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A New Statistical-Based Kurtosis Wavelet Energy

A New Statistical-Based Kurtosis Wavelet Energy
A New Statistical-Based Kurtosis Wavelet Energy

A New Statistical-Based Kurtosis Wavelet Energy Feature for Texture Recognition of SAR Images

Gholamreza Akbarizadeh

Abstract—In this paper,an ef?cient algorithm for texture recog-nition of synthetic aperture radar(SAR)images is developed based on wavelet transform as a feature extraction tool and support vec-tor machine(SVM)as a classi?er.SAR image segmentation is an important step in texture recognition of SAR images.SAR images cannot be segmented successfully by using traditional methods because of the existence of speckle noise in SAR images.The algorithm,proposed in this paper,extracts the texture feature by using wavelet transform;then,it forms a feature vector composed of kurtosis value of wavelet energy feature of SAR image.In the next step,segmentation of different textures is applied by using feature vector and level set function.At last,an SVM classi?er is designed and trained by using normalized feature vectors of each region texture.The testing sets of SAR images are segmented by this trained SVM.Experimental results on both agricultural and urban SAR images show that the proposed algorithm is effective for classi?cation of different textures in SAR images,and it is also insensitive to the intensity.

Index Terms—Fourth-order normalized cumulant,kurtosis wavelet energy(KWE),SAR image classi?cation,speckle,syn-thetic aperture radar(SAR).

I.I NTRODUCTION

I N MANY applications,such as the global monitoring for the

environment,recognizing and tracking special objects,map-ping the Earth’s resources,and developing military systems, it is often bene?cial to have an imaging system which is able to provide broad-area imaging at high resolutions and acquire images in inclement weather or during night as well as day. Synthetic aperture radar(SAR)imaging system can provide these requirements.SAR enhances optical imaging abilities because of the unique reactions of xerographic targets to radar frequencies and because of the minimum restrictions on time of day and atmospheric situations.

SAR imaging systems are known as the most popular remote sensing technique greatly used in the past decades because of their capability to be utilized in all weather conditions,day and night photography time,and the high spatial resolution[1].In order to recognize and identify selected objects,SAR can pro-vide high-resolution images to distinguish terrain features[2]. However,SAR image processing is extremely dif?cult because

Manuscript received September29,2010;revised March27,2011,May14, 2011,July21,2011,and November10,2011;accepted April1,2012.Date of publication May23,2012;date of current version October24,2012.This work was supported by the Shahid Chamran University of Ahvaz as a research proposal with code901.

The author is with the Electrical Engineering Department,Engineering Faculty,Shahid Chamran University of Ahvaz,Ahvaz61357-831351,Iran (e-mail:g.akbari@scu.ac.ir).

Color versions of one or more of the?gures in this paper are available online at https://www.doczj.com/doc/976380975.html,.

Digital Object Identi?er10.1109/TGRS.2012.2194787of the speckle noise[2]–[5].The speckle noise is a fully developed noise which usually affects SAR images.Speckle phenomenon can be described as multiplicative noise,with standard deviation equal to pixel re?ectivity value[6].The probability density function(PDF)of the pixel intensities in SAR images is also impressed by speckle noise.This phe-nomenon can be expressed by the nonlinear intensity inhomo-geneity in SAR images[2].As a result of speckle noise effect on pixel intensities in SAR images,this is one of the main reasons for SAR imaging,which is a crucial issue for accurate segmentation and classi?cation.Accordingly,traditional seg-mentation and classi?cation methods based on intensity cannot be used to SAR image processing because of speckle noise effect on pixel intensity in SAR images.The main scope of the image segmentation and classi?cation is to categorize pixels into obvious image regions that are easier to analyze.Thus, segmentation can be used for image partitioning problems,and classi?cation can be used for object recognition purposes. Active contour models or snakes have been used as one of curve-evolution-based methods for global image segmen-tation[7].These methods are classi?ed into two major cate-gories:edge-based methods[8]–[10]and region-based methods [11]–[13].These methods either have weak performances in facing with weak object boundaries in SAR images,or they are sensitive to the location of initial contour and pixel intensities. Furthermore,these methods are only useful to segment the extended areas such as rivers and urban and agricultural areas. On the other hand,nonlinearity of intensity inhomogeneity,as mentioned earlier,often occurs in SAR images from different modalities.Intensity inhomogeneity can be addressed by some active contour models which are widely known as piecewise smooth models[14]–[17].Recently,Li et al.[18]have proposed a region-based active contour model by de?ning a region-scalable?tting energy function that locally approximates the image intensities on two sides of a contour.This model relies on the intensity inhomogeneity.Nevertheless,nonlinear intensity inhomogeneity which is usually attendant with SAR images cannot be addressed in all of these methods.

However,recent SAR image segmentation models have been developed containing generic segmentation procedure[19], [20],spectral data clustering algorithms[3],[21]–[23],fuzzy clustering algorithms[24],and level set methods[2],[4].

A generic segmentation method,which is conformed to gamma PDF of SAR images,is proposed by Galland et al.

