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Numer. Math. Theor. Meth. Appl. Vol. 12, No. 2, pp. 370-402doi: 10.4208/nmtma.OA-2017-0149 May 2019

Proximal ADMM for Euler’s Elastica Based ImageDecomposition Model

Zhifang Liu1, Samad Wali1,∗, Yuping Duan2, Huibin Chang3,Chunlin Wu1 and Xue-Cheng Tai4

1 School of Mathematical Sciences, Nankai University, Tianjin, China2 Center for Applied Mathematics, Tianjin University, Tianjin, China3 School of Mathematical Sciences, Tianjin Normal University, Tianjin, China4 Department of Mathematics, Hong Kong Baptist University, Hong Kong, China

Received 5 December 2017; Accepted (in revised version) 27 February 2018

Abstract. This paper studies image decomposition models which involve functional re-lated to total variation and Euler’s elastica energy. Such kind of variational modelswith first order and higher order derivatives have been widely used in image processingto accomplish advanced tasks. However, these non-linear partial differential equationsusually take high computational cost by the gradient descent method. In this paper,we propose a proximal alternating direction method of multipliers (ADMM) for totalvariation (TV) based Vese-Osher’s decomposition model [L. A. Vese and S. J. Osher,J. Sci. Comput., 19.1 (2003), pp. 553-572] and its extension with Euler’s elastica regu-larization. We demonstrate that efficient and effective solutions to these minimizationproblems can be obtained by proximal based numerical algorithms. In numerical ex-periments, we present numerous results on image decomposition and image denoising,which conforms significant improvement of the proposed models over standard models.

AMS subject classifications: 68U10, 90C25, 49M37Key words: Alternating direction method of multipliers, total variation, Euler’s elastica, proximalmethod, image decomposition, image denoising.

1. Introduction

Assume that Ω ⊂ R2 is a bounded, open, and connected subset (usually a rectangle inimage processing). The image decomposition task is to decompose a given image f : Ω→R as the sum of two components

f = u+ v,

where u is geometric part or cartoon component, and v is an oscillating one. In general,u models homogeneous regions with sharp boundaries and v contains oscillating patterns

∗Corresponding author. Email addresses: [email protected] (Z. F. Liu), [email protected] (S. Wali), [email protected] (Y. P. Duan), [email protected] (H. B. Chang),[email protected] (C. L. Wu), [email protected] (X. C. Tai)

http://www.global-sci.org/nmtma 370 c©2019 Global-Science Press

Proximal ADMM for Euler’s Elastica Based Image Decomposition Model 371

such as texture and noise; See, e.g. [3,15,25,26,34,39,44,45]. The origins of these ideasare the remarkable book by Meyer [34], in which the author showed that the well-knownRudin-Osher-Fatemi (ROF) model [37] does not always represent texture or oscillatorydetails well.

An eligible and successful choice to detect textures is the generalized functions spaceG = G(Ω) [3,30,34,46], where

G(Ω) =¦

v =∇ · g= ∂x g1+ ∂y g2 : g= (g1, g2), g1, g2 ∈ L∞(Ω,R2)©

endowed with the norm

|v|G = infn

‖|g|‖L∞(Ω,R2) : v =∇·g, g= (g1, g2), g1, g2 ∈ L∞(Ω,R2), |g|=Æ

g21 + g2

2

o

.

The (BV, G) model proposed by Meyer in [34] is to solve the problem

inf(u,v)∈BV (Ω)×G(Ω)

¨∫

Ω

|u|BV + β |v|G , f = u+ v

«

,

where BV (Ω) is the bounded variation functions space.There is no standard calculation of the associated Euler-Lagrange equation due to the

term coming from an L∞-norm (in the G-norm), which maps a series of work out to over-come this difficulty; See, e.g., [2,4,45,46].

In [45, 46], Vese and Osher proposed to model oscillatory components v as first or-der derivatives of vector fields in Lp, (1 ≤ p < ∞) (approaching to the L∞-norm) toapproximate Meyer’s (BV, G) model. As the first practical image decomposition model,total variation (TV) based Vese-Osher’s decomposition model solves the following convexminimization problem

minu,g

∫

Ω

|∇u|+α

2

∫

Ω

| f − u− v|2+ β∫

Ω|g|ρ

1ρ

, (1.1)

where α, β > 0, are tuning parameters, v = ∇ · g, g = (g1, g2)T and 1 ≤ ρ < ∞. Themodel (1.1) was solved by sequential descent approach to its Euler-Lagrange equation. Ithas been shown that the advantage of the model (1.1) is that it is not sensitive to the choiceof ρ. The authors recommended to set ρ = 1 to yield faster calculations per iteration [45].In fact if we generate the model (1.1) to the case ρ = ∞, then we can easily handle theL∞-norm in the discrete setting by the ADMM method proposed in this paper.

In [12], the authors showed that TV regularization suffers from the undesirable s-taircase effect for image denoising application, which also exists in image decompositionproblems. To overcome this, high order models have been proposed [13, 32, 49]. Euler’selastica is one of the higher order energy functionals, which has a number of interestingapplications in elasticity, computer graphics and in image processing. To improve the qual-ity of an image in the sense of cartoon u and texture v and improve other applications like

372 Z. F. Liu, S. Wali, Y. P. Duan, H. B. Chang, C. L. Wu and X. C. Tai

image denoising, we can use a non-convex and non-linear regularization such as curvaturebased regularization.

Euler’s elastica energy, which was based on the curvature of the level curve of im-age, can effectively discriminate local behavior and global trends of the geometry hidingin certain image [14]. Euler’s elastica energy was first introduced into computer visionby Mumford [35] and successfully applied to a number of applications, such as imagerestoration [1,6], image segmentation [21,33,36,52] and image inpainting [7,9,40].

In recent years, fast numerical optimization methods, such as split Bregman method[27] and augmented Lagrangian method [38, 47, 48], have been widely used to solvevariational methods arising from image processing. In [43], Tai, Hahn and Chung proposea fast and efficient algorithm for Euler’s elastica model via augmented Lagrangian method.The algorithms for curvature based variational model, one can see [18, 42, 50, 51] fordetails.

1.1. Contribution

1. In this paper, we use the Euler’s elastica as regularization for geometric componentu and propose the following minimization problem

minu,g

∫

Ω

a+ b

∇ ·∇u

|∇u|

2

|∇u|+α

2

∫

Ω

| f − u− v|2+ β∫

Ω

|g|ρ

1ρ

(1.2)

where α > 0 and β > 0 are tuning parameters, v =∇ · g, g= (g1, g2)T and ρ ≥ 1.

2. We propose alternating direction method of multipliers (ADMM), to solve the Vese-Osher’s decomposition model (1.1). We show one subproblem can be easily solvedwith a closed form solution and the other subproblems can be transformed into aFourier linear system and solved by block Gaussian elimination. Besides, we usethe proximal method to further reduce the computational costs. Similarly, we im-plement same proximal ADMM for Euler’s elastica model (1.2). Owing to the useof suitable proximal step and fixed-point technique, each subproblem can be solvedeasily. Numerical results illustrate the effectiveness and efficiency of the proposedmethod.

