A remote sensing ship recognition method based on dynamic probability generative model

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Contents lists available at ScienceDirect

Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa 5 6

A remote sensing ship recognition method based ondynamic probability generative model

3 4 7

Q1

Weiya Guo a,b,⇑, Xuezhi Xia b, Wang Xiaofei b a

8 9

b

10 11

College of Information Technology, Harbin Engineering University, Harbin, China Wuhan Digital Engineering Research Institute, Wuhan, China

a r t i c l e

1 2 3 3 14 15 16 17 18 19 20 21 22

i n f o

Keywords: Ship recognition Saliency Image segmentation Entropy e-Local neighborhood information Probability generative model

a b s t r a c t Aiming at detecting sea targets reliably and timely, a novel ship recognition method using optical remote sensing data based on dynamic probability generative model is presented. First, with the visual saliency detection method, prior shape information of target objects in put images which is used to describe the initial curve adaptively is extracted, and an improved Chan–Vese (CV) model based on entropy and local neighborhood information is utilized for image segmentation. Second, based on rough set theory, the common discernibility degree is used to compute the significance weight of each candidate feature and select valid recognition features automatically. Finally, for each node, its neighbor nodes are sorted by their e-neighborhood distances to the node. Using the classes of the selected nodes from top of sorted neighbor nodes list, a dynamic probability generative model is built to recognize ships in data from optical remote sensing system. Experimental results on real data show that the proposed approach can get better classification rates at a higher speed than the k-nearest neighbor (KNN), support vector machines (SVM) and traditional hierarchical discriminant regression (HDR) method.  2014 Elsevier Ltd. All rights reserved.

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1. Introduction

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Target detection and recognition from remote sensing image play a critical role in various applications of pattern recognition, such as fishery management, vessel traffic services, and maritime activities. In particular, with the naval strength keeps developing at high speed in the world, ship detection and recognition has become increasingly important for effective and efficient ship monitoring to form the marine combat intelligence. In recent years, with the rapid development of earth observation technology, satellite remote sensing has entered an unprecedented new stage, and the sea reconnaissance and target surveillance are provided with abundant data source by a number of high spatial resolution, short revisit circle imaging satellites. For example, taking the French SPOT-5 satellite full-color image into account, its point resolution has access to 2.5 m, taking the American Quick bird full-color image into account, its resolution is 0.6 m, and revisit circle is 1–3.5 days and also taking the most advanced military spy satellite of America into account, its resolution is 0.05 m. Moreover, with the promotion and implementation of the earth satellite imaging system, in future, there will be some

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⇑ Corresponding author at: College of Information Technology, Harbin Engineer-

Q2

ing University, Harbin, China. Tel.: +86 13031666128. E-mail address: [email protected] (W. Guo).

better performance, higher resolution, shorter revisit circle of earth observation satellite, and the available satellite remote sensing image data also increases by explosive growth. In the face of such massive remote sensing image data, due to the low efficiency, high cost, long information acquisition cycle and other defects, the traditional way such as artificial visual interpretation cannot satisfy the requirements of the modern society efficient information. How to quickly and accurately extract automatically and recognize ship target from massive remote sensing data has become an urgent need to solve the problem. Recently, different kinds of features have been proposed for optical remote sensing target recognition, and they appear most promising for the recognition performance. However, these methods mainly have two shortcomings: (1) they are generally depended on experiences to select feature vectors, and the selected vectors are high dimensional and redundancy, it is difficult to improve recognition accuracy and speed; (2) its classifier needs global search, and the time complexity is high, which cannot satisfy the requirements of real-time processing. To overcome the first aforementioned shortcoming, in the last century 90’s, Siedlecki and Sklansky (1989), Siedlecki and Sklansky (1998) introduced genetic algorithm to apply to feature selection, which obtained good results, Zheng (2009) proposed an improved feature selection method based on genetic algorithm, (Shang, Hu, Jiao, and Bai (2012) employed NSGA-II algorithm to

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optimize the classification learning framework (MSCC) (Cai, Chen, & Zhang, 2010). Besides, based on rough set theory (Pawlak, 1991), Chun, Sun, Li, et al. (2012) used the idea of attributes reduction to compute the significance weight of each candidate feature and select valid recognition features. Moreover, the improved Zernike moment invariant was used to recognize large warship in air remote sensing image (Lan & Wan, 2009). In addition, an efficient and accurate face detection using feature selection was proposed in Pan, Zhu, and Xia (2013). Later, a novel linear function combining pixel and region characteristics by (Yang, Li, Ji, Gao, and Qizhi (2014) are employed to select ship candidates, which has optimized the detection performance. Despite they have enabled the recognition accuracy to improve, most of them have the lower time complexity. To overcome the second aforementioned shortcoming, depending on the existing various methods, image recognition efficiency have improved gradually. A task oriented facial behavior recognition method was presented (Gu, Zhang, & Ji, 2005), despite the recognition rate was high, it cost a lot of computation time. Support Vector Machine (SVM) technology was applied to the target recognition of Quick Bird satellite data (Zhang, 2009) and SAR images (Anagnostopoulos, 2009), and SVM also used in face recognition (Wei, Jianqi, & Xiang, 2011; Wen, 2012). Whereas SVM method was derived from statistical theory, which caused a preference for two class problems, it performs poorly in multi-class classification tasks. An improved k neighbor algorithm to realize the supervision of the remote sensing image classification was presented in (Luis, Andras, and Karsten (2008), and likelihood function based on the Bayesian criterion was designed to recognize the ship satellite image (Antelo, Ambrosio, & Gonzalez, 2009). These methods were successfully applied in ship target recognition. However, they were improved based on traditional algorithms, they needed global traversal optimization in the process of classification, and did not have the ability of online learning, namely, and it cannot effectively use the distribution of the training samples. (Akakın and Sankur (2011) proposed a method of robust classification of face and head gestures in video, although the recognition rate is as high as 98%, it costs a lot of computation time. In addition, hierarchical discriminating regression (HDR) (Shiuan and Juyang (2000) was already investigated, which unified the classification and regression problems to the issue of regression, the efficiency was greater, the time complexity was O(d log n), it was suitable for high-dimensional image data processing. Based on HDR method, (Weng and Hwang (2007) proposed an incremental discriminant regression technique, it allowed us to obtain a significantly better classification accuracy, but the computation time was slightly larger than the traditional HDR algorithm. Besides, for ship detection, a new classification approach using shape and texture was introduced (Uma Selvi & Suresh Kumar, 2011), which attained a good classification rate. And another method (Panagiotakis, Kokinou, & Sarris, 2011) provided a robust enhancement and detection of mostly line structures in2-D gray-scale images. Although, these methods relatively have a high detection or recognition accuracy, with respect to the time consumption, they are not suitable for the requirements of real-time ship detection or recognition. In recent years, the application of probability generative model in classifier becomes popular, for example, the approach (Zhu, Zhou, Wang, & Guo, 2010) classified ship candidates by using their class probability distributions rather than the direct extracted features, which was effective in distinguishing between ships and non-ships. For ship recognition, the higher recognition rate is acceptable, the lower time consumption is also desirable. In view of the above-mentioned facts, in this paper, we are concerned with recognition efficiency, taking more than 4 m resolution of optical remote sensing image data as the research object, based on class propagation distribution, a general ship target recognition method using

probability generative model is proposed, which has made necessary improvements in image segmentation, feature selection and the design of classifier. The rest of the paper is organized as follows. Section 2 roughly provides a description of the method overview. Section 3 briefly introduces an active contour model based on region saliency and describes how we use them to segment images. Section 4 sketches our features extraction and based on rough set theory how we apply attribute reduction to features selection. Section 5 is devoted to target classification of dynamic probability generative model based on the classes of neighbor nodes. Section 6 gives experiments and comparative results in detail, and Section 7 summarizes our contributions and sketches future work.