[19]for SAR image segmentation.This parametric approach is on the basis of a polygonal grid model with a hypothetical unknown number of regions.In this approach,the number of regions of the partition is approximated by minimizing the stochastic complexity.However,when the gray values of SAR images are not correctly described by gamma PDF,like in

0196-2892/$31.00?2012IEEE

textured speckle images,this technique will fail.Furthermore, since a parametric noise reconstruction of the segmentation procedure is regarded,the parameters of regions need to be adjusted,particularly when the data differ from gamma PDF such as in textured regions.Thus,this parametric procedure only illustrates a unique class of textured data which can lead to analogous limitations to those obtained with the gamma PDF. The same gamma PDF distributed-based method as that in[19]was exerted with different noise models by Delyon and Réfrégier[20].As a comparison with the segmentation algorithm proposed in[19],this approach has some evident advantages.It is established on a polygonal grid which can have an arbitrary structure,and its region number and normalcy of its borders are acquired by minimizing the stochastic com-plexity of a determined quantity version on Q levels of the image[20].Unlike in[19],this approach reaches to a standard model without parameters that need to be tuned by the user. However,the proper value of Q,which minimizes the number of misclassi?ed pixels,could not be computed automatically in that procedure.Also,this procedure may fail due to the nonlin-ear intensity inhomogeneity phenomenon,which is mentioned earlier,if a texture region of a SAR image has other attributes of parametric noise models such as Poisson or Gaussian.

In[3]and[21]–[23],various schemes of spectral clustering algorithms were developed.These spectral clustering algo-rithms have very evident advantages compared with the tradi-tional clustering algorithms.Some of these spectral clustering algorithms can identify the clusters of irregular shapes and ob-tain the globally optimal solutions in a relaxed continuous do-main by eigendecomposition[3].However,the computational complexity problem is an imperfection that exists in these meth-ods,and they are computationally expensive because of the use of a coherence matrix?xed by the similarity of each pair of pix-els.Thus,the method needs to compute the eigenvectors of the coherence matrix.Furthermore,spectral clustering algorithms need to allocate a parameter,namely,the scaling parameterσin the Gaussian radial basis function(RBF).Appropriate allocat-ing ofσis a crucial issue to obtain good segmentation results in spectral clustering algorithms.Unfortunately,it is dif?cult to select the appropriateσvalue,and it is always set manually.The improper value ofσcan corrupt the abilities because spectral clustering algorithms are highly sensitive toσ,and different values ofσmay lead to extremely different results[3].

In[4]and[24],two new SAR image segmentation models were presented based on intensity homogeneity in each region as well as no purpose for segmenting the special objects in SAR images.On the other hand,nonlinearity of intensity inhomogeneity often occurs in SAR images as discussed earlier. These two new methods cannot address the nonlinear intensity inhomogeneity phenomenon.Moreover,these methods need an initial curve to be created by the user.

In this paper,we?rst propose an ef?cient method of SAR image segmentation by de?ning a new energy function,which is based on fourth-order normalized cumulant concept,named kurtosis wavelet energy(KWE).We demonstrate that KWE energy function can be used as an ef?cient feature for texture discrimination in SAR images.In other words,this statistic-based energy function is a good texture feature for SAR image segmentation problem.Kurtosis is a fourth-order normalized cumulant concluded by working toward the higher order statis-tics(HOS)in statistics subjects.We derive the KWE energy function by implementing the wavelet coef?cient energy extrac-tion algorithm and level set functions.Then,an SVM classi?er is designed and trained by using normalized feature vector composed of wavelet energy feature,KWE feature,and gray values of eight neighborhood of SAR image.This normalized feature vector is formed for each region texture of SAR image. Statistical properties of texture of each region in SAR image can be extracted well by this normalized feature vector because of using the higher order cumulant(fourth order)in feature formulation design.We will show in this paper that,whenever the order of cumulant as a feature for a SAR image increases, this feature will give the more statistical properties of a speci?c region from a SAR image,and subsequently,it will outperform the accuracy of texture recognition process of SAR image. Note that our feature extraction section method,termed as KWE,is also related to the skewness wavelet energy(SWE) model which is proposed in[2]where the skewness value of the wavelet coef?cient energy of the local intensity values in each region of SAR image is derived and the SWE values of the whole regions are used as the minimizers of the wavelet energy function.In[2],the lower order of cumulant(third order)is used as well as there is not any classi?er scheme,and only the segmentation process is done.On the other hand,in this paper, a local texture of each region of a SAR image is segmented well,and then,a classi?er is developed for texture recognition purposes.

This paper is organized as follows.In Section II,the feature extraction step for SAR image segmentation is performed by computing the KWE formulation as an effective feature for texture segmentation and classi?cation in SAR images.In Section III,derivation of the level set formulation with KWE energy is presented.In Section IV,the recognition step of our algorithm is performed.For this purpose,an SVM classi?er is designed and trained by using the KWE extracted feature for texture discrimination.The implementation results of our method on both agricultural and urban SAR images are given in Section V.Finally,Section VI draws some conclusions.

