A New Statistical-Based Kurtosis Wavelet Energy Feature for Texture

0 downloads 0 Views 1MB Size Report
May 9, 2005 - vide high-resolution images to distinguish terrain features [2]. However, SAR ...... to local binary fitting (LBF) and SWE, the results of different.
4358

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012

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

Abstract—In this paper, an efficient algorithm for texture recognition of synthetic aperture radar (SAR) images is developed based on wavelet transform as a feature extraction tool and support vector machine (SVM) as a classifier. 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 classifier 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 classification 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 classification, speckle, synthetic aperture radar (SAR).

I. I NTRODUCTION

I

N MANY applications, such as the global monitoring for the environment, recognizing and tracking special objects, mapping the Earth’s resources, and developing military systems, it is often beneficial 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 provide high-resolution images to distinguish terrain features [2]. However, SAR image processing is extremely difficult because

Manuscript received September 29, 2010; revised March 27, 2011, May 14, 2011, July 21, 2011, and November 10, 2011; accepted April 1, 2012. Date of publication May 23, 2012; date of current version October 24, 2012. This work was supported by the Shahid Chamran University of Ahvaz as a research proposal with code 901. The author is with the Electrical Engineering Department, Engineering Faculty, Shahid Chamran University of Ahvaz, Ahvaz 61357-831351, Iran (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2012.2194787

of 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 reflectivity value [6]. The probability density function (PDF) of the pixel intensities in SAR images is also impressed by speckle noise. This phenomenon can be expressed by the nonlinear intensity inhomogeneity 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 classification. Accordingly, traditional segmentation and classification 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 classification is to categorize pixels into obvious image regions that are easier to analyze. Thus, segmentation can be used for image partitioning problems, and classification 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 segmentation [7]. These methods are classified into two major categories: 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 defining a regionscalable fitting 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 © 2012 IEEE

AKBARIZADEH: NEW STATISTICAL-BASED KWE FEATURE FOR TEXTURE RECOGNITION

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 complexity 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 misclassified pixels, could not be computed automatically in that procedure. Also, this procedure may fail due to the nonlinear 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 algorithms have very evident advantages compared with the traditional clustering algorithms. Some of these spectral clustering algorithms can identify the clusters of irregular shapes and obtain the globally optimal solutions in a relaxed continuous domain by eigendecomposition [3]. However, the computational complexity problem is an imperfection that exists in these methods, and they are computationally expensive because of the use of a coherence matrix fixed by the similarity of each pair of pixels. Thus, the method needs to compute the eigenvectors of the coherence matrix. Furthermore, spectral clustering algorithms need to allocate a parameter, namely, the scaling parameter σ in the Gaussian radial basis function (RBF). Appropriate allocating of σ is a crucial issue to obtain good segmentation results in spectral clustering algorithms. Unfortunately, it is difficult to select the appropriate σ value, and it is always set manually. The improper value of σ can corrupt the abilities because spectral clustering algorithms are highly sensitive to σ, and different values of σ may 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 first propose an efficient method of SAR image segmentation by defining 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 efficient feature for texture discrimination in SAR images. In other words, this statisticbased 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-

4359

tics (HOS) in statistics subjects. We derive the KWE energy function by implementing the wavelet coefficient energy extraction algorithm and level set functions. Then, an SVM classifier 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 specific 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 coefficient 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 classifier 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 classifier 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 classification 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 classifier 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 VALUE 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 EX {ejωX }

(1)

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

ΦX (ω) = EX {e

jωX

+∞ }= ejωx .fX (x) dx. −∞

(2)

4360

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012

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 function ΨX (ω) defined 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: n 

(jω)k k!

