Vol. 9, No. 4, 2018 Convex Hybrid Restoration and ......[6]. Chan Vese (CV) model [4] is the special...

(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 9, No. 4, 2018

Convex Hybrid Restoration and Segmentation Modelfor Color Images

Matiullah∗, Samiullah Khan†, Noor Badshah‡, Fahim Ullah§ and Ziaullah¶∗‡§Department of Basic Sciences and Islamiat, ¶ Department of Computer System Engineering

University of Engineering and Technology, Peshawar-Pakistan†Institute of Business Management Sciences, The University of Agriculture Peshawar-Pakistan

Abstract—Image restoration and segmentation are importantareas in digital image processing and computer vision. In thispaper, a new convex hybrid model is proposed for joint restorationand segmentation during the post-processing of colour images.The proposed Convex Hybrid model is compared with the existingstate of the art variational models such as Cai model, Chan-VeseVector-Valued (CV-VV) model and Local Chan-Vese (LCV) modelusing noises such as Salt & Pepper and Gaussian. Additionalfour experiments were performed with increasing combinationof noises such as Salt & Pepper, Gaussian, Speckle and Poissonto thoroughly verify the performance of Convex Hybrid Model.The results revealed that the Convex Hybrid model comparativelyoutperformed qualitatively and has successfully removed thenoises and segment the required object properly. The ConvexHybrid model used the colour Tele Vision (TV) as a regularizerfor denoising of the corrupt image. The Convex Hybrid Modelis convex and can get global minima. The PDEs obtained fromthe minimisation of the Convex Hybrid Model are numericallysolved by using explicit scheme.

Keywords—Color images; image restoration; image segmenta-tion; noise

I. INTRODUCTION

Image restoration works to restore an image in digitalimage processing. Noise in an image is the random variationof intensity or colour in images (unwanted signals), which isusually produced by the sensor or digital camera. Restorationis the process of denoising an image. Many variational modelsare proposed for restoration of images. Fundamental varia-tional model for denoising of grey level images is proposedby Rudin et al. [10]. Later on, Blomgren et al. [1] proposed amodel restoration of a colour image which is based on colourtotal variation (CTV). The advantage of the TV-based model isthe preserving edges during denoising. TV-based model createsthe staircase effects. Much higher order’s derivatives basedmodels are proposed in [10]–[14] to avoid the staircase effect.

Segmentation is divided into distinct subdomains basedon some criterion like homogeneity in intensity, colour andtexture. Segmentation of noisy images is an issue in the fieldof computer vision. Many variational models are developedfor segmentation of images to solve the issue of segmentationof noisy images. These models are classified into two cat-egories, i.e. Edge-Based Models and Region-Based Models.Edged based models use edge information for segmentationof images, such as snake model and geodesic active contourmodel [9]. Snake model and geodesic active contour modeluse edge information due to which they are not applicable innoisy images. Region-based models use region information for

segmentation of images such as Mumford-Shah (MS) model[6]. Chan Vese (CV) model [4] is the special case of MSmodel. The CV model is not convex, and it might stick tolocal minima. CV model cannot segment noisy images dueto non-convexity. Bresson et al. [2] proposed Fast GlobalMinimization (FGM) model for the convex formulation ofthe active contour model for grey level images. This modelalso does not segment images having heavy noise and is moredependent on the tradeoff parameters [15]–[17].

Image restoration is the first field of image processingtechnology which improves the quality of an image. X-Cai[3] proposed a Cai model which works jointly for imagerestoration and segmentation. Cai models restore the degradedimage and then segment the restored image. Cai model is anon-convex. Segmentation results of Cai model are not verysatisfactory in an image having intensity inhomogeneity.

In this paper, a new Convex Hybrid model is proposedwhich works jointly for restoration and segmentation of colourimages. The Convex Hybrid model is based on colour totalvariation (CTV) of restoration and global minimization forsegmentation [2].

II. RELATED VARIATIONAL MODELS

A brief overview of some image denoising and segmenta-tion models are presented in this section.

