当前位置：文档库 › A variational formulation for physical noised image segmentation(物理噪声图像分割的变分公式)

A variational formulation for physical noised image segmentation(物理噪声图像分割的变分公式)

Appl.Math.J.Chinese Univ.

2015,30(1):77-92

A variational formulation for physical noised image

segmentation

LOU Qiong1PENG Jia-lin2KONG De-xing3,?Abstract.Image segmentation is a hot topic in image science.In this paper we present a

new variational segmentation model based on the theory of Mumford-Shah model.The aim of our model is to divide noised image,according to a certain criterion,into homogeneous and smooth regions that should correspond to structural units in the scene or objects of interest.The proposed region-based model uses total variation as a regularization term,and di?erent?delity term can be used for image segmentation in the cases of physical noise,such as Gaussian,Poisson and multiplicative speckle noise.Our model consists of?ve weighted terms,two of them are responsible for image denoising based on?delity term and total variation term,the others assure that the three conditions of adherence to the data,smoothing,and discontinuity detection are met at once.We also develop a primal-dual hybrid gradient algorithm for our model.Numerical results on various synthetic and real images are provided to compare our method with others, these results show that our proposed model and algorithms are e?ective.

§1Introduction

Image segmentation is a fundamental problem in image processing.The main goal of seg-mentation is to recover an object of interest from a given dataset by partitioning it into disjoint compartments.As we known,edge detection and image smoothing are distinct techniques de-voted to di?erent goals.Basically,the former commonly concerns the location of sharp changes in image brightness or color intensity,whereas the latter tackles the need to reduce noise from the data.Classical edge detection approaches are based on local di?erential properties of an edge such as the?rst and the second derivatives of the image.In general,edge-based([3,9,12]), region-based([5,14,24])and edge-region-based([13,26,29])methods are common and useful in

Received:2013-09-15.

MR Subject Classi?cation:65K10,68U10,49M30.

Keywords:image segmentation,variational method,image denoising,primal-dual hybrid gradient algorithm, non-Gaussian noise.

Digital Object Identi?er(DOI):10.1007/s11766-015-3217-7.

Supported in part by the NNSF of China(11301129,11271323,91330105,11326033)and the Zhejiang Provincial Natural Science Foundation of China(LQ13A010025,LZ13A010002).

?Corresponding author.

Appl.Math.J.Chinese Univ.Vol.30,No.1

image segmentation.Image smoothing is commonly achieved through the use of weighting functions known as smoothing ?lters.Types of ?lters based on both linear and non-linear methods are available,such as moving average ?lter and the Savitzky-Goaly ?lter ([20])are two of the most commonly used ?lters.Since many kinds of ?lters are available,the smoothing ?lter that best minimizes the disturbances of the statistical properties of the original data should be chosen ([25]).

1.1The Mumford and Shah functional

This subsection presents a brief overview of main theoretical aspects of Mumford-Shah model which is necessary for its analytical treatment.Let Ω?R 2be a bounded open connected set,Γbe a compact curve in Ω,and f :Ω→R be a given image.Without loss of generality,the intensity level of image f can be considered as a function f :Ω→[0,1],and hence f ∈L ∞(Ω).

[14]provided Mumford-Shah (MS)model to solve the segmentation problem by minimizing the following energy:E MS (u,Γ)=12 Ω(f ?u )2d x +μ2 Ω\Γ

|?u |2d x +βLength (Γ),(1)where μand βare positive real parameters that control the relative strength of the three terms of MS,and u :Ω→R be a continuous or even di?erentiable in Ω\Γbut may be discontinuous across Γ.The function u represents a smooth approximation of the given image f ,whereas the set Γis a set of curves coming from the sharp discontinuities of f .

As we know that the ?rst two terms in Eq.1represent two surface energies whereas the third term represents a line energy concentrated on the unknown discontinuity set Γ.Later on,the set Γwill be replaced by the discontinuity set of the smooth function u that smooths the data.This new set remains unknown and it is not possible to assume this set to be composed of only closed boundaries.This makes the MS problem a free discontinuity and not a free boundary problem and explains the use of the “free”and of the noun “discontinuity”in the de?nition of this kind of variational problem ([1,18]).Exactly,the ?delity term Ω(f ?u )2d x forcing the solution u to be as close as possible to the given image f ;the smoothing term Ω\Γ|?u |2d x forcing the solution u to be as smooth as possible everywhere except along the image discontinuities;the geometric term Length (Γ)forcing the total length of the edges to be as short as possible.Thus minimizing the function equation (1)implies the minimization of a weighted combination,where f has sharp changes a discontinuity is introduced if this is more convenient than allowing u to have a large gradient,so to obtain a smaller value of E MS (u,Γ).

Over the years,people have tried hard to simplify the MS model (1).When we restrict ?u ≡0on Ω\Γ,(1)becomes a piecewise constant Mumford-Shah model.Recently,[17]pro-posed a convex relaxation approach to solve MS.The two-phase piecewise constant MS image segmentation model [10,14]is one of the most important image segmentation model,and has been studied extensively in the last two decades.The convex segmentation model based on MS

[2]can be regarded as an image restoration model.[2]uni?es the image processing works of image segmentation and image restoration.

LOU Qiong,et al.A variational formulation for physical noised image segmentation 79

1.2Total variation regularization

In [21],a region-based variational segmentation framework to segment images incorporating physical noise was proposed.This paper discuss the cases of Gaussian,Poisson and multiplica-tive speckle noise intensively.When come to the experiment of synthetic data with anatomical structures of di?erent size,they choose parameter of the length of edge according to di?erent criterions.Actually,generally acquired images are subject to di?erent kinds of physical noise.While there are several algorithms that can be considered state of the art for a particular noise model,typically the adaptation of such specialized algorithms to handle noise models has been proven to be either severely di?cult or just plain impossible ([18]).It is in this regard that regularization methods stand atop due to their ?exibility to use any given noise model,we focus on the total variation (TV)regularization method ([19]).Though the original TV regulariza-tion method targeted image denoising under Gaussian noise,it has evolved into a more general technique for inverse problems ([4])while retaining its edge preserving property ([22]).

