Vessel attachment nodule segmentation using integrated active contour model based on fuzzy speed function and shape–intensity joint Bhattacharya distance

Signal Processing 103 (2014) 273–284

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Signal Processing journal homepage: www.elsevier.com/locate/sigpro

Vessel attachment nodule segmentation using integrated active contour model based on fuzzy speed function and shape–intensity joint Bhattacharya distance Kan Chen, Bin Li n, Lian-fang Tian, Wen-bo Zhu, Ying-han Bao School of Automation Science and Engineering, South China University of Technology, GuangZhou city 510640, Guangdong, China

a r t i c l e i n f o

abstract

Article history: Received 31 May 2013 Received in revised form 2 September 2013 Accepted 7 September 2013 Available online 20 September 2013

Because of similar intensity in adjacent regions, intensity inhomogeneity and fuzzy boundary, vessel attachment nodule segmentation is a difficult task. In this paper, a novel integrated active contour model based on fuzzy speed function and shape–intensity joint Bhattacharya distance is proposed for vessel attachment nodule segmentation. First, the regularization component of the energy function is formulated according to the fuzzy speed function. The fuzzy speed function is calculated by the degree of membership based on intensity information and shape information. Second, the data component of the energy function is formulated according to the shape–intensity joint Bhattacharya distance between probability density of local region and probability density of the internal and external contour curve. To calculate the estimated probability density, intensity information and shape information are utilized. The proposed integrated active contour model is experimented on both the synthetic and clinical vessel attachment nodules for evaluating its performance. Experimental results demonstrate that the proposed integrated active contour model can accurately segment vessel attachment nodules. Crown Copyright & 2013 Published by Elsevier B.V. All rights reserved.

Keywords: Vessel attachment nodule Image segmentation The integrated active contour model Fuzzy speed function Shape–intensity joint Bhattacharya distance

1. Introduction Image segmentation is one of the first stages in many image analysis applications. In the domain of biomedical image processing, correct image segmentation would aid physicians a lot in identification of diseases [1]. In Computed Tomography (CT) images, the intensity of nodule is usually very similar to their adjacent vessels, and nodules have the feature of irregular shapes and intensity inhomogeneity, the accurate segmentation of vessel attachment nodule is still an extremely difficult problem. Active contour models have been extensively applied to image segmentation. There are several desirable advantages of active contour models over classical image

n

Corresponding author. Tel./fax: þ86 20 87110719. E-mail address: [email protected] (B. Li).

segmentation methods, such as edge detection and region growth [2]: (1) active contour models can achieve subpixel accuracy of object boundaries; (2) active contour models can be easily formulated under a principled energy minimization framework, and allow the incorporation of various prior knowledge, such as shape and intensity distribution, for robust image segmentation; (3) the active contour models can provide smooth and closed contours as segmentation results, which are necessary and can be readily used for further application, such as shape analysis and recognition. These active contour models can be categorized into three major classes [3]: the edge-based active contour models [4–10], the region-based active contour models [11–15] and the integrated active contour models [16–17]. Edge-based models use local edge information to attract the active contour toward the object boundaries. Regionbased models aim to identify each region of interest by

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using a certain region descriptor to guide the motion of the active contour. The integrated active contour models introduce the edge-based active contour models into the region-based active contour models. Those traditional active contour models tend to rely on difference intensity in each of the regions to be segmented. Recently, to solve the problem caused by intensity inhomogeneity, Li et al. proposed a local region intensity information-based active contour model [2]. Because of using local region intensity information, specifically local intensity mean, this model can cope with intensity inhomogeneity. Some related active contour models were proposed in Refs. [18–25] which have similar capabilities of handling intensity inhomogeneity. However, intensity information-based active contour models often face difficult challenges such as similar intensity in adjacent regions [26]. To solve the problem caused by similar intensity in adjacent regions, Yang et al. combined shape information in the active contour model [26]. Some related active contour models were proposed in Refs. [26–32]. Because of using shape information, those active contour models can solve segmentation problem of similar intensity in adjacent regions. However, the gradient informationbased speed functions of those active contour models do not approach zero at fuzzy boundary, so the problem of leakage may be caused. To accurately segment vessel attachment nodules, a novel integrated active contour model based on fuzzy speed function and shape–intensity joint Bhattacharya distance is proposed in this paper. The main contributions of the paper are summarized below: (1) The fuzzy speed function is incorporated into the integrated active contour model. In this integrated active contour model, the speed function is not calculated using gradient information, but using fuzzy characteristics based on intensity information and shape information. (2) The Bhattacharya distance between probability density of local region and probability density of the internal and external contour curve is incorporated into the integrated active contour model. In the proposed integrated active contour model, the data component of the energy function is not calculated using intensity