The paper is organized as follows. In the next section, we present the ADMM andproximal ADMM for Vese-Osher’s decomposition model with total variation and Euler’selastica. In Sect. 3, we explain numerical discretization of the subproblems associatedwith ADMM and proximal ADMM, and give the convergence analysis in case of TV. InSect. 4, we show some numerical experiments to illustrate the efficiency and effectivenessof our proposed algorithm. A concluding remark and future works are given in Sect. 5.


2. Proximal ADMM for Vese-Osher’s decomposition model

In current paper, we study a variational decompositional model that is based on a priormodel for plane curve, i.e. Euler’s elastica. A remarkable feature of elasticas revealed byMumford [35], which positively support the role of elasticas in image and vision analy-sis as an interpolation tool. It also shed light on the choice of "2" for curvature power inthe Euler’s elastica energy formula. In brief, our proposed elastica decomposition schemecombines both Vese-Osher’s model and Euler’s elastica energy. It thus provides a theoret-ical foundation for earlier works on PDE based image decomposition. In particular, weassume that our proposed fast proximal method for both TV and Euler’s elastica baseddecomposition models will be feasible.

In this section, we propose a fast and efficient algorithm for minimization problemsrelated to Vese-Osher’s decomposition model. We assume that the image domain Ω isnormally taken as a rectangle or a grid for a rectangle domain, i.e., Ω = [1, N]× [1, M].For mathematical notation, we will use the standard inner product ⟨u, v⟩ =

∫

Ωu · v and

norm ‖u‖=∫

Ω|u|2 in L2 space. The adjoint operator of∇ denote by∇∗, i.e. ∇∗g=−∇·g.

2.1. ADMM for TV based Vese-Osher’s decomposition model

In order to use ADMM, we should transfer the unconstrained minimization problem(1.1) to an equivalent constrained minimization problem by using an operator-splittingtechnique. We introduce two new variables p=∇u and q= g and reformulate the problem(1.1) to be the following constrained minimization problem

minu,g

∫

Ω

|p|+α

2

∫

Ω

| f − u−∇ · g|2+ β∫

Ω

|q|ρ

1ρ

,

s.t. p=∇u, q= g.

(2.1)

For simplicity of presentation, we let x =

u

g

, y =

p

q

and A =

∇ 00 I

with

identity maps I . The adjoint operator of A is given by A ∗ =

∇∗ 00 I

. It is s-

traightforward to see that the minimization problem (2.1) is equivalent to the followingminimization problem:

minx,y

H(x) + J(y)

,

s.t. y=A x,(2.2)

where

H(x) =α

2

∫

Ω

| f − u−∇ · g|2, J(y) =

∫

Ω

|p|+ β∫

Ω

|q|ρ

1ρ

.


We define the augmented Lagrangian function of (2.2) as

L (x,y;λ) = H(x) + J(y) + ⟨λ,y−A x⟩+γ

2‖y−A x‖2, (2.3)

where λ = (λp,λq)T is the Lagrange multiplier and γ is a positive constant. Then theADMM for solving (2.2) can be described in Algorithm 2.1.

Algorithm 2.1 ADMM for TV based decomposition model.

Initialization: x0,y0,λ0;while stopping criteria not satisfied do

xk+1 = argminxL

x,yk;λk, (2.4)

yk+1 ∈ argminyL

xk+1,y;λk, (2.5)

λk+1 = λk + γ

yk+1−A xk+1. (2.6)

end

The x−sub problem (2.4) is a quadratic optimization problem,

xk+1 = argminxL

x,yk;λk= arg minx

§

H(x)− ⟨λk,A x⟩+γ

2

yk −A x

2ª

,

whose optimality condition gives a linear equation

∇H(x)−A ∗λk + γA ∗(A x− yk) = 0 (2.7)

with the periodic boundary condition. Since we have

∇H(x) =∇Hu(x)∇Hg(x)

=

α(u+∇ · g− f )α∇( f − u−∇ · g)

, (2.8)

the equation (2.7) can be written explicitly as

α(u+∇ · g− f )α∇( f − u−∇ · g)

−

∇∗ 00 I

λkp

λkq

+ γ

∇∗ 00 I

∇u− pk

g− qk

= 0.

Note that ∆=∇2 =∇ ·∇=−∇∗∇, we have

α(u+∇ · g− f ) +∇ ·λkp+ γ∇ · p

k − γ∆u= 0,

α∂x( f − u− ∂x g1− ∂y g2)−λkq1+ γ(g1− qk

1) = 0,

α∂y( f − u− ∂x g1− ∂y g2)−λkq2+ γ(g2− qk

2) = 0.

(2.9)

In the linear system (2.9), taking the Fourier transform of both sides, we will have a Fourierlinear system. With an FFT implementation, the solutions of the Fourier linear system can


be obtained via block Gaussian elimination in discrete setting, which will be showed inSection 3.2.

For the y−sub problem (2.5), it has a closed form solution. We can separate the mini-mization problem (2.5) into two minimization problems independently, that is,

minp

∫

Ω

|p|+γ

2

∫

Ω

p−∇uk+1+λk

p

γ

2

, (2.10a)

minq

β

∫

Ω

|q|ρ

1ρ

+γ

2

∫

Ω

q− gk+1+λk

q

γ

2

. (2.10b)

The minimizers of (2.10a) and (2.10b) (ρ = 1) can be obtained as follows:

p=max

0,1−1

γ|ζ|

ζ and q=max

0, 1−β

γ|η|

η,

where ζ=∇uk+1−λkp/γ and η= gk+1−λk

q/γ.

2.2. Proximal ADMM for TV based Vese-Osher’s decomposition model

The Algorithm 2.1 is costly in practice due to the linear system (2.9). Thus we modifythe (2.4) by taking a proximal step and propose the proximal ADMM scheme to solve(2.2). We will show that the solutions of each subproblem of proximal ADMM scheme canbe obtained inexpensively.

Let

S =

τI α∇∗

α∇ τI −α∇∇∗

, (2.11)

where τ > α(λmax(∇∇∗)+λmax(∇∗)+λmax(∇)). According to the definition (2.11), S is aself-adjoint positive semidefinite operator. We define the induced norm ‖x‖S = ⟨S x,x⟩1/2.Our proximal ADMM for solving (2.2) is described in Algorithm 2.2.

In fact, the difference between ADMM scheme (2.4)-(2.6) and proximal ADMM scheme(2.12a)-(2.12c) is only the x−sub problem, where (2.12a) can be solved more efficientlyin computation owing to the proximal term 1

2‖x− xk‖2S .

The x−sub problem (2.12a) is the following quadratic optimization problem,

xk+1 = argminxL (x,yk;λk) +

1

2‖x− xk‖2S

= argminx

H(x)− ⟨λ,A x⟩+γ

2‖yk −A x‖2+

1

2‖x− xk‖2S

,

whose optimality condition gives a linear equation

∇H(x)−A ∗λ+ γA ∗

A x− yk+S

x− xk= 0, (2.13)


Algorithm 2.2 Proximal ADMM for TV based decomposition model.

