Bio-Medical Materials and Engineering 24 (2014) 539–547 DOI 10.3233/BME-130840 IOS Press
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Fuzzy Speed Function Based Active Contour Model for Segmentation of Pulmonary Nodules Kan Chen, Bin Li* , Lian-fang Tian, Wen-bo Zhu and Ying-han Bao School of Automation Science and Engineering, South China University of Technology, Guangzhou, Guangdong, China
Abstract. Pulmonary nodules are potential manifestation of lung cancer. Accurate segmentation of juxta-vascular nodules and ground glass opacity (GGO) nodules is an important and active area of research in medical image processing. At present, the classical active contour models (ACM) for segmentation of pulmonary nodules may cause the problem of boundary leakage. In order to solve the problem, a new fuzzy speed function–based active model for segmentation of pulmonary nodules is proposed in this paper. The fuzzy speed function incorporated into the ACM is calculated by the degree of membership based on intensity feature and local shape index. At the boundary of pulmonary nodules, the fuzzy speed function approaches zero and the evolution of the contour curve will stop, so the accurate segmentation of pulmonary nodules can be obtained. Experimental results on juxta-vascular nodules and GGO nodules show that the proposed ACM can achieve accurate segmentation. Keywords: Image segmentation, boundary leakage, active contour models (ACM), fuzzy speed function, fuzzy clustering
1. Introduction Lung cancer is a leading cause of cancer death in the world [1]. Pulmonary nodules are potential manifestation of lung cancer. To improve the chance of survival, an early detection of cancer is crucial [2]. In CT imaging, the intensities of juxta-vascular pulmonary nodules are usually very similar to their adjacent vascular [3], and GGO pulmonary nodules have faint contrast and fuzzy margins [4], consequently, accurate segmentation of the two kinds of pulmonary nodules is still an extremely difficult problem. The ACM are successful models for image segmentation. An advantage of the ACM for image segmentation is that it partitions an image into sub-regions with closed and smooth boundaries. These ACM can be categorized into three major classes [5]: the edge-based ACM [6-8], the region-based ACM [9-12] and the integrated ACM [13-15]. The edge-based ACM utilize image gradients to identify object boundaries. However, the models have many disadvantages: sensitive to the noise, difficult to detect the weak boundaries, highly dependent on the initial contour placement in segmentation. The region-based ACM not only utilize image information near the evolving contour, but also image statis*
Address for correspondence: Bin Li, School of Automation Science and Engineering, South China University of Technology, Guangzhou, 510640, Guangdong, China. E-mail: [email protected]. Telephone and Fax Number: +86-20-87110719. 0959-2989/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved
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K. Chen et al. / Fuzzy speed function based active contour model for segmentation of pulmonary nodules
tics inside and outside the contour. The models have many advantages over the edge-based active contour. First, they do not depend on the image gradient, and satisfactorily segment the objects with weak boundaries. Second, by utilizing the global region information, they are generally robust to the noise. Nonetheless, in the region-based ACM, the image intensities are assumed to be statistically homogeneous in each region. However, the assumption does not hold for some general images, which limits its applications. The integrated ACM integrate the two models mentioned above, thus combining the advantages of both, but it may still cause boundary leakage. Recently, to solve the boundary leakage problem, the local region information-based ACM have been employed in image segmentation [16-20]. Using local region information, these models can cope with the problem of boundary leakage, which has high contrast with their adjacent tissue. However, these models only use the intensity information, so these models often face such challenges as poor contrast or fuzzy boundaries [21]. Therefore, the boundary leakage problem may exist in this case. In this paper, a new fuzzy speed function – based active model for segmentation of pulmonary nodules is proposed. The main contributions of the paper are summarized as follows: (1) The fuzzy speed function is incorporated into the ACM, where the speed function is not calculated using gradient information but by fuzzy characteristics at the boundary of pulmonary nodules. When the active contour is at the boundary of pulmonary nodule, the fuzzy speed function values approach zero and the evolution of the contour curve will stop, so the accurate segmentation of pulmonary nodules can be obtained. When the active contour is far away from the boundary of pulmonary nodule, the fuzzy speed function values get bigger and bigger, and the evolution of the contour curve is very fast. (2) The degree of membership in fuzzy speed function is calculated using the fuzzy clustering algorithm based on intensity and local shape index. 2. ACM based on fuzzy speed function 2.1. Energy function of the proposed ACM Let ȐӨR2 be image domain and I:ȐЍR be a given gray level image. Image segmentation is achieved by finding a contour C which separates the image domain Ȑ into disjoint regions. Ȑ1 and Ȑ2 represent the inside region and outside region of the contour C, respectively. For a given pixel pѮȐ, the energy function of ACM based on the fuzzy speed function is defined as Eq. (1):
E (C ,c1 ,c2 ) = ³
L (C )
0
V3 (|C (s )|)ds + ³ V1 (p)|I (p)-c1|2 dp + ³ V2 (p)|I ( p)-c2 |2 dp Ω1
Ω2
(1)