information, but the data component of the energy function is calculated using shape–intensity joint probability density. The proposed integrated active contour model is experimented on both the synthetic and clinical vessel attachment nodules for evaluating its performance. Experimental results demonstrate that the proposed integrated active contour model can accurately segment vessel attachment nodules. The remainder of this paper is organized as follows. Section 2 introduces the proposed integrated active contour model. Section 3 presents experimental results using synthetic and clinical vessel attachment nodules, followed by some discussions in Section 4. In Section 5, the conclusion of the proposed integrated active contour model is provided. 2. The proposed integrated active contour model To accurately segment vessel attachment nodules, a novel integrated active contour model based on fuzzy speed function and shape–intensity Bhattacharya distance is proposed in this paper. The architecture of the proposed integrated active contour model is shown in Fig. 1, which consists of four main steps: (1) the regularization component of the energy function is calculated according to the fuzzy speed function; (2) the data component of the energy function is calculated according shape–intensity joint Bhattacharya distance between probability density of local region and probability density of the internal and external contour curve; (3) the punishing component of the energy function is calculated according to the fuzzy speed function; (4) the contour is evolved using the proposed integrated active contour model. 2.1. Overview of the proposed integrated active contour model Let Ω  R2 be the image domain, V be the feature domain, which is comprised of two-dimension feature vector that contains shape index [33] and intensity. The shape index describes the shape information. The segmentation of vessel attachment nodules is achieved by finding a contour C, which separates Ω into disjoint regions (the inside regions Ω1 and

The data component of energy function is calculated

Vessel attachment nodules

Obtaining lung parenchyma

The regularization component of energy function is calculated

The contour is evolved using the proposed integrated active contour model

The punishing component of energy function is calculated

Fig. 1. The architecture of the proposed integrated active contour model.

The rule-based filtering

Vessel attachment nodule segmentation results

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outside regions Ω2 of the contours C). ϕ is a level set function in image region, the contour C is represented by the zero level set. The level set function ϕ takes positive and negative values outside and inside the contour C, respectively. The energy function of the proposed integrated active contour model is defined as EðϕÞ ¼ ED ðϕÞ þμER ðϕÞ þ vEP ðϕÞ

ð1Þ

where ED ðϕÞ is the data component for modeling the image information of different regions. ER ðϕÞ is the regularization component for controlling the length of the boundaries and preventing over-segmentation. EP ðϕÞ is the punishing component for controlling the evolution of ϕ. The predefined constants μ and v are the weights. 2.2. Regularization component in the proposed integrated active contour model For handling fuzzy boundary leakage and preventing over-segmentation, the fuzzy speed function is incorporated into the regularization component of the proposed integrated active contour model. For all points q A Ω, the regularization component ER ðϕÞ of integrated active contour model is defined as Z ð2Þ ER ðϕÞ ¼ Sðj∇Hðϕðq; tÞÞjÞdq Ω

where S is the fuzzy speed function of evolution based on intensity and shape index. In traditional integrated active contour models, the speed function of evolution is calculated using gradient information. Due to the fuzzy boundary, the speed function cannot approach zero and the problem of boundary leakage is caused. Therefore, when we define the energy function of the proposed integrated active contour model, we should ensure that the speed function of evolution approaches zero at the boundary to handle boundary leakage. In order to ensure that the proposed speed function of evolution approaches zero at the boundary of vessel attachment nodules and speed up the convergence of the active contour, the proposed fuzzy speed function of evolution (S) based on intensity information and shape information has the following properties: (1) S A ½0; 1. (2) If the active contour is at the boundary of vessel attachment nodules, S  0. (3) If the active contour is close to the boundary of vessel attachment nodules, S gets smaller and smaller. (4) If the active contour is far away from the boundary of vessel attachment nodules, S gets bigger and bigger. At all points q A Ω, the proposed fuzzy speed function is defined as SðqÞ ¼ 1 þexp½tðUðqÞ  0:5Þ2 

ð3Þ

where U(q) is the degree of membership, t is a parameter, which controls the speed of the curve evolution. The degree of membership is calculated by using the fuzzy clustering algorithm [34] based on intensity information and shape information.

