Noname manuscript No.(will be inserted by the editor)

New Variational Formulations for Level Set Evolutionwithout Reinitialization with Applications to ImageSegmentation

Chunxiao Liu · Fangfang Dong · Shengfeng Zhu · Dexing Kong ·Kefeng Liu

Received: date / Accepted: date

Abstract Interface evolution problems are often solvedelegantly by the level set method, which generally re-quires the time-consuming reinitialization process. Inorder to avoid reinitialization, we reformulate the vari-ational model as a constrained optimization problem.Then we present an augmented Lagrangian method anda projection Lagrangian method to solve the constrainedmodel and propose two gradient-type algorithms. Forthe augmented Lagrangian method, we employ the Uzawascheme to update the Lagrange multiplier. For the pro-jection Lagrangian method, we use the variable split-ting technique and get an explicit expression for theLagrange multiplier. We apply the two approaches tothe Chan-Vese model and obtain two efficient alternat-ing iterative algorithms based on the semi-implicit ad-ditive operator splitting scheme. Numerical results onvarious synthetic and real images are provided to com-pare our methods with two others, which demonstrateeffectiveness and efficiency of our algorithms.

Keywords Level set method · Reinitialization · Aug-mented Lagrangian method · Projection Lagrangianmethod · Chan-Vese model · Additive operator splitting

C. Liu · K. Liu

Center of Mathematical Sciences, Zhejiang University, Hangzhou310027, P.R. China.

C. Liu

E-mail: xxliu198431@126.comF. Dong

School of Statistics and Mathematics, Zhejiang Gongshang Uni-versity, Hangzhou 310018, P.R. China

S. Zhu · D. Kong

Department of Mathematics, Zhejiang University, Hangzhou310027, P.R. China.

1 Introduction

Interface evolution occurs in a wide variety of settingssuch as imaging precessing, computer vision, shape op-timization and geometric inverse problems. Most tra-ditional explicit front tracking algorithms place markerpoints along the interface and advance the position ofthese points through an evolution equation. However,expensive reparametrization is necessary during the evo-lution and the topology of the subregions separated bythe interface cannot change automatically. The levelset method (LSM) originally proposed by Osher andSethian [1] for interface tracking can overcome thesetwo main drawbacks. Effective and efficient numericalschemes can be implemented on fixed grids. Moreover,it can handle topological changes such as merging, split-ting and forming sharp corners. The essential idea of theLSM is to implicitly embed the propagating interface asthe zero level set of a higher dimensional level set func-tion (LSF). With this representation, the motion of theinterface is described by a time dependent Hamilton-Jacobi type evolution equation of the LSF. The LSMhas been applied in many fields including optimal shapedesign [2,3], reservoir simulation [4], inverse problems[5,6], image processing [7,8], etc. For more details onthe LSM and its applications, we refer to see [9–11].

During the level set evolution, regularity is oftenneeded to impose on the LSF to prevent it to be toosteep or too flat near the interface. This is normallydone by requiring it to be a signed distance function.The process is commonly known as reinitialization, whichis used extensively as a numerical strategy for maintain-ing interface evolution stable (cf. [9,10]). However, thereare some drawbacks for reinitialization (cf. [9,12,13]).For the CV model in image segmentation task, whetherthe reinitialization is processed affects the topology of

2

the resulting image for some special images [13]. It stillremains to be a problem when and how to implementreinitialization [12,14].

There are several effective ways to implement reini-tialization. Sethian [15,16] developed the fast marchingmethod to efficiently calculate the signed distance func-tion by solving the Eikonal equation on both sides of theinterface. Another efficient algorithm for solving thisequation is the fast sweeping method (cf. [17]). Sussmanet al. [4] proposed a time dependent PDE based iter-ative algorithm to satisfy the Eikonal constraint equa-tion. For some cases requiring velocity extension, thefast marching method can be used to construct a ve-locity field, which simultaneously avoids reinitialization(cf. [18]). In [19], a PDE based fast algorithm imple-mented on a narrow band tube of the interface wassuggested to extend velocity and implement reinitial-ization. As interesting variants of the traditional LSM,level set algorithms of piecewise constant type [13,20–22] also eliminate the need of reinitialization. Thesemethods require to handle additional piecewise con-stant constraints.

To avoid reinitialization, Li et al. [14] proposed avariational formulation and applied it to the geodesicactive contour model for image segmentation. They addeda quadratic penalty term into the original energy func-tional in order to force the LSF to be a signed dis-tance function during evolution. Therefore, the algo-rithm they provided is essentially based on a classicalquadratic penalty method for constrained optimization(cf. [23]). By forcing the weight of the constraint termto infinity, one can penalize the constraint violationswith increasing severity. That is, the larger weight oneuses, the closer the LSF is to a signed distance function.However, the Courant-Friedrichs-Lewy (CFL) stabilitycondition for explicit discretization of the gradient de-scent flow does not allow the weight value to be large,or else the time step has to be small and the iteration israther slow. Therefore, there is a contradiction betweenthe accuracy of the constraint and the choice of largetime steps. In numerical experiments of [14], large timesteps were employed at the expense of small weight val-ues.