II.C UMULANTS AND K URTOSIS V ALUE AS A

D EFINITION FOR SAR T EXTURES

In statistics and probability theory,it can be shown that the cumulant generating function of a random variable X is expressed by the natural logarithm of the moment generating function of a random variable X as follows:

ΨX(ω)=ln(ΦX(ω))=ln

E X{e jωX}

(1)

in whichΦX(ω)is the moment generating function of a random variable X,E X{.}represents the mathematical expectation of that random variable,andωis the frequency variable of the Fourier transform.The moment generating functionΦX(ω)is also given by the Fourier transform of the PDF of the random variable X as follows:

ΦX(ω)=E X{e jωX}=

+∞

?∞

e jωx.

f X(x)dx.(2)

From the Fourier transform properties,it is obvious that the moment generating function takes the same information regarding the principal random variable as does the PDF.The natural logarithm functionΨX(ω)de?ned in(1)is usually mentioned as the cumulant generating function,and it is widely used in HOS.The cumulant generating function can be denoted by the cutoff Taylor series expansion as follows:

ΨX(ω)=

n

k=1

c X(k).

(jω)k

k!

(3)

where c X(k)is the coef?cient of the Taylor series and it is called the k th-order cumulant.It can be shown that the impression of cumulants gives a speci?cally powerful means for characterizing the nature of textures in images as stationary random series.Thus,it seems that cumulants can be good features to the description of textures.

It is reported that,whenever the order of cumulant as a feature for a SAR image increases,this feature will give the more statistical characteristics of a speci?c region from a SAR image[2].However,implementation of higher order cumulants will impose more calculations and is time consuming.Thus, we should perform a tradeoff between the higher order and the complexity of calculations.In[2],the third-order normalized cumulant named skewness was proposed to be used as a texture feature for segmentation of SAR images.In this paper,we propose to use the fourth-order normalized cumulant named kurtosis as a texture feature for segmentation step of our proposed algorithm.We show that this selection gives more statistical information of each region,and it has also more ef?cient implementation than the skewness.

It is feasible to represent the cumulants as functions of the moments of the random variable under analysis by using standard differentiation and mathematical identities.Note that the?rst-order moment is the mean moment,and the second-order moment is the central moment.For example,the?rst four cumulants as functions of the moments can be expressed by c(1)=m(1)=M=mean

c(2)=m(2)?[m(1)]2=σ2=covariance

c(3)=m(3)?3m(1)·m(2)+2[m(1)]3

c(4)=m(4)?3[m(2)]2?4m(1)·m(3)

+12[m(1)]2·m(2)?6[m(1)]4(4) where M andσ2represent the mean and covariance of the distribution of the related random variables,respectively.In order to clarify,the subscript X has been eliminated in these expressions.In most practical applications,the PDF of a ran-dom variable is unknown,and the cumulants must be computed from several comprehensions of the random variable.

The?rst step of our approach is segmenting different textures in SAR images using the kurtosis value of wavelet coef?cients of texture of each region as a texture feature.The formulation of third-and fourth-order cumulants c(3)and c(4),given by (3)and(4),respectively,can be used as such texture feature. In applications developed for segmentation of SAR images, a bene?cial feature is the feature that it is invariant to image size[2].However,the cumulant formulation as described in(4)is essentially dependent on the size of the generating random variable and also on the image size.For the purpose of this paper,we use a normalization of the higher order cumulants to make them invariant to image size.With this point of view, it is interesting to consider the normalized third-order and the normalized fourth-order cumulants with a desired random variable as de?ned respectively by

NC(3)=

c(3)

[c(2)]32

=

c(3)

[σ2]32

=

c(3)

σ3

(5)

NC(4)=

c(4)

[c(2)]2

=

c(4)

σ4

.(6)

Using this de?nition of cumulant,it directly follows that this normalized cumulant as a texture feature is image size invariant for any nonzero size of a SAR image.In(5),NC(3)is named skewness which is the third-order cumulant normalized with covariance.In(6),NC(4)is named kurtosis which is the fourth-order cumulant normalized with covariance.The skewness cumulant indicates the value of symmetry of the PDF histogram of a SAR image,and the kurtosis cumulant represents the sharpness of the PDF histogram of a SAR image. In other words,the kurtosis as a feature is the slope decreasing value of the PDF histogram curve of a SAR image.

It can be found that,whenever the order of cumulant as a feature for a SAR image increases,this feature will give the more statistical characteristics of a speci?c region from a SAR image[2].However,implementation of higher order cumulants will impose more calculations and is time consuming.Thus, we should perform a tradeoff between the higher order and the complexity of calculations.In[2],the third-order normalized cumulant named skewness is proposed to be used as a texture feature for segmentation of SAR images.In this paper,a texture representation of each region in SAR images with wavelet transform and kurtosis value concepts is de?ned.Our goal is to design and extract an ef?cient feature for texture segmentation of SAR images.For this purpose,we?rst apply a wavelet transform up to the possible last level on a SAR image,get the wavelet coef?cients,and compute the energy of the resulted wavelet coef?cients.Then,the kurtosis value of the wavelet coef?cient energy is calculated by computing the?rst four order moments of the wavelet coef?cient energies m(1)–m(4)and the four-order cumulant c(4)and at last applying(6)to get the value of the kurtosis NC(4).