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2.3. Data component in the proposed integrated active contour model As mentioned before, the data component of the energy function ED ðϕÞ is formulated according to the shape–intensity joint Bhattacharya distance between probability density of local region and probability density of the internal and external contour curve. When ED ðϕÞ is minimized, the shape–intensity joint Bhattacharya distance between probability density of local region and that of the internal contour curve are minimized. The same is true with the shape–intensity Bhattacharya distance between probability density of local region and that of the external region. The data component ED ðϕÞ of integrated active contour model is defined as Z ð4Þ ED ðϕÞ ¼ ∑i ¼ 1;2 λi Bðpi ; pl ÞMi ðϕðq; tÞÞdq Ω

where the pre-defined constants λ1 and λ2 are the weights, M 1 ðϕÞ is equal to HðϕÞ and M 2 ðϕÞ is equal 1  HðϕÞ. HðϕÞ is the Heaviside function [2]; p1 is the probability density of the internal contour curve; p2 is the probability density of the external contour curve; pl is the probability density of local region with size of r  r centered on the observed pixel q. The shape–intensity joint Bhattacharya distance Bðpi ; pl Þ between two probability densities pi and pl , with α A V, is defined as[35] Z ½pi ðαÞ1=2 ½ pl ðαÞ1=2 dα; i ¼ 1; 2 ð5Þ Bðpi ; pl Þ ¼  ln The two-dimension feature vector that contains shape index and intensity satisfies Gaussian distribution, and the shape index has been trained as some shape-prior models [26]. The probability density pi ðaÞ is defined as     1 1 pi ðaÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  ða  ui ÞT Τ i 1 ða  ui Þ ; iA 1; 2; l 2 2 ð2πÞ jΤ i j ð6Þ where u1 and Τ 1 are the mean value and covariance of two-dimension feature vector of the internal contour curve, respectively; u2 and Τ 2 are the mean value and covariance of two-dimension feature vector of the external contour curve, respectively; ul and Τ l are the mean value and covariance of two-dimension feature vector of local region, respectively; Introducing Eq. (6) into Eq. (5), the shape–intensity joint Bhattacharya distance can be rewritten as   1 Τ þΤl Bðpi ; pl Þ ¼ ðui ul ÞT i ðui  ul Þ 8 2   1 ð1=2ÞðΤ i þ Τ l Þ þ ln  ð7Þ   1=2 ; i ¼ 1; 2 2 jΤ i jΤ l 

2.4. Punishing component in the proposed integrated active contour model To regularize the evolution of the level set function, the punishing component EP ðϕÞ is utilized. For all points q A Ω, the punishing component of the integrated active contour

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model is defined as Z 1 ðj∇ϕðq; tÞj  1Þ2 dq EP ðϕÞ ¼ Ω2

3.1. Performance evaluation and parameter estimation ð8Þ

The punishing component prevents the evolving level set function from being too flat or too steep, and with this component, a larger time step can be utilized to accelerate the evolution to reach the convergence. 2.5. Minimization of the energy function Introducing Eqs. (2), (4) and (8) into Eq. (1), the energy function of the proposed integrated active contour model EðϕÞ can be rewritten as Z EðϕÞ ¼ ∑i ¼ 1;2 λi Bð pi ; pl ÞM i ðϕðq; tÞÞdq Ω

Z Z 1 ðj∇ϕðq; tÞj 1Þ2 dq þμ Sðj∇Hðϕðq; tÞjÞdq þν Ω Ω2

ð9Þ

The standard gradient descent method is adopted [18] to minimize the energy function EðϕÞ of the proposed integrated active contour model. The gradient flow of the energy function is used as the level set evolution equation. The level set evolution equation of contour is written as ∂ϕ ¼  δðϕÞðλ1 Bðp1 ; pl Þ  λ2 Bðp2 ; pl ÞÞ ∂t