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2. Method overview

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Image segmentation is a key step in image analysis and interpretation tasks. The more precise this step, the more accurate any further processing. During the past few decades, there has been substantial progress in the field of image segmentation and its application such as: (Felzenszwalb and Huttenlocher (2004) Q3 introduced an efficient method for image segmentation based on pair wise region comparison, which used the image grid to define a local neighborhood between image pixels. Additionally, (Felzenszwalb and Huttenlocher (2004) proposed a generic unsupervised feature classification method and a new image segmentation framework, which combined edge features, region features and topological constraints for obtaining a visual grouping conform to the principles of proximity, similarity and continuation. However, the approach constructing is relatively complex, and it is too difficult to determine the size of the initial regions. At present, among the current various algorithms, the active contour models based on partial differential equations have been widely used in image segmentation. The existing active contour model can be grouped into two kinds: one is the parameter active contour model, the other is the geometric active contour model (Osher & Sethian, 1988). Due to geometric active contour model using level set method, it implicates the ability of topological transformation, it has the advantages on image segmentation of complex structure images. In geometric active contour models, the most representative is the CV (Chan–Vese) model (Chan & Vese, 2001), which is based on MS (Mumford–Shah) model (Mumford & Shah, 1988). It does not depend on gradient information, it has the ability to resist noise, but has a poorer effect on image segmentation of the heterogeneity regions and complex. Based on this, a lot of research work obtained certain achievements: (Li et al., 2012) improved the CV model by local binary fit- Q4 ting energy (LBF). (Bai, Wang, and Liang (2012) proposed an active contour model based on region saliency for image segmentation, (Lu, Zheng, and Feng (2013) presented Gaussian regularizing CV model based on entropy and local neighborhood information, which is only suitable to the circular objects detection and segmentation. However, these algorithms also cost a long time. Recently, segmentation algorithms based on actively contours have been given wide attention by many internal and foreign researchers due to their variable forms, flexible structure and excellent performance. However, most available active contour models suffer from lacking adaptive initial contour and prior information of target region. Our goal is to develop an efficient approach to image segmentation, therefore, a novel improved CV model using entropy and local neighborhood information is presented. Firstly, an initial curve adaptively is extracted with the visual saliency detection mechanism. And then, in the cost function of the model, the interior and exterior energies are weighted by the entropy.

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On the other hand, under normal circumstances, a general optical remote sensing target recognition algorithm requires a few conditions: (1) features should have better robustness and invariance such as scaling, translation, rotation angle, scale change; (2) different features and their combinations for the contribution of correct target recognition is not the same, in order to improve recognition accuracy and speed, we should extract the most valid recognition features and combinations. (3) Classifier should have lower calculation complexity, and satisfy the requirements of real-time processing. For condition (1), more distinguished features are extracted. For condition (2), in recent years we witness a rapid grow of interest in rough set theory and its applications, especially for data reduction, so on the basis of the extracted invariant features, based on rough set theory, the common discernibility degree is used to compute the significance weight of each candidate feature and select valid recognition features automatically. Based on common discernibility degree, the attribute reduction algorithm can be directly applied to both complete and incomplete information system to reduce attributes without pretreatment, especially it can ensure the relatively high reduction rate and simultaneously reaches fairly low time complexity in incomplete information system. For condition (3), the classifier based on probability generative model is designed. Probability generative model is an effective classification method, which does not require training process, it can directly utilize iterative formula to obtain the classes of the nodes whose classes are unknown. The model first proposes the concept of class propagation distribution, and use the concept to define the probability of a node’s neighbors belonging to each class. Then based the class distribution, the dynamic probability generating model is presented. Finally, by fitting the model, the class of node whose class is unknown is got. Fig. 1 below illustrates the flow diagram of the proposed method procedure. Taking the ship target recognition as an example, at first, the initial curve adaptively is extracted with the visual saliency detection mechanism, based on the above, the improved CV model based entropy and local neighborhood information is utilized to image segmentation. Next, on the basis of the extracted

features, based on rough set theory, the common discernibility degree is adopted to compute the significance weight of each candidate feature and select valid recognition features automatically. Then, the classifier of probability generative model based on class distribution is designed. At last, considering the input image valid features, use the dynamic probability generative model to obtain the recognition results.

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3. An active contour model based on region saliency for image segmentation

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CV model is an effective segmentation model of improving the topology adaptive ability of the evolving curve. However, generally it mainly includes initial contour manual selection and slow speed of curve evolving two problems. In view of these problems, we focus our attentions on the two aspects: initial contour extraction based on visual saliency prior shape information of target objects and the improved model based on entropy and local neighborhood information.

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3.1. Initial contour extraction based on visual saliency prior shape information of target objects

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Recently, segmentation algorithms based on actively contours have been given wide attentions by many internal and foreign researchers due to their variable forms, flexible structure and excellent performance. However, most available active contour models suffer from lacking adaptive initial contour and prior information of target region. In this paper, based on visual saliency detection (Itti, Koch, & Niebur, 1998), the prior shape information of target objects in put images is used to describe the initial curve adaptively, which has reduced the influence of initial contour position. Different from the ICPS method (Initial Curve Based on Prior Shape) (Bai et al., 2012), we use Ostu (also known as the maximum between-class variance method) (Ostu, 1979) method, of which the threshold = 0.5, because of abandoning computing the threshold, which reduces the method calculation time.

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Extraction and selection of target candidates An active contour model based on region saliency for image segmentation Ship targets

Initial contour extraction based on visual saliency prior shape information

CV model based entropy and local neighborhood information

Features extraction

Features selection based on rough set theory Targets candidates Recognition results

Dynamic probability generative model based on the classes of neighbor nodes classification of target candidates

Features extraction of target candidates

Classification of target candidates Fig. 1. The proposed method flow diagram.