In this paper, we propose two more accurate con-strained optimization approaches, an augmented La-grangian method and a projection Lagrangian method,to get rid of reinitialization in the level set evolution.For the projection Lagrangian method, we constructa new scheme to update the Lagrange multiplier us-ing the variable splitting technique by introducing anauxiliary variable [24–26]. Then we apply our meth-ods to the well-known Chan-Vese (CV) model (cf. [27–29]), which is an especially useful model when the im-

age to be segmented can be approximated by piece-wise constant functions. The additive operator splitting(AOS) scheme is employed in solving the obtained non-linear diffusion equation. Numerical results show thatour methods are effective and less sensitive to noise.Comparing with the LSM with reinitialization and thepenalty method of [14] applying to the CV model, ouralgorithms work much faster.

The rest of this paper is organized as follows. InSection 2, some existing results related to the LSM andreinitialization process are briefly stated. In Section 3,our new methods are proposed in detail. Then we applyour methods to the CV model and devise two robustalgorithms in Section 4. Numerical examples are pre-sented in Section 5. Finally, we give a brief summaryand outline the future work in Section 6.

2 Related Works

2.1 Traditional LSM

We first recall the basic level set formulations. Let Ω ⊂Rd (d=2,3) be an open bounded domain and Γ (t) | t >0 be a series of moving closed interfaces in Ω withvelocity v = x′(t). For some t, Ω1(t) and Ω2(t) aretwo subregions separated by Γ (t). Define a Lipschitzcontinuous LSF φ(x, t) satisfying

φ(x, t) < 0, x ∈ Ω1(t),φ(x, t) = 0, x ∈ Γ (t),φ(x, t) > 0, x ∈ Ω2(t).

(1)

The evolution equation of Γ (t) can be transformed intothat of φ(x, t). Specifically, differentiating the equationφ(x(t), t) = 0 with respect to t reads:

φt + v · ∇φ = 0. (2)

By restricting the front to propagating along its normaldirection with speed vn, equation (2) turns to the so-called level set equation

φt + vn|∇φ| = 0. (3)

The velocity vn can be a function of the normal direc-tion, mean curvature, etc.

In order to keep stability in numerical implementa-tion, often regularity is imposed on the LSF to ensure0 < c 6 |∇φ| 6 C, for some constants c and C. Ac-tually, it is very desirable to require it to be a signeddistance function, i.e.,

φ(x, t) = −d(Γ (t), x), x ∈ Ω1(t),φ(x, t) = 0, x ∈ Γ (t),φ(x, t) = d(Γ (t), x), x ∈ Ω2(t),

(4)

3

where d(Γ (t), x) denotes the Euclidian distance from x

to Γ (t). An equivalent constraint to (4) is the Eikonalequation

|∇φ(x, t)| = 1. (5)

In order to satisfy (5), the authors of [4] used aniterative reinitialization scheme to solve the followingequation to steady state:

φt + sign(φ0)(|∇φ| − 1) = 0 in Ω × R+,φ(x, 0) = φ0 in Ω,

(6)

where φ0 is the function to be reinitialized and sign(φ0)denotes the sign function of φ0. But this expensive reini-tialization approach may cause the interface to moveappreciably after many iterations and it cannot reini-tialize the LSF which is far away from a signed distancefunction. These issues can be resolved well by arbitraryinterface preserving schemes [30] and an efficient nar-row band based algorithm [19]. In the filed of imagesegmentation, for some images without clear edges, thesegmentation results are largely dependent on the man-ner of reinitialization [12,14].

2.2 Variational Level Set Method (VLSM)

The VLSM proposed in [31] offers us a way to embedthe LSF directly into the energy functional.

Assume that the minimization problem with respectto interfaces is in the following general form:

minΓF(Γ ). (7)

Then the VLSM embeds the LSF into the energy func-tional F(Γ ) by utilizing the following facts about theHeaviside function and Dirac function:∫

Ω1

fdx =∫

Ω

f(1−H(φ))dx,

∫

Ω2

fdx =∫

Ω

fH(φ)dx, (8)

|Γ | :=∫

Γ

ds =∫

Ω

δ(φ)|∇φ|dx,

where f is some given function defined on Ω. The Heav-iside function H(x) is defined as

H(x) =

1, x ≥ 0,0, x < 0.

(9)

The Dirac function δ(x) is actually the derivative of theHeaviside function in the distributional sense, i.e.,

δ(x) =∞, x = 0,

0, x 6= 0.(10)

By the VLSM, the minimization problem (7) can bereformulated as

minφF(φ). (11)

The Euler-Lagrange equation of (11) with respect to φis much easier to be derived than that of (7) with re-spect to Γ . We can use the gradient descent methodor other efficient schemes to get an optimal solution of(11). Moreover, we can incorporate any additional in-formation or constraints on the LSF into the variationalenergy functional.