In order to characterize a texture of each region in a SAR image by the wavelet coef?cient energy,we consider RT(i,j) as a region texture of a SAR image as follows:

RT(i,j)=

1

MN

m

n

W Aφ(y0,m,n)φy

,m,n

(i,j) +

x=H,V,D

?1

y=?y0

m

n

W xψ(y,m,n)ψx y,m,n(i,j)

?

?(7)

where?is the scaling function,ψis the wavelet function,y0is the order of the decomposition,W Aφis approximation wavelet

coef?cients,and W H

φ

,W V

φ

,and W D

φ

represent detail wavelet

coef?cients.The W H

φ

,W V

φ

,and W D

φ

wavelet coef?cients

are measured along the different directions as follows:W H

φalong columns(horizontal direction),W V

φ

along rows(vertical

direction),and W D

φalong diagonals.

We have tested several kinds of mother wavelets and several

wavelet coef?cients W A

φ,W H

φ

,W V

φ

,and W D

φ

.After exam-

ining the results,we have selected to use the Haar wavelet as mother wavelet type and the energy of approximation wavelet coef?cients in our proposed method.The energy of the approx-imation wavelet coef?cients in each subband can be obtained by the following sequence:

W A

φ(y,m,n)

2

,?y0≤y≤?1,m,n∈Z

.(8)

Aujol et https://www.doczj.com/doc/976380975.html,puted experimentally(see[26])that the dis-tribution of the square of the wavelet coef?cients in a subband of any image(also for SAR images)follows a generalized Gaussian law of the form

P X2(y)=

K

2

y

exp

?

y

α

β

y≥0

(9)

where K,α,andβare the texture parameters.IfΓ(t)and N represent the Gamma function and the total number of pixels of the given SAR image,respectively,then we have

K=

αΓ

1

β

.(10)

To de?ne a texture feature based on kurtosis,we should

compute the fourth-order moment of the wavelet coef?cient energy distribution m(4)|W

φ|2

as follows:

m(4) |Wφ|2=E

|Wφ|8

=

+∞

(Wφ)8h(Wφ)d(Wφ)

=

Kα9

β

·Γ

9

β

.(11)

https://www.doczj.com/doc/976380975.html,parison between the curves of F?1(kurtosis),F?1(skewness),

and F?1(second moment).

Now,we can obtainαas follows:

α=

m(4)

|Wφ|2

m(3)

|Wφ|2

Γ

7

β

Γ

9

β

.(12)

Also,from c(4)and NC(4)de?ned by(4)and(6),respec-

tively,NC(4)can be rewritten as(13),shown at the bottom of

the page.

Thus,the formulation of function F(x)=NC(4)is then

obtained by substitution of the?rst-to fourth-order moments,

namely,m(1)|W

φ|2

,m(2)|W

φ|2

,m(3)|W

φ|2

,and m(4)|W

φ|2

,with their corre-

sponding values given in(14),shown at the bottom of the page.

Then,the value of classi?cation parameterβis calculated by

β=F?1(kurtosis)=F?1

?

??C(4)|Wφ|2

C(2)

|Wφ|2

2

?

??.(15)

The curve of F?1(kurtosis)is shown in Fig.1.As shown in

Fig.1,the curve of F?1(kurtosis)is approximately stable over

a wide range of kurtosis.In order to do a comparison between

the three methods,second moment,skewness,and kurtosis,we

have depicted the?gure of these methods in one plot as shown

NC(4)=C(4)

|Wφ|2

(σ2)4/2

=

m(4)

|Wφ|2

?3

m(2)

|Wφ|2

2

+4m(1)

|Wφ|2

·m(3)|W

φ|2

+12

m(1)

|Wφ|2

2

·m(2)|W

φ|2

?6

m(1)

|Wφ|2

4

σ4

(13)

F(x)=Γ3

1

β

Γ

9

β

?3·N·Γ2

1

β

·Γ2

5

β

?4·N·Γ2

1

β

·Γ

3

β

·Γ

7

β

Γ

1

β

·Γ

5

β

?NΓ2

3

β

2

+q

12·N2·Γ

1

β

Γ2

3

β

·Γ

5

β

?6·N3·Γ4

3

β

Γ

1

β

·Γ

5

β

?NΓ2

3

β

2(14)

Fig.2.De?ned segmentation problem.

in Fig.1.As shown in Fig.1,the curve of F?1(kurtosis)is

more stable than the F?1(skewness)and F?1(second moment)

which were proposed in[2]and[26],respectively.The curve

of F?1(kurtosis)is approximately stable over a wide range of

kurtosis.From Fig.1,as it is expected,the value ofβis about

0.25in all the ranges of kurtosis.Thus,if kurtosis is selected

as a texture feature for SAR image segmentation,it will be

an ef?cient feature because of its stability.As it is expected,

the kurtosis feature extracts more statistical information due

to its HOS.This is exactly the same thing that is required

to face with texture regions in SAR images.Thus,a texture

discrimination of each region in SAR images can be done by

the KWE explained in this section.In the next section,we

apply this feature in a level set function and develop it for

segmentation of each region in SAR images.