The algorithms based on total variation regularization for images corrupted with di?erent noise models are commonly used.The TV regularized functional always has the form E T V (u )=

Ω?log p (f |u )d x +λ

Ω|?u |d x,(2)where Ω?log p (f |u )d x is the data ?delity term which can be derived via a maximum a-

posteriori probability (MAP)estimation [21],it depends on the noise model.The scalar λis the regularization parameter. Ω|?u |d x represents the total variation of solution u ,written as ?u 1or |u |T V .Throughout this paper we will consider that the discrete version of ?u 1as n ∈S

((?x u n )2+(?y u n )2) 1,where n ∈S ={1}in grayscale images case or n ∈S ={1,2,3}

in color images case.We should note that S can represent an arbitrary number of channels,?x and ?y denote the horizontal and vertical discrete derivative operators respectively.

As we known that TV regularization based models have piecewise constant e?ect which also known as staircase e?ect [15].Considering the theory of MS and TV,we seek a new variational model which can force piecewise smooth image to piecewise constant data,the minimization of proposed functional obtain an approximation of given noisy image and a set of curves coming from the sharp discontinuities of image.ideal segmentation when come to the case of inhomogeneous image with high Gaussian noise.In [21],the most frequently used squared H 1?seminorm regularization and the Fisher information were investigated,which allow to incorporate a-priori information of possible solutions in the proposed segmentation framework.We study the recent advances of variational model [16]and region-based model [2].We joint image segmentation and restoration together in this paper,and present a new variational model for image segmentation.In this model,we use the total variation term as a regularization.As for the ?delity term,we introduce approximate image via a maximum a-posteriori probability estimation.As for the numerical realization of our model,we develop an e?cient and fast primal-dual hybrid gradient (PDHG)method [30,31].This descent type algorithm alternates between primal and dual formulations and exploits the information from both primal and dual variables.The performance of the proposed model is illustrated by experimental results on

80Appl.Math.J.Chinese Univ.Vol.30,No.1

synthetic and real data.

The paper is organized as follows.In Section2,we describe our variational segmentation model via constrained minimization problem for di?erent physical noise,and solve the con-strained problem with the quadratic penalty method.The PDHG algorithm corresponding to our model is introduced in Section3.In Section4,we give some experimental results and corresponding discussions.Finally,a brief conclusion is presented in Section5.

§2Variational segmentation model

Our variational segmentation model uses tools from statistics in cases of di?erent physical noise conditions.

2.1Basic operators

LetΩ?R2is the image domain,and f is the given image which needs to be segmented. P2(Ω)is an partition of image.

For simplicity,we restrict our model to a two-phase segmentation problem,and then we introduce the following notations.First,we assume that we want to segment the image domain Ωinto a background and a target subregion,which we denote withΩb andΩt,respectively.

P2(Ω)∈{(Ωb,Ωt)}.(3) Subsequently,we introduce an indicator functionχin order to represent both subregions in the

form that

χ(x)=

1,if x∈Ωt,

0,else.

(4)

Finally,we use the well-known relation between the Hausdor?measure and the total variation of an indicator function,which implies that

H d?1(Γ)=|χ|T V(Ω),(5) where|·|T V(Ω)denotes the total variation of a function inΩ.

2.2Statistical formulation for segmentation

A majority of works on image segmentation implicitly assume the given image is biased by additive Gaussian noise.Since this assumption is not suitable for a variety of problems,in this paper,we discuss image segmentation in the cases of Gaussian,Poisson and multiplicative speckle noise.

Following[6],the partition P2(Ω)ofΩcan be computed via a maximum a-posteriori prob-ability estimation.Since we also want to restore an approximation u of the original noise free image,we have to maximize a-posterior probability density as discussed below.In order to give precise statements on probability densities,we use discrete formulation with N representing the number of pixels.In our discussion,we have to maximize an a-posteriori probability density

LOU Qiong,et al.A variational formulation for physical noised image segmentation81

p(u N,P N i(Ω)|f N),which can be written as

p(u N,P N i(Ω)|f N)∝p(P N i(Ω))p(u N|P N i(Ω))p(f N|u N,P N i(Ω)).(6) Di?erent from[21],we consider the following energy functional

E(u,u b,u t,Ωb,Ωt)=

i (

Ωi

?log p i(f|u)d x+ξi R i(u,u i))+βH d?1(Γ),(7)

where i∈{b,t},R i(u,u i)is a discretized version of a non-negative energy functional of u and u i,ξi is a positive parameter,d=2throughout our experiments.u in(7)is the approximation of the given image f,u b and u t are the smooth approximation of u in the regionΩb andΩt respectively.It is clear that we consider u in the?delity term rather than u i in[21],this may help to get a better approximation of f.

For the sake of simplicity and since we are only interested in the formulation in(7),we will write p i(f(x)|u(x))which has to be interpreted as the value of pixels in the sense of a correct modeling in the following sections.

2.2.1Additive Gaussian noise

A majority of works on image segmentation implicitly assume the given image to be biased by additive Gaussian noise.Since

p(f(x)|u(x))∝e?12σ2(u(x)?f(x))2,(8) thus,this Gaussian noise model leads to the following negative log-likelihood function in the energy functional E in(7),

?log p(f(x)|u(x))=1

2σ2(u(x)?f(x))2.(9)

It’s easy to?nd that the?delity term of MS model is derived from(9).Since additive Gaussian noise is the most common form of noise,segmentation methods,such as MS model,are successful on a large class of images.The factorσ2in(9)is neglected in the course of this work because it can be scaled by the regularization parametersξi andβin the energy functional(7).

2.2.2Poisson noise

Poisson noise is signal-dependent and appears in a wide class of real-life applications,e.g., in positron emission tomography[23,27]and astronomical images[11].For Poisson noise one indeed counts natural numbers as data,so also the image in the discrete modeling needs to be quantized.The density function of Poisson is

p(f N=k|u N=λ)=λk

e?λ.(10)

Thus,this model leads to the following negative log-likelihood function for the energy functional E in(7),

?log p(f(x)|u(x))=u(x)?f(x)log u(x)?log f!.(11) Since f denotes the given image,then the term log f!of(11)is a constant.Thus log f!can be neglected in the course of our work.

Appl.Math.J.Chinese Univ.Vol.30,No.1

2.2.3Multiplicative speckle noise Image biased by multiplicative speckle noise has the form f =u +√uη,where ηis a

Gaussian-distributed random variable with mean 0and variance σ2.Conditional model

p (f (x )|u (x ))∝(u (x ))?12e ?12σ2(u (x )?f (x ))2

u (x )(12)

leads to the following negative log-likelihood function in the energy functional E in (7),given by

?log p (f (x )|u (x ))=(u (x )?f (x ))22σ2u (x )+12

log u (x ).(13)As for additive Gaussian noise,we can multiply the right-hand side of (13)by 2σ2and incor-porate this scaling factor in the regularization parameter ξi and βin (7).