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 defined respectively by

(3)

N C(3) =

where cX (k) is the coefficient of the Taylor series and it is called the kth-order cumulant. It can be shown that the impression of cumulants gives a specifically 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 specific 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 efficient 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 first-order moment is the mean moment, and the secondorder moment is the central moment. For example, the first four cumulants as functions of the moments can be expressed by

N C(4) =

ΨX (ω) =

cX (k).

k=1

c(1) = m(1) = M = mean c(2) = m(2) − [m(1)]2 = σ 2 = 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 and σ 2 represent 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 random variable is unknown, and the cumulants must be computed from several comprehensions of the random variable. The first step of our approach is segmenting different textures in SAR images using the kurtosis value of wavelet coefficients 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 beneficial feature is the feature that it is invariant to image size [2]. However, the cumulant formulation as described in (4)

c(3) [c(2)]

3 2

=

c(3) 3

[σ 2 ] 2

c(4) c(4) 2 = σ4 . [c(2)]

=

c(3) σ3

(5) (6)

Using this definition 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), N C(3) is named skewness which is the third-order cumulant normalized with covariance. In (6), N C(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 specific 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 defined. Our goal is to design and extract an efficient feature for texture segmentation of SAR images. For this purpose, we first apply a wavelet transform up to the possible last level on a SAR image, get the wavelet coefficients, and compute the energy of the resulted wavelet coefficients. Then, the kurtosis value of the wavelet coefficient energy is calculated by computing the first four order moments of the wavelet coefficient energies m(1)–m(4) and the four-order cumulant c(4) and at last applying (6) to get the value of the kurtosis N C(4). In order to characterize a texture of each region in a SAR image by the wavelet coefficient energy, we consider RT (i, j) as a region texture of a SAR image as follows:   1 RT (i, j) = √ WφA (y0 , m, n)φy0 ,m,n (i, j) MN m n ⎤ −1     x + Wψx (y, m, n)ψy,m,n (i, j)⎦ (7) x=H,V,D y=−y0 m

n

where ϕ is the scaling function, ψ is the wavelet function, y0 is the order of the decomposition, WA φ is approximation wavelet H V coefficients, and Wφ , Wφ , and WφD represent detail wavelet coefficients. The WφH , WφV , and WφD wavelet coefficients

AKBARIZADEH: NEW STATISTICAL-BASED KWE FEATURE FOR TEXTURE RECOGNITION

4361

are measured along the different directions as follows: WφH along columns (horizontal direction), WφV along rows (vertical direction), and WφD along diagonals. We have tested several kinds of mother wavelets and several wavelet coefficients WφA , WφH , WφV , and WφD . After examining the results, we have selected to use the Haar wavelet as mother wavelet type and the energy of approximation wavelet coefficients in our proposed method. The energy of the approximation wavelet coefficients in each subband can be obtained by the following sequence:



WφA (y, m, n) 2 , −y0 ≤ y ≤ −1, m, n ∈ Z .

(8)

Aujol et al. computed experimentally (see [26]) that the distribution of the square of the wavelet coefficients in a subband of any image (also for SAR images) follows a generalized Gaussian law of the form √   y β K (9) PX 2 (y) = √ exp − 2 y α y≥0

where K, α, and β are the texture parameters. If Γ(t) and N represent the Gamma function and the total number of pixels of the given SAR image, respectively, then we have K=

Nβ . αΓ β1

(10)

To define a texture feature based on kurtosis, we should compute the fourth-order moment of the wavelet coefficient (4) energy distribution m|Wφ |2 as follows: (4) m|Wφ |2



= E |Wφ |

8



+∞ = (Wφ )8 h(Wφ )d(Wφ ) 0

 9 Kα ·Γ = . β β 9

(4)

N C(4) =

Γ3 F (x) =

C|Wφ |2 (σ 2 )4/2

=

(11)

Fig. 1. Comparison between the curves of F −1 (kurtosis), F −1 (skewness), and F −1 (second moment).

Now, we can obtain α as follows:   (4) 7 m Γ β  |Wφ |2 . α =  (3) m|Wφ |2 Γ β9

(12)

Also, from c(4) and N C(4) defined by (4) and (6), respectively, N C(4) can be rewritten as (13), shown at the bottom of the page. Thus, the formulation of function F (x) = N C(4) is then obtained by substitution of the first- to fourth-order moments, (1) (2) (3) (4) namely, m|Wφ |2 , m|Wφ |2 , m|Wφ |2 , and m|Wφ |2 , with their corresponding values given in (14), shown at the bottom of the page. Then, the value of classification parameter β is calculated by ⎛ ⎞ (4) ⎜ C|Wφ |2 ⎟ β = F −1 (kurtosis) = F −1 ⎝ (15) 2 ⎠ . (2) C|Wφ |2 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 figure of these methods in one plot as shown