Initialization: x0,y0,λ0;while stopping criteria not satisfied do

xk+1 = argminxL

x,yk;λk+1

2‖x− xk‖2S , (2.12a)

yk+1 ∈ argminyL

xk+1,y;λk, (2.12b)

λk+1 = λk + γ

yk+1−A xk+1. (2.12c)

end

by the periodic boundary condition. Taking (2.8) and (2.11) into consideration, we obtainthe explicit form of (2.13) as follow

α(u+ divg− f )α∇( f − u− divg)

−

∇∗ 00 I

λkp

λkq

+ γ

∇∗ 00 I

∇u− pk

g− qk

+

τI α∇∗

α∇ τI −α∇∇∗

u− uk

g− gk

= 0,

i.e.(

α

u+ divgk − f

+ divλkp+ γdivpk − γ∆u+τ

u− uk= 0,

α∇

f − uk − divgk−λkq+ γ

g− qk+τ

g− gk= 0.(2.14)

Because of the variables u and g are independent in the linear system (2.14), we can easilyobtain the solution with less cost in discrete setting, which will be showed in Section 3.3.

2.3. Proximal ADMM for Euler’s elastica based Vese-Osher’s decompositionmodel

The Euler’s elastica energy based minimization problem (1.2) is composed of its totalvariation semi-norm and a curvature term. In fact, elastica is a combination of total varia-tion suppressing oscillation in the gradient direction, and a curvature regularization termthat penalizes non-smooth level set curves. Due to high non-linearity in elastica model, itis difficult to minimize and require high computational cost. To tackle the non linear con-straint arising in the model, a proximal based approach is proposed so that all subproblemseither is linear or has a closed-form solution. In this subsection, we propose a proximalADMM for Euler’s elastica based Vese-Osher’s decomposition model. We first introducefour new variables p,q, h and n, then reformulate the problem (1.2) to be the following


equality constrained problem

minu,g,p,q,h,n

∫

Ω

(a+ bh2)|p|+α

2

∫

Ω

| f − u−∇ · g|2+ β∫

Ω

|q|ρ

1ρ

,

s.t. p=∇u, p= |p|n, h=∇ · n, q= g.

(2.15)

For simplify of notations, we denote

x=

ugn

, y=

pqh

, A =

∇ 0 00 I 00 0 −∇∗

.

It follows that

A x=

∇ug∇ · n

, A ∗ =

∇∗ 0 00 I 00 0 −∇

.

Then we rewrite (2.15) as

minx,y

H(x) + J(y)

,

s.t. y= A x, p= |p|n,(2.16)

where

H(x) =α

2

∫

Ω

| f − u−∇ · g|2,

J(y) =

∫

Ω

(a+ bh2)|p|+ β∫

Ω

|q|ρ

1ρ

.

The augmented Lagrangian functional of (2.16) is defined as

L (x, y; λ,λn) = H(x) + J(y) + ⟨λ, y−A x⟩+γ

2

y−A x

2

+ ⟨λn,p− |p|n⟩+γn

2‖p− |p|n‖2, (2.17)

where λ = (λp,λq,λh)T , λn are the Lagrange multipliers and γ, γn are the positive con-stants. Our proximal ADMM for solving (1.2) is described in Algorithm 2.3.

In the Algorithm 2.3, we let S is a self-adjoint positive semidefinite operator, which isdefined as

S =

τ1I α∇∗ 0α∇ τ1I −α∇∇∗ 00 0 τ2I − γ∇∇∗

, (2.19)

where τ1 > α(λmax(∇∇∗) +λmax(∇∗) +λmax(∇)) and τ2 > γλmax(∇∇∗).


Algorithm 2.3 Proximal ADMM for Euler’s elastica based decomposition model.

Initialization: x0,y0,λ0,λ0n;

while stopping criteria not satisfied do

xk+1 = arg minxL

x, yk; λk,λk

n

+1

2

x− xk

2S , (2.18a)

yk+1 ∈ argminyL

xk+1, y; λk,λk

n

, (2.18b)

λk+1 = λ

k + γ

yk+1−A xk+1,

λk+1n = λk

n+ γn

pk+1− |pk+1|nk+1.(2.18c)

end

In the following, we show how to solve the subproblems in the each iteration of Algo-rithm 2.3. For the x−sub problem (2.18a), it is a quadratic optimization problem,

xk+1 = arg minxL (x, yk; λ

k,λk

n) +1

2‖x− xk‖2S

= arg minx

¨

H(x)− ⟨λk,A x⟩+

γ

2‖yk −A x‖2− ⟨λk

n, |pk|n⟩

+γn

2‖pk − |pk|n‖2+

1

2‖x− xk‖2S

«

,

whose optimality condition gives a linear equation∇H(x)−A ∗λ+ γA ∗(A x− yk)

−

00

|pk|λkn− γn|pk|(|pk|n− pk)

+ S (x− xk) = 0 (2.20)

by the periodic boundary condition. Combining (2.19) and using the fact that

∇H(x) =

∇Hu(x)∇Hg(x)∇Hn(x)

=

α(u+∇ · g− f )α∇( f − u−∇ · g)

0

,

we rewrite the equation (2.20) explicitly as

α(u+∇ · g− f )α∇( f − u−∇ · g)

0

+

∇ ·λkp

−λkq

∇λkh

+ γ

−∆u+∇ · pk

g− qk

∇(hk −∇ · n)

−

00~z

+

τ1I α∇∗ 0α∇ τ1I −α∇∇∗ 00 0 τ2I − γ∇∇∗

u− uk

g− gk

n− nk

= 0,


where ~z = |pk|λkn− γn|pk|(|pk|n− pk). Thus, we have

α(u+∇ · gk − f ) +∇ ·λkp+ γ∇ · p

k − γ∆u+τ1(u− uk) = 0,

α∇( f − uk −∇ · gk)−λkq+ γ(g− qk) +τ1(g− gk) = 0,

γ∇(hk −∇ · nk) +∇λkh − |p

k|λkn

+ γn|pk|(|pk|n− pk) +τ2(n− nk) = 0.

(2.21a)

(2.21b)

(2.21c)

Note that in the linear system (2.21), since the variables u, g and n are independent, wecan get the solution from (2.21a), (2.21b) and (2.21c) easily.

For the y−sub problem (2.18b), it is difficult to solve due to the non-differentiabilityelement |p| in the quadratic term and the variables p and h are strong coupled together.Following [18, 43], we replace p = |p|nk+1 by p = |pk|nk+1 in the quadratic penalty termin (2.18b) with considering to utilize the fixed-point formulation to the nonlinear con-straint p = |p|n, and separate the minimization problem (2.18b) into three minimizationproblems, i.e.

minp

¨∫

Ω

c|p|+ γ∫

Ω

p− ζ

2«

, (2.22a)

minq

β

∫

Ω

|q|ρ

1ρ

+γ

2

∫

Ω

q−η

2

, (2.22b)

minh

¨∫

Ω

(a+ bh2)|pk|+∫

Ω

λh · h+γ

2

∫

Ω

|h−∇ · nk+1|2«

, (2.22c)

where

c = a+ b(hk)2−λkp · n

k+1, ζ=γn|pk|nk+1+ γ∇uk+1−λp−λn

γ+ γn, η= gk+1−

λkq

γ.

The minimization problems (2.22a) and (2.22b) (ρ = 1) have closed form solutions, whichread

p=max

0, 1−c

(γ+ γn)|ζ|

ζ and q=max

0,1−β

γ|η|

η.

When ρ = ∞, problem (2.22b) can also be solved explicitly; See [23, Lemma 4.1, 4.2].The optimality condition for the minimization problems (2.22c) gives a linear equation

2b|pk|h+λh+ γ

h− divnk+1= 0

with a closed form solution

h=γdivnk+1−λh

2b|pk|+ γ.