In numerical implementation, we use the regular-ized version with parameter ε > 0 to approximate theoriginal nondifferentiable function H(x) and δ(x) re-spectively as

Hε(x) =12

+1π

arctanx

ε, (12)

and

δε(x) =1π

ε

x2 + ε2. (13)

As pointed out in [27], this kind of smooth approxima-tions have the tendency to lead a global minimizer ofthe algorithm.

2.3 Variational Level Set Method WithoutReinitialization (VLSMWR)

Using the VLSM and considering the constraint (5), wecan formulate the general interface evolution problemas the following constrained minimization problem

minφ

F(φ) subject to |∇φ| = 1. (14)

In order to avoid reinitialization in the geodesic ac-tive contour model, the authors of [14] added a penal-ization term into the minimization functional after ap-plying the VLSM and obtained the following uncon-strained minimization problem

minφ

L(φ) = F(φ) +

µ

2

∫

Ω

(|∇φ| − 1)2dx

. (15)

The first-order necessary condition leads to

∂L

∂φ= F ′(φ)− µ

[∆φ−∇ ·

( ∇φ

|∇φ|)]

= 0. (16)

They used the explicit Euler scheme

φk+1 = φk −∆t∂L

∂φ(φk). (17)

4

to solve the PDE

∂φ

∂t= −∂L

∂φin Ω × R+,

φ(x, 0) = φ0(x) in Ω.

(18)

It was pointed out in [14] that the time step ∆t >0 and the penalization parameter µ > 0 must satisfyµ∆t < 0.25 for stability. If one choose a relatively largeµ, ∆t must be rather small in order to satisfy the CFLcondition, which means that more iterations are neededto solve (18) to the steady state. On the other hand, wecan easily see that this method is essentially a kind ofquadratic penalty method for constrained optimization[23]. If µ is too small, we cannot penalize the constraintviolation effectively. In order to improve the accuracyand stability, we introduce the following two methods.

3 Two Constrained Optimization Methods

We know from [23] that the quadratic penalty functionis not an exact penalty function. The constraint will befulfilled only when the penalty parameter is sufficientlylarge. However, too large penalty parameters may re-sult in instability. In the following, we use two exact

methods: the augmented Lagrangian method and theprojection Lagrangian method, to force the LSF to beclose to a signed distance function so that we can avoidthe complicated and expensive reinitialization process.Based on the piecewise constant and binary level setframe, the two methods have been applied to imagesegmentation [13,20] and shape optimization [21,22].

3.1 Augmented Lagrangian Method

Firstly, we use the augmented Lagrangian method tosolve (14). The augmented Lagrangian method reducesthe possibility of ill-conditioning by introducing explicitLagrange multiplier estimates at each step into the min-imization function. There is a simple but efficient up-dating scheme, i.e., the Uzawa algorithm. Moreover, theconvergence of this algorithm can be guaranteed with-out increasing µ to a very large value as the penaltymethod.

Define K(φ) = |∇φ| − 1, then the augmented La-grangian functional of (14) is

Lµ(φ, λ) = F(φ) +∫

Ω

λK(φ)dx +µ

2

∫

Ω

K2(φ)dx, (19)

where λ ∈ L2(Ω) is the Lagrange multiplier. The pe-nalization parameter µ > 0 should be chosen properly.

A saddle point of Lµ requires that

∂Lµ

∂φ= 0 and

∂Lµ

∂λ= 0. (20)

In fact, we minimize Lµ with respect to φ and max-imize Lµ with respect to λ. From the definition of Lµ

in (19), we have

∂Lµ

∂φ= F ′(φ)−∇ ·

(λ∇φ

|∇φ|)

− µ

[∆φ−∇ ·

( ∇φ

|∇φ|)]

, (21)

∂Lµ

∂λ= K(φ). (22)

We update φ, λ alternatively to find the saddle pointstarting from the initial guesses φ0 and λ0.

The updating of φ is done by introducing an artifi-cial time variable t and moving in the steepest descentdirection by

∂φ

∂t= −∂Lµ

∂φ, (23)

where ∂Lµ/∂φ is given by (21). When t → ∞, (23)reaches the steady state ∂φ/∂t = 0, which implies that∂Lµ/∂φ = 0. In numerical implementation, we cansolve equation (23) by the explicit scheme

φ[k,n+1] = φ[k,n] −∆t[k,n] ∂Lµ

∂φ(φ[k,n], λk),

φ[k,0] = φk,

(24)

where n = 0, 1, · · · , N − 1 in the superscript [k, n] de-notes the nth inner iteration corresponding to the kthouter iteration and ∆t[k,n] > 0 is the time step, whichshould be small enough to ensure the stability of thenumerical scheme. A line search method can be ap-plied to find the optimal ∆t[k,n] at each iteration. Al-ternatively, considering the time consuming of the linesearch strategy, we can choose a small fixed ∆t bytrial and error. We can also employ the semi-implicitschemes to improve the stability and eliminate the timerestriction. After performing N inner iterations, we setφk+1 = φ[k,N ] to approximately solve (23).

Then we employ the Uzawa algorithm to update theLagrange multiplier λ by

λk+1 = λk + µK(φk+1). (25)

When the iteration of λ converges, we get K(φ) = 0.In conclusion, we incorporate all the above schemes

into the following inner-outer iterative Algorithm 1.