2 2 4 (4) (2) (1) (3) (1) (2) (1) m|Wφ |2 − 3 m|Wφ |2 + 4m|Wφ |2 · m|Wφ |2 + 12 m|Wφ |2 · m|Wφ |2 − 6 m|Wφ |2 σ4

1 9 3 7 2 1 2 5 2 1 β Γ β −3·N ·Γ β ·Γ β −4·N ·Γ β ·Γ β ·Γ β 2 N · Γ β1 · Γ β5 − N Γ2 β3 12 · N 2 · Γ β1 Γ2 β3 · Γ β5 − 6 · N 3 · Γ4 β3 +q 2 N · Γ β1 · Γ β5 − N Γ2 β3

(13)

(14)

4362

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012

In the distributional sense, when α → 0, we have Hα → H. Lk (x) is introduced as level set function. If Lk (x) is calculated for any point x and we get the sign of Lk (x), we can determine if x is in the region rk or not. For example, if Lk (x) > 0, then H(Lk (x)) = 1; thus, x ∈ k. Let x ∈ r be an arbitrary pixel and I(x) : r → R1 be a given vector of SAR image, where 1 is the dimension of the vector I(x). Dimension of SAR images corresponds to the dimension of gray-level images. We define the following function: F KWE (L1 , L2 , . . . , LK , f1 , f2 ) = εKWE (L1 , L2 , . . . , LK , f1 , f2 ) + μρ(L) x Fig. 2. Defined 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 of β is about 0.25 in all the ranges of kurtosis. Thus, if kurtosis is selected as a texture feature for SAR image segmentation, it will be an efficient 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 rk ), k is a parameter that shows the region index, x is a pixel of region, Rkl is the interface between region rk and region rl , and the image is a function considered as I : r → R. We define the region Rek = {x ∈ r|x belongs to the region k}. This defined segmentation problem is shown in Fig. 2. We denote that, for all k = 1, . . . , K, Rek is an open set rk given by a Lipschitz function Lk : r → R so that ⎧ x ∈ rk ⎨ Lk (x) > 0 if (16) Lk (x) = 0 if x ∈ Rk ⎩ Lk (x) < 0 if otherwise where Rk is the boundary of rk and Lk is the signed distance function to Rk . We can determine rk using the sign of Lk and the Heaviside distribution function H. This function is approximated by ⎧1 β if |β| ≤ α + π1 sin πβ ⎨2 1+ α α (17) Hα (β) = 1 if β > α ⎩ 0 if β < −α.

(18)

where μ is a positive constant, εKWE is the KWE, f 1 and f 2 are x the functions that minimize the εKWE , and ρ(L) is the deviation x of the level set function L from a signed distance function. The kurtosis value of the wavelet coefficient energy is proposed to be used as contour energy εKWE in (18). p E Now, for a given pixel x ∈ r, the KWE εKW is followed p by the distribution form of the kurtosis energy of the wavelet coefficients in a subband of each region. For each point x ∈ r, the KWE energy proposed in this paper is εKWE (C, k1 (x), k2 (x)) x  = γ1 K(x−y)|I(y)−k1 (x)|2 dy inside contour(C)



= γ2

K(x−y)|I(y)−k2 (x)|2 dy (19)

outside contour(C)

where C is a contour in the image region r, γ1 and γ2 are two constant numbers, K is a kernel function with a kurtosis property, and k1 (x) and k2 (x) are two functions that fit 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 benefit 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 √   β x K (20) K|Wφ |2 (x) = √ exp − α 2 x where the constant parameter K is obtained by (10) and the segment parameters α and β are given by (12) and (15), respectively. IV. R ECOGNITION OF S EGMENTED T EXTURES IN SAR I MAGES W ITH SVM C LASSIFIER The classifier plays an important role in image classification and recognition. After the segmentation of textures in SAR images is done, the classification of each texture in SAR image should be achieved by using an efficient classifier and suitable texture features. The theory of support vector machine (SVM), as a tool of pattern classification and recognition, is based on statistical learning theory and the principle of structural risk minimization. SVMs are a set of affiliated supervised learning methods which analyze data and recognize patterns. SVM is used for

AKBARIZADEH: NEW STATISTICAL-BASED KWE FEATURE FOR TEXTURE RECOGNITION

4363

statistical classification 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 infinite dimensional space which can be used for classification, 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 classifier. The SVM classifier belongs to a family of generalized linear classifiers, but there are some ways to create nonlinear SVM classifier by applying the kernel trick to maximum margin hyperplane. A linear SVM uses a systematic approach to find 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 classifiers. 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 classification problem is a nonlinear form, hence, the optimal decision function can be considered as m   f (x) = sgn ai yi K(x, xi ) + b (21) i=1

where K(x, xi ) is the kernel function. In the SVM classifier, 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 kernels in SVM research. They are defined as follows: 1) Linear kernels K(x, xi ) = x • xi .