 ∇ϕ ∇ϕ þv ∇2 ϕ  div þ μδðϕÞdiv S j∇ϕj j∇ϕj

ð10Þ

where δðϕÞ is the smoothed Dirac delta function [2]. 3. Experimental results The performance of the proposed integrated active contour model for vessel attachment nodule segmentation has been validated on both synthetic image and clinical image [36]. Those images are split into training images (15 synthetic vessel attachment nodule images and 15 clinical vessel attachment nodule images) and testing images (16 synthetic vessel attachment nodule images and 16 clinical vessel attachment nodule images). Unless otherwise specified, the parameters (λ1, λ2, μ and ν) for the local region intensity information-based active contour model [2], the shape–intensity based active contour model [26] and the proposed integrated active contour model are usually assigned to 1, 1, 1 and 0.65. Besides, the parameter (w) for the shape information based active contour model is usually assigned to 0.8 [26]. Figs. 3 and 4 show the experimental results of synthetic vessel attachment nodules, respectively. Figs. 5 and 6 show the experimental results of clinical vessel attachment nodules, respectively. In Figs. 3–6, the yellow curves are the results of segmentation using experienced radiologist delineation, the red curves are the results of segmentation using the local region information-based active contour model, the green curves are the results of segmentation using the shape information-based active contour model and the blue curves are the result of segmentation using the proposed integrated active contour model.

In this paper, several approaches and metrics are utilized for evaluating the segmentation results of the synthetic and clinical vessel attachment nodule images. Three overlapping area error metrics [37]: the true positive ratio (TP), the false positive ratio (FP) and the similarity (SI) were utilized. Let Aa be the pixel set of the tumor region selected by the active contour model and Am be the pixel set of the corresponding real tumor region, the three error metrics are 8 > > > < > > > :

TP ¼ jAmjA\mAj a j Am j FP ¼ jAm [jAAma  j

SI

ð11Þ

jAm \ Aa j ¼ jA m [ Aa j

when the TP ratio is higher, more real tumor regions are covered by the generated tumor regions. When the FP ratio is lower, fewer normal tissue regions are covered by the generated tumor regions. When the SI ratio is higher, the generated tumor region is more similar to the radiologist's delineation, the overall performance is better. The training datasets are used to estimate the parameter of the proposed integrated active contour model. The training datasets include the synthetic vessel attachment nodule images and clinical vessel attachment nodule images. In the fuzzy speed function, t is an important parameter. For larger value t, the integrated active contour model renders contour to be too smooth, and the burr of nodules is eliminated. For a small value t, the integrated active contour model renders contour to be too rough. So an appropriate value for the parameter t is selected for segmentation of vessel attachment nodule images. The performance of the proposed active contour model with different t is tested. Results of experiments on the training images are shown in Fig. 2(a). Fig. 2(a) shows the error rate of the results with t ranging from 0.1 to 1. While the error rate increases as t decreases, it approaches a stable value when t is less than 0.5. For this reason, t¼0.5 can be chosen as the default value in this paper. In the data component of the proposed integrated active contour model, r is an important parameter. The integrated active contour model with a small scale r has the capability of dealing with intensity inhomogeneity, but may be sensitive to noise. For larger scale r, the proposed integrated active contour model can be robust to noise; however, it may not segment images with intensity inhomogeneity accurately. An appropriate parameter r should be chosen according to the degree of intensity inhomogeneity. When the intensity inhomogeneity is severe, the parameter value r should be chosen small enough to enable the model to handle intensity inhomogeneity. For images with minor inhomogeneity, the parameter value r should be chosen a relatively large value to make the model more noise tolerant. The performance of the proposed integrated active contour model with different parameters r is tested. Results of experiments on the training images are shown in Fig. 2(b). Fig. 2(b) shows the error rate of the results with r ranging from 2 to 14. When

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Fig. 2. Segmentation error rate of the proposed integrated active contour model for different parameters t and r. (a) The errors rate of the results for different parameters t, with t¼ 0.1, 0.2, …,1; (b) the errors rate of the results for different parameters r, with r¼ 2, 3, …,14.