5

Algorithm 1 Uzawa Algorithm for VLSMWRChoose µ properly, fixed. Initialization: φ0, λ0, k = 0.

Step 1. Update φ using the explicit Euler scheme (24) or other

semi-implicit schemes, to approximately solve

Lµ(φk+1, λk) = minφ

Lµ(φ, λk).

Step 2. Update λ by (25).Step 3. Iterate again if necessary; k = k + 1.

3.2 Projection Lagrangian Method

We can also solve the problem (14) by the Lagrangemultiplier approach. Let the penalization parameter in(19) be zero. Then we get the following Lagrangianfunctional

L(φ, λ) = F(φ) +∫

Ω

λK(φ)dx. (26)

A saddle point of L requires that

∂L

∂φ= F ′(φ)−∇ ·

(λ∇φ

|∇φ|)

= 0, (27)

∂L

∂λ= K(φ) = 0. (28)

The constraint function in our problem contains thefirst order derivative of φ. With the divergence opera-tion upon λ in (27), we cannot use any similar projec-tion approach in [13] to get an explicit formula of λ by(28). Therefore, we use the variable splitting methodby creating a new vector variable, say p, to serve asthe argument of the functional K, under the constraintp = ∇φ. This leads to following constrained problem

maxλ

minφ,p=∇φ

F(φ) +

∫

Ω

λ(|p| − 1)dx

(29)

Relaxing the equality constraint p = ∇φ and penalizeits violation by the quadratic function, we obtain anapproximation of (29)

maxλ

minφ,p

L(φ,p, λ) = F(φ) +

∫

Ω

λ(|p| − 1)dx

+γ

2

∫

Ω

|p−∇φ|2dx

, (30)

where γ > 0 is the penalty parameter.The system of optimality conditions of (30) is

∂L∂φ

= F ′(φ)− γ(∆φ−∇ · p) = 0, (31)

∂L∂p

= λp|p| + γ(p−∇φ) = 0, (32)

∂L∂λ

= |p| − 1 = 0. (33)

The above three equations must be satisfied at a saddlepoint of the max-min problem (30).

We can directly optimize the functional in (30). An-other way is to solve the corresponding nonlinear sys-tem of optimality conditions (31)-(33).

The alternating iterative method of multipliers con-sists in minimizing L(φ,p, λ) with respect to φ and p,keeping λ fixed, then updating λ for fixed φ and p.

Firstly, for fixed λk,

(φk+1,pk+1) ∈ arg minφ,p

F(φ) +

∫

Ω

λk(|p| − 1)dx

+γ

2

∫

Ω

|p−∇φ|2dx

. (34)

To solve the problem (34), we separate it into the follow-ing two sub-problems and update φ and p alternatively.

φk+1 ∈ arg minφ

F(φ) +

γ

2

∫

Ω

|pk −∇φ|2dx

, (35)

pk+1 ∈ arg minp

∫

Ω

λk|p|dx +γ

2

∫

Ω

|p−∇φk+1|2dx

.

(36)

For updating φ, the optimality condition of subprob-lem (35) is actually (31). Then by the gradient descentmethod, we find a steady-state solution to the PDE

∂φ

∂t= −∂L

∂φ, (37)

where ∂L/∂φ is given by (31). The explicit scheme forsolving (37) reads:

φ[k,n+1] = φ[k,n] −∆t[k,n] ∂L∂φ

(φ[k,n],pk, λk). (38)

Then φk+1 is determined similarly as in Algorithm 1.The minimization with respect to p in subproblem

(36) can be done by obtaining the following closed form:

pk+1 =

∇φk+1 − λ∇φk+1

γ|∇φk+1| , if |∇φk+1| > λγ ,

0, else.(39)

The formulation (39) is the weighted shrinkage operatorthat can be computed in a similar way as the shrinkageoperator [24,32]. This weighted shrinkage is extremelyfast and requires only a few operations per element ofpk+1.

Finally, we derive the formula of λ by a projectionmethod. Multiplying p on both sides of equation (32)and using (33), we obtain the explicit expression of λas:

λ = γ(p · ∇φ− 1). (40)

Then with the updated values of φ and p, we have theupdating scheme of λ:

λk+1 = γ(pk+1 · ∇φk+1 − 1). (41)

6

Algorithm 2 Projection Lagrangian Algorithm forVLSMWR

Choose γ properly, fixed. Initialization: φ0, p0, λ0, k = 0.Step 1. Update φ by the explicit scheme (38) or semi-implicit

schemes, to approximately solve

L(φk+1,pk, λk) = minφL(φ,pk, λk).

Step 2. Update p by (39), to solve

L(φk+1,pk+1, λk) = minpL(φk+1,p, λk).

Step 3. Update λ by (41).Step 4. Iterate again if necessary; k = k + 1.

Now we present the inner-outer iterative projectionLagrangian Algorithm 2.