2.2.4TV-based regularization

TV-based regularization was originally introduced in image processing by Rudin,Osher and Fatemi in their pioneering work [19]for denoising.A signi?cant advantages of TV regularization is that it preserves edges in the solution.Since if we choose u T V as the term R i (u i )in Ωi ,then the lack of smoothness in the TV term makes the solution of (7)di?cult.In recent years,many algorithms were proposed to e?ciently solve the TV-based image processing problem,such as the split Bregman method [8],alternating direction method of multipliers (ADMM)

[7],Bregman operator splitting (BOS)method [28]and PDHG method [30].We apply PDHG algorithm for the numerical realization of our proposed model.

2.3The proposed model

In this section,we propose a new variational model for image segmentation.As we mentioned above,u in (7)is the approximation of given image f ,u b and u t are the smooth approximation of u in the region Ωb and Ωt respectively,we can formulate the energy function (7)as the following constrained minimization problem

min {u,u b ,u t ,Ωb ,Ωt }E (u,u b ,u t ,Ωb ,Ωt )

subject to ?????????u (x )=u b ,if

x ∈Ωb ,u (x )=u t ,if

x ∈Ωt ,?u b =0,if

x ∈Ωb ,?u t =0,if x ∈Ωt .(14)

The constrained minimization problem (14)is treated with a quadratic penalty min {u,u b ,u t ,Ωb ,Ωt } Ω?log p (f (x )|u (x ))d x + i λi Ωi |?(u ?u i )|d x + i αi 2 Ωi (u ?u i )2d x + i γi 2 Ωi |?u i |2d x +βH 1(Γ).

(15)Here the total variation term Ωi |?(u ?u i )|d x we used is a regularization somewhat like

re?ecting the sparsity of u ?u i .Then together with Eq.(4)and Eq.(5),the proposed model

LOU Qiong,et al.A variational formulation for physical noised image segmentation83

can be written as

min

{u,u b,u t,χ}

E(u,u b,u t,χ)

Ω?log p(f(x)|u(x))+λb(1?χ)|?(u?u b)|+λtχ|?(u?u t)|+αb

(1?χ)(u?u b)2

+αt

χ(u?u t)2+

γb

(1?χ)|?u b|2+

γt

χ|?u t|2d x+β χ T V.(16)

Remark.The model(16)consists of?ve weighted terms.The?rst three terms are responsible for image denoising,u is a piecewise constant approximation of f([19])and the?delity term is di?erent according to the noise type.The others assure the di?erent noise conditions of adherence to the data,smoothing and discontinuity detection are met at once,which help to get a proper segmentation.In fact that u i is a piecewise smooth approximation of u inΩi. Exactly,we force piecewise smooth data u i to be as close as possible to piecewise constant data u.The approximation help to smooth u and get relatively ideal segmentation result from the trade-o?process between segmentation and approximation.

§3Numerical realization

As we know that the total variation of u can also be de?ned as

T VΩ(u)=

Ω|?u|d x=max

p∈C10, p ≤1

?u·p=max

p ≤1

?u div p d x.(17)

Then the primal-dual formulation of the proposed model(16)is given by

min {u,u b,u t,χ}

max

q t ≤1

max

q b ≤1

max

p ≤1

E(u,u b,u t,χ,p,q b,q t)

Ω?log p(f|u)+λb(1?χ)q b?(u?u b)+λtχq t?(u?u t)+αb

(1?χ)(u?u b)2

+αt

χ(u?u t)2+

γb

(1?χ)(?u b)2+

γt

χ(?u t)2d x+β

p?χd x.(18)

From min-max problem(18),we can split it into separate subproblems as following problems,

(χk+1,p k+1)∈arg min

χmax

{

λb(1?χ)q k b?(u k?u k b)+λtχq k t?(u k?u k t)

+αb

(1?χ)|u k?u k b|2+

αt

χ|u k?u k t|2

+γb

(1?χ)(?u k b)2+

γt

χ(?u k t)2+βp?χd x}.(19)

(u k+1

b ,q k+1

)∈arg min

u b

max

q b

{

λb(1?χk+1)q b?(u k?u b)

αb

(1?χk+1)(u k?u b)2+

γb

(1?χk+1)|?u b|2d x}.(20)

(u k+1

t,q t)∈arg min

u t

max

q t

{

λtχk+1q t?(u k?u t)

αt

χk+1(u k?u t)2+

γt

χk+1|?u t|2d x}.(21)

84Appl.Math.J.Chinese Univ.Vol.30,No.1

u k +1∈arg min u { Ω?log p (f |u )+λb (1?χk +1)q k +1b ?(u ?u k +1b )+λt χk +1q k +1t ?(u ?u k +1t )+αb 2(1?χk +1)(u ?u k +1b )2+αt 2χk +1(u ?u k +1t )2d x }.(22)Thus the speci?c PDHG algorithm corresponding to (18)can be separated into four subal-

gorithms as follows.

We now give the PDHG algorithm that we use for subproblem (19).The algorithm is based on the following updates for the primal and dual variables:?????????????????p k +1=arg max p ∈X β Ωp ?χk d x ?12τp p ?p k 22,χk +1=arg min χ∈Φ Ωλb (1?χ)q k b ?(u k ?u k b )+λt χq k t ?(u k ?u k t )+αb 2(1?χ)|u k ?u k b |2+αt 2χ|u k ?u k t |2+γb 2(1?χ)(?u k b )2+γt 2χ(?u k t )2+βp k +1?χd x +12θχ

χ?χk 22,(23)where X ={p ∈R N ×N , p ≤1},Φ={χ∈BV (Ω;{0,1})},θχand τp represent the primal and dual step sizes corresponding to the regularization terms in (23).Due to the simple form for the quadratic term in (23),the iteration takes the form given in Algorithm 1.

Algorithm 1PDHG for subproblem (23).

p k +1=P X (p k +βτk p ?χk ),P X (p )=p max { p ,1}.

χk +1=P Φ(χk ?θk χ(?λb q b ?(u k ?u k b )+λt q t ?(u k ?u k t )?

αb 2 u k ?u k b 22+αt 2

u k ?u k t 22?γb 2 ?u k b 2+γt 2 ?u k t 2?βdiv p k +1)),Φ={χ∈BV (Ω;{0,1})}.

In Algorithm 1,the step size is updated by the rule τk p =0.2+0.08k ,θk χ=(0.5?