A. Rudin, Osher and Fatimi (ROF) Model

ROF model is one of the famous and commonly used modelfor the restoration and de-noising of corrupted images whichwas designed by Rudin, Osher and Fatemi (ROF) in 1992. LetΩ ⊂ R2 be a bounded, open, connected set, and v0(x, y) :Ω −→ R be given corrupted image and v(x, y) : Ω −→ R isthe desired clean image. Then the energy functional of ROFmodel can be written as:

EROF (v) = γ

∫Ω

(v0 − v)2dxdy +

∫Ω

|∇v|dxdy, (1)

where the regularization parameter is represented by γ andthe task of first term in (1) is that the clean image must beclose to the observed image. While in (1) the last term is thesmoothing or regularization term which is actually l1 norm.The corresponding Euler-Lagrange equation for ROF model isas:

www.ijacsa.thesai.org 342 | P a g e


∂

∂x

vx√v2x + v2

y

+∂

∂y

vy√v2x + v2

y

+γ(v − v0) = 0 in Ω, (2)

with ∂v∂n = 0 on the boundary of Ω = ∂Ω. For solving (2)

the following parabolic equation is considered:

∂v

∂t=

∂

∂x

vx√v2x + v2

y

+∂

∂y

vy√v2x + v2

y

+γ(v − v0) = 0 for t > 0. (3)

The solution of (2) can be obtained from the steady statesolution of (3). ROF model uses l1 norm instead of l2 normas a result this model perform better than other models whichutilizes l2 norm. The advantage of l1 norm over l2 norm is thatl1 norm smooth the image but keeps the boundaries of objectsharp, in contrast l2 norm also smooth the edges of the objectswhich is one of the main drawback to l2 norm. Moreover, theresults of ROF model is batter than linear smoothing but themain drawback of ROF model is that it create staircase effects.In order to overcome the limitation of ROF model, many othermodels have been developed [6].

B. Peter Blomgren TV Color Model

As ROF model uses the one dimensional TV norm for therestoration of scalar images. Therefore, to extend this modelfor the restoration of vector-valued images, Blomgren andChan proposed a new definition of TV norm for the restorationof color and vector-valued images. Which posses the propertiesof not penalizing discontinuities (edges) in the image and isrotational invariant [1]. TV-VV model proposed the followingenergy functional:

minv∈BV (Ω)

||v||TV−V V +γ

2||v − v0||22, (4)

where γ is Lagrange multiplier, |v|22 =∑nj=1 ||vj ||22 and

||.||TV−V V = TVm,n(v) and is defined as:

TVm,n(v) =

√√√√ n∑j=1

[TVm,1(vj)]2. (5)

Now clearly for n = 1 (14) turn out to TV norm forscalar valued function. Now the corresponding Euler-Lagrangeequation for the minimization problem (13) with ||.||TV−V V =TVm,n(v) can be obtained as:

TVm,1(vj)

TVm,n(v)∇.(∇vj||∇vj ||

)− γ(vj − v0,j) = 0. (6)

The solution can be computed by explicit time marchingscheme by considering the following PDE:

∂vj∂t

=TVm,1(vj)

TVm,n(v)∇.

(∇vj√

γ1 + ||∇vj ||2

)− γ(vj − v0,j), (7)

where γ1 is used to avoid division by zero and is regardedas small regularization parameter.

C. Active Contour without Edges (CV) Model

CV model was introduced by Chan and Vese, which isactually a special case of the Mumford-Shah (MS) energyfunctional [6] when restricted to only two phases [4], [7]. CVmodel tries to demonstrate the minimization energy functionalas:

ECV (e1, e2,Γ) = ν.(Length(Γ))

+γ1

∫inside(Γ)

|v0(x, y)− e1|2dxdy

+γ2

∫outside(Γ)

|v0(x, y)− e2|2dxdy, (8)

where the segmented and smooth curve is represented byΓ. ν, γ1 and γ2 are tuning positive parameters. While e1 ande2 denotes the mean intensities of v0 inside and outside of Γrespectively. CV model can be written in level set formulationas follow:

ECV (e1, e2, ϕ) = ν

∫Ω

δ(ϕ)|∇ϕ|dxdy

+γ1

∫Ω

|v0 − e1|2H(ϕ)dxdy

+γ2

∫Ω

|v0 − e2|2(1−H(ϕ))dxdy, (9)

where the level set function is represented by ϕ, H(ϕ) andδ(ϕ) are the Heaviside and dirac delta functions, respectively.CV model used the regularized form of Heaviside and diracdelta functions instead of the original ones because Heavisidefunction is not differentiable at the origin. The regularizedversions of Heaviside and dirac delta functions are denoted byHε(ϕ) and δε(ϕ), respectively and is defined So (9) is writtenas:

ECV (ϕ, e1, e2) = ν

∫Ω

δε(ϕ)|∇ϕ|dxdy

+γ1

∫Ω

|v0 − e1|2Hε(ϕ)dxdy

+γ2

∫Ω

|v0 − e2|2(1−Hε(ϕ))dxdy. (10)

Minimizing (10) with respect to e1 and e2 we have:

e1 =

∫Ωv0Hε(ϕ)dxdy∫

ΩHε(ϕ)dxdy

,

e2 =

∫Ωv0(1−Hε(ϕ))dxdy∫

Ω(1−Hε(ϕ))dxdy

, (11)



whereas, Euler-Lagrange equation for model (10) can beobtained by its minimization and leads to the following PDE:

δε

[νdiv

(∇ϕ|∇ϕ|

)−γ1(v0 − e1)2 + γ2(v0 − e2)2

]= 0 in Ω,

δε(ϕ)

|∇ϕ|∂ϕ

∂~n= 0 on ∂Ω.

(12)

The above PDE can be solved by numerical methods andthe values of e1 and e2 are updating at every iteration from(11).

D. Chan-Vese Vector-Valued (CV-VV) Model

CV model is capable to segment only gray scale images.Therefore, in [5] Chan, Sandberg and Vese extended the CVmodel to vector-valued images. As their are many imageswhich can be properly segmented in their vector representationrather than in its scalar representation. For instance, objectswith different missing parts in different channels are com-pletely detected (such as occlusion). Also, in color images,objects which are invisible in each channel or intensity can bedetected by our algorithm. Let v0,j be the given vector-valuedimage over the domain Ω, with j = 1, 2, ...,M channels, andΓ be the evolving curve. Then the energy functional of CV-VVmodel is written as:

ECV−V V (e1, e2,Γ) = ν.(Length(Γ))

+

∫inside(Γ)

1

M

M∑j=1

γ1,j |v0,j(x, y)− e1,j |2dxdy

+

∫outside(Γ)

1

M

M∑j=1

γ2,j |v0,j(x, y)− e2,j |2dxdy, (13)

where e1 = (e1,1, e1,2, . . . , e1,M ) ande2 = (e2,1, e2,2, . . . , e2,M ) are two unknown constant vectorsand γ1,j and γ2,j are positive parameters for each channel. Inlevel set formulation model 13 can be written as:

ECV−V V (e1, e2, ϕ) = ν.

∫Ω

δ(ϕ(x, y))|∇ϕ(x, y)|dxdy

+

∫Ω

1

M

M∑j=1

γ1,j |v0,j(x, y)− e1,j |2H(ϕ(x, y))dxdy +

∫Ω

1

M

M∑j=1

γ2,j |v0,j(x, y)− e2,j |2(1−H(ϕ(x, y)))dxdy, (14)

where H(ϕ) and δ(ϕ) are Heaviside and dirac deltafunctions, respectively as defined in CV model. The values ofconstants e1,j and e1,j can be obtain by from (14) by takingits partial derivatives.

The corresponding Euler-Lagrange equation can be findout by keeping e1,j and e1,j fixed and minimizing (14) withrespect to ϕ we have:

∂ϕ

∂t= δε

[ν.div

(∇ϕ|∇ϕ|

)− 1

M

M∑j=1

γ1,j(v0,j − e1,j)2

+1

M

M∑j=1

γ2,j(v0,j − e2,j)2]in Ω, (15)

where δε is the regularized version of the dirac deltafunction and with the boundary condition:

δε(ϕ)

|∇ϕ|∂ϕ

∂~n= 0,

on ∂Ω, and ~n represent the unit normal at the boundary ofΩ. The advantage of CV-VV model [5] over CV model [4]is that it can also segment those images which can not besegmented in any scalar representation. CV-VV is also able todetect those objects in vector valued images that are not visiblein each channel. However, the limitation of CV-VV model canbe seen in intensity inhomogeneous images because CV-VVmodel utilized the averages of CV model.