Remark 1: A useful stopping criterion should be de-vised according to the practical problem to be solved.Therefore, we have not and cannot present a unifiedstopping criterion for either of the above two algorithmsbased on the general frame of VLSMWR.

4 Applications to the Chan-Vese Model

As a piecewise constant case of the Mumford-Shah model[33], the CV model [27] is one of the classical activecontour models in image segmentation. It is usuallysolved by the VLSM. Therefore, the penalty approachbased VLSMWR in [14] can be naturally applied to CVmodel. In this section, we will apply our two proposedvariational methods to the CV model to get rid of reini-tialization.

Let I : Ω → R be an image to be segmented. TheCV model aims to find an interface which is optimal inthat it minimizes the following functional

F(Γ, c1, c2) = α

∫

Γ

ds + β1

∫

Γ out

(I − c1)2dx

+ β2

∫

Γ in

(I − c2)2dx, (42)

where Γ out and Γ in are respectively the subregions out-side and inside Γ . The constants c1 and c2 are the meanintensities of each region separated by Γ . Here, α > 0is the weight of the regularization term and β1, β2 > 0are the weights of the fidelity terms.

Using the variational level set frame, (42) can betransformed to

F(φ, c1, c2) = α

∫

Ω

δε(φ)|∇φ|dx

+ β1

∫

Ω

(I − c1)2(1−Hε(φ))dx (43)

+ β2

∫

Ω

(I − c2)2Hε(φ)dx.

We can update c1, c2 and φ alternatively.Minimizing F with respect to c1 and c2 for some

fixed φk, we obtain

ck1 =

∫Ω

I (1−Hε(φk))dx∫Ω

(1−Hε(φk))dx, (44)

ck2 =

∫Ω

I Hε(φk)dx∫Ω

Hε(φk)dx. (45)

Then we can apply our proposed methods in Section3 to the CV model. For the gradient descent flow (23)in the augmented Lagrangian method, we have

∂φ

∂t=µ∆φ +∇ ·

[(λ− µ)

∇φ

|∇φ|]

+ αδε(φ)∇ ·( ∇φ

|∇φ|)

+ δε(φ)[β1(I − c1)2 − β2(I − c2)2

]. (46)

In the projection Lagrangian method, the gradient flow(37) turns to

∂φ

∂t=γ∆φ + αδε(φ)∇ ·

( ∇φ

|∇φ|)

+ δε(φ)[β1(I − c1)2 − β2(I − c2)2

].

(47)

Noting the time step restrictions of the linear and non-linear diffusion terms in equations (46) and (47) if ex-plicit schemes are employed, we choose the semi-implicitAOS scheme to improve the stability and acceleratethe convergence. This scheme was first proposed in [34]and later applied to nonlinear diffusion filtering [35]. Itis unconditionally stable and does not suffer from thetime step restriction. The AOS scheme splits an arbi-trary d-dimensional spatial operator into a set of one-dimensional ones and computes implicitly in parallel bythe Thomas algorithm efficiently.

Specially for (46), we solve the following semi-implicitequation by the AOS scheme for n = 0, 1, · · · , N − 1:

φ[k,n+1] − φ[k,n]

∆t=

µ∆φ[k,n+1] +∇ ·[(λk − µ)

∇φ[k,n+1]

|∇φ[k,n]|]

+ αδε(φ[k,n])∇ ·(∇φ[k,n+1]

|∇φ[k,n]|)

(48)

+ δε(φ[k,n])[β1(I − ck

1)2 − β2(I − ck2)2

].

with φ[k,0] = φk. The solution of (47) by the AOSscheme is similar as (48). For full discretization, we usethe finite differences [27] to discretize the spatial partialderivatives.

Though it is not very expensive in calculation ofone semi-implicit step, the total computational effort ofone outer iteration requiring many inner steps can bevery huge. In order to reduce the computational effortwhile keeping effectiveness, we simplify the inner-outer

7

iterative framework of each algorithm by performingonly one step (i.e., N = 1) in the inner iteration usingthe AOS scheme. The resulting alternating algorithmsare very efficient from numerical experience.

The evolution can be stopped if some stopping cri-terion is satisfied. In our paper, we employ a criterionfor CV model recently proposed in [36]. The iterationwill be stopped automatically when the change of thecurve length keeps smaller than a prescribed thresholdθlength for a fixed threshold of iterations M.

Now we are ready to present the algorithms for theCV model:

Algorithm 3 Uzawa Algorithm for VLSMWR apply-ing to CV model

Choose µ properly, fixed. Initialization: φ0, λ0, k = 0.Step 1. Update c1, c2 by (44) and (45).

Step 2. Update φ by solving (46) using the semi-implicit AOSscheme (48) with N = 1.

Step 3. Update λ by (25).Step 4. Test whether the stopping criterion is satisfied. If yes,the algorithm is stopped. Otherwise, set k = k+1 and continue.

Algorithm 4 Projection Lagrangian Algorithm forVLSMWR applying to CV model

Choose γ properly, fixed. Initialization: φ0, p0, λ0, k = 0.