where I(p) is the gray value of pixel p, L(C) is the length of the contour C, c1 and c2 are two average gray values in Ȑ1 andȐ2, respectively. In the right hand side of Eq. (1), the first term is the smoothing term, which forces C to be smooth within each of the regions, the second term and the third term are the data terms, which force C to approach the boundary of pulmonary nodules and V1(p), V2(p) and V3(|C(s)|) are the fuzzy speed functions, respectively . Compared with classical ACM [18], the main improvements of the proposed active contour model are summarized as follows: (1) the degree of membership in fuzzy speed function is calculated using the fuzzy clustering algorithm based on intensity and local shape index. V1(p), V2(p) and V3(p) are calculated by the fuzzy membership degree, respectively. (2) V3(|C(s)|), which forces C to be smooth within each of the regions, is incorporated into smoothing term of ACM. (3) V1(p) and V2(p), which
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force C to be close to the boundary of pulmonary nodules, are incorporated into the data terms of ACM, respectively. 2.2. Fuzzy speed function in the proposed ACM In order to stop the evolution of the active contour at the boundary and speed up the convergence of the contour which is far away from the boundary, the three fuzzy speed functions, V1(p) , V2(p) and V3(p), need to satisfy the following properties: (1) V1(p) , V2(p) and V3(p) Ѯ[0,1]. (2) If the active contour is at the boundary, V1(p) V2(p) V3(p) 0. (3) The closer the active contour is to the boundary, the smaller V1(p), V2(p) and V3(p) will be, and vice versa. In the proposed ACM based on fuzzy speed function defined by Eq. (1), V1(p) , V2(p) and V3(p) are defined as Eq. (2), Eq. (3) and Eq. (4), respectively:
V1 (p) = |exp{-t1[Z ( p)-0.5]}-1|
(2)
V2 (p ) = |exp{-t2 [0.5-Z ( p )]}-1|
(3)
V3 ( p) = exp ª¬t3 ( Z ( p) - 0.5) 2 º¼ -1
(4)
where t1, t2 and t3 are three parameters, controlling the speed of the curve evolution, the degree of membership Z(p)Ѯ[0,1] is that of pixel p. The degree of membership Z(p) in fuzzy speed function is calculated by the fuzzy clustering algorithm based on intensity and local shape index. First, the two-dimension feature vector that contains local shape index [4] and intensity is constructed. Second, the fuzzy membership matrix is constructed using fuzzy clustering algorithm [22].The Z(p) has the following properties: (1) Z(p) approaches 0.5 at the boundary. (2) When the active contour is close to the boundary, the absolute value of difference between Z(p) and 0.5 gets smaller and smaller. (3) When the active contour is far away from the boundary, the absolute value of difference between Z(p) and 0.5 gets bigger and bigger. According to Eq. (2), Eq. (3) and Eq. (4), V1(p) , V2(p) and V3(p) satisfy basic properties of the fuzzy speed functions . At the locations from column 140 to 165 with row 71 being marked by the red line in Fig. 1(a), the fuzzy speed functions are plotted in Fig. 1 (b). At the location from column 85 to 100 with the row(128 ~148) being marked by the green line in Fig. 1(a), the fuzzy speed functions are plotted in Fig. 1(c). In Fig. 1(b) and Fig. 1(c), the red curve represents V2, the green curve represents V3 and the blue curve represents V1. As shown in Fig. 1(b), V1(a), V2(a), V3(a), V1(b), V2(b) and V3(b) approach zero at boundary R and S of pulmonary nodules, respectively. As shown in Fig. 1(b), V1(c), V2(c) and V3(c) approach zero in adhesion place between pulmonary nodule and vascular, respectively. 2.3. Level set formulation of the proposed ACM In the level set methods [16], the contour C is represented by the zero level set of a Lipschitz function φ:Ω → R , which is called level set function. In this paper, the level set function φ takes positive and negative values outside and inside the contour C, respectively. The energy function of the proposed ACM can be defined as Eq. (5):
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K. Chen et al. / Fuzzy speed function based active contour model for segmentation of pulmonary nodules
E (C ,c1 , c2 ) = ³ V3 (φ (p)) ∇H (φ (p )) dp + ¦ ³ Vi (p )|I ( p)-ci |2 M i (φ (p))dp Ω
i =1,2
Ω
(5)
where c1 and c2 are two constants that approximate the image intensity in outside(C) and inside(C), M1(.) is equal to H(.) and M2(.) is equal to 1-H(.). H(.) is Heaviside function.
Fig. 1. The fuzzy speed functions. (a) Original CT image; (b) The fuzzy speed functions at the locations from column 140 to 165 with row 71; (c) The fuzzy speed functions at the location from column 85 to 100 with from row 128 to 148.
The Euler-Lagrange variation method [18] is adopted to calculate the energy function of the proposed ACM. The energy function is minimized by using the standard gradient descent method. The proposed ACM can be rewritten as Eq. (6):
° ª ½° ∂φ ∇φ º = δ (φ ) ®div «V3 (p ) » + V3 (p ) ¾ + δ (φ ) {V2 (p) [ I (p )-c2 ] -V1 (p) [ I ( p)-c1 ]} ∂t ∇φ ¼ ¯° ¬ ¿°
(6)
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where δ (φ ) is Dirac function, ∇ is gradient operator and div(⋅) is divergence operator. 3. Experimental Results 3.1. Data sets A database of 80 thoracic CT scans [23] was used to evaluate the effectiveness of the proposed ACM based on fuzzy speed function. The data was split into training datasets (15 juxta-vascular pulmonary nodules and 6 GGO pulmonary nodules) and testing datasets (16 juxta-vascular pulmonary nodules and 6 GGO pulmonary nodules). Slice thickness varied from 1mm to 3mm and the total slice number of each scan varied from 79 to 396 with an average of 199/scan. The X-ray tube current ranged from 50mA to 160 mA. The pixel size of CT image varied form 0.5mm/pixel to 0.7mm/pixel. 3.2. Performance evaluation and parameter estimation In this paper, the well known Tanimoto/Jaccard error rate A(Cm,Co) [9] is adopted to measure the segmentation results of pulmonary nodules. A(Cm,Co) is defined as Eq. (7): A (C m ,C o ) = 1 -
³ ³
C0 Cm C0 *Cm
d xd y
(7)
d xd y
where Cm is the extracted and Co is the desired contours. In the fuzzy speed functions, t1, t2 and t3 are important parameter. For larger value t3, the proposed ACM renders contour to be too smooth, and the burr of nodules is eliminated. For a small value t3, the proposed ACM renders contour to be too rough. For large value t1 and t2 , the data terms can drive the active contour toward GGO nodule boundaries, and the contour is very rough. For a small value, the data terms can’t drive the active contour toward GGO nodule boundaries. So the different range of values for the parameter t1, t2 and t3 is selected for segmentation of pulmonary nodules.
