Research Article  Open Access
Liwei Liu, Xin Wang, Yang Li, Liping Wang, Jianghui Dong, "Adhesion Pulmonary Nodules Detection Based on DotFilter and Extracting Centerline Algorithm", Computational and Mathematical Methods in Medicine, vol. 2015, Article ID 597313, 11 pages, 2015. https://doi.org/10.1155/2015/597313
Adhesion Pulmonary Nodules Detection Based on DotFilter and Extracting Centerline Algorithm
Abstract
A suspected pulmonary nodule detection method was proposed based on dotfilter and extracting centerline algorithm. In this paper, we focus on the distinguishing adhesion pulmonary nodules attached to vessels in twodimensional (2D) lung computed tomography (CT) images. Firstly, the dotfilter based on Hessian matrix was constructed to enhance the circular area of the pulmonary CT images, which enhanced the circular suspected pulmonary nodule and suppresses the linelike areas. Secondly, to detect the nondistinguishable attached pulmonary nodules by the dotfilter, an algorithm based on extracting centerline was developed to enhance the circle area formed by the end or head of the vessels including the intersection of the lines. 20 sets of CT images were used in the experiments. In addition, 20 true/false nodules extracted were used to test the function of classifier. The experimental results show that the method based on dotfilter and extracting centerline algorithm can detect the attached pulmonary nodules accurately, which is a basis for further studies on the pulmonary nodule detection and diagnose.
1. Introduction
Pulmonary nodules are small masses of tissue in the lung, are prevalent findings on chest and abdominal CT scans, and can be cancerous, though most of them are benign [1]. Lung cancer is one of the biggest malignancy cancers among all kinds of cancers in our healthy life [2, 3] and is also the most common histological type in Aden carcinoma [4]. In recent years, the number of people suffering from lung cancer increases more and more rapidly. The early stage lung cancer is shown as lung nodules, which can be discovered and treated with the assistance of computeraided diagnostic technique in time, which will prolong the life of lung cancer patients [5, 6]. The computeraided diagnostic scheme can detect the nodules automatically in the pulmonary CT images and decrease the miss rate [5, 7], especially with the lowdose CT (LDCT) scanning [8].
To date, many researchers all over the world are devoted to the study of the detection of attached pulmonary nodules, for example, nodule attached to vessels and the pulmonary wall. However, limitations occur in lung cancer imaging of distinguishing nodules attached to vessels from the normal blood vessels, which infiltrate the vessels surreptitiously. Using the corrosion morphology and expansion to segment the pulmonary nodules from the vessels resulted in the corrosion of the nodule thorn, which is another important index for malignant nodule valuation [9]. A weighted fuzzy Cmeans clustering was developed for remotely sensed image classification but requires a given number of clustering and is easy to fall into local minimum rather than the global optimal solution [10]. A method based on EM and Meanshift or one of the two means was proposed to detect attached nodules, but there are many conditions need to be considered, and not entirely consistent with the actual situation [11, 12]. Algorithm Based on Fuzzy Integrated Active Contour Model and Hybrid Parametric Mixture Model to detect pulmonary nodules just extracts the adhesion nodules, but it did not exclude the false positives such as ends of the vessels. For the value of the pixel on the nodule which is close to that of pixel on the vessel, the gray threshold cannot work well and morphological operations cannot identify the adhesion nodules effectively [13–15]. Guo et al. developed a pulmonary nodule detection algorithm based on multiscale enhancement filtering of Hessian matrix and selecting of grads entropy, where Hessian matrix is relative to the gray scale of the pixel in the CT image, and grads entropy is also relative to gray scale of the pixel [16]. It worked well in the solitary pulmonary nodules detection, but it can only detect most suspect nodules and cannot exclude the false positives, especially the ends and the cross sections of the vessels or tracheas. Template matching method can be used to extract suspected nodules, but this will need more human intervention and prior information [17]. For solitary pulmonary nodules, regional growth can obtain good segmentation results [3]; for region growing segmentation results are part of vessel without separation and nodule. The method based on SVMS to detect the nodules, worked well, but it required a long processing time and lots of work [18–20].