(22)

2) Homogeneous polynomial kernels K(xi , xj ) = (xi • xj )d .

(23)

3) Inhomogeneous polynomial kernels K(xi , xj ) = (xi • xj + 1)d .

(24)

4) Gaussian RBF kernels

 xi − xj 2 K(xi , xj ) = exp − . 2σ 2

(25)

We use the RBF kernels as the kernel function and the artificial choice method to obtain the samples in our proposed method for classification of segmented textures in SAR images. After the original SAR image was filtered by wavelet transform, the energy values of wavelet coefficients, the kurtosis value of wavelet coefficient 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 classifier 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 classification 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 fitting (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 first show an experiment on a simulated

4364

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012

Fig. 5. SVM classifier 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. 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 coefficient A |2 from levels 1 to 9 of the “Washington, D.C.,” SAR image. energy |Wφ

three-look SAR image. The generation procedure of the simulated 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 segmentations obtained by different algorithms. Three algorithms are used for segmentation respectively: 1) the LBF model; 2) SWE; and 3) 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 from 4.68% to 3.33% by using SWE and to 1.46% 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 segmented. Compared 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 segmented is absent generally. In this case, the evaluation of the segmentation result is based on visual inspection of the segmented images.

Fig. 6. Test stage of the trained SVM classifier. Two different texture data sets of feature vectors of “Washington, D.C.,” SAR image is applied to the classifier, and the clustered data are classified 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 images are performed. Fig. 4(a) shows an original SAR image. It is a National Aeronautics and Space Administration (NASA) Goddard Space Flight Center image with 15-m resolution of Washington, D.C., acquired by LANDSAT 7 on May 9, 2005. This image is an urban X-band SAR image whose size is 512 × 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 first applied to the image to get the approximation wavelet coefficients WφA . Then, the values of the energy of the wavelet coefficients are computed by means of the expression obtained in (8). Note that, in order to extract all of the wavelet coefficients, the wavelet decompositions of the maximum level L have been used to compute all of the approximation wavelet coefficients WφA and then, the first-, second-, third-, and fourth-order moments of the energy distribution of these wavelet coefficients are calculated. For example, in SAR image,

AKBARIZADEH: NEW STATISTICAL-BASED KWE FEATURE FOR TEXTURE RECOGNITION

4365

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 (because 512 = 29 ). Fig. 4(c) shows the image representation of the approximation wavelet 2 coefficient energy |WφA | from levels 1 to 9 of the “Washington, D.C., ” SAR image. Now, we can extract the KWE feature from the approximation wavelet coefficient energy shown in Fig. 4(c) for X-SAR image of the “Washington, D.C.” The value of the kurtosis of the approximation wavelet coefficient energy shown in Fig. 4(c) is 6.514 × 1012 . It is obtained by (13). Now, we can get β = 0.2598 from the extended curve of Fig. 1 in the range [0, 12 × 1012 ] of only the kurtosis axis in detail. Also, we calculated α = 0.2552 and K = 2.1436 × 105 from (12) and (10), respectively. Note that the classification parameters β, α, and K are different for each SAR image. After extracting the texture feature, segmentation and classification 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 step τ = 0.1, μ = 1.0, and center point p = 1.0. Also, an SVM classifier is designed and trained by using the normalized feature vector. In Fig. 5, an SVM classifier 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 classifier is trained with two sets of feature vector. One, which is defined with +, is labeled with “1” and red color, and the other, which is defined 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 classifier is determined by a line which is obtained based on SVM principles and the nearest trained data which are labeled as support vectors.