III.C OMPUTING THE L EVEL S ET F ORMULATION

W ITH KWE E NERGY T ERM

In the previous section,we described a new texture feature

extraction method for texture segmentation in SAR images.

In this section,we apply this feature in a level set function

and develop it for segmentation of each region texture in

SAR images.

Suppose that r is an open subset of R2,K is the number

of segmented regions(number of r k),k is a parameter that

shows the region index,x is a pixel of region,R kl is the

interface between region r k and region r l,and the image is a

function considered as I:r→R.We de?ne the region Re k= {x∈r|x belongs to the region k}.This de?ned segmentation problem is shown in Fig.2.

We denote that,for all k=1,...,K,Re k is an open set r k

given by a Lipschitz function L k:r→R so that

??

?L k(x)>0if x∈r k

L k(x)=0if x∈R k

L k(x)<0if otherwise

(16)

where R k is the boundary of r k and L k is the signed distance function to R k.We can determine r k using the sign of L k and the Heaviside distribution function H.This function is approximated by

Hα(β)=?

?

?

1

2

1+β

α

+1

π

sinπβ

α

if|β|≤α

1ifβ>α

0ifβ

(17)

In the distributional sense,whenα→0,we have Hα→H.

L k(x)is introduced as level set function.If L k(x)is calculated

for any point x and we get the sign of L k(x),we can determine

if x is in the region r k or not.For example,if L k(x)>0,then

H(L k(x))=1;thus,x∈k.Let x∈r be an arbitrary pixel and

I(x):r→R1be a given vector of SAR image,where1is

the dimension of the vector I(x).Dimension of SAR images

corresponds to the dimension of gray-level images.We de?ne

the following function:

F KWE(L1,L2,...,L K,f1,f2)

=εKWE

x

(L1,L2,...,L K,f1,f2)+μρ(L)(18)

whereμis a positive constant,εKWE

x

is the KWE,f1and f2are

the functions that minimize theεKWE

x

,andρ(L)is the deviation

of the level set function L from a signed distance function.The

kurtosis value of the wavelet coef?cient energy is proposed to

be used as contour energyεKWE

p

in(18).

Now,for a given pixel x∈r,the KWEεKW E

p

is followed

by the distribution form of the kurtosis energy of the wavelet

coef?cients in a subband of each region.For each point x∈r,

the KWE energy proposed in this paper is

εKWE

x

(C,k1(x),k2(x))

=γ1

inside contour(C)

K(x?y)|I(y)?k1(x)|2dy

=γ2

outside contour(C)

K(x?y)|I(y)?k2(x)|2dy(19)

where C is a contour in the image region r,γ1andγ2are

two constant numbers,K is a kernel function with a kurtosis

property,and k1(x)and k2(x)are two functions that?t image

textures near the pixel x.We call the pixel x the center point of

the aforementioned equation,and we call the aforementioned

energy the KWE around the center point x.A bene?t kernel

function K(x)should be computed and used in the KWE

energy function derived in(19).We propose to use the kernel

function K(x)as the PDF.Thus,we have

K|W

φ|2

(x)=

K

2

x

exp

?

x

α

β

(20)

where the constant parameter K is obtained by(10)and the

segment parametersαandβare given by(12)and(15),

respectively.

IV.R ECOGNITION OF S EGMENTED T EXTURES IN

SAR I MAGES W ITH SVM C LASSIFIER

The classi?er plays an important role in image classi?cation

and recognition.After the segmentation of textures in SAR

images is done,the classi?cation of each texture in SAR image

should be achieved by using an ef?cient classi?er and suitable

texture features.The theory of support vector machine(SVM),

as a tool of pattern classi?cation and recognition,is based on

statistical learning theory and the principle of structural risk

minimization.

SVMs are a set of af?liated supervised learning methods

which analyze data and recognize patterns.SVM is used for

statistical classi?cation and regression analysis.Suppose that,in a set of training examples,each example is marked as belonging to one of two categories.An SVM training algorithm initiates a model that predicts whether a new example drops within one category or the other.

An SVM creates a hyperplane or a set of hyperplanes in a high or in?nite dimensional space which can be used for classi?cation,recognition,or other tasks.Intuitively,a good clustering is achieved by the hyperplane which has the largest distance to the nearest training data points of any class.This nearest data point is also called functional margin.In general,the larger margin leads to the lower generalization error of the classi?er.The SVM classi?er belongs to a family of generalized linear classi?ers,but there are some ways to create nonlinear SVM classi?er by applying the kernel trick to maximum margin hyperplane.A linear SVM uses a systematic approach to ?nd a linear function with the lowest vapnik–chervonenis dimension.For nonlinear separable data,the SVM can map the input to a high-dimensional feature space where a linear hyperplane can be found.Thus,a good generalization can be attained by the SVM compared to traditional classi?ers.

The kernel function in a linear SVM is just a simple dot product in the input space.However,for a nonlinear SVM,the training data can be mapped to a feature space of higher dimension via a nonlinear projecting function.Since our clas-si?cation problem is a nonlinear form,hence,the optimal decision function can be considered as

f (x )=sgn m

i =1

a i y i K (x,x i )+

b (21)

where K (x,x i )is the kernel function.