•(i, j) •(i+ 1, j)

•(i, j + 1) •(i+ 1, j + 1)

•(i+ 1, j − 1)•(i− 1, j − 1) •(i, j − 1)

•(i− 1, j + 1)

•(i− 1, j)

(i, j)

(i, j)

1

Figure 1: Grid definition. The rule of indexing variables in the augmented Lagrangian functional (2.17):u, h,λh are defined on •-nodes. The first and second components of g,p,q,n,λp,λq,λn are defined on-nodes and -nodes, respectively.

3. Numerical implementation and convergence analysis

In this section, we present the details of numerical algorithms for ADMM and proximalADMM. In Section 3.1, we introduce some basic notations. The numerical implementationsof Algorithm 2.1, Algorithm 2.2 and Algorithm 2.3 are described in Section 3.2, Section3.3 and Section 3.4, respectively.

3.1. Notation

Without loss of generality, let Ω = [1, N] × [1, M] be a set of N × M points in R2.For the convenience of description, we denote V as the Euclidean space RN×M and Q =V × V . Then for a given (i, j) ∈ [1, N] × [1, M], we denote u(i, j) ∈ R and p(i, j) =(p1(i, j),p2(i, j)) ∈ R2 for u ∈ V and p ∈ Q, respectively. We equip the standard Euclideaninner products as follows:

⟨u, v⟩V ≡∑

i, j

u(i, j)v(i, j) and ⟨p,q⟩Q ≡ ⟨p1, q1⟩V + ⟨p2, q2⟩V .

Note that the induced norm ‖ · ‖V is the `2− norm on the vector spaces V and Q. Inaddition, we mention that

|p(i, j)|= |(p1(i, j), p2(i, j))|=p

(p1(i, j))2+ (p2(i, j))2,

the usual Euclidean norm in R2.The discrete backward and forward differential operators for u ∈ V are defined with

periodic boundary condition as follows

∂ −x u(i, j) =

(

u(i, j)− u(i− 1, j), 1< i ≤ M ,

u(1, j)− u(M , j), i = 1,

∂ −y u(i, j) =

(

u(i, j)− u(i, j− 1), 1< j ≤ N ,

u(i, 1)− u(i, N), j = 1,


∂ +x u(i, j) =

(

u(i+ 1, j)− u(i, j), 1≤ i < M ,

u(1, j)− u(M , j), i = M ,

∂ +y u(i, j) =

(

u(i, j+ 1)− u(i, j), 1≤ j < N ,

u(i, 1)− u(i, N), j = N .

The discrete gradient operator ∇ : V →Q is given as follows: For u ∈ V ,

∇u(i, j) = (∂ +x u(i, j),∂ +y u(i, j)).

Considering inner products on V and Q, the discrete adjoint operator div : Q → V of −∇is given by

divp(i, j) = ∂ −x p1(i, j) + ∂ −y p2(i, j).

When a variable defined on -nodes (or -nodes, see Fig. 1) needs to be evaluated at(i, j) ∈-nodes (or -nodes), for example p= (p1, p2), we use the average operators:

Ai, j(p1) =p1(i, j+ 1) + p1(i− 1, j+ 1) + p1(i, j) + p1(i− 1, j)

4,

Ai, j(p2) =p2(i+ 1, j) + p2(i, j) + p2(i+ 1, j− 1) + p2(i, j− 1)

4,

where p1 and p2 are defined on -nodes and -nodes respectively. We need to define aspecial operator to measure the magnitude of p at (i, j) ∈ •-nodes using the followingoperator:

|A|•i, j(p) =

È

p1(i, j) + p1(i− 1, j)2

2

+

p2(i, j) + p2(i, j− 1)2

2

.

We compute the divergence of p at (i, j) ∈ •-nodes using the following operator:

div•i, j(p) = p1(i, j)− p1(i− 1, j) + p2(i, j)− p2(i, j− 1).

3.2. Solve the subproblems in Algorithm 2.1

For the x−sub problem (2.4) in Algorithm 2.1, we discretize (2.9) as follows:

α(u−γ

α∆u+ ∂ −x g1+ ∂

−y g2) = α f − div(λk

p+ γpk),

α(∂ +x u+ ∂ +x ∂−x g1−

γ

αg1+ ∂

+x ∂−y g2) = α∂

+x f − (λk

q1+ γqk1),

α(∂ +y u+ ∂ +y ∂−x g1+ ∂

+y ∂−y g2−

γ

αg2) = α∂

+y f − (λk

q2+ γqk2).

(3.1)

Applying Fourier transform to both side of (3.1), we have

KF (x) = F−B, (3.2)


where

K= α

I − γ

αF (∆) F (∂ −x ) F (∂ −y )

F (∂ +x ) F (∂ +x )F (∂−x )−

γ

αI F (∂ +x )F (∂

−y )

F (∂ +y ) F (∂ +y )F (∂−x ) F (∂ +y )F (∂

−y )−

γ

αI

,

F (x) =

F (u)F (g1)F (g2)

, F= α

F ( f )F (∂ +x )F ( f )F (∂ +y )F ( f )

, B=

F (div)F (λkp+ γp

k)F (λk

q1+ γq1)F (λk

q2+ γq2)

.

Clearly, the matrices F (∆), F (∂ +x ), F (∂+y ), F (∂

−x ), F (∂

−y ), F (div), K and F only need

to be computed once before iteration.The x−sub problem can be solved in three steps. Firstly, we apply discrete FFTs to both

sides of (3.1). Secondly, we solve the resulting systems (3.2) by block Gaussian eliminationfor F (x). Finally, we apply F−1 to F (x) to obtain a new x.

For the y−sub problem (2.5), we have the following closed-form solutions:

p(i, j) =max

0, 1−1

γ|ζ(i, j)|

ζ(i, j) and q(i, j) =max

0,1−β

γ|η(i, j)|

η(i, j),

where

ζ(i, j) =∇uk+1(i, j)−λk

p(i, j)

γ,

η(i, j) = gk+1(i, j)−λk

q(i, j)

γ.

At last, the update of the Lagrangian multiplier λ= (λp,λq) is as follows:

λk+1p1 = λ

kp1+ γ

pk+11 − ∂ +x uk+1

,

λk+1p2 = λ

kp2+ γ

pk+12 − ∂ +y uk+1

,

λk+1q1 = λ

kq1+ γ

qk+11 − gk+1

1

,

λk+1q2 = λ

kq2+ γ

qk+12 − gk+1

2

.


For the x−sub problem (2.12a) in Algorithm 2.2, we have the following discrete form:

(α+τ)u− γ∆u= α f −αdivgk − divλkp− γdivpk +τuk,

(γ+τ)g1 = λkq1+ γqk

1 +τgk1 −α∂

+x

f − uk − divgk,

(γ+τ)g2 = λkq2+ γqk

2 +τgk2 −α∂

+y

f − uk − divgk.

(3.3a)

(3.3b)

(3.3c)


The linear system (3.3) can be efficiently solved. First, we use Fourier transform to solvethe (3.3a),

u=F−1

αF ( f )−αF (div)F (gk)−F (div)F (λkp)− γF (div)F (pk) +τF (uk)

α+τ− γF (∆)

!

.