Step 1. Update c1, c2 by (44) and (45).Step 2. Update φ by solving (47) using the AOS scheme with

N = 1.Step 3. Update p by (39).Step 4. Update λ by (41).Step 5. Test whether the stopping criterion is satisfied. If yes,the algorithm is stopped. Otherwise, set k = k+1 and continue.

Remark 2: The proposed two algorithms can be gen-eralized to the multi-phase CV model [29] to eliminatethe need of reinitialization.

Remark 3: For all our implementation of the Uzawaalgorithm, we have set µ to be constant during the iter-ations. Better convergence behavior may be obtained ifµ is increased gradually. But instability may be causedif µ is increased too quickly, which is a common phe-nomenon when using the augmented Lagrangian ap-proach [23].

Remark 4: We have used variable splitting and thepenalty approach for the projection Lagrangian algo-rithm. Then one drawback exists that as γ becomesvery large, the intermediate minimization problems be-come increasingly ill-conditioned. Then numerical prob-lems will be caused [23,26]. Therefore, we should choosemoderate values of γ in simulation.

5 Computational Results

In this section, some numerical results for qualitativeand quantitative comparisons among different methodsare presented to demonstrate the effectiveness and ef-ficiency of our algorithms. We refer the Algorithm 3,Algorithm 4 and the penalty method of [14] applyingto CV model respectively as UA, PLA and Li’s methodin the following. We set β1 = β2 = 1 for generality. Theinitial LSF φ0 is a piecewise constant function with arectangular contour located in the middle of the im-age to be segmented for our methods and Li’s method.For the VLSM with reinitialization, the initial LSF isa signed distance function. The parameter α is usuallyformatted by α = η∗2552, η ∈ (0, 1). We set the spatialstep h = 1 and the parameter ε = 1.5. For the stoppingcriterion, we use θlength = 5 and M = 10 as in [36].

First, we use the natural Europe night-lights imageto illustrate the side effects of reinitialization for theCV model in Fig. 1. We can see from Fig. 1(b) and Fig.1(c) that the segmented images visually have differenttopologies for the VLSM with no or with reinitializationprocess using the same parameters, which demonstratesthat whether or not the reinitialization is done affectssegmentation results. For reinitialization, we have usedthe first order upwind scheme [30] to solve (6) with tentime marching steps and implement this procedure forevery fifth iteration. Then the segmentation results byLi’s method and our methods given in Fig. 2 show theeffectiveness of the three algorithms. Moreover, theyshow that Li’s method and the presented methodol-ogy incorporating the reinitialization constraint vari-ationally do not suffer from the same side effects asthe classical reinitialization. See Table 1 and Table 2for the corresponding iteration number and cost timefor segmentation of this example. For the VLSM withreinitialization, more than half of the time is spent onreinitialization. Li’s method has faster implementationalthough it requires more iterations than the VLSMwith reinitialization. We can see that our methods aremuch faster than both of the VLSM and Li’s method.

Then we compare our methods with Li’s method bya spiral image from an art picture. The original imageshown in Fig. 3(a) is processed by Li’s method andour methods, respectively. As shown in Fig 4, our twomethods can also detect features as Li’s method for thesame image. We observe from Table 2 that either ofour algorithms exceeds Li’s method in the convergencespeed and computational time.

We show further the robustness and accuracy of ourmethods by a synthetic binary image. The original bi-nary image with ground truth known a priori is pre-sented in Fig. 5(a). Then the image is imposed with

8

the Gaussian additive noise in Fig. 5(b). The Signal toNoise Ratio (SNR), which is defined as

SNR = 10 · log10

(Variance of DataVariance of Noise

),

for this image is 4.41.The degraded image is segmented by the VLSM

with reinitialization, Li’s method and the two proposedmethods, respectively. We use the same initial contouras shown in Fig. 5(b) for all these approaches. The in-termediate iterations and the final results are displayedin Fig. 6. From the satisfactory results, we can see thatall the four methods work for this noisy image. In Fig.7, we present quantitative comparisons among the fourmethods by giving the plots of the L2-Error (i.e., L2

distance between the characteristic function of the seg-mented object and that of the exact object) vs. theiteration number and the time spent, respectively. Weobserve from Fig. 7 that our methods converge fasterand give more accurate segmentation results. Then weinvestigate the influence of the parameter µ in Table 3by increasing µ gradually while fixing other parametersas α = 0.15 ∗ 2552 and ∆t = 0.02. We can see fromTable 3 that small values of µ lead to quantitativelybetter segmentations. Therefore, we need not use verylarge µ in the augmented Lagrangian method. Mean-while, the time reported in Table 1 and Table 2 showsa large reduction in the total computation effort by ouralgorithms.

In Fig. 8, we show the segmentation of an ultra-sound medical image by Li’s method and our methods,respectively. All the three algorithms can detect the cellin the image. Again, the time reported in Table 2 showsmore efficiency of our methods. Moreover, we use thisexample to investigate the role of the penalty param-eters µ and γ for our methods. In Fig. 9, we plot themean deviation of |∇φk| − 1, which measures the dis-tance between the computed LSF at the k-th iterationand the signed distance function of the same zero con-tour. We observe that the constraint |∇φ| = 1 can besatisfied better as µ or γ increases. Considering the seg-mentation accuracy, however, the time step should bedecreased accordingly.