In the SVM classi?er,the kernel function plays the important role of mapping the input samples into a feature space.At the present time,there is not any technique available to discover the structure of kernels.Typical choices of kernel function are the linear kernels,polynomial kernels,and Gaussian RBF ker-nels in SVM research.They are de?ned as follows:1)Linear kernels

K (x,x i )=x ?x i .

(22)

2)Homogeneous polynomial kernels

K (x i ,x j )=(x i ?x j )d .

(23)3)Inhomogeneous polynomial kernels

K (x i ,x j )=(x i ?x j +1)d .

(24)4)Gaussian RBF kernels

K (x i ,x j )=exp

?

x i ?x j 2

2σ2

.

(25)We use the RBF kernels as the kernel function and the arti?cial choice method to obtain the samples in our proposed method for classi?cation of segmented textures in SAR images.After the original SAR image was ?ltered by wavelet trans-form,the energy values of wavelet coef?cients,the kurtosis value of wavelet coef?cient energy,and gray values of eight neighborhood of that will be computed.These computed

values

Fig.3.(a)Three-look simulated SAR image (256×256).(b)Ground truth.(c)Segmentation obtained by LBF model (error rate:4.68%;the number of missegmented pixels:3070).(d)Segmentation obtained by SWE model (error rate:3.33%;the number of missegmented pixels:2185).(e)Segmentation obtained by KWE (error rate:1.46%;the number of missegmented pixels:954).

compose the feature vector of samples.Then,segmentation of textures is applied by using feature vector and level set function proposed in Section III.At last,an SVM classi?er is designed and trained by using normalized feature vectors of each region texture,and the testing sets of SAR image are divided by the trained SVM.According to the classi?cation results,the gray value whose category is “+1”was set at 255,and the gray value whose category is “?1”was set at 0.Thus,the segmentation of SAR image is realized.

V .I MPLEMENTATION AND T EST R ESULTS

To elucidate the relative advantages of the KWE with respect to local binary ?tting (LBF)and SWE,the results of different algorithms on simulated and real SAR images are presented.A.Segmentation of Simulated SAR Image

In order to evaluate the performance of the proposed method objectively,we ?rst show an experiment on a simulated

Fig.4.(a)X-SAR image of Washington,D.C.(512×512).(b)Aerial optical photograph of the same region as the one of the SAR image(adopted by Google Earth).(c)Representation of the approximation wavelet coef?cient energy|W A

φ

|2from levels1to9of the“Washington,D.C.,”SAR image.

three-look SAR image.The generation procedure of the sim-ulated SAR image was inspired by radar image formation phenomena.This is done by averaging three gamma-distributed realizations.The corresponding three-look noisy image,as shown in Fig.3(a),is generated by averaging three independent realizations of speckle.The ground truth image,as shown in Fig.3(b),is used to calculate the error rates of the segmenta-tions obtained by different algorithms.

Three algorithms are used for segmentation respectively: 1)the LBF model;2)SWE;and3)KWE.Fig.3(c)shows the segmentation result with the LBF model.Fig.3(d)shows the segmentation result of SWE,which is better than the LBF model.Fig.3(e)shows the best segmentation result according to the error rate.We found that the overall error rates are reduced from4.68%to3.33%by using SWE and to1.46% by using KWE.Therefore,SWE is better than LBF,and KWE outperforms the SWE.

Visually,the segmentation of the LBF,which is shown in Fig.3(c),is seriously spotty in consistent regions.Many pixels in two segments are confused.SWE performs better than the LBF,as shown in Fig.3(d).Therefore,SWE is more robust to the noise than the LBF.In the result,the missegmented pixels mainly locate in the white regions of Fig.3(a),and the black regions in Fig.3(a)are well https://www.doczj.com/doc/976380975.html,pared with SWE, KWE reduces the number of the missegmented pixels,as shown in Fig.3(e).

B.Segmentation of Real SAR Images

When we deal with real SAR image segmentation,the ground truth corresponding to the SAR images being seg-mented is absent generally.In this case,the evaluation of the segmentation result is based on visual inspection of the segmented

images.Fig.5.SVM classi?er with an RBF kernel which is trained by using the texture features of two different textures of the original“Washington,D.C.,”SAR

image.

Fig.6.Test stage of the trained SVM classi?er.Two different texture data sets of feature vectors of“Washington,D.C.,”SAR image is applied to the classi?er, and the clustered data are classi?ed in two different classes with violet and blue colors.

In this section,in order to verify the effect of the proposed

method,experiments on both agricultural and urban SAR im-ages are performed.