Fig. 3. Segmentation results of two synthetic images, which have fuzzy boundary. (a) and (e) the experienced radiologist delineation; (b) and (f) segmentation results obtained by the local region information-based active contour model; (c) and (g) segmentation results obtained by the shape information-based active contour model; (d) and (h) segmentation results obtained by the proposed active contour model. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

r approaches 7, it achieves the minimum value. For this reason, r ¼7 can be chosen as the default value in this paper. 3.2. Segmentation of synthetic vessel attachment nodule images In Fig. 3, the segmentation results of two synthetic attachment nodule images, which have fuzzy boundary, are presented and compared. The original images with manual delineation are shown in Fig. 3(a) and (e). The nodule regions generated by the local region informationbased active contour model are shown in Fig. 3(b) and (f). Compared to Fig. 3(a) and (e), it shows that the local region information-based active contour model cannot

handle adjacent nodules well. The nodule regions generated by the shape information-based active contour model are shown in Fig. 3(c) and (g), respectively. Compared to Fig. 3(a) and (e), it shows that nodule regions generated by the shape information-based active contour model cannot cover over the real nodule region. The segmentation results by the proposed integrated active contour model are shown in Fig. 3(d) and (h). Compared to Fig. 3(a) and (e), it shows that the generated nodule regions by the proposed integrated active contour model are much closer to the real nodule regions. In Fig. 4, the segmentation results of two synthetic vessel attachment images, which have intensity inhomogeneity, are presented and compared. As shown in Fig. 4, the local region information-based active contour model

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Fig. 4. Segmentation results of two synthetic images, which have intensity inhomogeneity. (a) and (e) the experienced radiologist delineation; (b) and (f) segmentation results obtained by the local region information-based active contour model; (c) and (g) segmentation results obtained by the shape information-based active contour model; (d) and (h) segmentation results obtained by the proposed active contour model. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Table 1 Three overlapping area error metrics for synthetic testing images. Method

TP ratio (%)

FP ratio (%)

SI ratio (%)

The local region information-based active contour model The shape information-based active contour model The proposed integrated active contour model

42.65 83.96 88.13

51.81 12.03 10.18

49.21 86.96 89.97

(shown as Fig. 4(b) and (f)) cannot handle adjacent nodules well, and the nodule regions generated by the shape information-based active contour model (shown as Fig. 4(c) and (g)) is also inaccurate and many normal tissues are mistakenly covered. However, the generated nodule regions by the proposed integrated active contour model (shown as Fig. 4(d) and (h)) are much closer to the real nodule regions. Three overlapping area error metrics on synthetic testing images are shown in Table 1. It shows that the TP ratio of the proposed integrated active contour model is higher, the FP ratio is lower and the SI ratio is higher, so the proposed integrated active contour model is superior to the local region information-based active contour model and the shape information-based active contour model, respectively. 3.3. Segmentation of clinical vessel attachment nodule images Figs. 5 and 6 show the segmentation results of clinical vessel attachment nodule images, which have fuzzy boundary. The original image with experienced radiologist

delineation of the nodule is shown in Figs. 5 and 6(a). The nodule regions generated by the local region informationbased active contour model and the shape informationbased active contour model are shown in Figs. 5 and 6(b) and (c), respectively. Compared to Figs. 5 and 6(a), it shows that the local region information-based active contour model (Figs. 5 and 6(b)) cannot handle adjacent nodules well, and that the nodule regions generated by the shape information-based active contour model (Figs. 5 and 6(c)) cannot cover over the real nodule region. The segmentation results by the proposed integrated active contour model are shown in Figs. 5 and 6(d). Comparing to Figs. 5 and 6(a), it shows that the generated nodule regions are much closer to the experienced radiologist delineation. Figs. 7 and 8(a) show the segmentation results of clinical vessel attachment nodule images, which have intensity inhomogeneity. The local region informationbased active contour model (Figs. 7 and 8(b)) cannot handle adjacent nodules well, and it did not converge on the real boundary of the nodule region. The shape information-based active contour model (Figs. 7 and 8 (c)) also did not generate accurate nodule regions. However, the proposed integrated active contour model