E. Fast Global Minimization (FGM) Model

Bresson et al. proposed a fast global minimization of theCV model [2]. In this model, first, they made CV model convexand then used dual formulation for minimization. As the CVmodel is non-convex, so local minima exist, and therefore, asmooth approximation of the Heaviside function is selected inthis model. Thus the steady state solution of (12) is same as:

∂ϕ

∂t= ν∇.

(∇ϕ|∇ϕ|

)− |v0 − e1|2 + |v0 − e2|2 (16)

Moreover, (16) is the partial differential equation obtainedfrom the model (8). It is non-convex and stuck at local minima.FGM model incorporated an edge detector function into theCV model (8) in order to make it convex and proposed thefollowing energy functional:

EFGM (e1, e2, ϕ) =

∫Ω

g(x)|∇ϕ|dxdy

+

∫Ω

(|v0 − e1|2 + |v0 − e2|2)ϕdxdy, (17)

where g(x) is an edge indicator function and the abovemodel is convex and provide us a global minima. Actually,it is homogeneous of degree 1 in ϕ and therefore, it hasno stationary solution. To get the optimal solution, someconstrained must be imposed on ϕ, then the model (17) canbe written as follows:

min0≤ϕ≤1

EFGM (e1, e2, ϕ) =

∫Ω

g(x)|∇ϕ|dxdy

+

∫Ω

(|v0 − e1|2 + |v0 − e2|2)ϕdxdy. (18)

The unconstrained form of minimization problem (18) is:



minϕEFGM (e1, e2, ϕ) =

∫Ω

g(x)|∇ϕ|dxdy

+

∫Ω

(|v0 − e1|2 + |v0 − e2|2)ϕ+ γv(ϕ)dxdy, (19)

where v = max

0, 2|ϕ − 12 | − 1

is an exact penalty

function provided that the constant γ is chosen a large numbersuch that:

γ >1

2||(v0 − e2

1) + (v0 − e2)2||.

Furthermore, using dual formulation the variational problem(19) is regularized and minimized as in CV model. This modelimproved the performance of the CV model when there islittle bit intensity change in the object to be detected and thebackground.

F. Xiaohao Cai Model

Image segmentation and restoration are closely relatedproblems to each other. Therefore, in [3] Cai proposed a jointimage segmentation and restoration model which is capable torestore as well as segment corrupted and degraded images.Cai et al. also provided a link between image restorationand segmentation in [17] while, in [18] the author provedthat the solution of CV model [4] for a certain γ can beobtained by thresholding the minimizer of the ROF model [8]using a proper threshold, which provides a clear link betweenimage segmentation and restoration. Cai model proposed thefollowing energy functional as:

ECai(v, ej , φj) = νΦ(v0,Aw) + γψ(v, ej , φj)

+

K∑j=1

∫Ω

|∇φj |dx, (20)

such that

K∑j=1

φj(x) = 1, φj(x) ∈ 0, 1,∀x ∈ Ω.

In model (20) the aim of first term is to restore the given cor-rupted image and comes from image restoration methods whilethe second term is designed for segmentation of images andthe third term is the regularization term. Whereas, A representthe problem related operator. Cai model is designed for thesegmentation of three types of images namely, blurry images,noisy images and images with missing pixels. Therefore theoperator A is to be tuned according to the given image. Nowby incorporating the data terms of ROF and CV model inmodel (20) i.e., by setting Φ(v0,Av) =

∫Ω

(v0 − Av)2dxdy

and ψ(v, ej , φj) =∑Kj=1

∫Ω

(v − ej)2φjdxdy, we have:

ECai(v, ej , φj) = ν

∫Ω

(v0 −Av)2dxdy

+γ

K∑j=1

∫Ω

(v − ej)2φjdxdy

+

K∑j=1

∫Ω

|∇φj |dxdy, (21)

where∑Kj=1 φj(x) = 1, φj(x) ∈ 0, 1. Now the above

energy functional (21) is minimized by the alternating mini-mization technique. For finding v keeping φj and ej fixed andminimizing (21) with respect to v, we have:

v = (νATA+ γ)−1(νAT v0 + γK∑j=1

ejφj), (22)

where A is considered to be linear operator. For finding ejkeeping φj and v fixed and minimize (21) with respect to ej ,thus:

ej =

∫Ωvφjdxdy∫

Ωφjdxdy

. (23)

Furthermore, for the minimization of energy functional(21) with respect to φj optimization methods can be used likeADMM (Alternating Direction Method of Mutiplying) method[19], [20] the max-flow approach [21] or the primal-dualalgorithm [22], [23]. Cai model is able to segment imagescorrupted from noise, blur or having missing pixels but onthe other hand the results of Cai model are not promising inimages suffered from intensity inhomogeneity. This is dueto the fact that Cai model used the data term of CV model,which uses the global intensity information of images ratherthan local one.

This section was devoted to the study of literature re-view of variational models used for image segmentation andrestoration. Each of the above discussed models play a crucialrole in image segmentation and restoration fields. However,these models have also some limitations as discussed above.Based on these techniques in the next chapter, we develop anew variational model for the joint image restoration and seg-mentation which is capable to segment images from intensityinhomogeneity.

III. CONVEX HYBRID MODEL

In this paper, a novel convex hybrid model is proposedfor both denoising and segmentation of colour images. In theconvex hybrid model, vector-valued TV is used for denoising.The energy functional of the convex hybrid model is givenbelow:



A. Numerical Scheme

The discretization of (8) using explicit finite differencescheme becomes:

∂ϕki,j∂t

= Ei,j −∇ ·(gi,j∇ϕi,j|∇ϕi,j |

),

(24)

where Ei,j =∑Mj=1(|vj − e1,j |2 − |vj − e2,j |2)

ϕk+1i,j − ϕki,j

M t= Ei,j −∇ ·

(gi,j∇ϕi,j|∇ϕi,j |

),

Simplifying we get,

ϕk+1i,j − ϕki,j

M t= Ei,j

−M t

h21

gi,j Mx−

Mx+ ϕki,j√(

Mx+ϕ

ki,j

h1

)2

+(

My+ϕ

ki,j

h2

)2

+ β

+M t

h22

gi,j My−

My− ϕ

ki,j√(

Mx+ϕ

ki,j

h1

)2

+(

My+ϕ

ki,j

h2

)2

+ β

, (25)

where the differences 4x+,4x−,4y+,4

y− are given by,

4x+ϕki,j = ϕki+1,j − ϕki,j , 4x−ϕki,j = ϕki,j − ϕki−1,j ,4y+ϕki,j = ϕki,j+1 − ϕki,j , 4y−ϕki,j = ϕki,j − ϕki,j−1,

(26)

ϕk+1i,j =M tEi,j + ϕki,j

−M t

h21

gi,j Mx−

Mx+ ϕki,j√(

Mx+ϕ

ki,j

h1

)2

+(

My+ϕ

ki,j

h2

)2

+ β

+M t

h22

gi,j My−

My− ϕ

ki,j√(

Mx+ϕ

ki,j

h1

)2

+(

My+ϕ

ki,j

h2

)2

+ β

, (27)


−M t

h21

gi,j Mx−

ϕki+1,j − ϕki,j√(

Mx+ϕ

ki,j

h1

)2

+(

My+ϕ

ki,j

h2

)2

dx+ β

+M t

h22

gi,j My−

ϕki,j+1 − ϕki,j√(

Mx+ϕ

ki,j

h1

)2

+(

My+ϕ

ki,j

h2

)2

+ β

,

where

Di,j =1√(

My+ϕ

ki,j

h1

)2

+(

My+ϕ

ki,j

h2

)2

+ β

varphik+1i,j =M tEi,j + ϕki,j

−M t

h21

gi,j

(Di,jϕ

ki+1,j

−Di−1,jϕki,j −Di,jϕ

ki,j

+Di−1,jϕki−1,j

)+M t

h22

gi,j

(Di,jϕ

ki,j+1 −Di,j−1ϕ

ki,j −Di,jϕ

ki,j +Di,j−1ϕ

ki,j−1

)(28)