Pulmonary nodules are similar to spherical objects, and the lung CT images are 2D. In order to enhance the dotlike regions and depress the linelike regions quickly and effectively, an algorithm named dotfilter was proposed by Li et al. [21]. However, when it was applied to detect pulmonary nodules, many false positives appeared, such as the ends and cross sections of the vessels and tracheas [16]. We found that the distances from the adhesion nodules center or false positives to the centerline of the vessel or tracheas were different. In this paper, starting from the relationship of their position, we combine dotFilter and algorithm of extracting centerline, using which to identify which is the end or head of the vessel and which the circle formed by the intersection of the lines. In this way, we can separate the nodules from vessels and tracheas effectively with fewer steps.
2. Materials and Methods
2.1. Algorithms of Adhesion Pulmonary Nodules Detection
The process of the algorithms used in this paper was shown in Figure 1. Firstly we removed the background noise from the initial CT images and then extract the lung parenchyma. Secondly we used the Gauss function to convolute the image and a smooth image can be obtained. After that we can use dotfilter to enhance the dotlike regions to obtain suspect nodules. At last, we used the extracting centerline algorithm to analyse the relationship of the position of the suspect nodules between the vessels and tracheas, which was used to recognize the adhesion pulmonary nodules.
2.2. Enhancement of Nodules by DotFilter
2.2.1. DotFilter Constructed by Hessian Matrix
To a medical CT image, the enhancement filter of local structure was used extensively which is based on the shape of organization. On a 2D image, we used the dot model conforming to Gauss distribution to represent a nodule [21, 22] as well as line model; the equation is expressed as
Here, denotes a dot expression expressed by a 2D Gaussian function; represents the dimension of the dot and the line. Because of the variety values of , we simulate the image of dots and lines shown in Figure 2(a).
(a)
(b)
(c)
Li et al. [21] proposed that dotfilter can be constructed by using Hessian matrix to effectively extract dotlike objects. For an original 2D image, we assume it has four second derivatives , , , and , where and its 2D Hessian matrix is
is the value of one of the pixels in the image. Suppose the Eigenvalues of are and and satisfied that abs is bigger than . If , exchange them. The and of the dot and line in the image satisfy the following expressions:
The enhanced dotfilter is expressed by the following expression [21]:
In CT images, if the semidiameter of one pulmonary nodule is, the nodule will account for 49.9% of the area of the Gauss function. If it is , it will account for 72.0% of the area of the Gauss function. And if , it accounts for 99.0% of the area of the Gauss function. Then, to a nodule of which the semidiameter is , we use one Gauss function with that equals to express it better [16]. For the position of the pulmonary nodules in CT images is different, the scale of the nodules is different between them. If the range of the scale of the nodules is , the in Gaussian function is in . In order to enhance all the goals in the range, we use different value of σ in Gaussian function to smooth a 2D CT image firstly; then we use the dotfilter constructed with Hessian matrix to enhance the goal area. The two steps above should be repeated times with increasing scale of from to to obtain enhanced CT images. If the range becomes bigger, will become bigger [16, 21]. In lung CT images, we find that the value of which equals 5 is better. In the range of , the algorithm to obtain the can be shown as follows:among which the equals . In each scale, we can obtain one most effective enhancement to the appointed nodule.
The steps of extracting dot with numbers of scales of dotfilter are as follows:(1)According to the range of scale of the nodules we compute the value of .(2)For every , repeat .(3)Using Gaussian function convolve with 2D .(4)For every pixel, repeat .(5)Compute and , .(6)Compute .(7)Stop computing.(8)Select the maximum of .
In order to prove better the effect of using dotfilter with variety value of to identify the dotlike shapes, we use Figure 2(a) as input, and the output is shown as Figures 2(b) and 2(c).