4366

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012

After training with the proposed KWE feature subspace, the SVM classifier 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 classifier. The test stage of the SVM classifier 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 classifier. The first cluster is the feature vectors of the texture of water regions which is classified in the first class. This class is labeled as “1(classified)” with violet color, and it is specified 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(classified)” with blue color, and it is specified with “•” signs. In the same way, the proposed SVM classifier can be trained by the texture features of the building regions of the “Washington, D.C.,” SAR image. Now, we can use this SVM classifier for segmentation and classification of different textures 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 first to segmentation and classification 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 proposed method are shown in Fig. 7(c). The magnified 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 obtained 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 defined. 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 magnified image of a selected 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 classification result, as shown in Fig. 7(c). The classification 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 identified as corresponding regions by using the KWE classification algorithm. Moreover, the boundaries of particular regions are well determined by KWE. Also, the proposed KWE classification 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). A |2 (c) Representation of the approximation wavelet coefficient energy |Wφ from levels 1 to 10 of 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 spacing of approximately 4.6 m in azimuth and 3.3 m 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 approxi2 mation wavelet coefficient energy |WφA | from levels 1 to 10 of the “rice-growing” SAR image is also shown in Fig. 8(c). This image consists of five 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 figure 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 magnified 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 identified as well. However, the rice area is segmented badly, and the urban area is not identified. The segmentation and classification of the proposed KWE shows an effective classification 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 identified as well [see the magnified image as shown in Fig. 9(f)]. VI. C ONCLUSION We have developed a new segmentation and classification algorithm based on kurtosis value of wavelet coefficient energy for segmentation of each region and recognition of each texture

AKBARIZADEH: NEW STATISTICAL-BASED KWE FEATURE FOR TEXTURE RECOGNITION

4367

Fig. 9. (a) Segmentation obtained by the LBF. (b) Segmentation obtained by SWE. (c) Segmentation and classification 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 performs 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 efficient for accurate segmentation and classification of several kinds of SAR images.

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. ACKNOWLEDGMENT The author would like to thank the Shahid Chamran University of Ahvaz for financial support.

VII. F UTURE W ORK

R EFERENCES

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 defined in R, but the energy is always a positive variable. A more appropriate transform to represent

[1] G. A. Rezai-Rad and G. Akbarizadeh, “A new readout circuit structure for SAR satellite imaging sensors,” Proc. IREE, vol. 5, no. 1, pp. 281–290, Feb. 2010. [2] G. Akbarizadeh, G. A. Rezai-Rad, and S. B. Shokouhi, “A new region-based active contour model with skewness wavelet energy for

4368

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

[18]

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012

segmentation of SAR images,” IEICE Trans. Inf. Syst., vol. E93-D, no. 7, pp. 1690–1699, Jul. 2010. X. Zhang, L. Jiao, F. Liu, L. Bo, and M. Gong, “Spectral clustering ensemble applied to SAR image segmentation,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 7, pp. 2126–2136, Jul. 2008. Y. Shuai, H. Sun, and G. Xu, “SAR image segmentation based on level set with stationary global minimum,” IEEE Trans. Geosci. Remote Sens., vol. 5, no. 4, pp. 644–648, Oct. 2008. S. Li, Z. Yanning, M. Miao, and T. Guangjian, “SAR image segmentation method using DP mixture models,” in Proc. IEEE Int. Symp. Comput. Sci. Comput. Technol., Dec. 2008, vol. 2, pp. 598–601. L. R. Varshney, “Despeckling synthetic aperture radar imagery using the contourlet transform,” 2004. [Online]. Available: http://web.mit.edu/~lrv/ www/cornell/publications/426%20Report%203.pdf C. Li, C. Kao, J. Gore, and Z. Ding, “Implicit active contours driven by local binary fitting energy,” in Proc. IEEE CVPR, Jun. 2007, pp. 1–7. C. Li, C. Xu, C. Gui, and M. D. Fox, “Level set evolution without re-initialization: A new variational formulation,” in Proc. IEEE CVPR, Jun. 2005, vol. 1, pp. 430–436. C. Xu and J. Prince, “Snakes, shapes, and gradient vector flow,” IEEE Trans. Image Process., vol. 7, no. 3, pp. 359–369, Mar. 1998. V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” Int. J. Comput. Vis., vol. 22, no. 1, pp. 61–79, Feb./Mar. 1997. N. Paragios and R. Deriche, “Geodesic active regions and level set methods for supervised texture segmentation,” Int. J. Comput. Vis., vol. 46, no. 3, pp. 223–247, Feb./Mar. 2002. C. Samson, L. Blanc-Feraud, G. Aubert, and J. Zerubia, “A variational model for image classification and restoration,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 22, no. 5, pp. 460–472, May 2000. R. Ronfard, “Region-based strategies for active contour models,” Int. J. Comput. Vis., vol. 13, no. 2, pp. 229–251, Oct. 1994. O. Michailovich, Y. Rathi, and A. Tannenbaum, “Image segmentation using active contours driven by the Bhattacharyya gradient flow,” IEEE Trans. Image Process., vol. 16, no. 11, pp. 2787–2801, Nov. 2007. L. Vese and T. Chan, “A multiphase level set framework for image segmentation using the Mumford and Shah model,” Int. J. Comput. Vis., vol. 50, no. 3, pp. 271–293, Dec. 2002. D. Mumford and J. Shah, “Optimal approximations by piecewise smooth functions and associated variational problems,” Commun. Pure Appl. Math., vol. 42, no. 5, pp. 577–685, Jul. 1989. A. Tsai, A. Yezzi, and A. S. Willsky, “Curve evolution implementation of the Mumford–Shah functional for image segmentation, denoising, interpolation, and magnification,” IEEE Trans. Image Process., vol. 10, no. 8, pp. 1169–1186, Aug. 2001. C. Li, C. Kao, J. Gore, and Z. Ding, “Minimization of region-scalable fitting energy for image segmentation,” IEEE Trans. Image Process., vol. 17, no. 10, pp. 1940–1949, Oct. 2008.