Fig.4(a)shows an original SAR image.It is a National Aeronautics and Space Administration(NASA)Goddard Space Flight Center image with15-m resolution of Washington,D.C., acquired by LANDSAT7on May9,2005.This image is an urban X-band SAR image whose size is512×512.Fig.4(b) shows an aerial optical photograph of the same region of the SAR image shown in Fig.4(a).In order to extract the texture features in different regions of SAR image,the wavelet transform is?rst applied to the image to get the approximation wavelet coef?cients W A

φ

.Then,the values of the energy of the wavelet coef?cients are computed by means of the expression obtained in(8).Note that,in order to extract all of the wavelet coef?cients,the wavelet decompositions of the maximum level L have been used to compute all of the approximation

wavelet coef?cients W A

φ

and then,the?rst-,second-,third-, and fourth-order moments of the energy distribution of these wavelet coef?cients are calculated.For example,in SAR image,

Fig.7.(a)Segmentation obtained by the LBF.(b)Segmentation obtained by SWE.(c)Segmentation obtained by the proposed KWE.(d)[respectively,(e)and (f)]Zoom of an area extracted from(a)[respectively,(b)and(c)].(g)Zoom of the same area extracted from Fig.5(b).

as shown in Fig.4(a),L=9(because512=29).Fig.4(c)

shows the image representation of the approximation wavelet

coef?cient energy|W A

φ|2from levels1to9of the“Washington,

D.C.,”SAR image.

Now,we can extract the KWE feature from the approxima-tion wavelet coef?cient energy shown in Fig.4(c)for X-SAR image of the“Washington,D.C.”The value of the kurtosis of the approximation wavelet coef?cient energy shown in Fig.4(c) is6.514×1012.It is obtained by(13).Now,we can get β=0.2598from the extended curve of Fig.1in the range [0,12×1012]of only the kurtosis axis in detail.

Also,we calculatedα=0.2552and K=2.1436×105 from(12)and(10),respectively.Note that the classi?cation parametersβ,α,and K are different for each SAR image. After extracting the texture feature,segmentation and classi-?cation algorithm of each texture should be carried out on SAR image with these extracted texture features.To implement our level set function with KWE texture feature,the parameters in the experimentation are supposed as follows:γ1=1.0,γ2= 1.0,ν=0.004×2552,c0=2(constant value of step function used as initial contour),time stepτ=0.1,μ=1.0,and center point p=1.0.Also,an SVM classi?er is designed and trained by using the normalized feature vector.In Fig.5,an SVM classi?er with an RBF kernel is designed and trained by using the feature vector of two different textures of the“Washington, D.C.,”SAR image.

As shown in Fig.5,the SVM classi?er is trained with two sets of feature vector.One,which is de?ned with+,is labeled with“1”and red color,and the other,which is de?ned with?, is labeled with“2”and green color.The class“1”is composed of the texture features of water regions,and the class“2”is composed of the texture features of vegetation regions of the “Washington,D.C.,”SAR image.This trained SVM classi?er is determined by a line which is obtained based on SVM principles and the nearest trained data which are labeled as support vectors.

After training with the proposed KWE feature subspace, the SVM classi?er is used to recognize the texture classes of water–vegetation subblocks of the input SAR image.For this purpose,another data set is used to test the trained SVM classi?er.The test stage of the SVM classi?er with two different texture data sets is shown in Fig.6.

In Fig.6,two different feature vectors of two different textures of the“Washington,D.C.,”SAR image are applied to the classi?er.The?rst cluster is the feature vectors of the texture of water regions which is classi?ed in the?rst class.This class is labeled as“1(classi?ed)”with violet color,and it is speci?ed with“+”signs.The second cluster of data sets is the feature vectors of the vegetation regions of the“Washington,D.C.,”SAR image which is labeled as“2(classi?ed)”with blue color, and it is speci?ed with“?”signs.

In the same way,the proposed SVM classi?er can be trained by the texture features of the building regions of the “Washington,D.C.,”SAR image.Now,we can use this SVM classi?er for segmentation and classi?cation of different tex-tures of SAR images.In order to examine the capability of the proposed method in this paper,two traditional methods such as SWE[2]and LBF[18]are selected to have some comparisons. The“Washington,D.C.,”SAR image as shown in Fig.4(a) is used?rst to segmentation and classi?cation operation.The experimental results of LBF model as a pixel-intensity-based method are shown in Fig.7(a).The experimental results of SWE model as a segmentation method based on skewness wavelet energy is shown in Fig.7(b).The results of the pro-posed method are shown in Fig.7(c).The magni?ed versions of the same selected area are shown in Fig.7(d)–(g).

The images,as shown in Fig.7,consist of three types of land cover:water,vegetation,and building.The water area is marked as region A,the vegetation area is marked as region B, and the building area is marked as region C in the results. These images are urban SAR images.The segmentation ob-tained by the LBF model is shown in Fig.7(a).One can see that the water area(lower left)is incorrectly segmented.In other words,the LBF model has failed while facing particular regions such as water.Furthermore,the boundary between the vegetation[region B in Fig.7(d)]and the building[region C in Fig.7(d)]is not correctly de?ned.The segmentation obtained by SWE,as shown in Fig.7(b),improves the uniformity in the water region.However,there is serious missegmentation in the vegetation region[see the magni?ed image of a se-lected area in Fig.7(e)].Furthermore,both LBF and SWE models need an initial contour created by the user.KWE gets the best segmentation and classi?cation result,as shown in Fig.7(c).

The classi?cation operation,obtained by KWE,improves the uniformity in the water region,and a local region of the region B [part of region B located in region C in Fig.7(f)]is consistently recognized as vegetation.Three types of land cover in Fig.7 are consistently identi?ed as corresponding regions by using the KWE classi?cation algorithm.Moreover,the boundaries of particular regions are well determined by KWE.Also,the proposed KWE classi?cation algorithm does not need an initial contour selected by the user.Thus,the KWE can be utilized in automatic processes.