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a1

b1

c1

279

d1

Fig. 5. Segmentation results of the clinical image, which have fuzzy boundary. (a) The experienced radiologist delineation, whereas (a1) is sub-images of (a); (b) segmentation result obtained by the local region information-based active contour model, whereas (b1) is sub-images of (b); (c) segmentation result obtained by the shape information-based active contour model, whereas (c1) is sub-images of (c); (d) segmentation result obtained by the proposed active contour model, whereas (d1) is sub-images of (d). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

a1

b1

c1

d1

Fig. 6. Segmentation results of the clinical image, which have fuzzy boundary. (a) The experienced radiologist delineation, whereas (a1) is sub-images of (a); (b) segmentation result obtained by the local region information-based active contour model, whereas (b1) is sub-images of (b); (c) segmentation result obtained by the shape information-based active contour model, whereas (c1) is sub-images of (c); (d) segmentation result obtained by the proposed active contour model, whereas (d1) is sub-images of (d). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

(shown as Figs. 7 and 8(d)) can handle adjacent nodules very well, and the generated nodule regions are very close to the experienced radiologist delineation.

Three overlapping area error metrics on clinical testing images are shown in Table 2. Some vessel regions are included by the local region information-based active

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b1

a1

d1

c1

Fig. 7. Segmentation results of the clinical image, which have intensity inhomogeneity. (a) the experienced radiologist delineation, whereas (a1) is subimages of (a); (b) segmentation result obtained by the local region information-based active contour model, whereas (b1) is sub-images of (b); (c) segmentation result obtained by the shape information-based active contour model, whereas (c1) is sub-images of (c); (d) segmentation result obtained by the proposed active contour model, whereas (d1) is sub-images of (d).

a1

b1

d1

c1

Fig. 8. Segmentation results of the clinical image, which have intensity inhomogeneity. (a) The experienced radiologist delineation, whereas (a1) is subimages of (a); (b) segmentation result obtained by the local region information-based active contour model, whereas (b1) is sub-images of (b); (c) segmentation result obtained by the shape information-based active contour model, whereas (c1) is sub-images of (c); (d) segmentation result obtained by the proposed active contour model, whereas (d1) is sub-images of (d). Table 2 Three overlapping area error metrics for clinical testing images. Method

TP ratio (%)

FP ratio (%)

SI ratio (%)

The local region information-based active contour model The shape information-based active contour model The proposed integrated active contour model

44.81 86.96 88.89

50.73 12.98 10.19

45.74 88.52 89.87

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contour model, so the TP ratio of this model is low (TP¼ 44.81%), the SI ratio of this model is low (SI¼ 45.74%), and the FP ratio of this model is high (FP ¼50.73%). Some vessel regions are excluded by the shape information-based active contour model, so the TP ratio of this model is high (TP¼ 86.96%), the SI ratio of this model is high (SI¼ 88.52%), and the FP ratio of this model is low (FP ¼ 12.98%). However, the TP ratio of the proposed integrated active contour model (TP¼88.89%) is a little higher than that of the shape information-based active contour model, the SI ratio of the proposed integrated active contour model (SI¼89.87%) is a little higher than that of the shape information-based active contour model, and the FP ratio of the proposed integrated active contour model (FP¼ 10.19%) is a little lower than that of the shape informationbased active contour model. This is due to the fact that the proposed integrated active contour model can handle the blurry regions well and find the real boundaries, and the generated nodule regions are much closer to the experienced radiologist delineation. It shows that the TP ratio of the proposed integrated active contour model is higher, the FP ratio is lower and the SI ratio is higher, so the proposed integrated active contour model is superior to the local region information-based active contour models and the shape information-based active contour models, respectively. 3.4. The processing speeds of the three active contour models In this section, the processing speeds of the local region information-based active contour model, the shape information-based active contour model and the proposed integrated active contour model are also studied with testing images (16 synthetic vessel attachment nodule images and 16 clinical vessel attachment nodule images). All active contour models are performed on a Core2 PC with 2GB RAM using MATLAB 7.0. Fig. 9 shows CPU times consumed by the three active contour models. The horizontal and the vertical axes denote the CPU times and images. The blue dotted line denotes the CPU times consumed by the proposed active contour model. The green dotted line denotes the CPU times consumed by the shape information-based active contour model. The red dotted line denotes the CPU times by the local region information-based active contour