[19] F. Galland, N. Bertaux, and P. Réfrégier, “Minimum description length synthetic aperture radar image segmentation,” IEEE Trans. Image Process., vol. 12, no. 9, pp. 995–1006, Sep. 2003. [20] G. Delyon and P. Réfrégier, “SAR image segmentation by stochastic complexity minimization with a nonparametric noise model,” IEEE Trans. Geosci. Remote Sens., vol. 44, no. 7, pp. 1954–1961, Jul. 2006. [21] C. Fowlkes, S. Belongie, F. Chung, and J. Malik, “Spectral grouping using the Nyström method,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 26, no. 2, pp. 214–225, Feb. 2004. [22] S. X. Yu and J. Shi, “Multiclass spectral clustering,” in Proc. 9th IEEE Int. Conf. Comput. Vis., 2003, vol. 1, pp. 313–319. [23] C. Ding, X. He, H. Zha, M. Gu, and H. Simon, “A min–max cut algorithm for graph partitioning and data clustering,” in Proc. 1st IEEE ICDM, 2001, pp. 107–114. [24] P. R. Kersten, J.-S. Lee, and T. L. Ainsworth, “Unsupervised classification of polarimetric synthetic aperture radar images using fuzzy clustering and EM clustering,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 3, pp. 519–527, Mar. 2005. [25] G. Davidson, K. Ouchi, G. Saito, N. Ishitsuka, K. Mohri, and S. Uratsuka, “Polarimetric classification using expectation methods,” in Proc. Polarimetr. Interferometr. SAR Workshop, Tokyo, Japan, Aug. 29–30, 2002. [26] J. F. Aujol, G. Aubert, and L. B. Feraud, “Wavelet-based level set evolution for classification of textured images,” IEEE Trans. Image Process., vol. 12, no. 12, pp. 1634–1641, Dec. 2003.

Gholamreza Akbarizadeh was born in Shiraz, Iran, on July 14, 1981. He received the B.S. degree from the Khajeh-Nassir Tousi University of Technology (KNTU), Tehran, Iran, in 2003 and the M.S. and Ph.D. degrees from the Iran University of Science and Technology, Tehran, in 2005 and 2011, respectively, all in electrical and electronics engineering. From 2003 to 2011, he has worked at the DSP R&D research laboratory as a Senior Researcher. He is currently an Assistant Professor and Faculty Member of the Engineering Department, Shahid Chamran University of Ahvaz, Ahvaz, Iran. He has more than 20 published 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 scientific societies such as Iranian Machine Vision and Image Processing (MVIP).