−M t

h21

gi,j

(Di,jϕ

ki+1,j − (Di−1,j +Di,j)ϕ

ki,j +Di−1,jϕ

ki−1,j

)+M t

h22

gi,j

(Di,jϕ

ki,j+1 − (Di,j−1 +Di,j)ϕ

ki,j +Di,j−1ϕ

ki,j−1

)(29)

ϕk+1i,j =M tEi,j + ϕki,j −

M t

h21

gi,j(Di,jϕ

ki+1,j − (Di−1,j +Di,j)ϕ

ki,j +Di−1,jϕ

ki−1,j

)+M t

h22

gi,j

(Di,jϕ

ki,j+1 − (Di,j−1 +Di,j)ϕ

ki,j +Di,j−1ϕ

ki,j−1

), (30)

where

Di−1,j =1√(

Mx+ϕ

ki−1,j

h1

)2

dx+(

My+ϕ

ki−1,j

h2

)2

dx+ β

Di,j−1 =1√(

Mx+ϕ

ki,j−1

h1

)2

dx+(

My+ϕ

ki,j−1

h2

)2

dx+ β

ϕk+1i,j =M tEi,j + ϕki,j −

M t

h21

gi,j

Di,jϕki+1,j + (Di−1,j +Di,j)

M t

h21

gi,jϕki,j +

Di−1,jM t

h21

gi,jϕki−1,j +

M t

h22

gi,jDi,jϕki,j+1

−(Di,j−1 +Di,j)M t

h22

gi,jϕki,j +Di,j−1

M t

h22

gi,jϕki,j−1 (31)

This



ϕk+1i,j =M tEi,j + ϕki,j −A1ϕ

ki+1,j +Aϕki,j −A2ϕ

ki−1,j

+B1ϕki,j+1 −Bϕki,j +B2ϕ

ki,j−1 (32)

where

A1 = Di,jM t

h21

gi,j , A = (Di−1,j +Di,j)M t

h21

gi,j ,

A2 = Di−1,jM t

h21

gi,j , (33)

and

B1 = Di,jM t

h22

gi,j , B = (Di,j−1 +Di,j)M t

h22

gi,j ,

B2 = Di,j−1M t

h22

gi,j , (34)

ϕk+1i,j =M tEi,j −A1ϕ

ki+1,j + (I +A−B)ϕki,j +A2ϕ

ki−1,j

+B1ϕki,j+1 +B2ϕ

ki,j−1 (35)

The matrix form of the above equation becomes:

ϕk+1 = Ek +Mϕk, (36)

where ϕk+1,ϕk and Ek are column vectors of size 1× n.M is a block Tri-diaognal matrix.

In this paper, a novel convex hybrid model is proposedfor both denoising and segmentation of colour images. In theconvex hybrid model, vector-valued TV is used for denoising.The energy functional of the convex hybrid model is givenbelow.

IV. RESULTS AND DISCUSSION

In this section, the proposed Convex Hybrid model is com-pared with X. Cai [3] and CV vector-valued model [5] usingsynthetic and real images. The results of these experiments arecompared using a qualitative approach.

Fig. 1 show ideal noise-free synthetic images of flower.Two different noise, i.e. “Salt and Pepper” and Gaussian wereadded to synthetic images as shown in Fig. 1 to evaluatethe proposed Convex Hybrid model thoroughly. Fig. 2 showssalt and pepper noised synthetic image with Gaussian noise(variance = 0.002).

In next phase, Cai, CV-VV, LCV and Convex Hybridmodels are applied to the noised synthetic images of bottles asshown in Fig. 2. Fig. 3 display the results of Cai model. Fig. 4revealed the results of CV-VV model. Fig. 5 shows the resultof LCV model. The balancing parameters such as µ1 = 10,µ = 0.0001, λ1 = 1 λ2 = 5 and λ3 = 15 are used in theConvex Hybrid Model.