From (3.3b) and (3.3c), we have

g1 =λk

q1+ γqk1 +τgk

1 −α∂+x ( f − uk − divgk)

γ+τ,

g2 =λk

q2+ γqk2 +τgk

2 −α∂+y ( f − uk − divgk)

γ+τ.

In Algorithm 2.2, the calculation y- sub problem (2.12b) and the update of the Lagrangianmultiplier λ= (λp,λq)T at each pixel (i, j) are similar to Algorithm 2.1.


The discretization scheme used in subsection 3.4 which has been inspired by varioustechniques in classical fluid dynamics, see in [40, 43]. It has a great advantage over con-ventional discretization scheme because this discretization scheme is more focus on local-ity of image, and focusing on locality is an important practice in numerical mathematics.In this computational scheme, we need to specify the half point values of the quantitiesg,p,q,n,λp,λq,λn. For example for x-half-point, we need to take (i + 1/2, j) and for y-half-point, take (i, j+1/2). Numerical experiments in Section 4 have shown the advantageof discretization scheme used for Euler’s elastica based decomposition. In the next subsec-tions, we briefly discuss about the solution of all other subproblems by using the staggeredgrid as shown Fig. 1.

3.4.1. The discretization for g-sub problem

From (2.21b), we have

g= (g1, g2) =λk

q+ γqk +τ1gk −α∇( f − uk − divgk)

γ+τ1.

Denote fixed variables uk,gk,λkq,qk by u, g,λq,q respectively. Since for g-sub problem, we

use the proximal method to minimize the computational cost and here g shows the value ofg at kth iteration. According to the rule of indexing variables in Fig. 1, the first and secondcomponent of g are defined on -nodes and -nodes, respectively. The discretization of gat (i, j) ∈ -nodes is obtained as follows:

g1(i, j) =1

γ+τ1

λq1(i, j) + γq1(i, j) +τ1 g1(i, j)−α∂ +x X (i, j)

,


whereX (i, j) = f (i, j)− u(i, j)− div•i, j(g).

Similarly, the discretization of g at (i, j) ∈-nodes is obtained as follows:

g2(i, j) =1

γ+τ1

λq2(i, j) + γq2(i, j) +τ1 g2(i, j)−α∂ +y X (i, j)

.

3.4.2. The discretization for n-sub problem

From (2.21c), we have

(n1, n2) =|pk|λk

n+ γn|pk|pk +τ2nk − γ∇(hk − divnk)−∇λkh

γn|pk|2+τ2.

Denote fixed variables pk,nk, hk,λkn,λk

h by p, n, h,λn,λh respectively. Similarly since forn-sub problem, we use the proximal method to minimize the computational cost and heren shows the value of n at kth iteration. According to the rule of indexing the dicretizationof n at (i, j) ∈ -nodes as follows:

n1(i, j) =|pi, j|λn1(i, j) + γn|pi, j|p1(i, j) +τ2n1(i, j)− γ∂ +x Y (i, j)− ∂ +x λh(i, j)

γn

|pi, j|2

+τ2

,

where Y = h(i, j)− div•i, j(n), and modulus of p at (i, j) ∈ -nodes is defined as

|pi, j| =q

p1(i, j)2+

Ai, j(p2)2.

Similarly the dicretization of n at (i, j) ∈-nodes as follows:

n2(i, j) =|pi, j|λn2(i, j) + γn|pi, j|p2(i, j) +τ2n2(i, j)− γ∂ +y Y (i, j)− ∂ +y λh(i, j)

γn

|pi, j|2+τ2

,

where modulus of p at (i, j) ∈-nodes is defined as

|pi, j| =q

Ai, j(p1)2+

p2(i, j)2.

3.4.3. The discretization for p-sub and q-sub problems

For the p- subproblem (2.22a) and q-sub problem (2.22b), we have

p(i, j) =max

0,1−c(i, j)

(γn+ γ)|ζ(i, j)|

ζ(i, j), q(i, j) =max

0,1−β

γ|η(i, j)|

η(i, j),


where

c = a+ b(hk)2 −λkp · n

k+1, ζ =γn|pk|nk+1+ γ∇uk+1−λk

p−λkn

γn+ γ, η = gk+1 −

λkq

γ.

Denote fixed variables uk+1,pk,gk+1,nk+1, hk,λkp,λk

q, λkn by u, p,g,n, h,λp,λq, λn respec-

tively. According to the rule of indexing for c, ζ and η at (i, j) ∈ -nodes is obtained asfollows:

c1(i, j) = a+ b

h(i+ 1, j) + h(i, j)2

2

−λn1(i, j)n1(i, j)− Ai, j(λn2)Ai, j(n2),

(γn+ γ)ζ11(i, j) = γ

u(i+ 1, j)− u(i, j)

+ γn|pi, j|n1(i, j)−λn1(i, j)−λp1(i, j),

(γn+ γ)ζ12(i, j) =

γ

2

u(i+ 1, j+ 1) + u(i, j+ 1)2

−u(i+ 1, j− 1) + u(i, j− 1)

2

+ γn|pi, j|Ai, j(n2)− Ai, j(λn2)− Ai, j(λp2),

η11(i, j) = g1(i, j)−

1

γλq1(i, j), η1

2(i, j) = Ai, j(g2)−1

γAi, j(λq2).

Similarly, the discretization of c, ζ and η at (i, j) ∈-nodes is obtained as follows:

c2(i, j) = a+ b

h(i, j+ 1) + h(i, j)2

2

− Ai, j(λn1)Ai, j(n1)−λn2(i, j)n2(i, j),

(γn+ γ)ζ21(i, j) =

γ

2

u(i+ 1, j+ 1) + u(i+ 1, j)2

−u(i− 1, j+ 1) + u(i− 1, j)

2

+ γn|pi, j|Ai, j(n1)− Ai, j(λn1)− Ai, j(λp1),

(γn+ γ)ζ22(i, j) = γ

u(i, j+ 1)− u(i, j)

+ γn|pi, j|n2(i, j)−λn2(i, j)−λp2(i, j),

η21(i, j) = Ai, j(g1)−

1

γAi, j(λq1), η2

2(i, j) = g2(i, j)−1

γλq2(i, j),

Therefore, we have the discretization of (2.22a) and (2.22b):

pd(i, j) =max

0,1−cd(i, j)

(γn+ γ)p

(ζd1(i, j))2+ (ζd

2(i, j))2

ζdd(i, j), d ∈ 1,2,

qd(i, j) =max

0, 1−β

γp

(ηd1(i, j))2+ (ηd

2(i, j))2

ηdd(i, j), d ∈ 1, 2.

3.4.4. The discretization for h-sub problem

The optimality condition for the minimization problems (2.22c) gives a linear equation

2b|pk|h+λh+ γ

h− divnk+1= 0,


with a closed form solution

h=γh div•i, j

nk+1−λkh

2b|A|•i, j

pk+ γ.