281

Table 3 Comparison of the average number of iterations and average CPU time on testing images (16 synthetic vessel attachment nodule images and 16 clinical vessel attachment nodule images). The local region information-based active contour model

The shape The proposed information-based integrated active active contour model contour models

The average number of iterations

The average CPU time (s)

The average number of iterations

The average CPU time (s)

The average number of iterations

The average CPU time (s)

350

28

300

24

110

13

model. Table 3 lists the average number of iterations (Niteration) and the required average CPU times (Nracputime) by the three active contour models. As shown in Fig. 9 and Table 3, the average number of iterations and the required CPU times of the local region information-based active contour model are high (Niteration ¼350 and Nracputime ¼ 28 s) and the average number of iterations and the required CPU times of the shape information-based active contour model are high (Niteration ¼300 and Nracputime ¼ 24 s). However, the average number of iterations and the required CPU times of the proposed active contour model are much less than those of the local region informationbased active contour model and the shape informationbased active contour model (Niteration ¼110 and Nracputime ¼ 13 s). The reason is that the fuzzy speed function, which can speed up the curve convergence, is incorporated into the proposed active contour model. In the local region intensity information-based active contour model, the shape–intensity based active contour model and the proposed integrated active contour model, the computation complexity is O(NiterationNpixel) for calculating the regularization component and the punishing component, and the time complexity is O(NiterationNpixelNwindow) for calculating the data component, where Npixel is the pixel number of lung pulmonary and Nwindow is the size of local region. Therefore, the computation complexity of the three active contour models is O(NiterationNpixelNwindow). In the three active contour models, Npixel and Nwindow have the same values, and Niteration has different values, so Niteration decides computation complexity. The fuzzy speed function can speed up the curve convergence and reduce Niteration, so the computation complexity of the proposed active contour model is much less than those of the local region information-based active contour model and the shape information-based active contour model. 4. Discussion 4.1. The fuzzy speed function

Fig. 9. The processing speed of the local region information-based active contour models, the shape information-based active contour models and the proposed integrated active contour models. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

It is well known that vessel attachment nodule has fuzzy boundary and similar intensity in adjacent region. The gradient-based speed function (g) [38] does not approach zero at the boundary of the nodule, so the problem of boundary leakage may be caused. In order to

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solve the problem of boundary leakage, the proposed fuzzy speed function (S) is incorporated into the integrated active contour model. The degree of membership (U(q)) of boundary pixels approaches 0.5, so the proposed fuzzy speed functions approaches zero according Eq. (3). In Fig. 10, the speed functions are shown in a clinical vessel attachment nodule image, which come from training datasets. At the locations from column 140 to 165 with row 71 being marked by the blue line in Fig. 10(a1), the proposed fuzzy speed function and the gradient-based speed function are plotted in Fig. 10(b). At the location from column 85 to 100 from row 128 to 148 marked by the red line in Fig. 10(a1), the proposed fuzzy speed function and the gradient-based speed function are plotted in Fig. 10(c). In Fig. 10(b) and (c), the red curve and the green curve represent the gradient-based speed function and the proposed fuzzy speed functions, respectively. As shown in Fig. 10(b), S(a) and S(b) approach zero at boundaries a and b of nodules, respectively. g(a) and g(b) do not approach zero at boundary a and b of nodules, respectively. As shown in Fig. 10(b), S(c) approaches zero in adhesion place between pulmonary nodule and vessels. g(c) does not approach zero in adhesion place between pulmonary nodule and vascular.

Introducing shape information can often solve this problem [26]. Intensity inhomogeneity often occurs in real images, local regions-based methods can often solve this problem. It is well known that probability density is related to statistical information in a region. The probability density difference, which is evaluated using Bhattacharya distance, is larger for two different objects, and smaller for two same objects. In this paper, the shape– intensity joint Bhattacharya distance between probability density of local region and probability density of the internal contour curve, and the shape–intensity Bhattacharya distance between probability density of local region and probability density of the external contour curve are used to distinguish the nodules from the vessels. In Fig. 11, the shape–intensity joint Bhattacharya distance is shown in a clinical vessel attachment nodule image, which comes from training datasets. As shown in Fig. 11(b) and (d), the disadvantage of just using intensity is that it cannot distinguish nodule from vascular tissues. However, shape–intensity joint Bhattacharya distance can distinguish between nodule and vessel. As shown in Fig. 11 (c) and (e), the shape–intensity joint Bhattacharya distance of nodule is less than 100, and the shape–intensity joint Bhattacharya distance of vessel is greater than 100.