Fig 2. Segmentation error rate of the proposed ACM for different parameters t1, t2 and t3. (a) The errors rate of the results for different parameters t1, with t1=0.25, 0.5, …,2.5; (b) The errors rate of the results for different parameters t2, with t2=0.25, 0.5, 2.5; (c) The errors rate of the results for different parameters t3, with t3=0.1, 0.2, …,1;
In this paper, the performance of the proposed ACM with different parameters (t1, t2 and t3), which are the most important parameters, is tested. The results of the experiments on the training datasets are shown in Fig. 2. The Fig. 2(a) shows the error rate of the results with t1 ranging from 0.25 to 2.5. The error rate decreases with the increase of t1 and it approaches a stable value when t1 is greater than 1.5. For this reason, t1 is set to 1.5 in this paper. The Fig. 2(b) shows the error rate of the results with t2
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K. Chen et al. / Fuzzy speed function based active contour model for segmentation of pulmonary nodules
ranging from 0.25 to 2.5. The error rate decreases as t2 increases, and it approaches a stable value when t2 is greater than 1.5. For this reason, t2 is set to 1.5. The Fig. 2(c) shows the error rate of the results with t3 ranging from 0.1 to 1. The error rate increases as t3 decreases, it approaches a stable value when t3 is less than 0.5. For this reason, t3 is set to 0.5. 3.3. Comparison with local region information-based ACM Unless otherwise specified, the parameters (1, 2, and ) for the proposed ACM and the local region information-based ACM are usually assigned to 1, 1, 1 and 65.025 [21], respectively. Besides, the proposed ACM and the local region information-based ACM use the same initialization. Fig. 3 and Fig.4 show the experimental results of the juxta-vascular pulmonary nodules, respectively. Fig. 5 and Fig. 6 show the experimental results of the ground glass opacity pulmonary nodule, respectively. In Fig. 3, Fig. 4, Fig. 5 and Fig. 6, the yellow curves are the results of segmentation described by a experienced radiologist , the green curves are the results of segmentation obtained by the local region information-based ACM and the red curves are the result of segmentation by the proposed ACM.
c1
b1
a1
(a) (a1) (b) (b1) (c) (c1) Fig. 3. Segmentation result of the juxta-vascular pulmonary nodules. (a) the experienced radiologist delineation; (b) segmentation results obtained by the local region information-basedACM; (c) segmentation results obtained by the proposed ACM. Whereas (a1)-(c1) are sub-images of (a)-(c), respectively.
a1
b1
(a)
(a1)
c1
(b)
(b1)
(c)
(c1)
Fig. 4. Segmentation result of the juxta-vascular pulmonary nodules. (a) the experienced radiologist delineation; (b) segmentation results obtained by the local region information-based ACM; (c) segmentation results obtained by the proposed ACM. Whereas (a1)-(c1) are sub-images of (a)-(c), respectively.
Fig. 3 and Fig.4 shows that the juxta-vascular pulmonary nodule regions generated by the local region information-based ACM (shown as Fig. 3(b1) and Fig. 4(b1)) are inaccurate and pulmonary nodules are not distinguished from vascular. The reasons are as follows: (1) The pulmonary nodules are difficult to be distinguished from vascular just using intensity information. (2) The speed function of evolution based on gradient information does not approach zero at the boundary of the GGO nodules, so the local region information-based ACM may cause the problem of boundary leakage. However, the juxta-vascular pulmonary nodule regions generated by the proposed ACM (shown as Fig. 3(c1) and Fig. 4(c1)) are much closer to the experienced radiologist delineation and pulmonary nodules are
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segmented from vascular structures. The reason is that the speed function of evolution based on fuzzy speed functions (V1(p), V2(p) and V3(p)) approach zero at the boundary of the GGO nodule. Fig. 5 and Fig. 6 show that the GGO nodule regions generated by the local region information-based ACM (shown as Fig. 5(b1) and Fig. 6(b1)) are inaccurate and many normal tissues are mistakenly covered. The reasons is that the speed function of evolution based on gradient information does not approach zero at the boundary of the GGO nodules, thus the local region information-based ACM and the integrated ACM may cause the problem of boundary leakage. However, the GGO nodule regions generated by the proposed integrated ACM (shown as Fig. 5(c1) and Fig. 6(c1)) are much closer to the experienced radiologist delineation, because the speed function of evolution based on fuzzy speed functions (V1(p), V2(p) and V3(p)) approach zero at the boundary of the GGO nodule.
a1
b1
(a)
(a1)
c1
(b)
(b1)
(c)
(c1)
Fig.5. Segmentation result of GGO pulmonary nodule. (a) the experienced radiologist delineation; (b) segmentation results obtained by the local region information-based ACM; (c) segmentation results obtained by the proposed ACM. Whereas (a1)-(c1) are sub-images of (a)-(c), respectively.
a1
(a)
b1
(a1)
(b)
c1
(b1)
(c)
(c1)
Fig.6. Segmentation result of GGO pulmonary nodule. (a) the experienced radiologist delineation; (b) segmentation results obtained by the local region information-based ACM; (c) segmentation results obtained by the proposed ACM. Whereas (a1)-(c1) are sub-images of (a)-(c), respectively.