515+k )/τk p

for improved e?ciency ([30]).Similarly,the PDHG algorithm that we use for subproblem (20)is based on the following updates for the primal and dual variables:?????????q k +1b =arg max q b ∈X Ωλb (1?χk +1)q k b ?(u k ?u k b )?12τq b q b ?q k b 22,u k +1b =arg min u b ∈U Ωλb (1?χk +1)q k +1b ?(u k ?u k b )+αb 2(1?χk +1)(u k ?u k b )2+γb 2(1?χk +1)|?u k b |2d x +12θu b

u b ?u k b 22,(24)where U ={u ∈R ,0≤u ≤1},θu b and τq b represent the primal and dual step sizes correspond-ing to the regularization terms in (24).Due to the simple form for the quadratic term in (24),the iteration takes the form given in Algorithm 2,where U ={u :u ∈R N ×N ,0≤u (x )≤1}.

In Algorithm 2,the step size is updated by the rule τk q b =0.2+0.08k ,θk u b

=(0.5?515+k )/τk q b for improved e?ciency ([30]).

LOU Qiong,et al.A variational formulation for physical noised image segmentation 85Algorithm 2PDHG for subproblem (24).

q k +1b =P X (q k b +λb τk q b (1?χk +1)?(u k ?u k b )),P X (p )=p max { p ,1}.

u k +1b =P U (u k b ?θk u b (?λb (1?χk +1)div q k +1b +αb (1?χk +1)(u k b ?u k )?γb (1?χk +1)div(?u k b ),P U (u )=max {0,min {u,1}}.

Thus by symmetry,the algorithm of (21)subproblem is as follows:where θu t and τq t represent Algorithm 3PDHG for subproblem (21).

q k +1t =P X (q k t +λt τk q t χk +1?(u k ?u k t )),P X (p )=p max { p ,1}.

u k +1t =P U (u k t ?θk u t (?λt χk +1div q k +1t +αt χk +1(u k t ?u k )?γt χk +1div(?u k t ),

P U (u )=max {0,min {u,1}}.

the primal and dual step sizes.In Algorithm 3,the step size is updated by the rule τk q t =0.2+0.08k ,θk u t =(0.5?515+k )/τk q t for improved e?ciency ([30]).

The gradient descent of subproblem (22)iteration takes the form given in Algorithm 4as follows.θu is the step size of iteration,F u (u,f )takes di?erent form based on original noised Algorithm 4Gradient descent for subproblem (22).

u k +1=P U (u k ?θk u (F u (u k ,f )?λb (1?χk +1)div q k +1b ?λt χk +1div q k +1t +αb (1?χk +1)(u k

?u k +1b

)+αt χk +1(u k ?u k +1t ))),P U (u )=max {0,min {u,1}}.

image,here we give its expression in the cases of Gaussian,Poisson and multiplicative speckle noise as shown in (25)F u (u,f )=?????u ?f,

Guassian ,1?f u ,Poisson ,1?f 2u 2+σ2u ,

multiplicative .(25)σis the parameter of multiplicative speckle noise,we set it 0.1in our experiments.

§4Experimental results and discussion

In this section,we test our variational model,and compared with the the models of MS [14]and AM [21].We report on experiments for ?ve test problems in image segmentation.The noisy images are generated by adding Gaussian,multiplicative speckle,and Poisson noise to the clean images using the MATLAB function imnoise .In order to implement the study,we

set αi =10,β=2,λk i =0.001+0.0001k ,γk i =0.001+0.0001k throughout the experiments.

86Appl.Math.J.Chinese Univ.Vol.30,No.1 The initialization is set as(26):

????????????

???????????χ0=χ0,

u0=f,

u0b=(1?χ0)u0,

u0t=χ0u0,

p0=0,

q0b=0,

q0t=0,

(26)

whereχ0is given automatically or arti?cially.

All data sets are normalized such that the intensities of reference images have range[0,1]. In our experiments,all algorithms are implemented in MATLAB,Version R2009b.

4.1Experimental results on synthetic data

In this subsection we validate the proposed variational segmentation framework on synthetic data.In order to segment the images discussed below,we use the algorithms proposed,for which 85iteration steps are su?cient in most cases.To terminate the inner iteration loops discussed in Sect.3,and we choose =10?6in(27).

u k+1?u k 22

u k+1 22< .(27) In Fig.1,we apply proposed Gaussian model to a blocky and noisy synthetic image Fig.1(a) of size190×190pixels.In Fig.1(b),we show the segmentation based on proposed Gaussian model,while Fig.1(c)shows the original image line with the contour of Fig.1(b).Fig.1(d)is the homogeneous approximation of Fig.1(a)based on its segmenting curve.Fig.1(e)and Fig.1(f) show u b and u t respectively,and Fig.1(g)is the di?erence between u and(1?χ)u b+χu t.It’s clear that most region of Fig.1(g)is almost dark,that means the di?erence is almost close to 0and the u i is the approximation of u in responding regions.

We investigate the experimental results on image with inhomogeneities.For this purpose we use an image of size160×160pixels with a simple object structure illustrated in Fig.2(a). In this image we put inhomogeneities inΩb andΩt covering the full range of intensities,such that the two regions have the same mean value.The challenge of this data is the grayscale values is the same in background and target,also with strong intensity changes at the border of the object structure.We can see that our model can get ideal segmentation results.

When Fig.2(a)biased by Poisson noise as shown in Fig.3(a),we also got better segmentation result than contrast models.

To evaluate the importance of a correct noise model in automated image segmentation we investigated images perturbed by physical noise forms described in Sect. 2.2.For the sake of simplicity we performed this experiment on piecewise constant images.We chose the objects to be segmented with respect to typical segmentation tasks from biomedical imaging [21].Fig.4(a)shows experimental data and its segmentation in Fig.4(b).Fig.4(c)shows the di?erence between Fig.4(a)and Fig.4(b)in order to see the di?erence intuitively.

LOU Qiong,et al.A variational formulation for physical noised image segmentation87

(a)synthetic data(b)proposed(c)segmentation(d)approximation

(e)u b(f)u t(g)di?erence

Figure1:Test the proposed Gaussian model on a blocky and noisy image

(a)synthetic data(b)ground truth(c)MS(d)proposed

Figure2:Comparison of proposed model with Mumford-Shah segmentation and ground truth on inhomogeneous data

(a)data perturbed by

Poisson noise

(b)MS(c)AM(d)proposed

Figure3:Comparison of segmentation for synthetic inhomogeneous data biased by Poisson noise.