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For image I of the domain X, we describe the process as followings:  Step1: Compute input image spectral residual:

RðIÞ ¼ LðIÞ  fn  LðIÞ

ð1Þ

where L(I) is the logarithmic spectrum by fast Fourier transform, f(n) is Gauss mean filter, ⁄ is a convolution operation.  Step2: Obtain the saliency map by the inverse Fourier transform reconstruction in the space: 2

SðIÞ ¼ iff t ½expðRðIÞÞ þ PðIÞ

ð2Þ

where P(I) is the phase spectrum by fast Fourier transform.  Step3: Use Ostu method (its threshold = 0.5), and make the saliency map S(I) binary processing, so the image is divided into saliency region and non saliency region, also named as the target region X1 and background region X2.  Step4: Apply morphology operations, such as: closed, labeled connected components and contour extraction to extract the binary image shape information in the saliency regions X1 and X2, and get the initial contour C of input image I.

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The environments of experiment as follows: the tool is Matlab7.0 R, and each original image solution is 400  300. Fig. 2 provides the results of the proposed algorithm, it is easy to see: the initial contour C was used as input image prior shape constraints to join the geometric active contour model, which can make good use of the visual features of the image.

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3.2. CV model based on entropy and local neighborhood information

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To better describe the algorithm, firstly, definitions below are stated as:

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3.2.1. Related concepts Definition 1: e-neighborhood Suppose (X, q) is a measurable space, x e X, for a given real e > 0, the set is defined as:

fy 2 Xjqðx; yÞ < eg

ð3Þ

Taking the Eq. (3) into account, which is written as B(x, e), and also called a circular neighborhood of the center x, and the radius e, for convenience, we call it x’s e neighborhood. Definition 2: Entropy (Shannon, 1948)

E¼

331

Z

þ1

pðX i ÞlogpðX i Þdx

ð4Þ

1

where pi(x) is the probability density function of event x. Entropy can not only determine the degree of probability of measure system, but also measure the degree of uniformity of gray. In image processing, generally speaking, the more uniform the gray is, the smaller the entropy value is, on the other hand, the more complex the gray level is, the greater the entropy is. Now assume that pi(I(y)) is the interior curve probability density function of gray I(y) in the pixel x’s e-neighborhood, and po(I(y)) is the exterior probability density function of gray I(y) in the pixel x’s e-neighborhood. Define the local entropies of image internal curve and external curve are chosen as shown in Eqs. (5) and (6) respectively:

Z

Ei ðIðyÞÞ ¼

pi ðIðyÞÞ log ðpo ðIðyÞÞÞdy

ð5Þ

X

Eo ðIðyÞÞ ¼

333 334 335 336 337 338 339 340 341 342 343 344 345

346 348 349

Z

po ðIðyÞÞ log ðpo ðIðyÞÞÞdy

ð6Þ

X

Definition 3: E-dominance Given the parameter E, if it satisfies the following conditions:

ðEi ðIðyÞÞ  Eo ðIðyÞÞ ¼ eÞ > 0

ð7Þ

351 352 353

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In the above Eq. (7), Ei(I(y)) is E-dominance to Eo(I(y)), denoted as Ei  eEo, otherwise Ei  eEo. If Ei  eEo, which means the gray distribution uniformity of external curve is superior to internal distribution uniformity, it needs to increase the proportion of the internal curve homogeneity. If Ei  eEo, which means the gray distribution uniformity of internal curve is superior to external distribution uniformity, it needs to increase the proportion of the homogeneity of external curve. Gradually adjust the above operating, until both local entropies are equal, energy achieves the minimum at this time.

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3.2.2. CV model Based on the above analysis, assume in image I(x), the pixel j’s neighborhood is Xj, which is divided into internal region and external region, and the average grays uj and vj are given as:

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R

uj ¼

IðxÞHð/ðxÞÞdx

Xj

R

Xj

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368 369 370

371

ð8Þ 373 374

R

vj ¼

Hð/ðxÞÞdx

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Xj

R

IðxÞð1  Hð/ðxÞÞÞdx Xj

ð1  Hð/ðxÞÞÞdx

ð9Þ

Fig. 2. Results of the proposed algorithm.

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Then the local region energy of neighborhood Xj is described as:

Eji ð/ÞjIðxÞ

2

2

 v j j ð1  Hð/ÞÞ

380

F j ð/Þ ¼

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When computing the internal energy F(/), we only consider the local energy pixels in the curve, ignore the grays heterogeneous inside of regions, so add the Dirac function d(/) in the above Eq. (10), based on CV model, the energy function becomes:

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eð/Þ ¼ l

Z

 uj j Hð/Þ þ

Ejo ð/ÞjIðxÞ

dð/Þjr/jdz þ m

X

Z

Hð/Þdz þ

Z Z

X

X

dð/ÞF j ð/Þdxdz

ð10Þ

ð11Þ

X

To avoid the level set re-initialization at each iteration, Gaussian kernel function is used to calculate the new level set:

/nþ1 ¼ /n  k

ð12Þ n+1

It can prove that |r/ | = 1. In Eq. (12), k is a template size of 5  5 of Gaussian kernel function. Gaussian kernel is used to regularize the level set function, which not only keeps the level set function smooth and stable, but also removes the traditional Euclidean length term and re-initialization. By Euler–Lagrange equation, gradually evolving the curve of Eq. (11), we can get the final results.

eð/Þ ¼ m

Z

Hð/Þdz þ X

Z Z X

dð/ÞF j ð/Þdxdz

ð13Þ

X

  Z @ ð/ðzÞÞ ¼ dð/ðzÞÞ m  r/ F j ð/ðxÞÞdx @t X

ð14Þ

3.2.3. Influences of parameter e on segmentation results Fig. 3 is the original image and image segmentation results of different e, as can be seen from Fig. 3, if the value of e is big, the proposed method concentrates on the boundary features, because the boundary features don’t reflect the fuzziness of infrared image, its effect is poor, if its value is small, the details of the target edge are lost badly. In the experiment, we take e = 5. 3.2.4. Segmentation results comparisons To compare the segmentation performance of the proposed model, CV model and LBF (Local Binary Fitting) model (Li et al., 2012) are applied to two synthetic images: Image1 (the first image of Fig. 2 in Section 3.1) and Image2 (the second image of Fig. 2) and

(a) Original image

(d) ε = 5

one natural image: Image3 (the fourth image of Fig. 2). Note that the experimental tool is Matlab7.0 R and the parameters setting of three models are as in Table 1. All three models can accurately converge to the object boundary, the proposed model has great advantages in the number of iterations and running speed, as shown in Fig. 4. In the premise of the initial contour is relatively close to each, the proposed method can improve the running speed of 7 8 times. More importantly, the proposed model avoids manual intervention, has automatically realized the image segmentation. Referring to Fig. 5, the first line images represent the initial contours of three models, to be fair, the initial contours of CV and LBF are also set a circle and a rectangular close to the object boundary. The second line images represent the segmentation results comparisons of three models. In view of the natural image with complex structure, high resolution, the numbers of three models segmentation iterations are100, 180 and 200 respectively. But as can be seen from Fig. 5, despite the target and background of the input image are not completely homogeneous, the proposed model can also get better effect of segmentation. Additionally, two remote sensing images (Image 4 and Image 5) that are intercepted from a Quick bird satellite image taken as examples are also shown as Fig. 6. Comparing with CV and LBF segmentation methods, the correct ratio of the proposed model has increased and the missed alarms ratio has reduced. In this section, it was easily found that, for the model in theory: ICPS is used to obtain the initialization automatically, which reduce the sensitivity to the initialization. The local information of the curve is considered rather than global image statistics, which reduces the impact of the hererogeneous gray inside of regions and improves the segmentation results. For the practical implications: The results of different experiments indicate that the proposed model has the advantages of high accuracy and strong robustness to segment the common images.