The Tanimoto/Jaccard error rate evaluated on testing datasets is shown in Table 1. The error rates of the proposed ACM are less than the local region information-based ACM. Table 1 The error rate evaluated on testing datasets (16 juxta-vascular pulmonary nodules and 6 GGO pulmonary nodules)
Method The error rate of juxta-vascular pulmonary nodule The error rate of the GGO pulmonary nodule
The local region information-based ACM
The proposed ACM
0.69
0.15
0.20
0.13
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K. Chen et al. / Fuzzy speed function based active contour model for segmentation of pulmonary nodules
4. Conclusion This paper proposes a new fuzzy speed function – based active model for segmentation of pulmonary nodules. The proposed fuzzy speed function – based active model can solve the problem of boundary leakage for segmentation of juxta-vascular pulmonary nodules and the GGO pulmonary nodules. The degree of membership in the fuzzy speed function is calculated by using the fuzzy clustering algorithm based on intensity and local shape index. The degree of membership function can enhance difference between pulmonary nodule and surrounding tissue. The fuzzy speed function is incorporated into the ACM. At the boundary of pulmonary nodules, the fuzzy speed function approaches zero and the evolution of the contour curve will stop. Experimental results show that the proposed algorithm can achieve accurate segmentation of juxta-vascular pulmonary nodules and GGO pulmonary nodules and is superior to the local region information-based ACM. 5. Acknowledgments This work is supported by National Natural Science Foundation of China (61273249, 61105062), the Natural Science Foundation of Guangdong Province, China (S2012010009886, S2011010005811), Guangdong-Hong Kong Technology Cooperation Funding under Grant(No. 2010Z11), Key Laboratory of Autonomous Systems and Network Control of Ministry of Education (SCUT of China), the National Engineering Research Center for Tissue Restoration and Reconstruction and the Guangdong Key Laboratory for Biomedical Engineering (SCUT of China). References [1]
S. Diciotti and G. Picozzi et al, 3-D segmentation algorithm of small lung nodules in spiral CT images, IEEE Transactions on Information Technology on Biomedicine 12 (2008),7-19. [2] J. Dehmeshki and H. Amin et al, Segmentation of pulmonary nodules in thoracic CT: a region growing approach, IEEE Transactions on Medical Imaging 27 (2008), 467-480. [3] X.J. Ye and G. Beddoe et al, Automatic graph cut segmentation of lesions in CT using mean shift superpixels, International Journal of Biomedical Imaging 2010(2010), 1-14. [4] X.J. Ye and X.Y. Lin et al, Shape-Based Computer-Aided Detection of Lung Nodules in Thoracic CT Images, IEEE Transactions on Biomedical Engineering 56 (2009), 1810-1820. [5] X.H. Xie, Active contouring based on gradient vector interaction and constrained level set diffusion, IEEE Transactions on Image Processing 19 (2010), 154-164. [6] S. Lankton and A. Tannenbaum, Localizing region-based active contours, IEEE Transactions on Image Processing 17(2008), 2029-2039. [7] C.M. Li and C.Y. Xu et al, Distance regularized level set evolution and its application to image segmentation, IEEE Transactions on Image Processing 19(2010), 3243-3254. [8] W.B. Tao, Iterative narrowband-based graph cuts optimization for geodesic active contours with region forces (GACWRF), IEEE Transactions on Image Processing 21(2012), 284-296. [9] S.Krinidis and V.Chatzis, Fuzzy energy-based active contours, IEEE Transactions on Image Processing 18(2009), 2747-2755. [10] D. Xiao and W.S. Ng et al, A region and gradient based active contour model and its application in boundary tracking on anal canal ultrasound images, Pattern Recognition 20(2007), 3522-3539. [11] C.M. Li and R. Huang et al, A level set method for image segmentation in the presence of intensity inhomogeneities with application to MRI, IEEE Transactions on Image Processing 20(2011), 2007-2015. [12] Y. Zhang and G.Y. Li et al, Geometric active contours without re-initialization for image segmentation, Pattern Recognition 42(2009), 1970-1976.
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[13] X. Xie and M. Mirmehdi, RAGS: region-aided geometric snake, IEEE Transactions on Image Processing 13(2004), 640-652. [14] F.Y. Shih and K. Zhang, Locating object contours in complex background using improved snakes, Computer Vision and Image Understanding 105( 2007), 93-98. [15] L. Chen and Y. Zhou et al, GACV: geodesic-aided C-V method, Pattern Recognition 39(2006), 1391-1395. [16] K.Q. Sun and Z. Chen et al, Local morphology fitting active contour for automatic vascular segmentation, IEEE Transaction on Biomedical Engineering 59(2012), 464-473. [17] S. Liu and Y. Peng, A local region-based Chan-Vese model for image segmentation, Pattern Recognition 45(2012), 2769-2779. [18] C.M. Li and C.Y. Kao et al, Minimization of region-scalable fitting energy for image segmentation, IEEE Trasactions on Image Processing 17(2008), 1940-1949. [19] L. Wang and L. He et al, Active contours driven by local Gaussian distribution fitting energy, Signal Processing 89(2009), 2435-2447. [20] C.J. He, Y. Wang and Q. Chen, Active contours driven by weighted regions-scalable fitting energy based on local entropy, Signal Processing 92(2012), 587-800. [21] J. Yang and J.S. Duncan, 3D image segmentation of deformation objects with joint shape-intensity prior models using level sets, Medical Image Analysis, 8(2004), 285-294. [22] C.C. Hung and S. Kulkarni et al, A new weighted fuzzy c-means clustering algorithm for remotely sensed image classification, IEEE Journal of Selected Topics in Signal Processing 5(2011), 543-552. [23] S.G. Armato, G. McLennan and L. Bidaut et al, The lung image database consortium (LIDC) and image database resource initiative (IDRI): a completed reference database of lung nodules on CT scans, Medical Physics, 38(2011), 915931.