Figure 2(a) is an image constructed by the expression (1) with variety scale of , and there are five dots and three lines. The scales of the among the dots are 2, 4, 6, 8, and 10 pixels. Figure 2(b) is the image, in which the better identified dot is enhanced by one dotfilter with the scale of 10 pixels. We also found that the lines is not identified and the dots smaller than 10 pixels do not have large output. Figure 2(c) is the image enhanced by four dotfilters, of which the dots equal to the scale of 2, 4, 6, 8, and 10 pixels all have large output, and the lines are depressed. According to Figure 2, we can prove that dotfilter can depress the linelike shapes and with variety value of it can extract all the goal areas better.
2.2.2. Application of DotFilter Constructed
As depicted above, we know that dotfilter can enhance the dotlike areas effectively. However, in the lung CT images, the ends and cross sections of vessels are also of dotlike shapes, which will be enhanced by using dotfilter, leading to many more false positives appearance. In order to prove that, we construct three types’ vessel models, such as single line model, type model, and type model shown in Figures 3(a), 3(d), and 3(g). The regions marked by red in the images are dotlike regions, which will be enhanced by dotfilter. Figures 3(b), 3(e), and 3(h) were smoothed by Gauss function. Figures 3(c), 3(f), and 3(i) were the images enhanced by dotfilter. The enhanced areas that we marked in Figures 3(a), 3(d), and 3(g) are also called suspect nodules. Due to many suspect nodules that appeared after the enhanced process by dotfilter we need to eliminate these false positives which may lead to much more computation works.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Now we will use the DotFilter constructed above based on Hessian matrix to lung CT images, and the result is shown in Figure 4. Lung CT images are given by one big hospital for lung nodules detection based on dotFilter and the algorithm of extracting centerline.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Figure 4(a) is an original pulmonary CT image. Figure 4(b) is the pulmonary segment extracted through segmentation of digital image. Figure 4(c) is the right pulmonary segment. In the image the blue arrow denotes a nodule identified by the doctor. From the image we can find that there are many dotlike areas such as solitary areas and dotlike areas attached to the vessels, which we take as false positives. Figure 4(d) is smoothed by Gauss function. It is not ideal to enhance the region of interest (ROI) instantly because there is much noise in the image, so we had better firstly use Gauss function to convolute with it. Figure 4(e) is the enhancement of the solitary nodules and other dotlike area by using dotfilter. In this image we cannot identify which is real nodule without any other assistance or algorithm, because dotfilter just enhances the dotlike areas. To effectively distinguish whether the dotlike areas are nodules or just the part of the vessels we will have to use the algorithm of centerline extracted. Figure 4(f) is a lung CT image without nodules and Figure 4(g) is pulmonary segment extracted; Figures 4(h) and 4(i) are the image smoothed by Gauss function and enhanced by dotfilter, respectively. According to Figures 4(e) and 4(i), we obtain that dotfilter can enhance the dotlike areas effectively but leads to many false positives appearing.
2.3. Algorithm of Extracting Centerline
There are many algorithms used to extract the central line [23–28], such as margin of linear least square fitting legitimate, symmetric moment fitting center method, and block cancroids least squares fitting. Compared with the algorithms used in this paper, they are not stable and accurate enough and have high computational complexity. As Figure 4(e) shows, dotfilter can enhance the dotlike area significantly with more false positive increased. To overcome this shortcoming, we will combine the algorithm of extracting centerline to reduce the false positive.
2.3.1. Principle
Different from the traditional algorithms of extracting central line, area skeleton can be defined by mean axle transforming (MAT). Describe an area whose profile is as follows: for every pixel in the , we search the nearest pixel in . If is bigger than the nearest pixel, we named centerline (skeleton) of , which obeys the following constraints: cannot delete the endpoint; cannot destroy connectivity; and cannot cause excessive corrosion of the area.