First,we perturbed the data with additive Gaussian noise as illustrated in Fig.5(a).In Fig.5(b)the data?delity term for additive Gaussian noise produces a satisfying segmentation

88Appl.Math.J.Chinese Univ.Vol.30,No.1

(a)synthetic data(b)ground truth(c)di?erence

Figure4:Synthetic data with anatomical structures of di?erent size.

(b)proposed Gauss(c)proposed Poisson(d)proposed speckle

(a)data perturbed by

Gaussian noise

Figure5:Comparison of segmentation for synthetic data with anatomical structures of di?erent size biased by additive Gaussian noise with variance0.007.

(b)proposed Poisson(c)proposed Gaussian(d)proposed speckle

(a)data perturbed by

Poisson noise

Figure6:Comparison of segmentation of synthetic data with anatomical structures of di?erent size biased by Poisson noise.

result.The results of Poisson and multiplicative speckle cases are illustrated in Figs.5(d),5(c), respectively.

Next,we perturbed the synthetic data with Poisson noise as presented in Fig.6(a).For this image we state that the Poisson data?delity term is an appropriate choice as can be seen in Fig.6(b).The results of Gaussian and multiplicative speckle cases are tested and illustrated in Fig.6(c)and Fig.6(d),respectively.

Finally,we investigated the case of data biased by multiplicative speckle noise as shown in Fig.7(a).The segmentation result use speckle data?delity term is presented in Fig.7(b).The segmentation results for the data?delity term of Gaussian and Poisson cases are presented in

LOU Qiong,et al.A variational formulation for physical noised image segmentation 89

(a)data perturbed by speckle noise

(b)proposed speckle (c)proposed Gaussian (d)proposed Poisson

Figure 7:Comparison of segmentation of synthetic data with anatomical structures of di?erent size by adding multiplicative speckle noise with variance 0.1.

(a)Cameraman (b)data perturbed by

Gaussian noise (c)AM (d)proposed Gauss

Figure 8:Comparison of segmentation of Cameraman by adding additive Gaussian noise with variance 0.01.

Figs.7(c),7(d).

4.2Experimental results on real data

In this subsection we report the results of the proposed algorithm for solving segmentation problem,and compare our algorithm with AM.

In the ?rst experiment,we consider the segmentation problem of the Cameraman image Fig.8(b)which is perturbed Fig.8(a)by additive Gaussian noise.The results generated by proposed algorithm and the AM are shown in Figs.8.The choice of the parameters follows the same rule used before.

In the second experiment,we add mammogram in Fig.9(a)with Poisson noise in Fig.9(b).The original mammogram shows cyst in the breast tissue,the aim of our segmentation is getting the edge of the cyst.It is clearly that the segmentation result for the proposed model in Fig.9(d)is better than the result for AM which is illustrated in Fig.9(c),the segmentation result in the region of interest of proposed Poisson model is shown as Fig.9(e).

From the experimental results,we can easily ?nd that our model is more robust to the noise than the AM model.In the present study,we force piecewise smooth image to piecewise constant image.The inner loops of our proposed algorithm is a trade-o?between segmentation and approximation.In conclusion,we emphasize that the incorporation of physical noise modeling

90Appl.Math.J.Chinese Univ.Vol.30,No.1

(a)cyst (b)data perturbed by

Poisson noise (c)AM (d)proposed Poisson (e)segmentation con-tour

Figure 9:Comparison of segmentation of cyst by adding Poisson noise.

for given data and trade-o?between image segmentation and denoise have a signi?cant impact on segmentation results and lead to improved accuracy in applications dealing with physical noise.

The number of iterations and CPU times (in seconds)for the experimental results of our model and AM in Fig.3,Fig.8and Fig.9are shown in Table.1.

Table 1:Comparisons on the computational e?ort between proposed model and AM.

Images (size)ball(160×160)Cameraman(128×128)cyst(140×140)Iterations Time (s)Iterations Time (s)Iterations Time (s)

proposed 120.4990.2676 3.04AM 5 3.9911 5.711612.62

§5Conclusion

In this paper,we propose a variational model for physical noise image segmentation.For the numerical realization of our model,we develop its PDHG algorithm.In particular,we implement synthetic data and real image with physical noise,especially,Gaussian noise,Poisson noise,and multiplicative speckle noise.We use the presented model for automated image segmentation.In our model,the ?delity term is always depend on the type of noise,the total variation of the di?erence between smoothed image and image in Ωb and Ωt can help getting better denoised image which is useful for segmentation.We utilize 2-norm of the di?erence between smoothed image and images in background or target subregion,in order to force u i to be as close as possible to u in Ωi .Experimental results on synthetic and real images show the necessity to adapt an algorithm to present conditions.

We plan to use wavelet frame instead of total variation term between piecewise smooth images and piecewise constant image.Exactly,this has theoretical fundament in [2].

LOU Qiong,et al.A variational formulation for physical noised image segmentation91

References

[1]L Ambrosio,N Fusco,D Pallara.Functions of Bounded Variation and Free Discontinuity Prob-

lems,Clarendon Press Oxford,2000.

[2]X Cai,R Chan,T Zeng.Image segmentation by convex approximation of the Mumford-Shah

model,2012,preprint.

[3]V Caselles,R Kimmel,G Sapiro.Geodesic active contours,Int J Comput Vision,1997,22:61-79.

[4]T Chan,S Esedoglu,F Park,A Yip.Recent developments in total variation image restoration,In:

Mathematical Models in Computer Vision,Springer Verlag,2005.

[5]T F Chan,L A Vese.Active contours without edges,IEEE Trans Image Process,2001,10:266-277.

[6]D Cremers,M Rousson,R Deriche.A review of statistical approaches to level set segmentation:

integrating color,texture,motion and shape,Int J Comput Vision,2007,72:195-215.

[7]D Gabay,B Mercier.A dual algorithm for the solution of nonlinear variational problems via?nite

element approximation,Comput Math Appl,1976,2:17-40.

[8]T Goldstein,S Osher.The split Bregman method for L1-regularized problems,SIAM J Imaging

Sci,2009,2:323-343.

[9]M Kass,A Witkin,D Terzopoulos.Snakes:active contour models,Int J Comput Vision,1988,1:

321-331.

[10]D KrishnanQ V Pham,A M Yip.A primal-dual active-set algorithm for bilaterally constrained to-

tal variation deblurring and piecewise constant Mumford-Shah segmention problems,Adv Comput Math,2009,31:237-266.

[11]H Lant′e ri,C Theys.Restoration of astrophysical images:the case of Poisson data with additive

Gaussian noise,EURASIP J Adv Signal Process,2005,15:2500-2513.