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4. Ship target feature extraction and selection

451

Feature selection as well as extraction is a challenging problem in areas such as pattern recognition, machine learning and data mining. In general, distinguished features are helpful for ship rec-

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(b) ε = 20

(c) ε = 10

(e) ε = 2

(f) ε = 0

Fig. 3. Original image and segmentation results of different e.

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473 Table 1 Parameters setting of three models. Parameter

The proposed method

px;y CV

5 0.5

l m

0.2

0.5 0.5  2552 0.2

ð16Þ 475

x¼1 y¼1

LBF

1 1

k1 k2 e Dt

8 X 8 X ¼ f ðx; yÞ= f ðx; yÞ

0.5 0.03  2552 0.2

where f(x, y) is the gray value of a point (x, y) in the neighbor window, and f(x, y)0, px,y is the probability distribution of f(x, y). (3) Shape features: bow shape F7 and stern F8, radius deviation F9, spindle length F10, and compactness F11, F11 is given as Eq. (17), compared to roundness, compactness gives full consideration to the impact on the average radius of the object boundary changes, which is better to describe complexity of target shapes.

2  2

X  L ¼ p r =A ¼ p

ii þ jj ðP AÞ 2

456 457 458

ognition, so in this section, we intend to find more effective features, and consider a consistency measure based on rough set theory (also called attribute reduction), which aims to retain the discriminatory power of original features.

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4.1. Feature extraction

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For optical remote sensing images, there are some differences in size, shape, texture, and so on. We extract features from the following items:

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(1) Size features: length F1, width F2, length–width ratio F3, and perimeter F4. (2) Texture features: image entropy F5 and smoothness F6. Within the 8  8-neighborhood- window, image entropy is computed as:

468 471

472 470

8 X 8 X   ME ðx; yÞ ¼  px;y log px;y

ð15Þ

x¼1 y¼1

(a) Initial curve of the prooposed model

(d) Result of the prropoosed model

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ð17Þ

ði;jÞ2X

455

477

485

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi P i2 þ j  j2 =P is the equivalent where r ¼ i  ði;jÞ2X

Fig. 4. Iteration comparisons of three models.

476

radius, P is the perimeter and A is the area, ði; jÞ represents its center, X expresses the target boundary points set. (4) Moment invariant features: 7 Hu moment invariants (F12 F18), the front 20 invariants of less than 8 Zernike moment (F19 F38) and 8 wavelet moment invariants (F39 F46). (5) Area ratio features: different with the area code of whole target (Chen, Ji, & Xing, 2012), this paper only extracts ship target area ratio of the bow F47 and stern F48, which is shown in Fig. 7:

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To Fig. 7, obtain the target slices circumscribed rectangle along the spindle rotation to its horizontal direction, then divide the rectangle into N equal parts, take the i’th part of target area as Si, each part is encoded according to Eq. (18) and ARC is defined in terms of Eq. (19).

C i16i6N ¼ floor P

16i6N Si

 10

C i ¼ ½C 1 ; C 2 ; . . . ; C N 

(e) Result of CV

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ð18Þ 508

ð19Þ

In view of the above-mentioned facts, N = 6, ARC code is C = [4, 8, 10, 10, 9, 7] in this paper, we just take C1 and C6, namely, F47 = C1, F48 = C6. Considering the majority of ship target, which bow and stern parts have occupied about 13 18 of the whole ship, so usually the divided number is 3 6 N 6 8,specifically, N = 6.

(b) Initial curve of CV

502

506

!

Si

501

(c) Initial curve of LBF

(f) Result of LBF

Fig. 5. Segmentation results comparisons of three models.

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(a) Origiinal Imaage 4

(b) Result of the proposed model

(c) Result of CV

(e) Original Image 5

(f) Result of the proposed model

(g) Result of CV

(d) Result of LBF

(h) Result of LBF

Fig. 6. Segmentation results comparisons of three models.

attribute set, D is called a decision attribute set. If $aj e A, and Vj includes ‘‘£’’, then S is called an incomplete information system. In consequence of the above assumption, firstly, we associate discernibility relation of the attribute set P # A as Eq. (20).

540 541 542 543

544 Fig. 7. Area code examples.

DISðPÞ ¼ fðx; yÞ 2 U  Uj9a 2 P; f ðx; aÞ – f ðy; aÞ; f ðx; aÞ – ;; ðy; aÞ – ;g

518

Summing up, obtain the feature vectors F = [F1, F2, , F48] from all the extracted features.

519

4.2. Feature selection

517

520 521 522 523 524 525 526 527 528 529 530 531

532 533 534 535 536 537 538 539

Insufficient prior knowledge often brings blindness to feature extraction, in this paper the relevance of extracted features is strong, due to the redundancy, and it’s easy to reduce recognition efficiency. In fact, ship target recognition is a multi-feature combination classification problem, different features combinations for recognition contributions are different, and our purpose is to find the most effective features combinations. Referring to the related algorithm (Chun, Sun, Li, & Teng, 2012), based on rough set theory (Pawlak, 1991), the common discernibility degree (Teng, Zan, Sun, & Tan, 2010) is utilized to compute the significance weight of each candidate feature and select valid recognition features automatically. 4.2.1. Common discernibility degree computing Formally, in rough set theory, an information system can be expressed by a system S, where, S = (U, A, V, f), U is an universe, and U = {x1, x2, , x|U|}, A is a set of all attributes. Each attribute aj S e A, and Vj is called the value set of aj. We let V ¼ aj 2A V j , and define an information function f:U  A ? V, which satisfies the conditions: "aj e A, f(x, aj) e Vj. If A = C [ D and C \ D = £, S = (U, C, D) is called a decision table, where C is called a condition

ð20Þ

546

The largest set of distinguishable objects with x can be written as:

547

DP ðxÞ ¼ fy 2 Ujðx; yÞ 2 DISðPÞg

548 550

ð21Þ

The attribute set P discernibility degree is viewed as:

jDISðPÞj ¼

jUj X

jDP ðxi Þj

551

552

ð22Þ

i¼1

|DIS(P)| in Eq. (22) measures the discernibility ability of attribute set P, which is equal to the number of the ordered elements. Further, we define the common discernibility relation between attributes P and Q:

DISðQ ; P Þ ¼ DISðQ Þ \ DISðPÞ

ð23Þ

554 555 556 557 558

559 561

Its corresponding common discernibility degree is described as:

562

jDISðQ ; PÞj ¼ jDISðQ Þj \ jDISðPÞj

563 565

ð24Þ

It can be seen from the above Eqs. (23) and (24), DIS(Q; P) belonging to attribute sets P and Q is an ordered set and can distinguish the domain U from another. Simultaneously, |DIS(Q; P)| is the number of elements of DIS(Q; P), which is used to measure the common discernibility abilities of P and Q.