Another experiment is carried out on an agricultural SAR image in Fig.8(a).This SAR image is a multilook C-band

SAR Fig.8.(a)C-SAR image of a rice-growing area near Okayama,Japan, obtained by JPL AirSAR(1024×1024).(b)Aerial optical photograph of the same region as the one of the SAR image(adopted by Google map).

(c)Representation of the approximation wavelet coef?cient energy|W A

φ

|2 from levels1to10of the“rice-growing”SAR image.

image of a rice-growing area near Okayama,Japan,obtained

by NASA/Jet Propulsion Laboratory AirSAR.This image is already multilooked nine times in azimuth to give a pixel spac-ing of approximately4.6m in azimuth and3.3m in range[25]. This image is an agricultural C-SAR image whose size is 1024×1024.Fig.8(b)shows an aerial optical photograph of the same region as the one of the SAR image.The approxi-

mation wavelet coef?cient energy|W A

φ

|2from levels1to10 of the“rice-growing”SAR image is also shown in Fig.8(c). This image consists of?ve types of land cover:water,urban, vegetation,rice,and wheat.

The segmentation obtained by the LBF method is shown in Fig.9(a),and a zoom of an area extracted from this?gure is shown in Fig.9(d).The water area on the down left is segmented badly,and one can see that two water local regions in the rice area[see the down middle image in Fig.9(d)]are mistakenly segmented with the vegetation,and a big part of the rice area is mistakenly segmented with the wheat area. Therefore,the LBF model is not effective for segmentation of this image.SWE improves the segmentation result to some degree,as shown in Fig.9(b).The magni?ed version of this image for the same area is shown in Fig.9(e).The uniformity in the water area is improved,and the vegetation area is identi?ed as well.However,the rice area is segmented badly,and the urban area is not identi?ed.The segmentation and classi?cation of the proposed KWE shows an effective classi?cation result in comparison with those of the LBF and SWE,as shown in Fig.9(c).The uniformity in the rice area and the water area is improved,and the urban area and the wheat area are identi?ed as well[see the magni?ed image as shown in Fig.9(f)].

VI.C ONCLUSION

We have developed a new segmentation and classi?cation algorithm based on kurtosis value of wavelet coef?cient energy for segmentation of each region and recognition of each texture

Fig.9.(a)Segmentation obtained by the LBF.(b)Segmentation obtained by SWE.(c)Segmentation and classi?cation obtained by the proposed KWE.

(d)[respectively,(e)and(f)]Zoom of an area extracted from(a)[respectively,(b)and(c)].(g)Zoom of the same area extracted from Fig.9(b).

of SAR images.A new energy named KWE is proposed to be used as a feature for texture discrimination of each region. In comparison with the LBF segmentation model,the KWE achieves better performance on the SAR images.It also per-forms better than the SWE in often cases because it extracts more statistical information of textures due to its higher order of cumulant.Furthermore,the KWE algorithm,proposed in this paper,avoids the selection of the initial contour by the user. Thus,the KWE algorithm can be used in automatic processes. Experimental results show that the proposed method is more ef?cient for accurate segmentation and classi?cation of several kinds of SAR images.

VII.F UTURE W ORK

In this paper,we have considered the kurtosis of the energy as the texture feature.The kurtosis is linked with the Fourier transform which is de?ned in R,but the energy is always a positive variable.A more appropriate transform to represent the energy will be the Mellin transform.In this case,the log cumulants are considered instead of classical moments.Thus, second kind statistics and the Mellin transform offer a better adapted formalism for positive random variables which could be a suitable topic for future work.Also,the reader can study the estimation methods such as Fisher information matrix and the Cramer–Rao bound to extract good features from SAR images in order to reach a better segmentation result.

A CKNOWLEDGMENT

The author would like to thank the Shahid Chamran Univer-sity of Ahvaz for?nancial support.

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2003.

Gholamreza Akbarizadeh was born in Shiraz,Iran,

on July14,1981.He received the B.S.degree from

the Khajeh-Nassir Tousi University of Technology

(KNTU),Tehran,Iran,in2003and the M.S.and

Ph.D.degrees from the Iran University of Science

and Technology,Tehran,in2005and2011,respec-

tively,all in electrical and electronics engineering.

From2003to2011,he has worked at the DSP

R&D research laboratory as a Senior Researcher.He

is currently an Assistant Professor and Faculty Mem-

ber of the Engineering Department,Shahid Chamran University of Ahvaz,Ahvaz,Iran.He has more than20published papers in different international electrical engineering conferences and journals.His research interests in pattern recognition,machine vision,image processing,and remote sensing analysis.

Dr.Akbarizadeh is member of some international scienti?c societies such as Iranian Machine Vision and Image Processing(MVIP).

带电作业作业步骤及注意事项修订稿

带电作业作业步骤及注 意事项 内部编号:(YUUT-TBBY-MMUT-URRUY-UOOY-DBUYI-0128)

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