[12]C Li,C Xu,C Gui,M D Fox.Level set evolution without re-initialization:a new variational

formulation,In:Proc of IEEE Conference on Computer Vision and Pattern Recognition,2005, 1:430-436.

[13]M Mueller,K Segl,H Kaufmann.Edge-and region-based segmentation technique for the extraction

of large,man-made objects in high-resolution satellite imagery,Pattern Recogn,2004,37:1619-1628.

[14]D Mumford,J Shah.Optimal approximations by piecewise smooth functions and associated vari-

atioanl problems,Comm Pure Appl Math,1989,42:577-685.

[15]S Osher,R Fedkiw.Level Set Methods and Dynamic Implicit Surfaces,AMS Vol153,Springer,

2003.

[16]J L Peng,F F Dong,D X Kong.Recent advances of variational model in medical imaging and

applications to computer aided surgery,Appl Math J Chinese Univ Ser B,2012,27:379-411.

[17]T Pock,D Cremers,A Chambolle,H Bischof.A convex relaxation approach for computing mini-

mal partitions,In:Proc of CVPR,2009:810-817.

[18]P Rodr′?guez.Total variation regularization algorithms for images corrupted with di?erent noise

models:a review,J Electrical Comput Eng,vol2013,2013,Article ID217021.

92Appl.Math.J.Chinese Univ.Vol.30,No.1

[19]L Rudin,S Osher,E Fatemi.Nonlinear total variation based noise removal algorithms,Phys D,

1992,60:259-268.

[20]A Savitzky,M J Golay.Smoothing and di?erentiation of data by simpli?ed least squares proce-

dures,Anal Chem,1964,36(8):1627-1639.

[21]A Sawatzky,D Tenbrinck,X Jiang,M Burger.A variational framework for region-based segmen-

tation incorporating physical noise models,J Math Imaging Vision,2012:1-31.

[22]D Strong,T Chan.Edge-preserving and scale-dependent properties of total variation regulariza-

tion,Inverse Problems,2003,19(6):S165.

[23]Y Vardi,L A Shepp,L Kaufman.A statistical model for positron emission tomography,J Amer

Statist Assoc,1985,80:8-20.

[24]L A Vese,T F Chan.A multiphase level set framework for image segmentation using the Mumford

and Shah model,Int J Comput Vision,2002,50:271-293.

[25]A Vitti.The Mumford–Shah variational model for image segmentation:an overview of the theory,

implementation and use,ISPRS J Photogramm,2012,69:50-64.

[26]M A Wani,B G Batchelor.Edge-region-based segmentation of range images,IEEE T Pattern

Anal,1994,16:314-319.

[27]M N Wernick,J N Aarsvold.Emission Tomography:The Fundamentals of PET and SPECT,

Access Online via Elsevier,2004.

[28]X Zhang,M Burger,S Osher.Bregmanized nonlocal regularization for deconvolution and sparse

reconstruction,SIAM J Imaging Sci,2010,3:253-276.

[29]Y Zhang,B J Matuszewski,L K Shark,C J Moore.Medical image segmentation using new hybrid

level-set method,In:BioMedical Visualization,2008,71-76.

[30]M Zhu,T Chan.An e?cient primal-dual hybrid gradient algorithm for total variation image

restoration,UCLA CAM Report08-34,2008.

[31]M Zhu,S J Wright,T F Chan.Duality-based algorithms for total variation image restoration,

Comput Optim Appl,2010,47:377-400.

1Center of Mathematical Sciences,Zhejiang University,Hangzhou310027,China.

2The School of Computer Science and Technology,Huaqiao University,Xiamen361021,China.

3Department of Mathematics,Zhejiang University,Hangzhou310027,China.

Email:dkong@https://www.wendangku.net/doc/2e1950662.html,

英语选修六课文翻译Unit5 The power of nature An exciting job的课文原文和翻译

AN EXCITING JOB I have the greatest job in the world. I travel to unusual places and work alongside people from all over the world. Sometimes working outdoors, sometimes in an office, sometimes using scientific equipment and sometimes meeting local people and tourists, I am never bored. Although my job is occasionally dangerous, I don't mind because danger excites me and makes me feel alive. However, the most important thing about my job is that I help protect ordinary people from one of the most powerful forces on earth - the volcano. I was appointed as a volcanologist working for the Hawaiian V olcano Observatory (HVO) twenty years ago. My job is collecting information for a database about Mount Kilauea, which is one of the most active volcanoes in Hawaii. Having collected and evaluated the information, I help other scientists to predict where lava from the volcano will flow next and how fast. Our work has saved many lives because people in the path of the lava can be warned to leave their houses. Unfortunately, we cannot move their homes out of the way, and many houses have been covered with lava or burned to the ground. When boiling rock erupts from a volcano and crashes back to earth, it causes less damage than you might imagine. This is because no one lives near the top of Mount Kilauea, where the rocks fall. The lava that flows slowly like a wave down the mountain causes far more damage because it

八年级下册3a课文

八年级下学期全部长篇课文 Unit 1 3a P6 In ten years , I think I'll be a reporter . I'll live in Shanghai, because I went to Shanfhai last year and fell in love with it. I think it's really a beautiful city . As a reporter, I think I will meet lots of interesting people. I think I'll live in an apartment with my best friends, because I don' like living alone. I'll have pets. I can't have an pets now because my mother hates them, and our apartment is too small . So in ten yers I'll have mny different pets. I might even keep a pet parrot!I'll probably go skating and swimming every day. During the week I'll look smart, and probably will wear a suit. On the weekend , I'll be able to dress more casully. I think I'll go to Hong Kong vacation , and one day I might even visit Australia. P8 Do you think you will have your own robot In some science fiction movies, people in the future have their own robots. These robots are just like humans. They help with the housework and do most unpleasant jobs. Some scientists believe that there will be such robots in the future. However, they agree it may take hundreds of years. Scientist ae now trying to make robots look like people and do the same things as us. Janpanese companies have already made robts walk and dance. This kond of roots will also be fun to watch. But robot scientist James White disagrees. He thinks that it will be difficult fo a robot to do the same rhings as a person. For example, it's easy for a child to wake up and know where he or she is. Mr White thinks that robots won't be able to do this. But other scientists disagree. They think thast robots will be able t walk to people in 25 to 50tars. Robots scientists are not just trying to make robots look like people . For example, there are already robots working in factories . These robots look more like huge arms. They do simple jobs over and over again. People would not like to do such as jobs and would get bored. But robots will never bored. In the futhre, there will be more robots everwhere, and humans will have less work to do. New robots will have different shapes. Some will look like humans, and others might look like snakes. After an earthquake, a snake robot could help look for people under buildings. That may not seem possibe now, but computers, space rockets and even electric toothbrushes seemed