566

4.2.2. Feature selection algorithm steps As mentioned in Section 4.2.1, the realized stages in this phase can be summarized as below:

571

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(1) Discretize the extracted ship features, construct feature decision table as follows.

ð25Þ

578 579 580 581 582 583 584 585 586 587 588 589 590 591

592 594

In the above Eq. (25), Ui = (Fi1, Fi2, , Fin) represents the i’th feature vector, U = {U1, U2, , Um} indicates the target set, Ci = (F1i, F2i, , Fmi)T means the i’th candidate features, C = {C1, C2, , Cm} indicates all candidate features set, D = {d1, d2, , dm} is the ship decision (category) set, Fij denotes the j’th feature of the i’th training sample, m, n represent the number of training samples and feature dimensions respectively. (2) Define the feature sets Q and T, and set Q = £, T = C, according to Eq. (24) calculate the common discernibility degree |DIS(Q; P)| between attribute sets C and D. (3) By Eq. (27) define feature set importance function, according to Eq. (28) calculate the most important feature Ck of set C Q, and update Q and T, namely, we set:

Q ¼ Q [ C k ; T ¼ T  fC k g;

1 6 i 6 jTj

ð26Þ

595

596 598

SGF ðC k ; Q ; DÞ ¼ jDISðD; ðQ [ fC k gÞÞj  jDISðD; Q Þj

ð27Þ

SGF ðC k ; Q ; DÞ ¼ maxSGF ðC i ; Q; DÞ

ð28Þ

599

600 602 603 604 605 606

607

C i 2T

(4) If |DIS(D; Q)| = |DIS(D; C)|, go to step 5, otherwise, go to step 3. (5) Take the final Q = {C1, C2, Cl} as the feature selection results, build the reduced training sample feature set:

ð29Þ

609 610 611 612 613

where xi is the i’th target reduced feature vector, l is the reduced feature dimensions number. To sum up, applications and performance of the feature extraction and selection method are discussed in subsequent Section 6.2.

614

5. Target classification and recognition

615

In this paper, the classifier is designed based on dynamic probability generative model. The main idea of the proposed method is derived from building a new generative model in an undirected graph, in which the edges of the graph are observed variables and the classes of the nodes whose classes are unknown are latent variables. The values of the latent variables can be calculated by fitting the generative model of the graph. The method does not require training process, it directly uses iterative formula to calculate the class of the unknown node. In this section, first we propose the concept of class propagation distribution, and use the concept to express the probability of a node’s neighbors belonging to each class. Then based the class distribution, the dynamic probability generating model is presented. Finally, by solving the model, the classes of the nodes whose classes are unknown are obtained.

616 617 618 619 620 621 622 623 624 625 626 627 628 629

5.1. Dynamic probability generative model based on the classes of neighbor nodes

630

5.1.1. Class propagation distribution In the networks with low homophily (WcPherson, SmithLovin, & Cook, 2001), there exists a majority of connected nodes whose classes are different from each other. So just depending on whether there is an edge between two nodes to judge whether they belonging to the same class are not enough. Therefore, this paper considers two nodes’ classes and their neighbor nodes’ classes at the same time. Assume there two nodes Vi and Vj, and Vj belongs to the class Lc, in the neighbor nodes of Vj, the more nodes belong to class Lc, the greater probability of edges exists between the nodes Vi and Vj. This paper differs from the class propagation distribution proposed in (Wang, Xiao, and Tan (2013), which only considered the neighbor nodes classes and had no limit on the number of neighbor nodes. It’s necessary to discuss a node’s own class contribution for improving classification accuracy, and in order to reduce the computing time, also discuss how many neighbor nodes should participate in computation. First, corresponding definitions are listed as follows: Definition 4: e- neighborhood distance In an undirected graph G = (V, E), in which there exists two nodes Vi and Vj, defines Vi’s e - neighborhood distance as below:

632

    De V i ; V j ¼ EðV i ; eÞ dist V i ; V j

654 656

ð30Þ

631

633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653

where dist(i, j) means the Euclidean distance between two nodes Vi and Vj, E(Vi, e) is the local entropy of the center Vi and the radius e(dist(Vi, Vj)/2 6 e 6 dist(Vi, Vj)). Then quickly sort the neighbor nodes of Vi in accordance with the e-neighborhood distance, and select Mi’th nodes from top of sorted neighbor nodes list (the value of Mi is about equal to 40%60% ratio of the all the nodes). Definition 5: class propagation distribution Suppose that there exists K classes, the node Vi has Mi neighbor nodes, in which there are Mic nodes belonging to class Lc, and the ratio of the node Vi itself and its neighbor nodes belonging to the class Lc is defined as:

657

 N 1 1a xi ic þ a ; xi ¼ ; K Mi K þ1

669

0
ð31Þ

For convenience, Eq. (31) can be formulated as:

658 659 660 661 662 663 664 665 666 667 668

671 672

673

hic ¼ xi

N ic 1 þa K Mi

ð32Þ

So the vector (hi1, , hic, , hiK) is called the class propagation distribution of Vi, which is usually denoted as hi. From the above definitions, for a node, despite its class propagation distribution is determined by a node’s own class and its neighbor nodes’ classes, by means of the subsequent Section 6.3, we can reasonably infer that when xi is close to 1 and a is close to 0, the better classification effect is obtained, so we change Eq.(32) into:

675 676 677 678 679 680 681 682

683

Nic hic ¼ Mi

ð33Þ

For simplicity, to a node, we only consider its neighbor nodes’ classes in the subsequent discussion, so this paper uses the class propagation distribution of Vi and the class of Vj to describe the probability of existing edge between Vi and Vj, which is the component of the class propagation distribution of Vi in the class of Vj.