We here give the mean of a refinement of twovalue algorithm region: we suppose the value of the pixel in the region is 1, and the values of the pixels on the background are 0. The value of the pixels in the edge of the region is 1 and at least there is one pixel of which the value is 0. As 8 neighborhoods shown in Figure 5(a), if it meets the following conditions (a)–(d), then (step 1) we take as the pixel that will be removed as follows:among which is the number of the nonzero adjacent pixels of ; in other words,among which is either 0 or 1, and is the frequency conversion from 0 to 1 in . For example, in Figure 5(b), and .
(a)
(b)
In Step 2, (a) and (b) remain unchanged, and (c) and (d) become
We apply step 1 to every pixel in the edge of the twovalue region. If we violate (a) or (b), the value of the pixel we talk about is unchanged. Otherwise, we take it as the pixel that will be removed after we handle all the pixels of the edge. Then, we use step 2 the same way as step 1 till there is no pixel needed to be removed any more and stop the algorithm.
Take Figure 6(a), which is processed by the mean of a refinement, for example, and the result is shown as Figures 6(b)~6(e).
(a)
(b)
(c)
(d)
(e)
Figure 6(a) is an image of a human chromosome by electron microscope magnified 30000 times and segmented using digital image processing algorithm. Figure 6(b) is the image after Gaussian smoothing. Figure 6(c) is the skeleton of the chromosome. Figure 6(d) shows skeletons after applying extinguishing the burr algorithm eight times. We found that on the skeleton there is much burr but less than that in Figure 6(c). Because this algorithm is related to the threshold of the pixel, we should increase the threshold value in the algorithm. Figure 6(e) presents seven more times for extinguishing the burr by using the algorithm.
If a line is expressed by , we will take and as the deviation [29]. If they are all very small, we will think the algorithm works better. We use other three algorithms for extracting center line to compare with the one used in this paper; the result is shown as Table 1.

According to Table 1 we found the algorithm in this paper can work effectively, though the symmetric moment fitting center method consumes the least time, but its deviation was the bigger. The algorithm used in this paper worked steadily and time consuming is not much. Combining all the factors, the algorithm used in this paper is better.
2.3.2. Application
In this paper, to accomplish the experiment combining dotfilter with the method proposed above we use six steps shown as follows.(1)We first selected three lung CT images after extracting the lung segment in Figures 7(a)~7(c); they were nodules attached to vessels, single vessel, and crossing vessel. There was one nodule attached to one end of the vessel noted by the doctor in Figure 7(a), shown as the arrow points to. In Figure 7(b), we can see that there is one vessel apparently and one of its ends is of dotlike shape, similar to the adhesion nodule in Figure 7(a). In Figure 7(c), the vessel is composed of two vessels and they are crossed.(2)As the value of vessels, tracheas, and nodules are bigger than the value of lung parenchyma, in order to decrease the computation, we extracted the soft tissue of the lung based on gray threshold which we defined as 130, which is obtained after many times of drawing histogram, shown in Figures 7(d)~7(f). For the low contrast nodules, we did not consider them in this paper. According to Figures 7(d)~7(f), we found that the soft tissue of the lung parenchyma was completely extracted.(3)To the tissues extracted in step , firstly we should extinguish the noise in the CT image. So we used Gauss filter to accomplish it. Then, we used the dotfilter constructed above to enhance the dotlike regions, in other words, we enhance the suspected nodules. The enhancement result was shown in Figures 7(g)~7(i).(4)We eliminated the tissues obtained in step from the images in step and then obtained the skeleton shown in Figures 7(j)~7(l). We considered the tissues obtained in step as the false positives; we should remove them and use the algorithm of extracting centerline to extract the skeletons. But for Figure 7(f), the vessels became three parts, which was not beneficial for us to use the algorithm of extract the center line, so we firstly supplied the lack, making the vessels become one connected vessel and then extracted the skeleton.(5)Firstly we worked out the center of every suspected nodule (false positive) and then computed the value of , . denotes the perpendicular distance from the center of mass of suspected nodules to the line of the skeleton near the suspected nodule obtained in step . denotes the minimum distance from the center of mass of suspected nodules to all pixels in the skeleton obtained in step .(6)The diameter of the nodules is 3 mm~30 mm, so we compared the diameter with obtained in . If is smaller than 1.5 mm and is smaller than 1.5 mm, the suspected nodule is treated as the intersection of the vessels or the end of the vessel. If was smaller than 1.5 mm and was bigger than 1.5 mm, the suspected nodule was treated as one end of the vessel. If was bigger than 1.5 mm, the suspected nodule was treated as the attached nodule shown in Figure 7(m).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)
(q)
(r)
Figures 7(p)~7(r) is the threedimension (3D) display of the nodule, the solitary vessel, and the crossing vessel. In Figure 7(p) the nodule with green edge is connected to the vessel in the yellow circle. Figures 7(q) and 7(r) are two kinds of different vessels in the green circle in order to be found easily. According to the three images we can easily find that the suspect nodules appeared in the process; they are actualized parts of vessels.