In Fig. 10, we present segmentation results of agalaxy image with scattered data shown in Fig. 10(a)using Li’s method to illustrate the influence of the pe-nalization parameter µ. Since the segmentation resultsare largely dependent on the weighted parameters ofthe regularization term and the data fitting terms, es-pecially for images with very smooth contours [27], weuse the same fixed value α = 0.01∗2552 and investigatethe role of the parameter µ. We observe that when µincreases from 0.01 to 10, the segmentation result be-

comes more visually pleasing, which coincides well withthe property of the penalty method. That is, we shoulduse relatively large penalty parameter in Li’s methodespecially for the segmentation of challenging imagessuch as this galaxy image. However, the time step (from1 to 0.02) must be small enough to ensure stability ofthe algorithm, which slows down the convergence of thealgorithm. In Table 2, we report the computational ef-fort for Fig. 10(d).

Fig. 11 displays segmentation of the galaxy imageusing UA. We use the same value for α as in Fig. 10 forcomparability. When µ changes from 10−4 to 1, the seg-mentation results are visually similar as that obtainedby Li’s method with larger µ = 10. Therefore, we con-clude that it is enough to choose a relatively small µ inthe augmented Lagrangian method as mentioned pre-viously. Fig. 12 shows the results using PLA. The iter-ation numbers for the two different parameters γ = 1and γ = 10 are 58 and 51, respectively. Therefore, thetotal time is almost the same. Once again, from Table2, we can see the superiority of our two algorithms toLi’s method in efficiency.

Since the AOS scheme we have employed needs thesolution of linear systems, the computational effort foreach iteration of our algorithms is higher than that ofLi’s method. However, we can see from Tables 2 that thetotal computational cost required by our approachesis much less than Li’s method due to the significantreduction in the iteration number.

Choosing the time step needs a practical considera-tion [35]. An unpractically large time step will influencethe accuracy of the segmentation and cause oscillations.Therefore, we do not choose very large time steps in ourexperiments even if the AOS scheme does not sufferfrom any time step restriction.

From the experiments, we conclude that both thetwo algorithms are very effective and efficient. It is hardto say which one is better. However, the Uzawa algo-rithm has a much simpler form and is easier for coding.

6 Conclusions and Future Work

We have eliminated the requirement of reinitializationin the VLSM by the augmented Lagrangian methodand the projection Lagrangian method for constrainedoptimization. For the augmented Lagrangian method,we employed the effective Uzawa type algorithm. Nu-merical results showed that we could get stable segmen-tation by choosing a small penalization parameter in alarge range. For the updating scheme of the projectionLagrangian method, we introduced an auxiliary vari-able to deal with the gradient term. In this way, we

9

obtained a simple and explicit updating formulation ofthe Lagrange multiplier. Furthermore, we applied theproposed methods to the CV model and employed thesemi-implicit AOS scheme, which improved the stabil-ity and reduced the total computational effort of ouralgorithms.

The proposed methods can generally be extendedto the multiphase case with multiple LSFs. Our fu-ture work includes applications of our methods to otherproblems using LSM with the reinitialization process,for example, topology optimization, shape reconstruc-tion and other imaging and vision tasks.

Acknowledgements The authors express their sincere thanks

to the reviewers for constructive suggestions and valuable com-ments, which greatly improved the original manuscript. This workwas supported in part by the NNSF of China (Grant No. 10971190)

and the Qiu-Shi Chair Professor Fellowship from Zhejiang Uni-

versity.

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Table 1 Comparisons on the computational effort between our methods and the VLSM with reinitialization.

Images (size)VLSM with reinitialization UA PLA

Iterations Time(s) R-Timea (s) Iterations Time(s) Iterations Time(s)

Europe night-lights (180× 195) 247 16.42 9.8 45 2.48 67 4.19

Synthetic image(128× 128) 30 4.26 3.51 18 0.47 13 0.37

a The time spent on reinitialization.

Table 2 Comparisons on the computational effort between our methods and Li’s method.

Images (size)Li’s method UA PLA

Iterations Time (s) Iterations Time (s) Iterations Time (s)

Europe night-lights (180× 195) 331 13.83 45 2.48 67 4.19Spiral (187× 227) 285 15.48 61 4.56 69 5.11Synthetic image(128× 128) 76 1.42 18 0.47 13 0.37

Ultrasound (202× 241) 184 11.02 24 1.96 18 1.53Galaxy (140× 179) 183 5.74 26 1.05 51 2.45

(a) (b) (c)

Fig. 1 Segmentation of Europe night-lights image using VLSM with no reinitialization and with reinitialization for α = 0.08∗2552 and

∆t = 0.01. (a) Original Europe night-lights image. (b) Processed image with no reinitialization. (c) Processed image with reinitializationfor every fifth iteration.

Table 3 Effect of µ on the converged value of L2-Error.