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W. Guo et al. / Expert Systems with Applications xxx (2014) xxx–xxx Table 2 Symbols interpretation. Symbol

Interpretation

S Sij VL VU Vi L

The adjacency matrix of the undirected graph G The element in the i’ th row and j’th column of the matrix S The set of nodes whose classes are known The set of nodes whose classes are unknown The i’th node in V The set of classes which are composed of the classes of the nodes in VL The c’th class in L The index indicating the position of the class of Vi in L The class propagation distribution of Vi The matrix which is composed of the class propagation distribution of all nodes

Lc yi hi h

691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713

714 716

5.1.2. Probability generative model based on class propagation distribution Before showing the usage of our model, we give the related symbols and their interpretations. We use an undirected graph G = (V, E) to express the class nodes, here, E is the set of edges, V is the set of nodes. Table 2 shows the interpretations of each symbol. Take yi as the class index of node Vi, namely, if the class index of Vi is yi, then the class of Vi is denoted by Lyi . If there is an edge between nodes Vi and Vj, then Sij = 1, otherwise Sij = 0. Assume in an undirected graph, there are N nodes belonging to K classes, and the creating edges steps are as follows: First of all, generate the class propagation distribution of each node from Dirichlet prior with parameter b. Then, for anode whose class is unknown, get an integer from the uniform distribution between 0 and K-1, and take the integer as the class index of the node, so assign a class to each node. Finally, the edges between Vi and Vj is generated from Bernoulli distribution, and the Bernoulli distribution parameter is the component hiyj of the class propagation distribution of Vi in class Lyj . Use y to express the set which is composed of the class indexes of the nodes whose classes are unknown, then the model’s joint probability distribution of observed variables and latent variables is shown as:

pðE; yjhÞ ¼ pðyÞpðEjy; hÞ

ð34Þ

727

In this paper, we apply the Gibbs sampling method to fitting the model. Gibbs sampling is a fast and efficient Markov chain Monte Carlo sampling method, which is commonly used in probability generative models (Zhang, Sun, & Ding, 2011). Usually Gibbs sampling is adopted iterative formula to get the values of latent variables by sampling from the posterior distribution of the latent variables. The model requires the posterior probability pðyu ¼ xjyu ; E; hÞ, here, yu represents the class indexes of the nodes except Vu.Fitting the model steps are as follows: First of all, expand Eq. (34):

730

jV j  a 1 pðyÞ ¼ K j¼1

717 718 719 720 721 722 723 724 725 726

728

U

ð35Þ

731

733

N Y K  N Y K W ij Y Y pðEjy; hÞ ¼ hiyj ¼ ðhic ÞC ic i¼1 j¼1

ð36Þ

i¼1 c¼1

737

In the above Eq. (36), Cic is the number of edges between Vi and the nodes whose class is Lc. In terms of the production process of the model, an estimator of the c’th element of parameter hi is obtained as:

740

hic ¼ PK

734 735 736

738

C ic þ b

k¼1 ðC ic

þ bÞ

ð37Þ

where hic is equal to the proportion of the nodes whose class is Lc in all neighbor nodes of Vi. The estimator can be used as the value of hi in calculation. The hic procedure is available in Appendix A. Finally, get posterior probability of Gibbs sampling:

2

33W iu ! 2   u u N Y C þ 1 þ b C þ 1 þ b ix ix 4 4 55 pðyu ¼ xjyu ; E; hÞ /  P   C uix þ b 1 þ Kk¼1 C uik þ b i¼1;i–u

741 742 743 744

745

 C uix

ð38Þ

747

where, apart from says the number of edges between Vi and the nodes whose class is Lc. The procedure is relegated to Appendix B. In this section, for the classification we use direct iteration method, and abandon the training process, so reduce the training time. According to Eq. (38) sampling, once the iteration is beyond burn-in period, go on with 20 iterations and end the processing. Besides, at each iteration, we record each unknown node class. So, to a node, in terms of Definition5, take the assigned highest number class as its final class.

748

6. Experimental results and analysis

758

6.1. Experiments settings

759

In order to test the efficiency of the proposed method, experimental data have been randomly selected from Google map’s aerial images. Firstly, extract the initial contour of images by active contour segmentation model based on regional significance as mentioned in Section 2.1. Secondly, segment images with improved CV model using entropy and local neighborhood information as discussed in Section 2.2. Finally, get target slices by GPAC (Graph Partitioning Active Contours) algorithm (Manjunath & Sumengen, 2006), as shown in Fig. 8. There are six vessels experimental data, and each vessel has 20 pieces, so there are a total of 120 pieces images.

760

6.2. Recognition feature extraction and selection

771

As explained in Sections 2.1 and 2.2, we get the 48-dimensional feature vector. Then take the ship training samples set as the domain U, its corresponding feature vectors as the condition attributes set C, the corresponding class as decision attributes set D. By Eq. (25), construct the decision table as below:

772

 Vu, C uix

2

F 11 6 . SðU; C; DÞ ¼ 6 4 ..

F 160

3

F 48 1 d1 .. .. .. 7 7 . . . 5 48

F 60 d60

750 751 752 753 754 755 756 757

761 762 763 764 765 766 767 768 769 770

773 774 775 776

777

ð39Þ 779

Where F ji represents the i’th feature value of the j’th training sample. As was pointed out in Section 2.1, we get the reduction result T Q = {C1, C3, C5, C43, C48}, where C i ¼ ðF i1 ; F i2 ; F i60 Þ is the feature vectors which are composed of the i’th feature of all the samples. With respect to Eq. (27), compute the significance weight of individual feature: W C k ¼ SGFðC k ; Q ; DÞ, for convenience processing, it can be modified as:

WC W ¼P k ; Ck W Ci

749

780 781 782 783 784 785 786

787

_

Ci 2 Q

ð40Þ

The selected valid features is Z = {F3, F4, F5, F11, F18, F29, F41, F44, F46, F47, F48}, namely, including the ratio of length to width, perimeter, entropy, compactness, the 7’th Hu moment invariant, the 11’th Zernike moment invariant, three of the most influential wavelet moment invariants: F41,F44, F46(F41 = ||W010||,F44 = ||W101||,F46 = ||W111||) and the first and last ARC.

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Fig. 10. Results of different a.

(a) Carrier

(b) Frigate

(d) Container

(e) Cargoship

(c) Destroyer

(f) Oil tanker

Fig. 8. Example of experiment data.

796 797 798 799 800 801 802

Fig. 9 shows the significance weight of the valid candidate features. It can conclude that based on rough set theory, the common discernibility degree is utilized to compute the significance weight of each candidate feature well and select valid recognition features automatically. 6.3. Influences of parameter a, b and burn-in period on classification results

810

In this experiment, we use Micro-F1 to evaluate the classification result. Micro-F1 is a real between 0 and 1, and the greater numerical value is, the better classification performance is. According to the method (Tang & Liu, 2011), calculate the value of MicroF1, as shown in Eq. (41). Where tic denotes the actual class of node Vi, yic represents the gained class by classification methods. If the actual class of Vi is Lc, then tic = 1, otherwise tic = 0. If the gained class is Lc, then yic = 1, otherwise yic = 0.