3. Results and Discussion
Table 2 depicts what are the attributes and where they come from the CT images used in this paper. 20 sets of CT images with less noise [25] were used in the experiments. They originated from LIDC database and Jida Hospital and each CT image has 512 512 pixels. Nodules in each CT image have been noted by doctors. The 20 true nodules and 20 false ones extracted were used to test the function of classifier. All experiments in this paper were based on the computer that consists of AMD CPU with a frequency of 2 GHz, 1.5 GB of RAM, and Windows XP operating system. Algorithm development code is developed on the platform of MATLAB.

For the CT images supported by the hospital it missed 3 adhesion nodules and missed none for the LIDC database. Table 3 shows the missing rate and runtime of each set of CT images. Literature [9] can better extract the solitary nodules but lacks the high capacity of extracting the adhesion lung nodules because it is based on the threshold value of the pixel and it needs much justice. Literature [12] uses the algorithm which has too much computation. If we use a method with only dotfilter, there will be more false positives appearance because dotfilter can enhance the ends and the intersections of the vessels meanwhile. By using dotfilter and the algorithm of extracting the center line, we have lower missing rate and less runtime. This is because dotfilter constructed with Hessian matrix can extract the dotlike region effectively and quickly. And after extracting the center line of the vessels, we can achieve a better understanding of the relationship between the nodules and vessels. According to the relationship, it is beneficial for us to extract the nodules. There are errors of this method because scale of some nodules is very small or the value of the pixels in nodules is very small, which can lead to the miss rate increases. The method in this paper has some limitations; for example, it cannot adapt to the lower contrast nodules and the nodules attached to the lung wall. It is just applied to the nodules attached to vessels and tracheas.
4. Conclusion
In this paper we first use 2D Hessian matrix to construct dotfilter constructed to extract dotlike region. In order to solve the problem that the dotfilter cannot detect attached pulmonary nodules, an algorithm based on extracting centerline was used. Results of experiment indicated that the method is easy and effective while extracting attached pulmonary nodules well. In the future, we will be devoted to extracting the lung nodules contacting pulmonary wall and ground glass opacity pulmonary nodules.
Conflict of Interests
The authors declared that they have no conflict of interests regarding this work.