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539
Fuzzy Speed Function Based Active Contour Model for Segmentation of Pulmonary Nodules Kan Chen, Bin Li* , Lian-fang Tian, Wen-bo Zhu and Ying-han Bao School of Automation Science and Engineering, South China University of Technology, Guangzhou, Guangdong, China
Abstract. Pulmonary nodules are potential manifestation of lung cancer. Accurate segmentation of juxta-vascular nodules and ground glass opacity (GGO) nodules is an important and active area of research in medical image processing. At present, the classical active contour models (ACM) for segmentation of pulmonary nodules may cause the problem of boundary leakage. In order to solve the problem, a new fuzzy speed function–based active model for segmentation of pulmonary nodules is proposed in this paper. The fuzzy speed function incorporated into the ACM is calculated by the degree of membership based on intensity feature and local shape index. At the boundary of pulmonary nodules, the fuzzy speed function approaches zero and the evolution of the contour curve will stop, so the accurate segmentation of pulmonary nodules can be obtained. Experimental results on juxta-vascular nodules and GGO nodules show that the proposed ACM can achieve accurate segmentation. Keywords: Image segmentation, boundary leakage, active contour models (ACM), fuzzy speed function, fuzzy clustering
1. Introduction Lung cancer is a leading cause of cancer death in the world [1]. Pulmonary nodules are potential manifestation of lung cancer. To improve the chance of survival, an early detection of cancer is crucial [2]. In CT imaging, the intensities of juxta-vascular pulmonary nodules are usually very similar to their adjacent vascular [3], and GGO pulmonary nodules have faint contrast and fuzzy margins [4], consequently, accurate segmentation of the two kinds of pulmonary nodules is still an extremely difficult problem. The ACM are successful models for image segmentation. An advantage of the ACM for image segmentation is that it partitions an image into sub-regions with closed and smooth boundaries. These ACM can be categorized into three major classes [5]: the edge-based ACM [6-8], the region-based ACM [9-12] and the integrated ACM [13-15]. The edge-based ACM utilize image gradients to identify object boundaries. However, the models have many disadvantages: sensitive to the noise, difficult to detect the weak boundaries, highly dependent on the initial contour placement in segmentation. The region-based ACM not only utilize image information near the evolving contour, but also image statis*
Address for correspondence: Bin Li, School of Automation Science and Engineering, South China University of Technology, Guangzhou, 510640, Guangdong, China. E-mail: [email protected]. Telephone and Fax Number: +86-20-87110719. 0959-2989/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved
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K. Chen et al. / Fuzzy speed function based active contour model for segmentation of pulmonary nodules
tics inside and outside the contour. The models have many advantages over the edge-based active contour. First, they do not depend on the image gradient, and satisfactorily segment the objects with weak boundaries. Second, by utilizing the global region information, they are generally robust to the noise. Nonetheless, in the region-based ACM, the image intensities are assumed to be statistically homogeneous in each region. However, the assumption does not hold for some general images, which limits its applications. The integrated ACM integrate the two models mentioned above, thus combining the advantages of both, but it may still cause boundary leakage. Recently, to solve the boundary leakage problem, the local region information-based ACM have been employed in image segmentation [16-20]. Using local region information, these models can cope with the problem of boundary leakage, which has high contrast with their adjacent tissue. However, these models only use the intensity information, so these models often face such challenges as poor contrast or fuzzy boundaries [21]. Therefore, the boundary leakage problem may exist in this case. In this paper, a new fuzzy speed function – based active model for segmentation of pulmonary nodules is proposed. The main contributions of the paper are summarized as follows: (1) The fuzzy speed function is incorporated into the ACM, where the speed function is not calculated using gradient information but by fuzzy characteristics at the boundary of pulmonary nodules. When the active contour is at the boundary of pulmonary nodule, the fuzzy speed function values approach zero and the evolution of the contour curve will stop, so the accurate segmentation of pulmonary nodules can be obtained. When the active contour is far away from the boundary of pulmonary nodule, the fuzzy speed function values get bigger and bigger, and the evolution of the contour curve is very fast. (2) The degree of membership in fuzzy speed function is calculated using the fuzzy clustering algorithm based on intensity and local shape index. 2. ACM based on fuzzy speed function 2.1. Energy function of the proposed ACM Let ȐӨR2 be image domain and I:ȐЍR be a given gray level image. Image segmentation is achieved by finding a contour C which separates the image domain Ȑ into disjoint regions. Ȑ1 and Ȑ2 represent the inside region and outside region of the contour C, respectively. For a given pixel pѮȐ, the energy function of ACM based on the fuzzy speed function is defined as Eq. (1):
E (C ,c1 ,c2 ) = ³
L (C )
0
V3 (|C (s )|)ds + ³ V1 (p)|I (p)-c1|2 dp + ³ V2 (p)|I ( p)-c2 |2 dp Ω1
Ω2
(1)