813

Micro  F1 ¼ 2 P

803 804 805 806 807 808 809

811

To test the influence of various parameters on the classification results, select 30, 40, 50, 60,70,80,90 samples from all the samples respectively. In definition 5, a is a weight coefficient. Referring to Fig. 10, if the value of a is too large, a node’s own class has greater influence on the classification results, if the value is too small, the classes of the neighbor nodes have greater influence on the classification results, however, it is easy to see that, the smaller the value of a is, the more suited the classification results appears, we can reasonably deduce a ? 0. In this paper, we take a = 0. b is a parameter of the prior distribution, before the actual data have been observed, it represents the estimator of the data distribution. Despite it can play a role in smoothing, if the value of b is too large, the prior distribution would weaken the actual data influence on the final results, which makes the classification performance decrease significantly. As can be seen from Fig. 11, if b changes in the vicinity of K1 , it has little impact on the classification performance. Burn-in period refers to the iterations of Gibbs sampling. Based on Gibbs sampling theory, once the sampling reaches burn-in period, sampling results would reach a steady state. It can be seen from Fig. 12, despite burn-in period increases, there is little change

P

i;c ðyic i;c ðyic

 t ic Þ þ t ic Þ

Fig. 9. Significance weight of candidate features.

ð41Þ Fig. 11. Results of different b.

Fig. 12. Results of different burn-in.

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W. Guo et al. / Expert Systems with Applications xxx (2014) xxx–xxx Table 3 Experimental data of algorithms (I). Method

Performance index

Group 1

Group 2

Group 3

Group 4

KNN

Average Average Average Average Average Average Average Average

65.4 0.491 60.5 0.075 61.8 0.050 68.7 0.043

68.9 0.462 70.7 0.076 78.1 0.048 80.1 0.041

69.6 0.423 72.3 0.079 84.0 0.046 86.2 0.038

75.8 0.361 76.1 0.076 86.2 0.045 88.1 0.036

SVM HDR The proposed method

recognition rate (%) time consumption (s) recognition rate (%) time consumption (s) recognition rate (%) time consumption (s) recognition rate (%) time consumption (s)

Table 4 Experimental data of algorithms (II). Method

Performance index

Group 5

Group 6

Group 7

Group 8

KNN

Average Average Average Average Average Average Average Average

78.1 0.299 78.3 0.071 87.0 0.041 87.8 0.039

79.6 0.263 80.1 0.065 88.7 0.039 89.1 0.035

82.8 0.241 83.2 0.057 89.1 0.037 88.9 0.029

84.8 0.215 86.1 0.046 90.2 0.029 90.1 0.021

SVM HDR The proposed method

recognition rate (%) time consumption (s) recognition rate (%) time consumption (s) recognition rate (%) time consumption (s) recognition rate (%) time consumption (s)

837

in classification performance, namely, once the iterations is 10, sampling results have reached the steady state.

838

6.4. Recognition performance comparisons

839

As was pointed out in Section 4.2.1, based on rough set theory, we use common discernibility degree to compute the significance weight of each candidate feature and select valid recognition features automatically, and take the valid recognition features as sample sets to run recognition tests. We use the average obtained from many experiments to verify the proposed algorithm recognition performance (average recognition rate, average time consumption) and compared it with the results performed by k-nearest neighbor (KNN) (Sun, 2008) method, support vector machine (SVM) method and hierarchical discriminant regression (HDR) method. In this paper, a total of 8 groups’ experiments have been done. In each experiment, randomly select 30, 40, 50, 60, 70, 80, 90, and 100training samples from 6 class cruisers. Repeat 10 times, for each group, take the average of 10 experiments data as the final recognition rate. The 8 sets of experiments are denoted: groups 1, 2, 3, 4, 5, 6, 7, and 8 respectively. Test data are shown in Tables 3 and 4. Based on the comparisons between Tables 3 and 4, it is well known that, under the same experimental conditions, the correct recognition rate of proposed algorithm is slightly higher than HDR classifier and is about 10% higher than other algorithms. Its time consumption is significantly lower than HDR classifier in each group and is at least 90% lower than other algorithms. As aforementioned, it can easily be found that the proposed method has advantages compared with the present traditional methods: KNN, SVM and HDR. KNN method has taken the nearest Euclidean distance samples classes as output, it can’t make use of the training sample distribution, which causes the lower recognition performance. SVM method has been proposed based on statistical learning theory, which is suitable for two classes’ recognition, to expand to multi-class recognition, the performance has declined. HDR has unified the classification and regression problems to return to the issue of regression, which is a tree classifier and has the ability of incremental learning, so with the samples increasing, it has a higher recognition rate, but it costs a long time.

836

840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873

The proposed method has an efficient recognition performance, because of the direct iteration method without the training process, and not all the neighbor nodes participating in computation. The proposed method uses direct iteration method without the training process, it’s helpful for classification, in addition, only a portion of all neighbor nodes are used to participate in computation, it lowers the computing time, so the proposed method is an efficient classification method.

874

7. Conclusion

882

This paper has proposed a novel approach for ship recognition using optical remote sensing data based on dynamic probability generative model. The proposed approach can be considered a reliable and timely recognition method for battlefield’s targets on the sea. To start with, image segmentation and simple shape analysis is adopted to obtain ship candidates. Image segmentation remains a challenging problem. However, we are beginning to make substantial progress through the introduction of CV model using entropy and local neighborhood information. Firstly, an initial curve is extracted with the visual saliency detection mechanism, which reduces the influence of initial contour position adaptively and automatically. And then, in the cost function of the model, the interior and exterior energies are weighted by the entropy, which improve the robust of the evolving curve, and the local information is considered rather than the global statistics, which reduces the impact of the heterogeneous grays inside of regions and improves the segmentation results. Secondly, an attribute reduction algorithm based on common discernibility degree is adopted to select recognition features automatically. In this section, on the basis of the extracted invariant features, based on rough set theory, the common discernibility degree is used to compute the significance weight of each candidate feature and select valid recognition features automatically. It is also worthwhile to mention in this context that the rough set theory ensures the relatively high reduction rate and simultaneously reduce time complexity. Finally, an efficient classifier with dynamic probability generative model is applied to recognizing the ship target. In this sec-

883

Please cite this article in press as: Guo, W., et al. A remote sensing ship recognition method based on dynamic probability generative model. Expert Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa.2014.03.033