选修6英语课本原文文档

高中英语选修 6 Unit 1 A SHORT HISTORY OF WESTERN PAINTING Art is influenced by the customs and faith of a people. Styles in Western art have changed many times. As there are so many different styles of Western art, it would be impossible to describe all of them in such a short text. Consequently, this text will describe only the most important ones. Starting from the sixth century AD. The Middle Ages(5th to the 15th century AD) During the Middle Ages, the main aim of painters was to represent religious themes. A conventional artistof this period was not interested in showing nature and people as they really were. A typical picture at this time was full of religious symbols, which created a feeling of respect and love for God. But it was evident that ideas were changing in the 13th century when painters like Giotto di Bondone began to paint religious scenes in a more realistic way. The Renaissance(15th to 16th century) During the Renaissance, new ideas and values gradually replaced those held in the Middle Ages. People began to concentrate less on

英语选修六课文翻译第五单元word版本

英语选修六课文翻译第五单元

英语选修六课文翻译第五单元 reading An exciting job I have the greatest job in the world. travel to unusual places and work alongside people from all over the world sometimes working outdoors sometimes in an office sometimes using scientific equipment and sometimes meeting local people and tourists I am never bored although my job is occasionally dangerous I don't mind because danger excites me and makes me feel alive However the most important thing about my job is that I heIp protect ordinary people from one of the most powerful forces on earth-the volcano. I was appointed as a volcanologist working for the Hawaiian Volcano Observatory (HVO) twenty years ago My job is collecting information for a database about Mount KiLauea which is one of the most active volcanoes in Hawaii Having collected and evaluated the information I help oyher scientists to predict where lava from the path of the lava can be warned to leave their houses Unfortunately we cannot move their homes out of the way and many houses have been covered with lava or burned to the ground. When boiling rock erupts from a volcano and crashes back to earth, it causes less damage than you might imagine. This is because no one lives near the top of Mount Kilauea, where the rocks fall. The lava that flows slowly like a wave down the mountain causes far more damage because it buries everything in its path under the molten rock. However, the eruption itself is really exciting to watch and I shall never forget my first sight of one. It was in the second week after I arrived in Hawaii. Having worked hard all day, I went to bed early. I was fast asleep when suddenly my bed began shaking and I heard a strange sound, like a railway train passing my window. Having experienced quite a few earthquakes in Hawaii already, I didn't take much notice. I was about to go back to sleep when suddenly my bedroom became as bright as day. I ran out of the house into the back garden where I could see Mount Kilauea in the distance. There had been an eruption from the side of the mountain and red hot lava was fountaining hundreds of metres into the air. It was an absolutely fantastic sight. The day after this eruption I was lucky enough to have a much closer look at it. Two other scientists and I were driven up the mountain and dropped as close as possible to the crater that had been formed duing the eruption. Having earlier collected special clothes from the observatory, we put them on before we went any closer. All three of us looked like spacemen. We had white protective suits that covered our whole body, helmets,big boots and special gloves. It was not easy to walk in these suits, but we slowly made our way to the edge of the crater and looked down into the red, boiling centre. The other two climbed down into the crater to collect some lava for later study, but this being my first experience, I stayed at the top and watched them.

人教版英语选修六Unit5 the power of nature(An exciting Job)

高二英语教学设计 Book6 Unit 5 Reading An Exciting Job 1.教学目标（Teaching Goals）: a. To know how to read some words and phrases. b. To grasp and remember the detailed information of the reading material . c. To understand the general idea of the passage. d. To develop some basic reading skills. 2.教学重难点: a.. To understand the general idea of the passage. b. To develop some basic reading skills. Step I Lead-in and Pre-reading Let’s share a movie T: What’s happened in the movie? S: A volcano was erupting. All of them felt frightened/surprised/astonished/scared…… T: What do you think of volcano eruption and what can we do about it? S: A volcano eruption can do great damage to human beings. It seems that we human beings are powerless in front of these natural forces. But it can be predicted and damage can be reduced. T: Who will do this kind of job and what do you think of the job? S: volcanologist. It’s dangerous. T: I think it’s exciting. Ok, this class, let’s learn An Exciting Job. At first, I want to show you the goals of this class Step ⅡPre-reading Let the students take out their papers and check them in groups, and then write their answers on the blackboard (Self-learning) some words and phrases：volcano, erupt, alongside, appoint, equipment, volcanologist, database, evaluate, excite, fantastic, fountain, absolutely, unfortunately, potential, be compared with..., protect...from..., be appointed as, burn to the ground, be about to do sth., make one’s way. Check their answers and then let them lead the reading. Step III Fast-reading 这是一篇记叙文，一位火山学家的自述。作者首先介绍了他的工作性质，说明他热爱该项工作的主要原因是能帮助人们免遭火山袭击。然后，作者介绍了和另外二位科学家一道来到火山口的经历。最后，作者表达了他对自己工作的热情。许多年后，火山对他的吸引力依然不减。 Skimming Ⅰ.Read the passage and answer: (Group4) 1. Does the writer like his job?( Yes.) 2. Where is Mount Kilauea? (It is in Hawaii) 3. What is the volcanologist wearing when getting close to the crater? (He is wearing white protective suits that covered his whole body, helmets, big boots and

Unit5 Reading An Exciting Job(说课稿)

Unit5 Reading An Exciting Job 说课稿 Liu Baowei Part 1 My understanding of this lesson The analysis of the teaching material:This lesson is a reading passage. It plays a very important part in the English teaching of this unit. It tells us the writer’s exciting job as a volcanologist. From studying the passage, students can know the basic knowledge of volcano, and enjoy the occupation as a volcanologist. So here are my teaching goals: volcanologist 1. Ability goal: Enable the students to learn about the powerful natural force-volcano and the work as a volcanologist. 2. Learning ability goal: Help the students learn how to analyze the way the writer describes his exciting job. 3. Emotional goal: Make the Students love the nature and love their jobs. Learn how to express fear and anxiety Teaching important points: sentence structures 1. I was about to go back to sleep when suddenly my bedroom became as bright as day. 2. Having studied volcanoes now for more than twenty years, I am still amazed at their beauty as well as their potential to cause great damage. Teaching difficult points: 1. Use your own words to retell the text. 2. Discuss the natural disasters and their love to future jobs. Something about the S tudents: 1. The Students have known something about volcano but they don’t know the detailed information. 2. They are lack of vocabulary. 3. They don’t often use English to express themselves and communicate with others.