where I(p) is the gray value of pixel p, L(C) is the length of the contour C, c1 and c2 are two average gray values in Ȑ1 andȐ2, respectively. In the right hand side of Eq. (1), the first term is the smoothing term, which forces C to be smooth within each of the regions, the second term and the third term are the data terms, which force C to approach the boundary of pulmonary nodules and V1(p), V2(p) and V3(|C(s)|) are the fuzzy speed functions, respectively . Compared with classical ACM [18], the main improvements of the proposed active contour model are summarized as follows: (1) the degree of membership in fuzzy speed function is calculated using the fuzzy clustering algorithm based on intensity and local shape index. V1(p), V2(p) and V3(p) are calculated by the fuzzy membership degree, respectively. (2) V3(|C(s)|), which forces C to be smooth within each of the regions, is incorporated into smoothing term of ACM. (3) V1(p) and V2(p), which
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force C to be close to the boundary of pulmonary nodules, are incorporated into the data terms of ACM, respectively. 2.2. Fuzzy speed function in the proposed ACM In order to stop the evolution of the active contour at the boundary and speed up the convergence of the contour which is far away from the boundary, the three fuzzy speed functions, V1(p) , V2(p) and V3(p), need to satisfy the following properties: (1) V1(p) , V2(p) and V3(p) Ѯ[0,1]. (2) If the active contour is at the boundary, V1(p) V2(p) V3(p) 0. (3) The closer the active contour is to the boundary, the smaller V1(p), V2(p) and V3(p) will be, and vice versa. In the proposed ACM based on fuzzy speed function defined by Eq. (1), V1(p) , V2(p) and V3(p) are defined as Eq. (2), Eq. (3) and Eq. (4), respectively:
V1 (p) = |exp{-t1[Z ( p)-0.5]}-1|
(2)
V2 (p ) = |exp{-t2 [0.5-Z ( p )]}-1|
(3)
V3 ( p) = exp ª¬t3 ( Z ( p) - 0.5) 2 º¼ -1
(4)
where t1, t2 and t3 are three parameters, controlling the speed of the curve evolution, the degree of membership Z(p)Ѯ[0,1] is that of pixel p. The degree of membership Z(p) in fuzzy speed function is calculated by the fuzzy clustering algorithm based on intensity and local shape index. First, the two-dimension feature vector that contains local shape index [4] and intensity is constructed. Second, the fuzzy membership matrix is constructed using fuzzy clustering algorithm [22].The Z(p) has the following properties: (1) Z(p) approaches 0.5 at the boundary. (2) When the active contour is close to the boundary, the absolute value of difference between Z(p) and 0.5 gets smaller and smaller. (3) When the active contour is far away from the boundary, the absolute value of difference between Z(p) and 0.5 gets bigger and bigger. According to Eq. (2), Eq. (3) and Eq. (4), V1(p) , V2(p) and V3(p) satisfy basic properties of the fuzzy speed functions . At the locations from column 140 to 165 with row 71 being marked by the red line in Fig. 1(a), the fuzzy speed functions are plotted in Fig. 1 (b). At the location from column 85 to 100 with the row(128 ~148) being marked by the green line in Fig. 1(a), the fuzzy speed functions are plotted in Fig. 1(c). In Fig. 1(b) and Fig. 1(c), the red curve represents V2, the green curve represents V3 and the blue curve represents V1. As shown in Fig. 1(b), V1(a), V2(a), V3(a), V1(b), V2(b) and V3(b) approach zero at boundary R and S of pulmonary nodules, respectively. As shown in Fig. 1(b), V1(c), V2(c) and V3(c) approach zero in adhesion place between pulmonary nodule and vascular, respectively. 2.3. Level set formulation of the proposed ACM In the level set methods [16], the contour C is represented by the zero level set of a Lipschitz function φ:Ω → R , which is called level set function. In this paper, the level set function φ takes positive and negative values outside and inside the contour C, respectively. The energy function of the proposed ACM can be defined as Eq. (5):
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E (C ,c1 , c2 ) = ³ V3 (φ (p)) ∇H (φ (p )) dp + ¦ ³ Vi (p )|I ( p)-ci |2 M i (φ (p))dp Ω
i =1,2
Ω
(5)
where c1 and c2 are two constants that approximate the image intensity in outside(C) and inside(C), M1(.) is equal to H(.) and M2(.) is equal to 1-H(.). H(.) is Heaviside function.
Fig. 1. The fuzzy speed functions. (a) Original CT image; (b) The fuzzy speed functions at the locations from column 140 to 165 with row 71; (c) The fuzzy speed functions at the location from column 85 to 100 with from row 128 to 148.
The Euler-Lagrange variation method [18] is adopted to calculate the energy function of the proposed ACM. The energy function is minimized by using the standard gradient descent method. The proposed ACM can be rewritten as Eq. (6):
° ª ½° ∂φ ∇φ º = δ (φ ) ®div «V3 (p ) » + V3 (p ) ¾ + δ (φ ) {V2 (p) [ I (p )-c2 ] -V1 (p) [ I ( p)-c1 ]} ∂t ∇φ ¼ ¯° ¬ ¿°
(6)
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where δ (φ ) is Dirac function, ∇ is gradient operator and div(⋅) is divergence operator. 3. Experimental Results 3.1. Data sets A database of 80 thoracic CT scans [23] was used to evaluate the effectiveness of the proposed ACM based on fuzzy speed function. The data was split into training datasets (15 juxta-vascular pulmonary nodules and 6 GGO pulmonary nodules) and testing datasets (16 juxta-vascular pulmonary nodules and 6 GGO pulmonary nodules). Slice thickness varied from 1mm to 3mm and the total slice number of each scan varied from 79 to 396 with an average of 199/scan. The X-ray tube current ranged from 50mA to 160 mA. The pixel size of CT image varied form 0.5mm/pixel to 0.7mm/pixel. 3.2. Performance evaluation and parameter estimation In this paper, the well known Tanimoto/Jaccard error rate A(Cm,Co) [9] is adopted to measure the segmentation results of pulmonary nodules. A(Cm,Co) is defined as Eq. (7): A (C m ,C o ) = 1 -
³ ³
C0 Cm C0 *Cm
d xd y
(7)
d xd y
where Cm is the extracted and Co is the desired contours. In the fuzzy speed functions, t1, t2 and t3 are important parameter. For larger value t3, the proposed ACM renders contour to be too smooth, and the burr of nodules is eliminated. For a small value t3, the proposed ACM renders contour to be too rough. For large value t1 and t2 , the data terms can drive the active contour toward GGO nodule boundaries, and the contour is very rough. For a small value, the data terms can’t drive the active contour toward GGO nodule boundaries. So the different range of values for the parameter t1, t2 and t3 is selected for segmentation of pulmonary nodules.
