(¢) denotes the derivative of
H, (¢) , and V' , div, an i V' 2 represent the gradient. divergence and Laplacian operators, respectively. Also, the functions el and e2 (given by the Euler-Lagrange equation for the first two terms of Eq. I) are obtained by (7)
)= 2
), In I K a (x - y) llI (y ) - / 1(· )1 M 1,e ( tp(y »dydx +
A.-z In IK (j( XvI
( 1)
2
y ) III ( y ) - / 2 (X)I M 2,£ (rp(y»)dydx +
Iv/I e ( tp(x»)1dx + ,uI ~ (Ivtp( )1- t) 2 ds ; x
2
where ¢ is the level-set function, )."
~ ,V
and
J.1
are four
constant coefficients. and y En represents the neighbors of the central pixel x. Also, we have ,'vi£ I (qJ(y ) = H £( tp(y »
(2) { M , :, (", yll = 1- H , (_ (y )),
where HE(¢) is a regularized Heaviside function defined as
I( 2
I IE (qJ ) = 2
tp )
I t - arc tan(-) . 1[
(3)
E
Also, the Gaussian kernel K a (with the standard deviation of (J > 0 ) is computed by (4)
In the LBF energy functional (Eq. I), th first two terms move the active contour toward the desired boundary, the third expression keeps the active contour continuous and derivable, and finally, the last term preserves the eli. tance function (¢) regularity during curve deformation, 3. Additive Operator Splitting Scheme
Additive operator splitting (A DS) [ 16] is an unconditionally stable numerical scheme which presents an effici nt approach for solving a curvature-based partial eli ITer ntial equation (PDF). In mor e detail, typical numerical implementations remain stable only with a small-enough time-step. Thus, they require a large number of iterations for convergence. However, ADS remains stable with a larger time-step size and in sequence. convergence is achieved with fewer iterations. Let us briefly explain A DS procedure already used to numerically implement the evolution equation of the geometric/geodesic active contours. In more detail, the curvatur e-based evolution equation of the level-set method is given by
Iv , 9'(0) = 100
utp = a (x) div . ( flex ) v tp ~ ~)
(8)
wh re a and fJ are two functions and the level-set function ¢ is primarily initialized by rAJ. For example, the above equation with ( a=g, fJ= I) and (a= I, fJ=g) yields the
o. /
Kltamechian et al.: Acce lerat ed Local Binary Fill ing Scheme f or Medical /mages Seg mentation
ALBF
geometric and geodesic active contours, respectively, where
cally f the ally
. . functi.on (e.g., g = s : R2 ~ (O. IJ )S a stopping
1
. ).
I +1'\7( K(j * 11
The explicit scheme is obtained by utilizing the matrix vector notation for Eq. (8) as (5) n +I n I!J. ., ( n ) 11 +1 tp = rp +T L- A l rp tp 1=1
(9)
(Initial curve)
LBF
(6)
"
where I is defined to indicate every dimension of the m dimensional coordinates system (e.g. for a 2-D level-set function, we have m = 2, x )=x and X2=Y)' Also, the superscript n indicates the iteration number (e.g. (l =f/A.J), I denotes the unit matrix, r is the time-step size, and
"J \ . \
~.
A I ((p») = [Qljl (¢ " )] is an operator matrix which indicates the interaction along the I-th direction; such that
(Iteration 95)
(Iteration 40)
~~
2a
l
ive of
('V; I (I V ; IJ:'
dient, o. the uation
/t;J
j e ·" / (i )
•
2
11
° ij/ (tp } = II1 E -C( . I
N 2:
I
(I I)"
(I. )( IVtp !1"J .. ~ fJ
fJ
/ 1
o
(7)
j
=i
m
Otherwise
(Iteration GO)
(Iteration 285)
~"=..:~-----): \~)
\;~«
l/~
where the set N!U) includes two neighboring pixels of the i th pixel along the I-th direction. However, the explicit scheme requires a small t for robust convergence. On the other hand, for solving Eq. (8), we can employ the semi-implicit scheme as
ations
\\ ith a ce IS
rpl1 + I
=[ J _ T 1/1 I
A 1 (rpl1 )
]-1
rpn
(II)
l=!
Although the semi-implicit scheme is unconditionally stable, inverting the large bandwidth matrix requires immense computational burden. To tackle this problem, we may take advantage of the AOS variant given by rp
11
+ I
=-I 11/ I
rn 1=1
[
J-
T III
11
A 1 (rp)
J-
J 11 rp
j'-'Sr (Iteration 80)
(Iteration 380)
Fig. 1. Comparing the convergence rate of (left column) ALBF and (right column) Lil F for the imago: vessel (of size 11 1y 110). The corresponding iteration number is quoted under each illustration.
The general deformation PDE of an active contour can be usually written as follows: drp
-
d
I
=
'\7 rp
( 13)
a div (/3 r,:;-:] ) + E , 1'\7 rpl
(J 2)
TIle above equation, which provides a solution to Eq. (8), can be efficiently implemented by Tomas' algorithm [1 6l
-to Proposed Algorithm (8)
/{~\-r
/'
( 10)
onally icient crential erical l-enough
l~
+
As aforementioned. the convergence rate of LBF is restricted by the time-step size. This difficulty can be addressed by utilizing AOS scheme in numerical implementation.
where the speed E is typically obtained from the external and regularization terms of the energy functional. In this case, the AOS scheme can be simply extended to rp n + 1 = - I m L 1/1
1=1
[ I - T m A I (rp) n
J-
J(
rp n + T E )
(14)
Specially, we can rewrite the deformation equation of LBF (Eq, 6) in the fonn of Eq. (13) by using
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LBF
ALBF
(Initial curve)
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ALBF
LBF
(Initial curve)
(Initial curve)
(Iteration 50)
(Iteration140)
(Iteration 75)
(Iteration 410)
(Iteration 110)
(Iteration 550)
(Initial curve)
(Iteration 50)
(Iteration250)
,r
\
j\
ij
~. ~ ~
t..~~
(Iteration 75)
(Iteration 750)
Fig. 3. Comparing the converge nce rate of (left column) AL BF and (right column) LBF for the image heart (o f size 152x 128).
, I
I
,,
I
l·. .. r\,~ '~. , ~j
)
r.
'
."\
..
-
"
~. ~- ii ,
. .---! ~
"
~
":'"
(Iteration 110)
,
- ' i'-',
(1
(15)
Therefore, the AOS scheme of LBF (referred to as ALBF) is computed by
(Iteration II 00)
Fig. 2. Comparing the conver gence rate of (left column) ALBF and (rig ht colu mn) LBF for the image fi nger bone (of size 236 x2 13).
=
5
Khamechian et al.: Accelerated Local Binary Fitting Scheme f or Medical Images Segmentation
ii) Evolution along the
ALBF
Xl =X direction by solving the following tri-diagonal set of equations using Tomas' algorithm to obtain " n ..-1
LBF
[I - 2
T
A , «1,'1 )) v l1 + 1=