References
 K. Alexander, H. Joly, L. Blond et al., “A comparison of computed tomography, computed radiography, and filmscreen radiography for the detection of canine pulmonary nodules,” Veterinary Radiology and Ultrasound, vol. 53, no. 3, pp. 258–265, 2012. View at: Publisher Site  Google Scholar
 T. Hiraki, H. Gobara, T. Iguchi, H. Fujiwara, Y. Matsui, and S. Kanazawa, “Radiofrequency ablation for earlystage nonsmall cell lung cancer,” BioMed Research International, vol. 2014, Article ID 152087, 11 pages, 2014. View at: Publisher Site  Google Scholar
 M. Inoue, S. Nakatsuka, and M. Jinzaki, “Cryoablation of earlystage primary lung cancer,” BioMed Research International, vol. 2014, Article ID 521691, 8 pages, 2014. View at: Publisher Site  Google Scholar
 W. D. Travis, E. Brambilla, M. Noguchi et al., “International Association for the Study of Lung Cancer/American Thoracic Society/European Respiratory Society international multidisciplinary classification of lung adenocarcinoma,” Journal of Thoracic Oncology, vol. 6, no. 2, pp. 244–285, 2011. View at: Publisher Site  Google Scholar
 W.J. Choi and T.S. Choi, “Genetic programmingbased feature transform and classification for the automatic detection of pulmonary nodules on computed tomography images,” Information Sciences, vol. 212, pp. 57–78, 2012. View at: Publisher Site  Google Scholar
 C. M. Niranjana and P. Deepa, “Nodule detection in lung intervention by using VDE and morphology techniques,” International Journal of Research in Computer Applications and Robotics, vol. 2, no. 5, pp. 114–123, 2014. View at: Google Scholar
 S. C. Park, B. E. Chapman, and B. Zheng, “A multistage approach to improve performance of computeraided detection of pulmonary embolisms depicted on CT Images: preliminary investigation,” IEEE Transactions on Biomedical Engineering, vol. 58, no. 6, pp. 1519–1527, 2011. View at: Publisher Site  Google Scholar
 F. Niknam, J. Chen, S. Napaki, and M. Aghmesheh, “Approach to multiple pulmonary nodules: a case report and review of literature,” The Scientific World Journal, vol. 11, pp. 760–765, 2011. View at: Publisher Site  Google Scholar
 W. J. Kostis, A. P. Reeves, D. F. Yankelevitz, and C. I. Henschke, “Threedimensional segmentation and growthrate estimation of small pulmonary nodules in helical CT images,” IEEE Transactions on Medical Imaging, vol. 22, no. 10, pp. 1259–1274, 2003. View at: Publisher Site  Google Scholar
 C.C. Hung, S. Kulkarni, and B.C. Kuo, “A new weighted fuzzy Cmeans clustering algorithm for remotely sensed image classification,” IEEE Journal on Selected Topics in Signal Processing, vol. 5, no. 3, pp. 543–553, 2011. View at: Publisher Site  Google Scholar
 S.S. Sun, H. Li, X.R. Hou, Y. Kang, and H. Zhao, “Pulmonary nodule segmentation based on EM and meanshift,” Journal of Image and Graphics, vol. 10, pp. 2016–2022, 2009. View at: Google Scholar
 K. Okada, D. Comaniciu, and A. Krishnan, “Robust anisotropic Gaussian fitting for volumetric characterization of pulmonary nodules in multislice CT,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 25, no. 2, pp. 281–288, 2003. View at: Publisher Site  Google Scholar
 V. Sudha and P. Jayashree, “Lung nodule detection in CT images by using thresholding and morphological operations,” International Journal of Emerging Science and Engineering, vol. 1, no. 2, 2012. View at: Google Scholar
 B. Chen, T. Kitasaka, H. Honma et al., “Automatic segmentation of pulmonary blood vessels and nodules based on local intensity structure analysis and surface propagation in 3D chest CT images,” International Journal of Computer Assisted Radiology and Surgery, vol. 7, no. 3, pp. 465–482, 2012. View at: Publisher Site  Google Scholar