4.2. Shape–intensity joint Bhattacharya distance

4.3. The analysis of the experimental results

It is well known that vessel attachment nodule has similar intensity in adjacent regions, so the segmentation method based on intensity often faces difficult challenges.

As shown in Figs. 3 and 4(b) and (f), Figs. 5–8(b), the local region information-based active contour model cannot handle adjacent nodules well. The main reason is that

a1

Fig. 10. The speed functions. (a) Original CT image with location label, whereas (a1) is sub-images of (a); (b) the fuzzy speed functions and the speed function based on gradient at the locations from column 140 to 165 with row 71 (blue line on original); (c) the fuzzy speed functions and the speed function based on gradient at the location from column 85 to 100 with from row 128 to 148 (red line on original). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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283

a1

Fig. 11. The shape–intensity joint local probability density. (a) Original CT image, whereas (a1) is sub-images of (a); (b) and (d) intensities of nodule and vascular, respectively; (c) and (e) shape–intensity joint Bhattacharya distance of nodule and vascular, respectively.

the intensities of nodule are usually very similar to their adjacent vascular tissues. In the local region informationbased active contour models, the motions of the active contours are guided by using only intensity information. As shown in Figs. 3 and 4(c) and (g), Figs. 5–8(c), the shape information-based active contour model can handle adjacent nodules well, and the problem of boundary leakage occurs. Nodules can be segmented from their adjacent vascular tissues in the shape-based active contour model. The reason is that shape information can handle adjacent nodules well. The problem of boundary leakage occurs in this model due to the facts that the nodule has fuzzy boundary. The speed function of evolution based on gradient information does not approach zero at the boundary of the nodule. As shown in Figs. 3 and 4(d) and (h), Figs. 5–8(d), the proposed integrated active contour model can accurately segment vessel attachment nodules. The fuzzy speed functions S are incorporated into the proposed integrated active contour model to segment vessel attachment nodules, which have fuzzy boundaries. As mentioned in Section 4.1, the advantage of the fuzzy speed function is that it can prevent evolution of active contour at boundary of nodule. The proposed fuzzy speed function approaches zero, and the evolution of the contour curve will stop. The shape–intensity joint local probability density is incorporated into the proposed integrated active contour model to segment vessel attachment nodules, which have similar intensity with adjacent vascular tissues. As mentioned in Section 4.2, the advantage of the shape–intensity joint local probability density is that it can distinguish nodule from vascular tissues and solve intensity inhomogeneity. Those are the main reasons that the proposed integrated active contour model can accurately segment vessel attachment nodules, which have fuzzy boundaries, intensity inhomogeneity and similar intensity in adjacent regions.

5. Conclusion In this paper, a novel integrated active contour model based on fuzzy speed function and shape–intensity joint Bhattacharya distance is proposed for vessel attachment nodules segmentation. In the proposed integrated active contour model, the regularization component of the energy function is formulated according to the fuzzy speed function, and the data component of the energy function is formulated according to shape–intensity joint Bhattacharya distance between probability densities of local region and probability densities of the internal and external contour curve. The standard gradient descent method is utilized for minimizing the energy function to accomplish the segmentation task. For evaluating the performance of the proposed integrated active contour model, the vessel attachment nodule regions generated by the proposed integrated active contour model are compared to the manual delineations of the nodule regions by the experienced radiologist. The overlapping area error metrics are utilized. The local region information-based active contour models, the shape information-based active contour model are also applied to the same vessel attachment nodules for comparison. The processing speeds of the proposed integrated active contour model, the local region information-based active contour model and the shape information-based active contour model are also studied. The average number of iterations and the required CPU times of the proposed active contour model are much less than those of the local region information-based active contour model and the shape information-based active contour model. The proposed integrated active contour model can achieve accurate segmentation of vessel attachment nodules, and it is superior to the local region information-based active

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