3.4.5. Update Lagrange multipliers

Finally, the update of the Lagrangian multipliers λ = (λp,λq,λh) and λn according to thealgorithm. Using the staggered grid as shown in Fig. 1, the discretized form for equations(2.18c) is written as

λk+1p1 = λ

kp1+ γ(p

k+11 − ∂ +x uk+1) at -nodes,

λk+1p2 = λ

kp2+ γ(p

k+12 − ∂ +y uk+1) at -nodes,

λk+1q1 = λ

kq1+ γ(q

k+11 − gk+1

1 ) at -nodes,

λk+1q2 = λ

kq2+ γ(q

k+12 − gk+1

2 ) at -nodes,

λk+1n1 = λ

kn1+ γn(p

k+11 − |pk+1

i, j |nk+1

1 ) at -nodes,

λk+1n2 = λ

kn2+ γn(p

k+12 − |pk+1

i, j |nk+1

2 ) at -nodes,

λk+1h = λk

h + γ(hk+1− div•i, j(n

k+1)) at •-nodes.

3.5. Convergence analysis

The TV based Vese-Osher’s texture model is a convex minimization problem. The con-vergence analysis of the ADMM scheme (2.4)-(2.6) and proximal ADMM scheme (2.12a)-(2.12c) have been widely studied, one can refer to [16,17,22,24,29].

For the sake of completeness, we give the convergence theorems for the ADMM scheme(2.4)-(2.6) and proximal ADMM scheme (2.12a)-(2.12c) in discrete setting but withoutproof, one can refer in [29] and [22, Appendix B] for more details.

Theorem 3.1. We assume that (x∗,y∗,λ∗) is a KKT solution of (2.2). Let (xk,yk,λk) begenerated from the ADMM scheme (2.4)-(2.6). Then the sequence (xk,yk) converges to anoptimal solution to (2.2) and λk converges to an optimal solution to the dual problem of(2.2), i.e. limk→∞ xk = x∗, limk→∞ yk = y∗, limk→∞λk = λ∗.

Since H(x) is twice differentiable convex function with a Lipschitz continuous gradient,there exists a self-adjoint positive semidefinite linear operators ΣH , such that for any x,x′,

ΣH W ∀W ∈ ∂ 2H(x),

where ∂ 2H(x) is the Clarke’s generalized Hessian at x.

H(x)≥ H(x′) + ⟨∇H(x′),x− x′⟩+1

2‖x− x′‖2ΣH

,

⟨∇H(x)−∇H(x′),x− x′⟩ ≥ ‖x− x′‖2ΣH.


(a) Barbara, size: 256×256

(b) Geometry, size:256× 256

(c) Fingerprint, size:251× 244

(d) Man, size: 256 ×256

(e) Floor, size: 300 ×400

(f) Flower, size: 250 ×250

(g) Wall, size: 210 ×236

Figure 2: Test Images.

Table 1: Numerical results of correlation Corr(u, v) for natural images.

Barbara Geometry Fingerprint ManCTV-L2 [20] 0.2581 0.1134 0.1462 0.2629BLMV [10] 0.0364 0.0117 0.0875 0.0875

AD-aBLMV-ADE [41] 0.0222 0.0055 0.0192 0.0492TV-L1 [31] 0.2775 0.0629 0.0589 0.1882

DF [11] 0.0758 0.02293 0.0539 0.1074ADMM-TV 0.0386 0.0146 0.023 0.1125PADMM-TV 0.0412 0.0148 0.0249 0.1125PADMM-EE 0.0155 0.0058 0.0167 0.0751

Theorem 3.2. We assume that (x∗,y∗,λ∗) is a KKT solution of (2.2) and ΣH +S +γA ∗Ais positive definite. Let (xk,yk,λk) be generated from the proximal ADMM scheme (2.12a)-(2.12c). Then the sequence (xk,yk) converges to an optimal solution to (2.2) and λk con-verges to an optimal solution to the dual problem of (2.2), i.e. limk→∞ xk = x∗, limk→∞ yk =y∗, limk→∞λk = λ∗.

4. Numerical experiments

In this section, we provide some numerical examples. We will present some resultsof proposed methods for image decomposition and image denoising. All the experimentswere performed using Windows 7 and MATLAB, R2013b (version 2.0) on a HP Z228 with


(a) u (b) zoom in (c) (d) 150+ v (e) u+ v

(f) u (g) zoom in (h) (i) 150+ v (j) u+ v

(k) u (l) zoom in (m) (n) 150+ v (o) u+ v

Figure 3: Barbara decomposition comparison between TV and Euler’s elastica based models. Fromtop to button, the results are form ADMM-TV, PADMM-TV, PADMM-EE, respectively. We chosethe parameters in the following way, for ADMM-TV we set α = 0.07,β = 0.8,γ = 0.35. For PADMM-TV,α = 0.07,β = 0.8,γ = 0.35,τ = 0.3. For PADMM-EE, α = 0.09,β = 0.99,γ = 0.04,γh = 0.016,γn =0.01,τ1 = τ2 = 1.5, a = 1.1, b = 100.

Intel(R) Core(TM) i7-4790 CPU @3.60GHz and 8GB memory.

4.1. Test images, practical implementation and stopping condition.

In this paper, we tested lots of images, for example the Barbara, Geometry, etc. (seeFig. 2). For all of the tested algorithms, we used the following stopping condition:

max

‖uk − uk+1‖‖ f ‖

,‖vk − vk+1‖‖ f ‖

≤ ε,

where ε > 0 is stopping tolerance, in our proposed algorithms, we set ε = 10−3. Forsimplicity of presentation, we denote our developed methods as follows:

• ADMM-TV: ADMM for TV based Vese-Osher’s texture model;

• PADMM-TV: proximal ADMM for TV based Vese-Osher’s decomposition model;


(a) p = 1, Corr(u, v) = 0.0128

(b) p =∞, Corr(u, v) = 0.0128

Figure 4: Image decomposition on clean images by PADMM-EE with p = 1 and p =∞.

Table 2: Numerical results of correlation Corr(u, v) for synthetic images.

Fig. 9(a) Fig. 9(b) Fig. 9(c) Fig. 9(d) Fig. 9(e)CTV-L2 [20] 0.2506 0.3724 0.2294 0.2195 0.1391BLMV [10] 0.1086 0.2747 0.0593 0.0247 0.0148

AD-aBLMV-ADE [41] 0.0214 0.0194 0.0196 0.0086 0.0147TV-L1 [31] 0.1469 0.18801 0.1733 0.1319 0.0697

DF [11] 0.1554 0.2901 0.0815 0.0268 0.0162ADMM-TV 0.0278 0.0232 0.0356 0.0377 0.0260PADMM-TV 0.0286 0.0229 0.0354 0.0273 0.0228PADMM-EE 0.0174 0.0134 0.0176 0.0053 0.0132

Table 3: Computational time (in Seconds) of the different methods for Barbara image (256× 256).

Computational timeCTV-L2 [20] 3.39BLMV [10] 0.23

AD-aBLMV-ADE [41] 5.694TV-L1 [31] 0.44ADMM-TV 0.7520PADMM-TV 0.3444PADMM-EE 1.1408

• PADMM-EE: proximal ADMM for Euler’s elastica based Vese-Osher’s decompositionmodel.


(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 5: Separation of cartoon and texture components (Barbara). (a) CTV-L2 [20]. (b) BLMV [10].(c) AD-aBLMV-ADE [41]. (d) TV-L1 [31]. (e) DF [11]. (f) ADMM-TV. (g) PADMM-TV. (h) ProposedPADMM-EE.