µ Converged L2-Error Iterations

10−3 5.3852 19

10−2 5.3852 1910−1 5.7446 19

1 5.8310 22

11

Fig. 2 Segmentation of Europe night-lights image using Li’s method, UA and PLA, respectively. First column: Segmentation result

using Li’s method with α = 0.15 ∗ 2552, µ = 1, ∆t = 0.02 and the piecewise constant approximation. Second column: Segmentationresult using UA with α = 0.05 ∗ 2552, µ = 0.01, ∆t = 0.05 and the piecewise constant approximation. Third column: Segmentation

result using PLA with α = 0.06 ∗ 2552, γ = 0.1, ∆t = 0.1 and the piecewise constant approximation.

(a) (b) (c)

Fig. 3 Segmentation of the spiral image using Li’s method. (a) The original spiral image. (b) Segmentation result using Li’s method

with α = 0.05 ∗ 2552, µ = 1, ∆t = 0.02. (c) The piecewise constant approximation.

12

Fig. 4 Segmentation of the spiral image using UA and PLA, respectively. First column: Segmentation result using UA with α =0.04 ∗ 2552, µ = 0.001, ∆t = 0.1 and the piecewise constant approximation. Second column: Segmentation result using PLA with

α = 0.05 ∗ 2552, γ = 1, ∆t = 0.03 and the piecewise constant approximation.

(a) (b)

Fig. 5 (a) Original synthetic image. (b) Degraded image and the same initial contour for all the methods.

13

(a) Iteration 1 (b) Iteration 4 (c) Iteration 6 (d) Final iteration 30

(e) Iteration 1 (f) Iteration 3 (g) Iteration 19 (h) Final iteration 76

(i) Iteration 1 (j) Iteration 2 (k) Iteration 3 (l) Final iteration 18

(m) Iteration 1 (n) Iteration 2 (o) Iteration 4 (p) Final iteration 13

Fig. 6 Segmentation of the noisy synthetic image using VLSM with reinitialization, Li’s method, UA and PLA, respectively. First

row: Evolution using VLSM with reinitialization with α = 0.3 ∗ 2552, ∆t = 0.01. Second row: Evolution using Li’s method with

α = 0.35 ∗ 2552, µ = 1, ∆t = 0.02. Third row: Evolution using UA with α = 0.15 ∗ 2552, µ = 1, ∆t = 0.02. Fourth row: Evolutionusing PLA with α = 0.15 ∗ 2552, γ = 1, ∆t = 0.1.

14

11 10 20 30 40 50 60 70 80

5

10

15

20

25

30

35

40

45

50

55

Iteration Number

L2 −E

rror

LSM with ReinitializationLi’s methodUAPLA

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

5

10

15

20

25

30

35

40

45

50

55

Iteration Time (s)

L2 −E

rror

LSM with ReinitializationLi’s methodUAPLA

Fig. 7 Evolution of L2-Error for segmentation of the synthetic image using different methods. Left: L2-Error vs. iteration number.

Right: L2-Error vs. iteration time.

(a) (b)

(c) (d)

Fig. 8 Segmentation of the ultrasound image using three different methods. (a) Original ultrasound image. (b) Processed image usingLi’s method with α = 0.05 ∗ 2552, µ = 1, ∆t = 0.1. (c) Processed image using UA with α = 0.05 ∗ 2552, µ = 0.01, ∆t = 0.2. (d)

Processed image using PLA with α = 0.1 ∗ 2552, γ = 1, ∆t = 0.02.

15

5 10 15 20 2510

0.2

0.4

0.6

0.8

1

1.2

1.4

Iteration Number

Mea

n D

evia

tion

of (

| ∇

φ | −

1 )

µ=0.01, ∆ t = 0.2µ=1, ∆ t = 0.1µ=10, ∆ t = 0.05

5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Iteration Number

Mea

n D

evia

tion

of (

| ∇

φ | −

1 )

γ=0.1, ∆ t = 0.02γ=1, ∆ t = 0.01γ=10, ∆ t = 0.005

Fig. 9 Evolution of mean deviation of (|∇φ| − 1) for the ultrasound image using UA (left) and PLA (right).

(a) (b)

(c) (d)

Fig. 10 Segmentation of the galaxy image using Li’s method with α = 0.01 ∗ 2552 but different values of µ. (a) Original image. (b)

Processed image with µ = 0.01, ∆t = 1. (c) Processed image with µ = 0.1, ∆t = 0.5. (d) Processed image with µ = 10, ∆t = 0.02.

16

Fig. 11 Segmentation of the galaxy image using UA with α = 0.01∗2552 but different values of µ. First column: Segmentation results

with µ = 10−4, ∆t = 0.2. Second column: Segmentation results with µ = 10−3, ∆t = 0.2. Third column: Segmentation results withµ = 10−1, ∆t = 0.1.

Fig. 12 Galaxy image processed using PLA with α = 0.01 ∗ 2552 but different values of γ. First column: Segmentation results with

γ = 1, ∆t = 0.1. Second column: Segmentation results with γ = 10, ∆t = 0.05.