Finally, the Convex Hybrid model results are revealed byFig. 5. It is clear from the results of the above figures thatConvex Hybrid model outperformed the Cai, CV-VV and LCV

Fig. 1. Original image of a bottle.

Fig. 2. Noisy image (having Salt and Pepper noise) of a bottle.

model as mentioned in Fig. 6. Convex Hybrid model segmentthe noised synthetic images of flower and bottle (Fig. 2)properly and successfully removed the noise as shown in Fig.6. The Convex Hybrid model does not depend on the positionof initial contour due to convexity.

For extensive evaluation, the performance of Convex Hy-brid model is verified using images with increasing combina-tion of noises such as Salt & Pepper, Gaussian, Speckle andPoisson. Four further experiments are performed as mentionedbelow.

A. Test Case 1: Salt & Pepper, Gaussian, Speckle and Poisson

In test case 4, four noises i.e. salt and pepper, Gaussian,Speckle and Poisson noises are added having variance 0.02



Fig. 3. Joint results of restoration and segmentation for colour images usingCai model.

Fig. 4. Joint results of restoration and segmentation for colour images usingChan-Vese Vector-valued model.

as shown in Fig. 7. Fig. 8 is the result of Convex Hybridmodel with smooth/denoised image having four noises. ConvexHybrid model with a segmented image is shown in Fig. 9.

V. CONCLUSION

In this paper, the Convex Hybrid model is proposed forrestoration and segmentation of images. Initially, Convex Hy-brid model is compared with three other models, i.e. Cai, CV-VV and LCV using noises such as Salt & Pepper and Gaussian.The results revealed that the Convex Hybrid model compara-tively outperformed qualitatively. Four additional experimentswere performed with increasing combination of noises suchas Salt & Pepper, Gaussian, Speckle and Poisson. The convex

Fig. 5. Joint results of restoration and segmentation for colour images usingLocal Chan-Vese Model.

Fig. 6. Joint results of restoration and segmentation for colour images usingConvex Hybrid model.

Fig. 7. Test Case 4: Noisy image having four noises, i.e. Salt & Peppernoise, Gaussian, Speckle noise and Poisson noise of variance=0.02.



Fig. 8. Test Case 4: Convex Hybrid model with smooth/denoised imagehaving four noises.

Fig. 9. Test Case 4: Convex Hybrid model with a segmented image havingfour noises.

Hybrid model successfully removed the noises and segmentedthe required object accurately. The Convex Hybrid model usedthe colour TV as a regularizer for denoising of the corruptimage. The Convex Hybrid Model is convex and can get globalminima. The PDEs obtained from the minimisation of theConvex Hybrid Model are numerically solved by using explicitscheme.

In future, this research work will be extended to 3Dimage reconstruction and segmentation. There is a need todeeply study and understand the relationship among imagesegmentation, restoration and enhancement to achieve morerobust models for image segmentation. Developing of new androbust numerical techniques for variational image segmenta-tion models are also our research field of interest for futurework.

REFERENCES

[1] P. Blomgren and T.F. Chan, ”Colour TV: Total variation methodsfor restoration of vector-valued images”, IEEE Transactions on ImageProcessing, Vol. 7, No. 3, pp. 304-309, March, 1998.

[2] X. Bresson, S. Esedoglu, P.Vandergheynst, J.P. Thiran and S. Osher,”Fast global minimization of the active contour/snake model”, Journalof Mathematical Imaging and Vision, Vol.28, No. 2, pp. 151-167, June,2007.

[3] X. Cai, ”Variational image segmentation model coupled with imagerestoration achievements”, Pattern Recognition, Vol.48, No. 6, pp. 2029-2042, June, 2015.

[4] T.F. Chan and L. A. Vese, ”Active contours without edges”, IEEE Trans-actions on Image Processing, Vol.10, No. 2, pp. 266-277, February,2001.