an exciting job 翻译

我的工作是世界上最伟大的工作。我跑的地方是稀罕奇特的地方，我见到的是世界各地有趣味的人们，有时在室外工作，有时在办公室里，有时工作中要用科学仪器，有时要会见当地百姓和旅游人士。但是我从不感到厌烦。虽然我的工作偶尔也有危险，但是我并不在乎，因为危险能激励我，使我感到有活力。然而，最重要的是，通过我的工作能保护人们免遭世界最大的自然威力之一，也就是火山的威胁。我是一名火山学家，在夏威夷火山观测站（HVO）工作。我的主要任务是收集有关基拉韦厄火山的信息，这是夏威夷最活跃的火山之一。收集和评估了这些信息之后，我就帮助其他科学家一起预测下次火山熔岩将往何处流，流速是多少。我们的工作拯救了许多人的生命，因为熔岩要流经之地，老百姓都可以得到离开家园的通知。遗憾的是，我们不可能把他们的家搬离岩浆流过的地方，因此，许多房屋被熔岩淹没，或者焚烧殆尽。当滚烫沸腾的岩石从火山喷发出来并撞回地面时，它所造成的损失比想象的要小些，这是因为在岩石下落的基拉韦厄火山顶附近无人居住。而顺着山坡下流的火山熔岩造成的损失却大得多，这是因为火山岩浆所流经的地方，一切东西都被掩埋在熔岩下面了。然而火山喷发本身的确是很壮观的，我永远也忘不了我第一次看见火山喷发时的情景。那是在我到达夏威夷后的第二个星期。那天辛辛苦苦地干了一整天，我很早就上床睡觉。我在熟睡中突然感到床铺在摇晃，接着我听到一阵奇怪的声音，就好像一列火车从我的窗外行驶一样。因为我在夏威夷曾经经历过多次地震，所以对这种声音我并不在意。我刚要再睡，突然我的卧室亮如白昼。我赶紧跑出房间，来到后花园，在那儿我能远远地看见基拉韦厄火山。在山坡上，火山爆发了，红色发烫的岩浆像喷泉一样，朝天上喷射达几百米高。真是绝妙的奇景！就在这次火山喷发的第二天，我有幸做了一次近距离的观察。我和另外两位科学被送到山顶，在离火山爆发期间形成的火山口最靠近的地方才下车。早先从观测站出发时，就带了一些特制的安全服，于是我们穿上安全服再走近火山口。我们三个人看上去就像宇航员一样，我们都穿着白色的防护服遮住全身，戴上了头盔和特别的手套，还穿了一双大靴子。穿着这些衣服走起路来实在不容易，但我们还是缓缓往火山口的边缘走去，并且向下看到了红红的沸腾的中心。另外，两人攀下火山口，去收集供日后研究用的岩浆，我是第一次经历这样的事，所以留在山顶上观察他们

英语选修六课文翻译第五单元

新目标英语七年级下册课文Unit04

新目标英语七年级下册课文Unit04 Coverstion1 A: What does your father do? B: He＇s a reporter. A:Really?That sounds interesting. Coverstion2 A:What does your mother do,Ken? B:She＇s a doctor. A:Really?I want to be a doctor. Coverstion3 A:What does your cousin do? B:You mean my cousin,Mike? A:Yeah,Mike.What does he do? B:He is as shop assistant. 2a,2b Coverstion1 A: Anna,doesyour mother work? B:Yes,she does .She has a new job. A:what does she do? B: Well ,she is as bank clerk,but she wants to be a policewoman. Coverstion2 A:Is that your father here ,Tony?

B:No,he isn't .He's working. B:But it's Saturday night.What does he do? B:He's a waiter Saturday is busy for him. A:Does he like it? B:Yes ,but he really wants to be an actor. Coverstion3 A:Susan,Is that your brother? B:Yes.it is A:What does he do? B:He's a student.He wants to be a doctor. section B , 2a,2b Jenny:So,Betty.what does yor father do? Betty: He's a policeman. Jenny:Do you want to be a policewoman? Betty:Oh,yes.Sometimes it's a little dangerous ,but it's also an exciting job.Jenny, your father is a bank clerk ,right? Jenny:Yes ,he is . Sam:Do you want to be a bank clerk,too? Jenny:No,not really.I want to be a reporter. Sam: Oh,yeah?Why? Jenny:It's very busy,but it's also fun.You meet so many interesting people.What about your father ,Sam. Sam: He's a reporter at the TV station.It's an exciting job,but it's also very difficult.He always has a lot of new things to learn.Iwant to be a reporter ,too

高中英语_An exciting job教学设计学情分析教材分析课后反思

人教版选修Unit 5 The power of nature阅读课教学设计 Learning aims Knowledge aims: Master the useful words and expressions related to the text. Ability aims: Learn about some disasters that are caused by natural forces, how people feel in dangerous situations. Emotion aims: Learn the ways in which humans protect themselves from natural disasters. Step1 Leading-in 1.Where would you like to spend your holiday? 2.What about a volcano? 3.What about doing such dangerous work as part of your job ? https://www.wendangku.net/doc/2e1950662.html, every part of a volcano. Ash cloud/volcanic ash/ Crater/ Lava /Magma chamber 【设计意图】通过图片激发学生兴趣，引出本单元的话题，要求学生通过讨论, 了解火山基本信息，引出火山文化背景，为后面的阅读做铺垫。利用头脑风暴法收集学生对课文内容的预测并板书下来。文内容进行预测，培养学生预测阅读内容的能力。同时通过预测激起进一步探究 Step2. Skimming What’s the main topic of the article?

新目标英语七年级下课文原文unit1-6

Unit1 SectionA 1a Canada, France ,Japan ,the United States ,Australia, Singapore ,the United Kingdom,China 1b Boy1:Where is you pen pal from,Mike? Boy2:He's from Canada. Boy1:Really?My pen pal's from Australia.How about you,Lily?Where's your pen pal from? Girl1:She's from Japan.Where is Tony's pen pal from? Gril2:I think she's from Singapore. 2b 2c Conversation1 A: Where's you pen pal from, John? B: He's from Japan, A:Oh,really?Where does he live? B: Tokyo. Conversation2 A: Where's your pen pal from, Jodie? B: She's from France. A: Oh, she lives in Pairs. Conversation3 A: Andrew, where's your pen pal from? B: She's from France. A: Uh-huh. Where does she live? B: Oh, She lives in Paris.