875 876 877 878 879 880 881

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tion, the classifier is designed based on dynamic probability generative model, its main idea is to build a new generative model in the undirected graph, in which the edges of the graph are observed variables and the classes of the nodes whose classes are unknown are latent variables. The values of the latent variables can be calculated by fitting the generative model to the graph. Consequently, the classes of the nodes whose classes are unknown are obtained. In feature extraction, besides, commonly used size, texture, shape and moment invariants features, an area ratio code has been introduced to enhance the representation ability of the feature set. Our experiments further prove the new feature to be a powerful feature for the target classification. In feature selection, based on rough set theory, the common discernibility degree is utilized to compute the significance weight of each candidate feature and select valid recognition features automatically. Experimental results in this paper also show that the attribute reduction algorithm based on common discernibility degree ensures the relatively high reduction rate and simultaneously reaches fairly low time complexity in incomplete information system, it can be a practical method for feature selection and helpful for ship recognition. In classification, the proposed method has shown to be more discriminative than KNN, SVM and HDR methods. Moreover, by the analysis and comparisons, experimental results show that the classification strategies based on dynamic probability generative model is helpful for achieving a good classification performance. It turns out to be more suitable for real-time target recognition application. Although our approach has shown promising results overall, several issues that necessitate our further improvement or refinement to enhance the performance of the proposed method still remain to be discovered. Firstly, the proposed method is heuristic, it consists of several steps containing a lot of thresholds, some of them should be further refined. A semi-supervised hierarchical strategy may be a better solution. Secondly, missing detection exists when the gray of a ship is very close to that of its neighbor. Suitable local segmentation and matching may be a solution. And then, false candidates, which mainly comprise ports and sea clutter, especially in the case of relatively lower resolution also exist. More effective features are needed to distinguish between them. For example, the information with regard to the context of an image should be helpful. Finally, the time consumption of the classifier is not ideal in multi-class tasks. As future work, to further our investigation, we pretend to explore different methods: better image segmentation, more effective feature extraction and detailed feature selection, efficient classification, the use of multispectral features, and a test of the proposed approach on a larger set of remote sensing images over a wide resolution range.

912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961

Appendix A.

971

A.1. Procedure of the estimator of c’th component hic of parameter hi

972

Posterior probability distribution of parameter hi:

Q5

Panagiotakis et al. (2011).

974

pðhi ; EjaÞ / pðhi ; EjaÞ ¼ pðhi jaÞpðEjhi Þ pðEjaÞ N  K  W ij C ik Y Y ¼ pðhi jaÞ hiyj ¼ pðhi jaÞ hiyj j¼1

Acknowledgment

965

The authors would like to acknowledge Wuhan Digital Engineering Research Institute for providing materials and equipments for this study, they also like to thank the anonymous reviewers and the associate editors for their constructive comments and suggestions. Furthermore, they should thank the America Dr Wang for giving useful advice for the related algorithm.

966 967 968 969 970

ðA:1Þ 976

k¼1

Because,

977

978

N K Y CðK aÞ Y a1 pðhi jaÞ ¼ Dirichlet ðhi jaÞ ¼ hik K C ð a Þ i¼1 k¼1

ðA:2Þ

We have:

980 981

982 K  C ik Y pðhi jaÞ hiyj / Dirichletðhi jbÞ

ðA:3Þ

k¼1

Where b = [a + Ci1, , a + Cic, , a + Cik], Cic says the number of edges between Vi and the nodes whose class is Lc. And Accordingly,

pðhi jEi ; aÞ / Dirichlet ðhi jbÞ

ðA:4Þ

The value of parameter hi can be estimated by the expectations of its posterior distribution. In fact, the estimator of the c’th component hic of hi is expressed as:

^hic ¼ P K

C ic þ a

k¼1 ðC ic

ðA:5Þ

þ aÞ

984 985 986 987

988 990 991 992 993

994 996

Appendix B.

997

B.1. Procedure of posterior probability of Gibbs sampling

998 999

1000

pðE; yu ¼ k; yu jhÞ pðyÞpðEjy; hÞ ¼ pðE; yu jhÞ pðyu ÞpðEjyu ; hÞ pðyÞ pðEjy; hÞ 1 pðEjy; hÞ ¼ ¼ pðyu Þ pðEjyu ; hÞ K pðEu jyu ; hÞpðEu jyu ; þÞ

pðyu ¼ xjyu ; E; hÞ ¼

/

pðEjy; hÞ pðEu jyu ; hÞ

ðB:1Þ

Eu is the set which is composed of all the edges connected by the nodes except Vu. According to Eqs. (26) and (27), we obtain the equation:

1002 1003 1004 1005

1006 N Y N  N Y K W ij Y Y pðEjy; hÞ ¼ hiyj ¼ ðhic ÞC ic i¼1 c¼1

i¼1 j¼1

¼

N Y K Y i¼1 c¼1

964

973

pðhi jE; aÞ ¼

8. Uncited reference

962 963

W. Guo et al. / Expert Systems with Applications xxx (2014) xxx–xxx

PK

C ic þ a

k¼1 ðC ic

!Cic

þ aÞ

ðB:2Þ 1008

Similarly, we also have Eq. (B.3):

1009

1010

pðEu jyu ; hÞ ¼

N Y

N  Y

hiyj

W ij

i¼1;i–u j¼1;j–u

1C uic  u C þ a ic @ A ¼ PK  u C þ a i¼1;i–u c¼1 ic k¼1 N K Y Y

0

ðB:3Þ

Please cite this article in press as: Guo, W., et al. A remote sensing ship recognition method based on dynamic probability generative model. Expert Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa.2014.03.033

1012

ESWA 9246

No. of Pages 13, Model 5G

18 April 2014 W. Guo et al. / Expert Systems with Applications xxx (2014) xxx–xxx

Given,

1013

1014

1016

K X ðC ik þ aÞ ¼ di ;

K   X  C uik þ a ¼ si

k¼1

k¼1

ðB:4Þ

Then,

1017

1018

QN QK Cic þaC ic pðyu ¼ xjyu ;E;hÞ /

i¼1

c¼1

di

QK Cu þaCu

QN

i¼1; i–u

c¼1

si

QK C ub þaCub QN b¼1

¼



QN

K  Y C

i¼1;i–u

du

i¼1; i–u

c¼1

ic þa

Cic

di

QK Cu þaC u c¼1

si

0 0 0 C ic 1 C ic 11W iu C ic þa C ic þa N Y K B N BY K B Y Y C CC B di C B B di CC / @  u Cu A ¼ @ @ u C uic AA C ic þa si

i¼1;i–u c¼1

i¼1;i–u

c¼1

C ic þa si

0 0 Cix Cic 11W iu C ix þa C ic þa N B K Y Y B CC B di B di CC ¼ @ u Cuix @ u Cuic AA i¼1;i–u

C ix þa si

C ic þa si

c¼1;c–x

0  u N B Y 1 þ a si B C ix þ ¼  u @ C þ a di ix i¼1;i–u

!C u



C uix þ 1 þ a di

!

0 Cic 11W iu C ic þa K Y B di CC B CC @ u Cuic AA

c¼1;c–x

C ic þa si

1020

ðB:5Þ

1021

With respect to the above Eq. (B.5), c cannot be equal to the class  index x of node Vu, C uic ¼ C ic .Correspondingly, we have the following equation: 0 !C uix !  u 1W iu   C N K Y C uix þ 1 þ a si C uix þ 1 þ a Y si ic A @ pðyu ¼ xjyu ;E;hÞ /  u di di C ix þ a di c–x i¼1;i–u

1022 1023

1024

N Y

0



u @ C ix þ 1 þ a /  C uix þ a i¼1;i–u

1026

11W iu  C uix þ 1 þ a A A PK   1 þ k¼1 C uik þ a ðB:6Þ

References

1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055

!C u 0 @

Q6

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