[5] T. F. Chan, B.Y. Sandberg and L.A. Vese, ”Active contours withoutedges for vector-valued images”, Journal of Visual Communication andImage Representation, Vol.11, No.2, pp. 130-141, June, 2000.

[6] D. Mumford, and J. Shah, ”Optimal approximations by piecewisesmooth functions and associated variational problems”, Communica-tions on Pure and Applied Mathematics, Vol.42, No. 5, pp. 577-685,July,1989.

[7] R. Ronfard, ”Region-based strategies for active contour models”, In-ternational Journal of Computer Vision, Vol.13, No. 2, pp. 229-251,October, 1994.

[8] L.I. Rudin, S. Osher and E. Fatemi, ”Nonlinear total variation basednoise removal algorithms”, Physica D: Nonlinear Phenomena, Vol.60No. 4, pp. 259-268, November, 1992.

[9] N. Paragios and R. Deriche, ”Geodesic active regions: A new frameworkto deal with frame partition problems in computer vision”, Journal ofVisual Communication and Image Representation, Vol.13, No.2, pp.249-68, March, 2002.

[10] J.F. Cai, R.H. Chan, Z.A. Shen, ”framelet-based image inpaintingalgorithm”, Applied and Computational Harmonic Analysis, Vol. 24,No.2, pp. 131-149, March, 2008.

[11] T. Chan, A. Marquina and P. Mulet, ”High-order total variation-basedimage restoration”, SIAM Journal on Scientific Computing, Vol. 22,No.2, pp. 503-516, July, 2006.

[12] X.F. Wang, D.S. Huang and H. Xu, ”An efficient local ChanVese modelfor image segmentation”, Pattern Recognition, Vol. 43, No. 3, pp.603-618, March, 2010.

[13] S.I. Setzer and G.A. Steidl, ”Variational methods with higher orderderivatives in image processing”, Approximation, Vol. 12, pp.360-86,January, 2008.

[14] X. Zhu, Z. Chen, C. Tang, Q. Mi and X. Yan, ”Application of twooriented partial differential equation filtering models on speckle fringeswith poor quality and their numerically fast algorithms”, Applied Optics,Vol. 52, No.9, pp.814-23, March, 2013.

[15] I. Csisz and G. Tusnady, ”Information geometry and alternating mini-mization procedures”, Statistics and Decisions, January, 1984.

[16] H. Ali, G. Murtaza and N. Badshah, ”Covariance Based Image Se-lective Segmentation Model”,Journal of Information & ComunicationTechnology, Vol.4, No.2, pp. 11-19, 2010.

[17] X. Cai, R. Chan, and T. Zeng. ”A two-stage image segmentation methodusing a convex variant of the mumfordshah model and thresholding”.SIAM Journal on Imaging Sciences, Vol.6, NO.1, pp. 368390, 2013.

[18] X. Cai and G. Steidl. ”Multiclass segmentation by iterated rof thresh-olding”. In International Workshop on Energy Minimization MethodsComputer Vision and Pattern Recognition, pages 237250. Springer,2013

[19] S. Boyd, N. Parikh, E. Chu, B. eleato, and J. Eckstein. ”Distributedoptimization and statistical learning via the alternating direction methodof multipliers”. Foundations and Trendsr in Machine Learning, Vol. 3,No. 1, pp. 1122, 2011

[20] Y.He, M. Y.Hussaini, J. Ma, B. Shafei, and G. Steidl. ”A new fuzzyc-means method with total variation regularization for segmentation ofimages with noisy and incomplete data”, Pattern Recognition, Vol. 45,No. 9, pp.34633471, 2012

[21] J. Yuan, E. Bae, X. Tai, and Y. Boykov. ”A continuous maxflowapproach to potts model”. Computer Vision, pages 379392, 2010.

[22] T. Pock, A. Chambolle, D. Cremers, and H. Bischof. ”A convexrelaxation approach for computing minimal partitions”. In ComputerVision and Pattern Recognition. pages 810817. 2009.

[23] A. Chambolle and T. Pock. ”A first-order primal-dual algorithm for con-vex problems with applications to imaging”. Journal of mathematicalimaging and vision, Vol.40, No.1, pp.120145, 2011.


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