Before we show our numerical results, we give some remarks on choosing parametersand stopping criterion. There are three parameters (α,β ,γ) coming from ADMM-TV, fourparameters (α,β ,γ,τ) coming from PADMM-TV, and eight parameters (α,β ,γ,γn,τ1,τ2, a,b) coming from PADMM-EE. The parameter α is regularization parameter associated withthe cartoon part of image. For decomposition case this parameter is need to be small forbetter subtraction of cartoon from the image. For the pure cartoon-texture decompositionproblem, we would like to remove the texture component without removing other key fea-tures such as edges, shading, etc. The parameter β , which is the texture norm coefficientin the energy is related to the texture components of image. γ and γn are associated withLagrange multipliers. All τ, τ1 and τ2 are the proximal parameters. These proximal pa-rameters play an important role for the effectiveness of the proposed methods. a and b


(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 6: Separation of cartoon and texture components (Geometry). (a) CTV-L2 [20]. (b) BLMV [10].(c) AD-aBLMV-ADE [41]. (d) TV-L1 [31]. (e) DF [11]. (f) ADMM-TV. (g) PADMM-TV. (h) ProposedPADMM-EE.

are Euler’s elastica coefficients. The ratio a/b indicates the relative importance of the totallength versus total squared curvature. For b = 0, the minimization problem (1.2) becomesTV problem (1.1). We used different setting for the parameters in different applications ofour proposed methods. We tuned all the parameters by the above stopping criterion withtolerance.

4.2. Image decomposition

The main objective of decomposition model is to decompose an image into its cartoonand texture part. Various approaches to image decomposition have been proposed in theliterature, among these approaches variational based image decomposition is one of the


(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 7: Separation of cartoon and texture components (Fingerprint). (a) CTV-L2 [20]. (b) BLMV [10].(c) AD-aBLMV-ADE [41]. (d) TV-L1 [31]. (e) DF [11]. (f) ADMM-TV. (g) PADMM-TV. (h) ProposedPADMM-EE.

successful approaches and it has numerous applications in the field of image processing andcomputer vision. Interesting application of image decomposition are explored in [8, 45].In this subsection, we illustrate the efficiency and effectiveness of the proposed algorithmsvia many examples in image decomposition.

As in the literature [5], we assume that the cartoon and texture parts in an image arenot correlated. We thus take the correlation between the cartoon u and the texture v,which is computed by

Corr(u, v) =Cov(u, v)σu.σv

to measure the quality of decomposition, where σ(·) and Cov(·, ·) refer to the standard


(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 8: Separation of cartoon and texture components (Man). (a) CTV-L2 [20]. (b) BLMV [10]. (c)AD-aBLMV-ADE [41]. (d) TV-L1 [31]. (e) DF [11]. (f) ADMM-TV. (g) PADMM-TV. (h) ProposedPADMM-EE.

deviation and covariance of given variables, respectively.For image decomposition, we do not have ground-truth generation in case of real im-

ages as well as synthetic images. Therefore we consider the best result for smallest correla-tion value. To evaluate the quality of different methods, we show the decomposition resultfor test images (see Figs. 2(a), 2(b), 2(c) and 2(d)). But we also evaluate numerically thecompeting methods on synthetic images (see Fig. 9). In our numerical experiments, weobserved that the most of the decomposition results for ρ = 1 and ρ =∞ in Euler’s elasticadecomposition model (1.2) are look similar (see Fig. 4). Therefore we decided to show andcompare the results with ρ = 1 in all experiments. We compared our proposed PADMM-TV and PADMM-EE methods with the following decomposition methods: CTV-L2 [19,20],AD-aBLMV-ADE [41], BLMV filters [10], TV-L1 [31], DF [11] and ADMM-TV. The codes


(a) (b) (c) (d) (e)

(f) (g) (h) (i) (j)

(k) (l) (m) (n) (o)

Figure 9: Synthetic images (256 ×256) used for numerical evaluation. Top row: original image. Middlerow: cartoon component. Bottom row: texture component.

for [41] was provided by author, for [10, 11, 20, 31], we used online demo provided onweb by IPOL†. We used them with best tuned parameters in each individual test case.

The most commonly used image for texture evaluation is the Barbara image, which hasprecise edges and it also favors the TV-based methods, especially ADMM-TV and PADMM-TV (see Fig. 3). However, some artifacts can be seen on cartoon image of TV-based decom-position, as the cloth on the table has some texture in cartoon part (see zoom-in imagesin Fig. 3). PADMM-EE can completely eliminate the texture from the table cloth. Fur-thermore, it is obvious that we can get better contour by using cartoon part extracted byPADMM-EE algorithm (see third column Fig. 3). For numerical evaluation, we used theparameters that gave the minimum correlation and also gave the best visual result. Ev-ery image has different kind of cartoon as well as texture, so the parameters need to bedifferent, for example the parameters are simulated for the Barbara (see caption in Fig. 3).

To see how strong the cartoon edges on the texture image and precise the cartoonpart, we used the Geometry image. All methods eliminate the texture part from cartoon,but methods [10, 11, 19, 20, 31], ADMM-TV and PADMM-TV bring some cartoon edges on

†Image Processing On Line. http://www.ipol.im/


(a) (b)

(c) u (d) v (e) u + v (f) u + v

(g) u (h) v (i) u + v (j) u + v

Figure 10: Synthetic images (108 ×108) compared between TV-Stokes [28] and Euler’s elastica decom-position model. (a) is the original image, (b) is the surface plot of the original image. (c) to (f) areresults of our proposed Euler’s elastica model while (g) to (j) are results of TV-stokes model.

the texture (see Fig. 6). For this image, adaptive image decomposition method [41] hasbest results, however Euler’s elastica-based decomposition has better results among all TV-based decomposition methods and its results are very closed to adaptive decompositionmethod [41].

Most of decomposition methods like [10,11,41] blur some parts of cartoon image (seeFigs. 5,7,8), and adaptive decomposition [41] method also sightly blur the texture part.But on the other hand Euler’s elastica-based decomposition has best results as compare toall TV-based methods [19,31], ADMM-TV and PADMM-TV and as well as other methods.

We report the numerical results for real and artificial images in Table 1 and Table 2,where the obtained correlations (Corr) of the decomposed two parts are reported forall decomposition methods. The computational time for PADMM-TV is increased com-pared to the fast BLMV method, but it is faster then CTV-L2, AD-aBLMV-ADE, TV-L1,and ADMM-TV. Similarly, the computational time for PADMM-EE is increased comparedto the BLMV, TV-L1, ADMM-TV and PADMM-TV methods, but it is faster than CTV-L2 andAD-aBLMV-ADE (see Table. 3).

Moreover, we also compared the decomposition results of generalized TV-stokes model[28] and our proposed Euler’s elastica-based model. We used TVS-L2+ TVn-L2 models for


(a) (b) (c)

(d) (e)

Figure 11: Plots of numerical energy, relative error of uk+1, correlation, residuals and relative error ofLagrange multipliers of our proposed Euler’s elastica-based decomposition model for Fig. 2(d).

Table 4: Number of iteration and CPU time for image denoising.

ImagesADMM-TV PADMM-TV EEALM [43] PADMM-EE

Nit Time Nit Time Nit Time Nit TimeFloor 19 0.9507 26 0.5054 30 6.1270 28 2.0494

Flower 17 0.4484 23 0.2765 30 2.9618 26 1.1291Wall 18 0.3712 24 0.2520 20 1.7888 29 1.0009

image decomposition from the general TV-stokes model. The TV-stokes model is aimedto separate the jump discontinuities of an image, where u contains all the discontinuitiesof f and v is a continuous function (see Fig. 10). These separated parts could be usedfor a large number of image processing purpose. Note that the TV-stokes model do notseparately obtain texture parts from an image. However, our proposed Euler’s elastica-based model can better decompose an image into its texture parts (see also Fig. 10).