Fig 2. Segmentation error rate of the proposed ACM for different parameters t1, t2 and t3. (a) The errors rate of the results for different parameters t1, with t1=0.25, 0.5, …,2.5; (b) The errors rate of the results for different parameters t2, with t2=0.25, 0.5, 2.5; (c) The errors rate of the results for different parameters t3, with t3=0.1, 0.2, …,1;
In this paper, the performance of the proposed ACM with different parameters (t1, t2 and t3), which are the most important parameters, is tested. The results of the experiments on the training datasets are shown in Fig. 2. The Fig. 2(a) shows the error rate of the results with t1 ranging from 0.25 to 2.5. The error rate decreases with the increase of t1 and it approaches a stable value when t1 is greater than 1.5. For this reason, t1 is set to 1.5 in this paper. The Fig. 2(b) shows the error rate of the results with t2
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ranging from 0.25 to 2.5. The error rate decreases as t2 increases, and it approaches a stable value when t2 is greater than 1.5. For this reason, t2 is set to 1.5. The Fig. 2(c) shows the error rate of the results with t3 ranging from 0.1 to 1. The error rate increases as t3 decreases, it approaches a stable value when t3 is less than 0.5. For this reason, t3 is set to 0.5. 3.3. Comparison with local region information-based ACM Unless otherwise specified, the parameters (1, 2, and ) for the proposed ACM and the local region information-based ACM are usually assigned to 1, 1, 1 and 65.025 [21], respectively. Besides, the proposed ACM and the local region information-based ACM use the same initialization. Fig. 3 and Fig.4 show the experimental results of the juxta-vascular pulmonary nodules, respectively. Fig. 5 and Fig. 6 show the experimental results of the ground glass opacity pulmonary nodule, respectively. In Fig. 3, Fig. 4, Fig. 5 and Fig. 6, the yellow curves are the results of segmentation described by a experienced radiologist , the green curves are the results of segmentation obtained by the local region information-based ACM and the red curves are the result of segmentation by the proposed ACM.
c1
b1
a1
(a) (a1) (b) (b1) (c) (c1) Fig. 3. Segmentation result of the juxta-vascular pulmonary nodules. (a) the experienced radiologist delineation; (b) segmentation results obtained by the local region information-basedACM; (c) segmentation results obtained by the proposed ACM. Whereas (a1)-(c1) are sub-images of (a)-(c), respectively.
a1
b1
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(a1)
c1
(b)
(b1)
(c)
(c1)
Fig. 4. Segmentation result of the juxta-vascular pulmonary nodules. (a) the experienced radiologist delineation; (b) segmentation results obtained by the local region information-based ACM; (c) segmentation results obtained by the proposed ACM. Whereas (a1)-(c1) are sub-images of (a)-(c), respectively.
Fig. 3 and Fig.4 shows that the juxta-vascular pulmonary nodule regions generated by the local region information-based ACM (shown as Fig. 3(b1) and Fig. 4(b1)) are inaccurate and pulmonary nodules are not distinguished from vascular. The reasons are as follows: (1) The pulmonary nodules are difficult to be distinguished from vascular just using intensity information. (2) The speed function of evolution based on gradient information does not approach zero at the boundary of the GGO nodules, so the local region information-based ACM may cause the problem of boundary leakage. However, the juxta-vascular pulmonary nodule regions generated by the proposed ACM (shown as Fig. 3(c1) and Fig. 4(c1)) are much closer to the experienced radiologist delineation and pulmonary nodules are
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segmented from vascular structures. The reason is that the speed function of evolution based on fuzzy speed functions (V1(p), V2(p) and V3(p)) approach zero at the boundary of the GGO nodule. Fig. 5 and Fig. 6 show that the GGO nodule regions generated by the local region information-based ACM (shown as Fig. 5(b1) and Fig. 6(b1)) are inaccurate and many normal tissues are mistakenly covered. The reasons is that the speed function of evolution based on gradient information does not approach zero at the boundary of the GGO nodules, thus the local region information-based ACM and the integrated ACM may cause the problem of boundary leakage. However, the GGO nodule regions generated by the proposed integrated ACM (shown as Fig. 5(c1) and Fig. 6(c1)) are much closer to the experienced radiologist delineation, because the speed function of evolution based on fuzzy speed functions (V1(p), V2(p) and V3(p)) approach zero at the boundary of the GGO nodule.
a1
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(a)
(a1)
c1
(b)
(b1)
(c)
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Fig.5. Segmentation result of GGO pulmonary nodule. (a) the experienced radiologist delineation; (b) segmentation results obtained by the local region information-based ACM; (c) segmentation results obtained by the proposed ACM. Whereas (a1)-(c1) are sub-images of (a)-(c), respectively.
a1
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Fig.6. Segmentation result of GGO pulmonary nodule. (a) the experienced radiologist delineation; (b) segmentation results obtained by the local region information-based ACM; (c) segmentation results obtained by the proposed ACM. Whereas (a1)-(c1) are sub-images of (a)-(c), respectively.