 A. O. de Carvalho Filho, W. B. de Sampaio, A. C. Silva, A. C. de Paiva, R. A. Nunes, and M. Gattass, “Automatic detection of solitary lung nodules using quality threshold clustering, genetic algorithm and diversity index,” Artificial Intelligence in Medicine, vol. 60, no. 3, pp. 165–177, 2014. View at: Publisher Site  Google Scholar
 W. Guo, Y. Wei, H. Zhou, and D. Xue, “Suspected pulmonary nodule detection algorithm based on Hessian matrix and grads entropy,” Chinese Journal of Scientific Instrument, vol. 30, no. 8, pp. 1702–1706, 2009. View at: Google Scholar
 Y. Lee, T. Hara, H. Fujita, S. Itoh, and T. Ishigaki, “Automated detection of pulmonary nodules in helical CT images based on an improved templatematching technique,” IEEE Transactions on Medical Imaging, vol. 20, no. 7, pp. 595–604, 2001. View at: Publisher Site  Google Scholar
 M. J. Gangeh, L. Sørensen, S. B. Shaker, M. S. Kamel, M. de Bruijne, and M. Loog, “A textonbased approach for the classification of lung parenchyma in CT images,” Medical Image Computing and ComputerAssisted Intervention, vol. 13, part 3, pp. 595–602, 2010. View at: Google Scholar
 S.S. Sun, H.Z. Ren, Y. Kang, and H. Zhao, “Lung nodule detection by GA and SVM,” Journal of System Simulation, vol. 23, no. 3, pp. 497–501, 2011. View at: Google Scholar
 J. S. Bie, Segmentation and recognition of lung nodules in CT images based on SVM [M.S. thesis], 2012.
 Q. Li, S. Sone, and K. Doi, “Selective enhancement filters for nodules, vessels, and airway walls in two and threedimensional CT scans,” Medical Physics, vol. 30, no. 8, pp. 2040–2051, 2003. View at: Publisher Site  Google Scholar
 Q. Li, F. Li, and K. Doi, “Computerized detection of lung nodules in thinsection CT images byuse of selective enhancement filters and an automated rulebased classifier,” Academic Radiology, vol. 15, no. 2, pp. 165–175, 2008. View at: Publisher Site  Google Scholar
 S. Kurugol, E. Bas, D. Erdogmus, J. G. Dy, G. C. Sharp, and D. H. Brooks, “Centerline extraction with principal curve tracing to improve 3D level set esophagus segmentation in CT images,” in Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society, vol. 2011, pp. 3403–3406, IEEE Engineering in Medicine and Biology Society, 2012. View at: Google Scholar
 J. Hai, G. Zhang, and J. Cheng, “Estimation of mass transfer coefficient in ozone absorption by linear least square fitting and Simplex search methods,” Journal of Central South University, vol. 19, no. 12, pp. 3396–3399, 2012. View at: Publisher Site  Google Scholar
 P.H. Tseng and K.T. Feng, “Derivation of CRLB for linear least square estimator in wireless location systems,” Wireless Networks, vol. 18, no. 7, pp. 735–747, 2012. View at: Publisher Site  Google Scholar
 L. M. Alawneh, C. J. Park, M. K. Jaradat, and B. Lee, “Burnup estimation for plate type fuel assembly of research reactors through the least square fitting method,” Annals of Nuclear Energy, vol. 71, pp. 37–45, 2014. View at: Publisher Site  Google Scholar
 S. Sergey, M. van der Kooij, and K. Tiampo, “A simultaneous inversion for deformation rates and topographic errors of DInSAR data utilizing linear least square inversion technique,” Computers and Geosciences, vol. 37, no. 8, pp. 1083–1091, 2011. View at: Publisher Site  Google Scholar
 W.P. Choi, K.M. Lam, and W.C. Siu, “Extraction of the Euclidean skeleton based on a connectivity criterion,” Pattern Recognition, vol. 36, no. 3, pp. 721–729, 2003. View at: Publisher Site  Google Scholar
 K. Wang, A. You, and L. Wang, “Algorithm of centerline extracted based on eigen decomposition of Hessian matrix,” High Power Laser and Particle Beams, vol. 25, no. 1, pp. 24–28, 2013. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2015 Liwei Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.