To show the convergence of our proposed method: PADMM-EE, we plot the numericalenergy, relative error in uk+1, correlation, residuals and relative error in lagrange multipli-ers verses number of iterations for the image Man (see Fig. 11). From these plots, it is easyto see the proposed algorithm for Euler’s elastica based decomposition has converged. Theplots of residuals and relative error in lagrange multipliers also give important informationabout tuning of parameters and one can see in Fig. 11 that they converge to zero fasterbefore 50 iterations.


(a) Noisy, σ=17, SNR:10.3215

(b) ROF,SNR:13.1478 (c) ADMM-TV, SNR:14.5366

(d) PADMM-TV, SNR:14.5457

(e) Noisy, σ=15, SNR:13.4393

(f) ROF,SNR:17.9577 (g) ADMM-TV, SNR:20.1119

(h) PADMM-TV, SNR:20.2018

(i) Noisy, σ=10, SNR:7.9816

(j) ROF,SNR: 10.8806 (k) ADMM-TV, SNR:11.3328

(l) PADMM-TV, SNR:11.3296

Figure 12: Image denoising caparison between TV based models. The results for ROF, ADMM-TVand PADMM-TV are shown from left to right respectively. Parameters for ADMM-TV, Floor (α =0.11,β = 10,γp = 0.1,γq = 0.1), Flower (α = 0.09,β = 9.5,γp = 0.1,γq = 0.1) and Wall ( α = 0.2,β =10,γp = 0.09,γq = 0.1). For PADMM-TV, Floor ( α = 0.11,β = 10,γp = 0.2,γq = 0.1,τ1 = 1), Flower(α= 0.09,β = 9.5,γp = 0.1,γq = 0.1,τ1 = 0.8) and Wall (α= 0.17,β = 10,γp = 0.12,γq = 0.1,τ1 = 1).

4.3. Image denoising

In this example firstly, we compare the very basic and pioneer ROF model [37] withVese-Osher’s decomposition model (1.1). During our experiments, we observed that al-though the ROF model is faster than the Vese-Osher’s decomposition model, but it removessmall texture and little stairs appeared in the denoised image see Fig. 12. Secondly, wecompare the Euler’s elastica-based model [43] (EEALM) with our proposed Euler’s elastica-based decomposition model (1.2). We observed that Euler’s elastica-based decompositionmodel has better results in the sense of SNR and also quality, see Fig. 13. In these exper-iments, we show the cartoon plus texture (u+ v) as a reconstruction or denoised imagefor decomposition model while in the original model the minimizer u is as a denoised im-age. We compare the CPU time and number of iteration of these models in Table 4, where


(a) Noisy, σ=17, SNR:10.3215 (b) EEALM, SNR:13.7963 (c) PADMM-EE, SNR:14.7520

(d) Noisy, σ =15, SNR:13.4393 (e) EEALM, SNR:18.6232 (f) PADMM-EE, SNR:20.4632

(g) Noisy, σ =10, SNR:7.9816 (h) EEALM, SNR:11.2021 (i) PADMM-EE, SNR:11.7696

Figure 13: Image denoising caparison between Euler’s elastica based models. The results for Euler’selastica [43] and PADMM-EE are shown from left to right respectively. Parameters for PADMM-EE,Floor ( α = 0.12,β = 12.5, a = 0.9, b = 10,γp = 0.15,γn = 0.3,γh = 0.01,γq = 0.09,τ1 = τ2 = 1.1),Flower(α = 0.1,β = 10.5, a = 0.9, b = 10,γp = 0.1,γn = 0.45,γh = 0.01,γq = 0.09,τ1 = τ2 = 1.2) and Wall(α= 0.17,β = 9, a = 0.6, b = 10,γp = 0.105,γn = 0.6,γh = 0.002,γq = 0.1,τ1 = τ2 = 1.5).

our proposed PADMM-TV is faster than the ADMM-TV. Similarly, PADMM-EE is more fasterthan EEALM.

The performance is evaluated by using the signal to noise ratio (SNR) and the SNR isdefined as:

SNR∆= 10 log10

‖u− u‖2

‖u− u‖2,

where u is the original signal, u is the mean intensity value of u and u is recovered sig-nal, in decomposition case the recovered signal becomes u+ v. For each experiment, the


(a) (b) (c)

Figure 14: Plots of numerical energy, relative error of uk+1 and SNR of our proposed Euler’s elastica-based decomposition model for Fig. 2(e).

parameters are estimated so that the best SNR is obtained.In our experiments, Gaussian white noise with zero mean and the standard deviation

17, 15 and 10 are used for images in Fig. 2(e), Fig. 2(f) and Fig. 2(g) respectively. Aswe have reduced the parameters associated with lagrange multipliers in Section 2 and 3,but for the better results of image denoising and for minimizing of lagrange multipliersindividually, we tuned all the parameters (see captions in Fig. 12 and Fig. 13). Finally, weplots numerical energy, relative error in uk+1 and SNR verses number of iterations for theimage Floor (see Fig. 14). As we can observe from these plots in Fig. 14, our proposedalgorithm for model (1.2) has converged.

5. Conclusions

Vese-Osher’s model is used for texture extraction and image denoising in the literatureof applied mathematics and image processing. Although the classical ADMM can be usedto solve these problems but it is very challenging to construct efficient algorithms whenminimization problem has non-linear high order derivatives. In this paper, we observedthat one can solve these problems more fast by proximal ADMM in which one more sub-problem has closed form solution. Firstly, we solved Vese-Osher’s decomposition modelby proximal ADMM in which we linearized g-subproblem to have a close form solution.Secondly, we solved Vese-Osher’s model having Euler’s elastica energy term instead of to-tal variation based on the proximal ADMM. All the subproblems are easy to solve and themethod is globally convergent at a linear rate for Vese-Osher’s model. We also presentsome preliminary numerical results, which indicate that our method is not only efficientbut it also can improve the quality of images in case of image denoising.

Acknowledgments The work of Zhifang Liu was supported by the Ph.D. Candidate Re-search Innovation Foundation of Nankai University. The work of Yuping Duan was sup-ported by National Natural Science Foundation of China (Grants 11701418). The work ofHuibin Chang was partially supported by National Natural Science Foundation of Chi-na (Nos. 11501413, 11871372), Natural Science Foundation of Tianjin (No. 18JCYB-JC16600), 2017-Outstanding Young Innovation Team Cultivation Program (No. 043-13520


2TD1703) and Innovation Project (No. 043-135202XC1605) of Tianjin Normal University,Tianjin Young Backbone of Innovative Personnel Training Program and Program for Inno-vative Research Team in Universities of Tianjin (No. TD13-5078). The work of ChunlinWu was supported by National Natural Science Foundation of China (Grants 11301289,11531013 and 11871035), Recruitment Program of Global Young Expert, and the Funda-mental Research Funds for the Central Universities. The research of Tai is supported byHKBU startup grant, RG(R)- RC/17-18/02-MATH, and FRG2/17-18/033.

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