The Tanimoto/Jaccard error rate evaluated on testing datasets is shown in Table 1. The error rates of the proposed ACM are less than the local region information-based ACM. Table 1 The error rate evaluated on testing datasets (16 juxta-vascular pulmonary nodules and 6 GGO pulmonary nodules)
Method The error rate of juxta-vascular pulmonary nodule The error rate of the GGO pulmonary nodule
The local region information-based ACM
The proposed ACM
0.69
0.15
0.20
0.13
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4. Conclusion This paper proposes a new fuzzy speed function – based active model for segmentation of pulmonary nodules. The proposed fuzzy speed function – based active model can solve the problem of boundary leakage for segmentation of juxta-vascular pulmonary nodules and the GGO pulmonary nodules. The degree of membership in the fuzzy speed function is calculated by using the fuzzy clustering algorithm based on intensity and local shape index. The degree of membership function can enhance difference between pulmonary nodule and surrounding tissue. The fuzzy speed function is incorporated into the ACM. At the boundary of pulmonary nodules, the fuzzy speed function approaches zero and the evolution of the contour curve will stop. Experimental results show that the proposed algorithm can achieve accurate segmentation of juxta-vascular pulmonary nodules and GGO pulmonary nodules and is superior to the local region information-based ACM. 5. Acknowledgments This work is supported by National Natural Science Foundation of China (61273249, 61105062), the Natural Science Foundation of Guangdong Province, China (S2012010009886, S2011010005811), Guangdong-Hong Kong Technology Cooperation Funding under Grant(No. 2010Z11), Key Laboratory of Autonomous Systems and Network Control of Ministry of Education (SCUT of China), the National Engineering Research Center for Tissue Restoration and Reconstruction and the Guangdong Key Laboratory for Biomedical Engineering (SCUT of China). References [1]
S. Diciotti and G. Picozzi et al, 3-D segmentation algorithm of small lung nodules in spiral CT images, IEEE Transactions on Information Technology on Biomedicine 12 (2008),7-19. [2] J. Dehmeshki and H. Amin et al, Segmentation of pulmonary nodules in thoracic CT: a region growing approach, IEEE Transactions on Medical Imaging 27 (2008), 467-480. [3] X.J. Ye and G. Beddoe et al, Automatic graph cut segmentation of lesions in CT using mean shift superpixels, International Journal of Biomedical Imaging 2010(2010), 1-14. [4] X.J. Ye and X.Y. Lin et al, Shape-Based Computer-Aided Detection of Lung Nodules in Thoracic CT Images, IEEE Transactions on Biomedical Engineering 56 (2009), 1810-1820. [5] X.H. Xie, Active contouring based on gradient vector interaction and constrained level set diffusion, IEEE Transactions on Image Processing 19 (2010), 154-164. [6] S. Lankton and A. Tannenbaum, Localizing region-based active contours, IEEE Transactions on Image Processing 17(2008), 2029-2039. [7] C.M. Li and C.Y. Xu et al, Distance regularized level set evolution and its application to image segmentation, IEEE Transactions on Image Processing 19(2010), 3243-3254. [8] W.B. Tao, Iterative narrowband-based graph cuts optimization for geodesic active contours with region forces (GACWRF), IEEE Transactions on Image Processing 21(2012), 284-296. [9] S.Krinidis and V.Chatzis, Fuzzy energy-based active contours, IEEE Transactions on Image Processing 18(2009), 2747-2755. [10] D. Xiao and W.S. Ng et al, A region and gradient based active contour model and its application in boundary tracking on anal canal ultrasound images, Pattern Recognition 20(2007), 3522-3539. [11] C.M. Li and R. Huang et al, A level set method for image segmentation in the presence of intensity inhomogeneities with application to MRI, IEEE Transactions on Image Processing 20(2011), 2007-2015. [12] Y. Zhang and G.Y. Li et al, Geometric active contours without re-initialization for image segmentation, Pattern Recognition 42(2009), 1970-1976.
K. Chen et al. / Fuzzy speed function based active contour model for segmentation of pulmonary nodules
547
[13] X. Xie and M. Mirmehdi, RAGS: region-aided geometric snake, IEEE Transactions on Image Processing 13(2004), 640-652. [14] F.Y. Shih and K. Zhang, Locating object contours in complex background using improved snakes, Computer Vision and Image Understanding 105( 2007), 93-98. [15] L. Chen and Y. Zhou et al, GACV: geodesic-aided C-V method, Pattern Recognition 39(2006), 1391-1395. [16] K.Q. Sun and Z. Chen et al, Local morphology fitting active contour for automatic vascular segmentation, IEEE Transaction on Biomedical Engineering 59(2012), 464-473. [17] S. Liu and Y. Peng, A local region-based Chan-Vese model for image segmentation, Pattern Recognition 45(2012), 2769-2779. [18] C.M. Li and C.Y. Kao et al, Minimization of region-scalable fitting energy for image segmentation, IEEE Trasactions on Image Processing 17(2008), 1940-1949. [19] L. Wang and L. He et al, Active contours driven by local Gaussian distribution fitting energy, Signal Processing 89(2009), 2435-2447. [20] C.J. He, Y. Wang and Q. Chen, Active contours driven by weighted regions-scalable fitting energy based on local entropy, Signal Processing 92(2012), 587-800. [21] J. Yang and J.S. Duncan, 3D image segmentation of deformation objects with joint shape-intensity prior models using level sets, Medical Image Analysis, 8(2004), 285-294. [22] C.C. Hung and S. Kulkarni et al, A new weighted fuzzy c-means clustering algorithm for remotely sensed image classification, IEEE Journal of Selected Topics in Signal Processing 5(2011), 543-552. [23] S.G. Armato, G. McLennan and L. Bidaut et al, The lung image database consortium (LIDC) and image database resource initiative (IDRI): a completed reference database of lung nodules on CT scans, Medical Physics, 38(2011), 915931.
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